Javier Morgado Brajones

Javier Morgado

Brajones Opto

OPTO-MECHA-SPIM : Dynamic and interactive imaging of mechanobiological phenomena in tissues

In recent years, tissue mimics (TMs) such as microtissues, spheroids, and organoid cultures have become increasingly important in life-science research, as they provide a physiologically relevant environment for cell growth, tissue morphogenesis, and stem cell differentiation. Selective Plane Illumination Microscopy (SPIM) is one of the most prominent microscopy modalities for three-dimensional tissue imaging, and a sine qua non tool to understand cell biology in TMs. However, while SPIM is regarded as a very powerful tool for TM imaging, optical microscopy suffers from certain limitations when imaging 3D samples. Indeed, scattering, absorption and First and foremost, I would like to thank my thesis supervisor, Corinne Lorenzo, for her guidance and support through these three years. Were it not for her, this thesis would have certainly not been possible. Many thanks also to my co-supervisor, Christophe Coudret, as well as the whole IDeAS team, who, even though my visits to the enigmatic world of chemistry were few and sparse, made me feel right at home. I would like to also thank the former members of our team, especially Martine Cazales and Théo Liu, whose assistance during the early stages of my stay

optical aberrations limit the depth at which useful imaging can be done, typically no more than 100 µm. Adaptive Optics (AO) is a technique capable of improving image quality at depth by correcting the optical aberrations introduced by the sample which is seeing increasing use in fluorescence microscopes.

For this thesis, I have incorporated a wavefront sensor AO scheme to SPIM, able to correct aberrations in optically thick TMs such as multi-cellular tumor spheroids (MCTS). Due to the low amount of light produced by non-linear guide stars such as the one used in our system, a custom high-sensitivity Shack-Hartmann wavefront sensor (SHWFS) was developed for our needs. In this work, I characterize the performance of this SHWFS and the ability of our system to correct aberration in various conditions, including inside TMs. I show unprecedented image quality improvements for in-depth imaging of MCTS, in regard of high-frequency detail and resolution. This allowed us to identify biologically relevant features at depths inaccessible to conventional SPIM.

Up-converting nanoparticles (UCNP) are rare-earth based particles that are able to undergo photon up-conversion when illuminated, emitting light of a shorter wavelength than that of the illumination. Guide stars made from UCNP are especially attractive due to the possibility of them being excited in the near-infrared while emitting visible light, reducing photodamage produced by the illumination light. The viability of using UCNP as guide stars in biological samples in explored in this thesis.

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Résumé

Les modèles en 3D occupent une place de plus en plus importante en biologie cellulaire car ils offrent de nombreux avantages pour la compréhension des processus physiopathologiques. La microscopie à feuille de lumière et plus particulièrement, le Selective Plane Illumination Microscope (SPIM), un de ses variants, représente un outil de choix pour observer et imager avec une haute résolution spatio-temporelle ces structures assimilées à des tissus (TM). Bien que le SPIM est un outil puissant, il souffre de phénomènes tels que la diffusion, l'absorption et les aberrations optiques qui limitent la profondeur à laquelle une imagerie utile peut être réalisée. L'optique adaptative (AO), de plus en plus utilisée dans le domaine de la microscopie, est une technique capable d'améliorer la qualité d'imagerie en profondeur en corrigeant les aberrations optiques introduites par l'échantillon.

Pour ce travail de thèse, j'ai implémenté dans un SPIM un système d'AO capable de corriger les aberrations dans les TMs optiquement épais tels que les sphéroïdes tumoraux multicellulaires (MCTS). Un capteur de front d'onde de type Shack-Hartmann (SHWFS) à haute sensibilité a été développé à façon afin de permettre la reconstruction du front d'onde à partir de la faible quantité de lumière produite par l'étoile guide non linéaire utilisé dans notre système. Au cours de ce travail, j'ai tout d'abord caractérisé les performances de ce SHWFS et évalué la capacité du système à corriger les aberrations dans diverses conditions, y compris à l'intérieur de TMs. J'ai observé des améliorations sans précédent de la qualité d'image en profondeur des MCTS, en termes de détails et de résolution haute-fréquence. Grâce à ces corrections, j'ai pu mettre en évidence des évènements biologiques tels que des mitoses non visibles sans corrections.

Les « up converting » nanoparticules (UCNP) sont des particules de terres rares capables d'émettre une lumière d'une longueur d'onde plus courte que celle de l'éclairage lorsqu'elles sont illuminées. Les étoiles guides en UCNP sont particulièrement intéressantes en raison de la possibilité qu'elles soient excitées dans l'infrarouge proche tout en émettant de la lumière visible, ce qui réduit le photodommage produit par l'éclairage. La possibilité d'utiliser des UCNP comme étoiles guides dans des échantillons biologiques a été explorée dans cette thèse. Table 1:

INTRODUCTION

Fluorescence microscopy is one of the most powerful and versatile tools for the observation and quantification of biological processes in a wide range of spatial and temporal scales: from single molecules to whole organisms, from microseconds to weeks. Its wide range of applications, combined with the ease of use of most instruments, has made fluorescence microscopes ubiquitous across biology laboratories.

The widefield fluorescence microscope is perhaps the most common of such instruments. While it is enormously valuable for imaging of thin samples such as monolayered cultures or thin slices of tissue, most three-dimensional information about the sample is lost due to its lack of optical sectioning. This inability to extract three-dimensional information from living samples has for a long time relegated the study of biological phenomena to their observation in monolayered cells cultures. Two-dimensional cultures allow the study of many biological processes, but they do not fully take into account the effects of the threedimensional organization of cells and extracellular matrix within tissues.

During the last decades, three-dimensional model systems of a large number of human organs and tissues have been developed to allow the study of processes sensitive to the three-dimensional structure of the tissue, such as organogenesis or disease spread. These tissue mimics provide a more physiologically relevant microenvironment for the study of tissue and organ development, presenting similar differentiation patterns as their in-vivo counterpart as well as a similar 3D network of cell-cell and cell-matrix interactions.

Still, adequate optical instruments are required to visualize three-dimensional processes in tissue mimics. Over the last decade, light sheet fluorescence microscopy (LSFM) has established itself as the tool of choice for three-dimensional imaging of tissue mimics. In the simplest LSFM variant, the selective plane illumination microscope (SPIM), a thin plane of the sample is illuminated by means of a light sheet placed in the focal plane of the detection objective. This way, three-dimensional imaging of the sample is possible by simply moving it through the light sheet. Since only a single plane is illuminated and all the emitted light is collected, photodamage and photobleaching are kept to a minimum, whereas acquisition frequency can be as high as tens of volumes per second. However, in-depth imaging of biological tissue is limited by absorption, scattering and optical aberrations arising from the interaction of light and the sample. These issues typically limit the maximum depth of subcellular-resolution imaging of tissue mimics to no more than a few dozen microns. Traditional methods aiming to improve in-depth imaging of such optically thick samples, such as sample clearing, are typically focused on changing the optical properties of the sample, which often involves the killing of the specimen and therefore have limited convenience for in-vivo studies.

Adaptive optics (AO) is a technique, first developed for astronomy, able to correct the aberrations induced by the sample and the optical system by modulating the wavefront of the acquired optical signal. In recent years, AO has seen increased use in different microscopy modalities. AO is a natural fit for LSFM, as it can substantially increase the maximum useful imaging depth, while providing improved resolution and contrast. Still, the use of AO in optically thick samples remains challenging, due to the low signal-to-noise ratio caused by the limited amount of ballistic light and the high background noise produced by scattered photons. This thesis is concerned with the improvement of three-dimensional imaging of TMs. For this, I have implemented an adaptive SPIM, able to correct the aberrations produced by both the sample and the optical system. I have fully characterized the microscope and the performance of the AO system, and demonstrated its use on TM imaging. Finally, I have explored the possibility of using up-converting nanoparticles, capable of producing anti-Stokes emission, as guide stars for AO correction.

Synopsis

The thesis is structured as follows:

• Chapter 2 presents the theoretical background behind tissue mimic imaging and adaptive optics, as well as the state-of-the-art of relevant topics. It starts with the fundamentals of image formation and the theoretical treatment of aberrations, followed by an introduction to tissue mimics and their optical properties. Next, I provide the basics of LSFM and its use on TMs. The last section offers a detailed explanation of adaptive optics, as well as the state-of-the-art of its use on microscopy.

• Chapter 3 refers to the implementation and characterization of our adaptive SPIM. A detailed description of the different subsystems and their implementation is provided. The performance of the AO corrections is determined, with focus on low photon flux conditions.

• Chapter 4 is devoted to TM imaging. Using multi-cellular tumor spheroids, I show improvements in image quality after AO correction. A detailed analysis of these improvements is provided.

• Chapter 5 discusses the use of up-converting nanoparticles (UCNP) as guide stars in TMs. First, I study the toxicity of UCNP and their effects in TM development. Then, the viability of UCNP for wavefront measurement is determined.

• Chapter 6 presents the conclusions of this thesis, as well as the perspectives for our research.

Nature of light

It is universally accepted that light behaves both as a particle, the photon, and as a wave. Since light is an electromagnetic wave, its behaviour can be directly derived from Maxwell's equations [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. The value of the electromagnetic field 𝑈𝑈 at any point 𝒓𝒓 may be represented as:

𝑈𝑈(𝒓𝒓) = 𝐴𝐴𝐴𝐴 𝑖𝑖(𝑘𝑘𝑘𝑘-𝜔𝜔𝜔𝜔) , (2-1)
where A is the amplitude of the wave, 𝑘𝑘 is its wave number, and 𝜔𝜔 is its angular frequency. 𝑘𝑘 and 𝜔𝜔 are related to the wavelength 𝜆𝜆 and frequency 𝜈𝜈 as follows:

𝑘𝑘 = 2𝜋𝜋 𝜆𝜆 , 𝜔𝜔 = 2𝜋𝜋𝜈𝜈.
In addition, the speed of light in vacuum 𝑐𝑐 is given by

𝑐𝑐 = 𝜆𝜆𝜈𝜈, (2-2)
and takes value 𝑐𝑐 = 2.998 × 10 8 𝑚𝑚/𝑠𝑠. The refractive index 𝑛𝑛 is the ratio between 𝑐𝑐 and the speed of light in a medium 𝑣𝑣:

𝑛𝑛 = 𝑐𝑐 𝑣𝑣 . ( 2-3)
The term

Φ = 𝑘𝑘𝑘𝑘 -𝜔𝜔𝜔𝜔 (2-4)
is the phase of the wave. A surface of constant phase is called a wavefront. The waves in which the wavefront takes the form of a sphere (such as those produced by a point emitter) are said to be spherical, and can be approximated to a plane wave after a long enough propagation.

When light forms an image, the magnitude of the signal is directly correlated to the intensity (power per unit area) of the electromagnetic wave at that point. The intensity or irradiance 𝐼𝐼 of a monochromatic wave propagating through a nonmagnetic medium is given by:

𝐼𝐼 = 𝑐𝑐𝑛𝑛𝜀𝜀 0 2 |𝑈𝑈(𝒓𝒓)| 2 , (2-5)
where 𝜀𝜀 0 is the vacuum permittivity. In calculations, the term 𝑐𝑐𝑛𝑛𝜀𝜀 0 /2 is usually omitted for simplicity.

Kirchhoff's diffraction theory

Image formation cannot be considered without taking into account the diffracting nature of the light. One of the firsts attempts to explain the propagation of light as a wave was made by Christiaan Huygens. According to Huygens, every element of the wavefront acts as the centre of secondary disturbances that give way to spherical wavelets, in such a way that the wavefront at a later time is the envelope of those secondary wavelets [START_REF] Huygens | Traite de la lumiere. Où sont expliquées les causes de ce qui luy arrive dans la reflexion, & dans la refraction[END_REF]. This principle, commonly referred as Huygens' construction, can explain propagation, refraction and reflection of light. However, diffraction phenomena cannot be adequately explained by it.

Fresnel later complemented Huygens' construction with the introduction of mutual interference between the secondary wavelets. This interference accounts for Huygens' postulate that the secondary wavelets only travel forward and can adequately explain diffraction. This combination of Huygens' construction with

Fresnel's ideas of interference receives the name of the Huygens-Fresnel principle, and is the foundation of physical optics.

A sound mathematical basis for this principle would have to wait for half a decade. From the wave equation and Green's second identity, Kirchhoff's derived the Fresnel-Kirchhoff integral theorem, of which the Huygens-Fresnel principle is a particular case. Let us consider a monochromatic wave originating from a point source 𝑃𝑃 0

propagating through an aperture in a plane screen, and let 𝑃𝑃 be the point at which the electromagnetic field is to be evaluated. The main result of the Fresnel-Kirchhoff integral theorem states that:

𝑈𝑈(𝑃𝑃) = - 𝑖𝑖𝐴𝐴 2𝜆𝜆 � 𝐴𝐴 𝑖𝑖𝑘𝑘(𝑘𝑘+𝑠𝑠) 𝑘𝑘𝑠𝑠 [cos(𝑛𝑛, 𝑘𝑘) -cos(𝑛𝑛, 𝑠𝑠)] d𝑆𝑆, 𝑄𝑄 (2-6)
where 𝑄𝑄 is the aperture, 𝑘𝑘 and 𝑠𝑠 are, respectively, the distances from 𝑃𝑃 0 and 𝑃𝑃 to a surface element the aperture, and (𝑛𝑛, 𝑘𝑘) and (𝑛𝑛, 𝑠𝑠) are the angles of 𝑘𝑘 and 𝑠𝑠 with the normal of the surface.

If 𝑃𝑃 sufficiently far away, the distance to the aperture can be approximated as a constant 𝑘𝑘 ≈ 𝑘𝑘 0 , and the previous expression can be simplified as:

𝑈𝑈(𝑃𝑃) = - 𝑖𝑖 2𝜆𝜆 𝐴𝐴𝐴𝐴 𝑖𝑖𝑘𝑘𝑘𝑘 0 𝑘𝑘 0 � 𝐴𝐴 𝑖𝑖𝑘𝑘𝑠𝑠 𝑠𝑠 𝑄𝑄 (1 + cos 𝜒𝜒) d𝑆𝑆, (2-7)
where 𝜒𝜒 = 𝜋𝜋 -(𝑘𝑘 0 , 𝑠𝑠).

Imaging with aberrations

Let us consider a rotationally symmetrical optical system with a point source of monochromatic light at 𝑃𝑃 0 , and let us assume that 𝑃𝑃 0 forms an image at 𝑃𝑃 1 * . We can take a cartesian system of axes, with origin at 𝑃𝑃 1 * , z-axis along 𝐶𝐶𝑃𝑃 1 * , where 𝐶𝐶 is the centre of the exit pupil, and y-axis along the meridional plane. Let 𝑊𝑊 be any wavefront that goes through 𝐶𝐶, and let 𝑄𝑄 and 𝑄𝑄′ be the points in which a given ray intersects the gaussian reference sphere and the wavefront 𝑊𝑊, respectively. The aberration function Φ(𝑄𝑄) represents the optical distance along the ray between the points 𝑄𝑄 and 𝑄𝑄′ and we define it positive as shown in Figure 2-2. Let 𝑠𝑠 be the distance between 𝑄𝑄 and an arbitrary point 𝑃𝑃, and let 𝑅𝑅 be the radius of the gaussian reference sphere. According to (2-7) and assuming small angles (cos 𝜒𝜒 ≈ 1):

𝑈𝑈(𝑃𝑃) = - 𝑖𝑖 𝜆𝜆 𝐴𝐴𝐴𝐴 -𝑖𝑖𝑘𝑘𝑖𝑖 𝑅𝑅 � 𝐴𝐴 𝑖𝑖𝑘𝑘(Φ+𝑠𝑠) 𝑠𝑠 d𝑆𝑆 𝐶𝐶 , (2-8)
where 𝐶𝐶 is the region of the reference sphere that approximately fills the exit pupil.

In polar coordinates, this equation becomes:

𝑈𝑈(𝑃𝑃) = - 𝑖𝑖 𝜆𝜆 𝐴𝐴𝑎𝑎 2 𝑅𝑅 2 𝐴𝐴 𝑖𝑖� 𝑖𝑖 𝑎𝑎 � 2 𝑢𝑢 � 1 0 � 𝐴𝐴 𝑖𝑖[𝑘𝑘Φ-𝑣𝑣𝑣𝑣 cos(𝜃𝜃-𝜑𝜑)-1 2 𝑢𝑢𝜌𝜌 2 ] 2𝜋𝜋 0 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌, (2-9)
where the circular pupil has radius 𝑎𝑎 and polar coordinates 𝜌𝜌 and 𝜌𝜌, the image plane has polar coordinates 𝑘𝑘 and 𝜑𝜑, and

𝑢𝑢 = 2𝜋𝜋 𝜆𝜆 � 𝑎𝑎 𝑅𝑅 � 2 𝑧𝑧, 𝑣𝑣 = 2𝜋𝜋 𝜆𝜆 � 𝑎𝑎 𝑅𝑅 � �𝑥𝑥 2 + 𝑦𝑦 2 (2-10)
are normalized coordinates in the focal plane. According to (2-5) and ignoring the constant factor, the intensity at 𝑃𝑃 is:

𝐼𝐼(𝑃𝑃) = |𝑈𝑈(𝑃𝑃)| 2 = � Aa 2 𝜆𝜆𝑅𝑅 2 � 2 �� 1 0 � 𝐴𝐴 𝑖𝑖[𝑘𝑘Φ-𝑣𝑣𝑣𝑣 cos(𝜃𝜃-𝜑𝜑)-1 2 𝑢𝑢𝜌𝜌 2 ] 2𝜋𝜋 0 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌� 2 .
(2-11)

If no aberrations are present, the intensity is then a maximum at the Gaussian image point (𝑃𝑃 1 * ; 𝑘𝑘 = 0): -12) The ratio between the intensity at the Gaussian image point of an aberrated and an unaberrated beam is called the Strehl Ratio (SR):

𝐼𝐼 * = 𝐼𝐼 Φ=0 (𝑃𝑃 1 * ) = 𝜋𝜋 2 � 𝐴𝐴𝑎𝑎 2 𝜆𝜆𝑅𝑅 2 � 2 . ( 2
𝑆𝑆𝑅𝑅 = 𝐼𝐼(𝑃𝑃 1 * ) 𝐼𝐼 * = 1 𝜋𝜋 2 �� 1 0 � 𝐴𝐴 𝑖𝑖[𝑘𝑘Φ-𝑣𝑣𝑣𝑣 cos(𝜃𝜃-𝜑𝜑)-1 2 𝑢𝑢𝜌𝜌 2 ] 2𝜋𝜋 0 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌� 2 .
(2-13)

Let Φ′ be a wavefront defined as:

Φ ′ = Φ + 𝑐𝑐 3 𝜌𝜌 2 + 𝑐𝑐 2 𝜌𝜌 cos 𝜌𝜌 + 𝑐𝑐 1 𝜌𝜌 sin 𝜌𝜌 + 𝑐𝑐 0 ,
where 𝑐𝑐 0 , … , 𝑐𝑐 3 are constants. It can be shown (ref) that there exists a point 𝑃𝑃 where:

𝑆𝑆𝑅𝑅 = 𝐼𝐼(𝑃𝑃 1 * ) 𝐼𝐼 * = 𝐼𝐼(𝑃𝑃) 𝐼𝐼 * . (2-14)
That is, the addition of an aberration in the form 𝑐𝑐 3 𝜌𝜌 2 + 𝑐𝑐 2 𝜌𝜌 cos 𝜌𝜌 + 𝑐𝑐 1 𝜌𝜌 sin 𝜌𝜌 + 𝑐𝑐 0 does not change the three-dimensional intensity distribution near focus, but causes the displacement of the distribution as a whole. This is equivalent to a shift of the Gaussian image point, i.e. by displacing our reference point, we can remove the quadratic and linear terms of the aberration. It will later be shown that an aberration of this form is a linear combination of piston, tilt and defocus aberrations. Therefore, no intensity improvements can be obtained from correcting these modes of aberration. If the remaining aberrations at the new Gaussian image point are represented by Φ 𝑣𝑣 , the Strehl Ratio can be written as:

𝑆𝑆𝑅𝑅 = 1 𝜋𝜋 2 �� 1 0 � 𝐴𝐴 𝑖𝑖𝑘𝑘Φ 𝑝𝑝 2𝜋𝜋 0 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌� 2 .
(2-15)

This important result shows how aberrations affect the degradation of the propagation. If the beam is unaberrated, Φ 𝑣𝑣 = 0 and 𝑆𝑆𝑅𝑅 = 1, and thus the intensity at focus will be diffraction limited. In the presence of any aberration at all, Φ 𝑣𝑣 > 0 and the Strehl Ratio will be less than unity [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF], [START_REF] Tyson | Principles of Adaptive Optics[END_REF].

