Eddie Maillard

Sandra Bosio

Philippe Deulat

Keywords: field microscopy techniques, illumination microscopy, resolution techniques, resolution methods, saturation microscopy

La microscopie joue un rôle important pour observer et étudier les petits objets. Parmi toutes les techniques d'imagerie de l'état de l'art, la microscopie optique en champ lointain est la plus largement utilisée en biologie, avec une bonne résolution à l'échelle des organites cellulaires. Parce qu'elle ne nécessite pas de préparation complexes des échantillons, son fonctionnement est assez simple, il est le seul moyen d'imager des échantillons épais en trois dimensions dans des conditions physiologiques. La fluorescence a joué un rôle important dans l'évolution de la microscopie optique en champ lointain. En filtrant spectralement les photons excitateurs et en ne collectant que les photons de fluorescence décalée vers le rouge, on peut obtenir des images de très bon contraste et on peut même détecter des nano-émetteurs uniques. Diverses techniques de microscopie à fluorescence telles que la microscopie confocale et la microscopie à deux photons ont été développés avec des avantages distincts en fonction des applications. Cependant la résolution d'un microscope optique est fondamentalement limitée par la diffraction, avec la résolution définie comme la plus petite distance nécessaire pour distinguer deux objets ponctuels. À cause de la diffraction, la lumière provenant d'un objet ponctuel ne converge pas vers un seul point du plan image, mais se répand dans l'espace selon un volume de taille fini, i.e. la fonction d'étalement du point (Point Spread Function ou PSF). L'étendue spatiale tri-dimensionnelle de la PSF détermine la plus petite dimension qui peut être résolu dans une image. Selon Émile Verdet, Ernst Abbe et Lord Rayleigh, la résolution d'un microscope optique est définie par 0.61λ/NA dans la direction latérale, et 2nλ/NA 2 dans la direction axiale, où λ est la longueur

la vitesse d'imagerie, l'amélioration de la résolution en trois dimensions, la profondeur d'imagerie dans l'échantillon et l'applicabilité à des échantillons vivants.

Au cours de cette thèse, nous avons développé deux technique de microscopies de super-résolution: Lattice-STED (optical lattice STED microscopy) et ESSat (excitation state saturation microscopy). Lattice-STED vise à améliorer la vitesse d'imagerie de la microscopie STED, et démontre pour la première fois sa parallélisation massive (jusqu'à 100 fois). La microscopie Essat étend l'application de la microscopie de super-résolution à la température de l'hélium liquide sur des molécules organiques uniques insérées dans une matrice solide. Nous avons atteint une résolution de moins de dix nanomètres à de faibles intensités lumineuses et montré que la microscopie Essat est capable de résoudre au-delà de la limite de diffraction des molécules uniques avec des résonances optiques proches.

Dans la première partie de ce manuscrit, je vais vous présenter Lattice-STED. STED repose sur la combinaison d'un faisceau d'excitation focalisé avec un faisceau de déplétion. Le faisceau de déplétion a une longueur d'onde décalée vers le rouge et généralement une forme de beignet, qui peut désexciter les émetteurs par émission stimulée à la périphérie du faisceau d'excitation. De cette manière, STED réduit l'émission de fluorescence à un volume inférieur à celui limité par diffraction. Pour obtenir une image, il nécessite le balayage des deux faisceaux laser sur l'ensemble du champ de vision pendant l'acquisition. Cette modalité d'imagerie séquentielle implique que le gain de résolution se fait au détriment de la vitesse d'imagerie, ce qui pourrait être une limitation sévère lors de l'étude des structures dynamiques dans les cellules vivantes.

Ainsi, STED doit être parallélisée afin d'améliorer cette résolution temporelle pour permettre l'imagerie rapide à grand champ. Dans ce contexte, nous proposons et démontrons la parallélisation massive du STED par des réseaux optiques. Les réseaux optiques sont des motifs périodiques qui peuvent être générés par des interférences multifaisceaux. Ils contiennent une matrice de cellules élémentaires, dont chacune a un min-imum d'intensité entouré par une distribution d'intensité à peu près isotrope. Chaque cellule élémentaire joue le même rôle que le faisceau en forme de beignet, i.e. limiter le volume d'émission de fluorescence à l'échelle sous-diffraction par effet STED. Pour obtenir une image super-résolue, l'échantillon ne doit être balayé que sur une maille élémentaire du réseau optique au lieu de l'ensemble du champ de vision. Avec cette méthode, nous avons atteint une résolution de 70 nm avec un champ de vision de 3 µ m * 3 µ m. La vitesse d'imagerie de notre microscope Lattice-STED est de 80 ms par image super-résolue, qui est seulement limitée par le temps de lecture de la caméra.

Dans la deuxième partie, je présenterai la microscopie Essat sur des molécules uniques à température cryogénique. Les molécules organiques insérées dans des matrices solides à la température de l'hélium liquide sont de bons candidats pour effectuer des expériences d'optique quantique, en raison de la réduction drastique du déphasage des transitions optiques liés aux phonons, du grand facteur de qualité de la raie zérophonon et de l'excellente photostabilité. Dans de nombreux cas, la mesure des distances nanométriques entre les molécules uniques est nécessaire. Par exemple, la dépendance cubique de la distance de l'interaction dipôle-dipôle entre deux molécules restreint ce phénomène à une échelle de quelques dizaines de nanomètres. En outre, l'extraction de la distance entre un émetteur unique situé dans le champ évanescent d'une nanostructure et la nanostructure elle-même nécessite une excellente précision. Pour effectuer l'imagerie de super-résolution sur des molécules uniques, on excite tout d'abord ces molécules avec un faisceau en forme de beignet, ce qui nous permet de démontrer une résolution de 4,4 nm (largeur à mi-hauteur) avec une intensité d'excitation de seulement 13 kW / cm 2 . L'application supplémentaire d'un faisceau Gaussien temporellement modulé avec une technique de démodulation offre un meilleur contraste pour des images d'échantillons très concentrés. Une résolution de 9 nm a été obtenue avec cette méthode.

Introduction Scientific context

Diffraction limit

Microscopy plays an important role in observing and studying small objects. Being one of the most popular microscopy techniques, far-field optical microscopy uses light to probe the sample and can be compatible with living specimens. It is therefore an essential tool to study living cells and to understand their fundamental functions. An optical microscope gives a magnified image of the sample, which is otherwise not resolvable by human eye. However, the magnification cannot be set arbitrarily high in order to observe the finer details of the sample, because an optical microscope has limited resolution ultimately given by the diffraction, with the resolution defined as the smallest distance needed to distinguish two point objects. Light originating from a point object does not converge to a single point at the image plane, but spreads out in space to a finite-sized volume, called the point spread function (PSF). The three-dimensional spatial extent of the PSF determines the smallest dimension that can be resolved in the image. According to Émile Verdet [START_REF] Verdet | Leçons d'optique physique[END_REF], Ernst Abbe [START_REF] Abbe | Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung[END_REF] and Lord Rayleigh [START_REF] Rayleigh | On the theory of optical images, with special reference to the microscope[END_REF], the resolution of an optical microscope is limited by diffraction as 0.61λ/NA in the lateral direction, and 2nλ/NA 2 in the axial direction, where λ is the light wavelength, NA = nsinα the numerical aperture of the objective, n the refractive index of the microscope medium and α the half-angle of the maximum cone of light that can enter or exit the objective. In the visible range and for an oil-immersion objective (typically of NA ∼ 1.4), the best lateral resolution is around 200 nm and the axial resolution is around 1 µm.

Achieving better resolution

In order to investigate the ultrastructure of biological specimens including microorganisms, cells and molecular assembling, several microscopy methods have been developed which offer better resolution than far-field optical microscopy.

One approach consists of using waves with shorter wavelength. X-ray [START_REF] Goldstein | Scanning electron microscopy and X-ray microanalysis. A text for biologists[END_REF] has typically the wavelength between 0.1 nm and 5 nm. However, the lack of high refractive index material sets a constraint on the achievable resolution. Up to this day, 10 nm resolution has been achieved with Fresnel zone plates [START_REF] Chao | Soft X-ray microscopy at a spatial resolution better than 15 nm[END_REF][START_REF] Vila-Comamala | Advanced thin film technology for ultrahigh resolution X-ray microscopy[END_REF][START_REF] Chao | Real space soft x-ray imaging at 10 nm spatial resolution[END_REF]. Because of the high energy applied on the sample, X-ray microscopy is not compatible with live cell imaging. Furthermore, the lack of available high energy sources limits its applications. Matter waves have even shorter wavelengths. For instance, electron beams have de Broglie wavelength at the order of 10 -4 nm and can be used to achieve sub-angstrom resolution [START_REF] Erni | Atomicresolution imaging with a sub-50-pm electron probe[END_REF]. Electron microscopy [START_REF] Goldstein | Scanning electron microscopy and X-ray microanalysis. A text for biologists[END_REF][START_REF] Williams | Transmission Electron Microscopy[END_REF] has been widely used to study biological specimens. However, similar to X-ray microscopy, it is not compatible with live cell imaging. Moreover, the sample preparation is elaborate [START_REF] Watson | Staining of tissue sections for electron microscopy with heavy metals[END_REF][START_REF] Palade | A study of fixation for electron microscopy[END_REF], often requiring fixing the sample chemically, cutting the sample into slices and marking them with contrast enhancing materials such as gold.

Another approach is scanning probe microscopy, such as scanning near-field optical microscopy [START_REF] Hecht | Scanning near-field optical microscopy with aperture probes: Fundamentals and applications[END_REF][START_REF] Betzig | Single molecules observed by near-field scanning optical microscopy[END_REF], atomic force microscopy [START_REF] Binnig | Atomic force microscope[END_REF][START_REF] Zhong | Fractured polymer/silica fiber surface studied by tapping mode atomic force microscopy[END_REF][START_REF] Meyer | Novel optical approach to atomic force microscopy[END_REF][START_REF] Giessibl | Advances in atomic force microscopy[END_REF] and scanning tunnelling microscopy [START_REF] Hansma | Scanning tunneling microscopy[END_REF][START_REF] Tersoff | Theory of the scanning tunneling microscope[END_REF][START_REF] Eigler | Positioning single atoms with a scanning tunnelling microscope[END_REF]. Near-field scanning microscopies use tips with sub-diffraction sizes (∼ 1Å to 100 nm) and only probe areas at the vicinity of the tips. Therefore the resolution is determined by the size of the probe-sample interaction volume and can achieve atomic scale resolution [START_REF] Ohnesorge | True atomic resolution by atomic force microscopy through repulsive and attractive forces[END_REF][START_REF] Giessibl | Atomic resolution of the silicon (111)-(7x7) surface by atomic force microscopy[END_REF]. However, they are intrinsically surface probing methods and fail to investigate the inner space of a sample.

Fluorescence microscopy

Among all the state-of-art imaging techniques, far-field optical microscopy is the most widely used microscopy in biology, with a good resolution at the scale of cellular organelles. Because it does not require elaborate sample preparations; its operation is rather simple; it is the only way to image thick samples under physiological conditions in three dimensions. Fluorescence [START_REF] Lichtman | Fluorescence microscopy[END_REF] has played an important role in the developments of modern far-field optical microscopy. By spectrally filtering out the exciting photons and collecting only the red-shifted fluorescence photons, one can obtain very good image contrast and can even detect single fluorescent nano-emitters [START_REF] Orrit | Single pentacene molecules detected by fluorescence excitation in a p-terphenyl crystal[END_REF][START_REF] Ambrose | Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation[END_REF]. Various fluorescence microscopy techniques [START_REF] Mertz | Introduction to optical microscopy[END_REF] such as confocal microscopy and two-photon excitation microscopy have been developed with distinct advantages depending on the applications. However, they share the same ultimate resolution limitation imposed by diffraction.

Super-resolution microscopy

By surpassing the diffraction limit and providing resolution down to a few nanometers, super-resolution fluorescence microscopy [START_REF] Godin | Super-resolution microscopy approaches for live cell imaging[END_REF] has revolutionized far-field optical microscopy in the last decade.

The use of the spectroscopic properties of the fluorescent emitters plays an important role in super-resolution methods. Due to diffraction limit, the fluorescence images of different emitters from a diffraction-limited volume overlap and are not distinguishable. Exploiting the photo-physical and photo-chemical properties of emitters allows making them fluoresce sequentially and thus become optically resolvable. The sequential selective imaging is achieved either in stochastic or deterministic ways. Stochastic super-resolution methods are based on the localization of isolated single emitters, such as (fluorescence) photo activated localization microscopy ((f)PALM) [START_REF] Betzig | Imaging intracellular fluorescent proteins at nanometer resolution[END_REF][START_REF] Hess | Ultra-high resolution imaging by fluorescence photoactivation localization microscopy[END_REF], (direct) stochastic optical reconstruction microscopy ((d)STORM) [START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy STORM[END_REF][START_REF] Heilemann | Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes[END_REF] and (universal) points accumulation for imaging in nanoscale topography ((u)PAINT) [START_REF] Sharonov | Wide-field subdiffraction imaging by accumulated binding of diffusing probes[END_REF][START_REF] Giannone | Dynamic superresolution imaging of endogenous proteins on living cells at ultra-high density[END_REF]. Deterministic CONTENTS methods are based on patterned excitation (or depletion) which restrict the fluorescence emission to pre-determined positions, such as stimulated emission depletion (STED) microscopy [START_REF] Hell | Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy[END_REF], ground state depletion (GSD) microscopy [START_REF] Hell | Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit[END_REF] and (saturated) structured illumination microscopy ((S)SIM) [START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF][START_REF] Heintzmann | Saturated patterned excitation microscopy-a concept for optical resolution improvement[END_REF][START_REF] Gustafsson | Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution[END_REF].

Super-resolution microscopy has showed great success in different fields especially in bioimaging. In 2014, the Nobel Prize in Chemistry was given to three researchers "for the development of super-resolved fluorescence microscopy" [START_REF]The Nobel Prize in Chemistry[END_REF]. Even nowadays, superresolution microscopy is still under rapid development. The challenge for researchers consists in improving the performances of super-resolution microscopy, such as the imaging speed [START_REF] Zhu | Faster STORM using compressed sensing[END_REF][START_REF] Grotjohann | rsEGFP2 enables fast RESOLFT nanoscopy of living cells[END_REF][START_REF] Huang | Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms[END_REF], the three-dimensional resolution enhancement [START_REF] Lee | The doublehelix microscope super-resolves extended biological structures by localizing single blinking molecules in three dimensions with nanoscale precision[END_REF][START_REF] Jones | Fast, three-dimensional super-resolution imaging of live cells[END_REF], the imaging depth into the sample [START_REF] Gould | Adaptive optics enables 3D STED microscopy in aberrating specimens[END_REF][START_REF] Urban | STED nanoscopy of actin dynamics in synapses deep inside living brain slices[END_REF] and the applicability to living samples [START_REF] Huang | Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms[END_REF][START_REF] Westphal | Video-rate far-field optical nanoscopy dissects synaptic vesicle movement[END_REF][START_REF] Nagerl | Live-cell imaging of dendritic spines by STED microscopy[END_REF].

During the course of this Doctorate work, we developed two super-resolution microscopies: optical lattice STED microscopy (Lattice-STED) and excitation state saturation (ESSat) microscopy. Lattice-STED aims at improving the imaging speed of STED microscopy, and demonstrates for the first time massive parallelization (up to 100-fold) of STED microscopy. ESSat microscopy extends the application of super-resolution microscopy to liquid helium temperature to image single organic molecules embedded in a solid host. We achieved sub-ten nanometer resolution at low light intensities and showed that ESSat microscopy is capable of resolving beyond the diffraction limit single molecules with close optical resonances.

In the first part of this thesis, I will present Lattice-STED. STED relies on the combination of a focused excitation beam with a depletion beam. The depletion beam has red-shifted wavelength and typically a doughnut shape, which can de-excite by stimulated emission fluorescent emitters at the periphery of the excitation beam. By this, STED restricts the fluorescence emission volume of the sample to a sub-diffraction scale.

To obtain an image, it requires sequential acquisition while scanning the two laser beams over the whole field of view. This sequential imaging modality implies that the resolution gain is at the expense of the imaging speed, which could be a severe limitation when studying dynamic structures in living cells. Fortunately STED microscopy only requires a limited number of fluorescence photons, which can be acquired at short pixel exposure times, to obtain the molecular density at the position of the emission. Therefore fast STED imaging has been achieved with raster scan for small fields of view [START_REF] Westphal | Video-rate far-field optical nanoscopy dissects synaptic vesicle movement[END_REF]. However, STED would need to be parallelized in order to enhance this temporal resolution to allow faster wide-field imaging. In this context, we propose and demonstrate large parallelization of STED by optical lattices. Optical lattices are periodic patterns and can be generated by multi-beam interferences. They contain an array of unit cells, each of which has a centred intensity minimum surrounded by nearly isotropic intensity distribution. Each unit cell plays the same role as the doughnut beam, i.e. restricting the fluorescence emission volume to a sub-diffraction scale by stimulated emission depletion.

To obtain a super-resolution image, the sample only needs to be scanned over a unit cell of the optical lattice instead of the entire field of view. With this method, we achieved a resolution of 70 nm with a field of view of 3 µm * 3 µm. The imaging speed of our Lattice-STED microscope is 80 ms per super-resolution image, which is only limited by the readout time of the camera.

In the second part, I will introduce cryogenic ESSat microscopy performed with single molecules. Organic molecules inserted in solid hosts at liquid helium temperature [START_REF] Orrit | Single pentacene molecules detected by fluorescence excitation in a p-terphenyl crystal[END_REF][START_REF] Ambrose | Detection and spectroscopy of single pentacene molecules in a p-terphenyl crystal by means of fluorescence excitation[END_REF] are good candidates to perform quantum optics experiments [START_REF] Trebbia | Indistinguishable near-infrared single photons from an individual organic molecule[END_REF][START_REF] Lettow | Quantum interference of tunably indistinguishable photons from remote organic molecules[END_REF], due to the drastic reduction of the dephasing related to phonons, the large quality factor of the zero-phonon line and the excellent photostability. In many cases, measurement of sub-diffraction distances between single molecules is necessary. For instance, the cubic dependence of the distance in the dipole-dipole interaction between two molecules [START_REF] Varada | Two-photon resonance induced by the dipole-dipole interaction[END_REF] restricts such phenomenon to a scale of tens of nanometers. Moreover, extracting the distance between a single emitter located in the evanescent field of a nanostructure and the nanostructure itself requires an excellent accuracy. To perform super-resolution imag-ing with single molecules, we first excite the molecules with a doughnut-shaped beam and demonstrate a resolution of 4.4 nm (FWHM) with an excitation intensity of only 13 kW/cm 2 . Application of a modulated Gaussian beam together with a demodulation technique offers a better contrast in images of highly concentrated samples. A resolution of 9 nm was obtained with this method.

