Sophia Imperato
email: sophia.imperato@espci.fr

Fabrice Harms

Antoine Hubert

Mathias Mercier

Laurent Bourdieu

Alexandra Fragola

Single-shot quantitative aberration and scattering length measurements in mouse brain tissues using an extended-source Shack-Hartmann wavefront sensor

Keywords: Adaptive Optics, Two-Photon Microscopy, Extended-Source Shack-Hartmann wavefront sensor, Scattering media, Neuroimaging

En premier lieu, je tiens à remercier l'ensemble des membres de mon jury d'avoir accepté d'évaluer mon travail, et pour l'échange qui s'en est suivi. Je remercie Sandrine Lévêque Fort et Martin Booth d'avoir rapporté cette thèse, ainsi qu'Eirini Papagiakoumou et Jean-Marc Conan pour leur évaluation bienveillante. Je remercie plus particulièrement Sylvain Gigan d'avoir présidé ce jury, et également de m'avoir suivie au cours de ses trois ans lors des comités de suivi de thèse.

A Alexandra, Laurent et Fabrice, je tiens à adresser de chaleureux remerciements pour la confiance que vous m'avez accordée, du début à la fin de ce projet. J'ai beaucoup appris à vos côtés, autant techniquement qu'humainement et vous suis très reconnaissante pour cette aventure.

Tout ce projet n'aurait pas pu voir le jour sans vous. Je vous souhaite une très belle continuation. Il n'existe de thèse sans équipe et quelle équipe formidable a été la SI2, ne serait-ce que par sa capacité à supporter ma voix criarde durant ces trois années. Je remercie l'ensemble des permanents Nicolas, Thomas, Sandrine, Céline et Vincent, d'avoir créé cette ambiance familiale qui m'a été si précieuse, pour votre bienveillance, votre passion des sciences et de la compétition, et enfin pour votre humour pas toujours fin mais qui a son charme.

Ne serait-il pas temps de mentionner nos très chers Docteurs, i.e grands frères et soeur de labo ? Ne voyez pas en cette question rhétorique autre chose que mon intention de satisfaire l'une des traditions de la SI2... Je souhaite remercier d'abord Antoine de m'avoir accueillie lors de mon premier jour ici, de m'avoir apporté conseils et support technique lors des deux années que nos avons eu en commun et d'avoir été disponible à ce sujet bien après ton départ. Viennent ensuite Mogha, Sheila et Subha, indissociables et pourtant si complémentaires, que je ne peux que qualifier d'exemples et des soutiens sans failles. Merci pour votre franchise, votre rage de vaincre, et votre détermination que vous avez su insufflé à nous autres les plus jeunes.

Grande adepte de montagnes, de plantes et de thé, et meilleure co-thésarde possible, je remercie Fanny, aux côtés de qui je suis très reconnaissante d'avoir pu partager cette folle aventure, et cette dernière année en particulier. Je pense en particulier à la confiance en moi que tu m'as inspirée, aux les chit chat thé et autres poses de yoga mémorables. Cette traversée de longue haleine n'aurait pas eu la même saveur sans toi.

De l'inclassable Charlie, à ma très chère Lina en passant évidemment par Corentin, merci à vous d'avoir été bien plus que des co-bureaux. Plus largement, je remercie Po et Yuzhou, Ningyuan, Henri, Alice, d'avoir contribué à l'ambiance du labo, vous êtes la relève, et quelle relève ! Je pense également à tous les stagiaires passés par là que ce soit en optique ou en chimie : Diego, Pierre, Méline, petit Nicolas, petit Antoine, Lucas, Sanya, Juishi, Simone. Bien évidemment, j'adresse des remerciements tout particuliers à Pauline, avec qui j'adoré travailler, et dont je suis encore aujourd'hui très fière de ce qu'on a pu faire ensemble ! Il est également très important pour moi de remercier l'ensemble de l'équipe de Laurent à l'IBENS, Cathie, Jeff, Walther, Anthony, Macha, Caio, Ombeline, Hugues, pour les échanges scientifiques très constructifs et pour vos regards éclairés ainsi que vos connaissances aiguisées en neuroimagerie. On ne s'est pas beaucoup croisés, mais je garde beaucoup de vos remarques pertinentes et vos questions incisives. Sans contestation possible les piliers du LPEM, je voudrais remercier Marie-Claude et Sophie sans qui le laboratoire ne pourrait tourner convenablement, et qui sont bien plus que des gestionnaires, pour leur réactivité ainsi leur écoute et leur soutien. Je remercie également Francis pour son précieux support en électronique, ainsi que Ricardo et Dimitri, tous deux directeurs du LPEM lors de mon passage ici, pour leur présence et leur volonté d'aider en toute situation. Outre-périphérique, l'ensemble de l'équipe d'Imagine Optic m'aura été d'une grande aide, et notamment Cynthia, Jérôme et Pauline avec qui ce fut un plaisir d'échanger et qui a fortement vi contribué à ma comprehension des tréfonds de l'optique adaptative et autres logiciels de correction de front d'onde.

Rien n'aurait été possible sans les financements attribués par le CNRS et l'Ecole Doctorale Physique en Ile de France ; au sein de ces institutions, je remercie plus particulièrement, Maria Chamarro, Nadine Yassine de l'EDPIF ainsi que Bako Forissier, Yves Abonnel de la délégation Paris centre du CNRS.

Dès lors, il ne me reste plus qu'à conclure ces remerciements par quelques mots plus personnels, à l'attention de tout ceux en dehors du labo qui ont supporté la partie la plus difficile de moi et à qui je dois tant.

Evidemment, un immense merci à ma famille pour leur soutien inconditionnel depuis le début, ainsi que pour l'organisation de ce pot de thèse, ainsi qu'à mes amis, de Clara, Lola, Caroline à Nico bien sûr. Merci aux copines de Supop avec qui j'ai eu une chance incroyable de traverser cette thèse ; on a toutes vécu cette aventure à notre façon et on s'est serré les coudes, alors bravo à nous ! Merci aussi à la team de l'ISMO pour tous les souvenirs à Mifobio, sûrement la conf la plus dépaysante à laquelle j'ai eu l'occasion d'aller, la preuve s'il en fallait une, que les conf à l'étranger, c'est surcôté.

Reste un râleur invétéré à qui je dédie ces dernière lignes, en espérant un retour sur investissement sous peu. Flegmatique voire nonchalant (i.e exaspérant), architecte dans l'âme et désormais pythonicien expérimenté, tu es la clé de vôute de cette thèse. À toi, grand fan de Lego, de dinosaures et de chaussettes à motifs douteux, à toi qui te reconnaitras et qui, je le sais, chercheras ces lignes avidement du regard en se demandant qu'est ce que j'aurais encore pu bien dire comme bêtises, merci pour tout, tout simplement. J'espère pouvoir t'apporter autant que ce que tu m'as donné (modulo la cuisine, faut pas abuser). vii 1.2. How to record cellular activity

Introduction

One of the longstanding goals in neuroscience is to obtain a three-dimension dynamic cartography of the brain in order to describe its structure-function relationship. This consists into linking the anatomic architecture of the brain (the structure), to its dynamical organization (function), and understand how neural activity is linked to behaviour. These researches are motivated by the will to understand the underlying mechanisms of brain pathologies, from strokes to depression, anxiety, schizophrenia or Parkinson, Alzheimer and so on... The neural understanding and description of cognitive processes such as memory formation, perception, learning, thinking, decision making are thus currently actively studied. In order to understand these mechanisms, the brain is to be imaged in activity driven by sensory stimuli or engaged in a behavioral task, but the field remains limited by the lack of tools to completely explore the intact brain of animals.

The discovery of the tandem aequorin/GFP in 1962 by Osumu Shimomura [1], followed by the cloning of the GFP protein in 1992 [2], has opened the way to the development of fluorescent reporters, genetically encoded light-sensitive proteins as GCaMP [3], that allow the recording of the neuronal activity, and has spread the hope to deepen our knowledge.

Alongside these developments, the old field of optical microscopy, dating back from the 17th century [4], adapted itself to the constraints imposed by these fluorescent reporters and has emerged as a major tool in neuroscience. Imaging the brain requires to follow precise constraints specific to the optogenetic tools used and to the brain structure of interest to achieve a single-cell spatial resolution over a large field of view, and with single action potential temporal resolutions.

Optical microscopy now allows to perform structural imaging and functional activity recordings within large volume of tissues with subcellular resolution.

In particular, non-linear microscopy enables the interrogation of neuronal activity in mammalian brains but remains limited in depth; scattering and optical aberrations, due to refractive index inhomogeneities, limit resolution and deteriorate progressively signal intensity. Indeed, aberrations encountered on the excitation path significantly affect imaging quality as depth increases.

For this purpose, in order to retrieve the microscope imaging quality while addressing important imaging depths, Adaptive Optics (AO), initially issued from the astronomical field, has been implemented on optical microscopy setups to push further depth limits. The current strategies in non-linear microscopy still suffer from either their sensitivity to strong scattering (as for the direct wavefront sensing methods) or from time-consuming iterative processes (indirect wavefront sensing) that are harder to handle with functional imaging signals.

My doctoral work during the last three years under the supervision of Pr. Alexandra Fragola in the team "Synthèse et Imagerie des Sondes Inorganiques (SI²)" at the Laboratoire de Physique et d'Etude des Matériaux (LPEM) of ESPCI, Pr. Laurent Bourdieu from the Institut de Biologie de l'Ecole Normale Supérieure (IBENS), "Cortical Dynamics and Coding Mechanisms team" in Paris, and Dr Fabrice Harms from Imagine Optic, has been dedicated to the implementation of a new wavefront sensing approach for two-photon microscopy for neuroimaging in the mouse, thus addressing scattering issues inherent to this animal model. Before the beginning of my studies, Antoine Hubert, former doctoral student in the LPEM, had designed the first Extended Source Shack Hartmann wavefront sensor and implemented it to a Light Sheet Fluorescence microscope, carrying ex vivo functional imaging of the neurons involved in the circadian clock of the Drosophila melanogaster. I extended this Adaptive Optics method to two-photon microscopy, in order to pave the way to Extended Source wavefront sensing for in vivo functional neuroimaging in the mouse.

This manuscript is organized into four chapters.

• In Chapter 1, are first introduced the constraints imposed by in vivo imaging, from the animal model to microscopy for in-depth functional neuroimaging. I present the state-ofthe-art of Adaptive Optics for microscopy and in particular for two-photon excited fluorescence microscopy.

• In Chapter 2, I develop the study of the resilience of Extended Source Wavefront Sensing to strong scattering conditions. Studies are conducted experimentally and numerically, and performances compared to the classical direct wavefront sensing approach based on centroid measurement.

• In Chapter 3, I detail the implementation of an Extended Source wavefront sensing-based Adaptive Optics loop on a home-made two-photon microscope. After characterizing the system in both scattering and non-scattering conditions, I describe the experimental comparison of the Extended Source wavefront sensing with the centroid-based method in scattering conditions on the two-photon microscope.

• In Chapter 4, I present results obtained through fixed brain slices from transgenic mouse, showing gain provided by this Adaptive Optics method.

Chapter 1

In-depth functional neuroimaging In this Chapter, I introduce the underlying considerations to our microscopy work. The motivations and the requirements for the study of neural networks through optical methods are first described, then the optical methods currently available to reach increasing penetration depth in the tissues and hence gather new information on the brain are presented. I finally introduce the motivation of our group to develop a new strategy as well as its previous implementation performed before the start of my doctoral work.

1.1 From the animal to the neurons

The animal model

So far, the majority of brain investigation can not be done on human as most are invasive. Thus, despite all the efforts from the scientific community, animals are still necessary for the exploration of structural and functional information. The current ethical legislation and state of mind consider acceptable animal experimentation when no other alternative is available and that the research outcome is valuable. Amongst the animal models relevant for neuroscience, the mouse is the most popular.

• The mouse is small, has an accelerated lifespan, with one mouse year equaling about 30 human years, and reaches adult age within 2 months. It also breeds very well (mouse's gestation only 19-21 days and delivering 6 to 10 baby mice), and constitutes an economical choice. Moreover, they are relatively easy to handle and transport.

• The mouse genome can be manipulated to virtually mimic any human disease or condition thanks to modern sequencing and genomic engineering technologies. The introduction of the precise mutations underlying human disease in the mouse leads to more accurate and crucial disease research data.

• Mice can be inbred in a consanguineous way to yield genetically identical strains after 20 generations. This uniformity allows for more accurate and repeatable experiments. The Jackson Laboratory now maintains more than 9,000 genetically defined strains of mice, after almost a century of development and study of mice lineages; thanks to their longstanding use, the mouse is very well understood.

• The mouse is biologically very similar to humans and get many of the same diseases, for the same genetic reasons. We share enough similarities to consider the mouse as a relevant model for neuroscience. Moreover, its brain structure is very similar in its organisation to that of humans, allowing us to study complex cognitive processes such as sensory perception or decision making as many of the processes and mechanisms involved are probably similar in humans.

All these elements make the mouse the foremost mammalian for neuroscience research. Note that an other rodent is getting more an more used: the rat. Until 2008, it was impossible to isolate stem cells, but now CRISPR-Cas9 discovery paves the way to a tremendous study development on this model, as rat is easier to perform surgery on, to add a catheter in blood vessel for drug self-administration, it is less aggressive than mouse, and has a social behaviour closer to human.

Anatomical elements on the cortical area and neuron description

The mouse brain is divided into several parts that are contributing to data processing (Figure 1.1). The most external part, the cortex, corresponds to the area where the highest level of data processing occurs. Thus, cortical areas are specifically dedicated to some tasks and senses, such as the visual cortex, or the auditory cortex. The regions are themselves subdivided in layers, from the surfacic layer L1 to the deepest L5 and L6, according to the geometry of the cells and of their projections that can be encountered. Now well-documented and easily accessible through online atlases such as the Allen Mouse Brain Atlas [6], some dynamic cartographic elements of the cortex are still missing. When anatomical structures were long discovered through usual histology, the now available voltage sensitive techniques allow the exploration of dynamic processes and the full recovery of the neuronal signal processing while the animal is performing a task. However these methods are still limited and a lot is to be explored.

In the cortex, as well as in all the other sections of the brain, the information is treated and transported by neurons. Identified by Santiago Ramón y Cajal as the elementary computational unit of the brain at the end of the 19th century, its description, as well as the neural network one, made him a pioneer of modern neuroscience and set the basis of the so called neuron doctrine.

A neuron is thus composed of a soma (≈ 10 µm), a single axon and one or more axon terminals (≈ 1 µm). The projections that receive the signal are dendrites. The synapse is the area where the signal is transmitted from one neuron to another (see Figure 1.3). As a example, in the human brain, the number of synapses is approximated between 10 14 and 10 16 , with around 2,000 synapses per neuron.

After years of neuronal studies taking for granted that the neuron was this structural and functional unit of the whole nervous system, the emergence of multineuronal recording techniques has paved the way to a new doctrine: the neural network doctrine [8]. Indeed, the simultaneous recording of many cells has revealed that ensembles of neurons were reacting together to form larger functional units and it is towards that direction that the studies are now conducted, in order to incorporate the knowledge acquired from this single-neuron perspective into a more global vision and to understand the underlying mechanisms of memory, behaviour or mental disease.

Spatiotemporal resolution requirements

The study to neural dynamics in the mouse can be performed though several methods, relying on the interrogation of different types of signal generated by the brain. Within all the available methods, an inevitable trade off between the spatial and temporal resolution, the Field-of-View (FOV), the Signal to Noise Ratio (SNR), the invasiness and the reachable imaging depth of these techniques requires the user to adapt beforehand its strategy. For example, functional Magnetic Resonance Imaging (fMRI) is a widely used technique to study brain activity at the scale of the whole brain, without being invasive but, with its millimetric resolution, it can not discriminate cells individually (see Figure 1.2). On the other side of the technical range, one can find the intracellular recordings, that exhibit a strong neuronal recording precision, fast and spatially well resolved, but at the great expense of the FOV. The basic spatiotemporal requirements are motivated below.

a. Spatial requirements

In order to study neuronal assemblies, the activity of many cells simultaneously needs to be recorded, with a single cell resolution, to untangle the signal coming from two closely located neurons for example. Moreover, we need to be able to focus at different depths in the the tissue to study the whole dynamics. Typically, a FOV of a up to several hundreds of neurons is to be considered, and in the cortex, a penetration depth of 500-600 µm is required to get access to the pyramidal cells in layer 5 for example, and image the cortical columns. A long standing goal is to get access to deeper layers while maintaining cellular resolution and wide FOV, and for these Adapted from [10] reasons, many developments have been carried out to push this limit even further. A single cell resolution implies a micrometric resolution; if the cell size is of approximately 10 µm, the often-used cytoplasmic labelling reveals the internal structure of typically a few microns. Some other objects such as neurites (the fine dendritic processes) can be observed and of interest to the community; set ups exhibiting a submicrometric resolution are thus useful. To combine all these spatial requirements, compromises are to be made and a flexible configuration has to be preferred in order to easily tune the spatial resolution and requirements to what we want to look at.

b. Temporal requirements

Neuronal signals are composed of short electrical pulses called Action Potentials (AP), or spikes, with a typical duration of 1-2 ms and and amplitude around 100 mV. These spikes constitute the elementary unit of signal transmission and propagate along the axon of the neuron through adjacent variations of membrane potential (at rest, it is around -70 mV) generated by ionic transmembrane currents. At the junction between the axon of one neuron and the dendrite of another is a synapse, a 20-40 nm gap between neurons, where the communication is made through the release of neurotransmitters, that, depending on their type, can either excite or inhibit this postsynaptic neuron. A sequence of APs generated by a neuron is called a spike train; these sequences contain the information in their frequency and timing. Thus, a millisecond resolution first seems necessary to record neuronal dynamics, but strongly depends on the type of activity reporter that is chosen, either electrical or optical. This point is further discussed in the following section.

How to record cellular activity

Only electrophysiological and optical methods allow to record neuronal activity in vivo, with a single cell resolution and of many cells simultaneously. The recordings can be based on two types of activity reporters: the voltage variation can be recorded through either electrophysiological methods via electrodes or with voltage-sensitive dyes coupled to optical microscopy to Chapter 1. In-depth functional neuroimaging follow the electrical dynamics. Or the variation of calcium concentration can be chosen as activity reporter and followed with calcium-sensitive dyes with optical microscopy techniques. Both methods are presented briefly here.

Electrophysiology

Electrophysiology is directly sensitive to the voltage variation occurring at the passage of an AP; both intracellular and extracellular methods are available.

Intracellular methods imply the insertion of an electrode directly into the cell; they present the highest sensitivity and allow a perfect detection of APs but remain limited to the recording of a few cells simultaneously. On the other hand, extracellular methods allow the assessment of large neuronal populations while extracting the activity of individual cells through spike sorting. These methods are particularly relevant when addressing deep regions, such as neuronal populations in the hippocampus, a region that is harder to access through optical methods.

High frequency fluctuations, characteristic to single APs and thus referring to the activity of individual neurons, can be detected, as well as low frequency fluctuations that result from the synchronous activity of many cells (between 10 2 and 10 6 cells).

Through spike sorting methods, the activity of individual neurons can be discriminated; however this can be reliably realized only for events occurring within the neighbouring region of the electrodes (< 50 µm). To overcome this limitation, multiple electrodes can be implanted, for example through an array of multiple microelectrodes spanned over the areas of interest, with one recording each site. As for the low frequency oscillations, they are recorded either via electroencephalography (EEG), that provides a high temporal resolution and resolves fast correlated events, electrocorticogram (ECoG) that presents a better sensitivity than EEG but at the expense of craniotomy, or Local field potential (LFP) measurements in which a single electrode of small size (10-100 µm) is inserted and around which the activity is recorded.

This fast-growing field, and more particularly extracellular recordings, is widely used and undergoing many developments [9] as they provide a high temporal resolution and great sensitivity, some major drawbacks can be cited: spike sorting can fail to sort neurons [11], the precise spatial localization of the cell is not obtained and chronic recordings are difficult to implement as the electrodes easily drift between sessions. Moreover, the technique leads to cell damage around the recorded area and lacks specificity; for example, we can not distinguish an inhibitory cell from an excitatory one, the cell type as defined by the genetics remains unknown and cell projections are not assessed.

Alternatively, fluorescence imaging has been benefiting from the major advances in fluorescent probes, and more particularly of the development of genetically encoded fluorescent proteins.

In the following, the basics of fluorescence imaging for neuroscience are presented.

Fluorescence imaging

Less invasive than electrode recording, with no mechanical disturbance when light penetrates into the tissue compared to electrodes, fluorescence imaging permits activity recording of the same neurons repeatedly over days and weeks.

Thanks to the difference between the excitation and the detection wavelength, the Stokes shift, the excitation can be spectrally separated from the emission. Several types of fluorescent probes are now available, with various absorption and emission spectra across the visible range, and with enhanced brightness and efficiency. Fluorescence imaging presents also the advantage of being a specific technique, allowing to address only a population of cells, and several labelling can be combined to provide multicolor imaging when relevant, or when the photon budget needs to be optimized.

Discovered in the early 1960s, the Green Fluorescent Protein (GFP) from the Aequorea victoria jellyfish, that emitted green fluorescence when excited in the ultraviolet range, paved the way to many engineering developments that led tho the production of various colored derivatives. Its fluorescent ability lies in the combination of a peptide triplet but positioned at the center of unique barrel structure folded into a tube. Nowadays, this collection has been completed with fluorescent proteins from other species, and spans across the full visible range. One of the utmost development of this multicolor strategy is undoubtedly the 'Brainbow' labelling, in which each neuron is labelled by an aleatory combination of three fluorophores [12] and that allowed multiscale brain imaging [13,14] and lineage tracking in the nervous system [15].

To report for neuronal activity, these fluorescent molecules can be linked to calcium sensitive ones. At rest, a cell exhibits a nominal intracellular concentration of Ca 2+ ions that rises to 10 to 100 fold when an AP occurs [16]. Indeed the voltage variation induces a rising phase in Ca 2+ concentration, followed by an exponential decay phase, with a time constant being around 50 to 70 ms [17].

a. Calcium imaging FIGURE 1.4: GCaMP scheme. From [18] Calcium imaging allows to measure indirectly the AP, with a simple model in which each AP corresponds to a determined calcium input that is almost always the same. In this model, a fixed calcium input at the time of the spike will correspond to a calcium transient and therefore to a fluorescence transient which is in principle stereotyped and will be characterised by its rise and fall time and its amplitude at the peak. One AP corresponds to 5 to 10% of ∆F F variation (see Section 1.2.2c.) Amongst the calcium imaging approaches, the Genetically Encoded Calcium Indicators (GECIs) are a tool of choice. They allow long term imaging, across weeks, and provide important flexibility in the labelling types: the neuronal population, its subregion or their projection type. Initially based on the GFP and its derivative, one can now find all the colored variant that were engineered. They became gold standard for the study of neuronal processes at the population level.

From the GECIs' family, the GCaMP indicators are composed of a fluorescent protein (Enhanced GFP, that is a circularly permuted GFP, called EGFP) and a protein that detect a change in calcium concentration, the calmodulin, that binds to the calcium ions. According to this change, the conformation of the molecule is modified and induces a change in fluorescence emission of EGFP [18]. This chemical reaction is, non exhaustively, characterised by its kinetics, the affinity of the probe, the sensitivity range, the dynamic range.

Chapter 1. In-depth functional neuroimaging

The formation of the complex [Ca 2+ -Indicator] in the presence of Ca 2+ is ruled by the equation reaction:

Ca 2+ + Indicator k on ⇄ k off [Ca 2+ -Indicator]
The reaction kinetic is dictated by the constants k on and k o f f , and their half decay time τ on = 1/k on and τ o f f = 1/k o f f . These binding and unbinding reactions must be fast enough to allow the accurate recording of spike trains. The dynamic range of the indicator is also a key parameter: it corresponds to the ratio between the maximal intensity fluorescence intensity and the minimal fluorescence intensity. These values are measured in vitro, respectively in the presence of saturated Ca 2+ solution, and in its absence. The in vivo dynamic range differs significantly from these conditions as the cell exhibit a non-zero Ca 2+ concentration in the absence of AP.

In the recent years, many works have been conducted to increase the sensitivity, and to adjust the kinetic of the GCaMP indicators, resulting in a wide variety of GCaMP indicators. After several GCaMP generation, the latest one jGCaMP8 [19] is progressively taking over the previously used jGCaMP7 family [20], that was providing enough sensitivity to detect single APs, and with three declination (jGCaMP7s (sensitive but slow), jGCaMP7f (fast) and jGCaMP7b (brighter baseline fluorescence)) to be chosen from depending on the wished kinetics. The jGCaMP8 is also declined in three versions (jGCaMP8s sensitive, jGCaMP8f fast and jGCaMP8m medium) and provides enhanced performances, some of its characteristics such as its average response to one AP and its half decay time are plotted in Figure 1.5. Let us note that red-shifted GECIs are developing fast too, to meet the needs of the scientific community to image deeper.

b. Voltage imaging

Faster than the GECIs, the Genetically Encoded Voltage Indicators (GEVIs), typ. 2 ms for the activation and the inactivation, are less invasive than electrophysiology and open the access to the study of fast kinetics neuronal events. If this field is rapidly growing, some challenges arise too, on both the excitation and detection sides. A sampling rate of several kHz is required and if it is almost impossible to reach with line-scanning imaging, the solution may come from Accousto-Optic Deflector (AOD) based scanning that allow to reach 20 kHz scanning rate per volume [21,22].

Indeed these probes with fast kinetics require to be coupled with highly efficient photon collection systems in order to detect individual spikes, as less photons are emitted over a shorter period of time than with GECIs. Further developments are expected to improve the brightness of the GEVIs too, but their potential generalization will not go without an improvement of the optical imaging techniques.

c. Signal processing issues

If all these available probes open the way to great neural recordings, however, major problems arise from the density of neural processes in the brain when recording activity [23]. To illustrate the main challenges in signal detection and issues in signal processing, typical calcium traces are simulated in the presence of increasing background signal, and are represented in Figure 1.6. This background can be due to several factors, that are discussed further in this manuscript (see Section 1.3.3).

Here, A is the calcium transient amplitude, F 0 the baseline fluorescence signal from the neuron of interest, and F the total fluorescent signal perceived i.e the Raw Signal. In the absence of background, ∆F F 0 is equal to A F 0 . For AP detection, it corresponds to 5 to 10%. If we first consider the influence of a non-null constant background B 0 as represented in Figure 1.6a, the variation of fluorescence to which the optical setup is sensitive decreases from A F 0 to A F 0 +B 0 . Moreover, the noise on the raw signal, following a Poisson law and inherent to a photon flux, increases from √

A + F 0 to √ A + F 0 + B 0 due to the increasing number of photons collected. But this constant background is not the only one: fluctuating background also occurs. In the simulation (Figure 1.6b) and corresponding Table 1.1, it is referred to as B f . This time-varying background is particularly annoying as it fluctuates similarly to the signal of interest, and adds its own noise σ f to the received signal that is, this time, not a simple Poisson noise. Eventually when considering spike trains, i.e, exactly the information we want to get, the addition of signal contributions on top of one another, occurring because of the slow decay of the fluorescence, in the case of a high background, that fluctuates itself, can lead to missed spikes or false alarms, as represented in Figure 1.6b. In this example, the colored calcium traces are generated by out-offocus neurons for example, that can spike in response to a stimulation; their contribution is of lower intensity but can ultimately lead to errors pointed out with black arrows. It induces the users into associating events that are not linked, or on the contrary, to not see the correlations.

B = 0 B = B 0 B = B 0 + B f ∆F F A F 0 A F 0 +B 0 A F 0 +B 0 +B f σ √ A + F 0 √ A + F 0 + B 0 A + F 0 + B 0 + σ 2 B f SNR A/F 0 √ A+F 0 A/F 0 √ A+F 0 +B 0 A/F 0 A+F 0 +B 0 +σ 2 B f

Conclusion

The emergence of activity recording techniques greatly widened the realm of possibility for neurosciences. In spite of the remaining inherent challenges, a better understanding of the neural processes with its dynamics is now possible. The later described fluorescent probes are not sufficient to record activity on their own; they must be coupled to fluorescence microscopy techniques. This filed underwent major developments too in parallel to the biochemistry side of the probes to satisfy the specific spatiotemporal requirements for the indicators assessment as well as to face the inherent challenges of imaging in depth of scattering media. In the following part of this Chapter, a closer look is given to fluorescence microscopy for neuroimaging, its challenges and developments to push neuroimaging limits further.

Fluorescence microscopy for neuroimaging 1.3.1 The basics of fluorescence microscopy a. Light interaction with the tissue

In a tissue, light propagation is affected by two phenomena: absorption and scattering. Both follow a Beer-Lambert law, and thus lead to an intensity decay, at a given depth z of:

I(z) ∝ e -(µ s +µ a )z (1.1)
The characteristic grandeurs are respectively the scattering coefficient µ s and the absorption coefficient µ a , in cm -1 .

Scattering occurs when light encounters a particle exhibiting a refractive index different from the propagation medium, and gets deviated from its initial trajectory. In tissues, the overwhelming majority of the scattering events are elastic scattering: the incident photon is absorbed and re-emitted without energy loss but a change in direction. Therefore, the particles behave as secondary sources.

Rayleigh scattering Mie scattering

Incident light a ~ λ a < λ/10 Depending on the size of the particle compared to the wavelength, one can distinguish two types of scattering: Rayleigh scattering and Mie scattering. Rayleigh scattering occurs when light encounters very small particles compared to its wavelength (a ≪ λ): the scattering strongly depends on the wavelength of the incident light (the intensity I(λ) ∝ 1/λ 4 ) and is isotropic. On the other hand, Mie scattering regime occurs when particle size is comparable to or larger than the wavelength (a ≥ λ). In this regime, the scattering is anistropic and light is mainly scattered forward (the bigger the particle, the more forward the scattering).

The resulting angular distribution eventually depends on the particle size, its refractive index, its size, its shape, the wavelength. In a scattering medium, we define the average distance between two scattering event as l s = 1/µ s , the scattering length, and its characteristics and influence on microscopy signals are further discussed in Section 2.1.1. Here, let's just precise that in the visible and near IR region, in the brain, l s ranges between a few tens of microns to approximately 200 microns and increases with the wavelength (see Figure 2.1, and [24,25,26]). Moreover, within biological tissues, the re-emission has a high probability to be in the forward direction for each scattering event as Mie scattering is predominant. This effect is quantified by the anisotropy of the medium, g, which is around 0.9 in biological tissues. Eventually, multiple scattering events still lead to randomization of the propagation direction and finally to the complete loss of information. This is characterized by the diminution of the Signal to Background Ratio (SBR) with depth, that is to say, the amount of ballistic photons over the background contribution decrease (see Figure 1.8 for typical scattered signal from fluorescent emitters in depth).

Chapter 1. In-depth functional neuroimaging Absorption appears conjointly to scattering. Light absorption by surrounding molecules to the fluorophore depends on the wavelength, and the molecules. For example, in a biological tissue mainly composed of water and blood, light is more strongly absorbed in the UV and in the visible because of hemoglobin, and in the IR because of water. In between meddles the transparency window of biological tissues (see Figure 1.9), between 700-1100 nm, where the absorption in minimal. This window is to be aimed at when performing fluorescence microscopy; in biological tissues, the absorption length l a is in the millimeter range, hence absorption is less limiting than scattering in depth.

Wavelength (µm)

Transparency window FIGURE 1.9: Absorption in biological tissues. Adapted from [27] Therefore, these phenomena set important limitation on fluorescence microscopy; in the following Section 1.3.2, we will see how techniques such as multiphoton microscopy bypass these issues to conserve optimal resolution or at least to keep image deterioration to the minimal when addressing in-depth signals.

Optical damages Yet the interaction between light and tissue is not limited to these effects: some optical damages can arise and constitute a major drawback to fluorescence microscopy (Figure 1.10).

As described, the excitation beam interacts with the tissue through absorption; where it is fundamental for the fluorophore to absorb incident light in order to re-emit fluorescence, some FIGURE 1.10: Main mechanisms leading to photo-induced damages. From [28] unwanted phenomena can ensue firstly, and on top of that, absorption becomes even more problematic when light gets absorbed by random molecules in the tissue. In any case, light absorption by a molecule, will it be of one or several photons, brings the absorbing molecule to an excited state. From now on, it can de-excitate itself through several pathways [29,28]:

• A non-radiative energy transfer between the molecule and the bulk can happen, leading to local heating and ultimately to photothermal damage.

• Several mechanisms can lead to Reactive Oxygen Species (ROS) formation such as dissociation, change in redox state or the reaching of the triplet state, exhibits a longer lifetime than a singlet state, it makes it more likely to interact with other molecules while being in it. These ROS then trigger biochemical damages in cascade, from DNA alteration to change in redox state of the cell cytoplasm and mitochondria [30]. This is why ROS are the main cause of phototoxicity for live imaging.

• The photon absorption of the absorber once in its excited state can result in its ionization, and an avalanche effect ultimately creates a low density plasma, inducing photochemical damage. The risk is to generate optical breakdown, a state in which the plasma is not confined, and that causes shock waves destroying the tissue and the cells.

• For a fluorophore, these de-excitation sideways cause it to never go back to its ground and brings a definitive stop to its fluorescence cycle: that is photobleaching. While theoretically a fluorophore can cycle indefinitely between the ground state and the excited singlet state, hence emit fluorescence for ever, real life conditions limit its lifetime to 10,000 to 40,000 cycles [31]. On top of the definitive bleaching, some more subtle photoxicity effects have to be considered such as the reduction of the dynamic range of calcium pulses in neurons in brain slices [30]. Note that the presence of the fluorophore into cells as labels or probes become additional sources of ROS too, thus linking narrowly phototoxicity and photobleaching [32].

These damages vary along many parameters such as the pulse length when performing multiphoton microscopy [33], and that constitute as many parameters to optimize for a gentler imaging. Using the right setup for the considered sample is key in order not to minimize imaging Chapter 1. In-depth functional neuroimaging duration and energy delivered to the tissue. Moreover, some common sense good practices can help minimize these effects, and are reviewed extensively in [30], the common denominator being: use no more light than necessary!

b. Optical features of a microscope

Optical sectioning. The optical sectioning ability of a system is defined by its capacity to reject background from out-of-focus planes. This ability is crucial when imaging thick samples. Whereas widefield microscopy does not provide any sectioning, several other optical microscopy techniques exhibiting optical sectioning can be mentioned. It is the case of confocal microscopy: the optical sectioning is obtained by the combination of a tightly focused excitation beam, scanned across the sample, and the discrimination of the photons emitted in focus from the out-of-focus ones by a pinhole positioned in front of the detector. Light Sheet Fluorescence Microscopy (LSFM) is also part of these but with sectioning provided at the excitation, by the generation of a light sheet that illuminates the sample orthogonally to the detection path. Exciting only the fluorescence from the in focus fluorophores, it drastically reduces photobleaching and photodamage, while still providing a fast acquisition rate, as the whole plane is illuminated simultaneously, it has become one of the most widespread technique for in vivo 3D imaging, but remains limited in depth of scattering sample [34,35]. Lastly, multiphoton microscopy provides optical sectioning and is to be discussed in Section 1.3.2.

Point Spread Function.

As light propagates according to the wave optics rules, optical system are limited by diffraction. The Point Spread Function is thus the 3D image of a point source through the system. It defines the diffraction limit of the system and is given by Equation 1.2 in the case of an aberration-free lens, with a circular aperture.

PSF(u, v) = 2 1 0 P(ρ)J 0 (νρ)e i 2 uρ 2 ρdρ 2 (1.2)
whereJ 0 is the zero-order Bessel function, v = k(NA)r is the radial normalized coordinate and u = k(N A) 2 z is the axial one, the wave number being given by k = 2π/λ [36]. NA stands for the Numerical Aperture of the limiting element, usually the objective: it sets the maximum angle for the propagation of the rays within the system (NA = n sin θ max ). The optical resolution is then given by the Full Width at Half Maximum (FWHM) of the PSF in the axial and lateral dimensions.

1.3. Fluorescence microscopy for neuroimaging

Two-photon microscopy

Since the end of the twentieth century, two-photon microscopy has emerged as a major tool in neuroscience as it is the only technique allowing to perform cellular imaging hundreds of microns deep in several organs. This technique based on the phenomena of Two-Photon Excited Fluorescence (2PEF) discovered by Maria Goeppert-Mayer in 1929 [37], and then observed 30 years later by Kaiser et al. [38]. In this 3rd order non linear incoherent process, two photons are simultaneously absorbed, inducing the transfer of an electron from the fundamental electronic state to a higher excited state. This electron then non radiatively relaxes through internal conversion and vibrational relaxation to the lower bound of the excited state, from which it can then relax radiatively to its fundamental state, emitting a higher energy photon than the two exciting ones (see Figure 1.11 This non linear process is proportional to the square of the excitation wave intensity I ω , to the density of fluorophores N, and tho the two-photon absorption cross-section σ 2P , η the quantum efficiency of the fluorophore [39]:

I 2PEF ∝ Nσ 2P η I 2 ω (1.3)
Only in 1990 was this technique implemented in microscopy by Denk et al. [START_REF] Denk | Two-Photon Laser Scanning Fluorescence Microscopy[END_REF] and compared to the performances obtained with confocal microscopy. The typical layout and main elements of a two-photon microscope are represented in Figure 1.12. Due to the low probability of the simultaneous absorption of two photons by a molecule, that is to say, both of them being within the absorption cross-section σ 2P of the molecule, the generation of 2PEF signal relies on a double confinement of the excitation to reach a density of excitation signal high enough to get fluorescence: a spatial confinement and a temporal confinement (see Figure 1.13).

a. Experimental requirements

To get temporal confinement, pulsed lasers are required to concentrate photons within time scales τ of the order of magnitude of the femtoseconds, with the corresponding repetition rate 1/T, of tens of MHz, T being the interval between two pulses.