The Maréchal approximation

When the aberrations are small, it is possible to express the intensity in terms of the mean square value of the beam aberration. Let Φ 𝑃𝑃 be the aberration referred to a reference sphere centred on a point 𝑃𝑃 in the image region. The exponential function in (2-15) can be expanded as a Taylor series:

𝑆𝑆𝑅𝑅 = 1 𝜋𝜋 2 �� 1 0 � 𝐴𝐴 𝑖𝑖𝑘𝑘Φ 𝑝𝑝 2𝜋𝜋 0 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌� 2 = 1 𝜋𝜋 2 �� 1 0 � (1 + 𝑖𝑖𝑘𝑘Φ 𝑣𝑣 + 1 2 �𝑖𝑖𝑘𝑘Φ 𝑣𝑣 � 2 +. . . ) 2𝜋𝜋 0 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌� 2 . (2-16)
Let 𝑥𝑥̅ denote the average value of any variable 𝑥𝑥 over the pupil:

𝑥𝑥̅ = ∫ 1 0 ∫ 𝑥𝑥𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 2𝜋𝜋 0 ∫ 1 0 ∫ 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 2𝜋𝜋 0 = 1 𝜋𝜋 � 1 0 � 𝑥𝑥𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 2𝜋𝜋 0 . (2-17)
If we assume that the aberrations are small, we can truncate the Taylor series expansion at the second order term. Using identity [START_REF] Huygens | Traite de la lumiere. Où sont expliquées les causes de ce qui luy arrive dans la reflexion, & dans la refraction[END_REF][START_REF] Tyson | Principles of Adaptive Optics[END_REF][START_REF] Zernike | Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode[END_REF][START_REF] Bhatia | On the circle polynomials of Zernike and related orthogonal sets[END_REF][START_REF] Noll | Zernike polynomials and atmospheric turbulence*[END_REF][START_REF] Lakshminarayanan | Zernike polynomials: a guide[END_REF][START_REF] Nijboer | The diffraction theory of aberrations[END_REF][START_REF] Andilla | Imaging tissue-mimic with light sheet microscopy: A comparative guideline[END_REF][START_REF] Shamir | Three-dimensional organotypic culture: experimental models of mammalian biology and disease[END_REF][START_REF] Jacques | Optical properties of biological tissues: a review[END_REF][START_REF]Optical-Thermal Response of Laser-Irradiated Tissue[END_REF][START_REF] Ntziachristos | Going deeper than microscopy: the optical imaging frontier in biology[END_REF][START_REF] Nichols | Click-Assembled, Oxygen-Sensing Nanoconjugates for Depth-Resolved, Near-Infrared Imaging in a 3 D Cancer Model[END_REF][START_REF] Xu | Fractal mechanisms of light scattering in biological tissue and cells[END_REF][START_REF] Schwertner | Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry[END_REF][START_REF] Sutherland | Cell and Environment Interactions in Tumor Microregions: The Multicell Spheroid Model[END_REF] in the truncated series:

𝑆𝑆𝑅𝑅 ≈ �1 + 𝑖𝑖𝑘𝑘Φ 𝑣𝑣 ���� - 1 2 𝑘𝑘 2 Φ 𝑣𝑣 2 ���� � 2 = 1 -� 2𝜋𝜋 𝜆𝜆 � 2 �Φ 𝑣𝑣 2 ���� -�Φ 𝑣𝑣 ���� � 2 � . (2-18)
The term in brackets corresponds to the variance of the wavefront:

�ΔΦ 𝑣𝑣 � 2 = ∫ 1 0 ∫ �Φ 𝑣𝑣 -Φ 𝑣𝑣 ���� � 2 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 2𝜋𝜋 0 ∫ 1 0 ∫ 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 2𝜋𝜋 0 = Φ 𝑣𝑣 2 ���� -�Φ 𝑣𝑣 ���� � 2 , (2-19)
and, thus, we can express the Strehl Ratio as:

𝑆𝑆𝑅𝑅 ≈ 1 -� 2𝜋𝜋 𝜆𝜆 � 2 �ΔΦ 𝑣𝑣 � 2 ≈ e -� 2π λ � 2 �ΔΦ 𝑝𝑝 � 2 (2-20)
This formula was first derived by Maréchal and provides a simple but powerful method of evaluating the quality of the beam for small aberrations by taking into account only the variance of the wavefront. The standard deviation of the wavefront ΔΦ 𝑣𝑣 is often called the root mean square (RMS) wavefront error, phase error or simply wavefront error and it is usually the main magnitude to be evaluated when assessing the quality of an aberrated instrument.

The Maréchal criterion establishes that an image is diffraction-limited when ΔΦ 𝑣𝑣 < 𝜆𝜆/14, which roughly corresponds to 𝑆𝑆𝑅𝑅 > 0.8. This is one of the most commonly used performance criteria, and will be the one employed throughout this thesis.

Modal representation of the wavefront

The function Φ has been historically represented in many different ways. Of special importance are the representations as a complete set of orthogonal polynomials over the unit circle. These representations are denominated modal representations, of which the most prominent representatives are the Zernike and Lukosz polynomials. Since most contemporary works employ Zernike polynomials, the explanation about the Lukosz representation will be omitted.

The Zernike polynomials

This set of polynomials was first introduced by Frits Zernike in 1934 1 [START_REF] Zernike | Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode[END_REF]. They are polynomials in two variables that are usually expressed in polar coordinates. In their real form they are defined as [START_REF] Noll | Zernike polynomials and atmospheric turbulence*[END_REF]:

� 𝑍𝑍 𝑛𝑛 𝑙𝑙 (𝜌𝜌, 𝜌𝜌) = �2(𝑛𝑛 + 1)𝑅𝑅 𝑛𝑛 𝑙𝑙 cos 𝑙𝑙𝜌𝜌 𝑍𝑍 𝑛𝑛 -𝑙𝑙 (𝜌𝜌, 𝜌𝜌) = �2(𝑛𝑛 + 1)𝑅𝑅 𝑛𝑛 𝑙𝑙 sin 𝑙𝑙𝜌𝜌 𝑚𝑚 ≠ 0 𝑍𝑍 𝑛𝑛 0 (𝜌𝜌, 𝜌𝜌) = √𝑛𝑛 + 1𝑅𝑅 𝑛𝑛 0 𝑚𝑚 = 0 (2-21)
where 𝑛𝑛 ≥ 𝑙𝑙, 𝑛𝑛 -𝑙𝑙 = even, and 𝑛𝑛 and 𝑙𝑙 are non-negative integers. Let 𝑙𝑙 = |𝑚𝑚|, the radial polynomials 𝑅𝑅 𝑛𝑛 𝑚𝑚 are defined as:

𝑅𝑅 𝑛𝑛 𝑚𝑚 (𝜌𝜌) = � (-1) 𝑠𝑠 (𝑛𝑛 -𝑠𝑠)! 𝑠𝑠! � 𝑛𝑛 + |𝑚𝑚| 2 -𝑠𝑠� ! � 𝑛𝑛 + |𝑚𝑚| 2 -2� ! 𝜌𝜌 𝑛𝑛-2𝑠𝑠 𝑛𝑛-|𝑚𝑚| 2 𝑠𝑠=0 , (2-22)
and they are normalized so that for all 𝑛𝑛 and 𝑚𝑚:

1 The derivation of the Zernike polynomials involves a lengthy and rather abstract departure into the world of group theory, and it is not considered of interest for this dissertation. The original derivation can be found in [START_REF] Zernike | Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode[END_REF] (in German), while an alternative approach can be found in [START_REF] Bhatia | On the circle polynomials of Zernike and related orthogonal sets[END_REF].

𝑅𝑅 𝑛𝑛 𝑚𝑚 (1) = 1.

(2-23) The polynomials 𝑍𝑍 𝑛𝑛 𝑚𝑚 (𝜌𝜌, 𝜌𝜌) are said to be even when 𝑚𝑚 ≥ 0 and odd when 𝑚𝑚 < 0.

The Zernike polynomials form an orthonormal basis over the unit circle [START_REF] Lakshminarayanan | Zernike polynomials: a guide[END_REF], i.e. any circular wavefront can be expressed as a linear combination of Zernike polynomials that is unique. The orthogonality in the radial part reads:

� 𝑅𝑅 𝑛𝑛 𝑚𝑚 (𝜌𝜌)𝑅𝑅 𝑛𝑛 ′ 𝑚𝑚 (𝜌𝜌)𝜌𝜌𝜌𝜌𝜌𝜌 1 0 = 1 2(𝑛𝑛 + 1) 𝛿𝛿 𝑛𝑛𝑛𝑛 ′ , (2-24)
where 𝛿𝛿 𝑖𝑖𝑖𝑖 is the Kronecker delta. The orthogonality in the angular part can be proven by examining the four possible products:

� cos(|𝑚𝑚|𝜌𝜌) cos(|𝑚𝑚 ′ |𝜌𝜌) 𝜌𝜌𝜌𝜌 2𝜋𝜋 0 = 𝜋𝜋(1 + 𝛿𝛿 𝑚𝑚0 )𝛿𝛿 |𝑚𝑚|�𝑚𝑚 ′ � , � sin(|𝑚𝑚|𝜌𝜌) sin(|𝑚𝑚 ′ |𝜌𝜌) 𝜌𝜌𝜌𝜌 2𝜋𝜋 0 = 𝜋𝜋𝛿𝛿 |𝑚𝑚|�𝑚𝑚 ′ � , � cos(|𝑚𝑚|𝜌𝜌) sin(|𝑚𝑚 ′ |𝜌𝜌) 𝜌𝜌𝜌𝜌 2𝜋𝜋 0 = 0, � sin(|𝑚𝑚|𝜌𝜌) cos(|𝑚𝑚 ′ |𝜌𝜌) 𝜌𝜌𝜌𝜌 2𝜋𝜋 0 = 0.
(2-25)

The fulfilment of the condition for orthonormality follows from (2-24) and (2-25):

∫ ∫ 𝑍𝑍 𝑛𝑛 𝑚𝑚 (𝜌𝜌, 𝜌𝜌)𝑍𝑍 𝑛𝑛 ′ 𝑚𝑚 ′ (𝜌𝜌, 𝜌𝜌)𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 2𝜋𝜋 0 1 0 ∫ ∫ 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 2𝜋𝜋 0 1 0 = 𝛿𝛿 𝑛𝑛𝑛𝑛 ′ 𝛿𝛿 𝑚𝑚𝑚𝑚 ′ . (2-26)
The aberration function Φ can be expressed as [START_REF] Nijboer | The diffraction theory of aberrations[END_REF]:

Φ(𝜌𝜌, 𝜌𝜌) = � � 𝐴𝐴 𝑛𝑛𝑚𝑚 𝑛𝑛 𝑚𝑚=-𝑛𝑛 ∞ 𝑛𝑛=0 𝑍𝑍 𝑛𝑛 𝑚𝑚 (𝜌𝜌, 𝜌𝜌). (2-27)
The coefficients 𝐴𝐴 𝑛𝑛𝑚𝑚 are called the Zernike coefficients and constitute a simple way to evaluate the contribution of each type of aberration to the total phase of the wavefront, as well as a way to calculate the total wavefront variance. If all the 𝐴𝐴 𝑛𝑛𝑚𝑚 are known, the wavefront error can be easily calculated from the orthonormality relations (2-24) and (2-25):

(ΔΦ) 2 = � � 𝐴𝐴 𝑛𝑛𝑚𝑚 2 2(𝑛𝑛 + 1) 𝑛𝑛 𝑚𝑚=-𝑛𝑛 ∞ 𝑛𝑛=1
.

(2-28)

With this, we conclude the introduction of the mathematical tools required to work with aberrations. Since the mathematical derivations have been purposely kept to a minimum, this evidently does not provide a complete view of aberrations from a physical optics point of view. The interested reader is suggested to consult [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF] and [START_REF] Tyson | Principles of Adaptive Optics[END_REF] for further context. In the next section, the properties of a kind of highly aberrating biological samples, the tissue mimics, are discussed.

Tissue mimics

Cell cultures are one of the most powerful tools biologist have at their disposition to study issues such as cell growth or differentiation. Although two-dimensional monolayered cultures allow the study of many biological processes, they do not fully take into account the effects of the three-dimensional organization of cells and extracellular matrix within tissues. Since mammalian tissue is often inaccessible to invivo observation, the study of processes sensitive to the three-dimensional structure of the tissue such as organogenesis or tumour development has historically relied on model systems.

Tissue mimics (TMs) are 3D models of biological tissue that provide a more physiologically relevant microenvironment for the study of tissue and organ development. They typically consist of roughly spherical clusters of cells with a size of a few hundred microns that present similar differentiation patterns as their in-vivo counterpart, as well as a similar 3D network of cell-cell and cell-matrix interactions [START_REF] Andilla | Imaging tissue-mimic with light sheet microscopy: A comparative guideline[END_REF].

Over the last decades there has been a steady development of the TM field, and TMs of most tissues and organs (presenting partial or total function of the mimicked tissues) have been developed [START_REF] Shamir | Three-dimensional organotypic culture: experimental models of mammalian biology and disease[END_REF].

Optical properties of tissue mimics

The optical properties of a TM are crucial when attempting to image it, as they govern the behaviour of the optical signal entering and leaving the tissue. For this reason, samples that interact heavily with light must be regarded as yet another element of the optical system, and must be taken into account when designing an experiment.

In summary, the optical properties of a tissue can be described in terms of the refractive index of the tissue 𝑛𝑛, the absorption coefficient 𝜇𝜇 𝑎𝑎 , the scattering coefficient 𝜇𝜇 𝑠𝑠 and the scattering function 𝑝𝑝(𝜌𝜌, 𝜓𝜓), where 𝜌𝜌 is the deflection angle of scatter and 𝜓𝜓 is the azimuthal angle of scatter [START_REF] Jacques | Optical properties of biological tissues: a review[END_REF].

Photons in a turbid medium such as a TM may be scattered or absorbed, in such a way that, when a photon propagates over an infinitesimal distance 𝜌𝜌𝑠𝑠, the probability of scattering and absorption are 𝜇𝜇 𝑠𝑠 d𝑠𝑠 and 𝜇𝜇 𝑎𝑎 d𝑠𝑠, respectively. We can define another useful property of tissues: the mean free path (MFP) of a photon between scattering or absorption events. It is defined as:

𝑀𝑀𝑀𝑀𝑃𝑃 𝑠𝑠 = 1 𝜇𝜇 𝑠𝑠 , 𝑀𝑀𝑀𝑀𝑃𝑃 𝑎𝑎 = 1 𝜇𝜇 𝑎𝑎 .
(2-29)

Hence, the probability that a photon is scattered (similar for absorption) in some distance 𝑥𝑥 < 𝑠𝑠 is given by the probability distribution function [START_REF]Optical-Thermal Response of Laser-Irradiated Tissue[END_REF]:

𝑀𝑀 𝑠𝑠 (𝑠𝑠) = 1 -𝐴𝐴 -𝜇𝜇 𝑠𝑠 𝑠𝑠 . (2-30)
Since both absorption and scattering affect imaging, sometimes the total attenuation (or extinction) coefficient 𝜇𝜇 𝑘𝑘 is used to account for their combined effect.

It is defined as -31) and it determines the possibility of a photon undergoing any of a scattering or absorption event. In most tissues 𝜇𝜇 𝑠𝑠 ≫ 𝜇𝜇 𝑎𝑎 so one may approximate 𝜇𝜇 𝑘𝑘 ≈ 𝜇𝜇 𝑠𝑠 [START_REF] Ntziachristos | Going deeper than microscopy: the optical imaging frontier in biology[END_REF].

𝜇𝜇 𝑘𝑘 = 𝜇𝜇 𝑠𝑠 + 𝜇𝜇 𝑎𝑎 . ( 2
When the tissue is illuminated by a collimated beam with intensity 𝐼𝐼 0 , the amount of light entering the tissue is reduced by several different mechanisms: a portion of the light is reflected at its surface, while another portion is scattered and absorbed by the tissue. If 𝜇𝜇 𝑘𝑘 is uniform, the total intensity at any point 𝑥𝑥 inside the tissue along the path of the beam is then given by Beer-Lambert's law:

𝐼𝐼(𝑧𝑧) = 𝐼𝐼 0 (1 -𝑅𝑅)𝐴𝐴 -𝜇𝜇 𝑟𝑟 𝑥𝑥 , (2-32)
where 𝐼𝐼 0 is the intensity of the incoming beam, 𝑅𝑅 is the reflectance of the surface, and 𝜇𝜇 𝑘𝑘 is the attenuation coefficient. As seen above, the attenuation of a beam inside a tissue depends as much on the depth inside the tissue as it depends on the attenuation coefficient. For this reason, the thickness of a tissue 𝜌𝜌 is not a good metric to evaluate ease of imaging. Instead, the optical thickness or optical depth (𝜏𝜏) is usually preferred, and is defined as:

𝜏𝜏 = 𝜇𝜇 𝑘𝑘 𝜌𝜌. (2-33)
Single scattering is a complex problem, and many different scattering functions 𝑝𝑝(𝜌𝜌, 𝜓𝜓) have been proposed to describe it. When many scattering events are present, like in TMs, the random orientation of scattering structures causes the contribution of the azimuthal angle of scatter 𝜓𝜓 to average out and, thus, the dependence of 𝜓𝜓 can be ignored. Multiple scattering also means that the contribution of 𝜌𝜌 can be described by an average parameter:

𝑔𝑔 = ⟨cos 𝜌𝜌⟩, (2-34)
called the anisotropy of scatter. 𝑔𝑔 accounts for the relative forward versus backward direction of scattering.

The situation in which scattering is much more probable than absorption (𝜇𝜇 𝑎𝑎 ≪ 𝜇𝜇 𝑠𝑠 ) is called the diffusion regime. In this case, it usually is useful to use the reduced scattering coefficient and the transport mean free path, defined as:

𝜇𝜇 𝑠𝑠 ′ = 𝜇𝜇 𝑠𝑠 (1 -𝑔𝑔), (2-35)
𝑇𝑇𝑀𝑀𝑀𝑀𝑃𝑃 = 1 𝜇𝜇 𝑠𝑠 = 𝑀𝑀𝑀𝑀𝑃𝑃 1 -𝑔𝑔 . (2-36)
The propagation of a beam between 0 and 1 MFP inside a tissue is largely ballistic. Scattering increases photon diffusion up to 1 TMFP, after which almost no ballistic light is present [START_REF] Ntziachristos | Going deeper than microscopy: the optical imaging frontier in biology[END_REF].

In mammalian tissue, microstructures include organelles (0.2 -0.5 µm), mitochondria (1 -4 µm), nuclei (3 -10 µm) and whole cells (10 -30 µm). Most structures have a size comparable or greater than the wavelength of the light used, which situates TM imaging in both the Rayleigh and Mie scattering regime.

Scattering and absorption depend on the wavelength of the signal. Notably, scattering diminishes with increasing wavelength, while the absorption profile depends on the medium. For water, there is a near-infrared optical imaging window, from 700 nm to 1400 nm, in which absorption is minimal and scattering is significantly reduced. This window is commonly exploited for optimal imaging of thick samples [START_REF] Nichols | Click-Assembled, Oxygen-Sensing Nanoconjugates for Depth-Resolved, Near-Infrared Imaging in a 3 D Cancer Model[END_REF].

Soft tissues present refractive indexes around 𝑛𝑛 = 1.35, with variations of about 0.04 -0.10 within the tissue [START_REF] Xu | Fractal mechanisms of light scattering in biological tissue and cells[END_REF]. Since TMs are commonly immersed in water for imaging (𝑛𝑛 = 1.33), the refractive index difference between the tissue and the surrounding medium, as well as its variance within the tissue, produces optical aberrations.

Aberrations are a major cause of image degradation on thick biological samples.

Typically, low order aberrations such as astigmatism, coma and spherical are the most prevalent [START_REF] Schwertner | Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry[END_REF]. Since refractive index vary irregularly across the specimen, sample-induced aberrations are not constant across the field of view (FOV). The regions in which the aberrations can be considered as constant for a given purpose are said to be isoplanatic. The mean isoplanatic patch size of a sample depends on its heterogeneity, being smaller for highly heterogeneous tissue. One of such highly heterogeneous TMs of particular interest are the multi-cellular tumor spheroids, widely used to study cancer development, which present significant challenges for modern optical imaging.