Plan of the manuscript

This thesis is structured as follows:

The first chapter introduces the principles of fluorescence microscopy and superresolution microscopy. Firstly, it discusses the bases of several fluorescence microscopies such as confocal microscopy and two-photon excitation microscopy. Then, it gives a brief description of super-resolution microscopies, both deterministic and stochastic methods.

The second chapter presents Lattice-STED. Optical lattices generated by threeand four-beam interferences are discussed. The experimental realization and results of Lattice-STED are then presented.

The third chapter demonstrates cryogenic ESSat super-resolution microscopy. The optical saturation of single molecules and the principle of ESSat microscopy are introduced. The experimental setup and results are then presented.

The last chapter summarizes this thesis and discusses an outlook of further experiments and improvements of both Lattice-STED microscopy and ESSat microscopy.

C H A P I T E R 1

Fluorescence microscopy and super-resolution microscopy 1.1 Fluorescence microscopy techniques

Over the last decades, researchers have developed a large number of fluorescent markers which are now available for labelling virtually any biological systems. Immunofluorescence [START_REF] Raff | Two distinct populations of peripheral lymphocytes in mice distinguishable by immunofluorescence[END_REF][START_REF] Franke | Different intermediate-sized filaments distinguished by immunofluorescence microscopy[END_REF] uses the specificity of antibodies to their antigen to target fluorescent dyes or nanoparticles to specific biomolecules within cells, in order to visualize the distribution of the target molecule through the sample. It can be performed on cells and tissue sections that are fixed by a variety of methods. Many synthetic ligand systems [START_REF] Griffin | Specific covalent labeling of recombinant protein molecules inside live cells[END_REF][START_REF] Keppler | A general method for the covalent labeling of fusion proteins with small molecules in vivo[END_REF][START_REF] Los | HaloTag: a novel protein labeling technology for cell imaging and protein analysis[END_REF] have also been developed to target fluorescent markers to biomolecules in living cells.

Fluorescent proteins [START_REF] Lippincott-Schwartz | Development and use of fluorescent protein markers in living cells[END_REF][START_REF] Verkhusha | The molecular properties and applications of Anthozoa fluorescent proteins and chromoproteins[END_REF] such as green fluorescent protein [START_REF] Tsien | The green fluorescent protein[END_REF][START_REF] Chalfie | Green fluorescent protein as a marker for gene expression[END_REF] can be fused to proteins of interest to study living cells or animals. Overall, fluorescence microscopy is a flourishing field allowing investigations of biological systems.

Wide-field epifluorescence microscopy

The energy diagram of a fluorescent emitter is shown in Figure 1.1. Upon excitation with a laser beam, the emitter can be pumped from the ground state S 0 to a higher vibrational level of the excited state

Confocal microscopy

Invented by Minsky in 1955 [START_REF] Minsky | Memoir on inventing the confocal scanning microscope[END_REF], confocal microscopy [START_REF] Conchello | Optical sectioning microscopy[END_REF] is one of key advances in microscopy. The principle is based on focused light illumination (as opposed to widefield illumination) and the placement of a pinhole in the detection path. The invention of laser [START_REF] Maiman | Stimulated optical radiation in ruby[END_REF] We define h exc (r) and h det (r) as the excitation and detection PSFs, i.e. the normalized excitation intensity distribution in the focal plane of the objective and and the normalized fluorescence intensity distribution in the plane of the pinhole. The effective PSF of confocal microscopy is therefore given by [START_REF] Dudás | Confocal line-scanning microscope with modified illumination[END_REF]:

h conf (r) = h exc (r) • [h det (r) ⊗ D(r)] (1.1)
where D(r) is the aperture function of the pinhole and ⊗ is the convolution sign.

The excitation and detection wavelengths being approximately the same, the two PSFs are approximately identical. In practice, the diameter of the pinhole is chosen to be equal or smaller than the full-width-at-half-maximum (FWHM) of the excitation and detection PSFs, hence we have: The excitation PSF is given by [START_REF] Zhang | Gaussian approximations of fluorescence microscope point-spread function models[END_REF]:

h conf (r) h exc (r) 2 . (1.2)
h exc = 2J 1 (k NA r) k NA r 2 (1.3)
where J 1 is the Bessel function of the first kind of order one, k = 2π/λ is the wavenumber, NA is the numerical aperture and r is the radial distance from the optical axis.

Approximating the excitation PSF by a Gaussian distribution, the lateral resolution of confocal microscopy compared to that of wide-field fluorescence microscopy is improved by a factor of 1.4. In the following, we approximate the confocal PSF by a

Gaussian distribution with a FWHM d c and an amplitude equalling unity as:

h conf (r) = exp(-4ln2 r 2 /d 2 c ).
(1.4)

Two-photon excitation microscopy

With few exceptions biological tissues strongly scatter light, especially in the visible range. While confocal microscopy can reject the out-of-focus fluorescence in general, it fails to do so in deep tissues, where strongly scattered out-of-focus fluorescence photons may still pass through the pinhole. In addition, in-focus fluorescence photons may also be scattered and using a pinhole causes a net reduction of the detected signal. Therefore confocal microscopy is not readily applicable to thick tissues.

Two-photon excitation microscopy [START_REF] Denk | Two-photon laser scanning fluorescence microscopy[END_REF][START_REF] Helmchen | Deep tissue two-photon microscopy[END_REF] has overcome this limitation and provides an imaging depth more than 500 µm in tissues [START_REF] Svoboda | In vivo dendritic calcium dynamics in neocortical pyramidal neurons[END_REF][START_REF] Kleinfeld | Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex[END_REF][START_REF] Helmchen | In vivo dendritic calcium dynamics in deep-layer cortical pyramidal neurons[END_REF][START_REF] Theer | On the fundamental imaging-depth limit in twophoton microscopy[END_REF]. A nonlinear absorption process involving two photons [START_REF] Peticolas | Double photon excitation in organic crystals[END_REF] is used (see Figure 1.4), which requires higher laser intensities for excitation than that in the single-photon case. Two-photon excitation microscopy only became possible with the emergence of pulsed lasers [START_REF] Svelto | Principles of lasers[END_REF] which can provide high peak intensities with reasonable average intensity. The probability of absorbing two photons being proportional to the square of the light intensity, the fluorescence excitation volume is restricted to the focal area of the beam. For commonly used fluorophores, two-photon absorption requires near-infrared wavelength excitation (700 -1000 nm).

Interestingly, near-infrared light also penetrates deeper into scattering tissue than visible light. Similar to confocal microscopy, two-photon excitation microscopy uses a focused beam on the sample and requires sample scanning to form an image. However, twophoton excitation microscopy does not require a pinhole on the detection path since out-of-focus excitation is minimal with the two-photon process. Assuming that the excitation PSF is h exc and the detector is much bigger than the diffraction limit, the effective PSF of two-photon excitation microscopy is given by:

1-photon excitation 2-photon excitation

h T P (r) = h exc (r) 2 . (1.5)
Note that although Equation 1.5 is similar to the expression of the confocal PSF (Equation 1.1), the excitation wavelengths are different in the two cases. Therefore the resolution of two-photon excitation microscopy remains approximately the same as wide-field microscopy.

Improving the axial resolution

Both confocal microscopy and two-photon excitation microscopy have axial resolutions of about 1 µm, much larger than their lateral resolutions (about 200 nm). Several other methods have been developed to tackle this problem. Total internal reflection (TIRF) microscopy [START_REF] Ambrose | A surface contact microscope for the study of cell movements[END_REF][START_REF] Axelrod | Cell-substrate contacts illuminated by total internal reflection fluorescence[END_REF] is a wide-field technique which uses illumination light with incident angle higher than the critical angle, so that the light is reflected at the interface of the microscope media and the sample. Only evanescent wave penetrates into the sample.

Since the evanescent wave has a penetration depth of about 100 nm, only fluorescent objects close to the interface within this distance are illuminated. TIRF microscopy is well suited to study structures close to the interface, such as cell adhesion [START_REF] Ezratty | Microtubuleinduced focal adhesion disassembly is mediated by dynamin and focal adhesion kinase[END_REF] and membrane dynamics [START_REF] Leake | Stoichiometry and turnover in single, functioning membrane protein complexes[END_REF]. Other methods such as 4Pi microscopy [START_REF] Hell | Properties of a 4Pi confocal fluorescence microscope[END_REF][START_REF] Hell | Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation[END_REF] and I 5 M [START_REF] Gustafsson | Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses[END_REF][START_REF] Gustafsson | I5M: 3D widefield light microscopy with better than 100 nm axial resolution[END_REF],

rely on PSF engineering with two opposing objectives, which generate a PSF with a nearly spherical symmetry. 4Pi microscopy is based on confocal microscopy and I 5 M is a wide-field technique. They both achieve an axial resolution of about 100 nm.

The aforementioned fluorescence microscopies have extended far-field optical microscopy to many new applications, however still with limited resolution, about 200 nm in all three dimensions. In order to study the stuctures of the sample at nanoscales, one needs to come up with new solutions providing better resolution. This was for a long time thought impossible due to diffraction limit, until the rise of super-resolution microscopy [START_REF] Hell | Toward fluorescence nanoscopy[END_REF][START_REF] Hell | Far-field optical nanoscopy[END_REF][START_REF] Huang | Super resolution fluorescence microscopy[END_REF] during the past two decades.

Super-resolution optical far-field microscopy techniques

To date, super-resolution methods can be separated into two categories [START_REF] Jensen | Coordinate-targeted and coordinatestochastic super-resolution microscopy with the reversibly switchable fluorescent rrotein dreiklang[END_REF]. The first one concerns deterministic methods based on patterned excitation (or depletion), including stimulated emission depletion (STED) microscopy [START_REF] Hell | Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy[END_REF], ground state depletion (GSD) microscopy [START_REF] Hell | Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit[END_REF] and (saturated) structured illumination microscopy ((S)SIM) [START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF][START_REF] Heintzmann | Saturated patterned excitation microscopy-a concept for optical resolution improvement[END_REF][START_REF] Gustafsson | Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution[END_REF].

The second one concerns stochastic methods based on single molecule localization, such as (fluorescence) photo activated localization microscopy ((f)PALM) [START_REF] Betzig | Imaging intracellular fluorescent proteins at nanometer resolution[END_REF][START_REF] Hess | Ultra-high resolution imaging by fluorescence photoactivation localization microscopy[END_REF], (direct) stochastic optical reconstruction microscopy ((d)STORM) [START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy STORM[END_REF][START_REF] Heilemann | Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes[END_REF] and (universal) points accumulation for imaging in nanoscale topography ((u)PAINT) [START_REF] Sharonov | Wide-field subdiffraction imaging by accumulated binding of diffusing probes[END_REF][START_REF] Giannone | Dynamic superresolution imaging of endogenous proteins on living cells at ultra-high density[END_REF].

STED was proposed in 1994 [START_REF] Hell | Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy[END_REF] and was experimentally realized in 1999 [START_REF] Klar | Subdiffraction resolution in far-field fluorescence microscopy[END_REF].

Thereafter, new super-resolution methods continue to emerge. SIM was introduced in 2000 [START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF], which offers a resolution enhancement of a factor of two compared to the diffraction limit. Based on the same principle as SIM, SSIM [START_REF] Heintzmann | Saturated patterned excitation microscopy-a concept for optical resolution improvement[END_REF][START_REF] Gustafsson | Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution[END_REF] can further provide theoretically unlimited resolution. Super-resolution method based on single molecule localization was proposed in 1985 [START_REF] Burns | Strategies for attaining superresolution using spectroscopic data as constraints[END_REF] and was first realized in 1998 with single molecules in a solid at low temperature [START_REF] Van Oijen | 3-Dimensional super-resolution by spectrally selective imaging[END_REF] and later in 2006 with biological samples [START_REF] Betzig | Imaging intracellular fluorescent proteins at nanometer resolution[END_REF][START_REF] Hess | Ultra-high resolution imaging by fluorescence photoactivation localization microscopy[END_REF][START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy STORM[END_REF].

These are the three most important super-resolution methods nowadays, which will be presented hereafter.

Stimulated emission depletion microscopy

Principles

The initial realization of STED microscopy [START_REF] Clausen | Pathways to optical STED microscopy[END_REF] During STED pulses, i.e. for 0 < t < t p , the relative populations P S1 of S 1 and P *

S0

of S * 0 evolve with time as:

∂ ∂t P S1 = -ΓP S1 - σλI p hc (P S1 -P * S0 ) ∂ ∂t P * S0 = -Γ vib P * S0 + σλI p hc (P S1 -P * S0 ) (1.6)
where Γ = τ -1 is the relaxation rate of S 1 , Γ vib is the vibrational relaxation rate from S * 0 to the lowest vibrational level of S 0 , λ and I p are respectively the wavelength and peak intensity of the STED laser, σ is the cross-section of the fluorophore at λ, h is the Planck constant and c the speed of light in vacuum. At t = 0, P S1 (0) = 1 and all other states are unpopulated. Because t p 1/Γ vib , the depopulation of S * 0 is fast during the STED pulses. P S1 therefore approximately writes as:

P S1 (t) = exp(-Γt) exp(- σλI p hc min(t, t p )). (1.7)
The first and second exponentials describe the de-excitation of S 1 due to respectively spontaneous and stimulated emissions.

Since t p is much shorter (a few hundreds of picoseconds) than τ , the fluorescence emitted at 0 ≤ t ≤ t p can be neglected. The probability of the emission of one fluores-cence photon during one excitation cycle can be expressed as:

P em (I) = η rad Γ Tp 0 P S1 (t)dt ≈ η rad Γ Tp tp P S1 (t)dt ≈ η rad exp(-Γt p ) exp(- σλI p hc t p ) = η rad exp(-Γt p ) exp(- ln2 I I S ) (1.8)
where η rad is the quantum yield of the fluorophores, I = I p t p /T p is the average STED intensity, I S = ln2 hc/(σλT p ) is the average saturation intensity at which half of the fluorescence is depleted [START_REF] Harke | Resolution scaling in STED microscopy[END_REF]. Obviously, I S depends on the spectroscopic properties of the fluorophores and the repetition rate of the STED laser. A STED laser with the same average power but lower repetition rates is therefore more efficient for depletion. At low excitation intensities, the effective PSF of STED microscopy can be then written as:

h ST ED (r) = h exc (r) P em (I donut (r)) [h det (r) ⊗ D(r)] ≈ h conf (r) P em (I donut (r)) (1.9)
where I donut (r) is the intensity distribution of the doughnut beam. It can be described by the Laguerre-Gaussian mode LG(0,1) and writes as [START_REF] Pampaloni | Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: a primer[END_REF]:

I donut (r) = eI ST ED 4r 2 d 2 exp(-
4r 2 d 2 ) (1.10)
where I ST ED is the maximal average intensity of the doughnut beam and d is its diameter, i.e. the distance between two opposing intensity maxima. The intensity profiles of the excitation and STED beams are shown in Figure 1.8A. The STED resolution depends strongly on the STED intensity. The higher the STED intensity is, the narrower the STED PSF becomes, as shown in Figure 1.8B.

Calculation of the resolution of pulsed-STED

For simplification, we approximate the doughnut beam with a sinus function near the optical axis [START_REF] Westphal | Nanoscale resolution in the focal plane of an optical microscope[END_REF]:

I donut (r) = I ST ED sin 2 ( πr d
).

(1.11) In the vicinity of the zero-intensity position, the STED profile is a parabola and its intensity distribution can be approximated as:

I donut (r) ≈ I ST ED ( πr d ) 2 .
(1.12)

Using Equation 1.8 and Equation 1.9, the PSF of pulsed-STED microscopy writes:

h p (r) = h conf (r) η rad exp(-Γt p ) exp(- π 2 r 2 d 2 σλ hc I ST ED T p ). (1.13)
The resolution of STED microscopy can be defined as the FWHM of its PSF h p (r)

and is given by: [START_REF] Rittweger | STED microscopy reveals crystal colour centres with nanometric resolution[END_REF][START_REF] Wildanger | Solid immersion facilitates fluorescence microscopy with nanometer resolution and sub-Ångström emitter localization[END_REF].

∆r p = d c 1 + π 2 d 2 c 4d 2 σλ ln2 hc I ST ED T p = d c 1 + π 2 d 2 c 4d 2
I ST ED I S . ( 1

Calculation of the resolution of gated continuous-STED

Pulsed-STED microscopy provides the most efficient fluorescence depletion, since it provides high laser peak intensities and the laser sequences can be optimized for efficient STED action. However, this configuration usually requires careful synchronization of the excitation and STED lasers and the use of complex and costly laser assemblies such as mode-locked laser systems [START_REF] Klar | Subdiffraction resolution in far-field fluorescence microscopy[END_REF][START_REF] Donnert | Macromolecularscale resolution in biological fluorescence microscopy[END_REF]. A simplification to STED microscopy has therefore been realized with continuous-wave (CW) lasers for both excitation and depletion [START_REF] Willig | STED microscopy with continuous wave beams[END_REF]. However, in CW-STED, the STED energy is stretched out and the depletion occurs gradually after the excitation of molecules. Consequently, a non-negligible part of the molecules emit fluorescence before having been sufficiently exposed to the STED light. This will lead to residual fluorescence outside the zero-intensity point of the STED light which adds a pedestal in the effective STED PSF, resulting in increased image blur. A remedy to this is using a time-gated detection scheme, referred to as "gated-STED" [START_REF] Moffitt | Time-gating improves the spatial resolution of STED microscopy[END_REF][START_REF] Vicidomini | Sharper low-power sted nanoscopy by time gating[END_REF]. In this case, a CW STED laser is used in combination with a pulsed excitation laser, the detection begins after a delay time T g from the arrival of the excitation pulses, as shown in Figure 1.10.

To explain why time-gated detection can improve the performances of CW-STED, In the case of gated-STED, the temporal evolutions of the populations of S * 0 and S 1 follow the same equations as pulsed-STED, with the only difference that t p equals to T p .

For 0 ≤ t ≤ T p , the temporal PSF of gated-STED is given by:

hg (t, r) = h conf (r) η rad Γ exp(-Γt) exp(- π 2 r 2 d 2 σλ hc I ST ED t). (1.16)
In Figure 1.11, we show simulations of the temporal PSFs in the gated-STED and pulsed-STED configurations. In pulsed-STED, the STED pulses reach the focal plane virtually simultaneously or within a few picoseconds after the excitation pulses so as to instantly inhibit fluorescence emission from excited molecules. In gated-STED case, the STED process takes place during the whole temporal interval between excitation cycles.