For example, let's consider a pulsed laser, with a pulse duration τ = 100 f s, and a repetition rate R = 100MHz, that is to say that the interval between two pulses is T = 1/R = 10ns here. If its intensity during one pulse is equal to I ω , the resulting mean intensity will thus be:

< I ω >= τ T I ω (1.4)
Chapter 1. In-depth functional neuroimaging Furthermore, the fluorescence emitted during one pulse is given by:

F 2PEF = 1 2 σ 2P I 2 ω (1.5)
The mean fluorescence is then:

< F 2PEF >= τ T F 2PEF = 1 2 σ 2P < I ω > 2 T τ (1.6)
In this configuration, the emitted signal is enhanced by a two-photon gain of T τ = 10 5 , compared to continuous illumination.

To get spatial confinement, high NA objectives are privileged tools to focus photons within a volume inferior to the µm 3 . This confined excitation is then scanned rapidly over the FOV to obtain images.

The ability of a fluorophore to absorb two photons simultaneously and then to emit fluorescence via 2PEF is set by its two-photon absorption cross-section. It constitutes major point to look at when selecting fluorophores for two-photon microscopy. This technique benefits from the relatively good two-photon absorption cross-sections of usual fluorescence emitters in microscopy. Because the two-photon cross-section is difficult to measure directly, the two-photon action cross section is usually measured instead. It corresponds to the product of the absorption cross-section σ 2P and the fluorescence quantum yield Φ f . This value is usually given in Goeppert-Mayer (GM), where 1GM = 10 -50 cm 4 .s. Some examples of two-photon action crosssection spectra of common endogenous and exogenous fluorophores are plotted in Figure 1.14. When intrinsic molecules exhibits very low action cross-section (NADH : Φ F σ 2P ≈ 10 -1 GM), some extremely efficient fluorophores such as Quantum Dots (QD) present action cross-section ranging from 10 4 to 5.10 5 GM. Besides, intrinsically fluorescent proteins (FP), such as GFP, possess large action cross-section (ranging from 100 to 200 GM at their optimum), making them particularly suitable for two-photon microscopy. Eventually, one can note that calcium sensitive dyes such as GECIs GCaMPs have been continuously improved to increase their action crosssections. Chapter 1. In-depth functional neuroimaging

b. Main advantages of 2PEF microscopy

Intrinsic optical sectioning. Because of the non-linearity of the two-photon absorption process, the 2PEF phenomena only occurs near the focus of the focalisation objective, allowing intrinsic tri-dimensional images of thick sample. Contrarily to widefield fluorescence microscopy, there is no out-of-focus fluorescence emitted from planes that are further than the axial dimension of the PSF and a 3D cartography of thick scattering samples can thus be obtained by successively illuminating planes. The excitation intensity of the focus is linked to the resolution of the system by the following equations; due to the non-linearity, the PSF of a 2PEF microscope becomes:

PSF 2PEF = PSF 2 illumination (1.7)
At a given depth z:

PSF 2PEF (0, z) ≈ 1 z 4 as PSF illumination ≈ 1 z 2 (1.8)
The optical sectioning that corresponds to the integral of the PSF over the plane of interest is thus:

PSF 2PEF (ρ, z) d 2 ρ ≈ 1 z 2 (1.9)
The Full-Width at Half Maximum (FWHM) of the PSF, i.e. the spatial resolution of the system, can thus be approximated respectively in the lateral and in the axial dimension, when N A > 0.7 [START_REF] Warren R Zipfel | Nonlinear magic: multiphoton microscopy in the biosciences[END_REF]:

FW HM x,y = 0.325 2 ln(2) λ N A 0.91 for the lateral resolution (1.10)

FW HM z = 0.532 2 ln(2) λ n - √ n 2 -N A 2
for the axial resolution (1.11) This resolution is theoretically worse than the confocal microscope one, similarly obtained by the multiplication of the illumination PSF with the detection PSF this time, as the wavelength used is twice the one used in confocal (see Figure 1.15). However, confocal microscopy resolution is strongly dependent on the effective opening of the detection diaphragm, and is sensitive to the scattering on the detection path, contrarily to 2PEF microscopy as we see further, and thus is ultimately outperformed by 2PEF in depth of scattering tissues.

Penetration capacity. The main advantage of two-photon microscopy, and more generally of multiphoton microscopy, is its ability to penetrate deeper within scattering tissues than confocal microscopy. As seen in Section 1.7 and discussed in Section 2.1.1, scattering length increases with the wavelength. Thus, this ability to image deeper can be explained by the excitation wavelength in the near-IR, that are twice higher than the ones used in single photon fluorescence microscopy. As the ballistic decay of the photons in depth follows a Beer-Lambert law, and of the quadratic dependence is 2PEF, the fluorescence signal decays as:

F 2PEF (z) ∝ e -z/l s (λ IR ) 2 (1.12)
leading to a typical decay constant of l s /2, whereas, for 1PEF microscopy, The coupling of the 2-fold larger decay constant, combined to the tissue scattering properties evolution more advantageous to 2PEF leads to a much better penetration ability of the excitation signal for 2PEF compared to single-photon excited fluorescence (1PEF).

F 1PEF (z) ∝ e -z/l s (λ vis ) (1.13)
Moreover, thanks to the intrinsic optical sectioning performed at the excitation and maintained even in strong scattering conditions, all fluorescence photons generated are known to originate from the focus. Thus, the detection strategy consist in maximizing the collection efficiency to collect as many photons as possible as even the strongly scattered ones contribute to the signal. This differs considerably from the 1PEF confocal microscopy here, where a pinhole rejects instead the scattered photons while blocking out-of-focus signal, hence reducing dramatically the amount of photons contributing to the image formation. Let us precise here that even if, on a first level of approximation, the fluorescence signal is not affected by scattering, in reality, the resulting light coming from the focal spot will emerge from the sample with a larger angle because of multiple scattering. Eventually, the radiating region at the surface will be equal to 1.5 time the focal depth [START_REF] Beaurepaire | Epifluorescence collection in two-photon microscopy[END_REF].

Photoxicity and photobleaching reduction. The excitation wavelength being within the biological transparency window, considerably limits the absorption of the molecules and the associated linear phototoxic mechanisms, compared to 1PEF microscopy. Thus, out-of-focus phototoxicity is drastically reduced, as well as photobleaching, compared to confocal microscopy: only the molecules within the plane of interest are excited, so all the generated signal is to be used for the image formation. Two-photon microscopy does not induce photothermal effect in typical imaging conditions, as usually, the studied tissue does not exhibit strong one-photon absorption within the NIR (unless when pigmented cells are considered for example [START_REF] Barry | Mitigating thermal mechanical damage potential during twophoton dermal imaging[END_REF]) . However some phototoxic effects subsist: single photon absorption can still occur above and below the focal plane, as well as heating, and higher photobleaching caused by higher-order photon interaction the focus is possible [30].

Chapter 1. In-depth functional neuroimaging Hence, two-photon microscopy reveals itself as particularly well suited for in-depth imaging of scattering tissues.

c. Drawbacks and how to deal with them

As two-photon microscopy is a scanning technique, users are confronted to major speed limitations. Imaging consists in a permanent trade off between acquisition speed, FOV and volumetric resolution. Many developments are now proposed to increase the acquisition rate, including random-access multiphoton (RAMP) microscopy. Using Acousto-Optic Deflectors (AOD), fast access to 2D and 3D neuronal target is now possible [START_REF] Bullen | High-speed, random-access fluorescence microscopy: I. High-resolution optical recording with voltage-sensitive dyes and ion indicators[END_REF][START_REF] Gaddum | Fast three-dimensional laser scanning scheme using acousto-optic deflectors[END_REF][START_REF] Salomé | Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors[END_REF] and in vivo functional recordings have been performed [START_REF] Gaddum | Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity[END_REF][START_REF] Katona | Fast two-photon in vivo imaging with three-dimensional randomaccess scanning in large tissue volumes[END_REF], but suffered from a great sensitivity to motion artefacts. New methods based on AODs have also recently been proposed to generate holographic patterns (grids of points) to illuminate targeted neurons in order to reduce the sensitivity of the method to sample motion while still assessing 3D neuronal signals at very high sampling rate (> 100 kHz) [START_REF] Akemann | Fast optical recording of neuronal activity by three-dimensional custom-access serial holography[END_REF].

Two-photon fluorescence microscopy limits in depth a. The fundamental limit

Two-photon microscopy allows to image deeper but still reaches a limit: 5-8 l s [START_REF] Theer | On the fundamental imaging-depth limit in twophoton microscopy[END_REF]. In this model, the maximum imaging depth is set by the signal decrease down to a point where it reaches the minimal detectable signal.

Consequently to the exponential decay of the excitation in depth caused by scattering, at a given depth z, the measured fluorescence signal F 2PEF (z) is given by:

F 2PEF (z) = 1 2 σ 2P T τ . I 0 e -z/l s (λ ex ) 2 .Φ(z) (1.14)
where Φ(z) corresponds to the fluorescence collection. The maximum imaging depth is given by [START_REF] Oheim | Two-photon microscopy in brain tissue: parameters influencing the imaging depth[END_REF]:

z max ∝ l λ ex s ln αI 0 T τ Φ(z max ) (1.15)
α a parameter taking into account the efficiency of the fluorophore and the detector one. Here, let us highlight the major importance of collection efficiency: the design of the detection path, the size of the detector, must be carefully adapted to optimize in-depth photon collection, and to limit the restriction of the FOV in depth.

Thus, the maximum achievable imaging depth is proportional to the scattering mean-free-path and depends logarithmically on available laser power, two-photon gain and collection efficiency [START_REF] Oheim | Two-photon microscopy in brain tissue: parameters influencing the imaging depth[END_REF].

Eventually, the maximum imaging depth is limited by the generation of out-of-focus background signal created by surface emitters, that progressively leads to a loss of the sectioning [START_REF] Ying | Spatial distribution of two-photon-excited fluorescence in scattering media[END_REF][START_REF] Ying | Effect of scattering on nonlinear optical scanning microscopy imaging of highly scattering media[END_REF]. For an almost uniformly labelled sample (for example, an extensive GFP labelling in mouse brain), Theer et al. [START_REF] Theer | Two-photon imaging to a depth of 1000 mm in living brains by use of a Ti:Al2O3 regenerative amplifier[END_REF] derived the following relation for the ratio between the in focus signal (S) and the out-of-focus fluorescence background (B): This depth limit defined by Equation 1.13 can be reached with regenerative amplifiers, but it can even be exceeded. Reached depths of 1 mm in in vivo cortex were reported by Theer et al. [START_REF] Theer | Two-photon imaging to a depth of 1000 mm in living brains by use of a Ti:Al2O3 regenerative amplifier[END_REF] and regenerative amplifiers allowed to perform functional imaging in L5 [START_REF] Mittmann | Two-photon calcium imaging of evoked activity from L5 somatosensory neurons in vivo[END_REF]. By reducing the rate R at which pulses are occurring, while keeping the average power to the same level, higher energy pulses can be obtained; in imaging, setting one pulse per pixel which is the minimal required, results in a rate reduction of a factor 1,000, and thus a peak power enhancement of 10 3 [START_REF] Beaurepaire | Ultra-deep two-photon fluorescence excitation in turbid media[END_REF]. This prevents from being limited in depth by the available power. As deeper regions are accessed, the limit set by the emission of fluorescence at the surface of the tissue is reached as predicted and demonstrated experimentally by Theer and Denk [START_REF] Theer | On the fundamental imaging-depth limit in twophoton microscopy[END_REF].

S B = 2π (
An other way to circumvent depth limit in 2PEF is to increase the excitation wavelength to 1100-1300 nm, which means reducing the attenuation of the excitation, and to smartly adapt the probes, reaching 1.6 mm depth in vivo while imaging cortical vasculature labelled with Alexa680-Dextran [START_REF] Kobat | In vivo two-photon microscopy to 1.6-mm depth in mouse cortex[END_REF].

Ultimately, three-photon excited fluorescence (3PEF) microscopy can be envisaged. The excitation wavelength benefits from the second transparency window of biological tissues, around 1700 nm, and the even larger scattering length, and most of all from the enhanced optical sectioning. Indeed, the PSF becomes:

PSF 3PEF = PSF 3 illumination (1.17)
and, with at a given depth z:

PSF 3PEF (0, z) ≈ 1 z 6 and PSF 3PEF (ρ, z) d 2 ρ ≈ 1 z 4 (1.18)
Thus, no more signal is generated from the surface layers and deeper regions can be addressed. The probability of the simultaneous recombination for three photons is very small, so the pulse energy has to be increased by lower the laser repetition rate. In vivo imaging was performed with this technique by Chris Xu's group too, first reported in [START_REF] Nicholas | In vivo three-photon microscopy of subcortical structures within an intact mouse brain[END_REF] and later allowed calcium imaging at 1 mm depth, within an intact brain [START_REF] Dimitre | In vivo three-photon imaging of activity of GCaMP6-labeled neurons deep in intact mouse brain[END_REF]. More recently, a strategy based on a combined excitation of 2PE and 3PE, to reach the second excited state instead of the first one was proposed, in order to perform multicolor imaging with only one single excitation wavelength, while still reaching 1.2-mm depth [START_REF] Hontani | Multicolor three-photon fluorescence imaging with single-wavelength excitation deep in mouse brain[END_REF].

Chapter 1. In-depth functional neuroimaging

b. The actual limits for neuroimaging

In the case of functional imaging, this background generation and the concomitant sectioning loss is particularly annoying, as that surface signal fluctuates exactly like the signal of interest, and makes it particularly hard to untangle (see Figure 1.6).

We saw that scattering sets a maximum imaging limit. At great depth and in a 3D labelled sample, the image loses contrast and a background signal from the surface is emitted. This constant background therefore limits the imaging depth in structural imaging. Its effect is even more troublesome in functional imaging, as it becomes a fluctuating signal related to activity. Therefore, at depth, functional imaging signals are contaminated by a fluctuating signal from the surface that is difficult to remove and that creates artefactual correlations between neurons, for example. It is therefore necessary to use algorithms that estimate this contamination by measuring the activity in the ROIs around the neurons and then find out how to subtract this signal from the one measured at the neuron's soma. This correction is used by all teams but is never perfect.

Another effect limits the imaging depth and decreases the SNR in depth. In addition to scattering, optical aberrations can deteriorate the PSF and enlarge the focal volume. Both signal and resolution are affected [START_REF] Ji | Adaptive optical fluorescence microscopy[END_REF].

To conclude, at a given depth, one of the only remaining way to enhance imaging quality is to try to correct for these aberrations; probe, labelling, laser wavelength and repetition rate set, no other possibility is left.

In depth, the enlarged focal volume and reduced focal intensity are two consequences of the scattering and of aberrations encountered by the excitation beam that lead to background generated by surrounding emitters. Effects are combined and drastically limits depth and resolution for functional imaging.

These effects strongly depend on the distribution of fluorophores: imaging into a densely labelled tissue will get limited early than in a sparse labelled one, and on other tissue properties (e.g. microvasculature, cell body arrangement) [7]. Estimations of the impact of the labelling density is detailled in [START_REF] Theer | On the fundamental imaging-depth limit in twophoton microscopy[END_REF].

c. How to overcome depth limit

Several strategies have been proposed to overcome these issues. One can first think of acting on the already obtained signal, by signal extraction techniques and ROI segmentation.

An other alternative is to modify the set up and to add adaptive elements that will balance the undesired effects. One can use a Deformable Mirror to reject out of focus fluorescence: by adding aberrations on the illumination beam that preferentially degrade 2PEF signal emitted in focus while leaving the background unchanged, both acquisitions can be subtracted afterwards to each other to retrieve only the in focus signal [START_REF] Leray | Rejection of two-photon fluorescence background in thick tissue by differential aberration imaging[END_REF]. However, this method becomes less adapted when motion occurs in the sample, which is very current for neural recordings, and leaves the issues linked to the broadening of the spot in depth not tackled. Otherwise, Adaptive Optics can be chosen to compensate for the aberrations generated by the sample and that increases in depth, in order to recover for a diffraction-limited spot. The chosen adaptive element will modify the incoming wavefront to a shape that perfectly compensates for the wavefront distortion generated by the sample so that the resulting wavefront in depth will be perfectly spherical.

As aberrations affect resolution, that is essential in both structural and functional imaging of small dendrite-axon structures, but also and above all signal in 2PEF microscopy, it is of great importance to correct for them while imaging in depth. Having more signal in functional imaging leads to the reduction of photon noise and therefore potentially detects small signal variations better.

In the next section, we develop the implementation of Adaptive Optics in microscopy of it is the chosen solution to address these issues by our team.

Increasing imaging 2PEF performance in depth with Adaptive Optics

Initially developed for astronomy in order to correct for atmospheric turbulence for star observation, Adaptive Optics (AO) has later been introduced in microscopy to correct for sample induced aberration and recover diffraction-limited images deep within the tissues; assessing deeper layers of neuronal tissues is still an on-going quest. The beginnings of AO for multiphoton microscopy date back to the early 2000s [START_REF] Neil | Adaptive aberration correction in a two-photon microscope[END_REF] and has continuously been addressed since, by the development of various aberration measurement and correction strategies.

In the following section, we first introduce aberrations and quality criteria that will be used in the manuscript. Then, the essential elements for AO, as well as the two main implementation strategies are reviewed. The AO contribution for neuronal imaging is discussed, and we conclude with the description of our original AO strategy for neuroimaging to introduce the thesis work.

Brief reminder of aberrations

Two types of aberrations exist: chromatic aberrations, generated by the difference in refractive index according to the wavelength, leading to longitudinal and transverse aberrations, and geometric aberrations. The later arise from deviations to the paraxial approximation, when angles between the optical axis and the propagating rays become too important, or when the rays are propagating too far from the optical axis. In the following, the formalism from [START_REF] Smith | Modern Optical Engineering: The Design of Optical Systems[END_REF] is selected.

The geometrical aberrations are due to the light propagation though a medium of inhomogeneous refractive index. They can be described by the Zernike polynomials that are orthogonal on a unit disk, this disk corresponding to a circular pupil in a classical optical system. These polynomials are given by the following relations, respectively corresponding to the even and odd polynomials:

Z m n (ρ, φ) = -2(n + 1)R |m| n (ρ) cos(mφ) if m ≤ 0 (1.19) Z m n (ρ, φ) = 2(n + 1) 1 + δ m0 R |m| n (ρ) sin(mφ) if m > 0 (1.20)
where φ is the azimuthal angle, ρ the radial distance comprised between 0 and 1, δ m0 Kronecker delta (

δ ij = 1 if i = j, δ ij = 0 if i ̸ = j)
, m an integer and n a positive integer with m ≤ n , and R n m (ρ) the radial polynomials given by:

R n m (ρ) = n-m 2 ∑ k=0 (-1) k (n -k)! k!( n-m 2 -k)!( n+m 2 -k)! ρ n-2k (1.21)
Chapter 1. In-depth functional neuroimaging

The most used Zernike polynomials are listed in Table 1.2, whilst the usual representation of the associated wavefront, ordered in a pyramidal way is presented in Figure 1.17.

The overall aberrated wavefront W can then be expressed as a sum of the polynomials Z m n :

W(ρ, φ) = ∑ (n,m) c nm Z m n (ρ, φ) (1.22)
The wavefront deformation can be expressed through the Root Mean Square (RMS) amplitude, noted σ W . It corresponds to the deviation of the wavefront from a reference sphere; when the wavefront is decomposed over the Zernike base, an amplitude (c nm ) is associated to each aberration coefficient, expressed in fraction of the wavelength, or in microns. The RMS is then obtained by the following formula:

σ W = (W -W) 2 = ∑ (n,m)̸ =(0,0) c 2 nm (1.23)
This can be translated into optical quality of the set up. In general, the Strehl ratio, defined as the ratio between the intensity peak of an aberrated point image and the aberration-free point image, in the case of a diffraction limited system. This Strehl ratio is:

S ≈ 1 - 4π 2 λ 2 σ 2 W (1.24)
In 1843, Rayleigh first established a visual criterion on spherical aberration; he noticed that a wavefront deformation inferior to λ/4 on the pupil edge, was inducing a decrease of the irradiance on the image of less than 20%, leading to a still visually diffraction limited spot.

This criterion was later extended to the other aberrations by Maréchal (1947): considering that the system was diffraction-limited if the decrease of the irradiance of the image was inferior to 20%, he obtained :

S > 0.8 ⇔ σ W < √ 0.2λ 2π ≈ λ 14 (1.25)
This relation is now commonly referred to as Maréchal's criterion.

In the following, we will mostly refer to the RMS of the wavefront to quantify the wavefront deformations before and after AO loops as well as for correction quality assessment.

Aberration correction with Adaptive Optics

Two elements are constitutive of an AO system : -the wavefront sensor, that allows to measure the aberrations induced by the system and the sample, -the wavefront corrector, that is the active element compensating for these aberrations. The AO principle in schematized in Figure 1.18. The microlens array is optically conjugated with the microscope objective back pupil, through which all the emitted light rays are passing, no matter their propagation angle. Thus, all the signal-contributing rays are intercepted by the WFS and the corresponding wavefront is properly sampled. Each microlens intercepts a small part of the wavefront, with its local slope, and focalizes the associated rays on the detector. In most cases, these WFS are used in combination of a guide star, positioned within the sample. In the case of a planar wavefront, each microlens will form an image of this guide star at its focal point (grey dots on Figure 1.20). On the other hand, in the case of a distorted wavefront, exhibiting various non-null local tilts, the images of the guide star will be shifted within the focal plane of the microlenses, according to the local slopes, and leading to the red dots on Figure 1.20. These shifts δ x,y are linked to the local wavefront slope a x,y by: Amongst other original wavefront sensing methods for microscopy one can cite the technology reported by [START_REF] Li | Conjugate adaptive optics in widefield microscopy with an extendedsource wavefront sensor[END_REF], named PAW after Partitioned Aperture Wavefront sensing. It is compatible with extended scenes, and requires uncollimated light. It is quite similar to a pyramidal wavefront sensors except that is is achromatic. The latter sensor uses a pyramidal prism to split the beam into several parts at the focus of a lens. Without aberrations, a four-faceted prism will generate four identical pupil images while with aberrations, the intensity distribution among the pupils will be modified. Thus, the local wavefront slopes can be obtained by recording the distribution of intensity in the pupil images.

a x,

b. Wavefront corrector

Two main phase modulators can be distinguished: the Spatial Light Modulator (SLM) and the Deformable Mirror (DM), and are schematized in Figure 1.21.

The Spatial Light Modulators are liquid crystal based devices, that can either work in transmission or in reflection, and that can tune the amplitude, the polarisation, and the phase of the incoming light. In microscopy, phase SLMs are the most useful. The application of a voltage generates the rotation of the liquid crystal optical axis and thanks to its birefringent properties, the refractive index can be locally tuned, leading to local phase shifts. In microscopy SLMs are used to shape the excitation path to recover perfect focusing through strongly scattering media [START_REF] Vellekoop | Focusing coherent light through opaque strongly scattering media[END_REF], or to obtain optical tweezers, or correct for aberrations on the excitation path [START_REF] Wang | Rapid adaptive optical recovery of optimal resolution over large volumes[END_REF].

However in microscopy, and more particularly in AO, SLM are not very suitable to closed-loop configuration in which the fluorescence signal, over which the wavefront distortion is measured, goes back through the SLM: only the polarized component will be kept, leading to major photon losses.

The Deformable Mirrors are either composed of a flexible reflective membrane positioned on top of actuators, or by micro-mirrors, that can be driven mechanically or by induction, depending on the dimensions of the DM. For continuous DM (or membrane DM), resolution of the phase deformation that can be achieved is way lower than the performances of the SLMs, typical membrane DM contain 50 to 100 actuators for 5 to 10-mm diameter as opposed to the thousands of pixels on the SLMs. On the other hand, fragmented DM can count up to 2000 actuators within 20-mm aperture diameter, but their segmentation generates diffraction effects. DM are not sensitive to polarization and achromatic, making them suitable to be combined to fluorescence signals. Moreover, the DM are faster than the SLMs. DM based on electrodynamic processes, are now gold standards for microscopy: they exhibit a good compromise between speed, resolution, dynamics required to correct for encountered aberrations in biological tissues and to the low hysteresis, high dynamic range and linearity [START_REF] Fernández | Adaptive optics with a magnetic deformable mirror: applications in the human eye[END_REF]. In this thesis, a DM commercialised by ImagineEyes will be used (Mirao52e).

c. Direct wavefront sensing

Direct wavefront sensing AO combines the use of a wavefront sensor (WFS) with an adaptive element. Performing the wavefront measurement raises a couple of supplementary difficulties, the main one being the insertion/creation of a guide star in the sample to measure the wavefront on it. The most straightforward method, that was first proposed, consisted in injecting beads directly into the sample, to measure the wavefront from these guide stars and then correct the aberrations. Azucena et al. [START_REF] Azucena | Wavefront aberration measurements and corrections through thick tissue using fluorescent microsphere reference beacons[END_REF] chose 1 µm fluorescent beads, solution proposed in the lab as well [START_REF] Vermeulen | Adaptive optics for fluorescence wide-field microscopy using spectrally independent guide star and markers[END_REF]. However, this remains invasive and it is hard to control both the density and the precise location of the guide stars in the sample, to match the regions of interests. Later, and consequently benefiting from the development of fluorescent proteins as labels, less invasive methods were proposed to use their signal instead to perform wavefront measurement on [START_REF] Tao | Adaptive optics microscopy with direct wavefront sensing using fluorescent protein guide stars[END_REF]. Autofluorescence from the sample itself was also considered as a guide star source, while exciting it with non linear process [START_REF] Tao | Adaptive optical two-photon microscopy using autofluorescent guide stars[END_REF]. This utilization requires a femtosecond laser to generate the guide star, but presents the considerable advantage of guarantying a single star in the ROI, even though it is quite complex and costly to add a pulsed laser when not performing multiphoton microscopy initially. This fluorescence signal can be descanned before reaching the wavefront sensor; this method is particularly relevant for 2PEF microscopy as the imaging system is already reliant on the scan of the excitation spot across the sample to form an image. Moreover, Wang et al. [START_REF] Wang | Rapid adaptive optical recovery of optimal resolution over large volumes[END_REF] showed that the wavefront measurement accuracy was way better in the field of view for a descanned excitation than for a not descanned one (see Supplementary Information [START_REF] Wang | Rapid adaptive optical recovery of optimal resolution over large volumes[END_REF]): in the example given on C elegans embryo, the local complexity of the aberrations leads to speckle patterns on the SH sensor and to inaccurate wavefront measurement. This effect is completely smoothed by the descanning of the fluorescence over the FOV.

This methods provides high photon and time efficiency, and by injecting exogenous dyes into the cortex to form the guide star, enabled the determination of spine geometry down to 600 µm below the pia and the functional imaging of spines down to 500 µm below the pia [START_REF] Wang | Rapid adaptive optical recovery of optimal resolution over large volumes[END_REF]. This approach paved the way to dual labelling strategies to optimize the photon budget, by either using neuronal labelling with red-shifted probes, such as tdTomato (λ peak = 580 nm), or by direct injection of towards near IR shifted probes to minimize scattering, such as Indocyanine Green (ICG (λ peak = 810 nm)) [START_REF] Wang | Direct wavefront sensing for high-resolution in vivo imaging in scattering tissue[END_REF]. A clever labelling strategy was lately introduced and consisted in using the brain vasculature as the host for the probes. By retro-orbital injection, a near-IR probe (cyanine5.5-conjugated 2-MDa dextran (Cy5.5-dextran)) was injected in brain microvessels, leading to a uniformly well repartied signal all over the cortical area and creating a micro-vascular based guidestar when excited by two-photon excitation, perfectly suitable for wavefront sensing [START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF]). Contrarily to the direct injection of probes in a region of the brain, the retro-orbital injection continuously replenish the fluorophore, so bleaching is minimized. Moreover, the non toxicity of the probe makes it very suitable for chronical studies. Thanks to this method, infragranular neurons and spines 850 µm below the pia were observed and their activity recorded, showing an important benefit of this labelling strategy for AO correction. However, it is important to note that in both cases, the laser wavelength has to be modified to be able to excite the guidestar (to 1020 nm for IGC, and 1250 nm for Cy5.5-dextran), thus the AO correction and the functional recordings had to be performed sequentially.

As for now, the fluorescence from the non-linear guide star is descanned. Through descanning, aberrations are averaged over the FOV. However, a too large FOV for wavefront measurement may deteriorate the correction quality. In the mouse brain, in vivo, Park et al. showed that, instead of averaging aberrations over a FOV of 450 x 450 µm², it was beneficial to correct independently for nine subregions of 150 x 150 µm², each of those addressed by a portion of a SLM [START_REF] Park | Large-field-of-view imaging by multi-pupil adaptive optics[END_REF]; a signal enhancement of a factor 1.5 to 2 was obtained over the whole FOV compared to the use of the entire descanned image for wavefront sensing.

Other microscopy methods can benefit from this AO strategy based on descanned non linear guidestar and in particular Structured Illumination Microscopy (see Figure1.23) This combination was first proposed by [START_REF] Zheng | Adaptive optics improves multiphoton super-resolution imaging[END_REF] More recently, Na Ji's team proposed the OS-SIM (for Optical-Sectioning SIM) with AO that consisted in adding an AO module based on direct wavefront measurement to a SIM to perform fast in vivo imaging of mouse brain and Drosophila larva at a high imaging rate thanks to the widefield imaging technique, but with an enhanced resolution in depth brought by AO [START_REF] Li | Fast widefield imaging of neuronal structure and function with optical sectioning in vivo[END_REF].

This method is compatible with Light Sheet Microscopy too: Betzig's team proposed the implementation of an AO module on a Lattice Light Sheet Microscopy to study a wide variety of 3D dynamic processes with a subcellular resolution, ranging from endocytosis to the migration of metastatic cancer cells in vivo [START_REF] Liu | Observing the cell in its native state: Imaging subcellular dynamics in multicellular organisms[END_REF].

While implementing direct wavefront sensing AO, one interrogation remains: choosing between closed-loop configurations [START_REF] Tao | Adaptive optics microscopy with direct wavefront sensing using fluorescent protein guide stars[END_REF][START_REF] Zheng | Adaptive optics improves multiphoton super-resolution imaging[END_REF] and open-loop configuration [START_REF] Wang | Rapid adaptive optical recovery of optimal resolution over large volumes[END_REF][START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF]. This consideration will be further discussed in Chapter 3.

Summary: Direct wavefront sensing

Advantages:

• High measurement accuracy • Fast • Usable within a closed-loop configuration: we can reach the optimum of the correction Disadvantages:

• Based on imaging on the wavefront sensor: suffers from scattering on the emission • Complex implementation • Expensive photon budget

d. Indirect wavefront sensing

When it comes to indirect methods, two main approaches can be distinguished: the pupilsegmentation approach developed by Na Ji's group, and the modal sensorless approach, initially introduced by Martin Booth's team, and further applied to multiphoton microscopy by Delphine Débarre and Emmanuel Beaurepaire. Indirect wavefront sensing is the most popular approach: it gets rid of the direct wavefront sensing guidestar issue by using the microscope image instead, and only an adaptive element and associated optics need to be added to the microscope. Moreover, no signal is lost by redirecting some photons towards a wavefront sensor.

Modal optimisation approach.

Modal optimization is based on an iterative procedure that aims at optimizing the DM correction in order to optimize an image quality metric M, typical metrics being the intensity of the image, contrast, frequencies... First applied to a simple incoherent transmission microscope coupled to the optimization of a low spatial frequencies [START_REF] Debarre | Image based adaptive optics through optimisation of low spatial frequencies[END_REF], it was rapidly coupled to other microscopy methods, and in particular multiphoton microscopy (used for Third Harmonic Generation (THG) [START_REF] Olivier | Dynamic aberration correction for multiharmonic microscopy[END_REF], two photon-microscopy [START_REF] Débarre | Image-based adaptive optics for two-photon microscopy[END_REF]), and allowed 3D aberration mapping in thick scattering tissues [START_REF] Zeng | 3D resolved mapping of optical aberrations in thick tissues[END_REF].

Modal optimisation was not only used for multiphoton microscopy, it has proven its relevance and efficiency also for structured illumination. A first example was given by [START_REF] Débarre | Adaptive optics for structured illumination microscopy[END_REF] and later, it was applied to HiLo microscopy, a technique in which a widefield image and a structured image are fused to get a better contrast and remove low-frequency content. It allowed the enhancement of the grid contrast and resolution 50-µm deep inside the Drosophila brain, that led to a 5-fold enhancement on the signal [START_REF] Pedrazzani | Sensorless adaptive optics implementation in widefield optical sectioning microscopy inside in vivo Drosophila brain[END_REF]. More recently, this method was efficiently applied to superresolution microscopy (STED) for replication sites imaging in HeLa cell nuclei, or actin imaging in COS7-cells, showing in both cases a 2-fold enhancement in signal [START_REF] Antonello | Multi-scale sensorless adaptive optics: application to stimulated emission depletion microscopy[END_REF].

-a c +b -b M M z M - M + 1 2 3
aberration free

FIGURE 1.24: Modal sensorless AO algorithm

There are two main ways in which the measurement and correction is carried out when correcting multiple modes through this approach. Either all modes are measured first, with the correction applied to the corrector at the end, referred to as the 2N + 1 algorithm (two measurements per mode: biases of +b and -b and one measurement when no aberrations are applied). Or the correction of each mode is applied sequentially, still with two measurements per mode to which one measurement when no aberrations are applied is added. In this case, a new 'zero bias' measurement is required after each aberration measurement and correction overall leading to 3N measurements. In the case of large aberrations, the 3N algorithm was proven to be more accurate [START_REF] Facomprez | Accuracy of correction in modal sensorless adaptive optics[END_REF]. The benefit of correction is obtained sooner, as updates are implemented sequentially rather than simultaneously, but requires more measurements overall. In practice, in both cases, to improve robustness in low signal-to-noise situations, more than the minimum number of 2N + 1 or 3N measurements, is often required (4N, 5N and so on methods are used). The basic formula to calculate the correction for each mode is pertinent to fitting the parabola for each mode with three points [START_REF] Jesacher | Sensorless adaptive optics for microscopy[END_REF]: where a is the derived aberration magnitude and the correction amplitude to apply is c = -a, the metrics value M Z , M + and M -being respectively obtained for a zero-aberration bias, for a +b bias, and for a -b bias.

a = - b(M + -M -) 2M + -4M Z + 2M - (1.28)
Sample dependence of the modal sensorless approach when combined to two-photon microscopy but also with THG, led to further algorithmic developments. Indeed, for some spatial distributions of fluorophores, the most common algorithms can fail [START_REF] Teixeira | Development of a new sensorless wavefront sensing approach for two photon microscopy: application to in vivo imaging of the hippocampus[END_REF]. Thus, Champelovier et al. [START_REF] Champelovier | Image-based adaptive optics for in vivo imaging in the hippocampus[END_REF] proposed a new image-based AO method designed for heterogeneously labeled samples such as the pyramidal cell layer of the hippocampus. It relies on a specific metric which consists in filtering out low and high spatial frequencies of the image before calculating the intensity variance, to lock on the optimal layer depth before performing aberration estimation. They showed that it was relevant and efficient for neural activity recording in the CA1 pyramidal cells whereas other modal optimization methods are not optimal because of the highly inhomogeneous labeling.

Pupil segmentation approach. The pupil segmentation method was first introduced by Ji et al. [START_REF] Ji | Adaptive optics via pupil segmentation for highresolution imaging in biological tissues[END_REF]. The rear pupil of an objective lens is segmented into subpupils, each matching an area of the SLM, and the rays are directed sequentially towards each subpupils to measure, by image shift, the deviation for each group of rays as they converge towards the focus and encounter specimen inhomogenities. These deviations result in a displacement of the imaged formed by the group of rays, these shifts are thus related to the local slopes of the wavefront, and in a way this method can get down to a Shack Hartmann wavefront sensing method. The deviations can thus be compensated for with the SLM by adding local tilts, so all the rays focus again at the same spot and a diffraction-limited image can be recovered in depth. As an example, it allowed to recover diffraction-limited performances 450 µm down inside the cortex of a living mouse [START_REF] Ji | Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex[END_REF].

Further developed towards a multiplexed aberration measurement [START_REF] Wang | Multiplexed aberration measurement for deep tissue imaging in vivo[END_REF], the system was parallelized to increase its speed; a Digital Micromirror Device (DMD) being now conjugated with the SLM. It led to the imaging of synaptic structures in deep cortical and subcortical areas of the mouse brain, and demonstrated high-resolution imaging of neuronal structures and somatosensoryevoked calcium responses at great depths in vivo when coupled to two-and three-photon microscopes (respectively 500 µm and 750 µm below the pia in the cortex, and down to 950 µm, reaching the hippocampus with the three-photon microscope) [START_REF] Rodríguez | An adaptive optics module for deep tissue multiphoton imaging in vivo[END_REF].