Multi-Cellular Tumor Spheroids

Multi-cellular tumour spheroids (MCTS) are three-dimensional cultures of tumour cells grown in a roughly spherical shape. This kind of TM has proven invaluable in the study of cancer progression and treatment as they present many of the features of in-vivo tumours. Spheroids can be grown from established tumour cell lines or even directly from primary tumours

The outer layers of an MCTS are rapidly dividing, while the proportion of quiescent cells increases in the inner regions. Due to the lack of vascularization, metabolite gradients are significative between the inner and the outer layers of an MCTS, and given enough growth it will eventually develop a necrotic core as a consequence of hypoxia and build-up of metabolic waste [START_REF] Sutherland | Cell and Environment Interactions in Tumor Microregions: The Multicell Spheroid Model[END_REF]. MCTS are notoriously difficult to image due to their heterogeneity. Andilla et al.

report scattering coefficients for fixed MCTS made from the HCT116 human colon carcinoma cancer cell line to range from 5,3 mm -1 for a 532 nm wavelength to 2,5 mm -1 for 1040 nm. This corresponds to an MFP of 0.19 mm and 0.40 mm, respectively, meaning the chance of multiple scattering within a medium-sized 400 µm spheroid is likely for visible light [START_REF] Andilla | Imaging tissue-mimic with light sheet microscopy: A comparative guideline[END_REF].

Regarding healthy colon submucous tissue, Bashkatov et al. report scattering coefficients of approximately 10 to 7.5 mm -1 for 500 -1000 nm wavelengths, and 32 to 21 mm -1 in the case of mucous tissue [START_REF] Bashkatov | Optical properties of human colon tissues in the 350 -2500 nm spectral range[END_REF]. The optical properties of non-healthy tissue are similar. According to Wei et al., the scattering coefficient of adenomatous human colon mucosa and submucosa are 22.3 mm -1 for a 532 nm wavelength, while that of healthy mucosa/submucosa is 20.8 mm -1 [START_REF] Wei | Determination of optical properties of normal and adenomatous human colon tissues in vitro using integrating sphere techniques[END_REF]. They also report absorption coefficients of 0.52 mm -1 and 0.33 mm -1 for adenomatous and healthy tissue, respectively.

Hence, while healthy and tumoral human colon tissue seem to have similar optical properties, fixed MCTS are considerably less opaque, with a scattering coefficient four times lower.

MCTS derived from other human tissues have also been characterized. Work done by Hargrave et al. provides scattering coefficients ranging from 90 to 40 mm -1 for wavelength in the 650 -1000 nm range in fixed MCTS made from the POC human small-cell lung cancer cell line [START_REF] Hargrave | Optical properties of multicellular tumour spheroids[END_REF], as well as absorption coefficients of 0.2 to 0.1 mm -1 for the same wavelengths. This is again in dissonance with the properties of invivo healthy lung tissue, for which Beek et al. provide scattering coefficients of 20 to 35 mm -1 for a 632.8 nm wavelength, depending of lung volume [START_REF] Beek | The optical properties of lung as a function of respiration[END_REF]. Is worth of note that, as opposed to colon MCTS, lung cancer fixed MCTS are more difficult to image than the in-vivo healthy tissue. Unfortunately, to the knowledge of the author no data is present in the literature about the optical properties of human lung tumours, so it is not possible to assess whether or not optical properties of MCTS differ from those of tumoral tissue.

So far, we have only discussed properties of fixed MCTS. While it is well known that fresh MCTS are more difficult to image, this fact has attracted little attention by the scientific community. Indeed, the only research to date that compares the optical properties of fixed and fresh spheroids is the previously mentioned work of

Hargrave and co-workers regarding lung MCTS [START_REF] Hargrave | Optical properties of multicellular tumour spheroids[END_REF]. There, they report scattering coefficients of approximately 160 to 90 mm -1 for wavelengths in the 650 -1000 nm range for fresh MCTS, i.e., 30% to 100% higher than those their fixed counterparts.

Absorption coefficients range from 0.4 mm -1 in the visible to less than 0.01 mm -1 in the NIR. MCTS made from the HCT116 cell line have a refractive index between 1.35 and 1.39 [START_REF] Hari | Optical coherence tomography complements confocal microscopy for investigation of multicellular tumour spheroids[END_REF], which is noticeably different from that of water (𝑛𝑛 = 1.33). Optical imaging of MCTS suffers from major aberration induced by the highly heterogeneous refractive index across the spheroid. Coma, astigmatism and trefoil are the most prominent aberrations in MCTS [START_REF] Jorand | Deep and Clear Optical Imaging of Thick Inhomogeneous Samples[END_REF], [START_REF] Masson | High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM[END_REF].

The highly scattering and aberrating nature of MCTS is of special relevance when choosing an optical imaging method, especially given the limitations imposed by their intrinsic three-dimensional nature. The following section provides an overview of the available alternatives for MCTS imaging, as well as an in-depth dive into our tool of choice, light sheet fluorescence microscopy.

Light sheet Fluorescence Microscopy

Tissue mimics and microscopy

Due to its unmatched ability to determine their morphology, optical imaging is essential in the study of TM behaviour [START_REF] Rios | Imaging organoids: a bright future ahead[END_REF]. However, imaging of thick samples such as TMs is a non-trivial task due to scattering-and aberration-related image degradation.

A straightforward way to acquire 3D images while avoiding such degradation is physical sectioning of the sample into slices thin enough to be imaged without loss of information. This technique, combined with immunostaining, has been successfully applied in the study of brain organoids [START_REF] Lancaster | Cerebral organoids model human brain development and microcephaly[END_REF]. Still, while providing useful insight, the destruction of the sample has obvious shortcomings in live imaging.

Since physical slicing is out of the question for work with live samples, researchers have usually favoured optical sectioning for TM imaging. Confocal microscopes are the most popular instruments to allow precise optical sectioning. By using a pinhole, out-of-focus light can be blocked, providing subcellular optical sectioning over suitable FOVs [START_REF] Jamieson | Derivation of a robust mouse mammary organoid system for studying tissue dynamics[END_REF], [START_REF] Leary | Quantitative Live-Cell Confocal Imaging of 3D Spheroids in a High-Throughput Format[END_REF]. Since most of the light is blocked out, the sample has to be exposed to high amounts of light, leading to high levels of phototoxicity and photobleaching. This issue makes confocal microscopy a less-than-ideal approach for sensitive live imaging.

Two-photon microscopy is another well-known method for high-resolution 3D imaging. It works by exploiting the low probability of occurrence of two-photon excitation event in a fluorophore, which requires very high intensities to reliably take place. By focusing a near-infrared laser into a fluorophore, these high intensities will only be present near its focus, thus providing a way to precisely confine excitation in three-dimensions. The laser can be scanned across the sample to provide thin plane images deep into the sample [START_REF] Dekkers | High-resolution 3D imaging of fixed and cleared organoids[END_REF]. Since NIR light is weakly absorbed by biological tissue and no fluorescence light is discarded, two-photon excitation is a much gentler approach than confocal microscopy for living samples.

In recent years, another TM imaging method, the light sheet fluorescence microscopy, has become increasingly popular owing to its negligible photodamage and fast acquisition times, which make it an ideal tool for long term live imaging of thick samples. A detailed (though by no means exhaustive) overview of LSFM and its different modalities is presented below.

Basics of LSFM

LSFM is a powerful technique for three-dimensional live imaging of biological samples that permits imaging large specimens over long periods of time. Typically, LSFM involves illuminating the sample with a beam, shaped into a thin sheet (the light sheet), in such a way that only a plane with thickness equivalent to that of the light sheet is excited. The fluorescence light emitted from this plane is collected by a perpendicular objective and registered by a camera as the sample moves through the light sheet, producing a three-dimensional image of the sample. This selective illumination pattern has the advantage that no out-of-focus light has to be filtered, yielding high signal-to-noise ratio (SNR) and precise optical sectioning, while requiring no pinhole schemes. As the whole light emitted from the sample is registered, LSFM can image with low acquisition times, resulting in a very low phototoxicity for the sample [START_REF] Power | A guide to light-sheet fluorescence microscopy for multiscale imaging[END_REF]. When imaging with LSFM, a zebrafish embryo is exposed to as much as 5000 times less energy than with confocal microscopy [START_REF] Keller | Reconstruction of Zebrafish Early Embryonic Development by Scanned Light Sheet Microscopy[END_REF]. The first light sheet device to come to existence was the Orthogonal-Plane Fluorescence Optical Sectioning microscope (OPFOS) developed by Voie et al. [START_REF] Voie | Orthogonal-plane fluorescence optical sectioning: Three-dimensional imaging of macroscopic biological specimens[END_REF].

This instrument was able to provide optical sectioning by using a light sheet, but was limited to an almost macroscopic resolution. In 2002, Fuchs et al. developed their Thin Light Sheet Microscope (TSLM), which was capable of resolving single bacteria [START_REF] Fuchs | Thin laser light sheet microscope for microbial oceanography[END_REF]. However, TSLM does not provide optical sectioning, as its intended use is the identification of microorganisms in sea water as opposed to 3D imaging of biological tissue. It was in 2004 that Huisken et al. developed the Selective Plane Illumination Microscope (SPIM) [START_REF] Huisken | Optical Sectioning Deep Inside Live Embryos by Selective Plane Illumination Microscopy[END_REF], employing it for in-vivo imaging of GFP-labelled Medaka fish Oryzias latipes and embryogenesis of the fruit fly Drosophila melanogaster.

Embryogenesis remains one of the most popular applications of LSFM.

SPIM was the first light sheet microscope to achieve wide awareness and provided the cornerstone that set the LSFM field in motion. Since then, the push for ever higher resolution and speeds has driven the scientific community to develop a plethora of LSFM schemes for imaging in any resolution range, from the macroscopic to super-resolution.

Resolution and speed in LSFM

Due to the decoupling of the illumination and detection paths, LSFM resolution and field of view depend on the NA of both the illumination and detection objectives.

High NA illumination objectives will produce a thinner light sheet, improving axial resolution, while a high NA detection objective provides high lateral resolution.

However, a common issue in high-resolution instruments is the mechanical constraints limiting objective placement, as high NA objectives are typically bulky and have short working distances. Since they collide with each other when placing them at short distances, it is usually impossible to arrange a system in which the sample is in the focal plane of both orthogonal high NA objectives.

Many schemes have been developed to overcome this problem. A simple solution is to use a low NA illumination objective, permitting longer work distances and more distance between the objectives [START_REF] Zanacchi | Live-cell 3D super-resolution imaging in thick biological samples[END_REF]. Obviously, this approach limits optical sectioning and axial resolution. In single molecule detection microscopes, this loss of depth information can be compensated by means of depth-dependent astigmatism [START_REF] Huang | Three-dimensional superresolution imaging by stochastic optical reconstruction microscopy[END_REF], but in many applications where axial resolution is critical an alternative solution has to be sought. Custom-made objectives can reduce the spatial constraints to an extent [START_REF] Chhetri | Whole-animal functional and developmental imaging with isotropic spatial resolution[END_REF], [START_REF] Chen | Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution[END_REF], but their cost can be prohibitive for most microscopists.

Despite the mechanical constraints, super-resolution can be achieved in LSFM using off-the-self components by modifying the geometry of the system. Gebhardt et al. developed the first of such systems, the reflected light sheet microscope (RLSM), using two opposed objectives with a 45° mirror cantilever between them [START_REF] Gebhardt | Single-molecule imaging of transcription factor binding to DNA in live mammalian cells[END_REF]. The cantilever is used to place it in the focal plane, in a similar way to RLSM [START_REF] Galland | 3D high-and super-resolution imaging using single-objective SPIM[END_REF]. In this approach, the sample placement is fixed, and the light sheet is horizontally scanned through the tilted mirror, displacing it axially through the sample. To keep the light sheet in the focal plane, the objective lens is also translated. This entails a horizontal displacement of the light sheet waist, which must be compensated by a tunable lens.

Another single objective setup is the Axial-Plane Optical Microscope (APOM) developed by Li et al. APOM uses the objective to creates a light sheet illuminating a plane parallel to the optical axis while a tilted mirror further down the optical path rotates the image over the axis into a lateral image that is registered by a camera.

This way, APOM can simultaneously produce a widefield image and an axial SPIM image of the sample using an NA=1.4 oil-immersion objective [START_REF] Li | Axial Plane Optical Microscopy[END_REF]. The fact that all the required optics for the axial image are remote makes this design easy to implement in a conventional widefield microscope as a module. LSFM systems have been pushed not only to the limits of spatial resolution, but also to the limits of imaging frequency. Since the light sheet must be placed in the focal plane of the objective, sample translation is usually required for volume imaging. This limits the amount of volumes that can be imaged per second, since fast shacking of the sample will destroy it. By keeping the sample still and scanning the light sheet through it, fast imaging can be achieved. However, the light sheet must be kept in the focal plane through the process in order to allow imaging.

The first of such approach to use this strategy was the Objective-Coupled Planar Illumination microscopy (COPI), developed by Holekamp et al. [START_REF] Holekamp | Fast Three-Dimensional Fluorescence Imaging of Activity in Neural Populations by Objective-Coupled Planar Illumination Microscopy[END_REF]. In this microscope, both the light sheet and the objective move by means of a piezoelectric manipulator, keeping the light sheet always on focus. Speeds of 5 volumes per second were achieved for calcium imaging of mouse vomeronasal organ.

Fahrbach et al. employed an electrically tunable lens to maintain focus without moving the objective [START_REF] Fahrbach | Rapid 3D light-sheet microscopy with a tunable lens[END_REF] . This way, they achieved imaging at up to 100 volumes per second. Another high-speed system is the previously mentioned SCAPE design, which uses a descanning mirror to keep focus and allow speeds of up to 20 volumes per second [START_REF] Bouchard | Swept confocally-aligned planar excitation (SCAPE) microscopy for high-speed volumetric imaging of behaving organisms[END_REF].

Olarte et al. avoid the use moving parts by employing a phase mask coupled with image deconvolution to extend the depth of field of the objective so the light sheet is always focused. In this way, they achieve imaging of up to 70 volumes per second [START_REF] Olarte | Decoupled illumination detection in light sheet microscopy for fast volumetric imaging[END_REF].

Ultimately, the speed of these systems is limited not by their optics, but by camera framerates. Hence, further improvement in LSFM speed is unavoidably linked to the development of faster commercial cameras.

The light sheet

As the characteristic element of LSFM, the light sheet plays an essential role in not only optical sectioning and axial resolution, but also in imaging depth. The ideal light sheet should be as thin as possible at the waist, while maintaining a thin profile over a large FOV. It is evident, however, that these two properties are mutually exclusive: a thin waist is only achievable by using a high NA objective, but this means a more acute divergence of the light sheet, which translates to a smaller effective FOV.

In addition, light sheet degradation is a concern for LSFM. Inhomogeneities in the sample cause aberration and scattering, resulting in a broadening and loss of quality of the light sheet as it passes through the sample. This leads to a compromised optical sectioning and loss of axial resolution, as well as the apparition of artifacts.

The last issue that needs to be considered for light sheet engineering is phototoxicity. One of the most attractive properties of LSFM is the need for low exposures of the sample, and it is generally taken as given. However, the required exposure and peak intensity delivered to the sample depends greatly on the modality of illumination and it can be far from harmless. Hence, the choice of the illumination system should always be addressed with consideration of not only the resolution and penetration needs, but also the effects of imaging on the biological system under study.

Light sheet properties

In this section, the most relevant properties of a Gaussian light sheet (the one used in our system) will be explained. The axial resolution of a LSFM system is given by both the NA of the detection objective and the thickness of the light sheet. Since the light sheet is focused by an objective, it has a hyperbolic profile, which means that there is one point of the beam at which its thickness is minimal. This point, placed at a focal distance of the objective, is called the waist, and its thickness 𝑤𝑤 0 is perhaps the most relevant property of a light sheet. The beam waist thickness is given by:

𝑤𝑤 0 = 𝑛𝑛𝜆𝜆 𝜋𝜋𝜋𝜋𝐴𝐴 , (2-37)
where 𝑛𝑛 is the refractive index of the medium, 𝜆𝜆 is the wavelength of the beam and 𝜋𝜋𝐴𝐴 is the numerical aperture of the objective. In most applications, the light sheet is thinner than the depth of field of the detection objective, meaning that the axial resolution 𝑅𝑅 𝑎𝑎𝑥𝑥 depends on the optical sectioning capability of the system and can be approximated as the thickness of the light sheet, i.e. 𝑅𝑅 𝑎𝑎𝑥𝑥 = 2𝑤𝑤 0 . It is evident from (2-37) and (2-38) that the FOV of a system can be increased by the user of a lower NA objective. However, a lower NA will reduce the optical sectioning capability of the system, since the waist thickness will also increase. This places a fundamental limit on Gaussian light sheets, as a large FOV and high sectioning capability cannot be achieved at the same time.

Illumination modalities

The simplest and most common kind of light sheet is the one created by a

Gaussian beam passing through a cylindrical lens, commonly referred as the Gaussian light sheet and first used by Voie et al. [START_REF] Voie | Orthogonal-plane fluorescence optical sectioning: Three-dimensional imaging of macroscopic biological specimens[END_REF]. Cylindrical lenses focus light in just one dimension, resulting in a plane of light that is then focused by the illumination objective and introduced into the sample. Though easy to create and practical, static

Gaussian light sheets' optical sectioning is typically not better than 1µm. In addition, shadow artifacts are created as the light sheet encounters opaque obstacles in its path, creating a stripe pattern characteristic of LSFM images. Another issue with light sheets in general is the fact that scattering will broaden the light sheet as it passes through the sample, compromising optical sectioning on the far side of it.

Huisken and Stainier addressed both issues of shadow artifacts and asymmetrical optical sectioning over the FOV with their multidirectional selective plane illumination microscope (mSPIM) [START_REF] Huisken | Even fluorescence excitation by multidirectional selective plane illumination microscopy (mSPIM)[END_REF]. In this setup, they illuminate the sample with two opposing light sheets, reducing the effect of scattering, and pivot them vertically over a 10° arc. This way, they severely reduce the stripe artifacts, while increasing axial resolution by a factor of √2 . By imaging the sample at different angles, a well resolved 3D image can be acquired. Of course, the several acquisitions entail an increased exposure time along with its associated issues of phototoxicity and photobleaching.

MuVI-SPIM (Krzic et al. [START_REF] Krzic | Multiview light-sheet microscope for rapid in toto imaging[END_REF]) and SIMView (Tomer et al. [START_REF] Tomer | Quantitative high-speed imaging of entire developing embryos with simultaneous multiview light-sheet microscopy[END_REF]) take this approach a step further, incorporating a second opposed detection objective. This way, the need for rotation of the sample is reduced and the temporal resolution is increased by a factor of 2 over mSPIM.

Digitally scanned light sheet microscopy (DSLM) provides an alternative and more effective approach to stripe artifact reduction, as shown by Keller et al. [START_REF] Keller | Reconstruction of Zebrafish Early Embryonic Development by Scanned Light Sheet Microscopy[END_REF]. In DSLM, a Gaussian beam is rapidly scanned through the sample, creating a virtual light sheet. Since the pixel dwell time decreases, an increased peak illumination intensity is required in order to achieve a comparable SNR. It has been reported that DSLM operates in a linear photobleaching regime and this does not result in increased photodamage to the sample compared to SPIM [START_REF] Keller | Reconstruction of Zebrafish Early Embryonic Development by Scanned Light Sheet Microscopy[END_REF]. Still, there is strong evidence that this is not the general case (since photodamage is in general not linear) and peak intensity is a better indicator of photodamage than total exposure [START_REF] Chen | Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution[END_REF], [START_REF] Murray | Evaluating performance in three-dimensional fluorescence microscopy[END_REF].

Even in DSLM, Gaussian light sheets are not able to achieve subcellular resolution over large FOVs. Non-diffracting beams provide an alternative approach to this problem. Though the laws of electromagnetism (and thus, diffraction) still govern these kinds of beams, they behave as non-diffracting over a small region. As a result, they maintain an invariable profile over this region, usually several times the size of the FOV of a Gaussian beam.