Molecules can in principle emit fluorescence before having been de-excited by the STED light, which enlarges the effective STED PSF. A time-gated detection can therefore improve the resolution of CW-STED by rejecting the early fluorescence, where the temporal PSF is large.

More precisely, the effective PSF of gated-STED writes:

h g (r) = h conf (r) η rad Γ Tp Tg P S1 (t)dt ≈ h conf (r) η rad exp(-ΓT g ) exp(- π 2 r 2 d 2 σλ hc I ST ED T g ) (1 + π 2 r 2 d 2 σλ hc I ST ED τ ) -1 (1.17)
In the absence of gating (T g = 0), the PSF of CW-STED has a Lorentzian shape.

When a gated detection is applied (T g > 0), the PSF consists of a Gaussian term and a Lorentzian term. The Gaussian term reflects the fluorescence depletion due to the STED action before the detection. By replacing the Lorentzian term with a Gaussian term having the same FWHM, the FWHM of the gated-STED PSF is given by:

∆r g = d c 1 + π 2 d 2 c 4d 2 σλ ln2 hc I ST ED (ln2 τ + T g ) (1.18)
Note that the resolution of CW-STED without time-gated detection can be obtained from Equation 1.18 by setting T g = 0. For pulsed-STED the resolution depends on the pulse energy I ST ED T p , whereas for gated-STED it depends on I ST ED (ln2 τ + T g ).

Therefore higher laser powers have to be applied in CW-STED to achieve the same resolution as in pulsed-STED, typically by about a factor of five (T p /(ln2 τ + T g )) .

A sharpening of the PSF of gated-STED is obtained with increasing time delay T g , at the expense of a decrease in fluorescence signal. The PSF of gated-STED can also be regarded as a weighted sum of different Gaussian distributions with decreasing FWHMs and decreasing weights represented by the temporal PSF. Collecting the photons after a time delay Tg from the excitation pulse, i.e. performing time-gated detection, removes the early temporal PSFs with larger FWHMs and therefore improves the effective resolution at the expense of a loss in overall signal, as outlined in Figures 1.11.

Other developments in STED microscopy

Besides CW-STED, another development for setup simplification was proposed under the acronym easySTED microscopy [START_REF] Wildanger | STED microscope aligned by design[END_REF]. It uses a special phase plate which selectively modulates the phase of the STED beam but leaves the excitation beam unaffected, so that the excitation and STED beams can share the same optical path. The alignment requirements and the mechanical drifts are both reduced in this case.

Dual-color STED is an important development to determine the relative sites and proximities of different molecules in specimens. For this purpose, two pairs of excitation/depletion beams, each pair exciting and depleting one specific emitter, can be used [START_REF] Donnert | Two-color far-field fluorescence nanoscopy[END_REF][START_REF] Meyer | Dual-color STED microscopy at 30-nm focal-plane resolution[END_REF]. This approach has several drawbacks due to its complexity, cost and the cross-photobleaching of the more red-emitting dyes by the STED lasers with shorter wavelength. A preferable approach for dual-color STED is to use fluorophores with partially overlapping emission spectra so that only one depletion beam is needed for both fluorophores. In this case, the two fluorophores are either excited at different wavelengths [START_REF] Tønnesen | Two-color STED microscopy of living synapses using a single laser-beam pair[END_REF][START_REF] Schmidt | Spherical nanosized focal spot unravels the interior of cells[END_REF] or detected in two spectral channels [START_REF] Pellett | Two-color STED microscopy in living cells[END_REF].

In many STED microscopy applications, imaging with two-dimensional resolution enhancement is good enough to reveal details of molecular assemblies. Still, many applications require imaging with super-resolution in all three dimensions. Two-photon excitation STED allows imaging in thick brain tissues [START_REF] Ding | Supraresolution imaging in brain slices using stimulated-emission depletion two-photon laser scanning microscopy[END_REF][START_REF] Moneron | Two-photon excitation STED microscopy[END_REF][START_REF] Bethge | Two-photon excitation STED microscopy in two colors in acute brain slices[END_REF], however with the axial resolution being diffraction-limited. To improve resolutions along all spatial directions, different phase plates have been adapted in STED microscopy to create the desired fluorescence restriction [START_REF] Harke | Three-dimensional nanoscopy of colloidal crystals[END_REF][START_REF] Hell | Diffraction-unlimited threedimensional optical nanoscopy with opposing lenses[END_REF][START_REF] Wildanger | A compact STED microscope providing 3D nanoscale resolution[END_REF]. The implementation of adaptive optics has further allowed three-dimensional super-resolution imaging in strongly aberrating specimen [START_REF] Gould | Adaptive optics enables 3D STED microscopy in aberrating specimens[END_REF]. More complex setups for three-dimensional STED have been realized in combination with 4Pi microscopy [START_REF] Schmidt | Spherical nanosized focal spot unravels the interior of cells[END_REF][START_REF] Dyba | Focal spots of size λ/23 open up far-field florescence microscopy at 33 nm axial resolution[END_REF] or with light-sheet microscopy [START_REF] Friedrich | STED-SPIM: stimulated emission depletion improves sheet illumination microscopy resolution[END_REF].

STED microscopy has showed applications in immuno-stained fixed samples [START_REF] Dyba | Immunofluorescence stimulated emission depletion microscopy[END_REF],

lithography [START_REF] Westphal | Stimulated emission depletion microscopy on lithographic nanostructures[END_REF][START_REF] Wollhofen | 120 nm resolution and 55 nm structure size in STED-lithography[END_REF], condensed matter [START_REF] Wildanger | Diffraction unlimited all-optical recording of electron spin resonances[END_REF] and living cells [START_REF] Westphal | Video-rate far-field optical nanoscopy dissects synaptic vesicle movement[END_REF][START_REF] Nagerl | Live-cell imaging of dendritic spines by STED microscopy[END_REF]. It has been also combined with fluorescence correlation spectroscopy, which allows a more detailed investigation of molecular interaction at small spatial scales [START_REF] Kastrup | Fluorescence fluctuation spectroscopy in subdiffraction focal volumes[END_REF][START_REF] Blom | Fluorescence fluctuation spectroscopy in reduced detection volumes[END_REF]. is a general term which represents such methods. Ground state depletion (GSD) microscopy [START_REF] Hell | Ground-state-depletion fluorscence microscopy: A concept for breaking the diffraction resolution limit[END_REF] depletes the ground state of the emitter either by pumping it to the excited state [START_REF] Rittweger | Far-field fluorescence nanoscopy of diamond color centers by ground state depletion[END_REF] or to a metastable dark state [START_REF] Han | Metastable dark states enable ground state depletion microscopy of nitrogen vacancy centers in diamond with diffraction-unlimited resolution[END_REF]. RESOLFT methods using reversible photoswitchable proteins have been also demonstrated [START_REF] Grotjohann | rsEGFP2 enables fast RESOLFT nanoscopy of living cells[END_REF][START_REF] Hofmann | Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using re-versibly photoswitchable proteins[END_REF][START_REF] Schwentker | Wide-field subdiffraction RESOLFT microscopy using fluorescent protein photoswitching[END_REF]. To be switched "on"

1.
and "off", reversible photoswitchable proteins require low light intensity, which reduces the bleaching of the fluorophores and the photo-toxicity to the sample. However, protein switching is a relatively slow on-off process ( 10 ms), which sets a limit to the imaging acquisition rate.

Structured illumination microscopy

Structured illumination microscopy (SIM) uses spatially modulated (e.g. sinusoidal)

intensity patterns to illuminate the sample. The period patterns can be obtained by two-beam interference, as shown in Figure 1.12A. An object containing fine details can be decomposed in Fourier components where the finer details are encoded in the higher frequencies. Suppose that an object, which has a single Fourier component, is illuminated by a period pattern, the resulting image contains "moire fringes" at lower frequencies than the original object, and can become observable by the microscope even if the original object is too fine to resolve (see Figure 1.12B). The illumination pattern needs to be scanned within a period (by changing the relative phases between the two beams) and be rotated, to render isotropic lateral resolution. A camera acquires successive wide-field images during the scan and the rotation. Post data-processing is required in order to retrieve an image of the object, with resolution improved by a factor of two compared to the diffraction-limited resolution. Multicolor and three-dimensional resolution enhancement with SIM has been realized [START_REF] Schermelleh | Subdiffraction multicolor imaging of the nuclear periphery with 3D structured illumination microscopy[END_REF]. Live cell imaging has also been demonstrated [START_REF] Kner | Super-resolution video microscopy of live cells by structured illumination[END_REF]. Instead of the period patterns, SIM can be also performed with unknown speckle illumination [START_REF] Mudry | Structured illumination microscopy using unknown speckle patterns[END_REF]. SIM has been combined with tunable surface plasmon interference as an illumination source to offer higher resolution in the near-field [START_REF] Wei | Plasmonic structured illumination microscopy[END_REF].

Saturated structured illumination microscopy (SSIM) applies modulated patterns with higher excitation intensities to saturate the fluorescence emission of the fluorophores.

Under such strong excitation, most emitter molecules are pumped to the excited state, additional intensity increases will not yield proportional increases in the emission rate.

The effective applied modulation pattern is not sinusoidal any more, but contains higher frequency components. The saturation effect therefore can further improve the resolution. From another viewpoint, SSIM can be seen as 'negative' imaging methods. The super-resolution information is from the valleys of the illumination pattern where the fluorophores are not excited and therefore do not fluoresce. SSIM simply confines the regions where fluorophores do not emit fluorescence to a smaller volume which offers better resolution. Since SSIM applies high laser intensities on the fluorophores, its implementation was long impeded by the photobleaching of the fluorophores. The first experimental realization of SSIM was then based on photoswitchable proteins [START_REF] Rego | Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution[END_REF].

Localization based super-resolution techniques

Principles

An image of a sample labelled with fluorophores is essentially defined by the spatial coordinates of each emitter convoluted by the PSF of the microscope. If one can determine the position of each emitter individually with precision better that the diffraction-limited resolution, a super-resolution image could be reconstructed by assigning corresponding coordinates to the fluorophores. Indeed, for an isolated emitter, although its image is a finite spot (see Figure .1.13A), its position can be determined with high precision by fitting the image with a two-dimensional Gaussian function (approximation of the PSF).

For a shot noise limited detection, the localization precision in the transverse plane is given by:

σ x,y ≈ ∆r √ N (1.19)
where ∆r is the FWHM of the PSF and N is the photon number. Given the fact that more than 10 4 photons can be detected from a single emitter, the localization precision can reach ∼ 1 nm [START_REF] Yildiz | Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization[END_REF][START_REF] Qu | Nanometer-localized multiple single-molecule fluorescence microscopy[END_REF].

A B 1 2 3 4
-300

-300 -300 -300

x (nm) y (nm)

Different approaches to sequentially image isolated single molecules

Different approaches have been implemented, distinguishing how fluorescent molecules are stochastically isolated from nonfluorescent ones.

The development of PALM [START_REF] Betzig | Imaging intracellular fluorescent proteins at nanometer resolution[END_REF][START_REF] Hess | Ultra-high resolution imaging by fluorescence photoactivation localization microscopy[END_REF] is closely linked to the advent of photoactivatable proteins, which allows the optical control of the density of fluorescing proteins in each raw image. Although it is restricted to expression systems, preventing the study of endogenous proteins in their native environment, PALM has the advantage of being versatile and specific and allows the study of intracellular biomolecules. By tracking the movement of individual proteins, PALM can be used to measure local diffusion properties in living cells on short timescales [START_REF] Manley | High-density mapping of singlemolecule trajectories with photoactivated localization microscopy[END_REF][START_REF] Rossier | Integrins β1 and β3 exhibit distinct dynamic nanoscale organizations inside focal adhesions[END_REF][START_REF] Gudheti | Actin mediates the nanoscale membrane organization of the clustered membrane protein influenza hemagglutinin[END_REF] and cellular structural changes in three dimensions on longer timescales [START_REF] Fu | In vivo structure of the E. coli FtsZ-ring revealed by photoactivated localization microscopy (PALM)[END_REF]. STORM [START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy STORM[END_REF] and dSTORM [START_REF] Heilemann | Subdiffraction-resolution fluorescence imaging with conventional fluorescent probes[END_REF] use switchable organic fluorophores placed in specific buffers (e.g. with reducing properties) instead of using fluorescent proteins as in PALM.

Those probes can be targeted on genetically encoded or endogenous proteins using adequate ligands. STORM was first demonstrated using Cy3-Cy5 pairs [START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy STORM[END_REF] but was quickly extended in dSTORM to other synthetic fluorophores which can be stochastically and reversibly switched in imaging buffers. STORM and dSTORM are particularly pow-erful for fixed cell applications [START_REF] Huang | Three-dimensional superresolution imaging by stochastic optical reconstruction microscopy[END_REF][136][START_REF] Löschberger | Super-resolution imaging visualizes the eightfold symmetry of gp210 proteins around the nuclear pore complex and resolves the central channel with nanometer resolution[END_REF]. Analogously to dSTORM, where stochastic photoswitching is applied to control the number of emitting fluorophores, ground state depletion microscopy followed by individual molecular return (GSDIM) [START_REF] Fölling | Fluorescence nanoscopy by groundstate depletion and single-molecule return[END_REF] employs the transition between the fluorescent singlet state and the metastable triplet state as a stochastic on-off switch. More precisely, the majority of fluorophores are switched to the long-lived triplet state by using high excitation intensities together with an imaging buffer (triplet quencher). The triplet state lifetime is long enough (∼ 100 µs -100 ms) to leave only a few fluorophores at singlet states at any time in each image. Under these conditions, GSDIM has been employed for imaging living cells using both fluorescent protein tags [START_REF] Lalkens | MRT letter: Nanoscopy of protein colocalization in living cells by STED and GSDIM[END_REF] (as in PALM) or various organic fluorophores (as in STORM) that selectively bind to tagged proteins [START_REF] Lukinavičius | A near-infrared fluorophore for live-cell super-resolution microscopy of cellular proteins[END_REF].

In contrast to PALM, STORM and GSDIM, which are based on the stochastic photoswitching of the emitters, (u)PAINT [START_REF] Sharonov | Wide-field subdiffraction imaging by accumulated binding of diffusing probes[END_REF][START_REF] Giannone | Dynamic superresolution imaging of endogenous proteins on living cells at ultra-high density[END_REF] captures real-time molecular interactions to control the density of fluorescent emitters in each image. In uPAINT, target molecules are individually imaged when binding to a specific ligand coupled to a fluorescent dye.

Unbound ligands freely diffuse in the imaging buffer with typical diffusion constants of ∼ tens of µm 2 /s, in contrast to bound ligands, which diffuse together with a membrane receptor with typical diffusion constants of <1 µm 2 /s. Due to an oblique illumination, unbound ligands are not excited in an efficient manner and are therefore not detected efficiently by a detector operating at a typical video rate. With uPAINT, any binding entity conjugated to fluorophores having high specificity toward a target molecule (e.g., natural/synthetic ligand, antibody) can be used as fluorescent probes to reveal the target molecule. uPAINT has been applied to image and track endogenous receptors such as glutamate receptors in neurons [START_REF] Giannone | Dynamic superresolution imaging of endogenous proteins on living cells at ultra-high density[END_REF] and epidermal growth factor receptors (EGFRs) at high densities in culture living cells [START_REF] Winckler | Identification and super-resolution imaging of ligand-activated receptor dimers in live cells[END_REF].

Other developments in localization-based techniques

Three-dimensional super-resolution has been achieved with localization based techniques by also localizing the positions of the fluorophores along the z-axis, with the engineering of the detection PSF. This can be done by using a cylindrical lens to introduce astigmatism [START_REF] Huang | Three-dimensional superresolution imaging by stochastic optical reconstruction microscopy[END_REF], dual focal-plane imaging [START_REF] Juette | Three-dimensional sub-100 nm resolution fluorescence microscopy of thick samples[END_REF] and double-helix PSF imaging [START_REF] Pavani | Three-dimensional, singlemolecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function[END_REF].

Multi-color imaging can be achieved either by using fluorophores with different activation wavelengths [START_REF] Bates | Multicolor superresolution imaging with photo-switchable fluorescent probes[END_REF][START_REF] Huang | Whole-cell 3D STORM reveals interactions between cellular structures with nanometer-scale resolution[END_REF][START_REF] Andresen | Photoswitchable fluorescent proteins enable monochromatic multilabel imaging and dual color fluorescence nanoscopy[END_REF] or (and) with different emission wavelengths [START_REF] Bossi | Multicolor far-field fluorescence nanoscopy through isolated detection of distinct molecular species[END_REF][START_REF] Subach | Photoactivatable mCherry for high-resolution two-color fluorescence microscopy[END_REF].

The fluorophores are dipoles with different orientations, thus they are either fixed in the specimen or have certain degree of rotational mobility. While the emission PSFs of rotating fluorophores can be well fitted with a two-dimensional Gaussian function [START_REF] Backlund | The role of molecular dipole orientation in single-molecule fluorescence microscopy and implications for super-resolution imaging[END_REF], those of fluorophores with fixed orientation cannot, which induces localization bias [START_REF] Engelhardt | Molecular orientation affects localization accuracy in superresolution farfield fluorescence microscopy[END_REF].

When imaging molecules with quasi-fixed orientations, several methods [START_REF] Backer | Singlemolecule orientation measurements with a quadrated pupil[END_REF][START_REF] Backer | A bisected pupil for studying single-molecule orientational dynamics and its application to three-dimensional super-resolution microscopy[END_REF] allow measuring the molecular orientation and subsequently correcting the localization bias.

Other super-resolution methods

Besides these three major super-resolution techniques, other methods based on distinct mechanisms have also been proposed and realized.

Saturated excitation microscopy

Saturated excitation in confocal microscopy has been used to improve the spatial resolution [START_REF] Fujita | High-resolution confocal microscopy by saturated excitation of fluorescence[END_REF][START_REF] Yonemaru | Saturated excitation microscopy with optimized excitation modulation[END_REF]. Similar as in SSIM, high excitation intensity is applied and the fluorescence signal is saturated. Consequently, the excitation PSF gets flattened and widened, which contains high spatial frequency components. Since these high spatial frequency components occur first at the peak of the laser excitation, they carry structural information on a scale that is smaller than the PSF and thus, may be used to improve resolution in all three dimensions. In contrast to SSIM, the high spatial frequency is extracted by modulating the excitation intensity at a frequency (ω) and by demodulating the flu-orescence intensity at the corresponding harmonic frequencies (2ω, 3ω, . . .). When the excitation intensity is high enough to generate saturation in fluorescence excitation, harmonic frequencies appear in the modulated fluorescence signal because the relationship between fluorescence and excitation intensity becomes nonlinear. This effect can be seen prominently at the center of the PSF of excitation, and can be used to detect fluorescence intensity at the position close to the center of PSF. This technique has been recently extended to other saturation effects to perform super-resolution, such as the scattering saturation of single gold nanoparticles [START_REF] Chu | Saturation and reverse saturation of scattering in a single plasmonic nanoparticle[END_REF][START_REF] Chu | Measurement of a saturated emission of optical radiation from gold nanoparticles: application to an ultrahigh resolution microscope[END_REF].