Summary: Indirect wavefront sensing

Advantages:

• Based on the 2PEF formed image: as resilient to scattering as 2PEF • Easy to implement Disadvantages:

• Slow • Hardly compatible with functional imaging • More likely to induce photobleaching

Discussion on AO contribution for 2PEF structural and functional imaging

We saw that AO methods have shown their efficiency for aberration correction in depth. By focusing tighter the excitation beam, an enhanced signal is obtained and important side benefits are obtained too, such as the reduction of the excitation power: less photodamage and less photobleaching are observed. Let's now address more specifically the benefits of AO for 2PEF, in regard of the specificities of the non-linear imaging method.

a. AO contribution for 2PEF structural imaging

The impact of AO for structural imaging strongly depends on the size of the objects. In nonlinear microscopy coupled to AO, the benefits of AO is not identical for all the structures: it is much more efficient at enhancing signal from small structures than for larger ones. Indeed, larger structures partly compensate for the loss of focal intensity by having more chromophores excited within the enlarged volume of an aberrated focus. Typically AO will lead to a 2-fold enhancement on the signal for soma, whereas a 5-fold gain in signal is observed on dendrites. As a general rule, smaller structures show larger signal gain upon AO correction than larger structures: the enlarged focal volume caused by aberration allows more fluorophores to be excited in a large fluorescent structure like a soma, thus partly compensating for the decreased focal intensity; for small fluorescent features such as micron-sized beads and dendrites, the full impact of reduced focal intensity is felt more acutely. This phenomenon has been studied by [START_REF] Ji | Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex[END_REF] (see Figure 1.27), where the authors were correcting aberrations introduced by a cranial window, and obtained 5.8× signal increase for a 0.5-µm bead, 3.8× increase for a 2-µm bead, and 1.8× signal increase for a fluorescent sea sample. This observation is concomitant with the analytical description from [START_REF] Sinefeld | Adaptive optics in multiphoton microscopy: comparison of two, three and four photon fluorescence[END_REF].

b. AO contribution for 2PEF functional imaging

Functional imaging with fluorescence microscopy techniques strongly suffers from the signal generated by the neuropil. This space between neurons, containing dendrites, axons, synapses, microvasculature and the extension of glial cell, is also widely labelled when a specific neural As discussed previously, some subtracting methods exist, but are complex to implement and not always reliable, therefore, it becomes of tremendous interest to start data analysis from signals as less background contaminated as possible. We can also add that because noise increases with background, the sensitivity of the measurement is consequently increased by the background reduction.

Let us now try to explain the improvements obtained with AO for neuroimaging. With calcium indicators, the labelling is usually quite heterogeneous. If all goes as expected, the stronger labelled structures (soma or neurites) are the structures of interest while all the surrounding axons, dendrites or even more weakly labelled neuron constitute potential background sources. As discussed by Wang et al. [START_REF] Wang | Direct wavefront sensing for high-resolution in vivo imaging in scattering tissue[END_REF], when recording activity linked to visual stimuli, the neuropil activity contains an averaged response of many neurons and is not selective to the orientation of the visual stimuli in this example. When the structure of interest is a fine neurite (Figure 1.28), the laser focus being larger than the structure, signal from both the neurite and the surrounding neuropil is recorded. This is particularly critical for an enlarged focus in depth because of aberrations, and of reduced intensity. In that situation, the neurite signal ends up being overwhelmed by that of the surrounding neuropil. In this situation, AO greatly helps recover information; by tightening the focus in depth, the reduction of the focal volume combined to the increase in focal intensity contributes to neurite signal enhancement while the surrounding neuropil background is reduced. This would also hold true when recording the somatic signal of a given neuron: if untuned signal from the neuropil is added, it will reduce the tuning strength measured for the neuron soma. From [START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF] To show the raw AO gain, the optimal way is thus to combine AO to signal extraction methods, such as segmentation, that are currently widely implemented. To our knowledge, for neuroimaging in the mouse, only Kleinfeld's team showed AO enhancement on functional imaging 500 µm below the pia, through thinned skull, in combination with segmentation (see Figure 1.29), revealing AO benefits, even when the data extraction algorithms are used.

c. A word on AO for 3PEF microscopy

AO can be coupled to higher order non linear fluorescence microscopy to increase their performances too. The results are even more spectacular, as the effect of aberrations on the signal increases exponentially with the order of non linearity in a thick fluorescent sample [START_REF] Sinefeld | Adaptive optics in multiphoton microscopy: comparison of two, three and four photon fluorescence[END_REF]. We can mention here works from Chris Xu's team pioneer in 3PEF microscopy for in depth neuroimaging [START_REF] Nicholas | In vivo three-photon microscopy of subcortical structures within an intact mouse brain[END_REF][START_REF] Dimitre | In vivo three-photon imaging of activity of GCaMP6-labeled neurons deep in intact mouse brain[END_REF]. This imaging technique is already on-going tremendous development and is getting popularized, as it benefits even more from the minimal scattering of the tissues at 1100-1200 nm and of the reduced background contamination due to better optical sectioning of three photons. Subsequently, reach the deepest layers of the cortex, even above the white matter and down to the hippocampus (1 µm depth). Recently, direct wavefront sensing AO was implemented on three-photon microscope coupled to remote focusing to perform in vivo imaging down to 750 µm below the pia, through the skull [START_REF] Qin | Deep tissue multi-photon imaging using adaptive optics with direct focus sensing and shaping[END_REF], and more is expected to come.

Conclusion on current AO methods for neuroimaging

As we saw, several AO methods are currently implemented, both with their advantages and drawbacks. Direct wavefront sensing provides accurate and fast wavefront correction but at the expense of the guidestar generation issue and suffers from scattering in depth. Indirect wavefront sensing, in spite of the enhancement of its speed, is still a slow method and inevitable photodamage arise after the long tissue exposure mandatory for aberration evaluation and correction. Hence, a technique resilient to scattering and still fast will be relevant for the neuroimaging community, and this is what is proposed by the InovAO project and its Extended Source wavefront sensor. Its principle and previous results are introduced in the next section.

1.5. Extended Source wavefront sensing for microscopy

Extended Source wavefront sensing for microscopy

The lack of a direct wavefront measurement without guidestar first led to pioneering studies in astronomy, more particularly for solar imaging and earth imaging from satellites [START_REF] Michau | Wavefront Sensing from Extended Sources[END_REF][START_REF] Michau | Shack-Hartmann wavefront sensing with extended sources[END_REF]. In both cases, the angular width of these objects was too important for the classical point-based methods. In microscopy, the lack of these types of sensors was first mentioned in [START_REF] Li | Conjugate adaptive optics in widefield microscopy with an extendedsource wavefront sensor[END_REF] but only in 2018 was the adaptation of extended-source wavefront sensor for microscopy reported by Lawrence et al. [START_REF] Lawrence | Scene-based Shack-Hartmann wavefront sensor for light-sheet microscopy[END_REF]. However, the first implementation of a Extended Source Shack Hartmann (ESSH) wavefront sensor within an AO-loop comes back to our group in 2019 [START_REF] Hubert | Adaptive optics light-sheet microscopy based on direct wavefront sensing without any guide star[END_REF]. Some other teams later developed their version of ESSH wavefront sensing for microscopy [START_REF] Ashida | Imaging performance of microscopy adaptive-optics system using scene-based wavefront sensing[END_REF][START_REF] Zhang | Laboratory quantification of a plenoptic wavefront sensor with extended objects[END_REF].

In a ESSH, the sampled wavefront does not originate from a single point source, but results instead of an average of all the emitters within the FOV of the microlenses. The considered FOV is further selected by a field diaphragm conjugated with the sample plane, and prevents the incoming signal from overlapping between microlenses (see Figure 1.30). The signal associated to each microlens into subimages undergoes 2D correlation with a reference subimage, chosen in the center. The obtained correlation peak is then equivalent to a point source for a usual Shack-Hartmann, and its position then gives the local slope of the wavefront. A couple of additional steps are necessary for the successful completion of the process such as a pre-filtering step and a sub-pixelic peak finding; more detail is given in Section 2.2.

The development of this sensor and its technical specifications were the results of several compromises tightly linked to microscopy requirements, such as the isoplanetic patch of the considered samples. The size of this patch, defined as the area over which the aberrations can be considered as constant, is of tremendous importance for the design. Indeed, it sets the maximum FOV of the microlenses. Its evaluation is not very well documented and strongly depends on the considered sample, however, some studies agree on a isoplanetic patch of the order of 150x150 µm² in the mouse brain [START_REF] Park | Large-field-of-view imaging by multi-pupil adaptive optics[END_REF][START_REF] Ji | Adaptive optical fluorescence microscopy[END_REF]. If this patch reveals itself to be too large for the considered sample, we can adjust the aperture of the field diaphragm before the ESSH to a smaller size and retrieve some flexibility. The design is also set by the size of the targeted objects: mainly soma and neurites, from sizes ranging between 1 to 10 µm, and overall by the typical FOV used in experiments; in functional imaging, FOV ranging between 150 x 150 µm² and 500 x 500 µm² are used. Wavefront sensor description and previous characterization are more extensively discussed in [START_REF] Hubert | Optique adaptative en source étendue pour la microscopie à feuille de lumière[END_REF], but let us just briefly summarize here its main characteristics for the sake of reading.

• Its wavefront measurement accuracy is λ/50.

• The square microlenses side dimension is d = 300 µm, and its focal length is f ′ = 5.2 mm. The resulting NA is thus 0.03.

• The pixel size is 6.9 µm (after binning).

• In front of each microlens a patch of 35 x 35 pixels² is selected for correlation, corresponding to a typical FOV of 120 x 120 µm² for our set-up configuration.

• For a pupil diameter matching the shortest dimension of the sensor, approximately 220 microlenses are illuminated and contribute to wavefront sensing.

• At 500 nm, and considering the squared aperture of the microlenses, the associated diffraction limit spot has a diameter of 16.7 µm, and is sampled over 2.4 pixels, matching Shannon's criterion. When coupled to a LSFM, AO with ESSH allowed signal and resolution enhancement of a factor two on circadian clock neurons in an ex vivo Drosophila brain [START_REF] Hubert | Optique adaptative en source étendue pour la microscopie à feuille de lumière[END_REF]. A typical result is shown on Figure 1.31, on jGCaMP7b-labelled neuron at 30 µm deep within the ex vivo Drosophila brain. In this proof of principle experiment, the exposure time was set to 100 ms, hence not enabling the resolution of calcium transients, the gain is evaluated on the time averaged signal.

Motivation to implement ESSH-AO for 2PEF microscopy

If ESSH-based AO showed its efficiency on LSFM and Drosophila brain, it is a good candidate for 2PEF. Its correlation-based algorithm for slope retrieval should be more resilient to scattering. Moreover, ESSH is also relevant to use even when we have a point source. Indeed, as even very low background, caused by scattering, or even inherent to the camera, can lead to very poor slope estimation with a centroid-based measurement, while on the other hand, correlation reveals itself far much less sensitive to the background [START_REF] Poyneer | Scene-based Shack-Hartmann wave-front sensing: analysis and simulation[END_REF]. Thus, implementing an AO loop with an ESSH on a twophoton microscope is relevant.

Summary and motivations

AO revealed itself as a very powerful tool for neuroimaging. When combined to optical sectioning techniques such as two-photon microscopy, it has help to push further the depth limit while increasing signal and resolution. Its contribution to the PSF narrowing and then the background reduction resulted in improved signal and resolution.

• In structural imaging, AO can provide a two-fold enhancement on signal from soma, and on fine structures of typically 0.5 µm, up to a five-fold signal improvement is obtained.

• In functional imaging, AO allows to detect calcium transients in depth from fine neural processes.

However, the current compromise between accuracy, speed and resilience to scattering led us to consider the development of a new wavefront sensing approach, basing the wavefront measurement on extended sources instead of point ones to release AO implementation constraints. While this method showed its efficiency when coupled to Light Sheet Microscopy, the aim of my work was too transpose this original wavefront measurement method to the privileged microscopy technique for neuroimaging in rodents: two-photon microscopy.

In depth signal on ESSH: wavefront measurement challenges

When addressing in depth imaging, and wavefront measurement at typical depths of hundreds of microns in biological tissues, the loss of ballistic signal turns out to be a drastic limitation to the penetration depth. In the case of two-photon microscopy, scattering leads to an exponential drop of the excitation power inducing a squared loss of emitted signal. This event is usually simply compensated for by increasing progressively the excitation power of the laser, eventually leading to two-photon emission at the surface of the tissue. This has long constituted the main reason why two-photon microscopy was limited in depth. Together with scattering, the lower order event of aberrations that broadens the focal volume, also forces the user to increase laser power at shallower depth than predicted, as discussed in Chapter 1.

In two-photon microscopy, scattering does not only impact the excitation beam; in particular in our AO strategy, the influence of scattering on the emitted fluorescence signal, on which the wavefront measurement is computed, could limit even earlier the wavefront measurement in depth. Indeed, as the use of wavefront sensor relies inherently on the formation of images, and not on photon counting as the scientific imaging part does, a preliminary study on the robustness of our wavefront-sensing technique had to be conducted beforehand; this issue being only encountered in direct wavefront sensing, and not in indirect methods. The loss of ballistic photons, the ones that are still containing the complete information from where they were emitted, and thus what we want to measure, leads to a drop in the signal to background ratio (SBR) and potential difficulties in the correlation process.

In this chapter, the different elements are discussed. First, the influence of the scattering on the sensor is studied, and tissue-related information are extracted from it. Then, a closer look is given to the wavefront measurement accuracy within these conditions; to ensure a complete evaluation of the situation, both numerical and experimental quantitative studies were conducted. The work presented in this chapter is related to the publication in Optics Express [START_REF] Imperato | Single-shot quantitative aberration and scattering length measurements in mouse brain tissues using an extended-source Shack-Hartmann wavefront sensor[END_REF], available at the end of the manuscript.

Ballistic signal evolution a. Definitions

In a medium, the distance between two scattering events is defined by the scattering length l s . Over it, the unscattered light decreases to 1/e of its initial power. Jointly with the absorption characterized by the absorption length l a , the total effective attenuation occurring within a tissue, and setting the corresponding attenuation length l e , can be obtained by the relation:

1 l e = 1 l s + 1 l a (2.1)
To evaluate the evolution of the ballistic decay in the mouse brain, and more particularly in the cortical area, Mie scattering simulations for a medium exhibiting similar scattering properties to in vivo tissues has been performed by Chris Xu's team [24,25], to deduce the effective attenuation over a wide spectrum (between 600 nm to 2000 nm). Some concordant experimental data were added to these plots, as we can see on Figure 2.1. These results highlight the fact that up to 1300 nm, the absorption, induced by water, is negligible compared to the scattering. Thus, in the following, only scattering will be considered here as we work in the visible and near infrared regions.

θ 0 = 0 θ 1 θ n <θ> FIGURE 2.2: Multiple scattering in a tissue
Besides, the scattering is not only defined by l s . The direction of the scattering, its anisotropy, is given by the anisotropy coefficient g =< cos θ >, with θ the deflection angle. A medium that strongly scatters in the forward direction will have an anisotropy coefficient close to 1 whereas for a medium scattering in all the directions, g will be close to 0. In the biological tissues, g is around 0.9 [START_REF] Min | Measurement of multispectral scattering properties in mouse brain tissue[END_REF]. Thus, a more global grandeur, that takes into account both the scattering and the anisotropy is the transport length l * , represented in Figure 2.2, and given by:

l * = l s 1 -g (2.
2)

It better reflects the loss of directionality occurring in an anisotropic medium and sets a characteristic scale to the scattering process. After l * , the scattered intensity becomes almost isotropic. On the other hand, these serpentine photons progressively contribute to the development of a speckle pattern around the ballistic spots. On a SH sensor for example, this effect is observed by Wang et al. [START_REF] Wang | Rapid adaptive optical recovery of optimal resolution over large volumes[END_REF] while performing aberration measurement on exogeneous point sources. With a single illumination spot, they observed a great deterioration of the precision of centroid localization, whereas descanning the fluorescence from several emitters led to an averaging of the speckle and to Gaussian-shaped points retrieval on the SH wavefront sensor.

In optical microscopy such as confocal or 2PEF microscopy, imaging is limited to 4-5 l s , that is to say of the order of (1/2)l * for g = 0.9); in this intermediate regime, some ballistic photons are still preserved. As we go further in depth and reach penetration depth superior to the transport length l * , that roughly corresponds to 10 l s , the photons loose completely the memory of their initial direction and propagate in the medium through random walk. Let us note that, in this multiple scattering regime, some methods have been proposed to retrieve focalisation by shaping the incident wavefront with spatial light modulators (SLM) [START_REF] Vellekoop | Focusing coherent light through opaque strongly scattering media[END_REF]. Even if these methods based on the analysis of the speckle pattern are currently undergoing drastic development, their application to neuroimaging is still not widespread due to the many constraints and especially the time required to recover focalisation (several tens of seconds). Moreover, the isoplanetism for one correction is limited to a few micrometers around the corrected point, hence requiring multiple correction to obtain a reasonable FOV for biological imaging (lateral dimension in the range of hundreds of microns). Still, major works have recently been published exhibiting improved speed [START_REF] Blochet | Focusing light through dynamical samples using fast continuous wavefront optimization[END_REF][START_REF] Blochet | Enhanced stability of the focus obtained by wavefront optimization in dynamical scattering media[END_REF], with wavefront shaping fast enough for in vivo transcranial imaging in mouse [START_REF] Blochet | Fast wavefront shaping for two-photon brain imaging with large field of view correction[END_REF][START_REF] Rauer | Scattering correcting wavefront shaping for three-photon microscopy[END_REF].

Thus, we aim at working within this intermediate regime, and while this depth limitation has been extensively studied for the signal amplitude in two-photon microscopy, no closer look was previously given to its impact on direct-wavefront sensing, and more particularly on how the wavefront sensing was affected within this regime. While trying to understand the causes and consequences to these scattering effects, we realized that more information could be extracted from it, the physical reasons being explained in the following.

b. Impact on Shack-Hartmann sensors

SH sensors are composed of an array of microlenses, positioned in front of a CCD or CMOS camera. Each microlens images the guidestar or the selected area in the sample plane; this area is defined by the size of the microlens and the camera portion in front of it, thus setting the small wavefront portion size. Considering the optical conjugation between the ESSH microlenses and the objective back pupil plane, the signal received by the outer microlenses propagates through a thicker slab of the scattering tissue than the one received by central microlenses. This effect can be transposed to any SH and is illustrated on Figure 2.3a. Thus, the thickness z p crossed to reach the subimage of index p (p varies between 0 (central subimage) and 7 (most external row of subimages) increases as:

z p = z c cos θ p (2.3)
where z c is the slice thickness and θ p is the angle over which the subimage p is seen, with

θ p = arctan p 7 tan θ max (2.4)
and θ max = arcsin N A objective /n = 48.8 • , θ max being the maximum angle collected by the objective (Olympus, XLUMPFLN20XW, 20X, NA = 1, f ′ = 9 mm, WD = 2 mm), and n its immersion medium index.

Moreover, the ballistic signal intensity decays in depth according to:

I(z p ) ∝ exp - z p l s (2.5)
On that account, within a single acquisition, we can follow the intensity drop from the center to the edges of the ESSH (see Figure 2.3b). This interesting property does not only help us to quantify the depth limit for wavefront sensing, we can also extract the scattering length value of the considered tissue by simply plotting the ballistic evolution as a function of the effective thickness seen z p . Moreover, let us note that the background signal drops too: above the transport length, the background decay is proportional to the inverse of the depth. Before, this background drop with depth is not characterized by any analytical expression. As we can see on Figure 2.3b, ballistic signal decays faster than the background, resulting into a Signal to Background (SBR) ratio that decays with the crossed thickness. Thus, the lower SBR on the pupil edges will limit wavefront sensing in depth. As the sample thickness increases, the SBR drops on all the subimages. In the following, we will refer to signal, background and signal to background ratio (SBR) according to the definitions below.

Signal and Background

In a subimage:

• The Signal S is defined as the average value of 0.5% of the brightest pixels (5 pixels). Solely the ballistic signal is kept, the PSF of the square microlens being spread over 2.5 pixels in both x and y. • The Background B is considered as the mean value of the 80% dimmest pixels.

• The Signal to Background Ratio is simply SBR = S/B.

Scattering length measurement of brain tissues

To assess the scattering properties of fixed brain slices, the propagation distance of the light through the medium has to be known. To measure this parameter, we chose to work with several thicknesses of fixed brain slices, comprised between 50 and 250 µm, from wild type mice. The fluorescence signal was provided by fluorescent emitters positioned under the slices. The resulting sample is thus simultaneously scattering and 2D, so that it can be imaged on a simple fluorescence microscope without optical sectioning. In this section, we first describe the emitters, then our calibration experiments, to conclude with the results obtained for fixed brain slices.

a. Sample preparation: fluorescent emitters and fixed brain slices

Fluorescence emitters. To obtain data from different region in the visible spectrum, we chose to use Quantum Dots (QD) based emitters. QD are colloidal semiconductor nanocrystals, composed of a core and a crown, and of typically 5 to 10-nm diameter, the crown is mainly added to enhance the quantum yield of the emitter and to prevent its decomposition. Our group in the LPEM holds a strong experience in their synthesis, characterisation and use as fluorescent probes for biomedical applications [START_REF] Muro | Small and stable sulfobetaine zwitterionic quantum dots for functional live-cell imaging[END_REF][START_REF] Muro | Comparing intracellular stability and targeting of sulfobetaine quantum dots with other surface chemistries in live cells[END_REF][START_REF] Tasso | Oriented bioconjugation of unmodified antibodies to quantum dots capped with copolymeric ligands as versatile cellular imaging tools[END_REF][START_REF] Tasso | Sulfobetaine-Vinylimidazole block copolymers: a robust quantum dot surface chemistry expanding bioimaging's horizons[END_REF][START_REF] Pons | In Vivo Imaging of Single Tumor Cells in Fast-Flowing Bloodstream Using Near-Infrared Quantum Dots and Time-Gated Imaging[END_REF]. Their specificity lies in their ability to exhibit a very broad absorption spectra (they can absorb light up to the excitonic peak), and in the fact that their emission wavelength is simply linked to the size of the nanocrystal: the broader the QDs, the redder their emission. The upper limit to their emission wavelength is the quantum confinement. These emitters are brighter than organic dyes and several populations of QDs with different sizes can be excited with the same laser source. The emission spectra of two types of QDs is represented in Figure 2.4. The near-IR QDs, composed of a ZnCuInSe core and of a ZnS shell were synthesized following the protocol reported by Pons et al. [START_REF] Pons | In Vivo Imaging of Single Tumor Cells in Fast-Flowing Bloodstream Using Near-Infrared Quantum Dots and Time-Gated Imaging[END_REF] and emitted at 800 nm, the visible QDs were synthesized according to the protocol described by Yang et al. [START_REF] Andrew | Synthesis of CdSe and CdTe Nanocrystals without Precursor Injection[END_REF], leading to emitters with an emission peak around 614 nm.

However to get bright separated emitters and to be able to resolve them on our set up, we had to proceed to a supplementary step and to form aggregates out of these emitters though a method of emulsion and evaporation, described in [122], and realized by Alexandra Bogicevic. 500-nm aggregates were obtained with respectively an emission peak at 614 nm (FWHM = 32 nm) and 805 nm (FWHM = 17 nm), a Transmission Electron Microscopy image of these aggregates is shown in Figure 2.4b.

Combining these QD-aggregates with the very bright commercially available 2-µm fluorescent beads (ThermoFischer, λ em = 515 nm) as emitters within the green range, we were able to get a multi-spectral acquisition, at the same localisation in the sample, without adding or changing the excitation source. This multi-spectral strategy is also used in Section 2.2.

Sample preparation

Samples were prepared with the three types of emitters. 4 µL of the 2-µm commercial beads, diluted 1/5000, were deposited first on a 170-µm-thick coverslip then dried up for 5 min at 60°C. The two types of QD aggregates were deposited on the same side of the coverslip (4 µL of each). On the other side of the coverslip a fixed brain slice was placed, with spacers around to control the thickness of the sample. Another coverslip was added on top of it to avoid the adhesion of the brain slice to the water-immersion objective during the experiment (represented on Figure 2.5).

Fixed brain slices.

Experimental procedures were conducted in accordance with the institutional guidelines and in compliance with French and European laws and policies. All procedures were approved by the 'Charles Darwin' local institutional ethical committee registered at the French National Committee of Ethical Reflection on Animal Experimentation under the number 05 (authorization number: APAFIS 26667).

For the scattering length measurements, as well as for the aberration measurements presented in Section 2.2, three 6-months old C57BL6 male mice were sacrificed by an overdose of Euthasol, after being placed under deep sedation by overdose of isoflurane (5 min at 5% isoflurane in an induction box). The extracted brain was then stored overnight in a solution of 4% paraformaldehyde and finally rinsed in Phosphate Buffer Solution (PBS). Coronal slices of different thicknesses ranging from 50 µm to 300 µm were then cut with a Microtome and stored in PBS.

b. Calibration experiments

To confirm the accuracy of the method, we turned ourselves towards 2-µm polystyrene beads solution to mimic the scattering properties of the tissue. In spite of the very common use of Intralipid solutions when scattering models are concerned, this medium is so not well characterized as it reaches the fundamental limits of scattering length measurement constraints of the usual 2ω autocorrelation. On the other hand, polystyrene beads and their scattering properties can be easily modelled by Mie scattering propagation simulation. The tuning of the scattering length of the calibration sample can thus be simply done by changing the concentration of the tested solution.

For this calibration experiment, 1 µL of the commercial solution, containing 10% of beads in mass was diluted 10 times to obtain a solution of 0.002 bead/µm 3 with a calculated scattering length of 74.9 µm. A 5-times diluted solution (0.005 bead/µm 3 ) with a scattering length of 29.9 µm was also prepared. The scattering coefficients were calculated with Mie-scattering theory, at 515 nm with the online calculator https://omlc.org/calc/mie_calc.html. For both solutions, 10 µL were deposited on the coverslip, surrounded by 130-µm-thick spacers, and recovered by another coverslip to retain the scattering droplet, instead of the brain slice here. The choice was made not to embed the scatterers in agarose gel, as even at very low concentration (typ 0.5%), agarose can impact the scattering properties of the medium [START_REF] Rochas | Scattering properties of agarose gels[END_REF]. We thus rely on the thickness precision of the spacers, and on our ability to acquire the image quickly to avoid the sedimentation of the beads. Green emitters are used as fluorescent sources and deposited as in Box 2.1.2a., sample illustrated in insets of Figure 2.5.

The resulting values are plotted on Figure 2.5. From these two semi-log plots, the slope coefficient gives the scattering coefficient, while the scattering length is the inverse value; hence for the 0.005 bead/µm solution, the measured scattering length is 26.5 ± 0.3 µm (29.9 µm simulated), and for the 0.002 bead/µm , we obtain 69.5 ± 1.4 µm (against 74.9 µm). As the results are in good agreement with the expected values, we now apply this method to fixed brain slices.

c. Results obtained through fixed brain slices

The scattering length of fixed mouse brain slices in the visible region is not so well documented. Indeed, the current employed methods to measure scattering lengths either necessitate very thin tissue slices (5 to 15 µm) such as methods based on quantitative phase imaging [START_REF] Ding | Measuring the scattering parameters of tissues from quantitative phase imaging of thin slices[END_REF][START_REF] Min | Measurement of multispectral scattering properties in mouse brain tissue[END_REF] or on Optical Coherence Tomography (OCT) [START_REF] Faber | Quantitative measurement of attenuation coefficients of weakly scattering media using optical coherence tomography[END_REF][START_REF] Lee | Performance of single-scattering model versus multiple-scattering model in the determination of optical properties of biological tissue with optical coherence tomography[END_REF]. Scattering lengths can also be accessed through full-field OCT, and has been implemented for in vivo measurements [START_REF] Binding | Brain refractive index measured in vivo with high-NA defocuscorrected full-field OCT and consequences for two-photon microscopy[END_REF]. Other methods rely on series of measurements at multiple depths, realised with multiphoton microscopy excitation beam, and for each acquisition point, the attenuation of the signal is recorded [24,[START_REF] Kobat | In vivo two-photon microscopy to 1.6-mm depth in mouse cortex[END_REF][START_REF] Nicholas | In vivo three-photon microscopy of subcortical structures within an intact mouse brain[END_REF]25]. A more flexible method could thus be of great help to the community.

Scattering measurements on the cortex of fixed mouse brain coronal slices, are performed at three different wavelengths (515 nm, 615 nm and 805 nm) to determine the chromatic dependence of the scattering length of the tissue, with the sample prepared as described in Section 2.1.2a.. To improve the measurement accuracy, three slice thicknesses are used at each wavelength: at 515 nm and 615 nm, 50-, 100-and 150-µm-thick slices and at 805 nm, 100-, 150-and 200-µm-thick slices. At each wavelength, we choose thicknesses that range from l s to 2.5l s as the effective thickness seen by the outer microlenses is 1.5 times larger than the one seen by the central subimage. Thus, it allows us to stay within thicknesses of 1 l s and 3.9 l s on the external edge, for the thickest slices; above this, the ballistic signal is too low to distinguish it from the background. Increasing the sample thickness is beneficial, until attenuation of the ballistic light is too strong to be separated from the background on each subimage.

As we concatenate data obtained from several slices this time, we need to normalize the intensity to plot and fit the data for all slices on a single figure. This is due to the illumination variations and intensity emissions as we move from one sample to another (we consequently have to increase the excitation power in order to still obtain signal for thick slices). Thus, we first plot on separated semi-log plots the data for each slice, linearly fit it, and extract the ordinate. The corresponding value is later subtracted to each acquisition in order to gather all data on a single plot. Let us note that this normalization is not so trivial. The obtained plot is presented in Figure 2.5.

The measured scattering coefficients are 38 ± 2 µm at 515 nm, 47 ± 2 µm at 615 nm and 77 ± 5 µm at 805 nm, increasing significantly with the wavelength as expected. These results have to be compared to the previously reported scattering length values in the literature. To our knowledge, no data was reported at these visible wavelengths on fixed brain tissues. Thus, comparative values summed up in Table 2.1 were obtained in the near-IR region, on fixed tissues and in vivo, but it was reported that the scattering length increases of a factor 2 to 3 between fixed tissues and fresh tissues [24,25]. Moreover, we could extrapolate linearly, on a logarithmic plot, the in vivo measurements to the wavelengths used in our study: the obtained fit is presented here on a linear scale (see Figure 2.6, blue line). On this fit are added the considered literature values (in blue). Note that this extrapolation is only valid under 1,300 nm where scattering widely dominates absorption [START_REF] Nicholas | In vivo three-photon microscopy of subcortical structures within an intact mouse brain[END_REF]25]. Then, we take into account the factor 2.4 reported by Kobat et al. [24] between the fixed tissues and the fresh ones to obtain the equivalent fitting curve on fixed tissues (brown line): we divide the coefficients from the linear fit of the logarithmic plot of fresh tissues data. Experimental data on fixed mouse brain from the literature are added to this curve (red dots). We finally add our measured values to the graph (orange dots).

Thanks to these, we can conclude that our data are in good agreement with the literature and that the ESSH sensor can provide a consistent measurement of l s . This single-shot measurement of l s is therefore both accurate and easy to implement on a wide range of samples. However, it relies on a planar sample configuration: it can be an obstacle to some applications in which the planicity can not be obtained (e.g skull would be more difficult to address as it is hardly flexible and its thickness can not well be controlled).

Note that both l s and g depend not only on tissue type but can change substantially with age and upon removal of the tissue from the animal [START_REF] Oheim | Two-photon microscopy in brain tissue: parameters influencing the imaging depth[END_REF]. As an example, the scattering length can increase by a factor two between baby mice and adult mice, due to the different level of myelin.

Sample λ (nm) l s (µm) Reference

Mouse in vivo 775 131 [

Assessing WF measurement accuracy in scattering conditions: experimental approach

The initial aim of our wavefront sensor is intrinsically to measure wavefront. As we saw in the previous section, scattering affects the signal and ultimately limits wavefront sensing in depth in biological tissues as the signal on the external subimages gets more and more deteriorated. Thus, we now assess the ability of our ESSH to measure accurately wavefront. For this, we implement the setup described in Section 2.2.1 in order to perform quantitative aberration measurements and to compare it with simulated values.

Setup and algorithm description a. Setup

A custom-made epifluorescence microscope, presented in Figure 2.7, is set up to perform aberration measurements within scattering conditions. Two pathways are implemented:

• The main pathway is dedicated to aberration measurements with the ESSH. The fluorescence excitation is ensured by a 488nm-laser (Cobolt 06-MLD), focused in the back focal plane of the objective with L1 (AC254-100-A, f ′ 1 = 100 mm, Thorlabs). The microscope objective is a water-immersion objective commonly used in two-photon imaging (Olympus, XLUMPFLN20XW, 20X, NA = 1, f ′ = 9 mm, WD = 2 mm). On the detection side, the objective is followed by a tube lens L2 (AC508-300-A, f ′ 2 = 300 mm, Thorlabs), resulting in an effective magnification of 33 between the sample and the first camera (Thorlabs) towards which half of the fluorescence signal is detected after being split up by the 50:50 beamsplitter (BSW10R, Thorlabs). This leads to a FOV of 130 x 100 µm². The other half of the signal is directed towards the ESSH: the area over with the correlation is computed is selected by a squared-field diaphragm (D1, Ealing). An area corresponding to 120 x 120 µm² in the sample is chosen to match the subimage dimensions and avoid

b. Algorithmic choices for slope retrieval

As opposed to centroid-based measurement usually performed combined to SH wavefront sensors, where solely a thresholding step is applied to the signal beforehand (see Section 2.3.2.b.), our approach requires several additional steps. A pre-filtering ahead of the 2D correlation is necessary to increase the signal to correlate on, and a sub-pixelic peak finding algorithm must be applied afterwards to the correlation map, in order to retrieve the slopes as discussed by Anugu et al. [START_REF] Anugu | Peak-locking centroid bias in Shack-Hartmann wavefront sensing[END_REF]. As for the cross-correlation C in the image space between the reference image I 0 and the considered one I S it is simply given by:

C[m, n] = N ∑ i=1 N ∑ j=1 I S [i + m, j + n]I 0 [i, j] (2.6)
In our situation, the filtering is essential to noise rejection and to high-frequency signal selection in order to obtain sharp correlation peaks and find precisely their location afterwards. The ESSH algorithms are based on cross-correlations on images of the sample, and the measurement of the position of the corresponding peaks. The shape of the correlation peaks without pre-filtering is related to the image content. For example, an image containing structures along a preferred direction will have correlation peaks elongated along one direction. Therefore, without prefiltering, the accuracy of peak location is likely to be significantly degraded. In order to minimise this effect, one solution is to pre-process the images in order to extract features for which the correlation will be less sensitive to the spatial characteristics of the image. Several methods are possible such as feature extraction, or the exploitation of specific frequency content. Here, we chose to extract the high frequency content (intensity transitions in the images), so as to minimise the dependence of the correlation on the spatial content of the image and detect its contours. This can be typically done by using a gradient filter (e.g. Sobel, Laplacian filters).

However, this approach alone has the disadvantage of increasing the importance of noise, particularly for images with low SBR. To minimise the impact of noise, we have combined the use of a gradient approach with low-pass Gaussian filtering. The two main ones used here are the Laplacian filter and the Laplacian of a Gaussian (LoG) filter; the Laplacian filter is solely a high-pass filter, based on the second-order derivative, that leads to sharp correlation peaks but is more sensitive to image noise. A possible kernel is Equation 2.7. On the other hand, the LoG filter is the combination of a Laplacian filter with a smoothing Gaussian filter. For a 5x5 kernel, the obtained matrix is Equation 2.7. It results in a compromise between the level of low-pass filtering set by the Gaussian blur and the Kernel size, and the precision of localisation of the correlation peak. A too wide Kernel leads to wide correlation peaks, hence reduces the precision of localisation.

Filter kernels

Laplacian =    0 -1 0 -1 4 -1 0 -1 0    and LoG =         0 - 0
        (2.7)
A comparison of these two filters is presented in Figure 2.9 to show how the Gaussian lowpass filter reduces the impact of the noise on in low SBR conditions. In this example, the ESSH signal is generated by several fluorescent beads positioned under a mouse brain slice, held between two coverslips. In this situation, we expect predominantly spherical aberration from the presence of strong refractive index changes at the water/glass/brain interfaces. Where this is observed with the LoG filter, it is not the case with the Laplacian filter. If we look closer at the slopes in the top left corner, we notice that some of them are aberrant when the Laplacian filter Chapter 2. Aberration measurements and wavefront sensing in scattering conditions is used; this area corresponds to a region where the SBR is low and the signal is quite noisy, as it can be seen on the raw signal. The pre-filtering insets reveal that instead of simply enhancing the signal, the Laplacian filter also enhances the noise, but this effect is less important with the LoG filter. This is translated in the following step: the correlation maps ensuing from the Laplacian is noisier than the LoG one, hence the hazardous peak finding, and the aberrant slopes.

Hence, the filter choice appears to be critical especially in low SBR conditions. In addition to this pre-filtering step, the choice of the sub-pixel peak-finding algorithm is important too; it was detailed in [START_REF] Hubert | Optique adaptative en source étendue pour la microscopie à feuille de lumière[END_REF] and amongst all the proposal listed by Löfdahl [START_REF] Löfdahl | Evaluation of image-shift measurement algorithms for solar Shack-Hartmann wavefront sensors[END_REF], we oriented ourselves to the pyramidal fit, for its speed and its accuracy, as well as its good resilience to low SBR conditions when being applied after a LoG pre-filtering step. This fit, for the estimation in the x direction, is given by:

s ′ x = x 0 + 0.5 C[x 0 -1, y 0 ] -C[x 0 + 1, y 0 ] min C[x 0 -1, y 0 ], C[x 0 + 1, y 0 ] -C[x 0 , y 0 ] (2.8)
with (s ′ x ) the sub-pixel displacement, C the correlation map, and (x 0 , y 0 ) the correlation map maximum coordinates. Thus, in the following, this combination of filter and sub-pixelic peak finding algorithm will be considered.