The Bessel beam is probably the most common of these beams. They are usually created by focusing a Gaussian beam with an axicon lens, and their cross-section consists on a central lobe surrounded by rings of diminishing intensity. One of the most interesting properties of Bessel beams is their self-healing nature: when the inner portion of the beam encounters an obstacle, the beam reforms at a short distance after it [START_REF] Fahrbach | A line scanned light-sheet microscope with phase shaped self-reconstructing beams[END_REF]. This effect increases penetration depth in thick tissue and makes Bessel beams a powerful tool when imaging heterogeneous samples. To address the first issue, they incorporated an optical sectioning algorithm (OS-SIM, Neil et al. [START_REF] Neil | Method of obtaining optical sectioning by using structured light in a conventional microscope[END_REF]) to their system, allowing them to remove the out-of-focus light at the cost of greatly increased phototoxicity. However, the strict validity of the reconstructed images resulting from the OS-SIM algorithm is unclear [START_REF] Gao | Noninvasive Imaging beyond the Diffraction Limit of 3D Dynamics in Thickly Fluorescent Specimens[END_REF].

Gao et al. improved upon this system by introducing Bessel beam superresolution structured plane illumination (Bessel plane SR-SIM) [START_REF] Gao | Noninvasive Imaging beyond the Diffraction Limit of 3D Dynamics in Thickly Fluorescent Specimens[END_REF]. This system uses an array of parallel Bessel beams, providing a 50% increase in resolution while requiring a lower peak intensity and diminishing the amount of photodamage to the sample. Modified from [START_REF] Chen | Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution[END_REF].

By coherent superposition of several beams, one can create a 2D periodic interference pattern called an optical lattice. Just like a Bessel beam, optical lattices are non-diffractive over a region, making them a suitable candidate for a light-sheet.

Moreover, optical lattices can be easily engineered in order to accommodate the needs of the system, such as lower background or higher axial resolution. where taken for each z plane, stepping the lattice light sheet a fraction of its period between images. These preliminary images could then be used to reconstruct a 3D image with resolution beyond the diffraction point (280 nm axial resolution compared to the diffraction-limited 370 nm in dithered mode).

Both modes can provide high temporal and spatial resolution, though the resolution gain of SIM requires slower acquisition and increased phototoxicity.

Overall, LLSM provides the best optical sectioning and resolution of all the LSFM variants.

The penetration problem

Volume imaging of large biological samples presents the unavoidable problem of image degradation. Due to inhomogeneities present in the sample, sample-induced aberrations, absorption and scattering degrade the light passing through it resulting in a loss of SNR and high-frequency detail which tends to worsen at greater depths. This is especially relevant in LSFM, as both the illumination and detection paths suffer from it. The light sheet will broaden when entering the sample, leading to a degraded the optical sectioning and loss of axial resolution, while the degradation in the detection path leads to a loss of lateral resolution.

Sometimes, highly scattering structures can simply be removed from the organism.

For example, dechorionation (removal of the outer opaque layers of an embryo) is a routine approach to zebrafish embryo imaging as it greatly improves light penetration while producing no substantial physiological effects [START_REF] Keller | In vivo imaging of zebrafish embryogenesis[END_REF].

When physical removal of scattering structures is not feasible, sample clearing is another popular approach, consisting in homogenising the refractive index of the sample using a chemical agent. While clearing significantly reduces scattering and aberration arising from the sample, it invariably requires killing it, making this technique incompatible with live imaging.

Since removal of aberrations and scattering at the sample level is not always feasible or even desirable, they should be lessened either by the illumination or after detection. Regarding illumination, this involves increasing penetration depth of the light sheet and reducing scattering underwent by it (and hence background fluorescence).

Both scattering and absorption are wavelength dependent. Scattering generally decreases for longer wavelengths, while the absorption profile depends on the medium. Very long wavelengths incur less scattering, but their use is not always desirable since they limit resolution and, more importantly, absorption coefficients of biological tissue increase dramatically after 1300 nm. Notably, for biological tissue there exists an optical window in the near-infrared range in which absorption is minimal and scattering is greatly reduced. This window can be employed to achieve greater penetration, generally at the cost of resolution. This method was further explored by Olarte et al. [START_REF] Olarte | Image formation by linear and nonlinear digital scanned light-sheet fluorescence microscopy with Gaussian and Bessel beam profiles[END_REF], who combined 2P-SPIM with Bessel beam illumination. This illumination method delivered a six-fold increase of the lateral field of view, but imposed a 40% loss of lateral resolution.

In the end, no matter how good the illumination system is, fluorescence light must pass through the sample to reach the objective, and this process will always degrade the image as a consequence of scattering and aberration. Fortunately, image quality can also be improved after collection. Indeed, by using adaptive optics, the aberrations induced by the sample and the system can be compensated, allowing undistorted imaging at depth.

Adaptive Optics

Introduction

Aberrations are an unavoidable effect of light passing through a heterogeneous medium. All across history, optical scientists have encountered the problems posed by them (Newton mentions the "tremulous motion of shadows" as far back as 1730 in his

Opticks [START_REF] Newton | Opticks Or, A Treatise of the Reflections, Refractions, Inflections, and Colours of Light[END_REF]), but little progress was made towards improving imaging conditions when aberrations were present until the second half of the 20th century with the apparition of adaptive optics (AO).

AO is a discipline that seeks to improve an optical signal travelling through a medium by manipulating its phase. This involves the use of a phase modulation device (like a deformable mirror or spatial light modulator) that is able to influence the wavefront of the signal in such a way that it compensates all the distortions produced by its passing through the medium. For this, three elements are usually required: a phase modulator, a wavefront sensor and a point emitter of light (the guide star).

A distinction is commonly made between active and adaptive optics. Some authors argue that everything having to do with actively controlling a beam of light is active optics, while adaptive optics is limited to the control of light in a real-time fashion by means of a feedback loop [START_REF] Tyson | Principles of Adaptive Optics[END_REF]. This distinction is often made based on other parameters such as the modes of aberration corrected [START_REF] Hardy | Active optics: A new technology for the control of light[END_REF] or the bandwidth of the system [START_REF] Wilson | Active Optics[END_REF]. For this dissertation, we will use the first definition, with the key element of the presence of a feedback loop being the difference between active and adaptive optics.

AO as a field has been developing over 7 decades since its first modern apparition in 1953, when Babcock presented the concept for an instrument based on a deformable optical element with feedback from a wavefront sensor for its use in large telescopes (that was never implemented) [START_REF] Babcock | The Possibility of Compensating Astronomical Seeing[END_REF]. Nowadays, AO is not only present in all of the largest telescopes, but it is also essential in applications such as beam shaping and optometry. In recent years, AO has been slowly finding its way into optical microscopy.

Booth and Wilson were the first to propose the use of an aberration correction device for microscopy in 2001 [START_REF] Booth | Refractive-index-mismatch induced aberrations in single-photon and two-photon microscopy and the use of aberration correction[END_REF]. A year later, they implemented AO in a confocal microscope, showing enhanced contrast and axial resolution [START_REF] Booth | Adaptive aberration correction in a confocal microscope[END_REF] and opening the way for AO utilization in other systems. Another later application of AO in confocal microscopy came from Azucena et al., who decoupled imaging and wavefront reading by using different wavelengths for each of them [START_REF] Azucena | Adaptive optics wide-field microscopy using direct wavefront sensing[END_REF].

The benefits of AO have not been limited to confocal microscopy. Sherman et al.

[70] demonstrated the use of AO in a two-photon excitation microscope imaging inert coumarin solutions. This approach was later used to improve imaging of fixed mouse embryos [START_REF] Débarre | Image-based adaptive optics for two-photon microscopy[END_REF] and in-vivo mouse brain [START_REF] Aviles-Espinosa | Measurement and correction of in vivo sample aberrations employing a nonlinear guide-star in two-photon excited fluorescence microscopy[END_REF]. Wang et al. employed an infrared guide star to further improve imaging depth of in-vivo mouse brain up to 700 µm [START_REF] Wang | Direct wavefront sensing for high-resolution in vivo imaging in scattering tissue[END_REF].

More recently, AO has been incorporated in super-resolution systems such as STED [START_REF] Gould | Adaptive optics enables 3D STED microscopy in aberrating specimens[END_REF] and STORM [START_REF] Tehrani | Point spread function optimization for STORM using adaptive optics[END_REF]. Thomas et al. developed an adaptive structured illumination microscope (SIM), in such a way that both the illumination beam and the collected imaged are shaped by a deformable mirror. This allowed them to achieve 140 nm resolution up to 35 µm deep into a C. elegans sample while substantially improving image reconstruction [START_REF] Thomas | Enhanced resolution through thick tissue with structured illumination and adaptive optics[END_REF].

LSFM is an excellent match for AO, since both the illumination and the detection paths of the microscope can be optimized [START_REF] Bourgenot | 3D adaptive optics in a light sheet microscope[END_REF], [START_REF] Wilding | Adaptive illumination based on direct wavefront sensing in a light-sheet fluorescence microscope[END_REF]. Recently, Hubert et al. employed

an adaptive LSFM to perform guide-star-less corrections in Drosophila melanogaster using extended sources for wavefront sensing [START_REF] Hubert | Adaptive optics light-sheet microscopy based on direct wavefront sensing without any guide star[END_REF].

Of especial interest to us is the use of adaptive LSFM in TM imaging. In conclusion, AO can substantially improve image quality in different optical systems when imaging thick samples, opening the door to precise imaging of dynamics that were previously unreachable by optical microscopy. But AO is a complex process. In this section, a basic overview of AO will be given, with focus on its two main steps: wavefront sensing and wavefront correction.

Wavefront sensing

In order to be able to correct an optical signal, the first logical step is to measure its quality. Ideally this sensing should be performed in real time, i.e., the sensing period should be lower than the time it takes for aberrations to change (which in some applications can be on the order of milliseconds). In microscopy, two main approaches are taken:

• Direct sensing: A wavefront sensor is used to measure the phase of the wavefront, determining it explicitly.

• Indirect sensing: No wavefront sensor is present. Instead, a figure of merit related to the magnitude of the aberrations (such as image sharpness) is used to compensate the wavefront.

Direct sensing provides the most accurate and fast corrections, but it is also the most complex and expensive approach. In this section, the physics behind direct wavefront sensing, as well as the most popular sensing device, the Shack-Hartmann wavefront sensor, will be explained.

Measuring the wavefront

The phase of a photon cannot be directly measured, and the only relevant information for wavefront sensing that we can extract from an optical signal is its intensity distribution. Hence, the whole problem of measuring the shape of the wavefront boils down to the problem of extracting phase information from intensity.

The wavefront is always continuous over the pupil, but it can be divided into a number 𝜋𝜋 of subapertures. If we can express the phase of each subaperture as a single number, then the wavefront is completely represented by the set of 𝜋𝜋 values. It is evident that a higher number of subapertures provides us with a more precise representation of the wavefront, which would be exact for 𝜋𝜋 = ∞. Moreover, each subpupil can provide more than a single value, such as tip and tilt of the wavefront over that region. In fact, the knowledge of tip and tilt over each region can be used to calculate the modal modes of the full aperture.

The Shack-Hartmann wavefront sensor (SHWFS) makes use of this fact to measure the wavefront. It was developed by R. Shack [START_REF] Platt | History and Principles of Shack-Hartmann Wavefront Sensing[END_REF], who improved upon a previous device conceived by J. Hartmann.

The SHWFS consists on a microlens array with an imaging sensor located at the focal plane of the microlenses. This way, each microlens creates a spot on the sensor, which has multiple pixels for each spot. By measuring the position of each spot on the sensor, tip and tilt can be calculated for their corresponding subapertures.

Calculating the spot centroid

The first step to calculate tilt is to locate the centroid of the Shack-Hartmann spot. Let us consider the spot produced by a subaperture of a SHWFS of size 𝜌𝜌. The spot has an intensity distribution 𝑃𝑃(𝑥𝑥, 𝑦𝑦) over a region of 𝑊𝑊 𝑣𝑣 × 𝑊𝑊 𝑣𝑣 pixels, which can be discretized as an array of the pixel intensity values 𝐼𝐼 𝑖𝑖,𝑖𝑖 . The centroid of 𝑃𝑃(𝑥𝑥, 𝑦𝑦) is located at (𝑥𝑥 0 , 𝑦𝑦 0 ). Let 𝜆𝜆 be the wavelength of the incoming wavefront. The parameter 𝜋𝜋 𝑠𝑠𝑎𝑎𝑚𝑚𝑣𝑣 = (𝜆𝜆/𝜌𝜌)/𝑝𝑝 compares the half-width of the diffraction-limited spot to the pixel angular size. The Nyquist sampling criterion is satisfied when 𝜋𝜋 𝑠𝑠𝑎𝑎𝑚𝑚𝑣𝑣 ≥ 2.

A noise-free, diffraction-limited spot will take the form of a Gaussian function:

𝑃𝑃(𝑥𝑥, 𝑦𝑦) = 𝜋𝜋 𝑣𝑣ℎ 2𝜋𝜋𝜎𝜎 2 exp �- (𝑥𝑥 -𝑥𝑥 0 ) 2 + (𝑦𝑦 -𝑦𝑦 0 ) 2 2𝜎𝜎 2 � , (2-39)
where 𝜋𝜋 𝑣𝑣ℎ is the total number of photons arriving at the sensor and 𝜎𝜎 2 is the variance of the intensity distribution. The FWHM of the spot is given by 𝑀𝑀𝑊𝑊𝐹𝐹𝑀𝑀 = 2√2𝑙𝑙𝑛𝑛2𝜎𝜎 = 2.3548𝜎𝜎 [START_REF] Thomas | Comparison of centroid computation algorithms in a Shack-Hartmann sensor: Comparison of centroid computation algorithms[END_REF].

Perhaps the simplest method to calculate the centroid of the spot is the Centre of Gravity (CoG) method. If we denote the calculated centroid by (𝑥𝑥 �, 𝑦𝑦 �) and the pixel coordinates by (𝑥𝑥 ′ , 𝑦𝑦 ′ ):

𝑥𝑥 � = ∑ 𝑥𝑥 ′ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ ∑ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ , 𝑦𝑦 � = ∑ 𝑦𝑦 ′ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ ∑ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ . (2-40)
This method is useful in near-ideal conditions, but it is imprecise in the presence of noise. It can be improved by thresholding so that the noise is filtered away. Let 𝐼𝐼 𝑇𝑇 be the threshold pixel intensity. The calculated centroid becomes:

𝑥𝑥 � = ∑ 𝑥𝑥 ′ �𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ -𝐼𝐼 𝑇𝑇 � 𝐼𝐼>𝐼𝐼 𝑇𝑇 ∑ �𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ -𝐼𝐼 𝑇𝑇 � 𝐼𝐼>𝐼𝐼 𝑇𝑇 , 𝑦𝑦 � = ∑ 𝑦𝑦 ′ �𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ -𝐼𝐼 𝑇𝑇 � 𝐼𝐼>𝐼𝐼 𝑇𝑇 ∑ �𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ -𝐼𝐼 𝑇𝑇 � 𝐼𝐼>𝐼𝐼 𝑇𝑇 . (2-41)
This is a more precise result, but still doesn't completely resolve the noise problem. Indeed, in the presence of high noise, a sizeable part of the signal will rest in pixels that fall under the threshold and it will be ignored. Instead, weights can be used so that the contribution of noisy pixels is attenuated as opposed to completely eliminated. Let 𝑤𝑤 𝑥𝑥 ′ ,𝑦𝑦 ′ be the discrete weight function. The calculated centroid coordinates are:

𝑥𝑥 � = 𝛾𝛾 𝑥𝑥 ∑ 𝑥𝑥 ′ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ 𝑤𝑤 𝑥𝑥 ′ ,𝑦𝑦 ′ ∑ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ 𝑤𝑤 𝑥𝑥 ′ ,𝑦𝑦 ′ , 𝑦𝑦 � = 𝛾𝛾 𝑦𝑦 ∑ 𝑦𝑦 ′ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ 𝑤𝑤 𝑥𝑥 ′ ,𝑦𝑦 ′ ∑ 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ 𝑤𝑤 𝑥𝑥 ′ ,𝑦𝑦 ′ , ( 2-42)
where 𝛾𝛾 𝑖𝑖 are normalization coefficients. There exist different approaches for the choice of the function 𝑤𝑤 𝑥𝑥 ′ ,𝑦𝑦 ′ , a comparison between them was performed by Thomas et al. [START_REF] Thomas | Comparison of centroid computation algorithms in a Shack-Hartmann sensor: Comparison of centroid computation algorithms[END_REF].

Spot centroid can also be calculated by a correlation algorithm. This is especially useful for use with extended light sources as their arbitrary shape may hinder centroid calculation by CoG-based methods. A correlation algorithm uses the crosscorrelation function 𝐶𝐶(𝑥𝑥, 𝑦𝑦) between the spot image 𝐼𝐼 and a template 𝑀𝑀:

𝐶𝐶(𝑥𝑥, 𝑦𝑦) = 𝐼𝐼 ⊗ 𝑀𝑀 = � 𝐼𝐼 𝑥𝑥 ′ ,𝑦𝑦 ′ 𝑀𝑀(𝑥𝑥 ′ + 𝑥𝑥, 𝑦𝑦 ′ + 𝑦𝑦) 𝑥𝑥 ′ ,𝑦𝑦 ′ . (2-43)
The spot centroid can be calculated by maximizing 𝐶𝐶(𝑥𝑥, 𝑦𝑦), which has arbitrarily high resolution. The template 𝑀𝑀 may be an analytical function, the spot of some subaperture or some function of the spots. The correlation method is especially robust against noise, but it is computationally slower than CoG-based methods.

Once the spot is properly located, information about the wavefront can be acquired from its position.

Calculating the slopes

A SHWFS reconstructs the wavefront in a zonal fashion, calculating its twodimensional slope in each subpupil from the position of the corresponding spot. Let (𝑥𝑥, 𝑦𝑦) be the centroid of a spot, and let (𝑥𝑥 * , 𝑦𝑦 * ) be the position of the same spot in the case of a flat, unaberrated wavefront. Δ𝑥𝑥 = 𝑥𝑥 -𝑥𝑥 * is then the displacement of the spot in the 𝑥𝑥 direction due to aberrations. This displacement is related to the local slope of the wavefront, tan 𝜌𝜌 𝑥𝑥 , by:

𝑠𝑠 𝑥𝑥 = tan 𝜌𝜌 𝑥𝑥 = Δ𝑥𝑥 𝑓𝑓 , (2-44)
where 𝑠𝑠 𝑥𝑥 is the average slope of the wavefront in the 𝑥𝑥 direction over the subaperture and 𝑓𝑓 is the focal distance of the microlens (Figure 2345678910111213141516). In terms of the phase Φ(𝒓𝒓):

� 𝜕𝜕Φ(𝒓𝒓) 𝜕𝜕𝑥𝑥 𝜌𝜌𝒓𝒓 𝑠𝑠𝑢𝑢𝑠𝑠𝑎𝑎𝑣𝑣𝑠𝑠𝑘𝑘𝜔𝜔𝑢𝑢𝑘𝑘𝑠𝑠 = 2𝜋𝜋𝐴𝐴Δ𝑥𝑥 𝜆𝜆𝑓𝑓 , (2-45)
where 𝐴𝐴 is the area of the subaperture. This way we calculate a set of slopes 𝑠𝑠 𝑖𝑖 𝑥𝑥 𝑗𝑗 (where {𝑥𝑥 𝑖𝑖 } = {𝑥𝑥, 𝑦𝑦}) that can be used to reconstruct the wavefront.

Reconstructing the wavefront

Each of the slopes is related to the phase at the ends of each aperture. For example, using the notation in Figure 2-17 (a):

𝑠𝑠 1 𝑥𝑥 = 𝜙𝜙 -1,1 -𝜙𝜙 0,1 , 𝑠𝑠 1 𝑦𝑦 = 𝜙𝜙 -1,1 -𝜙𝜙 -1,0 . (2-46)
Written in matrix form:

𝒔𝒔 = 𝑩𝑩𝑩𝑩, (2-47)
where 𝒔𝒔 is the slopes vector, 𝑩𝑩 is the phase vector and 𝑩𝑩 is the geometry or wavefront sensor interaction matrix. Hence, the wavefront phase can easily be calculated using the Moore-Penrose pseudoinverse of 𝑩𝑩: 𝑩𝑩 = (𝑩𝑩 𝑇𝑇 𝑩𝑩) -1 𝑩𝑩 𝑇𝑇 𝒔𝒔.

(2-48)

The geometry matrix 𝑩𝑩 depends on the spatial disposition and relations between

Calculating the Zernike modes

As seen in section 2.1.5.1, the wavefront can be exactly represented as a linear combination of Zernike polynomials. According to (2-27):

Φ(𝑥𝑥, 𝑦𝑦) = � � 𝐴𝐴 𝑛𝑛𝑚𝑚 𝑛𝑛 𝑚𝑚=-𝑛𝑛 ∞ 𝑛𝑛=0 𝑍𝑍 𝑛𝑛 𝑚𝑚 (𝑥𝑥, 𝑦𝑦), (2-49)
Where Φ(𝜌𝜌, 𝜌𝜌) is the phase of the signal, 𝑍𝑍 𝑛𝑛 𝑚𝑚 (𝜌𝜌, 𝜌𝜌) are the Zernike polynomials and 𝐴𝐴 𝑛𝑛𝑚𝑚 are the Zernike coefficients. The vector 𝑩𝑩 is a discrete representation of Φ(𝜌𝜌, 𝜌𝜌).