Super-resolution optical fluctuation imaging

Dertinger et al. introduced a concept for resolution enhancement referred to as superresolution optical fluctuation imaging (SOFI), which is based on higher-order statistical analysis of temporal fluorescence fluctuations recorded in a sequence of images (movie) [START_REF] Dertinger | Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI)[END_REF][START_REF] Dertinger | Superresolution optical fluctuation imaging with organic dyes[END_REF][START_REF] Geissbuehler | Comparison between SOFI and STORM[END_REF]. To perform SOFI, several requirements needs to be satisfied: (i) the fluorophores have to exhibit at least two different emission states; (ii) different emitters have to switch between states repeatedly and independently from each other in a stochastic way; (iii) the image should be acquired with pixels smaller than the diffraction limit. The pixel value of a SOFI image (of the order n) is obtained from the n th -order cumulant (a quantity related to the n th -order correlation function) of the original pixel time series. The signal in a pixel of conventional wide-field images is given by the sum of the fluorescence originating from different, nearby emitters. The n th -order cumulant filters this signal based on its fluctuations in such a way that only highly correlated fluctuations are left over. Since the fluorescence signal contribution of emitters to neighbouring pixels will nonlinearly yield lower correlation values, the higher order signal is limited to emitters within the pixel, which leads to an increased resolution. In contrast to the localization-based techniques, SOFI enables the separation of fluorophores with spatially overlapping diffraction patterns.

Super resolution by polarization demodulation

When excited with rotating linear polarized light, differently oriented fluorophores emit periodic signals peaking at different times. The measurement of the average orientation of fluorescent dyes attached to rigid sample structures mapped to regularly defined image nanoareas (typically 50 nm by 50 nm) can provide sub-diffraction resolution. Because the polarization angle range for effective excitation of an oriented molecule is rather broad and unspecific, this range can be narrowed by simultaneous irradiation with a second, deexcitation, beam possessing a polarization perpendicular to the excitation beam. The shortened periodic emission flashes allows better discrimination between molecules or nanoareas. This method has been implemented to standard wide-field microscopy with camera detection and to two-photon scanning microscopy, to image the fine structural details of neuronal spines [START_REF] Hafi | Fluorescence nanoscopy by polarization modulation and polarization angle narrowing[END_REF].

C H A P I T E R 2

Parallelization of STED microscopy with optical lattices

The aim of this chapter is to present the technique that we developed to massively parellelize STED microscopy. After discussing the motivations, I will describe the method of STED parallelization based on optical lattices. The experimental realization of the parallellized Lattice-STED microscopy is then detailed, followed by the presentation of experimental results. A summary and a discussion of the perspectives of Lattice-STED microscopy are given at the end of this chapter.

Motivations

Super-resolution optical microscopy methods [START_REF] Hell | Microscopy and its focal switch[END_REF] provide resolutions far beyond the diffraction limit, down to nanometer scale. The break of the diffraction barrier relies on the sequential selective imaging of fluorescent emitters, which are detected one after another in a diffraction-limited volume and consequently become resolvable. This sequential imaging modality implies that the resolution gain is at the expense of the imaging speed, which could be a severe limitation when studying dynamic structures in living cells. Indeed, many biological phenomena occur at timescales from milliseconds to seconds, such as the movement of synaptic vesicles in neurons [START_REF] Westphal | Video-rate far-field optical nanoscopy dissects synaptic vesicle movement[END_REF], the morphological deformation of the dendritic spines [START_REF] Nagerl | Live-cell imaging of dendritic spines by STED microscopy[END_REF] and the motion of the cell-surface receptors [START_REF] Huang | Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms[END_REF][START_REF] Jones | Fast, three-dimensional super-resolution imaging of live cells[END_REF].

Therefore, fast super-resolution microscopies are crucial for the study of such phenomena.

The trade-off between the spatial and temporal resolutions of super-resolution microscopy can be also explained in terms of sufficient sampling density. According to the Nyquist criterion [START_REF] Shannon | Communication in the presence of noise[END_REF], the size of the pixel of an image must be smaller than half of the desired resolution. Therefore compared to a diffraction-limited image, a super-resolution image needs N D times more pixels if a N-fold resolution improvement is achieved in D dimensions.

We now discuss the imaging speed of several super-resolution microscopy methods in two dimensions.

Localization-based methods are intrinsically parallelized. They use wide-field illumination together with a camera for detection, which records simultaneously fluorescence images of multiple emitters. An emitter should be well separated spatially from the others, in order that its PSF can be fitted independently to determine the centroid with high precision. To reveal the nanostructure of the sample, a sufficient number of emitters needs to be imaged and localized. Two factors need to be considered for the imaging speed of these techniques: the density of the fluorescing emitters per raw image and the stochastic imaging process. Naturally, the higher the density of fluorescing emitters is, the less the raw images are needed to reconstruct a super-resolution image, and therefore the faster the imaging speed is. Different algorithms [START_REF] Zhu | Faster STORM using compressed sensing[END_REF][START_REF] Huang | Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms[END_REF][START_REF] Holden | DAOSTORM: an algorithm for high-density super-resolution microscopy[END_REF] have been developed for fitting the positions of emitters, with densities up to 10 µm -2 [START_REF] Zhu | Faster STORM using compressed sensing[END_REF][START_REF] Huang | Simultaneous multiple-emitter fitting for single molecule super-resolution imaging[END_REF] corresponding to ∼ 0.5 emitters per focal spot (FWHM ∼ 250 nm). Compared with other super-resolution methods, the stochastic imaging techniques have the drawback of slow imaging speed. The structure of interest in the field of view can be regarded as pixelized with pixels, i.e. formed by pixels named "Nyquist-pixel" having the size determined by the Nyquist criteria. To form a super-resolution image of the structure, at least one localization event should take place at each Nyquist-pixel. Due to the stochastic character of the imaging process, more than one localization event could be registered at a Nyquist-pixel, which causes redundant sampling. However, the stochastic imaging process offers an advantage when imaging a sample of sparse sub-diffraction structures, such as transferrins in cells [START_REF] Jones | Fast, three-dimensional super-resolution imaging of live cells[END_REF]. Each localization event surely occurs at the position of the structures, only a few raw images are required to reveal the structures of the sample. As demonstrated in [START_REF] Jones | Fast, three-dimensional super-resolution imaging of live cells[END_REF] (see SIM is also a parallelized microscopy [START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF]. To illuminate the sample, it applies periodic patterns, which are scanned and rotated to render isotropic resolution. For each scan and rotation, the camera acquires one raw image. In total, 9 raw images are acquired to reconstruct a super-resolution image. It is possible to achieve video-rate imaging (11 Hz super-resolution image rate) with SIM, which however only offers a 2fold resolution enhancement [START_REF] Kner | Super-resolution video microscopy of live cells by structured illumination[END_REF]. SSIM can achieve better resolution [START_REF] Rego | Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution[END_REF], but at the expense of acquiring more raw images with a rather long exposure time, up to several hundreds of ms [START_REF] Rego | Nonlinear structured-illumination microscopy with a photoswitchable protein reveals cellular structures at 50-nm resolution[END_REF], in order to perform post-processing of raw data. Thus the imaging speed of SSIM is from several seconds to tens of seconds per super-resolution image.

STED microscopy is a point-scanning microscopy [START_REF] Hell | Toward fluorescence nanoscopy[END_REF]. It requires only a limited number of fluorescence photons, to obtain the molecular density at the position of the emission. It thus allows pixel dwell times down to several tens of µs [START_REF] Westphal | Video-rate far-field optical nanoscopy dissects synaptic vesicle movement[END_REF], significantly shorter than other super-resolution microscopies. However, a drawback of STED microscopy is that its detection is not parallelized so that its imaging speed depends strongly on the size of the desired field of view. The parallelization of STED microscopy will make it possible to achieve fast wide-field imaging.

A straightforward approach for STED parallelization would be the duplication of the exictation/STED beam pair. A four-fold improvement of the imaging speed has been achieved using this approach [START_REF] Bingen | Parallelized STED fluorescence nanoscopy[END_REF], with four separated excitation/depletion beam pairs, together with four corresponding point-detectors for detection (see Figure 2.2).

However, multi-doughnut STED faces a major limitation that the experimental setup will be very complicated for large parallelization.

Our approach is to use well-designed optical lattices to provide efficient depletion patterns for large parallelization of STED microscopy.

Optical lattices for STED parallelization

Optical lattices are periodic patterns generated by multi-beam interferences. They were first introduced for trapping cold atoms [START_REF] Grynberg | Quantized motion of cold cesium atoms in two-and three-dimensional optical potentials[END_REF][START_REF] Hemmerich | Two-dimensional atomic crystal bound by light[END_REF]. Their applications are then extended to other fields such as photonic crystal fabrication [START_REF] Campbell | Fabrication of photonic crystals for the visible spectrum by holographic lithography[END_REF] and microfluidics [START_REF] Macdonald | Microfluidic sorting in an optical lattice[END_REF]. Our target optical lattice should contain an array of zero-intensity minima, each of which surrounded by nearly isotropic intensity distribution. In Lattice-STED, each zero-intensity region plays the same role as the doughnut beam in doughnut-STED. To obtain a superresolution image, the sample only needs to be scanned over a unit cell of the optical To demonstrate this, we first write the total electric field E( r, t) of the interference pattern as the sum of the plane waves E j ( r, t):

E( r, t) = j=1,p E j ( r, t) = j=1,p E e j e i k j . r e -iωt e iφ j (2.1)
where E, k j , ω, φ j and e j are respectively the amplitude, the wave vector, the angular frequency, the phase and the polarization of the plane wave.

Equation 2.1 can be written as:

E( r, t) = Ee i kp. r e -iωt e iφp e p + j=1,p-1
e j e i( k j -kp). r e i(φ j -φp) .

(2.2)

Since the arbitrary choice of the time origin (t = t + t o ) allows cancelling the phase φ p , the number of independent relative phases in Equation 2.2 is p-1. Furthermore, the arbitrary choice of the space origin ( r = r + r o ) allows rewriting the sum of Equation 2.2 as:

j=1,p-1 e j e i( k j -kp). r e i[φ j -φp+( k j -kp). r 0 ] . (2.
3)

It clearly shows that the relative phase variations can be compensated by spatial translations of r 0 , if the following equations have solutions:

φ j -φ p + ( k j -k p ) • r 0 = 0. (2.4)
Equation 2.4 is a system of p -1 linear equations with n variables. For independent vectors k j -k p (1 ≤ j ≤ p -1), the system has a solution if:

p -1 = n. (2.5)
Two configurations have been used for the generation of two-dimensional optical lattices: three-and four-beam interference. The three-beam optical lattice has both the topographies of electric field and of intensity invariable to the beam phases. Although the four-beam optical lattice has the topography of electric field depending on the beam phases, we will show that in certain conditions its topography of intensity can still be invariable to the beam phases. The polarization e j and the wave vector k j of the transmitted beam can be expressed respectively as (see Appendix.A.1 for details):

Three-beam interference optical lattices

e j =        cos 2 α j cosθ + sin 2 α j cosα j sinα j (cosθ -1) -cosα j sinθ        , j = 1, 2, 3 (2.6)
and

k j =        -cosα j sinθ -sinα j sinθ cosθ        k, , j = 1, 2, 3 (2.7)
where

k = 2πn/λ, α 1 = α, α 2 = α + 120 • and α 3 = α + 240 • .
We approximate the field of a transmitted beam by a plane wave with amplitude E as:

E j (α j , θ) = E e j e i k j . r , j = 1, 2, 3. (2.8)
The three beams then overlap in the focal plane of the objective (sample plane) and interfere. The intensity distribution of the interference pattern is given by:

I( r) = ( E 1 + E 2 + E 3 ) • ( E 1 + E 2 + E 3 ) * = | E 1 | 2 + | E 2 | 2 + | E 3 | 2 + 2R( E 1 • E 2 * ) + 2R( E 2 • E 3 * ) +2R( E 3 • E 1 * ), (2.9)
with

R( E p • E q * ) = R E e p e i kp r • E e q e -i kq r = E 2 e p • e q • cos k p -k q • r (2.10)
where p, q = 1, 2, 3 and p = q.

The lattices have hexagonal structures with periodicity as a function of θ, given by (see AppendixA.2 for details): close to unity (98%).

p = 2λ 3n sinθ . ( 2
The topography of intensity of the interference pattern in the optimal experimental configuration, i.e. α = 30 • and θ = 70 • , is obtained and shown in Figure 2.6A. The pattern displays a hexagonal array of intensity "zeros", surrounded by quasi-isotropic intensity maxima. This is illustrated in Figure 2.6B where we plot the spatial variations of the optical intensity along two directions in the x-y plane (blue and green lines in the molecule can be regarded as isotropic. For a fast rotating molecule, the interaction between the dipoles and the electric field during the STED pulses can be viewed as an average effect between the electric field and a series of fixed molecules with different dipole moments. This interaction can be characterized by an effective intensity I f ast , i.e. the intensity "seen" by the molecule.

The dipole moment with an amplitude equalling unity of a fixed molecule can be described as:

d = sinθcosα e x + sinθsinα e y + cosθ e z (2.13)
where θ and α are the azimuthal and polar angles in a spherical coordinate system, with

0 ≤ θ ≤ 2π, 0 ≤ α ≤ π/2.
The electric field writes as:

E(t) = (E x e iφx e
x + E y e iφy e y + E z e iφz e z )e iwt (2.14) where φ x , φ y and φ z are phases.

Then the effective intensity for a fast rotating molecule is given by:

I f ast ≈ 1 2π 2π 0 π/2 0 | E t • d(α, θ)| 2 sinθdαdθ = 1 2π 2π 0 π/2 0 |E x e iφx sinθcosα + E y e iφy sinθsinα + E z e iφz cosθ| 2 sinθdαdθ = 1 3 (|E x | 2 + |E y | 2 + |E z | 2 ) = 1 3 I. (2.15)
According to Equation 2.15, the effective intensity is a third of the STED intensity regardless of the polarization of the electric field. The resolution of Lattice-STED microscopy for fast rotating molecules is thus directly related to the intensity distribution of the optical lattice. It will be isotropic, as the intensity gradient around the intensity minima of the optical lattice.

The effective intensity applied on a fixed dipole writes as:

I f ixed = | E(t) • d| 2 = |E x e iφx sinθcosα + E y e iφy sinθsinα + E z e iφz cosθ| 2 . ( 2

Four-beam interference optical lattices

A key point for faster Lattice-STED imaging is to generate lattices with the smallest possible unit cell. For this purpose, one can increase the incidence angle θ of the interfering beams in the three-beam configuration, but the back aperture of the objective sets a limit. Alternatively, one can use a four-beam interference configuration where a smaller unit cell can be obtained with the same angle θ. Although the topography of electrical field of a two-dimensional optical lattice generated by four-beam interference varies with the relative phases between the beams, we will show that the topography of intensity of the four-beam optical lattice could be independent to the relative phases under certain conditions. Indeed, if the four beams are composed by two orthogonally polarized pairs of beams, each pair of beams will generate a one-dimensional interference pattern independently from the other pair of beams. The intensity distribution of the final optical lattice is just the sum of the two one-dimensional interference patterns. A square optical lattice can be obtained with a periodicity p = λ/(2n sinθ).

Figure 2.9 shows the geometrical configuration of the four beams. One pair of beams is polarized along the x-axis and is propagating in the y-z plane; the other pair is polarized along the y-axis and is propagating in the x-z plane. Therefore all the beams transmitted by the objective maintain their initial polarization (s-polarization).

Writing the electric field of each beam as E j = e j Ee i( k j • r+φ j ) , j = 1 -4, with the four-beam optical lattice at the focal plane:

E tot = 2E e x e i( φ 1 +φ 2 2 ) cos( 2nπ λ xsinθ + φ 1 -φ 2 2
) + 2E e y e i( φ 3 +φ 4

2

) cos(

2nπ λ ysinθ + φ 3 -φ 4 2 ), (2.19
)

I tot = I max 2 (2 + cos( 4nπ λ xsinθ + φ 1 -φ 2 ) + cos( 4nπ λ ysinθ + φ 3 -φ 4 ) (2.20)
where

I max = 4E 2 .
The topography of intensity of the four-beam optical lattice is shown in along the diagonal direction is two times that along the y-axis, nevertheless the intensity gradients around the intensity "zeros" are about the same. Therefore the resolution is still isotropic for faster rotating molecules.

The topography of electric field depends on the relative phases, as is illustrated in We simulate the effective excitation PSFs of both a fast rotating molecule and a fluorescent bead containing 500 randomly oriented (in three-dimensional space) fixed molecules, shown in Figure 2.12.

Since the depletion of a fast rotating molecule depends only on the STED intensity, its fluorescence signal can be reduced to zero in all directions around the intensity minima, as shown by the fluorescence intensity profiles along the y-axis and the diagonal direction (45 • to the y-axis). The resolution is isotropic for

I max > 4I S .
The effective excitation PSF of the bead is shown in three cases for ∆Φ = 0, π/4 and π/2. For ∆Φ = 0, since the polarization of the field is linear everywhere, molecules are not depleted where their dipole moments are perpendicular to the local field. Therefore the fluorescence signal of the bead is not depleted down to zero even at higher STED intensities. The resolution along the y-axis (blue curve) is about 87% -90% of that along the diagonal direction (red curve) at I max > 10I S . For ∆Φ = π/2, the molecules are efficiently depleted where the polarization of the field is close to circular. Therefore the fluorescence signal of the bead can be depleted to near zero at higher STED intensities (see the fluorescence intensity profiles along the diagonal direction). However, the fluorescence signal of the bead is not reduced to zero along the y-(and x-) axis, where the polarization of the field is linear. The effective excitation PSF of the bead thus exhibits cross-like structures. The resolution along the y-axis (blue curve) is slightly larger (less than 10%) than that along the diagonal direction (red curve), at I max > 10I S . Moreover, the resolution of a fluorescent bead is about 1.5 times of that of a fast rotating molecule at I max = 80I S .

Resolution comparison between multi-doughnut STED and Lattice-STED

Besides the advantages of being able to easily generate a large array of intensity minima, the optical lattices also require less depletion power than a multi-doughnut STED setup, for the same number of intensity minima and the same target resolution. This is due to the fact that with interference one can achieve higher depletion intensities and betterconfined zero-intensity regions than those with isolated doughnuts.