Third order Spherical Aberration in depth

Now that the setup and corresponding algorithms are defined comes the choice of the criteria to assess the wavefront measurement accuracy in scattering conditions. As the wavefront is decomposed over the Zernike base, we can look straight at the aberration coefficients, and third order Spherical Aberration (SA3) is a good candidate. Generated by refractive index mismatch along ray propagation we expect it be predominant and linear to the first order. Moreover, the refractive index mismatch impacts the Optical Path Difference (OPD) which depends on the easily tuneable sample thickness t brain according to:

OPD = t brain (n brain -n water ) (2.9)
.

a. Spherical Aberration and Focus

SA3 coefficient is described in the Zernike polynomials as 6ρ 4 -6ρ 2 + 1. Thus, the focus variation, included in the ρ 2 term, has a strong impact on the SA3 value. Indeed, a best focus can be found to minimize the later (see Figure 2.10). This focus influence on SA3 is first verified experimentally (results shown in Figure 2.11). In this experiment, the sample stage axial position is manually adjusted to reach measured focus values with the ESSH from -1.5 µm to +1.5 µm, corresponding to stage shifts of -0.100 µm to +0.100 µm. This experiment is performed with three samples, similarly to the one used for scattering length measurement, but without any brain slice here (replaced by water); the same number of coverslips is kept, and so are the spacers.

As expected from Zernike description, a linear correlation between the SA3 value and the focus value is found, and is so for the three considered wavelengths. The overall SA3 coefficient varies of 0.100 µm, for Focus values ranging between -1 µm and +1 µm: this is even more than the SA3 values we expect to measure in the fixed brain slice (see Section 2.2b.). Moreover, when the focus is set to 0.0 ± 0.1 µm, the dispersion of SA3 is around 0.02 µm. This value must be compared to the precision of the ESSH: λ/50 ≈ 0.01 µm at 510 nm. Thus, if we set the focus to 0.0 ± 0.1 µm, the repeatability of the SA3 measurement falls within the precision of the sensor. This condition has to be satisfied in the following.

b. Spherical Aberration: simulation versus experiments

Let us now look at the SA3 value in depth, i.e as a function of the brain slice thickness crossed. We want to show here that the ESSH sensor provides a reliable wavefront measurement in strong scattering conditions. In order to eliminate the focus bias from our analysis, we minimize experimentally the associated aberration. Only the contribution of the brain slice is considered here: a reference wavefront, corresponding to the aberrations induced by the instrument and the coverslips, is subtracted at each wavelength. For each brain slice thickness, at least seven acquisitions are realised in different areas of the cortex, at the three considered wavelengths simultaneously, the sample preparation corresponding to the previously reported one in Section 2.1.2.a.

Experimentally, we use fixed slices of mouse brain, of 50-to 150-µm thicknesses within the visible range and up to 250-µm-thick in the near-IR to measure SA3. These thicknesses are in terms of scattering length, equivalent to 1.3l s to 3.9l s at 515 nm, to 1.0l s to 3.3l s at 615 nm and to 0.6l s to 3.2l s at 805 nm. The increase in thickness corresponds to an increase in OPD, that is the difference of refractive index between the tissue and water times the sample thickness.

Chapter 2. Aberration measurements and wavefront sensing in scattering conditions Thus, within these brain slice thickness range, the OPD is inferior to 10 µm. The results are presented in Figure 2.13: the SA3 increases linearly with the OPD as expected from the Zernike decomposition.

But how can we know that these values are correct? In the literature, very few details are usually given, and it is even worse for fixed brain tissue. Hence, we decide to simulate the optical system in OpticStudio (Zemax) in order to get an order of magnitude of the expected SA3 value consecutive to the refractive index mismatch due to the brain slices. More precisely, a simple model consisting in two additional interfaces inserted between the emitters' plane and the objective is chosen and represented in Figure 2.12. The objective and the tube lens are simulated as perfect lenses, which is consistent with the reference wavefront subtraction performed on the experimental data, as solely the contribution of the brain slice on the resulting wavefront is to be measured. The objective lens is the limiting aperture, with a diameter of 18 mm. A tube lens is added to ease the wavefront interpretation on the software and to avoid image location at infinity. All the distances are optimized first without the simulated brain slice to get a perfect imaging system. Then, a layer of refractive index of 1.368 is added to model the brain slice, this refractive index value previously experimentally measured on fixed mouse brain slices and reported by Lue et al. [START_REF] Lue | Tissue refractometry using Hilbert phase microscopy[END_REF]. The variation of the tissue refractive index between the different wavelengths is neglected. The thickness of the layer corresponds to the slice one.

Let us note that the refractive index is only 0.035 superior to the water one, to be put in perspective with the impact of the defocus on the SA3 value as well as with the precision of the sensor (see Section 2.2.2.a.). Thus, at low slice thickness (50 µm), the measured SA3 induced by the sample is within the measurement error values of the ESSH (8-9 nm is predicted by the Optic-Studio simulation). We repeat this simulation for all the brain slice thicknesses used in order to get comparative for each of the experimental ones, values are reported on Figure 2.13.

The experimental error bars represent the standard deviation of the distribution of the measured values, obtained from different locations under a single slice (ten each time except for the thickest one: six acquisitions for the 150 µm slices at 515 nm and 615 nm, four acquisitions for the 250-µm slice at 805 nm). In this simulation, errors arise from the water layer thickness: this thickness is adjusted to minimize the focus aberration value, the minimal obtainable value being 0.002 µm. For two water layer thicknesses different of 1 µm (smallest possible variation), a difference of 0.005 µm on the Spherical Aberration is measured. It thus sets the error on the SA3 simulations. The error coming from the evaluation of the refractive index reported by Lue at al. [START_REF] Lue | Tissue refractometry using Hilbert phase microscopy[END_REF] to be of 0.006 on the refractive index for fifty acquisitions in ten different fixed brain slices was not included here, and generate a supplementary bias. Indeed, we tried for the 250µm-thick slice only to consider a layer of 1.370 instead of 1.368, thus within the reported error: this layer was inducing 93 nm of SA3 against 47 nm, which is this time above the 70 ± 33 nm measured. This difference might explain why we tend to underestimate the SA3 value within our simulations. This aberration measurement method reaches its limits when the slice thickness becomes greater than 3.9l s with commercial beads as fluorescent emitters, that is to say 150 µm in the green range, and 3.3l s with the aggregates, corresponding to 250 µm in the near-IR and 150 µm in the red range. As the aggregates are less bright than the four-times larger commercial beads, we are not able to image as deep. At these slice thicknesses, the ballistic signal on the outer microlenses is too low for the algorithm. Indeed, these slice thicknesses correspond to effective thicknesses crossed on the outer microlenses of 4.8l s and 5.8l s ; only 0.25% of ballistic photons are remaining and the corresponding SBR is only 1.3.

Besides, the variability of the SA3 values for a single slice increases is a consequence of the loss of slope accuracy evaluation at these SBR and of the ballistic signal drop. Measuring aberrations with thicker slices could be achieved by restricting the pupil diameter over which the wavefront is calculated, but these results could not be compared to the previous ones with thinner slices, performed on the entire pupil.

From these measurements, in good agreement with the simulations, we can conclude that our ESSH allows quantitative aberration measurement through scattering slices that are up to 3.9l sthick.

Assessing WF measurement accuracy in scattering conditions: numerical approach

Now that we showed that our ESSH was able to perform quantitative aberration measurements in strong scattering conditions, here comes the moment to compare it with the currently longimplemented centroid-based wavefront measurement. Let us first begin with a couple of reminders on the wavefront computation based on centroid calculation. The centroid coordinates (C x , C y ) are given by:

C x = ∑ n k=1 a k x k ∑ n k=1 a k and C y = ∑ n k=1 a k y k ∑ n k=1 a k (2.10)
with, for a pixel k, a k corresponding to the pixel value, and (x k , y k ) its coordinates.

To optimize the calculation process and enhance the accuracy of the measurement, only the pixels neighbouring the one exhibiting the maximum value are taken into account. Usually, this corresponds to an array of 5x5 pixels. Moreover, a thresholding step is added. The pixel is taken into account in the calculation process only if its value a k satisfies the following condition:

a k < T(a max -a back ) + a back (2.11)
with T the threshold value, usually around 10%, a max and a back being respectively the maximum pixel value and the background value in the considered area.

The description of the wavefront slopes retrieval through both methods is detailed in Figure 2.14. Thus, to compare the results given by both methods, we can either focus on the total reconstructed wavefront, and look at the usual criteria as the RMS, or look ahead of this final calculation and instead focus on local shifts measurements. Indeed, the correlation peak position finding is meant to be equal to the centroid position.

Experimental comparison of centroid and ESSH WF measurements

Our first aim was too proceed to an experimental comparison of both methods, on the same signal, and to use the obtained wavefront as the comparison criterion. Hence, no artefacts due to the difference in optical paths between two sensors would be added. Moreover, the software used here is provided by Imagine Optic, with both the centroid-based measurement and the ESSH calculation from which we can switch to one another in post processing. We modified our sample preparation procedure to ensure larger and brighter emitters. Hence, we used the protocol developed by Subha Jana during his thesis [START_REF] Subha | Biodetection using fluorescence energy transfer from Quantum dot excited whispering gallery modes to fluorescent acceptors[END_REF], to form QD-loaded polystyrene beads of 10-µm diameter. The used ones were CdSe/ZnS QD emitting at 607 nm. The protocol is described below and typical images are presented in Figure 2.15.

Polystyrene beads loaded with QD: Synthesis steps 5 µL of the 10-µm polystyrene beads commercial solution, suspended in water and corresponding to ≈ 10 6 beads is first taken. The solvent is changed to ethanol (EtOH) by first adding 1 mL of it, shaking and then 5 s centrifugation. As the beads are very large and not soluble in organic solvent, they end up in the bottom of the Eppendorf tube. The liquid is removed.

In an other Eppendorf tube, 20 µL of QD are diluted in 1 mL of EtOH and then centrifuged for 3 min at 15,000 rpm. The liquid is removed. QD are then dissolved into 20 µL Chloroform and agitated. Once they are well dissolved, we add 330 µL of Chloroform to them and 670 µL of Butanol (ButOH) to the polystyrene beads. Under the action of ButOH, the beads are swelling, and nanopores are created. The QD solution is transferred to the beads one. As QD are hydrophobic, they will look for hydrophobic space and ultimately get into the beads through the nanopores. The solution is them mixed fro 30 min and then centrifuged 10 s. Three rounds of washing with EtOH are necessary to remove the remaining QD in suspension.

The obtained emitters are positioned on coverslips, following the same steps as described in Section 2.1.2. We compare the RMS values and the SA3 coefficient of the wavefronts calculated with the two methods: the centroid-based one, called centroid in the following, and the correlationbased method normally used along the ESSH sensor and simply abbreviated to ESSH in this section. The acquisitions and obtained wavefronts are presented in Figure 2.15. As the brain thickness increases, the signal gets deteriorated, as expected (see Figure 2.15a), it is reflected in the subimage profiles on the ESSH, that is broadened and noisier. For the thinner slice (Figure 2.15b), the two wavefronts calculated from the ESSH signal are very similar. This is reflected by the RMS values and the SA3 coefficient in Table 2.2. However, as we increase the brain slice thickness to 125 µm (Figure 2.15c), diverging results are obtained. We observe the apparition of a strong astigmatism on the centroid calculated wavefront, that is not present in the ESSH one. Moreover, while the SA3 is expected to be strongly predominant due to the two coverslips holding the brain slice, and without their contribution being taken into account here beforehand (no reference wavefront is subtracted), no major variation of it is supposed to happen while moving from a 50-µm to a 125-µm brain slice. Thus, the drop of 25% is SA3 coefficient is more than suspicious observed with the centroid method. We could then attribute this effect to a measurement artefact caused by the signal deterioration, and affecting only the centroid calculation. Even if this eventuality turns out to be in favour of our statement, that is to say that the ESSH calculation is in more agreement with what is expected, it remains difficult to affirm that a method is giving the accurate result while the other is wrong. How can we affirm which values correspond to the truth and which don't?

At this time, we feared a bias on the centroid based measurement linked to the software itself, in which several thresholds are implemented to satisfy a wavefront measurement accuracy of λ/100 for this method. These settings were not appropriate for our experimental conditions, in which the ballistic signal is low and the background strong, and were difficult to tune without a proper numerical simulation to predict their effect on the WF measurement. Hence, we decided to move to a different approach, oriented towards local slope retrieval instead of full wavefront measurement. Not only will this procedure allow a complete control of the shift, and remove any potential hazy interpretation, but also more parameters could be tested, such as the precise adjustment of the threshold value.

Comparison of centroid-based displacement measurement and correlation based displacement measurement on simulated data a. In-depth PSF simulation

We first simulate a well sampled PSF for a squared aperture. The steps are illustrated on Figure 2.16. For squared apertures as the microlenses, the PSF intensity profile along the x direction, in the focal plane of the microlenses, is given by:

I(x) = sinc 2 πax λ f ′ = sin πax λ f ′ πax λ f ′ 2 (2.12)
We choose matrices dimensions in order to obtain a final design that matches the ESSH one (3 pixels at the PSF FWHM with a 6.9-µm pixel, an overall subimage of 35x35 pixel²). For a 0.5 final sub-pixelic shift, the steps are applied as follow:

1. We generate a well sampled PSF, that is 10-times more sampled than the final one, and is represented in Figure 2.16a, the overall width of 350 pixels corresponds to 245 µm.

2. The total intensity is multiply by 50 before noise generation.

3. Background is added. A 350 x 350 pixel² matrix with a value corresponding to B is generated and added to the initial PSF. From these two final images, we can apply the process described in Figure 2.14 to retrieve the subpixelic shift with both methods.

However, the influence of the sub-pixelic shift value to retrieve is not trivial. Anugu et al. [START_REF] Anugu | Peak-locking centroid bias in Shack-Hartmann wavefront sensing[END_REF] showed that the sub-pixelic retrieval accuracy depended on the shape of the pattern, on the chosen peak-finding algorithm, and on the introduced shift. To confirm that the simulation is correct, we first evaluate the difference between the induced sub-pixelic shift and the measured one as a function of the initially introduced shift. In this process, we use the ESSH peak finding algorithm (LoG pre-filtering, 2D correlation and sub-pixelic peak finding), with two peak finding-algorithms: the Pyramidal fit and the Gaussian fit, on a simulated signal with a good SBR of 2. The centroid-based algorithm suffers from the same behaviour. Hence, the sinusoidal shape previously described [START_REF] Anugu | Peak-locking centroid bias in Shack-Hartmann wavefront sensing[END_REF] for both fits is retrieved (see Figure 2.17), with a minimal error for an 0.5 pixel shift (and naturally for a entire pixel shift).

Thus, in the following, we assess the accuracy of the shift retrieval for 0.5 pixel shifts on the simulated sensor.

b. Influence of the threshold on the centroid based displacement measurement

Before proceeding to the numerical comparison of both methods for shift retrieval remains one last parameter to optimize: the threshold value for the centroid-based method. A too high threshold will exclude too many points in the centroid calculation, and will ultimately lead to the so-called sampling-thresholding bias. On the other hand, in the presence of a strong background, thus for low SBR, a too low threshold will not exclude low signal pixels strongly impacted by the noise from the centroid calculation, and will also lead to an error. A compromise has to be found.

In order to determine the best threshold for low SBR conditions, we run the following simulation; for seven SBR values, ranging between 1.1 to 2.5, fifty images are simulated and shifted of 0.5 pixel according to the process described in Section a.. For each image, the sub-pixelic shift is determined via the centroid process coupled to various thresholding values. The tested thresholds are comprised between 0.07 and 0. As we can see, for too high thresholds (T = 0.15 and T = 0.2), the method fails at high SBR, as pixels containing signal are rejected. It does not get any better when the SBR drops. For the lower threshold tried (T = 0.07), the displacement measurement is correct at high SBR, but from SBR of 2, we can see that the average value is good but the standard deviation of the measurements starts to increase. This effect is confirmed for lower SBR.

The optimum tried threshold values are T = 0.1 and T = 0.12. They are both giving similarly good results down to SBR of 1.4. The standard deviation of the displacement error then increases, and so does the mean of the corresponding distribution, but no better threshold is found.

Alongside these results on thresholds, one can note the constant performance of the ESSH algorithms, labelled as Correlation on the Figure . We detail this point in the next section to show that independently of the threshold on the centroid measurement, the shift retrieval performances of the ESSH algorithm remain better, in low SBR conditions.

c. Results and discussion

Let us now dig into a couple of simulations to prove our point. From now on, we consider two ways to represent the data: the first one is simply based on the representation of the mean value and the corresponding standard deviation of the distribution is plotted as error bars. The other way to plot the results uses the Standard Error of the Mean (SEM). This grandeur is defined as:

SEM = σ √ n (2.13)
It takes into account the number of data points in each set (n) and the Standard Deviation (σ), thus linking the variability of the mean value with the number of points. It consequently decreases with the number of points.

To confirm that ESSH measurements are resilient to strong scattering conditions, in particular in comparison to the centroid-based measurements, we conduct simulations to assess the accuracy of the displacement retrieval at low SBR for both methods. For the highest SBR, typically larger than 1.5, both methods are giving similarly good results, as the average displacement error is null in both cases. However, as the SBR decreases, the centroid method is more impacted. A large dispersion of the shift measurement was observed in the case of the centroid method, as shown by the large error bars on Figure 2.19a and SEM increase (Figure 2.19b). An other way to interpret this second point is to say that the number of images over which the displacement calculation is done has to be increased by a factor 25 at a SBR of 1.2 to achieve the same precision on the displacement measurement with the centroid approach as with the ESSH algorithm. This variability in the displacement measurement affects directly the quality of the aberration correction that can be achieved with an AO loop.

At SBR smaller than 1.2, the noise also impacts the estimate of the average displacement obtained through the centroid process. We can compare the resulting distributions with Student t-tests. The Student t-test is a statistical test that helps to quantify the difference between the mean values of two groups or between one group to a reference value. It takes into account the mean value as well as the variance of the distributions. The later is chosen here.

In the worst situation, corresponding to SBR = 1.09, the Student t-test (α = 0.05) gives a p-value of p = 0.009 for the displacement error distribution obtained with the centroid method. On the other hand, with the correlation-based method, Student t-test (α = 0.05) gives us a p-value of p = 0.355. We can conclude that the mean value of the shift evaluated by the ESSH algorithm is not significantly different from the one imposed numerically at all SBR values assessed whereas the mean value of the measured shift is affected.

Thus, these simulations show that a much more accurate measurement of the shifts between subimages induced by aberration can be obtained at low SBR with the ESSH algorithm rather than with the centroid-based method.

Comparison of centroid-based displacement measurement and correlation based displacement measurement on experimental data

If our previous simulation took into account low SBR, many other aspects influencing peak detection were left aside because of this full-simulation method. In this section, we propose an alternative to evaluate the performances of both peak-finding methods, this time on experimentally acquired subimages, in order to confront our full simulation to real data, and take into account real scattering conditions from biological samples.

a. Procedure

This time, we start with experimentally acquired subimages, with the ESSH sensor. The 35 x 35 pixels² images are first oversampled by a factor 100 with the cubic interpolation function in Matlab, and are then shifted by the equivalent of 0.5 pixel in the two directions. Note that as interpolation starts in the top left corner on the image and relies on, the values outside the interpolation domain are set to zero. After subsampling the shifted images, the shift retrieval between the initial subimages and the shifted ones is performed as described above. This process is illustrated on Figure 2.20.

We ensure that the oversampling-subsampling steps do not add a shift bias, by submitting images to the algorithm, but without any additional shift in between; the measured shift is effectively null.

b. Results

Overall, nine subimages corresponding to signal from green beads acquired under fixed brain slices ranging between 50 µm to 150 µm, in the centre and on the edges of the ESSH were considered (examples of used subimages in Figure 1.8). Note that, for the sake of illustration, the example of a neuronal signal is displayed, on which dendrites are visible; in the following, we only chose situations in which the signal was coming from a single emitter, to allow centroid measurement. The obtained results are reported on Figure 2.21, on top the previous fully simulated ones to compare the accuracy of these results to the expected one. On this graph, we can see that the displacement error consecutive to the shift retrieval for experimental images, with the correlation based method (represented with blue diamonds), is systematically closer to zero than the one obtained with the centroid-based method (represented with single green dots).

Identically good shift retrieval is obtained with the two methods for SBR larger than 2; however, when the SBR drops, the error on the measured displacement remains within ± 0.1 pixel with the correlation-based method, whereas the centroid-based measurements deviates from zero.

Chapter 2. Aberration measurements and wavefront sensing in scattering conditions This error increases significantly at SBR smaller than 1.3. Moreover, we note that even if the correlation-based method does not perform as well as expected, as the values are half of the time out of the error range of the fully simulated data, for the centroid results, this effect is way worse. It shows that real life noise more strongly affect the centroid measurement than the ESSH algorithm.

SBR
Hence, these results based on experimental data confirm the robustness of the shift retrieval for low SBR while using the ESSH algorithm whereas it fails with the centroid-based one. These experimental-based simulations are not free from criticism. One main limitation sits in it not taking into account the shape deterioration of the PSF and of the images, due to aberration and the difference in SBR between the two correlated images. Indeed, as we correlated all the subimage with the reference one, chosen in the center, some important pattern changes can occur amongst the pupil because of scattering and aberrations and finally put the correlation algorithm at a disadvantage. However, this effect can be balanced by the fact that most often, the ESSH signal is not solely composed of one emitter, considering that the sample preparation or the neuronal labelling is optimized. The balance between a too dense sample and a not enough one can be hard to find but is expected to ultimately lead to the best results. Thus, optimal experimental conditions should lead to this expected accuracy.

AO loop within scattering conditions

After an analysis of the wavefront measurement accuracy in depth with the ESSH sensor comes the time to implement it in a complete AO set up in order to perform closed-loop correction.

In this section, we first describe the setup and the samples used for these experiments. The obtained results as well as the inherent limits to this example are then discussed.

Setup and sample description a. Setup

To perform closed-loop correction on fixed brain slices, we use a different setup, initially built by Antoine Hubert during his thesis. This setup does not provide optical sectioning here, it must be coupled to a 2D sample; in the following chapters, a microscopy setup with optical sectioning is to be used coupled to 3D samples. In this epifluorescence microscope, the sample is illuminated by a 488nm-laser (Cobolt 06-MLD) focused in the back focal plane of the water immersion objective (Leica 25X, NA = 0.

b. Neuronal sample

To mimic the signal obtained in vivo, neurons in culture prepared by Marine Depp from IBENS were used as fluorescent emitters. These neurons were labelled with mVenus protein and their emission peak was at 530 nm. The neuronal culture was left to grow for 21 days to achieve a sufficient concentration of neurites for the experiments, with an initial concentration of 180,000 neurons per well. The preparation of the neurons is described in Appendix A. On top of these

Results and discussion

As ESSH sensor needs to be coupled to some optical sectioning, and while we wanted to stick to an epifluorescence configuration here for the sake of the implementation simplicity, the sectioning was provided by the sample itself, aiming at mimicking in-depth optical sectioning in a scattering biological medium. Aberration correction in a closed-loop AO setup with the ESSH sensor is tested here with a model sample that resembles in vivo biological samples in term of structures and photon flux. In both represented cases (Figure 2.23 and Figure 2.24), we correct for the microscope static aberrations ahead of any image recording. Hence, only sample-induced aberration are contributing to the deterioration of the image quality and are corrected for.

Within the sample, we can distinguish two aberration sources; indeed, aberrations are induced by the brain slice and by the coverslip placed on top of it. The latter is generating a 210 nm RMS of SA3, prevailing to the 10 to 20 nm SA3 generated by the brain slice, as shown in Section 2.13. Thus, the brain tissue layer mainly enables the introduction of significant scattering. The AO closed loop is operated at 2 Hz, and improves significantly the image quality.

For the thinnest considered sample, its thickness being equivalent to 1.2l s , the signal from the soma is increased by 28%, between is increased by 34% as visible on the corresponding insets. This contrast and signal enhancement reveals some dendrites that were not visible before the correction.

In the case of the thicker sample, corresponding to the results shown on Figure 2.24, the AO correction increases the signal and contrast of the image even in these conditions despite the very low amount of ballistic photons available for wavefront sensing. If the signal increase of the central neuron is of 14%, its total width is reduced by 24% (from 20.1 µm to 16.4 µm): the image resolution is thus increased. In this example, ballistic photons represent only 8% of the incident light on the sensor for this 100-µm fixed brain slice, where in the yellow range, it corresponds to 2.5l s .

In this proof of principle experiment, we can highlight the fact that the gain in image quality is significantly lower than the one expected using two-photon microscopy; in non-linear microscopy, image quality is not affected by scattering on the emission path as all the emitted photons are collected. Thus, a RMS value drop from 250 nm to 80 nm or 29 nm as it is observed on the wavefront of experiments here should lead to a higher gain on the imaging path, especially on the smaller structures. Here instead, they are even more strongly affected by scattering, hence reducing the signal gain.

Thus, we showed that an AO correction based on ESSH wavefront measurement, through a scattering tissue and using fluorescently-labelled biological samples, allows a significant image improvement.

Conclusions

In this Chapter, we showed that our direct wavefront measurement strategy based on the use of an Extended Source Shack-Hartmann was resilient to strong scattering conditions, typically encountered in depth while performing in vivo two-photon microscopy. By several methods the aberration measurement accuracy of this method was assessed and its efficiency proven.

• The ESSH allowed quantitative aberration measurements in strong scattering conditions. We confirmed the accuracy of the measurement by determining third order spherical aberration coefficient as a function of the considered tissue thickness and cross checking it with optical simulations.

• The peak-finding algorithm used along the ESSH sensor showed a stronger resilience to scattering than the classically used centroid-based algorithm. This was shown on simulated data as well as on experimental acquisition.

• Finally, we performed aberration correction on biological-like sample by correcting aberration on neuronal signal within scattering conditions. A significant gain was shown in spite of the not optimized setup configuration for in-depth imaging (epifluorescence microscope in the visible).

• Analysing the signal on the ESSH allows to measure the scattering length of biological tissues.

Therefore, all these conclusions strengthen our choice to implement Extended Source-based wavefront sensing on a two-photon microscope for neuroimaging.

Chapter 3

Implementing an extended-source based AO loop on a two-photon microscope In this Chapter, we describe the practical implementation of the adaptive optics loop with the ESSH as the wavefront sensor, on a home-built two-photon microscope, in a closed-loop configuration. The designing part of the whole system is first detailed, along with the constraints imposed by both AO and two-photon microscopy. A section is dedicated to the pupil conjugation choices. Then, the setup is characterized and corrections are performed on ideal and scattering samples.

Setup design

Technical specification a. Closed-loop versus Open-loop

The main difference between a closed-loop and open-loop system is the arrangement of the AO components with respect to each other. In a closed-loop system, the light returning from the sample passes via the DM before it reaches the WFS. Thus, the WFS provides feedback on whether the deformable mirror has reached the desired shape to obtain a planar wavefront. In an open-loop system, there is no feedback on whether the DM has acquired the correct shape, as the light does not pass via the DM when returning from the sample.

The advantage of using a closed-loop system is that it can be more accurate; any non-linearities in the movement of the mirror surface with applied voltage are inherently compensated, as are hysteresis and drifts in the shape of the DM corrected for. However, the open-loop system still has the advantage of bringing the WFS closer to the signal emission, limiting the loss of signal consecutive to optics crossing. These photon losses are even more important in a descan configuration as the light travels back through the galvanometer mirrors, and the in between lenses.

In this thesis, the choice of a closed-loop configuration was made in regard of our wish to assess quantitatively the quality of the correction along the loop behaviour, but also provides the best possible correction.

b. Design requirements and constraints

Now that we set ourselves on a closed-loop configuration, the choice of the optics is driven by several other constraints. The diameter of the DM, and the dimensions of the ESSH, as well as the objective with its focal length and its back focal plane diameter do not leave us with many possible configurations.

• The final design of our microscope is a result of these constraints. The setup is represented in Figure 3.1, and annotated pictures on Figure 3.2.

c. Technical elements

Optics All the optical relays are 4f magnifying relays in a telecentric configuration, to image the objective pupil onto the pupil of the DM and on the ESSH; so the magnification of the relay helps matching pupil sizes, and reliably reproduces both amplitude and phase. Starting from the objective (XLUMPFLN20X, Olympus), the first afocal relay is composed of a tube lens L1 (AC508-180-AB, f ′ 1 = 180 mm, Thorlabs), and followed by L2 (AC508-150-A, f ′ 2 = 150 mm, Thorlabs) resulting in an effective magnification of 0.83 between the objective exit pupil and the DM; their dimensions exactly matching. The incident beam angle is set to an angle of 11°to the normal (minimum accessible angle due to bulk constraints) so the DM operates almost identically to normal incidence.

The relay between the DM and the first galvanometer mirror is made by L3 (AC508-180-AB, f ′ 3 = 180 mm, Thorlabs) and L5 ( f ′ 5 = 50 mm). The choice of L5 is discussed more specifically in Section 3.1.2, as well as the one of the relay lenses between the two galvanometer mirrors (L6-L7), that are used to conjugate properly the exit pupil onto each of the scanning mirrors.

The excitation path is finally completed by L8 (AC254-500-B, f ′ 8 = 500 mm, Thorlabs) and L9 (AC254-150-B, f ′ 9 = 150 mm, Thorlabs) that expands the laser beam, so that it overfills the first scanning mirror. The choice was made not to underfill the back aperture of the objective, for the sake of technical ease and bulk, at the expense of power transmission as described in [START_REF] Oheim | Two-photon microscopy in brain tissue: parameters influencing the imaging depth[END_REF].

Between L3 and L5, the beam is split to redirect the fluorescence emission towards the ESSH of the AO loop thanks to the dichroic beamsplitter (FF775-Di01-25x36, Semrock). The final relay is made by L4 (AC254-60-A, f ′ 4 = 60 mm, Thorlabs) to get a 1/3-fold beam enlargement on the ESSH and to fill its entrance pupil. An additional short-pass filter is positioned in front of the ESSH to reject any stray light (720SP, Chroma).

As for the detection path, its optical and electronic elements are detailed in Section 3.1.3.

Miscellaneous

The near-IR pulsed laser is a Ti:Sapphire femtosecond laser (MaiTaiHP DeepSee, h537), tuneable in wavelength between 690 and 1040 nm, and with dispersion compensation. The pulsed width is inferior to 100 fs and the repetition rate is 80 MHz, with a beam waist of 1.2 mm. A Faraday optical isolator (ISO-05-800-BB, Newport) is positioned at the laser output to prevent feedback in the laser cavity. A half-wave plate (AHWP10M-980, Thorlabs) is mounted on a motorized rotating stage to control output power. A mechanical shutter (Uniblitz Electronics, Vincent Associates) is added after the first beam expander L9-L8 and its opening is controlled via the software to preserve the sample from photodamage when no image is recorded.

Scanning mirrors are both galvanometer scanners (8315K, Cambridge Technology), with a 5mm diameter effective mirror aperture and a maximum total scanning angle of 40°. For a sinusoidal command, their maximum scanning rate is 1.3kHz. The control is operated via two servo drivers (671, Cambridge Technology), that are externally powered by ± 28V supply.

The signal generation and data acquisition is driven by a Data Acquisition system DAQ (NI USB-6343, National Instruments). This design choice is motivated by several OpticStudio simulations: in the following, the excitation path design is first discussed, then the detection on the collection path is described.

Excitation path: optical simulation

As we saw in the previous sections, many constraints imposed by the AO loop elements restrict the choices and most of the focal lengths are already fixed. However, as the two-photon microscope is a scanning setup, one question remains open: the utilization of scan lenses or not, as L5 and L6-L7.

Scan lens

Scan lenses are specifically designed to produce a spot with minimal distortion while being scanned over the FOV in the image plane, i.e as the angle of the incident beam with respect to the optical axis of the lens varies. They are positioned right after the scanning system, where the lenses required are short focal length ones (typ. 50 mm) while wide angles are scanned (typ. ± 10°) leading to potential large field aberrations and great spot distortion.

Even if we have an AO loop at our disposition on the setup, and that we can correct for these aberrations, the aim is to minimize the static ones, induced by the setup, in order to keep DM correction dynamics to correct for sample induced aberrations. The considered scan lens studied here is a 50-mm effective focal length, with a broadband antireflection coating (SL50-CLS2, 450 to 1100 nm, Thorlabs), and available as a Black Box for Optic-Studio simulations; the three tested configurations are presented on Figure 3.4 and their performances on Figure 3.3, the number of scan lenses is the varying parameter between the layouts. In the first one, three scan lenses are added to the optical path: one after the second galvanometer mirror (L5) and two in between the galvanometer pair (L6-L7). This design is expected to exhibit the best performances. The second design possesses only one scan lens, L5, after the second galvanometer mirror, the optical relay between the galvanometer mirror is composed of doublet lenses. The final setup is the simplest one, with only doublet lenses and no scan lens. The quality of the designs are assessed regarding three features: the PSF on-axis and off-axis, the spot diagram in both cases and the corresponding encircled energy. For a scanning angle of ± 4.5°on the galvanometer mirror, the total FOV is 400 µm. Here the layouts are only 2D. These parameters are to be compared to the optical features of an ideal optical system, that would be diffraction limited, on and off-axis. As expected, setup a is strongly resilient to field aberrations: the system is very close to the diffraction limit, for on-and off-axis rays. A minimal amount of rays are missing the diffraction limit disk. However, this configuration is very bulky: the short effective focal length of the scan lenses combined to their size (53.9 mm, working distance 26.4 mm) make them pretty hard to handle with galvanometer mounted on translation stages. Indeed, they are designed to work with two galvanometer scanners on the same mount, and not to be used as optical relays in between.

In comparison, the final setup c performance are much worse. Even after systematic distance optimizations with OpticStudio tools, the on-axis performances are still deteriorated due to spherical aberration, and the off-axis propagating rays suffer from distortion and astigmatism. Only 80% of the total energy is encircled respectively within a 4-µm and 5.4-µm diameter circles in this configuration.

The intermediate configuration b offers great on-axis performances, but still not optimal off-axis ones. This is due to the limited optical performances of the two doublet lenses used as x 1 optical relay between scanning mirrors, with their short focal length and experiencing high angled rays, Therefore, this configuration b is selected for its reasonably good performances, and minimal bulk.

Detection path design a. Optics

In a multiphoton microscope, the aim is to collect as many photons as possible. For sufficient depth, multiple scattered fluorescence light leaves the sample from a diffusely radiating region on the surface, which has a FWHM intensity of about 1.5-times the focal depth, independent of the scattering length [START_REF] Beaurepaire | Epifluorescence collection in two-photon microscopy[END_REF]. To collect a maximum of photons, propagating at high angles, the detection path must be as close as possible to the sample plane. Our detection path design is inspired from [START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF]. The PMT used here is the H7422-40 model from Hamamatsu, with a GaAsP/GaAs photocathode. Several physical constraints have to be considered, and are represented in Figure 3.5a and b:

• The epicollection implies the presence of a dichroic beamsplitter, and thus a containing cube. This cube is the closest to the objective output as possible.

• The PMT sets two additional constraints: the presence of the input window limits the entrance of the rays, with its opening diameter (Phi window = 7.2mm), as well as its distance from the photocathode d (d = 7.4 mm). The final limiting point is the size of the photocathode (Phi photocathode = 5 mm).

Note that new models are now available with wider windows and photocathode (up to 15 mm in diameter). The setup is optimized so that a maximum of rays entering the system can hit the photocathode (simulated as the final surface). The relay is then composed of two lenses, one condenser lens (L2 on Figure 3.5, and L11 on Figure 3.1: ACL3026-UA, f ′ = 26 mm, Thorlabs), and one singlet lens (L1 on Figure 3.5, and L10 on Figure 3.1: LA1265-A, f ′ = 75 mm, Thorlabs). On this simulation, three different ray propagation angles are tested: on-axis rays in blue, rays propagating at a 1.2°a ngle from the normal incidence in green, and at a 2.55°angle in red. These propagation angles respectively correspond to a lateral FOV dimension of 400 µm and 800 µm in a transparent medium. Yet, in a scattering medium, where the photons seem to originate from an extended source in the focal plane instead of from a spot, it is roughly equivalent to observing a FOV of 200 µm at a 400-µm imaging depth as the photon angular distribution at the entrance of the objective is spread [START_REF] Beaurepaire | Ultra-deep two-photon fluorescence excitation in turbid media[END_REF].

b. Electronics

To avoid saturation of the detection, the choice was made to work in an integrator configuration instead of photon counting. The output current of the PMT is converted to a voltage thanks to a preamplifier unit (C648-01, Hamamatsu), chosen for its high current-to-voltage conversion factor and speed. The latter is powered by an external power supply (AL991s, ELC), delivering ± 5 V. The dynamic range of the DAQ is adapted to the maximum and minimum voltage output of the amplifier (0 to 1V). Aberrant negative voltage values, due to electrical noise, are set to zero. Limits Let us note here that this acquisition is very likely not to be sufficient for functional imaging and is to be modified in order to increase its speed and gain. Most of the neuroimaging teams implementing AO on multiphoton setups build their own amplification system, especially for analog acquisition. One can cite Kleinfeld's team proposal based on two alternative integrators, to increase the speed without loosing the photons emitted during the discharge of the capacitors [START_REF] Nguyen | A Computer System for Two-Photon Laser Scanning Microscopy with Concurrent Plasma-Mediated Ablation and Electrophysiology[END_REF].