Our interest now is to calculate the coefficients 𝐴𝐴 𝑛𝑛𝑚𝑚 from the vector 𝑩𝑩. For simplicity and ease of reading, we will denote the polynomials and their coefficients with a single index 𝑘𝑘. If we want to calculate the first 𝐾𝐾 modes and neglect higher order aberration:

Φ(𝑥𝑥, 𝑦𝑦) = � 𝐴𝐴 𝑘𝑘 𝑍𝑍 𝑘𝑘 (𝑥𝑥, 𝑦𝑦) 𝐾𝐾 𝑘𝑘 .
(2-50)

The slopes 𝑠𝑠 𝑚𝑚 𝑥𝑥 𝑗𝑗 of each aperture 𝑚𝑚 can then be expressed as:

𝑠𝑠 𝑚𝑚 𝑥𝑥 = 𝜕𝜕Φ 𝜕𝜕𝑥𝑥 � 𝑚𝑚 = � 𝐴𝐴 𝑘𝑘 𝜕𝜕𝑍𝑍 𝑘𝑘 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑥𝑥 � 𝑚𝑚 𝐾𝐾 𝑘𝑘 , 𝑠𝑠 𝑚𝑚 𝑦𝑦 = 𝜕𝜕Φ 𝜕𝜕𝑦𝑦 � 𝑚𝑚 = � 𝐴𝐴 𝑘𝑘 𝜕𝜕𝑍𝑍 𝑘𝑘 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑦𝑦 � 𝑚𝑚 𝐾𝐾 𝑘𝑘 , (2-51)
where the subscript 𝑚𝑚 denotes evaluation at the coordinates (𝑥𝑥 𝑚𝑚 , 𝑦𝑦 𝑚𝑚 ). This can be expressed as a matrix equation:

𝒔𝒔 = 𝒁𝒁𝒁𝒁, (2-52)
where 𝒔𝒔 is the slopes vector and 𝒁𝒁 is the coefficients vector. Supposing that half of the measurements are slopes in the 𝑥𝑥 direction and half are in the 𝑦𝑦 directions, the vectors 𝒔𝒔 and 𝒁𝒁 are:

𝒔𝒔 = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 𝜕𝜕Φ 𝜕𝜕𝑥𝑥 � 1 ⋮ 𝜕𝜕Φ 𝜕𝜕𝑥𝑥 � 𝑀𝑀 2 𝜕𝜕Φ 𝜕𝜕𝑦𝑦 � 1 ⋮ 𝜕𝜕Φ 𝜕𝜕𝑦𝑦 � 𝑀𝑀 2 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ , 𝒁𝒁 = ⎝ ⎜ ⎜ ⎜ ⎛ 𝐴𝐴 1 𝐴𝐴 2 ⋮ 𝐴𝐴 3 ⎠ ⎟ ⎟ ⎟ ⎞ , (2-53)
while the matrix 𝒁𝒁 takes the form:

𝒁𝒁 = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 𝜕𝜕𝑍𝑍 1 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑥𝑥 � 1 𝜕𝜕𝑍𝑍 2 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑥𝑥 � 1 ⋯ 𝜕𝜕𝑍𝑍 𝐾𝐾 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑥𝑥 � 1 ⋮ ⋮ ⋮ 𝜕𝜕𝑍𝑍 1 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑥𝑥 � 𝑀𝑀 2 𝜕𝜕𝑍𝑍 2 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑥𝑥 � 𝑀𝑀 2 ⋯ 𝜕𝜕𝑍𝑍 𝐾𝐾 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑥𝑥 � 𝑀𝑀 2 𝜕𝜕𝑍𝑍 1 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑦𝑦 � 1 𝜕𝜕𝑍𝑍 2 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑦𝑦 � 1 ⋯ 𝜕𝜕𝑍𝑍 𝐾𝐾 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑦𝑦 � 1 ⋮ ⋮ ⋮ 𝜕𝜕𝑍𝑍 1 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑦𝑦 � 𝑀𝑀 2 𝜕𝜕𝑍𝑍 2 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑦𝑦 � 𝑀𝑀 2 ⋯ 𝜕𝜕𝑍𝑍 𝐾𝐾 (𝑥𝑥, 𝑦𝑦) 𝜕𝜕𝑦𝑦 � 𝑀𝑀 2 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ . (2-54)
As before, the coefficient vector 𝒁𝒁 can be calculated as:

𝒁𝒁 = (𝒁𝒁 𝑇𝑇 𝒁𝒁) -1 𝒁𝒁 𝑇𝑇 𝒔𝒔. (2-55)
However, since a finite set of Zernike polynomials over a discrete pupil are neither complete nor orthogonal, the reconstruction will differ from the actual wavefront by an RMS error 𝜖𝜖, which depends on the number of samples and the geometry of the sensor [START_REF] Navarro | Direct and inverse discrete Zernike transform[END_REF], [START_REF] Ramos-López | Optimal sampling patterns for Zernike polynomials[END_REF]. The presence of noise hinders centroid calculation of the Shack-Hartmann spots and will further increase the error as a function of the SNR.

In addition, subsampling is an issue to take into account. If the sampling density of the slopes in not enough for the number of modes required, higher order modes will appear as perturbations on the lower order modes, leading to a higher 𝜖𝜖.

The guide star

Wavefront sensing usually involves the use of a point-like source of light called the "guide star", recalling the origins of AO in the astronomy field. Although extended sources of light can be used for wavefront measurement, they severely compromise contrast (due to overlapping images) and present complex relationships between wavefront sensor signal and wavefront shape [START_REF] Rimmele | Solar adaptive optics at the National Solar Observatory[END_REF], [START_REF] Michau | Shack-Hartmann wavefront sensing with extended sources[END_REF].

Guide stars are useful because the wavefront from a point source of light is easy to model, and point-like Shack-Hartmann spots allows us to accurately calculate the wavefront slopes. They are therefore used in almost every wavefront sensor AO scheme.

Unlike the sky, biological tissue doesn't present naturally occurring guide stars. A number of approaches for guide star placing inside tissue have been tried over the last decade. Perhaps the most straightforward of them is the placement of fluorescent beads inside the sample. Beads can be arbitrarily small, are commercially available in a wide variety of spectrums, and usually produce a large amount of fluorescence. The beads can be inserted into the tissue either by microinjection [START_REF] Träber | Polyacrylamide Bead Sensors for in vivo Quantification of Cell-Scale Stress in Zebrafish Development[END_REF] or by adsorption during tissue development [START_REF] Jorand | Deep and Clear Optical Imaging of Thick Inhomogeneous Samples[END_REF]. However, these methods either are intrusive or yield random bead placement, limiting the available regions in which corrections can be performed. Moreover, the possibility of beads interfering with cell physiology must be considered. Either a pinhole or single-bead excitation is required to isolate a single guide star.

A less intrusive approach is the use of small fluorescent structures expressed by the cells, such as the centrosome [START_REF] Tao | Live imaging using adaptive optics with fluorescent protein guidestars[END_REF]. This approach has the advantage of a more uniform coverage of the tissue by guide stars (depending on the structure of choice), as well as a simpler sample preparation. However, the fluorescence yield of such guide stars is limited, and, more importantly, tracking small structures in live samples in order to isolate a single guide star can be a challenge.

This issue can be solved by the use of a laser guide star. By focusing a laser into larger fluorescent structures, we can generate a single guide star with a limited lateral size [START_REF] Tao | Adaptive optics confocal microscopy using fluorescent protein guide-stars for brain tissue imaging[END_REF]. Nonetheless, the axial length of the guide star depends on the axial of the structure, limiting its use to thin structures.

Two-photon excitation (TPE) provides a mechanism by which a laser guide star can be localized in the axial direction. TPE requires the capture of two photons at the same time by a fluorophore, emitting a higher energy photon. Since the probability of this happening is extremely low, it will only occur in high photon density regions. By focusing a femtosecond laser into a fluorescent structure, TPE can be achieved in a small spherical region close to the focus, thus providing a nonlinear guide star (NGS) [START_REF] Aviles-Espinosa | Measurement and correction of in vivo sample aberrations employing a nonlinear guide-star in two-photon excited fluorescence microscopy[END_REF]. NGS can be placed at will in the FOV, don't depend on the size of the labelled structure and, since the excitation wavelength is in the IR, are less distorted than guide stars produced by visible lasers. They provide a non-invasive and versatile method for wavefront measurement in biological samples and are arguably the best guide star alternative for live imaging. The main drawback of NGS is their low fluorescence yield, requiring the use of accordingly sensitive sensors.

Wavefront correction

Wavefront correction is, in essence, the process of distorting the wavefront to improve the performance of the system, either by removing or by adding aberration.

Φ(𝜌𝜌, 𝜌𝜌) = Φ 0 (𝜌𝜌, 𝜌𝜌) -2Φ M (𝜌𝜌, 𝜌𝜌) .

(2-56)

So, by applying a mirror shape Φ 𝑀𝑀 = Φ 0 /2, we obtain a flat wavefront: Closed-loop systems measure the figure of merit after the wavefront has been shaped by a correction device, providing feedback about the quality of the correction.

Φ(𝜌𝜌, 𝜌𝜌) = 0 . ( 2

ADAPTIVE OPTICS 57

This way, errors in wavefront shaping can be corrected in subsequent iterations.

Closed-loop operation is the norm in wavefront sensor systems.

Correction devices

Any instrument able to shape the wavefront of the light can be used as a correction device. Segmented and deformable mirrors are the most popular of such instruments in imaging systems. Due to the impracticality of large deformable mirrors, segmented mirrors most popular in large telescopes, whereas in microscopy deformable mirrors are predominant. A brief description of these devices is presented below.

Segmented Mirrors

The segmented mirror is a device made from a matrix of smaller mirrors with piston or tilt (on one or two axes) capability. Each of the submirrors in placed on top of an actuator that can move it up and down and, in some implementations, tip or tilt it. Submirror shape is a relevant parameter [START_REF] Malakhov | Estimating the parameters of a multielement mirror of an adaptive optical system[END_REF], and square, hexagonal and circular mirrors have been used. Segmented mirrors are easily scalable, some mirrors having hundreds or thousands of segments [START_REF] Tyson | Principles of Adaptive Optics[END_REF], [START_REF] Hardy | Instrumental Limitations In Adaptive Optics For Astronomy[END_REF].

A notorious drawback of segmented mirrors is the presence of discontinuities between the segments. These gaps produce two main optical effects: a loss of energy through the gaps and a diffraction of energy from the central lobe of the diffraction pattern. Both of these effects are proportional to the ratio of gap to segment area.

For gap area less than 2% the impact of the gaps on overall performance are minor [START_REF] Hulburd | High-bandwidth long-stroke segmented mirror for atmospheric compensation[END_REF].

As each segment is operates independently, there is no cross-coupling between them. However, cross-coupling can be introduced in order to increase bandwidth.

Segmented mirrors with a bandwidth as high as 5 kHz have been reported [START_REF] Hulburd | Segmented Mirrors For Atmospheric Compensation[END_REF].

Deformable Mirrors

Deformable Damping can be reduced by placing the membrane in a partial vacuum, though the presence of some damping has been found necessary in large-stroke systems to avoid pinning the membrane to the electrodes [START_REF] Tyson | Principles of Adaptive Optics[END_REF]. This kind of mirrors present no hysteresis.

Electromagnetic mirrors

In this case, the reflective membrane has small magnets at the back of the reflective surface. The membrane can be deformed using voice coil (solenoid) actuators, which are used to both push and pull each magnet. They provide a very fast response time, present no hysteresis, and require low voltages to operate [START_REF] Seilly | HELENOID Actuators-A New Concept in Extremely Fast Acting Solenoids[END_REF],

making them particularly popular for small mirrors, although there exist large ones as well [START_REF] Andersen | A new concept for large deformable mirrors for extremely large telescopes[END_REF]. The Mirao 52-e DM used in our system belongs to this category.

Piezoelectric mirrors

A piezoelectric mirror consists on a glass or metal plate firmly attached to a continuous piezoelectric material wafer (unimorph mirror) or two piezoelectric wafers attached together (bimorph mirror). In an unimorph mirror the glass sheet is polished and coated to become the reflecting part of the DM, while on a bimorph mirror one of the piezoelectric wafers may be used as the reflective element [START_REF] Steinhaus | Bimorph piezoelectric flexible mirror[END_REF]. The piezoelectric wafers are polarized normal to their surface. Electrodes concentrically placed on the back of the piezoelectric sheets are used to deform them.

Bimorph mirrors permit the decoupling of low-order and high-order aberration correction, allowing better correcting of low-order aberrations by allocating a larger stroke budget to them. However, absorption, scattering and aberrations can severely limit its in-depth imaging capability. By using AO, aberrations can be corrected and image quality, as well as imaging depth, can be improved.

Sensor-based AO approaches require the use of guide stars which are not naturally present in organisms and have to be artificially generated, for example, by using beads or labelling small structures inside the sample. However, the process of introducing these guide stars in the sample is often intrusive and yields randomly located guide stars. The non-linear guide star is an interesting approach to this problem. A non-linear guide star is generated by means of two-photon excitation. It is a non-intrusive method that can place a guide star anywhere on the focal plane of the microscope. Still, depending on the sample NGLs can be notably faint.

We have developed a system, the Wavefront sensor Adaptive Optics SPIM (WAOSPIM) with the objective of improving imaging quality at depth. It is based on a conventional SPIM setup in which an AO subsystem has been implemented, as well as an NIR illumination arm for NGL generation.

Our WAOSPIM makes use of a custom-made high--sensitivity SHWFS made by Imagine Optic in order to be able to work with very faint guide stars when reading the wavefront.

A detailed description of the system and high-sensitivity SHWFS, as well as a characterization of its correction capability, will be provided in this chapter.

Description of the system

General view

WAOSPIM consists of three main optical paths:

• Visible illumination path: A visible laser beam (491, 532 or 595 nm) is focused into a light sheet by a cylindrical lens for SPIM illumination. A 900 nm laser is swapped in for IR illumination.

• Two-photon illumination path: A 780 nm femtosecond laser is directed by a pair of galvanometric mirrors and focused into the sample for NGL generation.

DESCRIPTION OF THE SYSTEM

63

• Detection path: Light is collected, corrected and split into two sub-paths.

The shorter wavelengths reach the SHWFS while the longer ones reach a dual wavelength imaging camera. The lateral resolution of the microscope is 0.64 µm. The axial resolution depends on the light sheet thickness, and is around 2 µm, depending on the illumination wavelength. A detailed account of the optical elements of the system is provided in Appendix 1.

Visible illumination path

An Errol compact laser launch generates the laser lines of 491, 532 and 595 nm which are then coaxially merged. An acousto-optic tuneable filter permits precise control over the intensity of each beam.

The output beam is collimated and expanded by a telescope. A cylindrical lens generates the light sheet, which is focused into the sample by a 10x, 0.25 NA air objective (Leica Microsystems). For experiments requiring NIR illumination, a 980 nm continuous laser was used instead. In these cases, lenses L1 and L2 were replaced by lenses of similar focal distance but coated for operation in the infrared.

The waist thickness and Rayleigh range of the Gaussian light sheet is provided in

Two-photon illumination path

A 780 nm femtosecond laser (Toptica Photonics) is directed by a pair of galvanometric mirrors to the sample. A 20x, 0.5 NA water immersion objective (Leica Microsystems), shared with the detection path, is used to focus the beam into the sample. The pair of galvanometric mirrors is conjugated with the back-pupil plane of the objective. A short-pass dichroic mirror (Semrock, <746/9 nm pass) is used to introduce the NIR beam into the detection path.

Detection path

Fluorescence light is collected by a 20x, 0.5 NA water immersion objective and expanded by a telescope. The back-pupil plane of the telescope is conjugated with a Mirao 52-e electromagnetic DM (Imagine Optic). A long-pass dichroic beamsplitter (Semrock, >520/8 nm pass) directs the 350-512 nm wavelengths to the SHWFS, while allowing the 528-750 nm wavelengths to reach the ORCA-D2 dual imaging camera (Hamamatsu Photonics). The ORCA-D2 permits simultaneous imaging in the 641 (CCD1) and 520 nm (CCD2) wavelengths.

Sample mounting

The sample is embedded in an agarose cylinder that is polymerized inside a glass capillary. This capillary is immersed in an aqueous solution while hanging from a translation stage that allows sub-micron three-dimensional displacement of the sample. The agarose cylinder is pushed out from the glass capillary and exposed prior to observation.

Electronic control

The main control software of WAOSPIM is the house-made AMISPIM software based on LavView. The AO system was driven by the WaveSuite software (Imagine Optic) while galvanometer mirrors driving the NGS were controlled by the iLas software (Roper Scientific).

The non-linear guide star

The NGS is generated by means of two-photon excitation. A focused 780 nm femtosecond pulsed laser excites a fluorophore (GFP or fluorescein in our case) that only emits light in a small region around the focus. This region is small enough to be used as a guide star for wavefront measurements. A full characterization of the NGS used in our system, outside and inside TMs, is provided in this section.

In order to decouple NGS shape from the TM's fluorophore spatial distribution, unlabelled MCTS were used. The MCTS were immersed in a fluorescein solution for an hour before imaging. Fluorescein is a soluble fluorophore with peak emission around 520 nm that easily permeates MCTS and distributes uniformly within them, permitting reliable NGS generation inside and outside MCTS. The lack of ability of our microscope to move the guide star in the z direction prevented us from assessing the axial FWHM. The guide star had a lateral diameter of around 2.2 µm (FWHM) in agarose. In order to be considered as a point source, the guide star has to be below the diffraction limit of the wavefront sensor. In our case, for a 520 nm wavelength, this limit is 8.62 µm, so the NGS is a suitable point emitter for wavefront reading. As expected, the observed quality of the NGS decreased with depth. The maximum observed intensity suffered an exponential decline inside the MCTS, while the lateral FWHM usually increased after 50 µm, with a fast broadening typically occurring around the 70 nm mark. Deterioration from scattering in both the illumination and detection paths made the NGS typically unusable for wavefront reading after 90-100 µm.

High-sensitivity Shack-Hartmann wavefront sensor

As stated before, our setup requires the use of an especially sensitive SHWFS.

Imagine Optic (Orsay, France) prototyped such a sensor with feedback from our team which it was implemented in the WAOSPIM system. Our SHWFS is based on a Photometrics Evolve 512 EMCCD camera. EMCCD (electron-multiplying chargecoupled device) sensors are some of the most sensitive available, with a quantum efficiency over 0.9 in the visible range. Their electromagnetic gain can be tuned in real time, allowing a trade-off between exposure time and signal-to-noise ratio. These characteristics make EMCCD a powerful and versatile tool when working with a low photon budget, and the logical choice for a high-sensitivity wavefront sensor. The

Evolve 512 EMCCD has a 1-300 EM gain range.

Wavefront measurement error depends on parameters such as photon flux, noise, number of subapertures and type of sensor (see section 2.4.2). More specifically, centroid calculation error has a strong dependence on both readout and photon noise.

Let 𝜎𝜎 𝜖𝜖 𝑝𝑝ℎ 2 and 𝜎𝜎 𝜖𝜖 𝑟𝑟 2 be the contributions to phase variance of the photon and readout noise, respectively. It can be shown that, for a centre-of-gravity algorithm [START_REF] Thomas | Comparison of centroid computation algorithms in a Shack-Hartmann sensor: Comparison of centroid computation algorithms[END_REF]:

𝜎𝜎 𝜖𝜖 𝑝𝑝ℎ 2 = 𝜋𝜋 2 2 ln 2 1 𝜋𝜋 𝑣𝑣ℎ � 𝑤𝑤 𝜋𝜋 𝑠𝑠𝑎𝑎𝑚𝑚𝑣𝑣 � 2 , (3-1)
𝜎𝜎 𝜖𝜖 𝑟𝑟 2 = 𝜋𝜋 2 3 𝜋𝜋 𝑘𝑘 2 𝑆𝑆 2 𝜋𝜋 𝑣𝑣ℎ 2 𝜋𝜋 𝑠𝑠𝑎𝑎𝑚𝑚𝑣𝑣 2 , ( 3-2)
where 𝜋𝜋 𝑣𝑣ℎ is the total number of photons detected by the sensor, 𝜋𝜋 𝑠𝑠𝑎𝑎𝑚𝑚𝑣𝑣 = (𝜆𝜆/𝜌𝜌)/𝑝𝑝 is the ratio of the half-width of the diffraction ration to the pixel angular size, 𝑤𝑤 is the FWHM of the spot in pixels, 𝜋𝜋 𝑘𝑘 is the readout noise and 𝑆𝑆 is the total number of pixels used in the CoG calculation.