We theoretically compare the resolution obtained in Lattice-STED and in multidoughnut STED microscopies. For simplicity, we consider the case of a fast rotating molecule. Since the effective intensity applied on the molecule is a third of the STED intensity regardless of the polarization, we will neglect the orientation effect of the molecule in the following discussions.

An analytical expression of the doughnut STED resolution ∆r d (FWHM) has been derived in Chapter 1:

∆r d = d c 1 + π 2 d 2 c 4d 2 I d I S (2.21)
where d c is the confocal resolution, d is the diameter of the doughnut beam and I d the maximal intensity of the doughnut beam.

We use the same theoretical approach to establish an analytical expression for Lattice-STED resolution. Here, we focus on the four beam optical lattice. The intensity distribution of the optical lattice I(x, y) at the sample plane writes as:

I(x, y) = I l sin 2 ( 2πnxsinθ λ ) + sin 2 ( 2πnysinθ λ ) . (2.22)
The resolution along the x-axis (for y=0) is given by: To compare the performance of Lattice-STED to multi-doughnut STED, we calculate the power needed to achieve a given resolution for a same number N of intensity "zeros" for the two experimental configurations. For this purpose, we first express the intensity maxima I d and I l as a function of the total STED powers P d and P l used for each configuration. In the doughnut STED, using a Laguerre-Gauss mode intensity profile:

∆r l ≈ d c 1 + π 2 d 2 c 4p 2
I l I S ( 2
I(r) = I d 4er 2 d 2 exp(-
4r 2 d 2 ), (2.24)
one obtains

I d = 4 eπd 2 N P d . (2.25)
While in the case of the four-beam Lattice-STED, one has:

I l = cosθ N p 2 P l .
(2.26)

The factor cosθ takes into account of the oblique incidence of the four beams on the interference plane.

Equalising Equation 2.21 and Equation 2.23, we obtain:

P d P l = eπ 4 d p 4 cosθ.
(2.27)

It clearly shows that Lattice-STED is more efficient than multi-doughnut STED. For θ= 60 • , p = 290 nm and d = 450 nm, multi-doughnut STED microscopy requires 6.2 times more depletion power than Lattice-STED microscopy.

STED with a single doughnut depletion beam

We have realized in the first place a home-built pulsed-STED microscope with a single doughnut depletion beam, which allows us to better understand the mechanism of STED microscopy and to prepare the experimental requirements for STED parallelization such as the laser synchronization.

Experimental setup

Pulsed laser sources and their synchronization

The doughnut-STED microscope is based on two pulsed laser sources, as shown in Figure The repetition rate of oscillator 2 is adjusted by changing its cavity length with a motorized mirror to match that of the oscillator 1. The pulse delay between the two oscillators are adjusted with a delay applied on the RF field of the acoustic-optic modulator (AOM) .

Optical setup

We show the optical setup of the doughnut-STED microscope in Figure 2

Characterizations of the pulse sequences of the lasers and of the spatial profile of the STED beam

Pulse sequences

The time delay ∆t (see We first numerically evaluate the depletion efficiency as a function of ∆t. For this purpose, we consider that the excitation pulses are extremely short. For -t p < ∆t < 0, the STED pulses arrives before the excitation pulses, the population of the electronic For 0 < ∆t < T p -t p , the STED pulses arrives after the excitation pulses and P S1 reads:

P S1 (t) = exp(-Γt) exp(- σλ hc I p min(max(t -∆t, 0), t p ). ( 2

.29)

Before the arrival of STED pulses, P S1 decreases slowly due to spontaneous emissions, whereas P S1 follows a rapid reduction with the arrival of STED pulse. The effect of delays on P S1 is illustrated in Figure 2.17A, which shows the simulated temporal evolutions of The probability of the emission of one photon during one excitation cycle is related to P S1 as:

P em (∆t) = η rad Γ T 0 P S1 (t, ∆t)dt. (2.30)
where η rad is the quantum yield.

Spatial profile of the STED beam

Careful characterization of the intensity distribution of the STED beam in the focal plane of the objective is very important for STED imaging. Achieving zero intensity at the central region of the doughnut beam is a crucial point since any residual light will lead to unwanted fluorescence depletion and a deterioration of the achievable resolution.

The intensity distribution should also be isotropic around the doughnut center in order to achieve isotropic resolutions. To characterize the intensity distribution, one can use a camera image of the doughnut beam reflection at the sample plane. However, if a tightly focused beam is incident from the optically denser medium and is reflected at the interface, the plane-wave components of the focused beam with angles of incidence smaller than the critical angle have real reflection coefficients, whereas all the components with larger angles undergo total internal reflection and phase shifts. The reflection of the beam is thus rather complex [START_REF] Novotny | reflected image of a strongly focused spot[END_REF] and does not give access directly to the beam intensity distribution at the reflection plane.

Therefore, we choose to probe directly the intensity distribution of the doughnut beam at the sample plane by scanning a gold nanoparticle across the beam focus and simultaneously recording the scattered light with an avalanche photodetector (APD) conjugated with the sample plane. For this purpose, we spin-coat an aqueous solution containing nanoparticles of 60 nm diameter on a glass coverslip. We then deposit on the nanoparticles a drop of immersion oil having a refractive index (1.515) nearly matching that of the glass coverslip. This index matching minimizes the reflection of the doughnut beam at the coverslip surface which can blur the light scattered by the nanoparticle and even damage the APD. The signal delivered by the APD is proportional to the intensity of the field propagating in the backward direction of the objective which writes as [START_REF] Lindfors | Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy[END_REF]:

I det = |E r + E s | 2 = |E i | 2 (r 2 + |s| 2 -2r|s|cosφ) (2.31)
where E i , E r and E s denote respectively the incident, reflected and backward scattered light fields. r is the field reflectivity at the glass-oil interface, s is proportional to the particle polarizability and φ is the phase shift between the two components (mainly determined by the Gouy phase shift). The term proportional to r 2 is simply the background intensity which originates from the reflection at the interface between the sample's surface and the immersion oil. The term proportional to |s| 2 is the pure scattering signal while the last term represents the interference between the reflected and scattered fields. In the case of good matching of refraction indices and large particles, r is much smaller than |s|, therefore the detected intensity I det is dominated by the scattered light and is proportional to the incident light intensity I i (= |E i | 2 ). Recording the detected signal while scanning a nanoparticle across the STED beam therefore allows us to reconstruct its intensity distribution. Figure 2.18 shows the intensity profile of the doughnut beam prepared using a SLM. In the focal plane, the diameter of the doughnut beam is 500 nm. The signal at the center of the doughnut is at the background level, suggesting that the doughnut beam has a intensity minimum close to zero.

Experimental results

In this section we will present the results obtained with our home-built doughnut-STED microscope. as 50 micrometres, with an average length of 25 µm. The outer diameter of a microtubule is about 24 nm while the inner diameter is about 12 nm. The microtubules were stained using a standard immunofluorescence protocol involving a primary antibody (anti beta-tubulin) and a secondary antibody labelled with fluorescent dyes (Atto647N).

When imaging nanoscale structures in biological samples, photobleaching of fluorescent dyes constitutes a serious problem. Illuminated with high STED intensities, fluorescent dyes can be driven into highly excited and reactive states which lead to efficient photolysis processes [START_REF] Dyba | Photostability of a fluorescent marker under pulsed excited-state depletion through stimulated emission[END_REF][START_REF] Eggeling | Molecular photobleaching kinetics of Rhodamine 6G by one-and two-photon induced confocal fluorescence microscopy[END_REF]. Furthermore, since STED microscopy offers better resolution than confocal microscopy, its sampling is more dense. For a same imaging area, the dye molecules are exposed longer to laser beams and therefore are more prone to photobleaching.

To limit the photobleaching effect, one has to increase the scanning speed (reduce the pixel dwell time), so that the laser beams shine on the sample for shorter times during the data acquisition. However, the response time of the piezoelectric translation stage limits the scanning speed to ∼ 1 ms per step. Therefore we used two AOMs in the setup to switch "on" and "off" the excitation and STED beams. For each scanning step, the light beams only shine on the sample for 200 µs (see Figure 2.20).

Generation of light patterns with a SLM

The SLM used in this study is a phase modulator, whose principle is illustrated in depending on the voltage map applied by the pixel electrodes.

For example, to obtain a doughnut beam, we apply a vortex mask to the incident beam (see Figure 2.24A). The vortex beam has a singular phase point in its center where the phase value is not defined, hence the intensity in the beam center is zero at planes conjugated to the SLM surface. We consider the electric field of the light at the SLM surface, the diffracted beams can be written as:

E j = E j e x = E e x e i 2π λ [cos(ψ j )sin(ζ)x+sin(ψ j )sin(ζ)y+φ j ] (2.32)
where j=1,2 and 3, φ j is the phase, ψ 1 =ψ, ψ 2 =ψ + 120 • and ψ 3 =ψ + 240 • .

In order to generate the three beams, the phase mask Φ SLM imprinted by the SLM to the incident light is given by:

Φ SLM (x, y, ψ, ζ) = Arg E 1 + E 2 + E 3 E in (2.33)
where E in is the electric field of the incident beam. phase mask as the total power of the three beams divided by the power of incidence on the SLM, and found ≈ 40 %.

The four beams can be generated in the same way. Figure 2.26B shows the corresponding phase mask.

Experimental implementation of four-beam optical lattice

Due to its geometrical simplicity, we first discuss the experimental implementation of four-beam optical lattice. x y rotated with two half-wave plates to be along the y-axis, in order to obtain the desired optical lattice (see Figure 2.27B).

Four-beam Lattice-STED

L1 objective L2 L3 f 1 P1 f 1 +f 2 f 2 +f 3 f 3 +f 0 L1 objective L2 L3 f 1 P1 f 1 +f 2 f 2 +f 3 f 3 +f 0 SLM WP1 WP2 P2 Sample Sample HWP1 HWP2
The diffraction efficiency of multi-beam generation with the SLM is about 40%. To minimize the STED power loss, we use two Wollaston prisms (WP) to generate the four beams, shown in Figure 2.27C and D. The initial polarization of the laser beam is linear along the x-axis. The WP1 splits the beam into two, with orthogonal polarizations, ±45 • to the x-axis. A half-wave plate (HWP1) rotates the polarization of the two beams by 45

• to be parallel to the x-or y-axes. The WP2 further splits the two beams into four.

At last, two half-wave plates (HWP2 and HWP3) together set the polarizations of the four beams as desired. The efficiency of the two Wollaston prisms to generate the four beams is about 70%, which can be further improved by using prisms with anti-reflection coating at 760 nm.

Measurement of the effective excitation PSF

In order to visualize the obtained optical lattice at the focal plane of the objective, we ). The period of the optical lattice is 290 nm. We obtained 100 intensity "zeros" with sufficient depletion, corresponding to a field of view of 2.9 µm * 2.9 µm.

Unit cell scanning procedure

To obtain a super-resolution image with the Lattice-STED microscope, the sample is scanned over a unit cell of the optical lattice. We make N * N scan steps inside a unit cell, the distance of each step is then p/N , with p the period of the optical lattice.

The scanning is performed with a calibrated piezoelectric scanner, which has relatively long response times (a few ms). In order that the scanner functions well at high speed (800 µs per step), we scan the sample in a zigzag way (see Figure 2.29), instead of a single-direction way.

To demonstrate that scanning the sample in zigzag is an optimal modality compared to single-direction scan, we measure the trajectories of a fluorescent bead for both scanning modalities. A fluorescence image of the bead is first acquired for each scan step. The image is then fitted with a two-dimensional Gaussian function to locate the centroid, which determines the position of the bead with several nanometer precision. bead exhibits deviations from the desired trajectory except for the first line, due to the response time of the piezoelectric scanner. To reduce these deviations, one can add several steps during the return of the scanner to the initial coordinates and (or) add a time delay before scanning of a new line. However, the acquisition time of a super-resolution image would be longer, thus undesirable. The zigzag scan produces the trajectory with deviations of less than ten nanometers from desired positions. Since the resolution of our Lattice-STED microscope is more than ten nanometers, these deviations are negligible.

Parallelized detection with CMOS camera

We use a fast CMOS camera to record the fluorescence signal during the scan. The camera has 2048 * 2048 pixels (6.5 µm * 6.5 µm pixel size), low readout noise (1.9 electrons, given in root mean square). A pixel of the CMOS image sensor is composed of a photodiode and an amplifier that converts charge into voltage. The voltage of each pixel is sent to an analog-to-digital converter and is read out subsequently (see Figure 2.31A). The top and the bottom halves of the sensor are read out independently, with two lines of column amplifier and analog-to-digital converter at the top and at the bottom, as showed in Figure 2.31B. The readout time of a centred region is therefore proportional to the number of lines divided by two. In our experiment, a centred region of 128 * 128 pixels is used, with a readout time of 800 µs.

The camera acquires one frame per scan step. To extract the fluorescence signal originating from each intensity "zeros" of the optical lattice, we overlay on these raw images a binary mask, shown in Figure 2.32B. The transparent parts correspond to the positions of intensity minima of the depletion pattern, i.e. the fluorescence maxima in Figure 2.32A. Therefore, the CMOS camera together with the digital mask act as an array of parallelized "point detectors", each of which records the fluorescence signal originating from corresponding intensity "zero". The magnification of the detection system is set to be 225, by combining the tube lens (X60) of the microscope, an internal and an A B external magnification systems (X1.5 and X2.5 respectively). Hence the corresponding size of the camera pixel on the focal plane of the objective is 30 nm.

The shape of the "point detectors" is close to a circle, whose size is optimized to obtain a better signal-to-noise ratio of the super-resolution images. The signal-to-noise ratio can be expressed as:

SN R = S e (N ) N N 2 r + S e (N ) (2.34)
where N is the number of camera pixels included in a "point detector", N r the readout noise of the camera pixels and S e the number of electrons detected which is a function of N .

If S e is much higher than the readout noise, the signal-to-noise ratio is dominated by the photon shot noise. A bigger "point detector" allows collecting more fluorescence, but also increases the number of camera pixels involved, i.e. the related readout noise.

Therefore a compromise needs to be found. We apply a binary mask with each "point detector" containing 13 pixels (see Figure 2.32C). The signal-to-noise ratio of the superresolution image is then about 14 -31, with N r = 1.9 and S r about 200 -1000 (determined from experimental Lattice-STED images, see Another possible issue concerning the size of the "point detectors" is that a "detector" might detect fluorescence of a fluorophore while the fluorophore is around a neighbour intensity minimum, i.e. in the region of a neighbour "detector". To estimate this "crosstalk" between neighbour "detectors", we simulate two "detectors" separated by 290 nm (the period of the optical lattice), as shown in 2.33A. A fluorophore is assumed to be at the center of "detector" 1. The ratio between the fluorescence signals delivered by "detector" 2 and "detector" 1 is obtained as a function of the radius of the "detectors" (see Figure 2.33B). When the radius of the "detectors" is 100 nm, the cross-talk is 2%. The cross-talk between the neighbour "detectors" in the diagonal direction is also considered, which is about 1% for "detectors" with the radius of 100 nm. Despite the small amount of cross-talk, the super-resolution images could have artefacts such as "ghosts", meaning that a fluorophore has a brighter image at the right coordinates and darker images appearing in adjacent areas. Therefore we apply a linear unmixing process to the signals delivered by the "detectors" to eliminate the cross-talk (see Appendix B for more details).

Three-beam Lattice-STED

The experimental implementation of three-beam optical lattice is similar to that of the four-beam optical lattice, and will be presented briefly.

We measure the effective excitation PSF by scanning a fluorescent bead in the presence of the three-beam optical lattice and a homogeneous excitation, as shown in Fig- ure 2.34A. The signal maxima correspond to the regions of minimal depletion, which occur at the zero-intensity positions of the optical lattice. As expected, we obtain a hexagonal lattice with the periodicity of 390 nm. The bright spots have slightly elongated shapes due to an angle of incidence (60 • ) smaller than the optimal angle (70 • ).

At the angle of incidence of 60 • , the intensity gradient of the optical lattice is steeper in the horizontal direction than in the vertical direction (see Figure 2.34B and C), there-fore the resolution is better in the horizontal direction, leading to elongated effective excitation PSF. The depletion intensity is also higher in the center of the field of view, this explains why the bright spot is smaller and darker there. Only the center region (inside the blue circle) with a diameter of 2.3 µm (6 * 390 nm) is used for the Lattice-STED microscope, containing 35 intensity "zeros".

The unit cell of the three-beam optical lattice is also scanned in a zigzag modality, as shown in Figure 2.35. The horizontal length of the unit cell is 390 nm and the vertical length is 338 nm. The number of scan steps are set to be 10 along the horizontal direction and 9 along the vertical direction. The resulting scan steps along the two directions are respectively 39 nm and 37.7 nm.

The camera records 10 * 9 raw images during the scan. We apply a binary mask (see mask act as an array of parallelized "point detectors". Due to the large distance (390 nm) between neighbour "detectors", the crosstalk between them is small (about 1%)

and is therefore negligible. Each "point detector" records a super-resolution sub-image.

The final super-resolved image is obtained by assembling all the sub-images together.

The sub-images have the parallelogram form as the unit cell of the optical lattice. In order to display the final image, we interpolate the super-resolution image pixels with the Delaunay triangulation [START_REF] Chew | Constrained delaunay triangulations[END_REF] based on their coordinates and fluorescence values.

Results of Lattice-STED microscope 2.5.1 Results of three-beam Lattice-STED

We apply the three-beam Lattice-STED microscope on a sample containing 20 nm fluorescent beads spin-coated on a coverslip. unit cell, and is 72 ms.