Control and acquisition software

Two configurations are implemented and described below and are to be adapted to the sample labelling used for AO.

a. For full-field imaging

The first acquisition configuration is based on the chronogram in Figure 3.6. Ahead of the beginning of the acquisition are generated the scanning instruction signals, that are to be sent to the two galvanometer mirrors, according to the users' wish. This part is largely based on Mathias Mercier's work, and has been made with his help.

The pixel time, the number of pixels in one line, and the pixel size are set by the user ahead of the acquisition start. The AO loop is controlled independently from the Imagine Optic software, WaveTune 4.3.1. The complete implementation of the AO loop control to the same software is currently under processing. This aims at greatly simplifying the acquisition and avoid time waste, and consequently limit photobleaching.

b. For patterned scan

One of the advantages of the ESSH wavefront sensor is its compatibility with arbitrary scanning. In a uniformly labelled sample for example, that would not exhibit any pattern to correlate on, we can generate it artificially by scanning the sample cleverly. As we currently dispose of galvanometer scanners, our pattering abilities are currently limited by their displacement speed, but still enables interesting and effective trials.

In this configuration, the ESSH acquisition as well as the patterned scanning are triggered with an external waveform generation. The pattern used in the following sections is a 'IO', note that a cross pattern has also been tried. So far, the patterned scan and AO correction can not be performed simultaneously as imaging, and requires sequential use, in which the AO scan is performed ahead of the scientific imaging. It is not particularly constraining as aberrations do not evolve temporally: Ji et al. showed that brain-induced aberrations were stable over hours in [START_REF] Ji | Characterization and adaptive optical correction of aberrations during in vivo imaging in the mouse cortex[END_REF].

Objective and pupil conjugation

Let us now get back to practical optical considerations that comes during the alignment: pupil conjugations. In most of the AO setups, the back aperture of the objective is conjugated to the AO elements (DM and if present, microlens array of the wavefront sensor (such as in [START_REF] Wang | Direct wavefront sensing for high-resolution in vivo imaging in scattering tissue[END_REF][START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF][START_REF] Facomprez | Accuracy of correction in modal sensorless adaptive optics[END_REF]). An other possible implementation, proposed for microscopy by Mertz et al. and named 'Conjugate AO', consists into conjugating the AO elements with the main source of the aberrations, i.e the sample plane, shows enhanced FOV performances [START_REF] Mertz | Field of view advantage of conjugate adaptive optics in microscopy applications[END_REF]. This multiconjugate strategy has been assessed numerically [START_REF] Wu | Numerical study of multi-conjugate large area wavefront correction for deep tissue microscopy[END_REF] and experimentally on widefield imaging coupled to direct wavefront sensing [START_REF] Li | Conjugate adaptive optics in widefield microscopy with an extendedsource wavefront sensor[END_REF], before its implementation on a two-photon microscope, for indirect wavefront sensing AO [START_REF] Paudel | Axial range of conjugate adaptive optics in two-photon microscopy[END_REF]. However, conjugate AO employs solely subapertures of the DM as the position of the excitation beam is translated on the DM during the scan, contrarily to back pupil conjugation in which the beam is fixed on the DM. It requires to be coupled to DM with a large number of actuators to partially compensate for this loss in correction resolution. If a wavefront sensor is added, it also suffers from these translations. Moreover, conjugate AO is for now mostly adapted to cases where aberrations are confined to a single layer, which is not the case in the majority of the volumetric biological samples. Therefore, we decided to conjugate the back aperture of the objective with the AO elements, the determination of its localization is described in the following: a brief introduction to pupil definition and issues precedes the experimental localisation determination.

Pupil conjugation issues

The pupil of a system, also called aperture stop, is defined as the physical diaphragm that limits the aperture, i.e the angle of the rays that can propagate through the system for an on-axis point. The image on this plane in the object plane is the entrance pupil while in the image plane, it corresponds to the exit pupil. All the higher angled rays lean on these pupil. In the case of a telecentric system on the exit side (respectively on the entrance side), the exit pupil (resp. entrance pupil) is located at infinity, guaranteeing an homogeneous light transmission.

For an objective, the exit pupil (also called back pupil) diameter depends on the numerical aperture of the objective (NA) and its effective focal length (f'), it is given by: Φ = 2NA f ′ . However, some other physical elements can limit the aperture set by the pupil for off-axis objects: this effect is called vignetting, and unfortunately, in complex systems such as microscope objectives, it is inevitable [START_REF] Zhang | Systematic design of microscope objectives. Part I: System review and analysis[END_REF]. In the case of objectives, vignetting might be the only option to limit important field aberrations, at the expense of ray transmission, and help to keep a uniform imaging quality in the whole FOV.

For AO microscopy, vignetting complicates the situation. Indeed, the wavefront measurement and correction have to address the same pupil, during the calibration as well as during the correction, and an untimely movement affects the measurement: the measured slopes that at time t will be addressed at time t + dt by the neighbouring subpupil. For direct wavefront sensing coupled to scanning methods such as two-photon microscopy, these shifts are way easier to handle when they do not exists, hence the necessity to conjugate properly the scanning mirrors, the DM and the wavefront sensor with the objective plane corresponding to the minimal translation when scanning.

Implemented strategy

Whilst the most straightforward method to locate this plane, usually referred to as the 'back pupil' of the objective, is to illuminate from the side a layer of strongly scattering medium at the sample plane, so that the light will be scattered at angles greater than the acceptance angle of the objective lens (set by its NA), this method fails when numerous edges appear as neat as one another on the pupil imaging camera. The existence of other limiting apertures, for off-axis points, is not so rare, forces us to switch to another approach, and to implement a specific setup. The latter being presented on Figure 3.7.

The main principle lies in the scanning of a point smaller than the PSF of the studied objective, in its focal plane. Simultaneously, the back pupil plane is imaged on camera Cam2 while the sample plane is imaged on camera Cam1. In order to untangle the excitation path from the determination of the objective pupil position, and thus avoid potential bias generated by excitation paths misalignment, the point is generated by an other objective (Obj2) with a larger NA than the objective of interest (Obj1 Obj2 is mounted on a cage compatible XY translator mount for 1" optics, with a 2-mm total displacement range in both direction, in order to move the spot laterally in the focal plane of Obj1, typically of 120 µm to the left and to the right of the FOV, center being taken as the reference position. Following the spot displacement in the FOV, the imaged plane on Cam2 moves too, as we can see on Figure 3.8. The distance between the two objectives is fixed so their focal planes are coinciding. The block composed of Obj1 and Obj2 is axially moved over a 90-mm range to find the optimal conjugation plane, that is to say, the distance between Obj1 and L3 for which the beam movement on Cam2 is minimal when the spot is translated in the FOV.

These pupil displacements might seem at first negligible, regarding the total pupil size, but considering the beam magnification during the propagation, on the ESSH, it corresponds to a shift of almost one microlens for a lateral FOV of 240 µm.

The pupil displacement values with the lateral spot movements of ± 120 µm are plotted in Figure 3.9 as a function of the axial position of the objective (Obj1). The edge pupil shift and For a spot displacement to the left in the FOV, the pupil shifts mainly to the right, on Cam2, causing a principal shift on its left edge, but the right edge is affected too. Along the experimental data are added linear fits to ease the reading. The zero objective position corresponds to the starting setting in which the back focal plane of the objective is imaged on Cam2, whereas at -44 mm, the rear of the objective is.

We choose to position ourselves in the plane corresponding to the axial position of -20 mm of the objective, as three of the four shifts are minimized; the sum of the absolute shifts for the four edges is minimal in this position, in spite of the slight asymmetry of the shifts. For a total FOV of 240 µm, this pupil shift corresponds to a shift of 180 µm on the microlens array of the ESSH, so approximately to half a microlens, which is reasonable, especially since the maximum FOV scanned for wavefront sensing is 130 x 130 µm² (set by the dimension of the microlens). For this FOV, it corresponds to a shift of approximately a quarter of a microlens. This pupil movement corresponds to low-pass filter on the wavefront measurement.

Characterisation of the AO loop on two-photon signal

From this complete setup described in the firsts parts of this chapter, we now dive into the AO loop in itself for our system and its performances on two-photon excited signal. Starting with an introduction to AO loop calibration and its adjustable parameters, we then address the fine settings required for the correct operation of AO in two-photon microscopy, such as differential aberrations and focus drifts. The performances of the Extended-source based AO on the twophoton microscope (AO-ESSH) are later detailed: we detail the imaging performances of our setup in ideal conditions, prior to challenging it in strong scattering conditions.

Brief description of AO loop parameters a. Calibration process

To use use the DM in an AO closed-loop, a first calibration step has to be done and is briefly described here. The calibration procedure first starts with the generation of the Influence Functions Matrix (IF). All the actuators are first set to zero; then for each actuator, a positive and a negative voltage value are applied sequentially, the mirror surface locally moves upwards and downwards in consequence to the applied voltage (see Figure 3.10a). During the process, the change in slope is calculated and stored as a column in the Influence Functions Matrix. Its element values are given by:

IF l,act = S +C lx -S -C lx 2C and IF n l +l,act = S +C ly -S -C ly 2C (3.1)
with l the microlens index, n l the total number of microlenses, act the actuator number, and S +C lx and S -C lx the slopes in the x-direction for the microlens l respectively corresponding to a positive control signal +C applied, and to a negative one -C, relatively to the middle-position 0 (same for the y-direction).

Once the influence functions matrix is obtained, the required control signals C can be derived from its pseudo-inverse IF + and the vector of the measured slopes S via:

C = (IF + )S (3.2)
The pseudo-inverse matrix IF + is obtained through Singular Value Decomposition (SVD); the correction ability of the mirror is considered as the sum of independent mirror modes. The number of modes is equal to the number of actuators. The SVD decomposition of IF is represented in Figure 3.10b, giving:

IF = USV T (3.3)
The rows of V T give the control signals that are to be applied to each actuator to generate a given surface shape: these are the mirror modes. The columns of U are the modes as the wavefront sensor sees them, here a column vector of slopes.

The IF matrix is then pseudo-inversed to obtain the command matrix

IF + IF + = VS -1 U T (3.4)
S -1 leads to the inversion of the singular values, that become the mode gains; from this process can be generated high mode gains, resulting in potential actuator saturation. To produce the corresponding mirror modes, large control signals are required and the actuator control signals will exceed their maximum value. These modes have to be removed from the final calculation of the control matrix, so the corresponding mode gains are set to zero. The number of corrected modes is thus defined.

Note that in this manuscript, only circular pupils are considered, but wavefront control is still possible with arbitrarily shaped pupil, but specific control methods have to be implemented in this case [START_REF] Dong | Wavefront control in adaptive microscopy using Shack-Hartmann sensors with arbitrarily shaped pupils[END_REF].

b. AO loop adjustable parameters

Once that the calibration step is done, aberration correction in the closed-loop system can be realised. The control signals are thus driven by:

C New = C Current + g(IF + )S Meas (3.5)
where C New are the new control signals, C Current are the current control signals, g is the gain (a value between zero and one to limit oscillations in the mirror surface), IFS + is the control matrix (pseudo-inverse) in term of slopes, and S Meas are the measured slopes.

Gain

The gain sets the amplitude of the correction. The matrix actually used in the control loop is therefore the control matrix IF + multiplied by the gain g. If the gain is set very close to 1, the voltages applied to the actuators attempt to correct the entire wavefront deformation at once. However, in this case, the control system may be unstable in the case of significant noise or errors in the WF measurement. Lower gain enables to increase the robustness of the AO loop, at the cost of more time required for optimal convergence.

Number of corrected modes

As described in Section 3.3.1a., the number of corrected modes can be fixed too, and a maximum limit setting is necessary in order to avoid divergence of the loop. In our setup, the DM possesses 52 actuators, thus setting the maximum possible number of corrected modes to 52. However, within our setup, we note that the mode gains associated to modes 48 to 52 are too important: these modes originate from small singular values, and forces some actuators to their pushing limits when calibrated. If taken into account in the loop, they are more likely to add noise to the correction than other modes and to risk diverging the loop. For 42 corrected modes in ideal signal-to-noise conditions on the ESSH, we reach wavefront RMS values of 10 nm, that corresponds to the inferior limit of achievable wavefront RMS considering our wavefront sensing accuracy. Increasing the number of corrected modes can no longer enhance the correction, and solely increase the mirror amplitude used for the correction. Thus, the number of corrected modes is set to 42 in most of our experiments, based on the latter empirical assessment.

First settings

After this description of the setup and of the AO loop, let us now focus on the latter's fine settings. As we saw, many parameters impact the loop. For the first corrections, realised in ideal conditions, we consider a gain of 0.6 and 42 modes to correct for, in a solution of fluorescein. The AO-loop resulting mirror shape thus corrects for the system aberrations, and becomes our mirror starting position. However, from this simple experiment already arises an issue: a correction drift appears when the loop is kept on continuously and that too many iterations occur; this happens after three to five iterations. This issue arise from the combined effects of chromatic aberrations, and coupling of spherical aberration with focus, and is further discussed in Section 3.3.2a..

The difference between measured aberration on the wavefront sensor and the ones experienced by the excitation beam is the second point to tackle and is addressed in Section 3.3.2b.

a. Addressing focus shifts

While imaging an area of interest, the first necessity is to stick to this plane, same apply for the AO correction. A focus shift is not only inconvenient for the imaging part on the PMT as the plane of interest is lost during the AO loop, but it is also harmful to the AO loop itself as the signal on which the wavefront measurement is initially computed is lost too.

This focus shift has first led us to important axial shifts during the correction. Within a typical experiment, involving a coverslip on top of the sample of interest, we observed a shift in focus (the aberration) of 600 nm on the measured wavefront, corresponding at the stage level, to an axial shift of 10 µm. Two linked sources of focus shifts are identified here:

• Spherical Aberration (SA) is composed of a focus part, and the first way to minimize it consists into modifying the focus. As SA is large when considering samples with coverslips (our objective is not corrected for it), it can consequently lead to important axial shifts.

• The DM corrects the aberrations on the excitation beam. However, the wavefront is measured on the emitted signal. For a typical fluorophore, as GFP, the excitation is at 920 nm whereas the emission is at 520 nm. From this large difference can originate important focus shifts. Mahou et al. find an axial chromatic shift of -3.8 µm when considering FOV of 400 µm between excitation wavelengths of 850 nm and 1100 nm, hence for a difference of 250 nm [START_REF] Mahou | Metrology of Multiphoton Microscopes Using Second Harmonic Generation Nanoprobes[END_REF].

The change in refractive index in a glass is characterized by the V-number; a high V implies a low dispersion. The chromatic focal shift is a result of that dispersion:

δK K = -δ f ′ f ′ = 1 V (3.6)
Using doublets instead of singlet lenses in a polychromatic system will minimize this effect but never will a null focal shift be obtained over the whole spectrum.

In the absence of focus control, the AO loop diverges (see Figure 3.11). If the objective is directly immersed into a fluorescent solution, we note that only for the lowest number of loop iterations (below 20) is the correction satisfying. The final RMS of the wavefront is below 15 nm, and the mirror amplitude is 13% (the starting flat mirror shapes involves 7% of the mirror dynamic). In these cases, the loop behaves as expected and the corrected wavefront satisfies Maréchal's criterion. Above, the loop diverges.

This drift is caused by the coupling of spherical aberration and focus, added to the difference in focus between the excitation wavelength and the emission one caused by longitudinal chromatic aberration. The focus measured by the ESSH is based on a zonal type measurement, in particular the focus is calculated by performing a plane fit on the measured slopes, as the derivative of a curvature is a plane. This measurement is different from the measurement that would be obtained by measuring the focus using a modal method (e.g Zernike decomposition). As a

b. Differential aberrations

In our closed-loop configuration, the AO detection path does not match completely the excitation path. More particularly, the rays emitted at the sample planes follow the same optical path than the excitation rays up to a dichroic mirror that redirects them to the ESSH. This effect typically gives rise to a measured astigmatism that is not present on the excitation path. This astigmatism is due to mechanical constraints impacting the beamsplitter mount, especially in reflection. It thus becomes important to quantify this shift for the following experiments. On the collection path to the PMT, the aberrations are not an issue.

A simple experiment is implemented here to measure the differential aberrations. Two types of emitters compose the sample: on a coverslip are deposited 0.5-µm beads emitting at 510 nm, and two-color larger emitters, 10-µm diameter polystyrene beads loaded with two types of QDs, with 520-nm and 605-nm emitting QDs. Their preparation procedure is described in 2.3.1. The green signal is directed towards the PMT path thanks to a dichroic mirror (FF560-Di02-25x36, Semrock), the red signal from the QD-loaded polystyrene beads is used to measure the aberrations on the ESSH.

The green emission from the large polystyrene beads allows us to position ourselves into an area where both emitter types are present, example given in Figure 3.12, on which the 0.5-µm beads signal can be observed (Figure 3.12a), as well as the green signal from the larger QD-loaded polystyrene beads observed 5 µm above the smaller beads plane (Figure 3.12b). The ring shape of the dual-labelled emitters is due to the preferential positioning of the QD on the inside of the beads. The image on Figure 3.12b matches the signal on the ESSH towards which the red emission is redirected (Figure 3.12d and inset).

We manually modify the wavefront with the BeamShaper tool from Imagine Optic software to maximize the smallest beads signal on the PMT. We estimate in these conditions that the This evaluation is pretty delicate: its sensitivity of the axial stage movement makes it hard to entangle from the sample tilt to the optical axis set by the objective.

An other aberration, that is not strictly a differential aberration but that can be compensated for, is the focus shift induced by the coverslip positioned on top of our samples in most of our experiments. After the AO loop and the correction of the noticeable amount of SA3 from the coverslip (typically 200 nm RMS), an amount of -600 nm of focus has to be added to retrieve the same focalisation plane than before. It corresponds to a 10-µm axial shift at the sample plane. Similar shifts were observed later in Chapter 4. This arises from the way the focus is measured on the wavefront sensor: it is not based on the Zernike decomposition but is an average value that takes into account the focus from all the Zernike modes.

AO-ESSH in ideal conditions

A couple of optical characterizations of the setup are described in this section. We first measure experimentally the PSF of our system, once setup and sample aberrations are corrected for, and differential aberrations included. The isoplanetic patch of the system is evaluated afterwards. The PSF of the setup is measured on 0.5-µm diameter beads in between coverslips after AO correction and differential aberrations. The 0.5-µm diameter beads are used: as they are wider than the theoretical PSF, we convolve the theoretical PSF with a top-hat function of a 0.5-µm width. The resulting FWHM is to be compared with the experimental one.

a. PSF evaluation

For ten different beads, intensity profiles are plotted in the lateral dimensions, and fitted with Gaussian profiles. The FWHM are then recorded. Results are reported in Table 3.1, against the theoretical values. The intensity of two typical neighbouring beads is represented as a 3D surface plot on Figure 3.13.

The performances of our setup match the theoretical ones fixed by the diffraction limit.

b. Assessment of the isoplanetic patch of the setup

The isoplanetic patch defines the area over which the aberrations are similar, hence over which the applied correction is optimal. Large scanning angles generate important field aberrations that are to be taken into account. The excitation beam is focused to a single spot in the sample, in the center of the FOV into a a solution of fluorescein at 1/100. Aberrations are corrected and the spot is then scanned in the microlenses FOV by 10 µm steps. For each position, the wavefront RMS is noted, giving the following RMS map (see Figure 3.14). • During the alignment procedure, the system was not perfectly centered, around the optical axis of the objective, causing an asymmetry that then propagates through the setup.

• The centering of the field diaphragm is not well adjusted.

• The numerical segmentation of the subimages doesn't properly match the illuminated area.

• The evaluation of galvanometer positions that corresponds to the centre of the FOV is slightly shifted from the optical axis.

c. Impact of the wavefront measurement method on the isoplanetic patch

While using an image to measure the wavefront, we already average the aberrations over the FOV, leading to an averaged correction of the aberrations over this area. We can wonder how the wavefront method influences the isoplanetic patch size. We first correct for setup and sample aberration with two different patterns:

• The Small IO pattern, that is 35 x 35 µm² in the sample plane.

• The Large IO pattern, that is 70 x 70 µm² in the sample plane.

Once this prior correction done, the excitation spot is then swept across the microlenses FOV.

For each position, the wavefront is measured with his spot as guidestar, and its RMS noted in order to reconstruct the maps presented in If we compare these acquisitions with the one obtained in Section 3.3.3b., at the center of the FOV, the measured RMS is respectively 10 nm and 11 nm for the point-based loop, and for the small pattern based loop, its value is slightly larger when the large pattern is used for the initial correction loop (15 nm). Indeed, the system aberrations are averaged over a wider FOV in the case of a large pattern from the beginning. Besides, as it can be noted from the summed up RMS distribution results gathered into Table 3.2, all the methods lead to an average RMS of 37 ± 1 nm, the distribution of these RMS differs slightly with the ESSH error. For the initial isoplanetic patch measurement, a lower median is associated to a wider data points distribution contrarily to the large pattern-based correction.

For all the configurations, 60% minimum of the FOV satisfies the Maréchal's criterion (38 nm RMS). If we relax a bit this constraints and set the limit to 50 nm instead, reasonably close to Maréchal's criterion regarding the λ/50 ≈ 10 nm precision of ESSH, we note that up to 84% of the points within this FOV are below this limit, for the small-patterned based AO. The patterning does not impact the isoplanetic patch. All the methods gives similar isoplanetic patch size here, the difference are below the precision of our wavefront sensor.

RMS

AO-ESSH in scattering conditions

Our system is not to be kept for transparent sample, the aim is to address strong scattering conditions encountered in depth in the mouse. Thus, additional characterizations in conditions mimicking in vivo imaging are essential to our purpose. We propose two experiments here in order to test ESSH wavefront measurement resilience to scattering.

Scattering sample preparation for in depth AO loops.

100 µL of the 2-µm polystyrene beads commercial solution (Sigma Aldrich), suspended in water and corresponding to ≈ 10 6 beads is first taken, and mixed to fluorescein diluted at 1/100e in water. The solution is embedded in 500 µL of agarose gel, prepared from low-melting agarose at 1%. A coverslip is positioned on top of the preparation.

The resulting sample has a scattering length of 70 µm at 520 nm according to Mie theory.

a. Limit in depth

Let us start by testing the ability of the ESSH wavefront sensor to measure and correct effectively aberrations in depth of a scattering media. The scattering sample is illuminated with the previously tested patterned illumination. In this subsection, the sample preparation is similar to the one describe in Box 3.3.4 above, except that scattering length is 140 µm instead due to a beads/agar ratio modification. Initial correction is performed right below the coverslip, as well as for each plane while going down in the sample by 50-µm steps. In these conditions, we can reach depths corresponding up to four-times the scattering length of the medium.

Let us precise that the photon flux is not limiting here, but more the distortion of the pattern on the edges due to scattering. The outer subimages are too blurry and different from the reference subimage to obtained a sharp correlation peak.

b. Influence of the pattern on depth limit

To see whether or not this depth limit depends on the patterning, we compare the pattern with a point illumination. As a distorted spot can also be a good signal to correlate on, we expect it to allow corrections to similar depth. On Figure 3.17 can be compared their in-depth performances.

At the surface of the sample, a first correction is performed, similar wavefront RMS are obtained.

As deeper regions are accessed, the ESSH signal starts getting deteriorated. As a consequence, the correction loops at these depth do not reach the optimum, set by Maréchal's criterion. However, this effect is noticed earlier in the absence of scan than for the patterned scan. Indeed, when a corrected wavefront of 18 nm-RMS is obtained with the patterned scan, 35-nm RMS are reached with the point source at 450 µm deep in the sample (≈ 3.2l s ). At 550 µm (≈ 3.9l s ), both configurations struggle to achieve optimal correction: if a RMS value close to Maréchal's criterion is still achieved with the patterned scan, it is not the case for the point illumination.

Thus, correlation effectively takes advantage from the distribution of the signal into several points in a scattering medium. This point is later discussed (see Section 3.4.3). 96 Chapter 3. Implementing an extended-source based AO loop on a two-photon microscope

Comparing Extended Source wavefront sensing to Descanned wavefront sensing in scattering conditions

After these various characterizations, one major question remains: how does this AO method based on extended source wavefront measurement performs compared to the centroid based wavefront measurement, in two-photon microscopy and in scattering conditions ?

The classical centroid based approach is usually implemented in a descanned configuration (see Section 1.4.2): the emitted signal hits back the galvanometer mirrors before reaching the wavefront sensor, recreating a guide star. In this configuration, the wavefront sampling results from the signal averaging over the imaged FOV too.

In this section, the implemented setup is first described, before some control experiments in non scattering conditions. The final part is dedicated to results obtained in scattering conditions and to the associated discussion.

Setup

To compare the ESSH approach with the descanned wavefront sensing approach (referred to Descanned SH in the following), two wavefront sensors are implemented simultaneously on the microscope (see Figure 3.18).

The chosen SH for the descanned wavefront measurement possesses exactly the same dimensions as the ESSH. Hence, the wavefront comparison becomes almost straightforward. The same camera coupled to the same microlens array ensures similar wavefront sampling: the same size pupil is decomposed over the same amount of microlenses, giving rise to the same wavefront sampling. On the Descanned SH, the local slopes are obtained through the centroid-based measurement described in 2. The wavefront is reconstructed over 52 Zernike modes, on a 2.265-mm pupil radius. As a reminder, the radius of the imaged objective pupil on the microlens array is 2.5 mm, we solely remove the most external subimage row, corresponding the pupil edge, on which rays are partly vignetted.

The fluorescence signal is split up between the two optical paths by a 50:50 beamsplitter (Throlabs, BSW26R), the first half reaching the ESSH while the second half is descanned by the galvanometer mirrors, before being reflected by the dichroic beamsplitter (FF775-DiO1-25x36, Semrock). An additional afocal relay, L10-L11, is added to conjugate the objective pupil on to the microlens array of the Descanned SH (AC254-150-A, f ′ 10 = 150 mm, Thorlabs) (AC508-180-A, f ′ 11 = 180 mm, Thorlabs). Ahead of the Descanned SH are added two filters, a shortpass filter and a bandpass filter (respectively 842SP, Semrock and 520/44, Semrock), The shortpass filter is kept in front of the ESSH (72SP, Semrock).

Lastly, to ensure that the same amount of photons is used for wavefront sensing on both sensors, in spite of the difference in optical path, and the difference in photon distribution on the subimages, neutral density filters are added in front of the ESSH. Choice is made for a single excited point in fluorescein and in non scattering conditions: density filters are adjusted so that the same level of camera saturation is obtained on both (Neutral density filters: ND = 0.4 and ND = 0.5, Thorlabs). 98 Chapter 3. Implementing an extended-source based AO loop on a two-photon microscope

In non scattering conditions

Let us start with a couple of control experiments to check that both methods are giving similar results in non scattering conditions. The sample is simply composed of a 200 µL of fluorescein at 1/100embedded in 800 µL of agar at 1%; on top of it is positioned a coverslip.

As the sample is intrinsically uniform in fluorescent emitters, the excitation has to be patterned in order to get something to correlate on. We choose the small IO pattern.

Acquisition parameters

For both wavefront sensors,

• An external trig at 2 Hz controls the acquisition start.

• Pattern duration: 100 ms.

• Acquisition duration: 100 ms.

• Averaged images: 2.

• Wavefront reconstruction: Pupil diameter of 2.265 mm, Modal reconstruction over 52 Zernike modes.

The sample stage is translated axially to start the acquisition right under the coverslip. At this position, the ESSH is used to measure and correct for the setup and sample aberrations. With a gain of 0.6 and 42 modes corrected, the resulting wavefront RMS is 16 nm. On the other path, the signal reaching the Descanned SH suffers from differential aberrations from the noncommon optical path. Hence, 86 nm RMS are measured on it after the AO correction with the ESSH. These two wavefronts are taken as references and subtracted in the following, as we aim at measuring the shift from them.

a. In depth

From this starting point, we first translate the stage to reach deeper zones in the sample, by steps of 200 µm. As the sample is uniform, no aberration is expected. Indeed, only 20 nm RMS is measured on the residual wavefront by both methods, while the sample has been moved of 1.8 mm (see Figure 3.19a). The variation of focus is sensibly the same: as 17 nm shift is measured by the Descanned SH against a 10 nm shift on the ESSH (see Figure 3.19b). Both differences, in RMS and in Focus, between the sensors are within the error range of the measurement (λ/50 ≈ 10 nm).

b. Adding sample aberrations

Over a second phase, we introduce voluntarily aberration at the sample level. Starting from an aberration corrected configuration as previously, we now add a 170-µm coverslip on top of the fluorescent sample, meant to add strong spherical aberration.

The obtained signals on both sensors are represented in Figure 3.20 along the measured wavefronts. We note here that the signals on the subimages are not deteriorated at all, as we can see on the insets, and that the intensity distribution is kept constant over the pupil in both cases, except the most external row that is limited by the objective aperture. Note that, due to pupil flips on mirrors and through optical conjugations, the wavefront between the sensors is flipped upside-down. If we look more closely at the measured aberrations, induced by the added coverslip, both sensors measure wavefronts with similar RMS (180 nm for the ESSH against 203 nm for the Descanned SH). The wavefront appearance is flipped consecutively to the pupil flip between the sensors leading to strong Coma but decomposed differently. Moreover, considering centrosymmetric aberration, hence not sensitive to this flip, we note that the obtained values with both methods are identical (see Table 3.3). 100 Chapter 3. Implementing an extended-source based AO loop on a two-photon microscope Thus, in ideal conditions, both wavefront sensing methods are giving similar results, no distortion of the spots or patterned are observed in the case of strong aberrations, we can shift to more complicated conditions.

Aberration

In scattering conditions

We now add scatterers to our sample. The commercial solution having changed from the previous experiments in scattering conditions, our sample is now composed of 200 µL of fluorescein at 1/100, of 400 µL of polystyrene beads from the commercial solution (Polysciences, Inc) 2.61% solids containing 5.68 × 10 9 particles/mL and of 400 µL of low-melting agarose gel at 1%. The top is sealed by a coverslip.

This sample exhibits a scattering length of 133 µm at 520 nm, and of 77 µm at 850 nm.

We correct from the setup and sample induced aberrations underneath the surface of the coverslip, subtract the corresponding reference wavefront to the next acquisitions, and translate the stage axially by 20 µm steps. We record the signal on both sensors, shown in Figure 3.21 as well as the resulting wavefront, represented in Figure 3.22.

We note that as we go deeper in the sample, the signal gets deteriorated, and more particularly on the external row of subimages of both sensors. While this effect is expected, we can highlight the comet shape on the Descanned SH (see Insets on Figure 3.21). In this example, the elongation of the spot is mainly centrosymmetric, but the lower part of the pupil is more impacted.

a. First wavefront reconstructions

In the light of this remark, one can also note that the wavefront obtained with this sensor exhibits increasing deformation with depth, while the one measured with the ESSH stays quite similar, with mainly high order aberrations (5th, 7th order of the Zernike decomposition). On the other hand, the Descanned SH measures an increasing astigmatism in depth. This is the aberration mostly impacting the RMS, that is plotted for both methods in Figure 3.23.

Moreover, what is not visible on the RMS plots and on the wavefronts shown here is the variation in measured focus. This centrosymmetric aberration is not to change (Figure 3.24a). On this graph, some supplementary data are represented: after the vertical line, the scanning rate was reduced by a factor two in order to increase the signal, as the laser power could no longer be increased. The intensity variation is followed along the focus measured value on Figure 3.24b for the Descanned SH. We note a couple of things:

• Below this point, while solely the laser power was increased to compensate for the scattering at the excitation, the focus value measured by the ESSH was increasing from 20 to 70 nm. The focus measured by the Descanned SH was instead increasing much more, from 20 to 700 nm.

• At the turning point, on the Descanned SH, the focus value drops drastically whereas it is not impacted in the ESSH by the increased photon flux. This point corresponds to 250 µm below the sample surface.

• After this drop, the focus resumes its rise on the Descanned SH. Around 340 µm and 360 µm, the signal gets too weak for both wavefront measurements to be meaningful. At 360 µm, three scanning strategies are tried:

-The same scan is kept: ESSH measures 1505 nm while Descanned SH can't detect a complete pupil due to a too low signal.

102 Chapter 3. Implementing an extended-source based AO loop on a two-photon microscope -The scanning rate is doubled once more along the acquisition time, from 200 ms to 400 ms. 815 nm of focus is measured by the ESSH against 1174 nm for the Descanned SH.

-The scan is stopped: ESSH measures 79 nm and Descanned SH 980 nm.

It reveals a bias induced by the scattering medium and impacting more importantly the Descanned SH than the ESSH. As we saw in the previous section, the ESSH seems to perform better than the Descanned SH in these scattering conditions. However, one might ask whether this effect is due to the slope retrieval method, with the pre-filtering step and the 2D correlation, to the signal distribution over a pattern instead of a single spot, or to both combined effects.

Hence, we consider the same acquisitions, but instead of measuring the wavefronts with the classical centroid evaluation, the algorithms algorithms implemented on the ESSH are used on the Descanned SH signals. We refer to this combination as the Descanned SH -ES algorithms one. The corresponding wavefronts are represented in Figure 3.25, and compared to the previously shown wavefronts obtained from the ESSH acquisitions. Similarly, the evolution of the wavefront RMS, the focus and the Spherical Aberration with depth in the sample are respectively plotted in Figure 3.26a, 3.26b and 3.26c.

In this case, the RMS evolution in depth are similar until the final derivation around 340 µm were ESSH fails, and so does the Descanned SH coupled to the ES-algorithms. The focus evaluation is no longer sensitive to the intensity of the signal; no noticeable focus drop occurs this time. However, a stronger focus aberration value is still measured, most likely due to the more visible elongations on the external subimage, on the Descanned SH than on the ESSH. The evolution of the SA3 is the same as the one measured on the ESSH, considering the 10-nm measurement error of the sensors, until both of the sensors fail at 340 µm down the sample. As the average refractive index of the medium is very close to water and slightly increased by the presence of the polystyrene beads, we do not expect a strong variation (the total volume of polystyrene here is inferior to 10 8 mL, for a sample volume of 1 mL).

As spots are good scene no matter their size, it is not surprising to note here that the wavefront measurement on Descanned SH is more accurate when coupled to correlation-based algorithms.

Their relevance was previously explained in [START_REF] Poyneer | Scene-based Shack-Hartmann wave-front sensing: analysis and simulation[END_REF] on a similar looking objects, but this time in astronomy where the spot is elongated because of light propagation through atmosphere. This bias caused by the spot shape seems to arise from the scattering in the sample only, as we verified in the previous section 3.4.2 that it didn't occur in the absence of scatterers. This spot shape is also observed in other publications using Shack Hartmann wavefront sensors (e.g. in [START_REF] Wang | Direct wavefront sensing for high-resolution in vivo imaging in scattering tissue[END_REF]). The Gaussian-fit centroid algorithm used to measure the spot displacement in publication from Kleinfeld's team [START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF] might help reduce the bias but no further explanation is given. In the light of this results, a good compromise would be to descan the sample fluorescence onto the SH, and to couple the Descan SH with the correlation algorithms implemented in the ESSH to benefit from their resilience to scattering. This would avoid splitting the photons on the wavefront sensor while still averaging the aberrations over an FOV of 100 x 100 µm², gathering enough signal while still being close from the isoplanetic patch of the setup.

Remark on photon flux

Conclusions and perspectives

In this Chapter, we reviewed extensively our implementation of an Extended Source-based AO loop on a home-built two-photon microscope.

If at first, the best wavefront sensing strategy seems to use a Descanned SH coupled to correlationbased algorithms. This provides higher wavefront measurement accuracy than coupling this method to centroid-based algorithms. Moreover, the fact that all the photons are redirected towards one spot instead of being spread into a subimage seems to help the aberration measurement quality. For now, it is still unclear for us why so much photons were lost on the path between the ESSH and the Descanned SH, forcing us to add undesired density to compare equal incoming fluxes; the photon flux must be precisely assessed, and losses sources identified.

So far, ESSH performs better than Descanned SH mainly because its wavefront measurement is based on correlation calculation and not because it is based on an image rather than a centroid. The centroid calculation fails because the shape of the spots at the edges is distorted by scattering.

To complete the comparison of the Descanned SH (coupled to ES-algorithms) and the ESSH, several other patterns are to be tried to mimic different signal distributions, and see in a more detailed way how the photon distribution across the subimage affects wavefront sensing and correction. The ideal would be to repeat this experiment on a 3D labelled brain slice, with varying labelling densities, but this implies that lots of photons are to be lost, on both the excitation and the detection paths.

Let us precise that the results presented in the following chapter with the ESSH on biological tissues, were obtained before the experiments from Section 3.4. The drawn conclusions from the final section might have had us consider to try other configurations, in particular to try correcting for tissues aberrations with the Descanned SH too, coupled to ES algorithms. But at this period, this second wavefront sensor was not yet implemented.