Amplifying the signal 𝜋𝜋 𝑣𝑣ℎ detected by the sensor can reduce the contributions of both photon and readout noise. However, if EM gain is applied to achieve it, the reduced photon and readout noise come at the price of amplifying thermal and clock-induced noise. The optimal solution is to increase the photon flux that reaches the sensor, but that is not always feasible. Hence, a careful balance has to be stricken between amplification of signal and amplification of noise.

A model of wavefront error as a function of photon flux and EM was carried out by Imagine Optic. Their analysis concludes that at least 2 × 10 5 photons are required for an optimal wavefront measurement. With this amount of photons and an EM gain of 10 or more, the RMS wavefront error is kept below 7 nm. Additionally, using the maximum EM gain of 300 allows us to work with less than 5 × 10 4 photons while maintaining an error of less than 10 nm. In order to maximize the photon flux per subaperture, the only option is to reduce the number of subpupils so that the photons are less distributed between them.

However, the reduced amount of subapertures implies a reduced number of slopes, which will reduce sensor accuracy. Additionally, in a SHWFS there are many other parameters such as microlens focal length, sensor geometry, CCD pixel size, etc.

The design of a wavefront sensor is a complex multiparameter problem that requires a requires a fine tuning of all these parameters to be able to meet the application requirements. In essence, this it is a trade-off between accuracy, sensitivity and dynamic range. In our case sensitivity is a priority, as reflected by the final specifications of the SHWFS.

To help define optimal wavefront sensor design, a simulation (data not shown) of the wavefront measurement dynamic and accuracy as a function of the number of microlenses and photon flux was performed by Imagine Optic. As the objective is to maximize sensitivity, dynamic range was fixed to the minimum required for MCTS imaging. In section 2.2, it was discussed how the most significant aberrations in SPIM for MCTS, in minimized refractive index mismatch conditions, are astigmatism, coma, and spherical aberrations. Based on previous experiments carried out by our team, it was evaluated that a 3 µm peak-to-valley dynamic range is necessary on the third-order spherical aberration (𝑍𝑍 4 0 ) for a 555 nm wavelength in order to sufficiently characterize the aberrations produced by an MCTS.

λ

Photon number for optimal accuracy 200 000

Wavefront measurement accuracy λ/70

Working wavelength range (nm) 400-800

Readout speed (MHz) 5

Electron-Multiplying gain 1-300

Diffraction limit (µm) 9.20

The problem then becomes a trade-off between sensitivity and accuracy. Since wavefront correction error is dependent on both the wavefront measurement and wavefront modulation errors, wavefront measurement error must be kept smaller than the modulation error to ensure accurate wavefront correction by the DM. For our Mirao 52-e mirror, this error is no more than 20 nm RMS. This provides a rough approximation of the required wavefront sensing accuracy. It was determined that a 14 × 14 array of microlenses is able to provide a wavefront reconstruction that meets our accuracy and dynamic range requirements, while maximizing photon flux per subpupil.

System set-up

The dichroic beamsplitters D1 and D2 introduce a considerable amount of aberration in our system (especially astigmatism and coma). In addition, misalignments between the optical elements of the microscope can also induce a lesser degree of aberration. In order to allow diffraction-limited imaging, these aberrations must be compensated by the AO system. A preliminary system configuration was used with this objective.

Since the main sources of system-induced aberration are the dichroic mirrors, aberrations must be measured after transmission by both of them. The SHWFS was therefore placed after beamsplitter D2 as shown in Figure 3-10.

SYSTEM SET-UP 75

This way, the wavefront could be measured after passing through most of the system. Although there was still a differential path between the SHWFS and the imaging camera, the wavefront phase difference was considered negligible. The DM was then shaped to generate a flat wavefront in the SHWFS position (Figure 3-11), which was therefore also flat at the imaging camera. This DM shape compensated all system-induced aberration and was used as the default state of the system, denoted as "AO off" from here on.

The SHWFS was then placed in its dedicated path (Figure 3-1). With the DM locked in the "AO off" shape, the SHWFS measured aberrations equal to those produced by the differential path between its current position and its previous one.

The measured wavefront was saved and henceforth used as the target wavefront for all AO corrections. This process was repeated whenever mechanical drifts of the optical elements of the system (especially in the differential path between the camera and SHWFS) caused the apparition of measurable aberration.

Adaptive optics performance

The correction capability of WAOSPIM in transparent media was evaluated. For this, we corrected the image of green fluorescent beads using a closed-loop AO. Since our TM imaging imposes the restriction of a limited photon flux, performance was evaluated in both high and low intensity conditions as a function of available light and number of closed-loop iterations. A detailed description of the experiment is presented below.

Methods

In order to evaluate the adaptive optics correction capability of WAOSPIM, a set of experiments involving fluorescent beads in transparent media was performed. Green fluorescent beads (InSpeck™ Green, Thermo-Fisher) with a diameter of 2.5 µm and 515 nm emission peak were embedded in a 1% agarose cylinder, which was polymerized inside a glass capillary. The agarose cylinder could be pushed out of the glass capillary at will by a piston.

In a first step, a set of beads were observed in agarose, with the mirror in the "AO off" configuration. The sample was illuminated by a 491 nm light sheet with constant intensity.

A different set of the same kind of beads was then observed inside the glass capillary. The glass capillary acts as a source of aberration, whose main component is astigmatism and, to a lesser extent, trefoil and first order comma. This caused the image to appear distorted in the lateral direction. The wavefront was then corrected using an AO closed loop with a set number of iterations, 𝜋𝜋 = 15, and the beads were observed again. The RMS aberrations were calculated from the wavefront slopes and displayed in real time. The Strehl ratio was then calculated using the Maréchal approximation as shown in section 2.1.4.

To evaluate performance as a function of the amount of light, the number of photons reaching the sensor was estimated from the acquired signal and the physical properties of the sensor. The signal 𝐼𝐼 𝜔𝜔 (𝑥𝑥, 𝑦𝑦) detected by the sensor has three main sources:

• Signal photons from the guide star that reach the subapertures and are focused into a bright spot.

• Signal photons from the guide star diffracted by the subaperture grid that forms a diffraction pattern between the Shack-Hartmann spots.

• Noise and background signal, 𝐼𝐼 𝑠𝑠 (𝑥𝑥, 𝑦𝑦).

𝐼𝐼 𝜔𝜔 (𝑥𝑥, 𝑦𝑦) = 𝐼𝐼 𝑔𝑔𝑠𝑠 (𝑥𝑥, 𝑦𝑦) + 𝐼𝐼 𝑠𝑠 (𝑥𝑥, 𝑦𝑦), where 𝐼𝐼 𝑔𝑔𝑠𝑠 (𝑥𝑥, 𝑦𝑦) is the signal coming from the guide star. As our main interest is to determine how much light we need to collect from the guide star to perform an accurate correction, a step to remove background was performed. The background signal distribution 𝐼𝐼 𝑠𝑠 (𝑥𝑥, 𝑦𝑦) was determined to be approximately constant over the pupil, and was roughly estimated as equal to the median value of the total intensity distribution 𝐼𝐼 𝜔𝜔 (𝑥𝑥, 𝑦𝑦). The number of signal photons 𝜋𝜋 𝑣𝑣ℎ was then estimated as:

𝜋𝜋 𝑣𝑣ℎ = 𝐼𝐼 • 𝐶𝐶𝐶𝐶 𝐸𝐸𝑀𝑀 • 𝑄𝑄𝐸𝐸 , ( 3-3)
where

𝐼𝐼 = ∑𝐼𝐼 𝑔𝑔𝑠𝑠 (𝑥𝑥, 𝑦𝑦) (3-4)
is the sum of the pixel value over the image after removing background, 𝐶𝐶𝐶𝐶 is the conversion gain of the sensor, 𝐸𝐸𝑀𝑀 is the electron multiplying gain applied and 𝑄𝑄𝐸𝐸 is the quantum efficiency of the sensor.

Performance results

Favourable light conditions correction

In order to determine AO performance, green fluorescent beads were first observed in agarose. Beads showed a round shape, with a mean FWHM of 2.61 × 2.32 × 3.11 µm.

When imaging beads at 𝑧𝑧 = 100 µm through the glass capillary, the image appeared greatly distorted in the lateral direction. RMS error of the measured wavefront was 0.238 ± 0.014 µm before correction. After correction, the image of the bead was restored, with a similar shape to that of the previous, capillary-free observation. RMS error after correction was 0.011 ± 0.001 µm. Strehl ratio (estimated using the Maréchal approximation) improved from 0.007 ± 0.021 to 0.983 ± 0.005. Contrast was improved by an average 80% ± 29% (peak signal to background).

Performance in low-light conditions

A similar experiment as above, in which beads were imaged through a glass capillary, was performed with decreasing illumination intensity. During experiments, emission intensity was measured as a fraction of the saturation value of the SHWS, defined as the maximum readable intensity for each pixel. The number of captured photons was later calculated from the acquired SHWS images.

Corrected images typically achieved imaging near the diffraction limit (𝑆𝑆𝑅𝑅 > 0.8).

Good corrections were obtained with as few as 12 × 10 3 signal photons over the pupil, with correction performance increasing with intensity. For intensities below 12 × 10 3 photons (6.6% of SHWS saturation value), our software failed to measure wavefront aberrations due to low SNR, making it impossible to correct the image.

Beyond that limit, corrections typically allowed for imaging near the diffraction limit. Four closed loop iterations were necessary for a good correction in low intensity conditions (8% of SHWS saturation value, 14.5 × 10 3 to 16.5 × 10 3 signal photons), while three were enough under high intensity conditions (90% of SHWS saturation value, 290 × 10 3 to 305 × 10 3 signal photons).

Conclusion

We have implemented an AO-capable SPIM and characterized its correction ability. Since the intended application in TM imaging imposes the restriction of a low available photon flux for wavefront sensing, a custom made EMCCD-based SHWFS was developed by Imagine Optic with our feedback. An assessment of AO correction performance in low photon flux conditions was performed.

It was found that wavefront sensing was the limiting factor in the AO correction.

Even though 2 × 10 5 photons are required for an optimal wavefront reconstruction, as few as 1.2 × 10 4 signal photons (62 per SHWFS subpupil) are sufficient to measure the wavefront phase in a low-scattering environment. If the wavefront could be reconstructed, AO corrections typically permitted near diffraction-limited imaging (𝑆𝑆𝑅𝑅 > 0.8).

The alignment of the two-photon excitation path is crucial, since aberrations arising from its misalignment may cause the NGS to become distorted and appear as an extended light source (especially in the z direction) severely hindering centroid calculation of the Shack-Hartmann spots. In addition, misalignment of the twophoton excitation path causes the typically faint NGS to be even fainter, limiting correction ability in adverse conditions. Our system does not currently have the ability to acquire precise axial information of the NGS, making alignment a difficult process.

The NGS used in our system is 2.2 µm when unaberrated. However, NGS intensity decreases exponentially inside MCTS, and its observed diameter tends to increase after 50 µm, with fast deterioration after approximately 70 µm.

The SPIM architecture of our microscope, paired with its aberration correction capabilities with a low photon flux using non-invasive methods, makes it a powerful tool for TM imaging. AO corrections allowed us to resolve subcellular features that were not resolved before correction, as it is made clear by an increase of high spatial frequency features after correction. AO corrections allowed us to clearly discern biologically relevant features, such as mitotic cells, that couldn't be observed before correction.

Methods

The ability of WAOSPIM to correct aberration within TM was evaluated using MCTS made from the HCT116 cell line. The MCTS expressed H2B-GFP and ArrestRed (H2B-tKR) histones and were fixed at a diameter of around 300 µm.

ArrestRed has an emission maximum at 610 nm and was used for imaging, whereas H2B-GFP was used for both NGS generation and imaging purposes. The spheroids were embedded in 1% agarose cylinders.

An NGS was employed for wavefront reading. H2B-GFP was excited with a 780 nm femtosecond pulsed laser, with an emission peak around 510 nm. A full characterization of the NGS is provided in section 3.2.2. Corrections were performed at different points and depths using different exposures depending on NGS intensity.

MCTS were imaged using 100 ms exposures.

Improvements in image quality were quantitatively assessed in terms of spatial frequency [START_REF] Masson | High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM[END_REF]. Let 𝑢𝑢(𝑥𝑥, 𝑦𝑦) be an 𝑀𝑀 × 𝜋𝜋 image, and let 𝑢𝑢 ��𝑓𝑓 𝑥𝑥 , 𝑓𝑓 𝑦𝑦 � = ℱ�𝑢𝑢(𝑥𝑥, 𝑦𝑦)� be its By comparing 𝑈𝑈 � (𝑓𝑓 𝑘𝑘 ) before and after correction we can evaluate improvements in frequencies corresponding to specific feature sizes. We define the relative difference, 𝑅𝑅𝑅𝑅, of radial frequencies between images acquired before and after correction (𝑈𝑈 � 𝑜𝑜𝑜𝑜𝑜𝑜 (𝑓𝑓 𝑘𝑘 ) and 𝑈𝑈 � 𝑜𝑜𝑛𝑛 (𝑓𝑓 𝑘𝑘 ), respectively) as:

𝑅𝑅𝑅𝑅(𝑓𝑓 𝑘𝑘 ) = 𝑈𝑈 � 𝑜𝑜𝑛𝑛 (𝑓𝑓 𝑘𝑘 ) -𝑈𝑈 � 𝑜𝑜𝑜𝑜𝑜𝑜 (𝑓𝑓 𝑘𝑘 ) 𝑈𝑈 � 𝑜𝑜𝑜𝑜𝑜𝑜 (𝑓𝑓 𝑘𝑘 ) . ( 4-2)
Improvements in observed high-frequency detail can be readily assessed by evaluating 𝑅𝑅𝑅𝑅(𝑓𝑓 𝑘𝑘 ). The 3D images were split in several z-sections and a maximum intensity z-projection was done in each of them. 𝑅𝑅𝑅𝑅 was evaluated in each section for corrections performed at different depths.

Results

MCTS were observed while embedded in 1% agarose cylinders before and after AO correction at different depths. Typically, a maximum of five to ten closed loop iterations could be performed due to the fast bleaching of the fluorophore and the long exposure times required for wavefront measurements (usually from 0.3 to 1s, depending on the sample and location of the guide star). Corrections were all performed using 1s exposure. Scale bar: 25 µm.

RESULTS

91

The AO-corrected images show again improvements in contrast and resolution with respect to the uncorrected image. Subcellular sized structures that were not well resolved in the "AOs off" position can be resolved after correction.

Frequencies were evaluated by means of angular projections of the Fourier transform of the images, as described in section 4. Compared to the uncorrected case, there is a substantial increase (up to 79%) of frequencies in the range 0.1 to 0.8 µm -1 , corresponding to features of cellular and subcellular size (1.25 to 10 µm). Frequencies outside this range show a slight decrease. For the Z1-40µm section, the most efficient correction was realized at 30 µm, while at the Z41-80µm section the most efficient was realized at 50 µm. At Z81-120µm imaging depth, AO correction becomes unreliable and those performed at 30 and 80 µm showed a general worsening of image quality, although the one performed at 50 µm showed a slight improvement.

Discussion

We have shown how AO corrections can improve in-depth imaging in optically thick MCTS. Our system is capable of reducing the MCTS-induced aberrations by up to 87.6% (RMS), resulting in improvements in image quality regarding resolution and high-frequency detail. Corrections are more effective near the guide star, and result in greater image quality gains when performed at shallow depths. The reduction of aberrations allows the identification of biologically relevant features (such as chromosome distribution) inside MCTS were only blurry outlines could be previously observed 2 .

Correction depth is mainly limited by scattering. Signal-to-noise ratio degrades deeper into the sample due to the loss of ballistic photons and the increase of noise originating from scattered light. The loss of SNR translates into a loss of accuracy of the Shack-Hartmann spots centroid calculation, resulting in inaccurate wavefront readings, and the total inability to measure the wavefront at greater depths.

Although wavefront reading is typically impossible deeper than around 100 µm, corrections performed at shallower depths can still improve image quality after that point.

Another relevant issue when performing AO corrections is the fast bleaching of the NGS. Typically, between 5 and 10 iterations can be performed before the NGS becomes too faint to measure the wavefront. This is more prominent at greater depths, as scattering further limits the amount of available light. In its current state, our system is controlled by several different software and manual controls, resulting in the generation of an NGS for a few seconds while no correction is being performed and further limiting the maximum number of iterations per correction. A more integrated hardware and software could provide an increase of 1 to 3 iterations per correction.

To sum up, WAOSPIM permits imaging of features of subcellular size at depths inaccessible to standard SPIM setups. By using an NGS, we can perform corrections targeted at the desired region of interest while keeping intrusiveness to a minimum, making WAOSPIM a powerful and versatile instrument. The excitation by NIR light sources makes UCNP compatible with biological tissue imaging, as photodamage is kept to a minimum and autofluorescence is nearly absent. In addition, they present no blinking or bleaching upon continuous NIR irradiation over long periods [START_REF] Wu | Non-blinking and photostable upconverted luminescence from single lanthanide-doped nanocrystals[END_REF]. Hence, they are excellent tools for long-term invivo imaging of live samples [START_REF] Park | Upconverting nanoparticles: a versatile platform for wide-field two-photon microscopy and multi-modal in vivo imaging[END_REF].

UCNP are formed by a substrate, such as NaYF4 or NaGdF4, that is usually doubly doped by a sensitizer cation, typically Yb 3+ , and an activator cation, such as Yb 3+ , Er 3+ or Tm 3+ . The optical properties of the UCNP (e.g. emission and excitation wavelength, quantum yield, photostability) depend mainly on the cation pair of choice [START_REF] Shin | Two-Dimensional and Three-Dimensional Single Particle Tracking of Upconverting Nanoparticles in Living Cells[END_REF].

The advantageous optical properties of UCNP can be exploited for their use in adaptive optics. Given their photostability and intensity, a guide star made from such material could allow performing frequent and fast AO corrections comprising tens of closed-loop iterations with no photobleaching of the guide star while being completely innocuous to the sample. However, since individual particles are not bright enough for wavefront reading, small aggregates of them would have to be used as a guide star. In addition, while the optical compatibility of UCNP with biological tissue is remarkable, the toxicological compatibility is not so clear. Hence, in order to make useful guide stars, the UCNP must meet the following requirements: i. They must not interfere with cell physiology or tissue development.

ii. They must form aggregates that are small enough to be considered point-like (less than 10 µm) but large enough to provide a sufficient amount of photons for wavefront reading.

In this chapter, the adequacy of NaYF4:Yb,Tm-based UCNP to be used as guide stars for AO correction is evaluated using the criteria stablished above.

Methods

The UCNP employed in this chapter, denoted CR3936-69P1, consist on NaYF4:Yb,Tm nanoparticles coated in PEG6000-PAA6500, with a size of 36 ± 4 nm.

CR3936-69P1 has photon up-conversion emission peaks at 346, 362, 452 and 476 nm as well as a much greater peak at 801 nm. MCTS were mounted on 1% agarose cylinders for imaging. Firstly, widefield transmitted light images of the MCTS were taken, after which an NIR light sheet was used to excite the UCNP, which were imaged by conventional SPIM.

The viability of several UCNP solubilization strategies was studied. For this, MCTS were grown in the presence of both UCNP and a 1 mg/ml BSA solution.

UCNP concentrations of 0.1 µg/ml and 1 µg/ml were employed in each case. MCTS were measured and fixed after 5 days.

Results

Toxicity

The compatibility of CR3936-69P1 with biological tissue was determined. For this, MCTS were grown in the presence of CR3936-69P1, in concentrations ranging from 0.1 µg/ml to 100 µg/ml.

A concentration of 0.1 µg/ml produced MCTS with a mean size of 340 nm and 490 nm after 4 and 8 days, respectively. This is in accord with growth dynamics of unperturbed MCTS and, thus, this concentration did not appear to hamper MCTS growth. However, larger concentrations significantly affected MCTS growth. After 4 days, MCTS were 7.4% and 16.7% smaller when grown in 1 µg/ml and 10 µg/ml concentrations, respectively (Table 5-1, Figure 5-2). The size difference was reduced to 4.6% and 12.1% after 8 days (Table 5-2). Cells plated in the presence of 100 µg/ml CR3936-69P1 did not form spheroids after 4 days, arranging in irregular aggregates instead. These cell aggregates had dispersed by the 8 th day.