Results of four-beam Lattice-STED

We then try the four-beam Lattice-STED microscope with 20 nm fluorescent beads. Fig. 2.38A and B show respectively the images without and with the depletion optical lattice, with a field of view of 2.9 µm * 2.9 µm. The diffraction-limited PSF of the beads has a FWHM of 290 nm, while the Lattice-STED image offers better resolution and resolves closely packed beads. The FWHM of the intensity profiles of a bead gives the resolution of the Lattice-STED microscope (70 nm), as showed in Fig. 2.38C. The resolution of STED microscopy depends on the STED intensity, which makes the resolution of the Lattice-STED microscope better in the center where the STED intensity is higher. The resolution at the edge is about 1.4 times (100 nm) that measured in the center. We then apply the four-beam Lattice-STED microscope to image microtubules labelled with Atto 647N in fixed COS cells. The photobleaching of fluorophores causes fluorescence signal to decrease as a function of time and induces artefacts in the Lattice-STED image, as showed in Figure 2.39A. The photobleaching-induced artefact is mostly present in the vertical axis, which is the slow scan axis. To correct the signal decrease, we multiply a correction factor to the camera raw images. The correction factor is defined as e N ∆t/T bleach , where N is the number of the raw camera frame, ∆t the integration time per frame and T bleach the characteristic time of photobleaching. T bleach is found to be ∼ 800 ms for I exc = 2.5 kW/cm 2 , I max = 2.4 MW/cm 2 . The super-resolution image after photobleaching correction exhibits no longer discontinuous artefacts, as shown in

Reducing the photobleaching of Lattice-STED with structured excitation

The photobleaching of the fluorophores in Lattice-STED microscopy is a constraint when recording large number of super-resolution images. Lattice-STED microscopy illuminates the sample with a homogeneous excitation, molecules are therefore excited even when they are not in the intensity minima of the optical lattice. By exciting the molecules only when they are in the intensity minima, the photobleaching can be reduced. For this purpose, we use a structured excitation instead of a homogeneous one to excite the molecules. As showed in Figure 2.41, we generate an excitation optical lattice (Figure 2.41A) complementary to the depletion optical lattice (Figure 2.41B) (having maxima where the other has minima, and vice versa). Excitation of molecules occurs thus mainly at regions of zero depletion intensities and the probability that a molecule gets promoted to highly excited and reactive levels is reduced, causing less photobleaching [START_REF] Donnert | Major signal increase in fluorescence microscopy through dark-state relaxation[END_REF][START_REF] Vogelsang | A reducing and oxidizing system minimizes photobleaching and blinking of fluorescent dyes[END_REF]. The SLM allows us to tune the incident angle and the relative phases of the excitation beams. The tuning of incident angle adapts the period of the excitation optical lattice to that of the depletion optical lattice. And the tuning of the relative phases can translate the excitation optical lattice, so as to overlay the intensity maxima of the excitation optical lattice with the intensity minima of the depletion optical lattice.

To compare the photobleaching rates of the homogeneous and structured excitation configurations, we record successive Lattice-STED images of a highly concentrated sample of 20 nm fluorescent beads in both cases. We then integrate over all the pixels of the that of the first image. Figure 2.43 shows the corresponding temporal evolutions of the fluorescence together with their exponential decay fits. In the case of structured excitation, the decay time (44.1 s) is found to be two times longer than that of homogeneous illumination (20.7 s). Therefore, we can acquire two times more super-resolution images using Lattice-STED with structured excitation.

Super-resolution imaging of diffusing nanoparticles

The high-speed capability of the Lattice-STED microscope is demonstrated by imaging the movement of 20 nm fluorescent particles in a Carbopol gel (mass percentage 2%).

Carbopol polymers [START_REF] Carnali | The use of dilute solution viscometry to characterize the network properties of carbopol microgels[END_REF] are polymers of acrylic acid cross-linked with polyalkenyl ethers or divinyl glycol. The Carbopol resins are hydrophilic substances that are not soluble in water. Rather, these polymers swell when dispersed in water forming a colloidal, mucilage-like dispersion. Most of the beads are static in the Carbopol gel while some of them diffuse. We record successive Lattice-STED images with structured excitation t=0ms t=240ms t=160ms t=80ms t=320ms t=400ms at 12.5 frames per second, shown in Figure 2.44. Interestingly, we clearly show that recording of fast relative movement of two particles separated by a distance well below the diffraction limit is possible with our methodology. We measured the root-meansquare speed of one 20 nm particle in the gel, ∼ 450 nm 2 /s.

In all microscopes, the moving speed of the diffusing objects should not exceed a certain limit in order to be captured. Similarly, in the Lattice-STED microscope, the moving speed of the nanoparticles should not diffuse more than the effective excitation PSF (FWHM 70 nm) during the acquisition of one super-resolution image, otherwise the resolution is deteriorated. Moreover, if one nanoparticle diffuses more than the size of the unit cell, it will appear in more than one sub-images, inducing "ghosts" in the super-resolution images.

Conclusion

In this chapter, we presented the parallelization of STED microscopy using both threebeam and four-beam optical lattices. The multi-beam optical lattices contain a large number of intensity minima, allowing a large degree of STED parallelization. Using the four-beam optical lattice, we achieved super-resolution imaging with 70 nm resolution over a field view of 2.9 µm * 2.9 µm at a rate of 12.5 images per second, both with fluorescent beads and microtubules in fixed cells. We also demonstrated the capacity of Lattice-STED at tracking diffusing nanoparticles with super-resolution, which has the potential to perform fast super-resolution imaging in living cells, for example to study cell shaping and migration.

Motivations

Far-field super-resolution microscopy techniques are based on the sequential selective imaging of fluorescent emitters [START_REF] Hell | Microscopy and its focal switch[END_REF]. Single organic molecules inserted in solid hosts exhibit distinctive absorption lines at liquid helium temperature, thus are spectrally separable [START_REF] Tamarat | Ten years of singlemolecule spectroscopy[END_REF]. Achieving super-resolution using spectrally selective imaging was first proposed in 1985 [START_REF] Burns | Strategies for attaining superresolution using spectroscopic data as constraints[END_REF], and was experimentally realized in 1998 [START_REF] Van Oijen | 3-Dimensional super-resolution by spectrally selective imaging[END_REF]. However it did not draw much attention at that time. Then, with the success of modern super-resolution microscopies [START_REF] Betzig | Imaging intracellular fluorescent proteins at nanometer resolution[END_REF][START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy STORM[END_REF][START_REF] Hell | Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy[END_REF][START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF], researchers have started again to look for methods to develop super-resolution microscopies at cryogenic temperature.

One of the motivations is for biological applications [START_REF] Kaufmann | Fluorescence cryomicroscopy: current challenges and prospects[END_REF]. In order to preserve the biological structures in the best condition, living cells should be imaged, but this remains very challenging with super-resolution microscopy techniques [START_REF] Jones | Fast, three-dimensional super-resolution imaging of live cells[END_REF][START_REF] Westphal | Video-rate far-field optical nanoscopy dissects synaptic vesicle movement[END_REF]. Typically, at room temperature, the movements of the structures of interest in cells are immobilized by chemical fixation to achieve the best technical results, which however can cause structural change. The cryo-immobilization can preserve the structures in a near-native state in glass-like amorphous ice [START_REF] Bleck | Comparison of different methods for thin section EM analysis of Mycobacterium smegmatis[END_REF], and is frequently used in the fields of electron and X-ray cryo-microscopy [START_REF] Schneider | Cryo X-ray microscope with flat sample geometry for correlative fluorescence and nanoscale tomographic imaging[END_REF][START_REF] Hurbain | The future is cold: cryo-preparation methods for transmission electron microscopy of cells[END_REF]. However, oil-immersion objectives are not compatible with cryogenic temperature and solid-immersion based optical systems have limited fields of view (a few tens of microns) [START_REF] Trebbia | Efficient generation of near infra-red single photons from the zero-phonon line of a single molecule[END_REF][START_REF] Trebbia | Indistinguishable near-infrared single photons from an individual organic molecule[END_REF]. The lack of suitable high NA objectives therefore limits the resolution of a cryo-fluorescence microscope to ∼ 500 nm, stimulating the development of cryogenic super-resolution methods. For this purpose, localizationbased super-resolution microscopy [START_REF] Chang | Correlated cryogenic photoactivated localization microscopy and cryo-electron tomography[END_REF][START_REF] Kaufmann | Super-resolution microscopy using standard fluorescent proteins in intact cells under cryo-conditions[END_REF][START_REF] Weisenburger | Cryogenic colocalization microscopy for nanometerdistance measurements[END_REF] has been recently implemented at cryogenic temperature.

Another motivation for developing cryogenic super-resolution techniques is for quantum optical applications. In this context, single quantum emitters have been the subject of thorough investigations. They proved to be efficient and stable single photon sources [START_REF] Lounis | Single-photon sources[END_REF][START_REF] Lounis | Single photons on demand from a single molecule at room temperature[END_REF][START_REF] Lounis | Photon antibunching in single CdSe/ZnS quantum dot fluorescence[END_REF]; strong coherent coupling between single emitters [START_REF] Gaebel | Room-temperature coherent coupling of single spins in diamond[END_REF][START_REF] Hettich | Nanometer resolution and coherent optical dipole coupling of two individual molecules[END_REF] or between emitters and nanostructures [START_REF] Celebrano | Efficient coupling of single photons to single plasmons[END_REF][START_REF] Chang | A singlephoton transistor using nanoscale surface plasmons[END_REF] have attracted much attention in the applications to quantum computing and quantum communication. In 2002, Hettich et al.

achieved super-resolution by performing cryogenic laser spectroscopy under a scanning probe electrode that induces a local electric field [START_REF] Hettich | Nanometer resolution and coherent optical dipole coupling of two individual molecules[END_REF]. Two molecules separated by 12 nm underwent a strong coherent dipole-dipole coupling that produced entangled suband super-radiant states. Since the coupling strength scales with 1/r 3 , where r is the distance between the two dipoles [START_REF] Varada | Two-photon resonance induced by the dipole-dipole interaction[END_REF], a nanometric distance is required to achieve the strong coupling regime. Super-resolution method are thus required to observe this phenomenon. However, this method applies only to molecules displaying Stark shift and is experimentally difficult to realize.

During this thesis, we performed both direct-ESSat and modulated-ESSat microscopies with single fluorescent organic molecules in solid hosts at liquid helium temperature. Sub-ten nanometer resolution can be obtained with reduced laser intensities (few tens of kW/cm 2 ), which offers new opportunities to study coherent coupling and interactions between single molecules.

Single molecules at cryogenic temperature and optical saturation

In 1989, W.E. Moerner and L. Kador [START_REF] Moerner | Optical detection and spectroscopy of single molecules in a solid[END_REF] detected single molecules in a solid host at liquid helium temperature (T = 2 K) for the first time. Despite their technique of double modulation absorption, the signal-to-noise ratio was low. M. Orrit and J.

Bernard [START_REF] Orrit | Single pentacene molecules detected by fluorescence excitation in a p-terphenyl crystal[END_REF] have introduced single molecule detection by fluorescence excitation which offers an excellent signal-to-noise ratio. It has opened the way for many single molecule applications, such as single molecule spectroscopy in condensed matter [START_REF] Moerner | Illuminating Single Molecules in Condensed Matter[END_REF], imaging [START_REF] Huang | Super resolution fluorescence microscopy[END_REF] and quantum information [START_REF] Lounis | Single-photon sources[END_REF][START_REF] Lounis | Single photons on demand from a single molecule at room temperature[END_REF].

Physical principles

The energy levels of a molecule inserted in a host matrix can be depicted as in Figure 3.1.

The electronic states can be simplified as three states: the singlet ground state S 0 , the first singlet excited state S 1 , and the triplet excited state Optical transitions also exist between the ground state S 0 and vibronic states, i.e. exited vibrational levels of S 1 . After excitation into a vibronic state, rapid (in the picosecond timescale) non-radiative relaxation brings the molecule into the lowest vibrational level of S 1 , from which radiative relaxation to S 0 takes place. The linewidth of a vibronic transition is typically ∼ 1 cm -1 , three orders of magnitude broader than that of the zero-phonon line. After excitation to S 1 , the molecule can decay to S 0 by emitting a photon. The emission of a photon with the same energy as the absorbed photon is called the "resonance fluorescence". Decay to exited vibrational levels of S 0 occurs by emission of a red-shifted photon, which is called the "Stokes-shifted fluorescence". Further de-excitation of the molecule to the lowest vibrational level of S 0 takes place by non-radiative processes, converting the excess electronic energy into heat by the creation of phonons. Relaxation from S 1 to S 0 is also possible by the creation of a large number of phonons, called "internal conversion". Alternatively, a normally spin-forbidden transition from S 1 to the metastable triplet state T 1 can occur via a spin-orbit coupling between iso-energetic vibrational levels of the electronic states, called "inter-system crossing". Inter-system crossing to vibrational levels of T 1 is followed by a non-radiative decay to the triplet ground state, with the creation of phonons. From the triplet ground state, the decay to the singlet ground state takes place either by radiative transitions called "phosphorescence" or non-radiative transitions via back inter-system crossing. Since these transitions are spin-forbidden, they occur with low probability.

Decay rates are referred to as k ij , where i denotes the state from which the transition takes place and j the state into which the transition ends up. The lifetime of the lowest vibrational level of S 1 is on the order of nanoseconds, much longer than that of other vibrational levels (picoseconds). The triplet ground state has a much longer lifetime (from microseconds to seconds), which limits the fluorescence emission rate at high excitation intensities, referred to as the "triplet bottleneck".

The relative intensities of the vibronic lines and phonon sidebands with respect to the zero-phonon line, in an absorption or an emission spectrum, are determined respectively by the Franck-Condon and the Debye-Waller factors. The Franck-Condon factor is determined by the intermolecular properties and takes into account the vibrational levels.

The Debye-Waller factor is a measure of the relative intensity of the zero-phonon line and the phonon sideband. It decreases exponentially with temperature.

In fluorescence excitation experiments, the molecule is excited at the zero-phonon line. The Stokes-shifted fluorescence is detected by the photo-detector, with filters placed before to block the scattered excitation light.

Spectral selection of single molecule

One way to detect single molecules is to reduce the concentration of the impurity molecules. The concentration is sufficiently reduced when only one molecule is at resonance with the excitation frequency in the illumination volume. This method is referred to as "spatial selection". Due to the narrow zero-phonon lines at low temperature, the number of molecules at resonance with a single-frequency laser beam in the illumination volume is much less at cryogenic temperature than at room temperature, for a given concentration. Thus the zero-phonon lines give rise to an additional dimension to address single molecules, which is called "spectral selection". Due to various local host environments, the zero-phonon absorption lines of single molecules have different frequencies (see Figure 3.3). The ensemble spectrum, which is the sum of the homogeneous zero-phonon lines of all single molecules, displays therefore a broad "inhomogeneous band". The width of the inhomogeneous band reflects the distribution of interaction strengths between the impurity molecules and the host matrix.

A laser with a bandwidth narrower than the homogeneous linewidth is used to perform spectral selection, such that a molecule goes in and out of resonance when scanning the laser frequency. A fluorescence excitation spectrum over the inhomogeneous band of terrylene molecules embedded in a p-terphenyl single crystal is exemplified in Figure 3.4.

In the red wing of the inhomogeneous band, one can distinguish single molecules which are spectrally well isolated and display a signal to noise ratio better than ten. The FWHM of the homogeneous line of a single molecule in frequency units can be expressed as:

∆ν hom = 1 πT 2 = 1 2πT 1 + 1 πT * 2 (3.1)
where T 2 is the total dephasing time of the optical transition. The effective photon-capture area of a single molecule under resonant excitation is given by the absorption cross-section. In the limit of low laser intensities, the absorption cross section σ of the zero-phonon line of single molecules in a solid can be written as

σ = α F C α DW 3λ 2 2π T 2 2T 1,rad cos 2 θ (3.2)
where λ is the wavelength and T 1,rad the radiative lifetime of the excited sate. cm 2 [START_REF] Wiedmann | Excited-state absorption cross-sections in rhodamine dyes determined after molecular reorientation[END_REF][START_REF] Rittweger | Fluorescence depletion mechanisms in super-resolving STED microscopy[END_REF].

Optical saturation of single molecules

At low excitation intensity, the number of photons absorbed by the molecule varies linearly with the excitation intensity. Increasing the excitation intensity will eventually saturate the population of the excited state. Therefore the absorption cross-section of a molecule depends on the light intensity and can be described as [START_REF] Moerner | Single-Molecule Optical Detection, Imaging and Spectroscopy[END_REF]:

σ(I) = σ(0) 1 + I/I S (3.3)
where σ(0) is the cross section at low light intensity, I the excitation laser intensity and I S the saturation intensity. For a three-level system including S 0 , S 1 and T 1 , the saturation intensity is derived from optical Bloch equations and is given by [START_REF] Plakhotnik | Single molecule spectroscopy: maximum emission rate and saturation intensity[END_REF]: where R ∞ is the maximal emission rate and ∆ν hom (0) the homogeneous linewidth.

I S = hν 2 σ(0) T 1 1 1 + k 23 /2k 31 . ( 3
We have experimentally investigated the fluorescence saturation of single DBATT molecules in octadecane.

Principle of super-resolution microscopy based on excited state saturation

In this section, I will explain how to achieve super-resolution by exploiting the optical saturation of single molecules at cryogenic temperature. Two methods will be presented.

In the first one, named direct-ESSat microscopy, molecules are excited only with a doughnut-shaped beam. The resulting images of the molecules are "negative" superresolution images, in which molecules are localized by local fluorescence minima. The second one is called modulated-ESSat microscopy. The molecules are excited with both a high intensity doughnut beam and a temporally-modulated Gaussian beam. The modulated-ESSat microscopy gives "positive" images which offer a better contrast than that obtained with direct-ESSat microscopy.

Direct-excited state saturation microscopy

We aim at simulating the fluorescence image of a molecule in direct-ESSat microscopy.

The image is obtained by recording the fluorescence intensity emitted by the molecule while scanning the molecule across the doughnut beam in the focal plane of the objective.

The intensity distribution of the doughnut beam can be described as: We then simulate the fluorescence intensity distribution in an image of 50 randomly located identical molecules. Figure 3.10A shows a conventional confocal image obtained with a Gaussian beam having the intensity distribution:

I donut (r) = eI D 4r 2 d 2 exp(-
I Gaussian = I G exp(- 2r 2 w 2 0 ) (3.11)
where I G and w 0 are respectively the maximal intensity and the waist of the Gaussian beam. Its FWHM is 380 nm. The brightest spots suggest the contribution of more than one molecule to their fluorescence intensity.

Modulated-excited state saturation microscopy

In order to improve the image contrast, we introduce "modulated-ESSat microscopy".

A temporally modulated Gaussian beam is added to the doughnut beam to excite the molecules. A demodulation method is applied to extract the amplitude of the fluorescence intensity modulation. For simplicity, we consider a single cosine modulation of the Gaussian beam:

I Gaussian (r, t) = I G 1 + cos(2πνt) 2 exp(- 2r 2 w 2 0 ) (3.12)
where ν is the modulation frequency.

At moderate Gaussian beam intensity (I G ≤ I S ), the detected count rate of a single molecule can be modelled as: of a single molecule into a "positive" signal on a dark background. We then simulate images of 50 randomly located identical molecules, shown in Figure 3.12D-F. While both direct-ESSat and modulated-ESSat images offer super-resolution, the latter has a better contrast. Indeed, demodulated signals from single molecules do not overlap, as long as molecules are separated by a distance larger than the effective resolution.

S = S 0 + ∆S 1 + cos(2πνt) 2 ( 3
We will now discuss the resolution of modulated-ESSat microscopy as a function of intensities of the doughnut and Gaussian beams.