Chapter 4. Applications of Extended Source AO-2PEF to biological samples

In this Chapter, some applications to biological samples of the Extended Source AO loop on the two-photon microscope are presented. As the aim of this thesis was to provide AO for neuroimaging in the mouse, we focused ourselves on this model. Results on fixed brain tissues are presented here as a first step towards in vivo functional neuroimaging. Aberration sources are discussed as well as AO loop parameters. Several biological signal sources are tested for the wavefront sensing, from the simple signal labelling to the dual labelled strategy, with perspectives of refined AO strategy combining near Infra-Red Quantum Dots emitters for wavefront sensing to reach higher imaging depths. The experiments presented in this Chapter were conducted ahead of the comparison between the ESSH and Descan SH in 3.4.

Extended-Source AO-2PEF on fixed mouse brain slices

This first section gathers our first results obtained on single-labelled fixed tissues from transgenic mice. The setup configuration used corresponds to the one represented in Figure 3.1 and described in the first part of Chapter 3.

As only one labelling is present in the sample, we must proceed to two sequential acquisitions on the ESSH and on the PMT, as not enough signal is emitted and received by the two detection paths if a beamsplitter is added between the PMT path and the ESSH path. Thus the ROI is first located with the PMT signal: a dichroic mirror (FF560-Di02-25x36, Semrock) is introduced on the optical path to redirect all the emitted photons towards the PMT. Once the area is found and the image before correction is recorded, the dichroic mirror is removed. The wavefront is then measured and corrected. As the aberrations don't evolve in time over several minutes, the dichroic mirror can be put back in place in front of the PMT to allow to record the image after correction. Because of potential focus shift due to spherical aberration correction (see 3.3.2), a stack of a few microns is saved (20 µm around the plane of interest).

Note that the impact of the dichroic presence and removal on the correction quality is negligible. Its impact is assessed on a point focused in fluorescein, bright enough and with a large enough emission spectrum to be visible on the ESSH in spite of the dichroic mirror, we only measured a variation of 3 nm on the corrected wavefront RMS.

Sample description a. Transgenic mouse

The fixed brain slices are obtained from genetically-modified adult mice from the lineage GADGFP: the GABA (gamma-aminobutyric acid) neurotransmitter plays an inhibitory role in the neural system, and is expressed in the mouse in approximately 10% of the cortical neurons. It is synthesized from the Glutamic Acid Decarboxylase (GAD), hence the common name of GADGFP used here to refer to this lineage. This relatively sparse labelling is a good starting point for the biological experiments with Extended Source AO. This distribution allows us to get enough soma in the ESSH FOV to perform correlations on while the axons and dendrites, more sensitive to the AO gain, can be observed on the collection path.

b. Aberrations induced by brain

When considering fixed tissues, the compromise between accessible depths limited by scattering leading to excitation signal loss, and the amount of aberrations to measure is hard to handle. Indeed, as we can see on Figure 4.1, from Na Ji's group, the amount of aberrations induced by the brain slowly increases to 200 nm RMS of aberrations for depth around 400 µm in vivo. Yet, these depth are much harder to access on fixed tissues, as their scattering length is twice shorter than the fresh ones [24].

For this reason, in the majority of the conducted experiments presented in this Chapter, the 'No AO' labelled results are obtained with a mirror shape correcting for the setup static aberrations, and the 'AO' ones results in the correction of the coverslip and the brain slice aberrations. Note that, for fixed slices, this definition of AO has already been used (e.g [START_REF] Ji | Adaptive optics via pupil segmentation for highresolution imaging in biological tissues[END_REF]). Because of this intrinsic compromise between scattering and the amount of aberration to correct, most of the groups now present directly in vivo results to reach deeper layers.

AO correction loop results

Results obtained on these samples are now presented. After a brief summary of the acquisition and AO loop parameters, we sequentially look at the influence of the AO loop on the ESSH and the corrected wavefront, and later focus on the gain brought by AO on the full-field image acquired on the collection path.

For the experiments described in the following, the experimental parameters are listed in the Box below. Any difference from these reference parameters is mentioned in the text. be explained by the shape of the diagonally-elongated brighter neuron, in the top right corner of the PMT image. On the ESSH, its signal is more important than the signal emitted by the second soma, mainly because, it is in the plane of interest while the second one is slightly below. Hence, if both are seen on the ESSH, the correlation is made on the brightest and bias is generated (see Figure 4.4). The artefactual astigmatism measured in the direction of the elongation is to be corrected for numerically, to tackle this issue.

Note that the right and left edges of the full-filled images appear to be of poorer quality: it is due to execution of the fast scanning in this direction, the galvanometer mirror speed varies in this region consequently to the direction change, and in spite of the triangle signal. We choose not to remove these edges here as even if the small details are lost, some soma are still visible. This deformation is to be corrected for numerically in the future.

Influence of the number of corrected modes

As we saw in Section 3.3.1, one of the tuneable parameter influencing final correction quality is the number of corrected modes. Previously, 36 mirror modes were corrected; in the experiment in Figure 4.5, we compare the gain provided by AO for more corrected modes on neuronal signal 120 µm under the sample surface.

The difference seems to arise from the correction of some residual 7th order and 9th order aberrations (for example: 7th order astigmatism at 0°passes from -13 nm to 0 nm, 6th order Coma at 90°, from 13 nm to 3 nm). As expected the 3rd order aberrations were already well corrected for and are not modified. The overall wavefront RMS is solely modified by 17 nm but its impact on the image quality enhancement is not negligible, as on the plot in Figure 4 the second peak present on the orange plot around 11 µm on the x-axis (corresponding to 42 corrected modes) is not visible when only 36 modes are corrected. This highlights the influence of the high order modes on the resolution quality. As for the gain in signal, it represents around 20% gain between 36 and 42 corrected modes. This effect has also been observed on Light-Sheet microscopy, in ex vivo Drosophila brain by Antoine Hubert [START_REF] Hubert | Optique adaptative en source étendue pour la microscopie à feuille de lumière[END_REF], where the increase of corrected modes was also leading to gradually enhanced performances. The upper limit of this is set by the divergence of the loop when modes associated to high modes gains (i.e low singular values) are trying to be corrected for (see Section

AO for 3D imaging: discussion on the isoplanetic patch

In this section, we initiate the study of the isoplanetic patch in fixed brain slices. Large volumes of tissue are imaged to open in particular the discussion on the axial resolution enhancement provided by AO. The aberrations are corrected from the wavefront measurement in a single plane.

A large volumetric acquisition is shown on Figure 4.7, before (on the left) and after AO correction (on the right): 3D rendering of the stacks are presented along the maximum intensity projections. Here, the correction in performed at at 50-µm depth under the coverslip. These 3D stack projections are saturated to the same intensity level for both before and after AO acquisitions, ahead of the Z color coding step, in order to enhance signal from the fine dendritic processes. What we can first notice is the gain in signal obtained on these small structures, and on the 3D renderings, one can see gain over the whole volume, up to 300 µm deep. For a better estimation of this gain, let us look now at Figure 4.8, on which the same stacks are sliced in the vertical dimension instead and the maximum intensity projection in this direction is done. On these plots, the gain on large structures such as soma, is estimated to a two-fold enhancement, in both the lateral (red line) and axial directions (purple line), at 200 and 240 µm below the coverslip. On finer details of typically 1 µm, the signal is enhanced by a factor 2.7 (see plot along the brown line).

To our knowledge, these images are the first demonstration of a gain in signal and resolution provided by AO based on Extended Source wavefront sensing on a two-photon microscope in a fixed mouse brain slice. This enhancement is visible over a FOV of 150 x 150 µm² at the depth where the wavefront is measured but also more than 200 µm below the corrected plane.

Next experiments

The complete characterization of the axial isoplanetism of brain slices requires wavefront measurements and correction at several planes while going in depth. If this experiments has already been tried, a couple of technical issues prevented us from concluding it; due to the PBS immersion of the brain slices, they tended to migrate during the acquisition, preventing us from repeating the 3D acquisitions with several correction planes. This is required to completely characterize the 3D gain provided by AO according to the plane of interest, and to optimize the number of planes to correct in a 3D acquisition. This means reaching an optimal gain in the whole volume while minimizing the AO duration and limiting photobleaching of the sample. A complete study will be carried out to assess the isoplanetic patch of the mouse cortex in both the imaging plane and the axial direction.

In this section, we showed the ability of the ESSH to measure and correct aberrations on biological tissues. These experiments proved the efficiency of our method on biological signal in depth of a scattering medium; however, this approach is not optimized so far, especially in terms of photon budget. Same labelling is used for wavefront sensing and scientific imaging leading to losses of photons of interest on the imaging path. To avoid this, dual-labelling strategies are now commonly implemented for in vivo experiments and is tested in the following section.

Dual-labelling for AO

A common AO strategy consists into using two fluorescent sources: one dedicated to wavefront sensing only, and the other to the biological study. It permits a better use of the photon budget, as photons emitted by neuronal activity will not be wasted on wavefront sensing for example. Furthermore, as the functional signal varies in time, it is not suitable for wavefront measurement; a stable signal such as the anatomical one, is required.

The used setup is represented in Figure 4.9 and has been adapted to a dual-labelling (green and red); it is almost identical to the one used in the previous section, at the exception of the now fixed DBS1 (Dichroic Beamsplitter FF560-Di02-25x36, Semrock) in front of the collection path.

The second dichroic beamsplitter, labelled DBS2 here, redirects the redder part of the signal towards the ESSH (DBS2, FF775-Di02-25x36, Semrock).

At first, we consider brain tissues from a dual-labelled mouse. In this section, two typical experimental results obtained are detailed. At first, data from a proof-of-principle experiment on this tissue is presented in the same conditions as the one used for single-labelled slice. Then, a deeper region is assessed; this time, the starting point of the correction loop already corrects for the aberrations induced by the coverslip, so the image gain will only come from the aberration correction of the brain slice.

Two-coloured transgenic mouse

The considered sample is a fixed brain slice, from a genetically modified mouse resulting from the crossbreeding of two lineages; a Cx3Cr1GFP mouse, and a PV-Cre R26 mouse expressing tdTomato protein (emission pk at 580 nm) in the Paravalbumine neurons. The microglia are nonneuronal cells, that provide support to the neurons. The Cx3Cr1GFP mice express Enhanced-GFP under the promoter Cx3Cr1, in the microglia. The Parvalbumine neurons are a subtype of inhibitory neurons.

Note that this cross-breading was an original trial, and that the labelling quality in the resulting mouse is neither guaranteed, nor controlled.

Proof of principle experiment

In this example, AO correction is performed on a 40 x 40 µm² area, at the center of the larger field images. The AO area is reduced in order to optimize the energy density and to get enough signal on the ESSH to correlate on as the red labelling of these first two-colour samples was not as strong as expected. In this proof-of-principle experiment, the loop is realized at shallow depth, only 50 µm within the fixed brain slice. The aim here is to show that this strategy is implementable.

At the sample, the laser power is 21 mW, 36 modes are corrected, with a gain of 0.6.

We chose to limit the acquisition time for the ESSH by scanning smaller area around the signal of interest (40 x 40 µm²), as it was coming from a single cell here, and with a smaller pixel size. This indeed modifies the area over which the aberrations are measured, but we find it of interest to this manuscript. In this acquisition, we note that the amount of aberrations and the wavefront decomposition is similar to what has been shown in 4.1. No major impact of the wavelength difference between the one for the PMT imaging and the one for wavefront sensing, as we can see from the third order aberration coefficients in Table 4.3, and on the wavefronts in Figure 4.10e. Moreover, by red-shifting the signal wavelength directed to the ESSH, we expect to limit the chromatic shifts by reducing the gap between the excitation and emission wavelength on the ESSH path, hence benefiting even more from this dual-labelled strategy. The correction of the aberrations provides signal and gain enhancement again on both the finest details and the soma of the cells. The impact of the correction on the neurites is visible on the zoomed parts Figure 4.10c-d, and is confirmed by the intensity profiles (Figure 4.10g), on which a signal enhancement of 1.8 to 2.6 is obtained. Moreover, the resolution is also enhanced: after the AO loop, the structure within the yellow cell soma, visible thanks to the cytoplasmic labelling, is now resolvable. The corresponding plots are shown on Figure 4.10h. Note that on its cytoplasm, a signal enhancement of a factor 1.6 is obtained with AO.

For now on, a dual-labelling strategy on fixed brain tissues allows to correct efficiently for sample-induced aberration.

Correction of brain slice only

Let us now take this experiment further. On the previous sample area, not only was a single cell used for wavefront sensing, but also, shallow depths were addressed. Let us interrogate deeper regions in the tissue and address wider areas for wavefront sensing. The loop results on Figure 4.11 are obtained 200 µm away from the surface, the starting mirror shape is set by a previous correction realized 20 µm below the sample surface, hence already correcting for the coverslip-induced aberrations; indeed, we can note the much smaller aberration amplitude to correct for (here 89 nm wavefront RMS instead of 210 to 240 nm on the reported data above, for the same pupil size). As a consequence, the aberration content is also modified: the strong SA3, combined to a lesser extent, with Coma, and caused by the coverslip is no longer to be corrected for. Instead, the third order aberration values before the correction are reported in Table 4.4. The brain slice seems to generate mainly low order aberrations, and more particularly, solely third order aberrations. The higher order aberration values are inferior to 10 nm. In this experiments, shallower gain is brought by the correction; this is the direct consequence of the fact that the aberrations induced by the coverslip is corrected ahead of this acquisition. Indeed, at 180 µm away from the corrected plane, the brain slice only introduces 89 nm RMS of aberrations, which is less than half of the previously tackled amount of aberrations. Consequently, the gain on the image is not as important. On the intensity profiles (Figure 4.11g-h), we note a signal enhancement of 1.5 on the small details (Figure 4.11h), this effect is visible on the zoomed parts too (Figure 4.11c-d). As for the larger structure such as the soma in the bottom of the FOV, we see that AO helps us to retrieve the structure induced by the cytoplasmic labelling (4.11g); resolution is enhanced too.

Here, we experiment a couple of issues on this type of sample; the red dyes are more sensitive to photobleaching and are less bright than the green ones. Hence, continuous illuminations of the area of interests can lead us to signal loss on red-labelled cell. Moreover, the low labelling density is not so beneficial to ESSH measurement. A dual-labelled strategy based on an other strategy for red emission would be of great help.

Conclusions and perspectives

Conclusions

In this Chapter, we showed firsts results of AO corrections on biological samples, when coupled to two-photon microscopy. For the first time to our knowledge, signal and resolution enhancement were obtained with an Extended Source based AO loop on two-photon signal.

Focusing ourselves on 3D labelled fixed brain slices, obtained from transgenic mice, we obtained significant image gain, typically of a factor two to three, but remained limited in depth penetration, due to the short scattering length of the considered tissue. Indeed, in this chapter the maximum imaged depth was 300 µm; considering the scattering length of the tissue at the excitation wavelength, it corresponds to 3.9l s . This is equivalent to depths of 600 µm in vivo in the green range, and to more than 700 µm if near-IR sources are used for wavefront sensing, corresponding to the state-of-the-art.

Further improvements. To push this setup further, some changes aside from the AO loop itself have to be done:

1. Reduce power losses. By simply modifying the beam expander at the output of the laser, a non-negligible amount of laser power could be saved for imaging. Initially chosen to overfill the galvanometer mirror and to keep only the top hat part of the beam on the DM, this choice can now be questioned. Currently, a twentieth of the laser power reaches the sample plane. Note that only 60% of the excitation power is transmitted through the objective at this wavelength.

2. Further develop the control software to increase user speed. The power control unit need to be motorized to automatically compensate for excitation scattering in depth. This would prevent greatly useless photobleaching at the surface, and would allow more optimized 3D acquisitions.

3. Adapt the acquisition electronics. To address functional imaging, the integration unit of the analog detection will have to be modified, to custom-made electronics to follow the required acquisition speed and enhance the detection sensitivity.

Imaging deeper with Near infrared Quantum Dots for AO

To push the dual-labelling strategy further, and get rid of red organic dyes constraints, a duallabelling strategy including near-IR QD for wavefront sensing signal generation is considered.

As the wavefront measurement on the ESSH is based on imaging, the measurement is impacted by scattering (as is the imaging of the descanned signal in the case of the classical SH). Thus, correcting for aberrations deeper in the tissue requires to increase the scattering length at the emitted fluorescence wavelength. As the tissue itself cannot be changed, the easiest way consists into shifting in wavelength. But organic fluorophores are not very bright in the near infrared. Therefore we propose to use QDs emitting around 700-800 nm, where scattering will be less but the analyser camera is still sensitive. These QDs are manufactured within the SI2 team of the LPEM, which has expertise in their synthesis but also in their use as probes in biology. In addition, these QDs have good quantum yields and a large 2-photon absorption cross-section, making them the probes of choice for labelling dedicated to the measurement of the wavefront.

In this section, the results of a proof-of-principle experiment on this labelling are reported. The QD were synthesized by Alice Guillaume-Manca during her Master's internship in the lab; she is to take this project over. The synthesis steps followed the protocol reported by Pons et al [START_REF] Pons | In Vivo Imaging of Single Tumor Cells in Fast-Flowing Bloodstream Using Near-Infrared Quantum Dots and Time-Gated Imaging[END_REF] and their emission peak is around 800 nm. The QD layer is deposited on the brain slice side opposite to the objective, i.e on the bottom part of the sample. As QD are hydrophobic, they do not migrate into the slice. As their nanometric size do not permit to image them individually on the ESSH, we create a pattern at the excitation, like in Section 3.4 to measure the wavefront, as we showed that this method is adapted to ESSH wavefront sensing.

The gain provided by AO is visible on Figure 4.12. In this example, a 250-µm fixed brain slice is positioned on top of the QD layer. During the loop, the wavefront RMS drops from 261 nm to 43 nm. Before and after AO, 3D stacks are saved, and the depth in the sample is represented through a color code on the maximum intensity projections. A couple of interesting regions are encircled on both images; they correspond to neurons in the lowest part of the sample, that is to say, to the ones closest to the QD layer, approximately 250 µm below the sample surface. We remark that their signal intensity was increased with the correction, confirming AO benefits in this situation too. The brightness of the QD combined to the patterned scan can hep reduce the ESSH acquisition time, down to 500 ms typically. As the loop converges in 4 to 5 iterations, and remained stable afterwards, the overall correction time can be reduced to 2 s (the limiting time parameter being the ESSH acquisition time): this is compatible with in vivo temporal requirements. Thus, this dual-labelling approach based on near-IR QD constitutes a promising and easy-toimplement strategy.

Perspectives. With the optimization of the photon budget via the bicolour setup, in vivo experiments become our main short-term objective. Anatomic imaging will first be addressed to progressively adapt ourselves to the constraints set by the animal itself. Over a second phase, functional signals will be considered. Our main objective will be to image deep: while going down in the animal brain, we will be able to correct several planes successively, with an adapted sampling, resulting from the study of the isoplanetic patch. If required, several strategies will be implemented, such as decreasing the number of corrected modes for a first correction loop, then increasing it back once the low-order aberrations are corrected.

123

General Conclusion Conclusion

During these three years, I had the opportunity to combine an adaptive optics loop using an original wavefront sensing method to two-photon microscopy. While the sensor had previously been characterized and implemented on a Light Sheet Microscope at the LPEM to perform ex vivo imaging on adult Drosophila brain, I took it onto a different set up, for a different biological subject: two-photon microscopy for in-depth neuroimaging in the mouse brain. Being at the interface of the practical optical consideration and the biology requirements imposed by the neuroimaging subject allowed me to develop a tool answering to biological needs. Moreover, the close collaboration with Imagine Optic has been crucial in terms of wavefront sensing understanding, development of specific algorithms and software, optimization of the adaptive optics loop and its implementation on a microscope.

The first half of my thesis was dedicated to the characterization of this extended source wavefront sensing method in scattering conditions corresponding to biological tissues. Building this set up, understanding the issues generated by scattering on the wavefront measurement accuracy, and finding the right criteria along the proper experimental conditions took me along for the beginning of my journey. These first studies permitted the publication of an article in Optics Express and were essential for the complete understanding of the phenomena at stake, as well as their impact on wavefront measurement. This work also led, unexpectedly, to the measurement of the scattering length l s from images of single emitters under the scattering tissue and saved on the wavefront sensor.

The second half was devoted to the design and the construction of the two-photon microscope set up from scratch along the coupling to the Extended Source based AO loop. Closed-loop corrections of aberrations in both test and biological samples were performed on two-photon signals.

The first experiments of in-depth wavefront measurements on calibrated scattering media were conducted in parallel to the ones on fixed mouse brain slices. The results I obtained are, to our knowledge, the first demonstration of a gain in two-photon microscopy by Adaptive Optics using an Extended Source wavefront measurement in a scattering biological sample. It was a major aspect to confront ourselves to typical photon fluxes of labeled biological samples of interest in order to prepare the in vivo experiments. These obtained results exhibiting gain in signal and resolution in depth, left us confident with the ability to provide gain for in vivo experiments with AO and address functional questions at larger depth than those commonly reached nowadays in two-photon neuroimaging of the mouse brain. Considering the gain brought by AO and ESSH up to 300 µm deep within the sample, corresponding to 3.9-times the scattering length of the fixed tissue at the excitation wavelength, we should be able to address areas down to 700 µm below the pia in vivo.

Miscellaneous

• Poster MiFoBio, November 2021, 'Single-shot quantitative aberration and scattering length measurements in mouse brain tissues using an extended-source Shack-Hartmann wavefront sensor'

• Mathias Mercier, Sophia Imperato, and Alexandra Fragola, Workshop MiFoBio, November 2021, 'Adaptive optics for Light-Sheet microscopy'

Introduction

Optical microscopy has emerged as a major tool in neuroscience as it allows to image the architecture of neuronal networks and to record their functional activity in several animal models as zebrafish, rodents, ferrets or primates through different methods including widefield fluorescence microscopy [1], confocal [2], one-photon [3,4] or two-photon light-sheet microscopy [5] and non-linear scanning microscopy [6][7][8]. Modern optical sectioning methods such as confocal or light-sheet can image a large part of the entire brain of relatively transparent animal models such as zebrafish, drosophila or C-Elegans. In mammalian brains, depth penetration is limited even at low depth by scattering and imaging can only be achieved using non-linear microscopy. However, at large depths, optical aberrations linked to the refractive index inhomogeneity of biological tissues still limit resolution and signal intensity in all microscopy modalities [9,10]. To overcome this difficulty, adaptive optics (AO) has been implemented on several linear and non-linear optical sectioning microscopy setups and currently provides a reliable live correction of the aberrations [11], enabling functional imaging of synaptic boutons [12], axons and spines [13] and soma [14] in infragranular layers of the mouse cortex in 2-photon microscopy, or enabling subcellular imaging of organelle dynamics in the early zebrafish brain in light-sheet microscopy [15], and deeper imaging of multicellular tumor spheroids [16,17]. The first strategy to implement AO in microscopy is based on a sensorless configuration [18][19][20][21], that does not make use of a wavefront sensor but analyses the variation of the fluorescence signal induced by a wavefront modulator to drive the correction of aberrations. This method minimizes instrumental complexity, and provides a good resilience to scattering. However, it relies on a time-consuming iterative approach requiring many acquisitions before reaching a good correction: typically, 30 seconds are necessary for a single iteration of a sensorless process, and a couple of iterations are required to reach an optimal correction. The approach is thus hardly compatible with time-varying aberrations and with photobleaching issues, in particular when multiple corrections in a volume are required, e.g. for a small isoplanetic patch which is often the case in depth [15]. The second approach to enable AO in microscopy is based on direct wavefront sensing, and makes use of a wavefront sensor to allow a fast and accurate convergence of the AO loop, with a better photon budget: direct wavefront sensing is thus a key method to evaluate and correct aberrations in vivo over large scales. Direct wavefront sensing for AO-enhanced optical microscopy has been reported over the last years through several approaches, using either Shack-Hartmann (SH) sensors [13][14][15][16]22,23], or partitioned aperture wavefront (PAW) sensors [24] as a variant of pyramid wavefront sensors applicable to microscopy. The use of PAW in fluorescence microscopy is severely limited, since it requires only a moderate spatial incoherence of the excitation source, so that its demonstration in fluorescence microscopy was restricted to widefield using a specific illumination geometry (Oblique Back Illumination) [24,25]. Direct wavefront sensing in microscopy is thus currently mostly based on SH sensors.

Early AO demonstrations based on SH [16,22] used extrinsic fluorescent beads injected into the sample to provide a guide star for centroid computation, which is usually unwanted. More recently, Wang et al. used two-photon excitation, scanned over a given field of view, to create a guide star outside the sample by descanning the fluorescence signal, the guide star being then used by a SH sensor based on a conventional centroid computation [14]. The approach was successfully used to drive AO in Lattice Light-Sheet [15], as well as in two-photon [14] and in Structured Illumination Microscopy (SIM) [26,27]. Recently, we demonstrated the use of an Extended-Source SH wavefront sensor (ESSH) to enable AO in light-sheet microscopy [23], by adapting pioneer work from astronomy to the constraints of fluorescence microscopy. The method relies on the cross-correlation of images of an extended source obtained through a microlens array. In fluorescence microscopy, when coupled to an optical sectioning method such as two-photon microscopy or light-sheet fluorescence microscopy, the ESSH uses the fluorescence signal as a guide plane, providing lower instrumental complexity and cost than the previous approach based on the scan and descan of a non-linear signal. Its efficiency has been proven when coupled to light-sheet for AO-enhanced neuroimaging in the adult drosophila brain in weekly scattering conditions [23]. However, direct wavefront sensing in scattering tissue is limited in depth penetration. Indeed, in pioneering SH wavefront measurements based on centroid estimation in the rodent brain, the strong scattering of the emitted fluorescence reduces significantly the capacity to measure the centroid position due do the decreasing ballistic signal and increasing background on images below each lenslet [13,14]. A method for direct wavefront sensing more resilient to scattering of the fluorescence emission would therefore improve the use of AO in optical microscopy. To this aim, we analyse here the performances of an ESSH in scattering tissues and show that it provides quantitative aberration measurements in highly scattering fixed mouse brain tissues, together with a fast and precise estimation of its scattering length. We compare our extended-source wavefront measurement approach to the Shack-Hartmann method based on centroid calculation, and show the benefit of the former in the case of scattering samples, and its improved accuracy at large depths in low Signal to Background Ratio conditions. In a proof of principle experiment, the ESSH method is finally used with a model sample consisting of neurons in culture placed under a fixed brain slice to illustrate the image quality improvement that AO, driven by ESSH, can provide in microscopy experiments in brain tissue at large depths, and in particular in the mouse cortex.

Material and methods

Setup

To perform aberration measurements within scattering conditions, an ESSH wavefront sensor was implemented on a custom-made epifluorescence set-up represented in Fig. 1(a), composed of a laser light source at 488 nm (Cobolt 06-MLD) focused in the back focal plane of the water immersion objective (Olympus 20X, NA = 1, XLUMPFLN20XW) by L1 (Thorlabs, AC254-100-A, f' = 100 mm). The fluorescence emitted was collected through the objective, its back focal plane being conjugated with the ESSH microlens array by a pair of achromatic lenses L2-L3 (L2, Thorlabs, AC508-300-A, f' = 300 mm and L3, Thorlabs, AC254-80-A, f' = 80 mm). The ESSH was similar to the device reported in our previous publication [23], and was composed of an array of 17 × 23 microlenses, with a focal length of 5.1 mm, the CMOS sensor being in the focal plane of the microlenses. A squared field diaphragm was placed in front of the ESSH, leading to a field of view of 120 × 120 µm 2 in the object plane, for each microlens, thus selecting the correlation area and avoiding crosstalk between adjacent thumbnails, defined as the undersampled images behind each microlens. A 50:50 beamsplitter (BS) separated the beam between a full-field camera (Thorlabs) and the ESSH path. The observation wavelength was selected by either two bandpass filters (525/50, 625/40 from Chroma) or a long-pass filter (LP715 from Semrock) allowing us to measure the wavefront at three different wavelengths using three types of emitters (see details in 'Sample preparation' section) being deposited on a single coverslip. A white-light transmission imaging path was added in order to check sample positioning, in particular to localize precisely the cortex area of the coronal slices for wavefront measurement. This path was composed of a white light source, followed by L4 (Thorlabs, AC254-150-A, f' = 150mm), and L5 (Thorlabs, AC508-250-A, f'= 250 mm), and an objective (Olympus 5X, NA = 0.15, MPLFLN5X) The sample was imaged by the ESSH microlenses onto the camera sensor, leading to undersampled thumbnails. A reference thumbnail was chosen centrally, and correlations were computed between each thumbnail and the reference one in order to retrieve the relative displacement of each thumbnail to the reference. A pre-processing step was required to avoid correlation errors induced by noise. Considering the low SBR on our thumbnails (see SBR definition below), a Laplacian of Gaussian filter was selected. A pyramidal fit was used to find the correlation peak with sub-pixelic displacement for each thumbnail, leading to the local displacement of the wavefront; this choice results in a trade-off between accuracy and time calculation. These steps followed the procedure and pseudo-code described in Anugu et al [29]. Finally, the wavefront reconstruction was achieved with a conventional zonal method. This process is summed up in Fig. 2. The closed-loop can reach a maximum rate of 10Hz, with the following computer configuration: processor IntelCore i5-8400 CPU@2.8GHz, 32Go RAM, Windows 64bits, and to which must be added the exposure time imposed by the sample labelling.

Sample preparation

Because of the low numerical aperture (NA) of the microlenses (NA = 0.03), the ESSH required to be coupled with an optical sectioning method to provide quantitative wavefront measurements in thick 3D samples. To demonstrate the performances of the ESSH in the case of a scattering sample, we used a simple configuration, where the sectioning was inherent to the samples: we used 2D fluorescent samples placed below unlabelled 3D scattering samples (fixed mouse brain slices) and achieved by this mean wavefront measurement on an epifluorescence microscope lacking optical sectioning. This approach allowed to easily control the scattering properties of the sample using slices of known thicknesses, which would be extremely challenging using intact biological samples, and relaxed the need for implementing optical sectioning in the instrument. Such a model has already been used in other AO-based experiments [30].

Experimental procedures were conducted in accordance with the institutional guidelines and in compliance with French and European laws and policies and with the ARRIVE guidelines [31]. All procedures were approved by the 'Charles Darwin' local institutional ethical committee registered at the French National Committee of Ethical Reflection on Animal Experimentation under the number 05 (authorization number: APAFIS 26667).

Three 6-months old C57BL6 male mice were sacrificed by an overdose of Euthasol, after being placed under deep sedation by overdose of isoflurane (5 min at 5% isoflurane in an induction box). The extracted brain was then stored overnight in a solution of 4% paraformaldehyde and finally rinsed in phosphate buffer solution (PBS). Coronal slices of different thicknesses ranging from 50 µm to 300 µm were then cut and stored in PBS.

Several fluorescent emitters were used; 2-µm fluorescent beads (ThermoFischer, λ em = 515 nm) were selected as emitters within the green range while quantum dots (QD) aggregates were used as emitters in the red and near-infrared ranges in order to keep the same excitation wavelength. ZnCuInSe/ZnS QD emitting at 800 nm were synthesized following the protocol reported by Pons et al [32] whereas those emitting at 614 nm were obtained from protocol described by Yang et al [33]. QD in hexane (4 nmol) were mixed with ethanol (1 mL) to precipitate QD by centrifugation. QD were resuspended in chloroform (250 µL) and CTAB (5 mL, 3 mM) was added. The solution was heated for 20 min at 80 • C to evaporate chloroform. The QD aggregates surrounded by hydrophobic ligands were precipitated by centrifugation (5 min, 8500 rpm) and then dispersed into ethanol. A second round of centrifugation was necessary to eliminate all non-aggregated QD. Finally, aggregates were dispersed into ethanol (1 mL). The aggregate solutions were diluted 10 times. 500-nm aggregates were obtained with respectively an emission peak at 614 nm (FWHM = 32 nm) for the visible ones, and 805 nm (FWHM = 17 nm) for the near-IR ones.

Samples were prepared with the three types of emitters. 4 µL of the 2-µm commercial beads, diluted 1/5000e, were deposited first on a 170-µm-thick coverslip then dried up for 5 min at 60 • C. The two types of aggregates were deposited on the same side of the coverslip (4 µL of each). On the other side of the coverslip a fixed brain slice was placed, with spacers. Another coverslip was added on top to avoid the adhesion of the brain slice to the water-immersion objective during the experiment.

Neuronal cultures containing a mVenus fluorescent tag were generated by an ex-utero electroporation in the sensory cortex of E15 murin embryos. A drop of DNA pH1SCV2_shRNA_control (0.7 µg/µl) was injected with a capillary in one hemisphere of the sensory cortex of the embryo and then electroporated. This region was then extracted from the whole brain and dissected in a medium containing 10% HBSS, 5% glucose and 2% HEPES. The piece of remaining cortical hemisphere was digested by chemical and mechanical digestions, first, under the action of papain for 15 min at 37 • C and then under pipet up and down movements, to obtain individual neurons. The neurons were plated onto a 18-mm coverslip coated with 80 µg/ml polyornithine in 12 well plates, at a density of 180,000 cells/well in MEM, 10% HS, 1% glutamax and 1% sodium pyruvate 100 mM and incubated for 2 hours at 37 • C, 5% CO2. When the neurons were completely adherent on the coverslip, the medium was replaced by Neurobasal, 2% B27, 1% glutamax and antibiotics. One third of the medium was changed every 5-6 days.

To obtain model scattering samples, fluorescent emitters (2-µm commercial beads diluted 1/5000e (4 µL), dried up for 5 min at 60 • C) were first deposited on a coverslip. On the other side, 2-µm polystyrene beads (Sigma-Aldrich, 78452) dispersed in water were used as scattering samples and deposited. 1 µL of the commercial solution, containing 10% of beads in mass was diluted 10 times to obtain a solution of 0.002 bead/µm 3 with a scattering length of 74.9 µm. Similarly, a 5-times diluted solution (0.005 bead/µm 3 ) with a scattering length of 29.9 µm was realised. The scattering coefficients were calculated with Mie-scattering theory, at 515 nm (online calculator https://omlc.org/calc/mie_calc.html). 10 µL of the diluted solutions were deposited on the coverslip, surrounded by 130-µm-thick spacers, and recovered by another coverslip to retain the scattering droplet.

2.3. Third-order spherical aberration (SA3) measurement and modelling SA3 coefficient is described in the Zernike polynomials as 6ρ 4 -6ρ 2 + 1. Thus, the focus variation has a strong impact on the spherical aberration value, as a best focus can be found to minimize the later. In order to eliminate this factor, we minimized experimentally the focus aberration to eliminate this bias from our analysis. Only the contribution of the brain slice was considered here: a reference wavefront, corresponding to the aberrations induced by the instrument and the coverslips, was subtracted at each wavelength. For each brain slice thickness, at least seven acquisitions were realised in different areas of the cortex, at the three considered wavelengths simultaneously.

We simulated the optical system in Zemax in order to obtain an order of magnitude of the refractive index mismatch-induced spherical aberration for mouse brain slices. A simple model consisting in two additional interfaces inserted between the emitters' plane and the objective was chosen. The brain slice was modelled by a layer of refractive index of 1.368, previously experimentally measured on fixed mouse brain slices and reported by Lue et al [34]. The process was repeated for the corresponding experimental thicknesses used.

Numerical simulation of the measured shift of a thumbnail for several SBRs

The SBR of a given thumbnail was defined as the ratio between the average signal value of the 1% brightest pixels, and the background mean value, considered as the mean value of the 80% dimmest pixels. A well-sampled point spread function corresponding to a squared aperture, at 600 nm, was first generated numerically. Poissonian noise was added with a variance equal to the square root of the signal plus the background value. The generated image was then undersampled on one hand to obtain our reference thumbnail, with dimensions matching to the experimental ones. On the other hand, the equivalent of a 0.5 pixel shift was introduced in the two directions and the shifted image was undersampled afterwards. As discussed in Anugu et al [29] it corresponds to the minimum bias error value in the shift retrieval, for the centroid process as well as for the correlation-based method. The induced shift was then retrieved by the two methods computed here. The centroid position was measured on the two thumbnails, on which a threshold corresponding to the background value plus 10% of the difference between the maximum and the background value was applied. The threshold choice was motivated by the results shown on Fig. 4(c) and Fig. 4(d) of the influence of the threshold choice on the shift retrieval in low SBR conditions. This simulation was run on fifty images for each SBR value. For the subpixelic shift retrieval on the experimental thumbnails, 35 × 35 pixels 2 images were first oversampled by a factor 100 and then shifted by the equivalent of 0.5 pixel in the two directions. After under-sampling the shifted images, the shift retrieval between the initial thumbnails and the shifted ones was performed as described above.

Results

SBR across the ESSH wavefront sensor in scattering tissues

The ESSH microlenses are conjugated with the back focal plane of the objective (Fig. 2(a)). Due to the scattering of light in biological samples, the images below each micro-lens are losing contrast (Fig. 2(b)) in depth, as in the case of the centroid-based measurement [13,14]. This loss in contrast ultimately limits the possibility to reliably measure a wavefront with Shack-Hartmann sensors at large depth in scattering tissues. More specifically, the intensity distribution on each thumbnail of the ESSH shows a decrease of the signal from the central thumbnail to the ones on the edges, with a rotational symmetry (Fig. 3(b)). At the same time, the background decreases, but slower, from the centre to the edge of the ESSH (Fig. 2(b)). Accordingly, the signal to background ratio (SBR) decreases significantly from the centre to the edge with the same rotational symmetry (Fig. 3(b)). On the contrary, in the absence of a scattering tissue and with a homogeneous fluorescent sample, the signal on the ESSH is uniform in intensity over all the thumbnails, except the ones on the most external row where the entrance of light rays starts being limited by the edge of the pupil (data not shown). This effect can be explained by the geometry of the sensor (Fig. 3(a)): when a thick scattering sample is positioned on top of the imaged fluorescence emitters, the emitted signal received by the outer microlenses is propagating through a larger distance in the scattering sample than the one received by central microlenses. Let us note that this effect is not specific to the ESSH but is present in SH with centroid-based estimation of shifts too [14]. When measuring aberrations through scattering media, this Signal to Background (SBR) deterioration across the sensor limits thus the size of the pupil over which the wavefront can be computed, with the limiting SBR found at the outer thumbnails.