RESULTS

Distribution inside MCTS

The viability of using UCNP as guide stars depends on their spatial distribution inside the spheroid, due to the requirement that guide stars must be point-like. A single nanoparticle acts as a point-like source of light, but its small size causes it not to be intense enough to be useful. Therefore, clusters of UCNP should be big enough to produce a workable signal-to-noise ratio but small enough to be considered pointlike. Furthermore, these clusters should be uniformly distributed over the MCTS in order to provide a versatile selection of regions of interest in which to perform the AO correction.

RESULTS

101

Due to toxicity of CR3936-69P1 at high concentrations, MCTS were grown in UCNP solutions at 0.1 µg/ml and 1 µg/ml. They were fixed and observed after 5

days. The nanoparticles formed aggregates with a size of up to tens of microns (Figure 5-3). These aggregates tended to cluster in on size of the spheroid, leading to a total lack of UCNP in most of the MCTS. Increasing nanoparticle concentration did not translate into a more uniform distribution. Instead, the UCNP formed a greater number of aggregates, which were larger and brighter. However, they still clustered in a side of the spheroids while not being present in the rest of their volume.

The distribution of UCNP into a single large cluster in each MCTS led us to believe that, during their development, the spheroids were not properly spreading the nanoparticles in the vertical direction, probably due to the short precipitation time of CR3936-69P1 (a few minutes). As the consequence, the UCNP rested at the bottom of the spheroids, producing a highly unbalanced distribution.

Bovine serum albumin (BSA) is a commonly used protein known to enhance solubility and promote surface biofunctionalization [START_REF] Muzi | Interactions and effects of BSA-functionalized single-walled carbon nanotubes on different cell lines[END_REF]. To improve UCNP distribution and avoid an accumulation of them at the bottom of the spheroid, a batch of MCTS were grown in presence of both CR3936-69P1 and BSA. The resulting nanoparticle distribution was similar to that obtained in the absence of BSA. UCNP formed aggregates that grouped into a single large cluster, presumably at the bottom of the MCTS.

Discussion

Up-converting nanoparticles are rare-earth-based particles that undergo photon up-conversion when excited, emitting photons of higher energy than the captured ones. This characteristic makes UCNP an attractive approach to guide stars, as they can be illuminated using NIR light while emitting in the visible, substantially decreasing photodamage to the sample. We have studied the viability of one of such UCNP, CR3936-69P1, to be used as guide stars in TMs.

One of the predicted possible issues with rare-earth based compounds is toxicity to the sample. We found that CR3936-69P1 hinders MCTS growth at concentrations greater than 0.1 µg/ml. However, effects on the physiology of the tissue at lower concentrations cannot be definitively discarded without further research.

While a concentration of 0.1 µg/ml provides enough light to be able to measure the wavefront in our setup, the three-dimensional spatial distribution of the nanoparticles makes this task impossible. The point-like nature of guide stars and the resolution of our SHWFS require that the sources of light do not surpass roughly 9 µm. However, the aggregates that are bright enough to be used as a guide star have diameters typically greater than 20 µm. However, our setup uses an inefficient illumination system, with an objective that is not optimal for NIR applications, blocking around 90% of the incoming beam intensity. The use of a better suited illumination objective would probably allow wavefront reading using much smaller UCNP aggregates, with sizes suitable for their use as guide stars.

A more worrying issue is the fact that the UCNP aggregates distribution is not uniform over the volume of the MCTS. The UCNP tend to form a large cluster in one side of the spheroid, while, at most, very small amounts are present anywhere else. Therefore, no corrections can be performed in most of the volume of the spheroid due to a lack of guide star. Since the fast precipitation of CR3936-69P1 was assumed to be the main factor in the uneven UCNP distribution, we tried to improve their solubility and cell adhesion by adding BSA to the MCTS growth medium.

However, the addition of BSA did not lead to a more uniform distribution.

In summary, CR3936-69P1 does not evidently affect MCTS development at low concentrations (< 0.1 µg/ml). This UCNP concentration can offer small nanoparticle aggregates, which are presumed to provide enough light for wavefront reading with an NIR-optimized illumination system. Therefore, the main detriment for the use of The main concern regarding NGS is the low amount of light emitted, especially when working with rather opaque samples. To be able to measure the wavefront in TMs, we employed a custom high-sensitivity wavefront sensor based on EM-CCD technology. We have fully characterized the system and shown that adequate corrections can be performed with as few as 12000 photons. Even so, long exposures (0.3 s -1 s) are required to collect that many photons from NGS in TMs. When coupled with fast bleaching, this severely limits the amount of correction steps that can be completed to between 5 and 10. However, further optimization of our system, especially the control software, seems straightforward and could somewhat alleviate this problem.

The ability of WAOSPIM to correct aberrations has been demonstrated in fixed MCTS, showing improvements in in-depth imaging regarding resolution and highfrequency detail. RMS aberration was reduced by up to 87,6%. Image improvements are greater near the guide star and when the correction is performed at shallow depths. AO correction allowed us to clearly identify previously-undistinguishable mitotic cells.

However, scattering limits the maximum depth at which useful corrections can be performed. Indeed, the loss of SNR usually precludes aberration correction deeper than 100 µm, even though shallower corrections can slightly improve image quality after this point.

The use of up-converting nanoparticles as guide stars could provide a way to increase the maximum depth accessible to AO. They can be illuminated with IR light, produce high emission intensities and show negligible photobleaching. However, their toxicity for TMs has never been ascertained, and their small size requires them to be clustered to provide enough light for wavefront reading. We proved that MCTS show slower development when in presence of UCNP concentrations upwards of 1 µg/ml.

A more severe issue with the use of UCNP is the lack of uniform coverage inside TMs. The nanoparticles tend to cluster at the bottom of the spheroid, with almost no presence in over half of it. We believe that this is a consequence of the lack of solubility of the tested UCNP. Exploratory trials with the objective of increasing their solubility in cell culture medium were unsuccessful.

Perspectives

Given the versatility of LSFM and the many innovations present in the literature, there is a vast amount of options for further development of our microscope.

Our WAOSPIM system can adequately correct aberrations in highly scattering samples, but the maximum depth at which we can improve imaging is limited by scattering to around 120 µm. While this is enough for small TMs, larger samples would benefit from deeper corrections. It is thus desirable to increase the SNR produced by the guide star in such conditions.

Beads and UCNP can provide more light than NGS, improving the SNR and allowing correction at greater depths [START_REF] Jorand | Deep and Clear Optical Imaging of Thick Inhomogeneous Samples[END_REF]. The use of UCNP is however more desirable than fluorescent beads, since they exhibit almost no photobleaching and IR illumination is less damaging to the sample. By optimizing the illumination path of the system for NIR light, we could potentially use small UCNP aggregates as guide stars in non-toxic concentrations. However, their uneven distribution inside MCTS makes this impractical. It is my belief that further research into the assimilation of UCNP by biological tissue, especially TMs, could lead to greater viability of their use as guide stars. Another approach worth of consideration to increase SNR is the use of longer wavelength guide stars, which are less sensitive to scattering [START_REF] Wang | Direct wavefront sensing for high-resolution in vivo imaging in scattering tissue[END_REF]. The longest accessible wavelength is, however, limited by the optics of the detection path of the 108 microscope, which are coated for the visible and thus block wavelengths longer than approximately 700 nm.

A scanned NGS approach has been proposed to decrease speckle-like artifacts in the wavefront sensor image [START_REF] Wang | Direct wavefront sensing for high-resolution in vivo imaging in scattering tissue[END_REF]. By scanning the NGS over a small region, scattering artifacts can be averaged out. This approach could somewhat improve wavefront reading accuracy at depth, but it would require a significant redesign of the microscope, as well as an increase of system complexity. We believe the small expected gains (since these artifacts are more prominent in high-resolution systems)

are not worth the significant undertaking.

The light sheet is also subject to scattering. Scattered illumination produces undesired fluorescence from out-of-focus regions, giving way to ghost image artifacts and severely decreasing resolution. A dual illumination approach would somewhat alleviate ghosting artifacts, but its implementation would require rethinking the geometry of our system due to the space constraints. A better option would be the use of non-diffractive beams.

Bessel beams have shown increased penetration depth and reduction of ghost artifacts, due to the self-healing nature of the beam [START_REF] Fahrbach | A line scanned light-sheet microscope with phase shaped self-reconstructing beams[END_REF], [START_REF] Fahrbach | Light-sheet microscopy in thick media using scanned Bessel beams and twophoton fluorescence excitation[END_REF]. This illumination method could further improve axial and lateral resolution to allow the detection of features deep into the sample. A scanned Bessel beam strategy is especially attractive for our system. By keeping the sample still, the light sheet can be scanned in the z direction while the AO system corrects the resulting defocus, opening the way to faster imaging and simpler sample mounting [START_REF] Fahrbach | Rapid 3D light-sheet microscopy with a tunable lens[END_REF].

Since AO correction is most effective near the guide star, the resulting image quality will vary greatly across the field of view. One of the advantages of NGS is that it can be positioned anywhere in the image, allowing correction near the objects of interest. An image can be repeatedly corrected placing the NGS at different points to later stitch the well-corrected regions together, obtaining a single image that is well corrected everywhere [START_REF] Liu | Observing the cell in its native state: Imaging subcellular dynamics in multicellular organisms[END_REF]. This approach could be easily implemented in our 109 microscope, since it would only require the development of a software algorithm.

However, it is of limited utility for TM imaging. Indeed, since an acquisition is required for each correction, several exposures are required for each final image. In highly heterogeneous tissue, given the small size of the isoplanatic patch, the number of required corrections per image could easily scale into the hundreds. The increase in acquisition time defeats the main advantages of SPIM: fast acquisitions and low photodamage. Hence, while easy to implement, this tiling method would be useful only in limited cases.

In any case, adaptive optics is a necessary technology for optical TM imaging.

While AO is still to be fully embraced by the microscopy community, it is a rapidly developing field that is rapidly expanding the range of applications of optical microscopy. The implementation of current and future ideas in our adaptive WAOSPIM system will lead to an even more powerful instrument for the imaging of tissue mimics.

FiguresFigure 2 - 1 :

 21 Figures

Figure 2 - 2 :

 22 Figure 2-2: Aberrated wavefront propagating through an aperture. ............................

Figure 2 - 4 :

 24 Figure 2-4: Spatial shape of the first 28 Zernike polynomials. ...................................

Figure 2 - 5 :

 25 Figure 2-5: Beam propagating through a TM. ...........................................................

Figure 2 - 6 :

 26 Figure 2-6: Structure of an MCTS .............................................................................

Figure 2 - 7 :

 27 Figure 2-7: SPIM setup. .............................................................................................

Figure 2 - 8 :

 28 Figure 2-8: Detail of the RLSM illumination scheme. ................................................

Figure 2 - 9 :

 29 Figure 2-9: OPM setup. .............................................................................................

Figure 2 - 10 :

 210 Figure 2-10: APOM microscope. ................................................................................

Figure 2 - 11 :

 211 Figure 2-11: Main properties of a light sheet. ............................................................

Figure 2 - 12 :

 212 Figure 2-12: Stripe artifact created by the blocking of the light by an obstacle. .......

Figure 2 -

 2 Figure 2-13: Self-healing nature of a Bessel beam. .....................................................

Figure 2 - 14 :

 214 Figure 2-14: Properties of Gaussian and Bessel beams, and square and hexagonal lattices.. .....................................................................................................................

Figure 2 - 15 :

 215 Figure 2-15: Working of a SHWFS. ...........................................................................

Figure 2 - 16 :

 216 Figure 2-16: Comparison between detected and actual intensity distribution over a sensor .........................................................................................................................

Figure 2 - 17 :

 217 Figure 2-17: Calculation of wavefront slope over a subaperture from the displacement of the displacement of the Shack-Hartmann spot. .....................................................

Figure 2 - 18 :

 218 Figure 2-18: Comparison between Hudgin and Fried geometries. ..............................

Figure 2 - 19 :

 219 Figure 2-19: Wavefront correction by phase conjugation. ..........................................

Figure 2 - 20 :

 220 Figure 2-20: Example of open and closed loop configurations. ...................................

Figure 2 - 21 :

 221 Figure 2-21: Electrostatic deformable mirror. ............................................................

Figure 2 - 22 :

 222 Figure 2-22: Electromagnetic deformable mirror. .......................................................

Figure 2 - 23 :Figure 3 - 1 :

 22331 Figure 2-23: Bimorph piezoelectric wafer deformable mirror. ....................................

Figure 3 - 2 :

 32 Figure 3-2: Detail of the illumination path. ...............................................................

Figure 3 - 3 :

 33 Figure 3-3: Detail of the two-photon excitation arm. .................................................

Figure 3 - 4 :

 34 Figure 3-4: Detail of the detection path. ....................................................................

Figure 3 - 5 :

 35 Figure 3-5: Mounting scheme of our system. .............................................................

Figure 3 - 6 :

 36 Figure 3-6: Guide star observed at different depths within an MCTS. .....................

Figure 3 - 7 :

 37 Figure 3-7: NGS observed maximum intensity as a function of depth inside an MCTS. .......................................................................................................................

Figure 3 - 8 :

 38 Figure 3-8: Simulated wavefront error as a function of the total number of photons reaching the SHWFS. ................................................................................................

Figure 3 - 9 :

 39 Figure 3-9: Simulated dynamic range of wavefront reconstruction. ...........................

Figure 3 - 10 :

 310 Figure 3-10: Alternative position of the SHWFS used for the correction of systeminduced aberrations. ...................................................................................................

Figure 3 - 11 :

 311 Figure 3-11: Wavefronts measured at the two different SHWFS positions with the mirror in the "AO off" shape. .....................................................................................

Figure 3 - 12 :

 312 Figure 3-12: Schematic of the experiment for the determination of AO performance . .............................................................................................................

Figure 3 - 13 :

 313 Figure 3-13: Maximum intensity projection and wavefront of green fluorescent beads imaged through a capillary glass before and after AO correction ..............................

Figure 3 -

 3 Figure 3-14: RMS and Strehl ratio values of wavefronts before and after AO correction as a function of photon flux and number of closed loop iterations ............

Figure 4 - 1 :

 41 Figure 4-1: Maximum intensity projection of an MCTS expressing H2B-GFP before and after correction at different depths. ....................................................................

Figure 4 - 2 :

 42 Figure 4-2: Measured wavefronts after AO corrections of an MCTS at different depths. .......................................................................................................................

Figure 4 - 3 :Figure 4 - 4 :

 4344 Figure 4-3: Maximum intensity projection of an HCT116 nucleus expressing H2B-GFP before and after AO correction at differet depths .............................................

Figure 4 - 5 :

 45 Figure 4-5: Maximum intensity projection of three regions of an MCTS before and after AO correction. ...................................................................................................

Figure 4 - 6 :

 46 Figure 4-6: Quantitative analysis of AO correction in MCTS. ...................................

Figure 5 - 1 :

 51 Figure 5-1: CR3936-69P1 up-conversion spectrum. ....................................................

Figure 5 - 2 :

 52 Figure 5-2: Widefield image of MCTS after 4 days of growth in presence of different concentrations of CR3936-69P1. . ..............................................................................

Figure 5 - 3 :

 53 Figure 5-3: Three-dimensional distribution of nanoparticles on MCTS as seen from different angles over the Y axis. ..............................................................................

Figure 5 - 4 :

 54 Figure 5-4: Effect of BSA on the three-dimensional distribution of nanoparticles on an MCTS. .....................................................................................................................

23 Table 2 : 73 Table 5 : 79 Table 6 :

 232735796 Scattering coefficients for healthy tissue, tumoral tissue and MCTS for visible and near-infrared wavelengths. ....................................................................... Waist thickness and Rayleigh range of the light sheet for all the possible illumination modes of the system ............................................................................... 65 Table 3: Observed lateral FWHM of the NGS at different depths inside an MCTS. ....................................................................................................................... 69 Table 4: Specifications of our Shack-Hartmann wavefront sensor for a 555 nm guide star. .................................................................................................................. Mean full width at half maximum, RMS aberration and Strehl ratio values for beads imaged through a glass capillary before and after correction. .......... Mean correction results for images of beads through a glass capillary for different emission intensities. ..................................................................................... 80

Figure 2 - 1 :

 21 Figure 2-1: Wave propagating through an aperture.

Figure 2 - 2 :

 22 Figure 2-2: Aberrated wavefront propagating through an aperture.

Figure 2 - 3 :

 23 Figure 2-3: Spatial shape of the first 28 Zernike polynomials. Modified from [7].

Figure 2 - 4 :

 24 Figure 2-4: Beam propagating through a TM.

Figure 2 - 5 :

 25 Figure 2-5: (a) Structure of an MCTS. Gradients of O2, nutrients and metabolic waste divide the MCTS into a proliferating rim, a quiescent zone, and a central necrotic core. (b) Widefield image of an MCTS. Scale bar: 150 µm.

- 1 )

 1 for healthy tissue, tumoral tissue and MCTS for visible and near-infrared wavelengths. (a) Colon tissue. (b) Lung tissue. the scattering contribution to attenuation far outweighs absorption. As seen above, absorption coefficients are one or two order of magnitude lower than scattering coefficients. It is worth noting that fixed spheroids have a much more stable absorption coefficient with respect to wavelength. While fresh lung MCTS 𝜇𝜇 𝑎𝑎 goes from 0.4 mm -1 in the visible to almost vanishing in the NIR, fixed MCTS keep a fairly stable profile after 650 nm, with an almost constant 𝜇𝜇 𝑎𝑎 of 0.1 mm -1 .

Figure 2 - 6 :

 26 Figure 2-6: SPIM setup.

  cantilever redirects the light sheet incoming from the illumination objective into the detection focal plane, allowing the use of NA=0.8 and NA=1.4 oil immersion objectives in the illumination and detection paths, respectively.

Figure 2 - 7 :

 27 Figure 2-7: Detail of the RLSM illumination scheme.

Figure 2 - 8 :

 28 Figure 2-8: OPM setup.

Figure 2 - 9 :

 29 Figure 2-9: APOM microscope. The tilted mirror reflects the axial image of the XZ plane into a lateral image, while a widefield image of the XY plane is registered by another camera.

Figure 2 -

 2 Figure 2-10: Main properties of a light sheet.

Figure 2 - 11 :

 211 Figure 2-11: Stripe artifact created by the blocking of the light by an obstacle.

Figure 2 -

 2 Figure 2-12: Self-healing nature of a Bessel beam. The beam is reformed shortly after encountering an obstacle.

Figure 2 - 13 :

 213 Figure 2-13: Properties of Gaussian and Bessel beams, and square and hexagonal lattices.

  phototoxicity by an order of magnitude when comparing to scanned Bessel beams.The hexagonal lattice was used in combination with SIM to provide superresolution images by taking advantage of the periodic light sheet. Several images

  Wang et al.[START_REF] Wang | Light-sheet microscopy in the near-infrared II window[END_REF] used NIR excitation and emission to image large chunks of mammal tissue. They employed a 785 nm light sheet to achieve a penetration of up to 2.1 mm in glycerol-cleared brain tissue, and different fluorescent probes (emitting in the 850 -1700 nm range) permitted cellular resolution imaging up to 2.5 mm deep.NIR excitation was also exploited by Truong et al., who combined two-photon excitation with SPIM (2P-SPIM)[START_REF] Truong | Deep and fast live imaging with two-photon scanned light-sheet microscopy[END_REF]. Instead of the usual light sheet, they scanned a focused NIR pulsed laser vertically through the sample. Due to the quadratic dependence of fluorescence emission with two-photon excitation intensity, precise optical sectioning could be achieved, delivering better axial resolution than both single-photon SPIM and two-photon scanned laser microscopy. Since scattered light is not intense enough to undergo two-photon excitation, 2P-SPIM produce little background light, improving lateral resolution over 1P-SPIM.