Using Equation 3.5, the amplitude of the fluorescence intensity modulation can be expressed as:

∆R(r) = R[I donut (r) + I Gaussian (r)] -R[I donut (r)] (3.14)
In the following, we consider the case where r d. Therefore I Gaussian (r) ≈ I Gaussian (0) = I G . In the center of the doughnut beam, the amplitude is maximal and equals to R(I G ).

The resolution can be defined as the FWHM of ∆R(r) and is given by:

∆r = 2d π A(I G ) I D /I S , ( 3.15)
with A is less than 1. However, the signal-to-noise ratio of modulated-ESSat images will also be lower. Therefore, in practice, I G is at the order of tens of I S or even higher and √ A ∼ 1.

A(I G ) = (1 + I G /I S ) + (1 + I G /2I S ) 2 -(1 + I G /2I S ). ( 3

Experimental setup

Sample

The molecule-matrix system used for the present study is the highly fluorescent diben-

Experimental results

Characterization of the doughnut beam

There are different types of optical aberrations in imaging systems [START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF]. In a confocal microscope, careful alignment of the optical setup assures minimal aberrations and provides diffraction-limited images. However the objective used in this experiment is not designed for cryogenic temperature applications. The optical elements of the objective have different thermal expansion coefficients, which induces aberrations, in particular astigmatism, once cooled down to 2 K. Such aberrations can affect the quality of the doughnut beam, leading to non-perfect "zero" in the doughnut center and anisotropic intensity distribution.

Optical aberrations can be expressed with Zernike polynomials [START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF], which are defined on a disk, thus well adapted to describe waves passing through a circular aperture.

Figure 3.17 shows the first fifteen wavefront modes for the corresponding Zernike polynomials.

Figure 3.17: Wavefront modes for the first fifteen Zernike polynomials. Extracted from [START_REF] Fuensanta | Topics in Adaptive Optics[END_REF].

To correct the aberrations, we apply a compensation phase mask with the SLM, together with the vortex mask and the blazed grating (see Fig 3.16). The compensation mask is a linear combination of the Zernike polynomials. The first eight Zernike polynomials are used, since they represent the most common aberrations in an imaging system (see Figure 3.17).

We developed an algorithm written in Labview which automatically determines this compensation mask. Figure 3.18 shows the interface of the program. An error function is defined as: After the launch of the algorithm, the changes of the three Zernike polynomials which represent the x-tilt, y-tilt and focusing actions, decrease the value of the error function rapidly. It points to location and size mismatches between the two images.

Err(a 1 , a 2 , . . . , a n ) = x,y (I exp (a 1 , a 2 , . . . , a n , x, y) -I cal (x, y)) 2 (3.17
Thereafter, the changes of the weights for polynomials representing the astigmatisms mostly affect the error function, suggesting that the main existing aberrations in our system are astigmatisms. A very important issue regarding the doughnut intensity distribution is the intensity in its center, where residual intensity will degrade the resolution performance of ESSat microscopy. The cross-sectional profile in Figure 3.20B shows that the fluorescence intensity at the center of the doughnut profile, due to background signal, is less than 1% of the peak intensity.

Results of direct-excited state saturation microscopy

Results of modulated-excited state saturation microscopy

In all modulated-ESSat experiments, the Gaussian beam is modulated at the frequency of 1 kHz, 500 µs "on" and 500 µs "off".

Conclusion

In this chapter, we presented a new super-resolution imaging method of single molecules at cryogenic temperature, based on excited state saturation with a doughnut beam.

A resolution of 4.4 nm was achieved with this method, approaching the size of the molecule. Due to extremely low saturation intensity of single molecules at liquid helium temperature, this resolution is obtained with a laser intensity of only a few tens of kW/cm 2 . Application of a modulated Gaussian beam together with a demodulation technique offers a better contrast in images of highly concentrated samples. A resolution of 9 nm was obtained with this method, due to longer integration times over which mechanical vibrations of the cryostat are integrated. The resolution of ESSat microscopy is limited by the residual intensity in the center of the doughnut beam, the mechanical stabilities of the cryostat and of the piezo scanner, which could be improved with future developments. ESSat microscopy is capable of sub-diffraction imaging of molecules with close optical resonance, which offers the possibilities to pursue different quantum optics applications, such as the entanglements and interactions between molecules by coherent dipole-dipole coupling [START_REF] Hettich | Nanometer resolution and coherent optical dipole coupling of two individual molecules[END_REF].

per second.

Increasing the field of view of Lattice-STED is another interesting challenge. Lasers with lower repetition rate can deplete molecules more efficiently [START_REF] Wildanger | STED microscopy with a supercontinuum laser source[END_REF]; nanoparticleassisted STED [START_REF] Sivan | Nanoparticle-assisted stimulated-emission-depletion nanoscopy[END_REF] uses metal nanoparticles to enhance both the excitation light and the fluorescence of the emitters. They all allow STED imaging with lower light level, thus can be implemented with Lattice-STED to achieve bigger fields of view.

Another possibility is to use reversible photoswitchable fluorescent proteins, which protein switching is a relatively slow on-off process (∼ 10 ms), which sets a limit to the imaging acquisition rate. Therefore, Lattice-STED and parallelized-RESOLFT can be regarded as complementary techniques: the first one offers higher imaging speed and the second one provides larger fields of view. We will combine the two techniques to pursue biological applications such as the study of fast cell shaping and migration.

ESSat microscopy is based on the optical saturation of the sharp zero-phonon line of single molecules inserted in solid hosts at 2K. By imaging molecules with a doughnutshaped beam, we demonstrated a resolution of 4.4 nm. Due to the extremely low saturation intensities of the zero-phonon line, this resolution is obtained with a laser intensity of only 13 kW/cm 2 . Application of a modulated Gaussian beam together with a demodulation technique offers a better contrast in images of highly concentrated samples. A resolution of 9 nm was obtained with this method, due to longer integration times over which mechanical vibrations of the cryostat are integrated.

Besides ESSat microscopy, one can also consider the implementation of STED microscopy at cryogenic temperature. STED microscopy at room temperature has demonstrated a resolution of 8 nm with single N-V centers in diamond with an oil-immersion objective [START_REF] Rittweger | STED microscopy reveals crystal colour centres with nanometric resolution[END_REF], then a 4.2 nm resolution with a solid immersion objective [START_REF] Wildanger | Solid immersion facilitates fluorescence microscopy with nanometer resolution and sub-Ångström emitter localization[END_REF]. However, it requires high laser intensity (up to several GW/cm 2 ), which is about 10 5 -10 6 times higher than the intensity applied in ESSat microscopy. Applying STED microscopy to single molecules at liquid helium temperature could reduce the required depletion intensity by three orders of magnitude. Since STED microscopy does not require the modulation of the Gaussian beam as in modulated-ESSat microscopy, pixel dwell times as short as several tens of µs are possible [START_REF] Westphal | Video-rate far-field optical nanoscopy dissects synaptic vesicle movement[END_REF]. Such short integration times are interesting to reduce the effects of mechanical vibrations during imaging.

Further improvement of the super-resolution microscopy at cryogenic temperature could be three-dimensional resolution enhancement. 3D super-resolution is crucial to determine the distance between single emitters in a thick sample. Studying spectrally and spatially close molecules offers the possibilities to pursue different quantum optics applications, such as the entanglements and interactions between molecules by coherent dipole-dipole coupling [START_REF] Hettich | Nanometer resolution and coherent optical dipole coupling of two individual molecules[END_REF].

We set two Cartesian bases as B 1 = ( e x , e y , e z ) and B 2 = ( e x , e y , e z ). The matrix for changing the basis from B 1 to B 2 is:

M 1→2 =        cosα sinα 0 -sinα cosα 0 0 0 1        (A.1)
The polarization of the incident beam is (1, 0, 0) in the basis B 1 . In the basis B 2 , it will be:

( e i ) B 2 = M 1→2 ( e i ) B 1 =        cosα sinα 0 -sinα cosα 0 0 0 1               1 0 0        =        cosα -sinα 0        (A.2)
The transmitted wave's polarization e f is the rotation of the incident wave's polarization of an angle θ around the axis e y . The rotation matrix is:

N =        cosθ 0 -sinθ 0 1 0 sinθ 0 cosθ        (A.3)
Therefore the transmitted wave's polarization in the basis B 2 is given by: ( e f ) B 2 = N ( e i ) B 2 (A.4)

We can then rewrite the transmitted wave's polarization in the basis B 1 as: where p is the period of the optical lattice and R is the radius of the "detectors".

( e f ) B 1 = M -1 1→2 N M 1→2 ( e i ) B 1 =       
P SF (x, y) is the emission PSF of a fluorophore and is given by:

P SF = 2J 1 (k NA r) k NA r 2 (B.2)
where J 1 is the Bessel function of the first kind of order one, k = 2π/λ is the wavenumber, NA is the numerical aperture and r is the radial distance from the optical axis.

Suppose that there is an array of P * Q "detectors", and cross-talk only exists between neighbour "detectors", the fluorescence signal F p,q of the detector (p,q) would be:

F p,q = F p,q + a(F p-1,q + F p+1,q + F p,q-1 + F p,q+1 ) + b(F p-1,q-1 + F p-1,q+1 + F p+1,q-1 + F p+1,q+1 ) (B.3) with 1 < p < P and 1 < q < Q. F p,q consists of the fluorescence originating from emitters in the region of the detector (p,q) (F p,q ) and cross-talks of fluorescences from emitters in its neighbour "detectors". A similar formula can be written for p=1,P or q=1,Q.

Set F l = F p,q with l = (p -1) * P + q, Equations B.3 writes: The fluorescence values of "detectors" corrected from cross-talk are therefore:

F l = F l +
F = M -1 F (B.6)
Abstract: This thesis presents the development of two new super-resolution microscopy techniques.

The first technique aims at improving the imaging speed of super-resolution microscopy at room temperature for biological applications. As a scanning technique, STED (Stimulated Emission Depletion) microscopy needs parallelization for fast wide-field imaging. Using well-designed optical lattices for depletion together with wide-field excitation and a fast camera for detection, we achieve large parallelization of STED microscopy. Wide field of view super-resolved images are acquired by scanning over a single unit cell of the optical lattice, which can be as small as 290 nm * 290 nm.

Lattice-STED imaging is demonstrated with a resolution down to 70 nm at 12.5 frames per second.

The second one extends super-resolution microscopy to liquid helium temperature for applications in quantum technologies. Optical resolution of solid-state single quantum emitters at the nanometer scale is a challenging step towards the control of delocalized states formed by strongly and coherently interacting emitters. ESSat (Excited State Saturation) microscopy operating at cryogenic temperatures is based on optical saturation of the excited state of single fluorescent molecules with a doughnutshaped beam. Sub-10 nm resolution is achieved with extremely low excitation intensities, more than million times lower than those used in room temperature STED microscopy. Compared to superlocalisation approaches, our technique offers a unique opportunity to super-resolve single molecules having overlapping optical resonance frequencies, paving the way to the study of coherent interactions between single emitters and to the manipulation of their degree of entanglement.

Keywords: super-resolution, microscopy, STED, single molecule Ré sumé : Cette thè se pré sente le dé veloppement de deux mé thodes de microscopie super-ré solue.

La première méthode vise à améliorer la vitesse d'imagerie de la microscopie super-ré solue à tempé rature ambiante pour des applications biologiques. En tant qu'une technique de scan, la microscopie STED a besoins d'être parallélisé pour faire de l'imagerie rapide en champ large. Nous avons obtenu une parallé lisation massive de la microscopie STED en utilisant les réseaux d'optique avec une excitation en en champ large et une camé ra rapide pour dé tection. Les images super-ré solues d'un champ de 3 µm par 3 µm sont acquises en scannant une maille é lé mentaire du ré seau optique, qui peut ê tre aussi petite que 290 nm * 290 nm. La microscopie Lattice-STED est dé montré e avec une ré solution allant jusqu'à 70 nm à une cadence de 12,5 images par seconde.

La deuxiè me mé thode é tend la microscopie super-résolue à la température de l'hélium liquide pour des applications aux technologies quantiques. Des ré solutions optiques à l'é chelle nanomé trique des é metteurs quantique est une é tape cruciale vers le contrôle des é tats dé localisé s formé s par les interactions fortes et cohé rentes entre des é metteurs. Dans ce contexte, nous avons dé veloppé une technique de microscopie à des tempé ratures cryogé niques, dé nommé e la microscopie Essat. Cette technique est basé e sur la saturation optique de l'é tat excité des molé cules fluorescentes uniques par l'excitation d'un faisceau en forme d'anneau. Une ré solution moins de 10 nm est obtenue avec de basses intensité s d'excitation, plus de millions de fois plus faibles que celles utilisé es dans la microscopie STED à la tempé rature ambiante. Par rapport aux approches basé es sur la superlocalisation, notre technique offre une occasion unique de ré soudre sous la limite de diffraction les molé cules uniques ayant des fré quences de ré sonance optiques qui se chevauchent. Ceci ouvre la voie à l'é tude des interactions cohé rentes entre é metteurs uniques et à la manipulation de leur degré d'intrication.

Mot-clé s: super-ré solution, microscopie, STED, molé cule unique

S 1 ,Figure 1 . 1 :FilterFigure 1 . 2 :

 11112 Figure 1.1: Energy diagram of a fluorescent emitter. An excitation laser excites the emitter from the ground state S 0 to a higher vibrational level of the excited state S 1 . The emitter can emit fluorescence from the lowest vibrational level of S 1 by spontaneous emission.

  largely facilitated this technique by making it possible to focus light into a diffraction-limited volume (the PSF). The diagram of a confocal microscope is shown in Figure 1.3. The focused light excites the fluorescent objects along its propagation both before and after the focal plane, inducing out-of-focus fluorescence. The role of the pinhole is to reject the out-of-focus fluorescence and only let through fluorescence originating from the focal volume. In order to form an image, the illumination volume is scanned through the sample while a point detector records the fluorescence signal.

Figure 1 . 3 :

 13 Figure 1.3: Diagram of a confocal microscope. An objective focuses the laser beam on the sample. A pinhole is set in the detection path and is conjugated with the focused laser spot to reject the out-of-focus fluorescence.

Figure 1 . 4 :

 14 Figure 1.4: Single-photon and two-photon excitations. In the single-photon excitation case, one photon of the energy hv can excite fluorophores. The excitation takes place along the propagation of the light, both before and after the focal plane. In the twophoton excitation case, two photons of energy hv/2 act together to excite fluorophores. The excitation occurs only in the focal volume.

Figure 1 . 5 : 1 pFigure 1 . 6 :

 15116 Figure 1.5: Principle of STED microscopy. (A) STED mechanism. An excitation laser excites an emitter from the ground state S 0 to a higher vibrational level of the excited state S 1 . The emitter can emit fluorescence from the lowest vibrational level of S 1 by spontaneous emission. A STED photon can deplete S 1 to a excited vibrational level of S 0 by stimulated emission. (B) Diagram of the STED microscopy setup. The excitation and STED lasers are combined by dichroic mirrors and focused by an objective on the sample. The spatial form of the STED beam is modulated by a phase plate. A detector records the fluorescence. (C) Enhanced resolution with STED microscopy. A STED beam with a doughnut form overlaps the excitation beam, and suppresses the fluorescence from the pheriphery of the excitation beam. This leads to a reduced effective excitation PSF compared to the diffraction-limited PSF. Figure adapted by the author from [84].

Figure 1 .Figure 1 . 7 :

 117 Figure 1.7 shows P em as a function of I/I S . At higher STED intensities (I > 7I S ), P em is well below 0.01 and the fluorescence is efficiently depleted.

Figure 1 . 8 :

 18 Figure 1.8: (A) Intensity profiles of the excitation Gaussian beam (green) and the depletion doughnut beam (red); (B) effective excitation PSFs of STED microscopy as a function of I ST ED /I S . Calculated for a Gaussian beam of the FWHM of 220 nm, a doughnut beam of the diameter of 500 nm and I ST ED /I S = 0, 5, 20 and 100.

Figure. 1 .Figure 1 . 9 :

 119 Figure.1.9 compares the intensity profiles of a doughnut beam calculated with respectively Equation 1.10 and Equation 1.11. The two profiles overlap well around the "zero" intensity (-d/2 ≤ r ≤ d/2), with an intensity difference within 10 %.

Figure 1 . 11 :

 111 Figure 1.11: Temporal evolution of the PSF of STED microscopy. (A) and (C) for CW-STED configuration; (B) and (D) for pulsed-STED configuration. (A) (or B) Lateral intensity distribution hg (t, r) (or hp (t, r)) (left panel: normalized by η rad Γ; right panel: normalization to 1 at each time). (C) (or D) FWHM (black, left axis) and the peak signal hg (t, 0) (or hp (t, 0)) (red, right axis) of the STED PSF as a function of time (excitation at time 0). Calculated for an objective of NA = 1.4, τ = 3.4 ns, T p = 12.5 ns, t p = 300 ps, excitation wavelength 635 nm, STED wavelength 760 nm, the same average intensities for both configurations, with I ST ED = 25I S . Figure extracted from [99].

Figure 1 . 12 :

 112 Figure 1.12: Principle of SIM and SSIM microscopy. (A) Diagram of the experimental setup. A grating diffracts the excitation beam into two beams which recombine on the sample to create one-dimensional periodic structures. SIM applies low excitation intensities whereas SSIM applies high intensities to saturate the fluorophores. A camera acquires fluorescence images. (B) Moire effect. Two overlapping fine patterns generate a beat pattern which contains lower frequencies than the original patterns. (C) The camera acquires successive wide-field images while the illumination pattern is scanned and rotated. Post data-processing is needed to reconstruct a super-resolution image.

Figure 1 . 13 :

 113 Figure 1.13: Principle of super-resolution methods based on single molecule localization. (A) Single molecule localization. Left side is a diffraction-limited image of a fluorphore. Right side is the localization precision of its centroid. (B) Super-resolution process. The sample shown in B.1 is labelled with thousands of fluorophores. Only a subset of the fluorophores is allowed to fluoresce and imaged each time, their positions are then localized (B.2 and B.3). Repeating this process until a large number of fluorophores are localized, one can reconstruct a super-resolution image (B.4). Figure adapted by the author from [84].

Figure 2 . 1 )Figure 2 . 1 :

 2121 Figure 2.1: STORM images of transferrin in live cells. (A) Wide-field two-dimensional STORM image. (B) Close-up STORM (top) and conventional (bottom) images of the boxed region in A. Scale bars, 2 µm (A) and 100 nm (B). Figure extracted from [44].