Scattering coefficient measurements

Thanks to this intensity profile across the ESSH, the scattering length of the medium can be measured. In a scattering sample, the intensity of the ballistic fluorescence signal decays with depth following

I(z) ∝ exp ( - z l s ) (1)
where l s is the scattering mean free path (or scattering length) of the sample and z its thickness.

In the geometry of the ESSH, the thickness z p crossed to reach the thumbnail of index p (p varies between 0 (central thumbnail) and 7 (most external row of thumbnails) increases as

z p = z c cos θ p (2)
where z c is the slice thickness and θ p is the angle over which the thumbnail p is seen, with

θ p = arctan ( p 7 tan θ max ) (3)
and θ max = arcsin ( NA objective /n ) = 48.8 • , θ max being the maximum angle collected by the objective.

Measurements on a calibrated scattering sample composed of 2-µm polystyrene beads and 130µm-thick are first performed to confirm the accuracy of the method, at two bead concentrations chosen to be within the range of l s values of fixed brain tissues. In these experiments, we position a single fluorescence emitter at 515 nm in the ESSH field of view. In Fig. 3(c), the intensity of the ballistic signal is plotted as a function of the effective thickness after the subtraction of the background on each thumbnail. This requires the ability to measure both the ballistic signal and the background level in each thumbnail, and thus to use samples with a relatively sparse distribution of emitters. From the slope of the semi-log plot, which is equal to 1/l s , we measure for a 0.005 sphere/µL concentration a scattering length of 26.5 ± 0.3 µm, very close to the expected value of 29.9 µm. Similarly, we measure a 69.5 ± 1.4 µm scattering length at 0.002 sphere/µL concentration where 74.9 µm was expected. This demonstrates that the scattering length can be measured using the ESSH sensor with an accuracy of 10%. Measurements on fixed mouse brain coronal slices, in the cortex, are performed at three different wavelengths (515 nm, 615 nm and 805 nm) to determine the chromatic dependence of the scattering length of the tissue. To improve the measurement accuracy, three slice thicknesses are used at each wavelength: at 515 nm and 615 nm, 50-, 100-and 150-µm-thick slices and at 805 nm, 100-, 150-and 200-µm-thick slices. The selected slice thicknesses correspond to ranges of l s to 2.5l s .

For each slice, the intensity is normalized to plot and fit the data for all slices on a single figure (Fig. 3(d)); the ordinate of the individual semi-log plot being subtracted to each acquisition. The measured scattering coefficients are 38 ± 2 µm at 515 nm, 47 ± 2 µm at 615 nm and 77 ± 5 µm at 805 nm, increasing significantly with the wavelength as expected. The obtained l s values at the three wavelengths are compared to previously reported scattering length values in the Discussion.

Spherical aberration measurements with ESSH

In this section, we show that the ESSH sensor provides a reliable wavefront measurement in very scattering conditions. We consider in particular third-order Spherical Aberration (SA3), which is mainly generated by refractive index mismatch and which depends on the sample thickness, a parameter that we can easily adjust. We use 50-to 150-µm-thick slices within the visible range and up to 250-µm-thick in the near-IR. These thicknesses correspond to 1.3l s to 3.9l s at 515 nm, to 1.0l s to 3.3l s at 615 nm and to 0.6l s to 3.2l s at 805 nm. The increase in thickness corresponds to an increase in Optical Path Difference (OPD), that is the difference of refractive index between the tissue and water times the sample thickness. The variation of the tissue refractive index between the different wavelengths is neglected. Comparisons of the experimental SA3 values to the one obtained by simulations (see Methods), presented in Fig. 3(a-c), show that our ESSH sensor is providing accurate aberration measurement and confirm quantitatively the increase of the spherical aberration with the brain slice thickness, in agreement with the simulation. At low slice thickness (50 µm), the SA3 induced by the sample is within the measurement error values of the ESSH (8-9 nm is predicted by the Zemax simulation). Our measurement method reaches its limits when the slice thickness became greater than 3.9l s with commercial beads as fluorescent emitters, and 3.3l s with the aggregates. Corresponding images on thumbnails are shown on Fig. 3(d). These slice thicknesses correspond to effective thicknesses crossed on the outer microlenses of 4.8l s and 5.8l s , for which the ballistic signal is considerably reduced at the pupil edges (typ. 0.25%), corresponding to a SBR of 1.3 only. Alongside this loss of ballistic signal, we observe a more important variability of the SA3 values for a single slice, as a consequence of the loss of slope accuracy evaluation at these SBR. Measuring aberrations with thicker slices could be achieved by restricting the pupil over which the wavefront is calculated, but these results could not be compared to the previous ones with thinner slices, performed on the entire pupil. Our ESSH thus allows performing quantitative aberration measurement through scattering slices that are up to 3.9l s -thick.

Comparison of ESSH/centroid computation at low SBR

To confirm that ESSH measurements are resilient to strong scattering conditions, in particular in comparison to the centroid-based measurements, we conducted simulations (described in Methods) to assess the accuracy of the displacement retrieval at low SBR for both methods. The chosen SBR corresponds to previous experimental SBR shown on Fig. 3(d). For the highest SBR, typically larger than 1.5, both methods are giving similarly good results, as the average displacement error is null in both cases (Fig. 5(a)). However, as the SBR decreases, the centroid method is more impacted. A large dispersion of the shift measurement was observed in the case of the centroid method, as shown by the large error bars (Fig. 5(a)) and Standard Error of the Mean (Fig. 4(b)). Thus, the number of images over which the displacement calculation is done has to be increased by a factor 25 at a SBR of 1.2 to achieve the same precision on the displacement measurement with the centroid approach as with the correlation method. This variability in the displacement measurement affects directly the quality of the aberration correction that can be achieved with an AO loop. At SBR smaller than 1.2, the noise also impacts the estimate of the average displacement obtained through the centroid process (at SBR = 1.09, Student t-test α = 0.05, p = 0.009). On the contrary, the mean value of the shift evaluated by the correlation-based measurement is not significantly different from the one imposed numerically at all SBR values assessed (at SBR = 1.09, Student t-test α = 0.05, p = 0.355). As the results provided in Fig. 4(a,b) were obtained for a specific threshold of 0.1 for the centroid-based measurement (see Methods), we checked if they depend on the threshold value. For this purpose, the same simulation is computed for thresholds varying between 0.07 and 0.2 and the standard deviation and the mean values of the retrieved displacement error are represented respectively on Fig. 5(c) and Fig. 5(d), as a function of the SBR. It shows that the threshold value of 0.1 was roughly optimal and that independently of the threshold on the centroid measurement, the shift retrieval performances of the correlation remain better, in low SBR conditions, whatever the threshold used in the centroid measurement. Our simulation proves thus that a much more accurate measurement of the shifts between thumbnails induced by aberration could be obtained at lower SBR with the correlation method rather than with the centroid-based method. We also tested the accuracy of subpixelic shift retrieval on experimentally acquired thumbnails to take into account real scattering conditions from biological samples (see Methods). Results are reported on Fig. 4(a). Identically good shift retrieval is obtained with the two methods for SBR larger than 2; however, when the SBR drops, the error on the measured displacement remains within ± 0.1 pixel with the correlation method, whereas the centroid-based measurements deviates from zero, and this error increases significantly at SBRs smaller than 1.3. These results based on experimental data confirm the robustness of the shift retrieval for low SBR using the correlation method.

ESSH allows closed-loop aberration correction for enhanced imaging through scattering media

We finally tested the possibility to correct aberration and improve image quality in a closed-loop AO set up using the ESSH sensor (see Methods), with a model sample that resembles in vivo biological samples in term of structures and photon flux. The sample consists in fluorescentlylabelled neurons placed under a fixed unlabelled mouse brain slice (Fig. 6(a) and Fig. 6(h), see The latter is generating a 210 nm SA3, prevailing to the 10 to 20 nm SA3 generated by the brain slice, the brain tissue mainly enabling the introduction of significant scattering. The geometry of the sample was chosen to mimic in depth optical sectioning in a scattering biological medium. The AO-loop, running at 2Hz, improves significantly the image quality. The initial images (Fig. 6(b) and Fig. 6(i)) are corrected for the microscope aberrations, so that only correction of the sample-induced aberrations is performed to retrieve the images in Fig. 5(c) and Fig. 6(j). For the thinnest sample, the signal from the soma is increased by 28%, between Fig. 5(b) and Fig. 5(c) and the signal from dendrites is increased by 34% (Fig. 6(f) and Fig. 6(g)). This contrast and signal enhancement reveals some dendrites that were not visible before the correction (Fig. 6(c)).

In the case of the thicker sample (thickness of 2.5l s ), the correction, which is achieved with a very low amount of ballistic photons (8% of the incident light), increases the signal and contrast of the image even in these conditions (Fig. 6(j)). In this proof of principle experiment, the gain in image quality is even significantly lower than the one expected using two-photon microscopy, as in the latter scattering on the emission path doesn't affect image quality. This experiment illustrates the fact that a correction based on the ESSH wavefront measurement obtained through a scattering tissue and using fluorescently-labelled biological samples allows a significant image improvement, that is crucial especially in functional imaging where sensitivity is a key parameter.

Discussion

We showed that our innovative Shack-Hartmann sensor, based on an extended-source wavefront measurement, is a reliable tool to measure quantitatively aberrations in scattering samples up to 3.9l s -thick. We have shown that wavefront measurements using Shack-Hartmann sensors is ultimately limited by the SBR on the thumbnails at the edge of the sensor, due to the increased scattering of the rays reaching them. However, by computing simulations at low SBR, corresponding to real scattering conditions, we obtained more accurate displacement measurements with the correlation-based method than with the centroid-based approach. This study demonstrates the robustness of the ESSH wavefront measurement approach in depth in biological samples with significant levels of scattering. This can be attributed to the joint use of correlation and Laplacian of Gaussian filter, instead of centroid to estimate shifts. Concerning correlation, we currently make use of conventional correlation algorithms (see Methods), whereas more advanced correlation algorithms such as phase correlation are known to provide better sensitivity and/or accuracy, depending on the characteristics of the signal, but at the cost of more computational complexity and time. There is as a consequence still some room for further improvement of the approach.

By digging into the characteristics of the ESSH signal, we have also shown that the use of the ESSH allows the measurement of the scattering lengths of these samples with a single image acquisition. Indeed, it allows to measure in a single shot using a single sample the attenuation through 7 effective thicknesses, corresponding to the 7 microlenses conjugated along the objective back focal plane radius. It facilitates therefore the measurement of l s as it can be obtained directly from a single measurement in one sample. Indeed, it doesn't require multiple measurements at multiple depths [35,36] or the use of multiple samples of different thicknesses, which would introduce additional uncertainties associated with the estimation of each sample thickness. This single shot measurement of l s is therefore both accurate and easy to implement on a wide range of samples. A specific design of the ESSH with dedicated number of microlenses and characteristics of the camera could be realized to optimize the performance of the l s measurement. The accuracy of the measurement will depend on the sample thickness, as the relative attenuation between the central and border thumbnails in our case decreases as exp (-0.5z c /l s ) -1. Therefore, increasing the sample thickness is beneficial, until attenuation of the ballistic light is too strong to be separated from the background on each thumbnail. In our hand thicknesses of the order of l s to 3l s in fixed tissues seemed optimal. The values obtained can be compared with the ones reported in the literature (see Table 1). Since scattering lengths measured in vivo are 2 to 3 times higher than the one realized on fixed tissues [35,36], we used for comparison values reported only in fixed slices [35] and extrapolated linearly their measurement to the wavelengths used in our study. Our measurements match well the values reported in literature and confirm the ability of the ESSH sensor to provide a consistent measurement of l s . The slight discrepancy can be related to differences in slice preparation or animal age. One possible limitation of the scattering measurement method is the possible impact of the geometry of the sample on the measurement, in particular its surface. A non-flat surface will lead to some measurement artifacts, different local heights leading to differences in measured local scattering values. A proper use of the technique is then conditioned to the use of samples with flattened surfaces, for examples using glass slides or coverslips, which corresponds to a significant part of experimental conditions, such as e.g. transcranial imaging in the rodent using optical windows, which minimizes movement artifacts.

In this study we realized quantitative aberration measurement and correction in scattering conditions with an extended-source wavefront sensor, paving the way to the combination of adaptive optics based on this method to optical sectioning microscopy, such as two-photon microscopy or light-sheet microscopy, for deep imaging. We have shown here that the ESSH allows to perform quantitative aberration measurement deep inside biological tissues up to 3.9l s -thick. Using standard GFP labelling (λ em = 514 nm), and since l s is of the order of 60 µm in vivo [36], the ESSH sensor would give access to quantitative aberration measurements, and then correction, up to 3.9 × 60 µm ≈ 250 µm deep inside a live mouse brain at this wavelength. To make ESSH-based wavefront measurement compatible with the maximum imaging depths in 2-photon microscopy (up to 800 µm in the mouse brain [13,14]) or in light-sheet microscopy, a dual-labelling approach is relevant. It allows to keep the commonly used GFP-based functional labelling in the visible range, and to shift the wavelength used for the wavefront measurement towards the near-IR. This strategy presents two advantages: first, the impact of scattering is minimized on the ESSH thumbnails; secondly, the photon budget is optimized as all photons from functional dyes are kept for the neuronal activity recording. Using dye emitting around 900 nm, where l s ≈ 170 µm (studies at 920nm are reporting l s between 170 µm [37] and 200 µm [38]), AO corrections at depths of 3.9 × 170 µm ≈ 700 µm are reachable.

There are however several limitations to the present study that will require further experimental confirmation. First of all, the experimental demonstration realized in this study only partially confirms the prediction based on simulation. Indeed, the approach used to prepare the sample did not allow a precise enough control of the SBR of the wavefront measurement to correspond to extreme values (about 1.7 in our experiment), enabling only partial confirmation of the capabilities of the ESSH measurement to work with very low SBR. Also, the sample used does not reflect the behavior of signal and background for both light-sheet and 2-photon microscopy, since for example the main effect of aberrations in non-linear microscopy is a decrease of signal. Considering this, the proof of principle experiment is more representative of the behavior of the AO loop in a light-sheet experiment. This particular aspect will have to be investigated using a dedicated 2-photon setup. The present experiment is nevertheless a clear demonstration of the capability of an ESSH-driven AO loop to provide a correction in low SBR situations, whatever the origin of such SBR is, and on real biological objects of interest such as neurons. Also, there was no consideration here of the impact of the isoplanetic patch with the use of SA3, since this topic was already discussed in detail in our previous work [23]. Finally, our ESSH sensor doesn't require descanning of the fluorescence, as in centroid measurement [14], nor requires the use of a specific scanning arrangement and of an ultrafast laser for the wavefront measurement when the technique is applied outside non-linear microscopy [22]. This is advantageous in term of photon budget, as the ESSH can be placed closer to the microscope objective. Furthermore, it makes it perfectly suited for advanced scanning methods as acousto-optic deflectors, where descanning is difficult to implement [39].

Résumé

La microscopie optique s'est imposée comme un outil majeur en neurosciences, car elle permet de réaliser des images structurelles et des enregistrements de l'activité fonctionnelle dans de grands volumes de tissus avec une résolution subcellulaire. De plus, la microscopie non linéaire permet d'étudier l'activité neuronale dans le cerveau de petits mammifères mais reste limitée en profondeur : la diffusion et les aberrations optiques, dues à l'inhomogénéité de l'indice de réfraction, limitent la résolution et font chuter l'intensité du signal. En effet, les aberrations rencontrées sur le trajet d'excitation affectent significativement la qualité de l'imagerie lorsque la profondeur augmente. Pour surmonter ces problèmes, des stratégies d'optique adaptative (OA) ont été mises en oeuvre pour améliorer la qualité d'imagerie du microscope même à des profondeurs importantes, permettant par exemple l'imagerie fonctionnelle à haute résolution avec un microscope à deux photons dans les couches corticales infragranulaires de la souris. Alors qu'une première stratégie d'OA, largement mise en oeuvre en microscopie non linéaire, repose sur une configuration sans capteur, et donc sur un processus itératif chronophage, une deuxième approche, basée sur la mesure directe du front d'onde à l'aide d'analyseur de front d'onde de type Shack-Hartmann, et utilisant la mesure de centroïdes, a prouvé son efficacité in vivo. Cependant, cette méthode échoue à grandes profondeurs dans les tissus en raison de la forte diffusion de la fluorescence utilisée pour la mesure du front d'onde. Une méthode de mesure directe du front d'onde plus résistante à la diffusion du signal émis permettrait donc d'améliorer l'utilisation de l'OA en microscopie optique pour l'imagerie d'échantillons biologiques épais. Cette thèse propose une méthode alternative de mesure directe du front d'onde qui tire parti des méthodes de marquage existantes des échantillons biologiques et qui repose sur l'intercorrélation des images d'une source étendue, obtenues à travers un réseau de microlentilles. Cet analyseur de front d'onde de Shack-Hartmann en source étendue (ESSH), précédemment couplé à la microscopie à feuille de lumière pour la neuroimagerie chez la drosophile adulte, est ici confronté à des régimes plus diffusifs. Nous montrons que cette méthode permet des mesures quantitatives d'aberrations à travers des tranches de cerveau fixées hautement diffusantes, à des profondeurs atteignant jusqu'à quatre fois la longueur de diffusion du tissu et montrons qu'elle est plus résistante à la diffusion que l'approche actuelle basée sur la mesure de centroïdes. Nous présentons ensuite sa mise en oeuvre sur un microscope à deux photons dans une configuration en boucle fermée pour la neuroimagerie en profondeur chez la souris et nous comparons ses performances dans des milieux diffusants à l'approche descannée en centroïde. Enfin, nous présentons des résultats d'imagerie en profondeur dans des tranches de cerveaux de souris fixées.

Mots clés

Optique Adaptative, Analyseur de front d'onde Shack Hartmann en Source Etendue, Microscopie de Fluorescence Biphotonique, Milieux diffusants, Neuroimagerie

ix List of Figures 1 . 1

 11 The mouse brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Representation of neuronal recording methods against their spatial resolution, temporal resolution and field of view . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Elements of neuronal system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 GCaMP GECI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 GCaMP characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Calcium signal evolution for increasing background . . . . . . . . . . . . . . . . . . 1.7 Rayleigh and Mie scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Signal evolution in depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Absorption in biological tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Main mechanisms leading to photo-induced damages . . . . . . . . . . . . . . . . . 1.11 2PEF energy diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Principle of two-photon microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Spatial focusing and temporal focusing for multiphoton microscopy . . . . . . . . 1.14 Two-photon action cross-section of common fluorophores . . . . . . . . . . . . . . 1.15 Point Spread Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Background signal generated by surface emitter when power is increased to reach deeper layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 First Zernike polynomials representation . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Principle of AO in microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19 One SH microlens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.20 Shack-Hartmann wavefront measurement principle . . . . . . . . . . . . . . . . . . 1.21 Wavefront correctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22 Direct wavefront sensing with microvessels as guidestar . . . . . . . . . . . . . . . 1.23 Direct wavefront sensing AO for super-resolution microscopy . . . . . . . . . . . . 1.24 Modal sensorless AO algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.25 Modal-based sensorless AO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.26 AO correction with pupil segmentation for in vivo imaging of neuronal structure in the mouse brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.27 Influence of the object size on AO-TPEF . . . . . . . . . . . . . . . . . . . . . . . . . 1.28 AO correction improving functional calcium imaging in depth on neurites . . . . . 1.29 AO correction improving functional calcium imaging in depth on soma . . . . . . 1.30 Extended-Source Shack-Hartmann wavefront measurement principle . . . . . .

2 . 19

 219 Numerical comparison of displacement measurement between centroid and correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20 Semi-numerical sub-pixelic shift simulation steps . . . . . . . . . . . . . . . . . . . 2.21 Comparison of correlation and centroid based displacement measurements on experimental subimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22 Closed-loop AO setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.23 AO correction on neurons under 50µm-fixed brain slice . . . . . . . . . . . . . . . . 2.24 AO correction on neurons under 100µm-fixed brain slice . . . . . . . . . . . . . . . 3.1 AO-2PEF setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Setup pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Optical performances of the simulated optical systems . . . . . . . . . . . . . . . . 3.4 Excitation design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Detection path design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Chronogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Determination of the objective pupil plane for AO: setup . . . . . . . . . . . . . . . 3.8 Pupil deformation across the FOV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Pupil displacement plot as a function of the axial position of the objective . . . . . 3.10 AO calibration steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Impact of the uncontrolled focus on the AO loop. RMS and mirror amplitude derivation as a function of the number of iterations . . . . . . . . . . . . . . . . . . 3.12 Differential aberrations evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Two 0.5-µm neighbouring beads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 RMS values in nm across the FOV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 RMS values in nm across the FOV: patterned scan . . . . . . . . . . . . . . . . . . . 3.16 In depth wavefront measurement in scattering conditions, patterned signal . . . . 3.17 Comparison of non scanned signal with scanned pattern in scattering conditions . 3.18 Comparing WF measurements with ESSH and SH: setup . . . . . . . . . . . . . . . xi 3.19 Comparison of wavefront measurements, in non scattering conditions, obtained with ESSH and descanned SH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Comparison of the ESSH and Descanned SH signals and wavefronts in presence of strong aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 Signals obtained on ESSH and Descanned SF as a function of the depth in the scattering sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.22 Wavefronts obtained with both methods as a function of the depth in the sample . 3.23 Measured wavefront RMS as a function of the depth on both configurations . . . . 3.24 Focus evolution in depth of a scattering medium . . . . . . . . . . . . . . . . . . . . 3.25 Wavefronts obtained with ESSH and Descanned SH coupled to ES-slope retrieval algorithms as a function of the depth in the sample . . . . . . . . . . . . . . . . . . 3.26 Comparison of the performances of the Descanned SH combined to the Extended Source algorithms for slope retrieval, with the performances of the ESSH in depth of a scattering sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Comparison of brain induced aberration with brain and cranial induced one . . . . 4.2 AO correction on GADGFP fixed brain slice at 100-µm deep on the ESSH . . . . . . 4.3 AO gain on maximum intensity projection . . . . . . . . . . . . . . . . . . . . . . . 4.4 One ESSH subimage of Figure 4.3, adjusted contrast. . . . . . . . . . . . . . . . . . 4.5 Influence of the number or corrected modes on the gain . . . . . . . . . . . . . . . . 4.6 Zernike aberration coefficients corrected by 36 and 42 modes . . . . . . . . . . . . . 4.7 AO for 3D resolution enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Maximum intensity projection along the xz plane . . . . . . . . . . . . . . . . . . . 4.9 Dual labelling setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 AO loop on dual labelled brain slice . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 AO loop on dual labelled brain slice, correction of brain slice only . . . . . . . . . . 4.12 AO correction on QD under a 250-µm fixed brain slice . . . . . . . . . . . . . . . . . xiii List of Tables

Contents 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5 1 . 6 4 Chapter 1 .

 11121314151641 From the animal to the neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The animal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Anatomical elements on the cortical area and neuron description . . . . 1.1.3 Spatiotemporal resolution requirements . . . . . . . . . . . . . . . . . . . How to record cellular activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Fluorescence imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluorescence microscopy for neuroimaging . . . . . . . . . . . . . . . . . . . . 1.3.1 The basics of fluorescence microscopy . . . . . . . . . . . . . . . . . . . . 1.3.2 Two-photon microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Two-photon fluorescence microscopy limits in depth . . . . . . . . . . . Increasing imaging 2PEF performance in depth with Adaptive Optics . . . . 1.4.1 Brief reminder of aberrations . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Aberration correction with Adaptive Optics . . . . . . . . . . . . . . . . 1.4.3 Discussion on AO contribution for 2PEF structural and functional imaging 1.4.4 Conclusion on current AO methods for neuroimaging . . . . . . . . . . Extended Source wavefront sensing for microscopy . . . . . . . . . . . . . . . Summary and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In-depth functional neuroimaging

1. 1 .FIGURE 1 . 1 :

 111 FIGURE 1.1: The mouse brain. (a) Dorsal view of fixed mouse brain. Adapted from Airey et al. [5]. (b) Coronal slice taken along the dotted line in (a), and corresponding brain areas labelled, taken from the Allen Mouse Brain Atlas [6]. (c) In vivo two-photon imaging from the neocortex. Maximum intensity projection of a fluorescence stack, down to layer 5, the corresponding area being boxed on (b), neurons being labelled with Clomeleon, a genetically encoded chloride indicator, and expressed preferentially in layer 5. From Helmchen et al. [7].

FIGURE 1 . 2 :

 12 FIGURE 1.2: Representation of neuronal recording methods against their spatial resolution, temporal resolution and FOV. INTRA, intracellular recordings; S/MUA, single or multiunit activity; LFP, local-field potentials; 2-PH, two-photon microscopy; MEA: multielectrode array; VSDI, voltagesensitive dye imaging; OI-IS, optical imaging of intrinsic signals; fMRI, functional magnetic resonance imaging; E/MEG, electro-or magneto-encephalogram; IDEAL, the ideal technique. From [9]

FIGURE 1 . 3 :

 13 FIGURE 1.3: A. Single neuron in a drawing done by Ramón y Cajal, inset showing an AP. B. Signal transmission from a presynaptic neuron j to a postsynaptic neuron i, synapse being circled.Adapted from[10]

FIGURE 1 . 5 :

 15 FIGURE 1.5: Structure of the GCaMP molecule: a circularly permuted GFP molecule is bonded to a calmodulin fragment and M13 peptide. Adapted from https://www.janelia.org/ open-science/gcamp.(b) jGCamp8f and jGCaMP7f responses to field stimulation pulses of increasing frequencies imaged in neuronal culture (c) Average responses of GCaMP indicators to a single AP in vivo, (d) Half decay times of the fluorescence in vivo of several GCaMP versions. From [19].

Chapter 1 .FIGURE 1 . 6 :

 116 FIGURE 1.6: (a) Calcium signal evolution for with and without constant background. Plot of the recorded raw intensity corresponding to an AP and of the relative fluorescence variation from which the signal is eventually extracted ∆F/F. (b) Top: Neuronal signal to extract (black) from activity of surrounding neurons (coloured traces). Bottom: plot of the raw noisy signal that is measured, resulting from the noised signal of interest summed to the noised background activity (in blue), and the signal to extract (in red), arrows showing the missed spikes. Simulations performed with k o n = 10k o f f = 30 ms, over 200 ms. B 0 = 20, B f

FIGURE 1 . 7 :

 17 FIGURE 1.7: Rayleigh and Mie scattering

FIGURE 1 . 8 :

 18 FIGURE 1.8: Effect of scattering on signal deterioration. A fluorescent emitter (λ = 515 nm) is positioned under several brain slices of increasing thicknesses. Increasing scattering leads to a signal decrease and a Signal to Background Ratio drop (SBR), values being listed in the table aside.

FIGURE 1 . 13 :FIGURE 1 . 14 :

 113114 FIGURE 1.13: Spatial focusing and temporal focusing for multiphoton microscopy. Localization of excitation by 2PEF in a fluorescein solution. (a) 1PEF by focused continuous laser excitation at 488 nm, emission from the excitation beam propagation cone. (b) 2PEF by focused femtosecond laser at 960 nm, emission limited to the focal volume. Comparison of the temporal distribution of photons between a continuous laser (a) and a pulsed laser (b). From [41].

FIGURE 1 . 15 :

 115 FIGURE 1.15: Axial and lateral PSF of one-photon widefield, confocal and two-photon fluorescence microscopy. From [45]

FIGURE 1 . 18 :FIGURE 1 . 19 :

 118119 FIGURE 1.18: Principle of AO in high resolution microscopy. (a) Within the sample, aberrations are induced by the specimen as light encounters refractive index inhomogeneities. This results in an aberrated focus. (b) The deformable mirror compensates for these aberrations on the illumination path, restoring diffraction limited focus within the sample. (c) The deformable mirror corrects the aberrations on the detection path. From [70]

FIGURE 1 . 21 :

 121 FIGURE 1.21: Wavefront correctors. (a) SLM scheme, (b) DM scheme, from https://www.alpao.com/products-and-services/deformable-mirrors/

  FIGURE 1.22: Direct wavefront sensing with microvessels as guidestar, adapted from [80]. (a) Steps invvolvd in the AO direct wavefront sensing, with the 3D visualization of the labelled microvessels withCy5.5-dextran, by retro-orbtital injection. Spots on the SH at 3 depths, and corresponding wavefronts. (b) Scientific two-photon images: neurons at 690 µm below the pia, before and after AO, and 3D volumetric rendering of mRuby-labelled pyramidal cells.

Chapter 1 .

 1 FIGURE 1.23: Direct WFS AO for super-resolution microscopy (SIM), with a two-photon descanned guidestar. (a) and (b) Lateral view of Rhodamine-phalloidin-stained actin in a primary mouse endothelial cell embedded in collagen matrix, 150 µm from surface of coverslip, before (a) and after (b) AO and deconvolution. Scale 5 µm. From [82]. (c) and (d) Maximum intensity projections of 8 µm stack between 21 and 29 µm in depth of Thy1-GFP line M brain slices, before (c) and after (d) AO (signal normalized to the AO one and then artificially increased 4 times on (c)). Image 40 x 40 µm². From [83].

FIGURE 1 . 25 :

 125 FIGURE 1.25: Modal-based sensorless AO. (a) Flowchart of the 2N+1 and 3N methods. (b) Example of two-photon imaging with modal based sensorless AO on mouse embryo, before (left) and after (right) AO. From [87](c) Drosophila brain imaging at 40 µm in depth, before (left) and after (right) AO, coupled to HiLo microscopy, from [90]. (d) Drosophila larva imaged with THG, before (top)and after (bottom) AO[START_REF] Olivier | Dynamic aberration correction for multiharmonic microscopy[END_REF].

FIGURE 1 . 26 :

 126 FIGURE 1.26: AO correction with pupil segmentation for in vivo two-photon imaging of neuronal structures in the mouse brain. (a) Functional imaging at 150-µm depth, and activity recordings with and without AO under visual stimulation. Scale bar 10 µm. Adapted from [98]. (b) Maximum intensity projection of planes from 490 µm to 513 µm under the dura, before and after correction. Scale bar 5 µm. Plot along the blue line (dotted before, full after AO), corresponding corrective wavefront. Insets: the Fourier transform of the images, dotted circle: diffraction limit. From [99].

FIGURE 1 . 27 :

 127 FIGURE 1.27: Influence of the object size on AO-TPEF reveling a five times greater signal enhancement for 0.5 µm fluorescent features than for uniformly labelled area. The green circles representing the fluorescent features are compared to the uncorrected and corrected focal volumes. From [97]

FIGURE 1 .

 1 FIGURE 1.28: AO correction improving functional calcium imaging in depth in mouse cortex. (a) Calcium responses in the primary visual cortex, induced by drifting-grating stimulation, before and after AO correction. Scale bar 20 µm. (b) Calcium transients before and after AO within the corresponding ROI i to vi. From [79]

Chapter 1 .FIGURE 1 . 29 :

 1129 FIGURE 1.29: AO correction enabling functional imaging through thinned skull window 520µm below the pia. (a) Imaging apical dendrites and spines labelled by jRGECO1a, before and after AO correction. (b) Calcium transients before and after AO within the corresponding ROI i to iv, ROI i corresponding to a soma. Neuropil responses being subtracted by using surrounding regions.From[START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF]

FIGURE 1 . 30 :FIGURE 1 . 31 :

 130131 FIGURE 1.30: Extended-Source Shack-Hartmann wavefront measurement principle

FIGURE 2 . 1 :

 21 FIGURE 2.1: Absorption and scattering in the brain. Theoretical model of scattering and absorption of mouse brain, based on water absorption and Mie scattering. Black stars: measured effective attenuation lengths in vivo. From [25].

FIGURE 2 . 3 :

 23 FIGURE 2.3: In-depth signal distribution on the ESSH. (a) The back focal plane of the objective is conjugated with the microlens array. The signal intensity drops progressively towards the external edges of the imaged pupil as light is collected through higher angles. (b) Typical acquisition through a 150-µm fixed brain slice, signal emitted at 515 nm, with a zoom on a subimage highlighting ballistic loss. The signal, background and SBR are plotted against the effective thickness crossed.

FIGURE 2 . 4 :

 24 FIGURE 2.4: Quantum dots and aggregates. (a) Illustration of the fluorescence emission shift towards higher wavelengths as the particle enlarges, typical sizes varying between 2 to 6 nm for emission wavelength peaks varying from 500 nm to approximately 700 nm. (b) TEM images of QD aggregates, diameter of 500 nm, taken by Alexandra Bogicevic. (c) Emission spectra of the QDs used in this study

Chapter 2 .FIGURE 2 . 5 :

 225 FIGURE 2.5: Scattering length measurements. Illustration of the sample preparation and plot of the signal logarithm as a function of the effective sample thickness crossed by the signal (a) On calibration measurements realised on polystyrene beads, for two beads concentrations. (b) Results obtained on fixed brain slices at three different wavelengths.

2. 2 .

 2 Assessing WF measurement accuracy in scattering conditions: experimental approach 53

FIGURE 2 . 7 :FIGURE 2 . 8 :

 2728 FIGURE 2.7: Aberration measurement setup. The setup has two optical pathways. The main one (in green) is dedicated to the aberration measurement through fixed brain slices, while the secondary one (in yellow) is used for sample positioning, allowing us to ensure that the measurements are made within the cortical area. L1-6: Lenses, D1: Squared filed Diaphragm, D2: Aperture Diaphragm, D3: Field Diaphragm, BS: 50:50 Beamsplitter, P1: Back focal plane of the objective conjugated to P2, located on the microlens array of the ESSH.

2. 2 .FIGURE 2 . 9 :

 229 FIGURE 2.9: Influence of the pre-filtering step on the wavefront calculation in scattering conditions. Raw signal obtained for red QD aggregates under a 100-µm fixed brain slice in the cortical area. For the two filtering processes, the filtered signal, the 2D-correlation maps, the corresponding slopes and the reconstructed wavefront are shown. In this example, SBR = 2.

FIGURE 2 . 10 :FIGURE 2 . 11 :

 210211 FIGURE 2.10: Decomposition of the 3rd order spherical aberration into a r 4 mode and a low-order mode in r 2 to emphasize defocus contribution in SA3. Illustration from [131]

1 FIGURE 2 . 14 :

 1214 FIGURE 2.14: Comparison of point-source based wavefront measurement on a classical Shack-Hartmann with extended source based wavefront measurement on ESSH.

FIGURE 2 . 15 :

 215 FIGURE 2.15: Experimental comparison of correlation-based WF calculation with centroid-based WF calculation. (a) Fluorescence emitters under brain slices with increasing thickness, from left to right: no brain slice, a 50-µm brain slice, and a 125-µm brain slice. (b) and (c) Wavefront calculation through 50-µm brain slice (respectively 125-µm brain slice), full ESSH signal with zoom on a subimage, and calculated wavefront with both methods.

2. 3 .FIGURE 2 . 16 :

 3216 FIGURE 2.16: Simulation steps. (a) Well-sampled PSF. (b) Background and Poisson noise are added. (c) The noisy PSF is shifted of 5 pixels. (d) The noisy PSF is subsampled of a factor 10. (e) The noisy and shifted PSF is subsampled of a factor 10 too.

4 . 6 .

 46 Poisson noise is added to mimic Shot noise. The parameter λ of the Poisson distribution is √ N, with N the number of photons. For each pixel, the noise value is randomly generated from a Poisson distribution, with λ equaled to √ N, the pixel initial value containing signal plus the background value. The noise matrix is added to the PSF with background one. The resulting image is shown on Figure 2.16b. 5. This image is then shifted of 5 pixels, in both horizontal and vertical directions, to get Figure 2.16c. Both 2.16b and 2.16c are subsampled to the final dimensions but summing the pixel values within squares of 10 x 10 pixel², respectively leading to Figure 2.16d and Figure 2.16e.

2 .

 2 Results are shown on Figure 2.18; the accuracy of the measurement being assessed via the Standard Deviation of the displacement error (Figure 2.18a) and the mean of the displacement error (Figure 2.18b).

FIGURE 2 . 20 :

 220 FIGURE 2.20: Semi-numerical steps. (a) Original subimage of a mVenus labelled neuron, SBR = 1.6. (b) Oversampled image by a factor 100. (c) Shifted oversampled image. (d) Subsampled image down to its initial sampling.

FIGURE 2 . 21 :

 221 FIGURE 2.21: Comparison of correlation and centroid based displacement measurements on experimental subimages. The results are added on top of the one obtained for fully simulated data to compare the difference between the expected results and the actual ones. Single dots and diamonds represent shift retrieval for experimental images with both methods.