  Jorand et al.[START_REF] Jorand | Deep and Clear Optical Imaging of Thick Inhomogeneous Samples[END_REF] used AO to correct significant aberration in fixed MCTS, improving image quality up to 197 µm deep using fluorescent beads as guide stars. Even deeper corrections can be made by previously clearing the sample. Indeed, Masson et al.[START_REF] Masson | High-resolution in-depth imaging of optically cleared thick samples using an adaptive SPIM[END_REF] developed a model of aberration produced by cleared MCTS and used it to improve resolution at depths of up to 300 µm. It is worth noting that, in this work, aberrations were estimated beforehand as opposed to measuring them during imaging.TM imaging with adaptive LSFM was further explored by Liu et al.[START_REF] Liu | Observing the cell in its native state: Imaging subcellular dynamics in multicellular organisms[END_REF]. They developed an adaptive LLSM, able to correct the aberrations of both the lattice light sheet and the image. Given the sensitivity of lattice light sheets to sample-induced aberrations, their correction with AO is fundamental to enhance penetration. In their work, they study in-vivo processes such as clathrin dynamics and organelle morphology, imaging them with diffraction-limited resolution inside zebrafish embryos. By making use of tiled corrections, they achieve consistent resolution over the whole field of view. Though most of their work is done in zebrafish, they briefly demonstrated the ability of their system to perform aberration correction up to 40 µm deep in human stem-cell derived organoids, which allowed them to clearly locate previously-unresolved subcellular features. This same system was employed by Schöneberg et al. to image human intestinal epithelial organoids with similar performance[START_REF] Schöneberg | 4D cell biology: big data image analytics and lattice lightsheet imaging reveal dynamics of clathrin-mediated endocytosis in stem cellderived intestinal organoids[END_REF].

Figure 2 - 14 :

 214 Figure 2-14: Working of a SHWFS. (a) An undistorted wavefront creates a regular array of spots on the imaging chip. (b) When the wavefront is aberrated, the focused spots are distributed in an irregular way. The displacement of each spot determines the slope of wavefront along the corresponding subaperture.

Figure 2 - 15 :

 215 Figure 2-15: Intensity distribution 𝑃𝑃(𝑥𝑥) over a linear array of pixels. The actual distribution is presented in red, while the measured one, 𝐼𝐼 𝑖𝑖 , is presented in blue. The actual and calculated centroids are designated by 𝑥𝑥 0 and 𝑥𝑥 �, respectively.

Figure 2 - 16 :

 216 Figure 2-16: Calculation of wavefront slope over a subaperture from the displacement of the displacement of the Shack-Hartmann spot.

  nodes. If there are 𝜋𝜋 subapertures, 𝑩𝑩 has dimension 𝜋𝜋, while 𝒔𝒔 has a dimension 𝑀𝑀 that depends on the geometry of the sensor. Hence, 𝑩𝑩 has dimension 𝜋𝜋 × 𝑀𝑀. Hudgin and Fried geometries, two common configurations in SHWFS, are shown in Figure 2-17.

Figure 2 - 17 :

 217 Figure 2-17: Comparison between (a) Hudgin and (b) Fried geometries. Hudgin geometry takes into account the slopes between spots, while in Fried geometry the slopes are considered from the middle of the segments joining the spots.

-57) 2 . 4 . 3 . 2 .

 2432 AO loop Usually, AO systems work continuously in an iterative fashion (called a loop) either to keep up with changing aberrations or to optimize correction through several passes. Based on the presence or not of feedback, a distinction is made between open and closed loops. In an open-loop system, the figure of merit (RMS aberration, intensity, image sharpness…) is measured before the correction is performed, in such a way that there is no feedback about the effect of the correction. Sensor-less AO systems often run in open-loop mode.

Figure 2 - 19 :

 219 Figure 2-19: Example of (a) open loop configuration and (b) closed loop configuration. In an open loop, the wavefront is measured before the correcting device, while, in the closed loop, it is measured after it.

  mirrors (DM) consist on a continuous reflective surface equipped with some mean of deformation. In microscopy DMs are usually the tool of choice due to their achromaticity, high reflectivity and, most importantly, speed. Many kinds of DMs have been prototyped over the last 50 years, and, thanks to Micro-Electro-Mechanical Systems (MEMS) technology, several models of DMs are now being massproduced at relatively inexpensive costs. The most commonly used are, depending of on their actuation mechanism:Electrostatic mirrorsA thin (0.5 µm to 1 µm) conductive membrane acts as the reflecting element of the mirror. It can be deformed by electrostatic forces by means of applying voltage differences between the membrane and the electrode actuators. These forces can only be attractive, so only pull movement is allowed. Movement in the upward direction can be enabled by placing an additional set of transparent actuators above the membrane[START_REF] Yellin | Using Membrane Mirrors In Adaptive Optics[END_REF],[START_REF] Bonora | Push-pull membrane mirrors for adaptive optics[END_REF].

Figure 2 - 20 :

 220 Figure 2-20: Electrostatic deformable mirror.

Figure 2 - 21 :

 221 Figure 2-21: Electromagnetic deformable mirror.

Figure 2 - 22 :

 222 Figure 2-22: Bimorph piezoelectric wafer deformable mirror.

Figure 3 - 1 :

 31 Figure 3-1: Schematic image of WAOSPIM (not to scale). L1-10: Lenses; M1-7: Mirrors; D1-3: Dichroic beamsplitters; CL: Cylindrical lens; DM: Deformable mirror; SHWFS: Shack-Hartmann wavefront sensor; GM: Pair of galvanometer mirrors; PC: Physiological chamber; TS: Translation stage.

Figure 3 - 2 :

 32 Figure 3-2: Detail of the illumination path.

Figure 3 - 3 :

 33 Figure 3-3: Detail of the two-photon excitation arm.

Figure 3 - 4 :

 34 Figure 3-4: Detail of the detection path.

Figure 3 - 5 :

 35 Figure 3-5: Mounting scheme of our system.

Figure 3 - 6 :

 36 Figure 3-6: Guide star observed at different depths within an MCTS. The MCTS was immersed in a fluorescein solution and the NGS imaged after 1h. (a) NGS outside spheroid. (b) Depth: 25 µm. (c) Depth: 50 µm. (d) Depth: 75 µm. Scale bar: 10 µm.

Figure 3 - 7 :

 37 Figure 3-7: NGS observed maximum intensity as a function of depth inside an MCTS.

Figure 3 - 8 :

 38 Figure 3-8: Simulated wavefront error as a function of the total number of photons reaching the sensor.

Figure 3 - 9 :

 39 Figure 3-9: Simulated dynamic range of wavefront reconstruction.

Figure 3 - 10 :

 310 Figure 3-10: Alternative position of the SHWFS used for the correction of system-induced aberrations.

Figure 3 - 11 :

 311 Figure 3-11: Wavefronts measured at the two different SHWFS positions with the mirror in the "AO off" shape. a) Preliminary position. b) Final position. This wavefront was used as the target for all subsequent corrections.

Figure 3 - 12 :

 312 Figure 3-12: Schematic of the experiment. Beads with a diameter of 2.5 µm were observed through (a) agarose and (b) a glass capillary filled with agarose.

Figure 3 -

 3 Figure 3-13: (a) Maximum intensity projection of green fluorescent beads embedded in 1% agarose and imaged through a capillary glass at a depth of 100 µm, both before and after AO correction. Scale bar: 5 µm. Beads were illuminated with constant intensity at 491 nm. (b) Measured wavefronts before and after correction, offset by the reference wavefront corresponding to an aberration-free system.

Figure 3 -

 3 Figure 3-14: (a) RMS and Strehl ratio values of wavefronts before and after correction for a range of light conditions. 15 closed loop iterations were used for each correction. Each pair of points (AO off and AO on) corresponds to a different fluorescent bead. (b) SR and RMS values after correction as a function of the number of iterations used in the closed loop, for fixed low (8% saturation) and high (90% saturation) emission intensity.

  Multi-cellular tumor spheroids are three-dimensional cultures of tumoral cells grown in a roughly spherical shape. They are a widely used TM that has proven invaluable in the study of cancer progression and treatment. However, MCTS are notoriously difficult to image in depth due to the high levels of aberration and scattering introduced by the sample.In this chapter, I show the ability of our WAOSPIM setup to improve the quality of images of MCTS up to 120 µm deep. For imaging and NGS generation purposes, we marked the H2B histone present in the nucleus with GFP and KillerRed fluorophores. As a histone, H2B has a role on structuring and condensing DNA during the different phases of the cell cycle and it is therefore commonly used to observe DNA distribution inside the nucleus[START_REF] Kanda | Histone-GFP fusion protein enables sensitive analysis of chromosome dynamics in living mammalian cells[END_REF].

Fourier

  transform, defined for �𝑓𝑓 𝑥𝑥 , 𝑓𝑓 𝑦𝑦 � ∈ [0, 𝑀𝑀 -1] × [0, 𝜋𝜋 -1]. Fourier representation allows us to determine the frequencies corresponding to an object of a size 𝜌𝜌 𝑘𝑘 . By angularly projecting the Fourier transform we can represent its value as a function of the size of the object, ignoring the influence of orientation. For this we construct a set of concentric rings Ω 𝑘𝑘 comprising frequencies |𝑓𝑓 𝑥𝑥 | = 𝑀𝑀/2𝜌𝜌 𝑘𝑘 in the x direction and �𝑓𝑓 𝑦𝑦 � = 𝜋𝜋/2𝜌𝜌 𝑘𝑘 in the y direction. By summing the value of 𝑢𝑢 ��𝑓𝑓 𝑥𝑥 , 𝑓𝑓 𝑦𝑦 � along Ω 𝑘𝑘 we obtain the angular projection of 𝑢𝑢 ��𝑓𝑓 𝑥𝑥 , 𝑓𝑓 𝑦𝑦 � as a function of the radial frequency 𝑓𝑓 𝑘𝑘 . Denoting this projection as 𝑈𝑈 � (𝑓𝑓 𝑘𝑘 ): 𝑈𝑈 � (𝑓𝑓 𝑘𝑘 ) = ∑ �𝑢𝑢 ��𝑓𝑓 𝑥𝑥 , 𝑓𝑓 𝑦𝑦 �

Figure 4 - 1 :

 41 Figure 4-1: Maximum intensity projection of an MCTS expressing H2B-GFP. The MCTS was imaged before AO correction and after corrections performed at 30, 40 and 60 µm deep with the guide star placed at the green spots in each case. Corrections were performed using 1s exposures. Scale bar: 25 µm. The highlighted feature is further analysed in Figure 4-3.

Figure 4 -

 4 Figure 4-1 shows a maximum intensity projection of an MCTS imaged up to 150 µm deep. Corrections were performed at depths of 30, 40 and 60 µm, with guide stars located at different suitable points for each correction. The AO-corrected images

Figure 4 - 2 :

 42 Figure 4-2: Measured wavefronts after AO corrections at different depths. a) Correction at 30 µm. b) Correction at 40 µm. c) Correction at 60 µm.

of interest highlighted in figure 4 - 1 .

 41 This ROI features a single cell nucleus located at a depth spanning approximately from z = 55 µm to z = 65 µm. XY and XZ projections of the cell before and after AO corrections are provided in figure 4-3. Improvements in image quality are greater for corrections performed at 40 µm and 60 µm, since the guide star is spatially closer to the feature. While the uncorrected image is too blurry to extract any useful information, AO correction allows us to clearly identify a spatial structure of H2B (and hence DNA). The ring-like distribution shows H2B forming chromosomes that are split apart at different ends of the nucleus, as it is characteristic of a cell undergoing metaphase.

Figure 4 - 3 :

 43 Figure 4-3: Maximum intensity projection of an HCT116 nucleus expressing H2B-GFP, located at a depth of approximately 60 µm inside the MCTS. The cell was imaged before AO correction and after corrections were performed at a depth of 30, 40 and 60 µm. Structured DNA can be easily discerned after corrections at 40 µm and 60 µm. Scale bar: 10 µm.

Figure 4 -

 4 Figure 4-4 shows an MCTS expressing ArrestRed (H2B-tKR) before AO correction and after correction at 30, 50 and 80 µm. The spheroid was imaged up to 120 µm deep each time. In order to quantitatively analyse image quality improvements at different depths, the 3D images were divided into three Z sections (Z1-40µm, Z41-80µm, Z81-120µm), after which a maximum intensity projection was

Figure 4 - 4 :

 44 Figure 4-4: Maximum intensity projections of an MCTS expressing the fluorescent histone ArrestRed before AO correction and after corrections were performed at 30, 50 and 80 µm.

Figure 4 - 5 :

 45 Figure 4-5: Maximum intensity projection of three regions of an MCTS before and after AO correction. NGSs were placed at the green dots in the respective corrections. Scale bar: 25 µm.

2 .

 2 Profiles along the paths highlighted in figure 4-5, along with relative differences of frequencies with respect to the uncorrected image are shown in Figure 4-6.

Figure 4 - 6 :

 46 Figure 4-6: Quantitative analysis of AO correction in MCTS. a) Intensity profiles along paths 1,2 and 3 of Figure 4-5. b) Relative differences of spatial frequencies between each of the corrections and the uncorrected case for each region of interest.

  the lanthanide cations embedded in the host matrix can convert incident NIR light to visible or ultraviolet light. Unlike other nonlinear processes such as second harmonic generation or two-photon fluorescence, photon upconversion involves real intermediate energy levels, with lifetimes of up to tens of milliseconds (compared to hundreds of attoseconds for two-photon fluorescence) [101]. The longer lifetimes of the intermediate states reduce the required excitation power by several orders of magnitude, allowing the use of lamps or continuous wave lasers as illumination sources.

Figure 5 - 1 :

 51 Figure 5-1: CR3936-69P1 up-conversion spectrum.

Figure 5 - 2 :

 52 Figure 5-2: Widefield image of MCTS after 4 days of growth in presence of different concentrations of CR3936-69P1. MCTS grown in 0.1 µg/ml concentrations exhibited a nominal development, while those grown in 1 µg/ml and 10 µg/ml presented a slower growth. Cells grown in 100 µg/ml concentrations did not ultimately form spheroids. Scale bar: 150 µm.

Figure 5 - 3 :

 53 Figure 5-3: Three-dimensional distribution of nanoparticles on MCTS as seen from different angles over the Y axis. UCNP formed aggregates with a typical size of up to tens of µm, which tended to be clustered in one size of the spheroid. MCTS grown in 1 µg/ml CR3936-69P1 presented larger and brighter aggregates of UCNP. Intensity data not shown.

Figure 5 - 4 :

 54 Figure 5-4: Effect of BSA on the three-dimensional distribution of nanoparticles on an MCTS. MCTS were grown in a 1 µg/ml CR3936-69P1, 1 mg/ml BSA solution. No substantial improvements caused by the addition of BSA could be observed.

  UCNP as guide stars is their uneven distribution inside the MCTS due to the low solubility of the nanoparticles. Further research in solubilization strategies for these UCNP with the objective of improving solubility or suspension time could open the CHAPTER 5 : UCNP AS GUIDE STARS 104 way for this application. Specifically, precipitation times longer than 4 or 5 days could provide a uniform UCNP distribution over the volume of the MCTS. Additionally, the infrared light used for the two-photon excitation suffers from minimal absorption and scattering, generating useful NGS up to 100 µm deep in TMs.

Table 7 :

 7 RMS aberration before and after AO correction at different depths. ...... 87

Table 8 :

 8 Size of MCTS grown in the presence of CR3936-69P1 after 4 days. ..... 100

Table 9 :

 9 Size of MCTS grown in the presence of CR3936-69P1 after 8 days. ..... 100 xii

Table 2 - 1 :

 21 Scattering coefficients 𝜇𝜇 𝑠𝑠 (mm

Table 3 -

 3

1.

Table 3 -

 3 1: Waist thickness and Rayleigh range of the light sheet for all the possible

	illumination modes.
	𝝀𝝀	𝒘𝒘 𝟎𝟎 (µm)	𝒛𝒛 𝑹𝑹 (µm)
	491 nm	0.83	4.41
	532 nm	0.90	4.78
	595 nm	1.01	5.29
	980 nm	1.66	8.80

Table 3 -

 3 2: Observed lateral FWHM of the NGS at different depths inside the MCTS.

	Depth (µm) FWHMX (µm) FWHMY (µm)
	0	2.2	2.2
	25	2.4	2.1
	50	2.4	2.3
	75	4.7	4.9
	100	8.7	7.9

Table 3 -

 3 3: Specifications of our Shack-Hartmann wavefront sensor for a 555 nm guide star.

	Aperture Dimension (mm 2 )	7.73 × 7.73
	Number of microlenses	14 × 14
	Microlens step (mm)	0.552
	Focus dynamic range (µm)	24.1
	Wavefront dynamic on 𝒁𝒁 𝟎𝟎 𝟒𝟒

Table 3 -

 3 4: Mean full width at half maximum (FWHM), RMS aberration and Strehl ratio (SR) values for beads imaged through a glass capillary before and after correction.

		AO off	AO on
	FWHMX (µm) 3.38 ± 0.18	2.61 ± 0.12
	FWHMY (µm) 4.68 ± 1.16	2.43 ± 0.28
	FWHMZ (µm) 3.49 ± 0.54	3.78 ± 0.46
	RMS (µm)	0.238 ± 0.014 0.011 ± 0.001
	SR	0.007 ± 0.004 0.983 ± 0.002

Table 3 -

 3 5: Mean correction results for images of beads through a glass capillary for different emission intensities.

			AO on
		7%	8%	9%	10%
	FWHMX (µm) 2.672 ± 0.019 2.695 ± 0.016 2.686 ± 0.018 2.682 ± 0.018
	FWHMY (µm) 2.343 ± 0.030 2.230 ± 0.045 2.277 ± 0.030 2.281 ± 0.025
	FWHMZ (µm) 3.491 ± 0.073 3.483 ± 0.065 3.495 ± 0.070 3.540 ± 0.063
	RMS (µm)	0.025 ± 0.002 0.016 ± 0.001 0.014 ± 0.001 0.013 ± 0.001
	SR	0.909 ± 0.011 0.961 ± 0.003 0.969 ± 0.003 0.973 ± 0.002
	AO performance as a function of the number of closed loop iterations employed in
	the correction for fixed emission intensity was evaluated in a different experiment.

Table 4 -

 4

	Correction depth	RMSoff (µm)	RMSon (µm)
	30 µm	0.096	0.037
	40 µm	0.086	0.025
	60 µm	0.129	0.016
	Image quality improvements after AO correction are especially clear in the region

1: RMS aberration before (RMSoff) and after AO correction (RMSon) at different depths.

Table 5 -

 5 1: Size of MCTS grown in the presence of CR3936-69P1 after 4 days.

	CR3936-69P1 concentration	MCTS size (µm)
	0.1 µg/ml	340.0 ± 32.5
	1 µg/ml	315.8 ± 17.4
	10 µg/ml	283.3 ± 24.6
	100 µg/ml	Irregular cell aggregates

Table 5 -

 5 2: Size of MCTS grown in the presence of CR3936-69P1 after 8 days.

	CR3936-69P1 concentration	MCTS size (µm)
	0.1 µg/ml	490.0 ± 14.1
	1 µg/ml	467.5 ± 25.6
	10 µg/ml	430.8 ± 42.7
	100 µg/ml	Few scattered cells

This work has been the subject of a publication, provided here as an appendix.

Acknowledgements

Hence, many of the mechanisms that cause wavefront distortion can be used to correct it. The physical process by which the wavefront is corrected and the most important correction devices will be presented in this section.

Phase conjugation

The principle behind every AO system is the phase conjugation. Let us consider a distorted wavefront. When this wavefront is reflected in a flat mirror, it preserves its shape while propagating in the opposite direction. But, if the mirror is not flat, the shape of the wavefront will be modified by that of the mirror. Indeed, it is easy to see that if Φ 0 (𝜌𝜌, 𝜌𝜌) is the phase of the incoming wavefront and Φ 𝑀𝑀 (𝜌𝜌, 𝜌𝜌) is the shape of the deformable mirror, the phase Φ(𝜌𝜌, 𝜌𝜌) of the reflected wavefront is given by: Chapter 6

CONCLUSION

Over the last years, TM have acquired increasing relevance in the study of biological processes, since they provide a physiologically relevant environment for cell growth, tissue morphogenesis and disease spread. A popular tool for TM imaging is light sheet fluorescence microscopy, as it provides fast 3D acquisition with minimal photodamage to the sample. However, in-depth optical imaging of TMs poses a challenge due to absorption, scattering and aberration arising from the sample. This thesis is devoted to the improvement of in-depth imaging of such challenging samples by combining LSFM with adaptive optics.

Summary

I have implemented an adaptive SPIM that makes use of a non-linear guide star and a custom high-sensitivity Shack-Hartmann wavefront sensor to correct aberrations arising from the sample. The NGS used in our system can be placed at will over the FOV, allowing the correction of aberrations in any region of interest.

Appendix 1: Detailed list of the optical elements of WAO SPIM

This appendix provides further detail about the components of WAOSPIM that the reader might find useful.