FIGURE 2 . 22 :

 222 FIGURE 2.22: Closed-loop AO setup. L: Lens, M: Mirror, D: Diaphragm, BS: Beamsplitter, P: Pupil plane.

  [START_REF] Champelovier | Image-based adaptive optics for in vivo imaging in the hippocampus[END_REF]) by L1 (L1, Thorlabs, AC254-100-A, f' = 100 mm). A first pair of achromatic relay lenses L2-L3, (L2, Thorlabs, AC508-150-A, f' = 150 mm, and L3, Thorlabs, AC254-125-A, f' = 125 mm) conjugates the back focal plane of the objective with the deformable mirror (Mirao52e, Imagine Eyes). A second pair of relay lenses L4-L5 (L4, Thorlabs, AC508-250-A, f' = 250 mm, L5, Thorlabs, AC254-80-A, f' = 80 mm) is added to conjugate the back focal plane on the ESSH. The scientific image is realised on an ORCA Flash V2 (Hamamatsu), on which the total FOV corresponds to an array of 400 x 400 µm² in the sample. The correlation computation is performed over the 130 x 130 µm² central zone of the full-field image.

FIGURE 2 . 23 :

 223 FIGURE 2.23: AO correction on mVenus labelled neurons under 50µm-fixed brain slice. Before (a) and after (b) AO loop. Insets signal enhancement on dendrites. Intensity profiles along the pink and black lines in (a) and (b) are respectively plotted in (c) and (d)

Figure 2 .

 2 23a and Figure 2.23b and the signal from dendrites

FIGURE 2 . 24 :

 224 FIGURE 2.24: AO correction on mVenus labelled neurons under 100µm-fixed brain slice. Before (a) and after (b) AO loop.

75 •

 75 The diameter of the DM aperture: 15 mm • The objective effective focal length: 9 mm • The objective back pupil plane diameter: Φ = 2NA f ′ = 18 mm • The ESSH lateral dimension: 5 mm • The ESSH microlenses focal length: f' = 5.2 mm On top of that, some elementary specifications are: • Field of view: 400 x 400 µm² 3.1. Setup design Acquisition minimal frequency for one pixel line: 1kHz • Resolution (for structural imaging): 0.2-0.5 µm

FIGURE 3 . 1 :

 31 FIGURE 3.1: AO-2PEF setup. L1-11: Lenses, D1: Squared field Diaphragm, DBS: Dichroic Beamsplitter, BS: 50:50 Beamsplitter, P1: Back pupil of the objective conjugated to P2, on the Deformable Mirror, P3, on the microlens array of the ESSH, P4 and P5 on the galvanometer scanners.

FIGURE 3 . 2 :

 32 FIGURE 3.2: Setup pictures. (a) From the side, (b) from the top, with excitation path, detection path for full-field imaging and detection path for AO loop identified.

FIGURE 3 . 3 :FIGURE 3 . 4 :

 3334 FIGURE 3.3: Optical performances of the simulated optical systems. For each system, the on and off-axis performances are shown, through 3D PSF plots, (top: on axis, bottom: off-axis), Spot diagrams and Diffraction encircled energy performances (right: on-axis, left: off-axis). In black: performances of a diffraction limited setup. (a) Three scan lenses (b) Only one scan lens (c) No scan lens.

FIGURE 3 . 5 :

 35 Photocathode Input window

  FIGURE 3.6: Chronogram

FIGURE 3 . 8 :

 38 FIGURE 3.8: Pupil deformation across the FOV. Top: diffraction limited spot on Cam1. Bottom: Corresponding pupil on Cam2, orange circle noting from the reference pupil shape and position associated to a centered spot. From left to right: Starting spot position and its displacement across the FOV, the pupil deformations are highlighted by blue arrows.

FIGURE 3 . 9 :

 39 FIGURE 3.9: Pupil displacement plot as a function of the axial position of the objective. (a) Pupil displacement scheme for a lateral displacement of the point source in the FOV as respectively observed on Cam2 and Cam1. (b) Relative displacement of the pupil edge for a 120 µm spot shift in the FOV to the right and to the left, for each of the pupil edge in both cases, as a function of the axial position of the objective. The zero objective position corresponds to the conjugation of the Back Focal Plane on Cam2.

3. 3 .FIGURE 3 . 10 :

 3310 FIGURE 3.10: AO calibration steps. Adapted from [139]. (a) For each actuator, voltages +C and -C are applied one after the other and the corresponding change in slopes per unit voltage is calculated. For each actuator, these measurements are stored in one column of the Influence Functions Matrix IF. (b) IF is decomposed in three matrices via SVD. Each row of V T corresponds to the control signals necessary to generate an independent mirror modes. Multiplication of the corresponding singular values (in S) with their corresponding columns in U results in each mirror mode as seen by the wavefront sensor.

FIGURE 3 . 11 :

 311 FIGURE 3.11: Impact of the uncontrolled focus on the AO loop. RMS and mirror amplitude derivation as a function of the number of iterations

FIGURE 3 . 12 :

 312 FIGURE 3.12: Differential aberrations. Full-field image of (a) the 0.5-µm green bead, (b) the 10-µm green and red beads, (c) and of the two merged images. (d) Signal of the 10-µm two-coloured beads on the ESSH with a zoom on one subimage.

FIGURE 3 . 13 :

 313 FIGURE 3.13: Two 0.5-µm neighbouring beads

  nm

FIGURE 3 . 14 :

 314 FIGURE 3.14: RMS values in nm across the FOV.

Figure 3 FIGURE 3 . 15 :

 3315 FIGURE 3.15: RMS values in nm across the FOV: patterned scan

FIGURE 3 . 16 :

 316 FIGURE 3.16: In depth wavefront measurement in scattering conditions, patterned signal. (a) At the surface (b) At 350 µm in-depth. l s ≈ 140 µm

FIGURE 3 . 17 :

 317 FIGURE 3.17: Comparison of non scanned signal with scanned pattern in scattering conditions.l s ≈ 140µm.

3. 4 .FIGURE 3 . 18 :

 4318 FIGURE 3.18: Comparing WF measurements with ESSH and SH: setup. L1-11: Lenses, D1: Squared field Diaphragm, DBS: Dichroic Beamsplitter, BS: 50:50 Beamsplitter, P1: Back pupil of the objective conjugated to P2, on the Deformable Mirror, P3, on the microlens array of the ESSH, P4 and P5 on the galvanometer scanners, P6 on the SH.

3. 4 .

 4 FIGURE 3.19: Comparison of wavefront measurements, in non scattering conditions, obtained with ESSH and descanned SH.

FIGURE 3 . 20 :

 320 FIGURE 3.20: Adding sample aberration, comparison of the ESSH and Descanned SH signals and wavefronts (a) ESSH (b) Descanned SH

FIGURE 3 . 21 :

 321 FIGURE 3.21: Signals obtained with both methods as a function of the depth in the sample. Top line: Signal on the ESSH, bottom line: signal on the descanned SH. Insets showing one subimage on the outer edge of the sensor.

3. 4 . 101 FIGURE 3 . 22 :

 4101322 FIGURE 3.22: Wavefronts obtained with both methods as a function of the depth in the sample

FIGURE 3 . 23 :

 323 FIGURE 3.23: Measured wavefront RMS as a function of the depth on both configurations

FIGURE 3 . 24 :

 324 FIGURE 3.24: Focus evolution in depth of a scattering medium. The vertical bar on (A) corresponds to the point after which the scanning time is doubled.

3. 4 .

 4 Comparing Extended Source wavefront sensing to Descanned wavefront sensing in scattering conditions 103 b. Decorrelating the algorithm choice from the signal on the Descanned SH

FIGURE 3 . 25 :FIGURE 3 . 26 :

 325326 FIGURE 3.25: Wavefronts obtained with ESSH and Descanned SH coupled to ES-slope retrieval algorithms as a function of the depth in the sample

2 R

 2 Because of the 50:50 Beamsplitter, major losses are induced at both the excitation and the detection. Due to the non linearity of the excitation, a loss of 50% of the excitation flux induces a loss of 75% on the emitted signal. At the detection, only 50% of the emitted photons are directed towards the ESSH, and the neutral density filters reduce again the amount of available photons for the wavefront measurement (this time, on purpose). Overall, the ratio of remaining fluorescent photons incident on the ESSH is: R = Signal emitted × Signal transmitted to ESSH × Density 1 × Density = 0.25 × 0.50 × 0.40 × 0.35 = 0.0175 (3.7) Hence, the wavefront measurements via the ESSH fails at shallower depths than what we obtained in Section 3.3.4, when we only had the ESSH, neither beamsplitter nor density filters, even though the amount of fluorescein has been doubled in between.

FIGURE 4 . 1 :

 41 FIGURE 4.1: Comparison of brain induced aberrations with brain and cranial induced ones, as a function of the imaging depth, in vivo. From [98]

ParametersFIGURE 4 . 2 :

 42 FIGURE 4.2: AO correction on neuronal signal from GADGFP fixed brain slice at 100-µm deep on the ESSH. (a) ESSH signal after correction (b) Zoom on the central subimage, showing a signal enhancement between before and after AO. (c) Corresponding wavefronts, before and after correction.

FIGURE 4 . 3 :

 43 FIGURE 4.3: AO gain on maximum intensity projection of 24 µm, step 2 µm. Full-field image before (a) and after (c) AO with corresponding wavefronts in insets. (b) Intensity profiles along the three signalized lines. Region at 180 µm below the coverslip.

FIGURE 4 . 4 :

 44 FIGURE 4.4: One ESSH subimage of Figure 4.3, adjusted contrast.

FIGURE 4 . 5 :

 45 FIGURE 4.5: Influence of the number or corrected modes on the gain. Signal from GADGFP slice, 120-µm deep (a) Before AO, (b) For 36 modes corrected, and (c) For 42 corrected modes. (d) Plot only the doted lines for the three acquisitions.

FIGURE 4 . 6 :

 46 FIGURE 4.6: Zernike aberration modes, before (top), after 36 modes corrected (middle) and after 42 modes corrected (bottom). In blue: 3rd order modes, in pink: 5th order modes, in black: 7th and 9th orders. The y-axis scale after correction (36 and 42 modes) is adapted to smaller Zernike values than before AO.

FIGURE 4 . 7 :

 47 FIGURE 4.7: AO for 3D resolution enhancement, for AO correction at a single plane (50 µm deep). Maximum projection of a 300-µm stack obtained within a fixed brain slice, labelled with GADGFP. (a) Before and (b) after correction. Insets: 3D rendering of the acquisitions.

FIGURE 4 . 8 :

 48 FIGURE 4.8: Maximum intensity projection along the xz plane for the 300-µm thick slice. Before (a) and after (c) AO. (b) Plots along the dotted lines shown on (a) and (c). AO correction at a single plane (50 µm deep)

FIGURE 4 . 9 :

 49 FIGURE 4.9: Dual labelling setup.

FIGURE 4 . 10 :

 410 FIGURE 4.10: Dual labelled brain slices. Maximum intensity projection in z, color coded, of a 70µm thick stack, z step 1 µm. Full-field image of brain slice, dual-labelling, before (a) and after (b) AO. ROI for AO correction squared in white. (c)-(d) Zoom on the the area in the orange box, before (c) and after (d) AO. (e) Corresponding wavefronts, before and after AO. (f) Signal on the ESSH. (g) Intensity profiles along the brown lines shown on (c) and (d). (h) Intensity profiles along the red lines shown on (a) and (b). Contrast is adjusted on the insets.

FIGURE 4 . 11 :

 411 FIGURE 4.11: Dual labelled brain slices, at 200 µm in depth. Maximum intensity projection in z, over 15 µm, z step 1 µm. Full-field image of brain slice, before (a) and after (b) AO. (c)-(d) Zoom on the center of the FOV before (c) and after AO (d). (e) ESSH signal for AO correction and zoom on the central subimage. (f) Measured wavefronts before and after correction. (g) Profile along the purple doted lines (h) Profile along the orange doted lines.

FIGURE 4 . 12 :

 412 FIGURE 4.12: AO correction on QD under a 200-µm fixed brain slice. Before (a) and after (b) AO

Fig. 1 .

 1 Fig. 1. Schematic of the set up. (a) Aberration measurement set up. L1-6 lenses, D1 squared field diaphragm, D2 aperture diaphragm, D3 field diaphragm, BS 50:50 beamsplitter, P1 back focal plane of the objective, conjugated to P2 located on the microlens array of the ESSH. (b) Closed-loop AO set-up. L1-5 lenses, D1 squared field diaphragm, BS 50:50 beamsplitter, P1 back focal plane of the objective, conjugated to P2 located on the Deformable Mirror, and to P3 on the microlens array of the ESSH.

Fig. 2 .

 2 Fig. 2. Flowchart representing the steps of the wavefront reconstruction, for the centroid based measurement and the correlation based-measurement used with our ESSH.

Fig. 3 .

 3 Fig. 3. Scattering length measurement on fixed brain slices. (a) The ESSH microlenses are conjugated with the back focal plane of the objective. Each microlens sees the same field of view but with a different angle. (b) Example of an ESSH image showing the decrease in intensity between the side-by-side thumbnails within the doted boxes (represented in (a)), with a zoom on one corresponding thumbnail, obtained through a 150-µm-thick brain slice at 515 nm. Corresponding plot of the signal, the background and the signal to background as a function of the effective thickness seen along the direction of the doted boxes (c) Calibration measurements with the corresponding sample scheme. (d) Measurements on brain slices, with sample scheme.

Fig. 4 .

 4 Fig. 4. Spherical Aberration measurement on scattering sample with ESSH. (a)-(c) 3rd order Spherical Aberration amplitude increases with the Optical Path Difference (i.e. the brain slice thickness) induced by mouse brain slice at 3 wavelengths, mean and standard deviation plotted. The experimental data are compared to simulation data (described in Methods). (d) Examples of signal on external thumbnails for 4 different thicknesses of brain slice, at 615 nm. The corresponding values of slice thicknesses are given in terms of number of scattering lengths.

Fig. 5 .

 5 Fig. 5. Comparison of shift retrieval accuracy using centroid and correlation method. (a) Numerical simulation of cross-correlation and centroid method to retrieve a subpixelic displacement induced numerically as a function of the SBR. Coloured dots with error bars correspond respectively to the mean and standard deviation for 50 simulations at each SBR. Single dots and diamonds represent shift retrieval for experimental images with both methods (b) Standard Error of the Mean (SEM) as a function of the SBR. The SEM values correspond to the simulated data in (a). (c-d) Standard deviation (c) and mean values (d) of the displacement error for 50 simulations at each SBR, for several thresholds (T) applied for the centroid method, T varying between 0.07 and 0.2

Fig. 6 .

 6 Fig. 6. ESSH based AO correction loop enables to retrieve microscope performances through scattering samples. (a) Fixed mVenus neuron culture placed under a 50-µm fixed mouse brain slice, corresponding to 1.2l s . Wide-field images before (b) and after the correction (c) with the corresponding wavefronts. Signal profiles along the black and pink lines in (b) and (c) are plotted in (f) and (g). Zoom on dendrites before (d) and after (e) correction, with adjusted contrast. (h) Same as (a) but using a 100-µm fixed mouse brain slice, corresponding to 2.5l s . Images before (i) and after (j) correction, with corresponding wavefronts.

  . . 1.31 Enhanced functional imaging on ex vivo Drosophila brain with Extended-Source wavefront sensing coupled to Light Sheet Microscopy . . . . . . . . . . . . . . . . . Quantum dots and aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Scattering length measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Scattering lengths l s : comparison of values from the literature (fixed and in vivo data) to our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Setup used for aberration measurements in scattering tissues . . . . . . . . . . . . . 2.8 Signal through a brain slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Influence of the pre-filtering step on the wavefront calculation in scattering conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Decomposition of SA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 3rd order spherical aberration against focus. . . . . . . . . . . . . . . . . . . . . . . 2.12 Modelling SA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 SA3 measurements on scattering sample . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Workflow describing ESSH and SH WF measurement . . . . . . . . . . . . . . . . . 2.15 Experimental comparison of correlation-based WF calculation with centroid-based WF calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Simulation steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Difference between the calculated displacement error and the induced subpixelic shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Threshold influence on centroid-based displacement measurement . . . . . . . . .

2.1 Absorption and scattering in the brain . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multiple scattering in a tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 In-depth signal distribution on the ESSH . . . . . . . . . . . . . . . . . . . . . . . .

x 2.4

  1.1 Influence of the background on the signal extraction . . . . . . . . . . . . . . . . . . 1.2 Most usual Zernike polynomials normalised to RMS and corresponding aberrations 2.1 Measured scattering lengths l s of fixed brain slices at different wavelengths with the ESSH analyzer compared to literature reported values. . . . . . . . . . . . . . . 2.2 RMS values of wavefronts calculated with the two methods, within scattering conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	xvi
	SVD	Singular Value Decomposition
	THG	Third Harmonic Generation
	WD	Working Distance
	WF	WaveFront
	WFS	WaveFront Sensor
	1PEF	Single Photon Excited Fluorescence
	2PEF	Two Photon Excited Fluorescence
	3PEF	Three Photon Excited Fluorescence
	AO	Adaptive Optics
	AOD	Acousto-Optic Deflector
	AP	Action Potential
	BS	Beam Splitter
	DM	Deformable Mirror
	DMD	Digital Micromirror Device
	DBS	Dichroic Beam Splitter
	ESSH	Extended Source Shack Hartmann
	FOV	Field Of View
	FP	Fluorescent Protein
	FWHM Full Width at Half Maximum
	GAD	Glutamic Acide Decarboxylase
	GABA Gamma-Aminobutyric Acid
	GECI	Genetically Encoded Calcium Indicator
	GEVI	Genetically Encoded Voltage Indicator
	GFP	Green Fluorescent Protein
	ICG	IndoCyanine Green
	IR	InfraRed
	LSFM	Light Sheet Fluorescence Microscopy
	NA	Numerical Aperture
	LoG	Laplacian of Gaussian
	OPD	Optical Coherence Tomography
	OPD	Optical Path Difference
	PAW	Partitioned Aperture Wavefront
	PBS	Phosphate Buffer Solution
	PMT	PhotoMultiplier Tube
	PSF	Point Spread Function
	QD	Quantum Dot
	ROI	Region Of Interest
	ROS	Reactive Oxygen Species
	RMS	Root Mean Square
	SA	Spherical Aberration
	SA3	3rd order Spherical Aberration
	SBR	Signal to Background Ratio
	SEM	Standard Error of the Mean
	SIM	Structured Illumination Microscope
	SNR	Signal to Noise Ratio
	SLM	Spatial Light Modulator
	SH	Shack Hartmann

3.1 PSF measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Distribution of RMS values across the FOV for the three considered AO signals . . 3.3 Aberration coefficient retrieval for both methods in depth. . . . . . . . . . . . . . . 4.1 Aberrations content before AO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Aberrations content before AO and after AO, for 36 and 42 corrected modes . . . . 4.3 Aberrations content before AO for the dual-labelled brain slice . . . . . . . . . . . . 4.4 Aberrations content before AO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Laser power at 920 nm in the sample plane . . . . . . . . . . . . . . . . . . . . . . . xv

List of Abbreviations

TABLE 1 . 1 :

 11 Influence of the background on the signal extraction

  ).

	Energy	S 1 S 0	Excited singlet state Vibrational relaxation Ground state 2PEF 1PEF Fluorescence Excitation Fluorescence hν ex 2 1 Excitation hν ex hν ex 1 2 hν em hν em
	FIGURE 1.11: 1PEF versus 2PEF Jablonski diagrams

TABLE 1 . 2

 12

: Most usual Zernike polynomials normalised to RMS and corresponding aberrations FIGURE 1.17: First Zernike polynomials representation. From

[START_REF] Hsieh | Integral-based parallel algorithm for the fast generation of the Zernike polynomials[END_REF]

  is the lateral dimension on the microlens, f ′ its focal length, θ x,y being the angle between locally tilted WF and the locally planar WF. In the small angle approximation we get:

	1.4. Increasing imaging 2PEF performance in depth with Adaptive Optics	29
	where d a x,y =	dδ x,y f ′		(1.27)
	Aberrated WF
			y δ
			x δ
	FIGURE 1.20: Shack-Hartmann wavefront measurement principle
	y = d sin(θ x,y ) = d sin arctan	δ x,y f ′	(1.26)

Chapter 2 Aberration measurements and wavefront sensing in scattering conditions Contents 2.1 In depth signal on

ESSH: wavefront measurement challenges . . . . . . . . . 2.1.1 Ballistic signal evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Scattering length measurement of brain tissues . . . . . . . . . . . . . . . 2.2 Assessing WF measurement accuracy in scattering conditions: experimental approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Setup and algorithm description . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Third order Spherical Aberration in depth . . . . . . . . . . . . . . . . . 2.3 Assessing WF measurement accuracy in scattering conditions: numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Experimental comparison of centroid and ESSH WF measurements . . . 2.3.2 Comparison of centroid-based displacement measurement and correlation based displacement measurement on simulated data . . . . . . . . . 2.3.3 Comparison of centroid-based displacement measurement and correlation based displacement measurement on experimental data . . . . . . . 2.4 AO loop within scattering conditions . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Setup and sample description . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TABLE 2 .

 2 1: Measured scattering lengths l s of fixed brain slices at different wavelengths with the ESSH analyzer compared to literature reported values.

	24]

s : comparison of values from the literature (fixed and in vivo data) to our results. Literature values corresponding to Table

2

.1. Fitting curve expression for the in vivo measurements: y = 35.5 exp (0.0016x), R² = 0.992. Fitting curve for the data in fixed tissue deduced from the fit and the 2.4 factor reported between in vivo and fixed tissues scattering lengths.

  2.2. Assessing WF measurement accuracy in scattering conditions: experimental approach 59

	a			Brain slice thickness (µm)		b				Brain slice thickness (µm)
		0.12		50	100 125 150	190	250			0.12		50	100 125 150	190	250
	SA3 amplitude (µm)	0.02 0.04 0.06 0.08 0.1		515 nm			Experimental Simulation	SA3 amplitude (µm)	0.02 0.04 0.06 0.08 0.1		615 nm	Experimental Simulation
		0									0
	-0.02	0	2	4 OPD (µm) 6	8	10	-0.02	0	2	4 OPD (µm) 6	8	10
								Brain slice thickness (µm)
						0.12		50	100 125 150	190	250
					SA3 amplitude (µm)	0.02 0.04 0.06 0.08 0.1	805 nm					Experimental Simulation
						0
					-0.02	0	2	4 OPD (µm) 6	8	10

c

FIGURE 2.13: Spherical Aberration measurement on scattering sample with ESSH. 3rd order Spherical Aberration amplitude increases with the Optical Path Difference (i.e. the brain slice thickness) induced by mouse brain slice at three wavelengths, mean and standard deviation plotted. The experimental data are compared to simulation data

Brain slice thickness RMS (µm) SA3 (µm)

	in µm	in l s	Centroid	ESSH	Centroid	ESSH
	50	1.1	0.447	0.471	0.99	1.02
	125	2.7	0.558	0.462	0.76	0.96

TABLE 2 .

 2 2: RMS values of wavefronts calculated with the two methods, within scattering conditions.

  Displacement error in pixel as a function of the SBR for both methods. (b) Standard error of the mean as a function of the SBR on the same simulated data set.

			Chapter 2. Aberration measurements and wavefront sensing in scattering conditions
	a										b
		0.5										0.05
		0.4						Centroid on simulated data Correlation on simulated data		0.045								Centroid Correlation
												0.04
	Displacement Error in pixel	0 0.1 0.2 0.3									Standard Error of the Mean	0.015 0.02 0.025 0.03 0.035
		-0.1
												0.01
		-0.2										0.005
		-0.3	1	1.2	1.4	1.6	SBR	1.8	2	2.2	2.4	0 1	1.2	1.4	1.6	SBR	1.8	2	2.2	2.4
	FIGURE 2.19: Numerical comparison of displacement measurement between centroid-based and
	correlation-based methods. (a)

Simulations are run on fifty images for each SBR value, with SBR corresponding to previous experimental ones; results are presented on Figure 2.19.

94 3.4 Comparing Extended Source wavefront sensing to Descanned wavefront sens- ing in scattering conditions

Contents 3.1 Setup design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.1.1 Technical specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.1.2 Excitation path: optical simulation . . . . . . . . . . . . . . . . . . . . . . 76 3.1.3 Detection path design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.1.4 Control and acquisition software . . . . . . . . . . . . . . . . . . . . . . . 81 3.2 Objective and pupil conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.1 Pupil conjugation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.2 Implemented strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3 Characterisation of the AO loop on two-photon signal . . . . . . . . . . . . . . 86 3.3.1 Brief description of AO loop parameters . . . . . . . . . . . . . . . . . . 86 3.3.2 First settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.3 AO-ESSH in ideal conditions . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3.4 AO-ESSH in scattering conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.2 In non scattering conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4.3 In scattering conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.5 Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 74 Chapter 3. Implementing an extended-source based AO loop on a two-photon microscope

  Determination of the objective pupil plane for AO: setup. L1-4: Lenses, BS: 50:50 Beamsplitter, P1: Back pupil plane of the objective to study, Obj1, conjugated to P2, on the pupillooking camera Cam2. Obj2: high NA objective to generate a point source at the focus of the Obj1.This point is imaged on the camera Cam1.

	z					BS	Cam1
	x	y
			L3
	Obj1+Obj2		Obj1	P1		L4
	Laser diode	Obj2	ϴ col	ϴ ill Obj2	Cam2 Obj1: Obj2:	P2 20X NA = 1 60X NA = 1.2
	L1	L2
	FIGURE 3.7:
							4 = 50
	mm) to reach Cam2.

). The back focal plane of Obj2 (Olympus, UPlanSApo, 60x, NA = 1.2) is filled by the illumination from a laser diode, expanded by the relay L1 -l2 (L1, f ′ 1 = -25 mm and L2, f ′ 2 = 400 mm). The sample plane, i.e the diffraction-limited spot, is imaged on Cam1 through L3 ( f ′ 3 = 180 mm) while the back pupil plane of Obj1 is relayed by L4 ( f ′

92

  Chapter 3. Implementing an extended-source based AO loop on a two-photon microscope

	PSF	Theoretical	Th * Bead	Experimental
	FW HM x,y (µm)	0.38	0.55	0.60 ± 0.05

TABLE 3 .

 3

1: PSF measurement

TABLE 3 .

 3

2: Distribution of RMS values across the FOV for the three considered AO signals

TABLE 3 .

 3

		ESSH (nm)	Descanned SH (nm)
	Focus	217.4	224.7
	SA3	141.0	140.8

3: Aberration coefficient retrieval for both methods in depth.

  .5d. For example,

	a	No AO		b		AO, 36 modes		c	AO, 42 modes
	50 µm	641	Intensity	65535
	350				350			350
	-350 nm			-350 nm			-350 nm
	RMS 206 nm	d		RMS 29 nm				RMS 12 nm
			50000	No AO 36 modes 42 modes
			40000
		Gray values (a.u)	30000 20000
			10000
			0	3	6	9	12	15	18
						Distance (µm)

TABLE 4 .

 4 2: Aberrations content before AO and after AO, according to the number of corrected modes. Values in nm. Accuracy: lambda/10 ≈ 10 nm

	3.3.1). A trade off is thus determined.
	Aberration	Ast 0°Ast 45°Coma 0°Coma 90°SA3
	Before AO	-10	14	32	80	193
	AO, 36 modes	6	0	0	-4	-2
	AO, 42 modes	2	6	0	3	0

TABLE 4 .

 4 3: Aberrations content before AO in nm.

	Aberration	Ast 0°Ast 45°Coma 0°Coma 90°SA3
	Value (nm)	-4	35	81	106	224

  Chapter 4. Applications of Extended Source AO-2PEF to biological samples Scanning the spot with 0.2 µm per pixel, an exposure duration of 2 µs per pixel over a FOV of 150 x 150 µm², allows us to deliver more power to the sample without increasing the laser power.

	Aberration	Ast 0°Ast 45°Coma 0°Coma 90°SA3
	Value (nm)	-54	41	31	17	47
	TABLE 4.4: Aberrations content before AO. Ast: Astigmatism. SA3: Third order Spherical Aberra-
		tion at 200 µm, in the dual-labelled fixed brain slice

Table 1 . Measured scattering lengths l s of fixed brain slices at different wavelengths with the ESSH analyzer compared to literature reported values.

 1

	Sample	λ [nm]	l s [µm]	Reference
	Fixed brain	775	55.2	Kobat et al. [35]
		1280	106.4
	Fixed slice	515 615 805	38 ± 2 46 ± 2 77 ± 5	Results

Chapter 1. In-depth functional neuroimaging

mm 20 µm

5

Acknowledgements

Acknowledgments. We thank Marine Depp, Ombeline Hoa, Alexandra Bogicevic, Rémi Carminati and Thomas Pons for technical assistance and fruitful discussion. This work was supported by grants from France's Agence Nationale de la Recherche (INOVAO, ANR-18-CE19-0002). This work has received also support under the program « Investissements d'Avenir » launched by the French Government and implemented by ANR with the references ANR-10-LABX-54 (MEMOLIFE), ANR-10-IDEX-0001-02 (Université PSL) and ANR-10-INSB-04-01 (France-BioImaging Infrastructure).

Funding. Agence Nationale de la Recherche (ANR-10-IDEX-0001-02 PSL* Université Paris, ANR-10-LABX-54 MEMOLIFE, ANR-18-CE19-0002 INOVAO).

Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Chapter 4

Applications of Extended Source AO-2PEF to biological samples Contents

We first note an increase in intensity of the soma on the insets between before and after the loop: the retrieved focalisation quality of the excitation beam through the sample increases the amount of fluorescence signal emitted as expected.

A second point concerns the quality of the correction: in this typical area, that presents an almost ideal neuron distribution for ESSH wavefront sensing, a wavefront RMS of 24 nm is obtained after 4-5 loop iterations for 36 corrected modes and is stable over time. As precised earlier, both coverslip and brain aberration contributions are corrected for here.

The amount of third order aberrations before the loop is in the Table 4 induced by the coverslip is the main aberration to be corrected (after the loop, its value drops to 10 nm). The coverslip also induces 5th and 7th order of Spherical Aberration (respectively 41 nm and -34 nm).

Therefore, ESSH wavefront sensor can efficiently measure and correct for sample-induced aberrations on a two-photon excited signal, and with photon flux comparable to in vivo ones. The closed-loop system guarantees us to reach an optimized result.

b. ESSH-based AO for enhanced two-photon imaging

Now that we saw that Extended Source based AO could be achieved on this type of signals, we interest ourselves on the signal and resolution gain provided to the full-field image.

The full-field lateral dimension of 150 µm is set according to the typical isoplanetic patch referenced in the literature [START_REF] Liu | Observing the cell in its native state: Imaging subcellular dynamics in multicellular organisms[END_REF][START_REF] Ji | Adaptive optical fluorescence microscopy[END_REF]; it is larger than the measured one from our setup itself, so we expect a variation of the correction quality in the FOV (see Section 3.3.3b.). The pixel lateral size is set to 0.3 µm, smaller than the FWHM of the lateral PSF (see Section 3.3.3a.) and guaranteeing two and a half pixels per PSF; it satisfies Shannon's criterion. This pixel size is good a compromise between the FOV and the total acquisition time for one frame, as well as with the maximal scanning speed of the galvanometer mirrors.

In this section, the typical laser power delivered at the sample plane is 8 mW for the PMT acquisition, for an excitation wavelength of 920 nm. For the AO correction, the power is increased to 33 mW, with a pixel size of 0.2 x 0.2 µm² (see Appendix B for the available power in our setup).

On Figure 4.3 are represented the maximum intensity projection of 24-µm stacks, recorded before and after the correction loop, 180 µm below the coverslip. On these images, we observe a gain in both signal and resolution. We note a signal enhancement of a factor 2 to 2.4 on the fine details, highlighted by the intensity profiles on Figure 4.3b; the signal enhancement on the soma is less important (increase of approximately a factor 1.6). Moreover, some fine details that could not be resolved without AO became visible; the resulting signal enhancement revealed two distinct peaks around 27 µm on the red plot, but also several other structures on the purple plot.

In this example, the wavefront RMS after the loop is still 55 nm, higher than expected. The remaining aberrations are astigmatisms (Ast 0°of -21 nm and Ast 45°of 29 nm) which can Chapter 4. Applications of Extended Source AO-2PEF to biological samples Finally, I added the Descanned Shark-Hartmann wavefront sensor to experimentally compare the in-depth performances of our extended source wavefront sensing approach to the stateof-the art measurement on the descanned fluorescence signal. If this part had already been engaged at the beginning of my third year, the numerous issues raised on the way forced me to modify the setup lately and reported these experiments to the final weeks of my thesis. The conclusions brought by these experiments are exciting for the following: the combination of descanned wavefront sensing with correlation-based algorithms, more resilient to scattering, could constitute an even more pertinent approach for AO in two-photon microscopy and indepth imaging.

Perspectives

My project has just been taken over by Alice Guillaume-Manca who started her thesis this October. Her aim is to take this setup to in vivo and then to functional neuroimaging in the mouse.

The transition from the fixed brain slice to the anaesthetised and then awake mouse will undoubtedly be a source of multiple challenges to be solved, from the instrumental point of view as well as from the analysis of the signals or the preparation of the sample. In a second phase, the challenge will be to push back the limits in terms of imaging depth. This work will be carried out in close collaboration with Thomas Pons at the LPEM, who has the expertise in synthesizing near infra-red Quantum Dots for their use in biology: these probes will be injected into the mouse brain via retro-orbital injection, or intravenous drip to create microvascular-based patterns for wavefront sensing. If this type of method has already been done by Kleinfeld et al [START_REF] Liu | Direct wavefront sensing enables functional imaging of infragranular axons and spines[END_REF], the near infrared dye injected forced them to switch the excitation wavelength as well as suffering from the poor brightness of the available dyes at these wavelength. The brightness of the QD, and the ability to coat them with a stable zwitterionic polymer surface chemistry, as done by the LPEM group in [START_REF] Pons | In Vivo Imaging of Single Tumor Cells in Fast-Flowing Bloodstream Using Near-Infrared Quantum Dots and Time-Gated Imaging[END_REF], opens the way to wavefront sensing labelling further in the infrared without the constraint of excitation wavelength tuning, and suffering from the lack of brightness of the dyes in this region of the spectrum.

Above this project, the Extended Source Shack Hartmann wavefront sensor has been implemented on an Axially Swept Light Sheet Microscope by Mathias Mercier, for Drosophila brain and zebrafish larvae imaging.

Both projects are to be continued at the Institut des Sciences Moléculaires d'Orsay, at Université Paris Saclay, always in close collaboration with the Institut de Biologie de l'Ecole Normale Supérieure and Imagine Optic.

Appendix A

Neuron culture preparation

Neuronal cultures containing a mVenus fluorescent tag were generated by an ex-utero electroporation in the sensory cortex of E15 murin embryos. A drop of DNA pH1SCV2-shRNA-control (0.7 µg/µl) was injected with a capillary in one hemisphere of the sensory cortex of the embryo and then electroporated. This region was then extracted from the whole brain and dissected in a medium containing 10% HBSS, 5% glucose and 2% HEPES. The piece of remaining cortical hemisphere was digested by chemical and mechanical digestions, first, under the action of papain for 15 min at 37•C and then under pipet up and down movements, to obtain individual neurons. The neurons were plated onto a 18-mm coverslip coated with 80 µg/ml polyornithine in 12 well plates, at a density of 180,000 cells/well in MEM, 10% HS, 1% glutamax and 1% sodium pyruvate 100 mM and incubated for 2 hours at 37•C, 5% CO2. When the neurons were completely adherent on the coverslip, the medium was replaced by Neurobasal, 2% B27, 1% glutamax and antibiotics. One third of the medium was changed every 5-6 days.

Appendix B

Laser power

For a lasing wavelength of 920 nm, used in Chapter 4, we summarize the typical laser power delivered in the sample plane according to the angle of the retarding plate positioned at the output of the femtosecond laser.

Angle (°)

Laser

Abstract

Optical microscopy has emerged as a major tool in neuroscience as it allows to perform structural imaging and functional activity recordings within large volume of tissues with subcellular resolution. In particular, non-linear microscopy allows the interrogation of neuronal activity in mammalian brains but remains limited in depth as scattering and optical aberrations, due to refractive index inhomogeneities, limit resolution and deteriorate signal intensity. Indeed, aberrations encountered on the excitation path significantly affect imaging quality as depth increases. To overcome these issues, Adaptive Optics (AO) strategies have been implemented to retrieve the microscope imaging quality while addressing important imaging depths, enabling e.g. high resolution two-photon functional imaging in the mouse infragranular cortical layers. Whilst one AO strategy implemented in non-linear microscopy relies on a sensorless configuration, and thus on a time-consuming iterative process, a second approach, based on direct wavefront sensing using centroid-based Shack-Hartmann wavefront sensors, proved its efficiency in vivo. However, this method fails at large depths because of the strong scattering of the emitted fluorescence used for wavefront sensing. A method for direct wavefront sensing more resilient to scattering of the fluorescence emission would therefore improve the use of AO in optical microscopy. This work proposes an alternative method of direct wavefront measurement that takes advantage of existing labelling methods of biological samples and that relies on the cross-correlation of images of an extended source obtained through a microlens array. This Extended-Source Shack-Hartmann wavefront sensor (ESSH), previously coupled to Light Sheet Fluorescence Microscopy for neuroimaging in the adult Drosophila brain, is confronted here to higher scattering conditions. We show that this method allows quantitative aberration measurements through fixed brain slices, up to four times their scattering length and demonstrate its stronger resilience to scattering compared to the current centroid-based approach. We present its implementation on a two-photon microscope within a closed-loop configuration for in-depth neuroimaging in the mouse and compare its performances in scattering media to the classical descanned centroid approach. Finally, we present imaging results on fixed mouse brain slices, in depth.