Figure 2 . 2 :

 22 Figure 2.2: Multi-doughnut STED setup realized by Hell and co-workers in 2011 (extracted from [165]). The STED and excitation beams follow the same optical path. Two Wollaston prisms split both beams in four parts. A special optical shaping device (SWP) converts the STED beams into doughnut-shaped and leaves the excitation beams unaffected. Four dedicated avalanche photodiodes, conjugated with the sample plane, are used to collect the fluorescence.

  To build the three-beam optical lattice in the focal plane of a high NA objective suitable for STED parallelization, we consider three beams propagating along the optical axis e z of the objective, with the same frequency ω, linear polarization along e x and the same intensity, as shown in Figure2.3. The beams intersect the objective at the vertices of a centred equilateral triangle. After passing through the objective they are deviated by an angle θ toward the focal region where they interfere and form an optical lattice. The beams transmitted by the objective have wave vectors k j . α j is the angle between the x-axis (polarization direction) and the projection of k j in the x-y plane. The intensity distribution of the optical lattice is therefore determined by the angles α j and θ. The incident beams are slightly focused Gaussian beams with beam waist located at the back focal plane of the objective. The transmitted beams are collimated and deviated by the objective and have their polarizations rotated.

Figure 2 . 3 :

 23 Figure 2.3: Geometrical configuration of the three beams arriving on the sample plane. The three beams have the same polarization (along the x-axis) and propagate parallel to the optical axis before the objective. They intersect the back focal plane of the objective at the vertices of a centered equilateral triangle. Red lines show the wave vectors of the three beams, blue arrows indicate the polarizations.

. 11 )Figure 2 . 4 :

 1124 Figure 2.4: Simulations of the intensity distribution of three-beam optical lattices as a function of α and θ. Each image is the sum of the squared electric field components along the x-, y-and z-axes, and is normalized to its maximal intensity value. Simulated with λ = 760 nm and n = 1.515.

Figure 2 .Figure 2 . 5 :

 225 Figure2.5A shows the three-dimensional plot of the contrast C(θ, φ) as a function of α and θ. Clearly, the highest contrasts are obtained when α=30 • . Figure2.5B shows the evolution of the contrast as a function of θ for α=30 • . The maximal contrast (more than 99.9%) is reached when θ=70 • . In practice, the angle of incidence is limited both by the size of the beams at the back aperture of the objective and the size of the back aperture. In our experiment, a high NA objective (NA=1.49) is used, which allows an angle of incidence up to θ=60 • , where the size of the beams is ∼ 1 mm and the size of the back aperture is 10 mm. At α = 30 • and θ = 60 • , the theoretical contrast is still

Figure 2 .Figure 2 . 6 :

 226 Figure 2.6A). Clearly, the two intensity curves show the same spatial evolution up to more than 150 nm away from the position of the zero intensity. The intensity distribution of the three electric field components |E x | 2 , |E y | 2 and |E z | 2 are shown in Figure 2.6C. The values of |E x | 2 and |E z | 2 are much higher than |E y | 2 . The polarizations of different combinations of field components are shown in Figure 2.6D. Since the field components |E x | 2 and |E z | 2 are more significant than |E y | 2 , we are particularly interested in the polarization of E x + E z . Around the intensity minima, its evolution is linear along the x-axis and along directions ±60 • to the x-axis and is elliptical elsewhere. Since molecules are dipoles, the STED efficiency depends on the dot product of the dipole moment and the electric field as | E(t) • d(t)| 2 , where d is the dipole moment. d(t)is a function of time and is related to the rotational mobility of the molecules. We define τ rot as the characteristic rotating time of the molecule. For timescales longer than τ rot ,

. 16 )Figure 2 . 8 :

 1628 Figure 2.8: Simulated effective excitation PSF of Lattice-STED, with a homogeneous wide-field excitation and the three-beam depletion optical lattice. The PSFs are obtained by scanning a fluorescent object over the field of view and recording its fluorescence signal. The fluorescent object is either a fast rotating molecule (A-C) or a bead containing 500 randomly oriented molecules (G-I). The cross-sectional profiles of the fluorescence images along the y-(D,J) and x-axes (E,K) are also shown. Blue, green and red lines correspond respectively to I max /I S = 5, 20 and 80. (F) and (L) show the FWHMs of the PSFs along the y-(blue) and x-axes (red) as a function of I max /I S . The FWHMs are obtained by fitting the profiles of the PSFs with a Gaussian function. The FWHMs of the bead are averaged over 20 independent simulations.

Figure 2 .

 2 8G, H and I show respectively the effective excitation PSF of the bead for I max /I S =4, 20 and 80. The cross-sectional profiles along the y-and x-axes of the effective excitation PSF are shown respectively in Figure 2.8J and K. Since the depletion of fixed molecules depends strongly on the polarization of the field, all molecules are efficiently depleted where the polarization is elliptical, whereas molecules oriented perpendicular to the field of linear polarization are not depleted. Therefore cross-sectional profiles along the y-axis does not decrease to zero even at high STED intensities. The FWHMs of the PSFs along the y-and xaxes (Figure 2.8L) show that the resolution exhibits slightly anisotropy, with relative differences between the FWHMs along the two directions being about 9 % -22 % for I max between 0 and 80I S . At I max = 80I S , the resolution along the y-(or x-) axis of a fluorescent bead is about 1.15 (or 1.38) times of the resolution of a fast rotating molecule.

e 1 , e 2 =Figure 2 . 9 :

 1229 Figure 2.9: Geometric configuration of the four beams arriving on the sample plane. Beams 1 and 2 have polarization along the x-axis, and beams 3 and 4 have polarization along the y-axis. The four beams intersect the back focal plane of the objective at the vertices of a centered square.

Figure 2 .Figure 2 . 10 :

 2210 Figure 2.10: Optical lattice generated by the four-beam interference. (A) is the topography of intensity, with contributions of both field components |E x | 2 and |E y | 2 . (B) shows the intensity plots along the blue and green lines in A. (C) presents the intensity distribution for different field components. Intensities in the unit of E 2 .

Figure 2 . 11 . 2 = 0 , 2 Figure 2 . 11 : 2 )

 2112022112 Figure 2.11. If the relative phase difference between the four beams satisfy the relation ∆Φ = φ 1 +φ 2 -φ 3 -φ 4 2

Figure 2 . 12 :

 212 Figure 2.12: Simulated effective excitation PSF of four-beam Lattice-STED, obtained with a homogeneous wide-field excitation and the depletion optical lattice. Images are simulated for a fast rotating molecule and for a bead containing 500 randomly oriented fixed molecules (in three-dimensional space). The fluorescence signal is in units of the maximal emission rate of one molecule. Effective excitation PSFs are shown for different values of ∆Φ and I max /I S . The profiles of the PSFs along the y-axis and the diagonal directions (45 • to the y-axis) are also given, where the blue, green and red curves correspond respectively to I max /I S = 5, 20 and 80. The FWHMs of the PSFs for both the fast rotating molecule and the bead are shown as a function of I max /I S , where the blue and red curves correspond respectively to the resolutions along the y-axis and the diagonal direction. The FWHMs are obtained by fitting the profiles of the PSFs with a Gaussian function. The FWHMs of the bead are averaged over 20 independent simulations. Color bar and scale bar apply to all figures.

. 23 )

 23 where p = λ/(2n sinθ) is the period. Compared to Equation 2.21, d and I d are simply replaced respectively by p and I l .

2 .Figure 2 . 13 :

 2213 Figure2.13: Laser synchronisation. Oscillator 1 is coupled with an optical parametric oscillator and delivers a train of pulses at a fixed repetition rate(76 MHz). The repetition rate of oscillator 2 is adjusted by changing its cavity length with a motorized mirror to match that of the oscillator 1. The pulse delay between the two oscillators are adjusted with a delay applied on the RF field of the acoustic-optic modulator (AOM) .

Figure 2 . 14 :

 214 Figure 2.14: Optical setup of the doughnut-STED microscope. The doughnut shape of the STED beam is obtained with a helix phase (see Figure 2.24) applied on the spatial light modulator (SLM). The excitation and STED beams are combined with two dichroic mirrors (DM)and focused on the sample plane with an oil-immersion objective (NA = 1.49). An avalanche photodiode (APD) is conjugated with the sample plane and records the fluorescence signal. A control camera is used to monitor the beam profile. Abbreviations: NPBS: Non-Polarising cube BeamSplitter; HWP: Half-Wave Plate; PBS: Polarizing Beamsplitter; AOM: Acousto-Optical Modulator; SMPM fiber: Single-Mode Polarisation-Maintaining fiber; TCF: Triplet Fiber Collimator; QWP: Quarter-Wave Plate; FI: Faraday Isolator; Flipping M: Flipping Mirror.

Figure 2 .Figure 2 . 15 :

 2215 Figure 2.15: Pulses sequences of the excitation and STED beams. T p is the laser repetition period, t p the STED pulse duration and ∆t the delay.

Figure 2 . 16 :

 216 Figure 2.16: Absorption and emission spectra of a crimson fluorescent bead. Green line: excitation wavelength at 560 nm; red line: depletion wavelength at 760 nm; yellow zone: fluorescence detection zone (600 nm -680 nm) corresponding to the transmission windows of the emission and dichroic filters.

Figure 2 . 17 :

 217 Figure 2.17: Effects of the delay between the excitation and STED pulses. (A) Temporal evolutions of the population level of S 1 of a molecule after an excitation pulse at t = 0 for delays of the STED beam of -2 ns, 0 ns and 2 ns. (B) Fluorescence of the molecule as a function of the delay. For t > 0, one recovers the lifetime of the excited state (3.4 ns). Data are calculated for τ = 3.4 ns, I ST ED = 2.3 I S , t p = 200 ps and T p = 12.5 ns. Population and fluorescence are normalized to 1. (C) Experimental data of fluorescence signal emitted by a fluorescent bead as a function of delay. Average intensities: I exc =100 kW/cm 2 and I ST ED = 50 MW/cm 2 . (D) Temporal evolution of the fluorescence of a single fluorescent bead while applying a periodic on/off STED beam together with a constant excitation beam. Average intensities: I exc = 100 kW/cm 2 , I ST ED = 300 MW/cm 2 , integration time per point 100 ms.

Figure 2 .

 2 Figure 2.17B shows P S1 as a function of the delay. At ∆t = 0, the fluorescence de-

Figure 2 .

 2 Figure 2.17C shows the fluorescence signal of a single bead recorded as a function of the delay, which is similar to the simulated data (Figure 2.17B). At ∆t=0, the depletion and excitation pulses arrive simultaneously on the sample and the depletion efficiency is maximal. No depletion occurs when the entire depletion pulse arrives before the excitation pulse. For increasing delays, the depletion efficiency decreases and the fluorescence signal increases with a time constant of the order of the excited state lifetime of the molecules.

Figure 2 .

 2 Figure 2.17D presents the temporal evolution of the fluorescence of an excited bead submitted to successive on/off time periods for ∆t = 0. The fluorescence signal, which is approaching 2000 c/s without depletion beam, drops to the background level 200 c/s when the depletion beam is switched "on" with an average depletion intensity 300 MW/cm 2 .

Figure 2 . 18 :

 218 Figure 2.18: Intensity profile of the doughnut beam. (A) Two-dimensional intensity distribution of the STED beam in the sample (focal) plane. This measurement was obtained by scanning a 60 nm gold nanoparticle across the focused beam and recording the scattered light. Pixel dwell time 10 ms. (B) Cross-sectional intensity profile.

Figure 2 .

 2 19A shows a 2 µm * 2 µm confocal image of 20 nm fluorescent beads spin-coated on a glass coverslip recorded without STED beam. Figure 2.19B is the corresponding STED image recorded with an average STED intensity of 75 MW/cm 2(average STED power 400 mW). The STED image clearly demonstrates resolutions beyond the diffraction limit. The images presented are raw images, without deconvolution.

Figure 2 .Figure 2 . 19 :

 2219 Figure 2.19C shows the confocal and STED profiles of an individual bead. The FWHM of the STED profile gives a resolution of about 40 nm, much smaller than that of the confocal profile (≈ 230 nm, diffraction limited).We then use our doughnut-STED microscope to image microtubules in fixed fibroblastlike COS cells. Microtubules are a component of the cytoskeleton, they can grow as long

Figure 2 . 20 :Figure 2 . 21 :

 220221 Figure 2.20: Illumination time for biological sample. The scanner moves the sample every 1 ms, whereas the illumination time is set by the AOM to 200 µs.

Figure 2 .Figure 2 . 22 :

 2222 Figure 2.22: Optical setup of the Lattice-STED microscope. The oscillator 1 and the OPO together deliver the excitation beam which provides a wide-field illumination (10 µm * 10 µm). The oscillator 2 delivers the STED beam, divided by a SLM into multiple beams which interfere on the sample plane to generate optical lattices of the size of 3 µm * 3 µm. A sCMOS camera records the fluorescent signal. Abbreviations: NPBS: Non-Polarising cube BeamSplitter; HWP: Half-Wave Plate; PBS: Polarizing Beamsplitter; SMPM Fiber: Single-mode Polarisation-Maintaining Fiber; TFC: Triplet Fiber Collimator; QWP: Quarter-Wave Plate; L: lens; FI: Faraday Isolator; DM: Dichroic Mirror.

Figure 2 . 23 .

 223 Figure 2.23. It contains an array of pixels (800 * 600). Each pixel is a square cell (20 µm by 20 µm) filled with liquid crystal. When an electric voltage is applied on a pixel, the liquid crystal molecules are tilted. As liquid crystal molecules show optical anisotropy,this tilt changes the refractive index of the medium (for incident polarization along the long display axis) which causes a modified optical path length within the liquid crystal cell. Therefore an electric voltage will correspond to a phase level. An incident light passes through two times the liquid crystal layer and is imprinted with a phase e iΦ(x,y)

Figure 2 .Figure 2 . 24 :Figure 2 . 25 :

 2224225 Figure 2.23: SLM Principle. An array of electrode applies on each pixel (filled with liquid crystal) a given voltage. This voltage locally modifies the orientations of the molecules and therefore modulates the refractive index of the medium. A phase pattern can be thus imprinted to the incident light when it passes through the liquid crystal layer.

Figure 2 .Figure 2 . 26 :

 2226 Figure 2.26: Images of phase masks applied on the SLM in grey scale (255 levels coded over 8 bits) for the three-beam (A) and four-beam (B) generations.

Figure 2 .

 2 Figure 2.27A and B show the generation of the four-beam optical lattice with the SLM.The SLM surface is conjugated with the sample plane by three lenses and the objective, with respectively the focal lengths f 1 , f 2 , f 3 and f 0 . The magnification of this optical system is f 2 f 0 /(f 1 f 3 ), which affects both the angle of incidence and the size of the optical lattice at the sample plane. The size of the optical lattice can be defined as the FWHM ∆r 1 of the interfering beams at the focal plane of the objective. It is determined by the available STED laser power and the desired resolution. In order to obtain an optical lattice of 3 µm * 3 µm, the size of the beams at the back aperture of the objective is ∆r 2 ≈ f 0 λ n ∆r 1 ≈ 0.55 mm, with f 0 = 3.3 mm, n = 1.515 and λ = 760 nm. The lens L3 is installed at the entrance of the microscope and its focal length is therefore set to 300 mm. The initial size (5 mm) of the four beams and the focal lengths of the lenses L1 (500 mm) and L2 (35 mm) are optimized in order to obtain the optical lattice with the target size. According to the magnification of the system, the angle ζ (between the wave vectors of the four beams originating from the SLM and the optical axis) is then set by the SLM to 2.8' to obtain an angle of incidence of 60 • at the focal plane of the objective. The four beams diffracted by the SLM initially have linear polarizations along the x-axis, thus the polarizations of two of the beams have to be

Figure 2 . 27 :

 227 Figure 2.27: Optical setup to produce the four beams (A, C: side views; C, D: transversal planes). The four beams are generated with either a SLM (A, B) or two Wollaston prisms (C, D). The SLM (or the two Wollaston prisms (WP)) are conjugated with the sample plane by a set of lenses (L1, L2 and L3) and the objective, with focal lengths of f 1 , f 2 , f 3 and f 0 , shown in (A) (or C). The polarization of each beam at given transversal planes is shown in (B) (or D). Black spots illustrate the intersections of the beams with the planes, blue double arrows show the polarizations. Yellow circles and squares indicate respectively the half-wave plates (HWP) and the Wollaston prisms. Dashed yellow lines show the beam splitting directions.

  scan a big fluorescent bead (∼ 100 nm) over the field of illumination in presence of both the STED optical lattice and a homogeneous excitation, while recording its fluorescence signal with the camera. For each scanning step, a camera image is acquired and the fluorescence intensity is integrated over the PSF of the bead image. The reconstructed fluorescence map is therefore the effective excitation PSF of the Lattice-STED microscope, shown in Figure 2.28. The bright spots correspond to the intensity "zeros" of the optical lattice (see Figure 2.10A), where the depletion is minimal and the fluorescence is maximal. The effective excitation PSF exhibits cross-like structures, similar to the simulated PSF of a fluorescent bead containing large numbers of fixed molecules when the lattice beams have relative phases ∆Φ = 0 (see Figure.

2 . 12

 212

Figure 2 . 1 0Figure 2 . 28 :Figure 2 . 29 :

 21228229 Figure 2.28: Effective excitation PSF of the four-beam Lattice-STED microscope. It is obtained by measuring the fluorescence emitted by a fluorescent bead with a CMOS camera while scanning the bead over a homogeneous excitation beam and the STED optical lattice (3 µm * 3 µm). I exc = 100 W/cm 2 , I max = 1.68 MW/cm 2 .

Figure 2 . 30 :

 230 Figure 2.30: Two ways of scanning the sample with the piezoelectric scanner. (A) and (B) are respectively the trajectories of a bead scanned in single-direction and zigzag modalities. A fluorescence image of the bead is acquired for each scan step, the position of the bead is determined by fitting the image with a two-dimensional Gaussian function to locate its centroid. Blue and red lines indicate respectively the desired and measured positions of the bead. The scans in both (A) and (B) are from top to bottom. Time per step 800 µs.

Figure 2 . 31 :

 231 Figure 2.31: Principle of the CMOS camera. A) Electronic structure of the camera. Each pixel is composed of a photodiode and an amplifier. B) Readout of the sensor. The high speed readout is performed by splitting the sensor in two parts where both half sensors are read out simultaneously. Each part is read out line by line successively at a rate of 9.7 µs/line. A region of interest of 128 * 128 pixels is used in our Lattice-STED microscope.

Figure 2 . 32 :

 232 Figure 2.32: Binary mask for the four-beam Lattice-STED microscope. (A) Effective excitation PSF of the four-beam optical lattice combined with a homogeneous excitation. (B) Corresponding binary mask, with black and white parts being respectively opaque and transparent. Each white mask comprises 13 camera pixels and overlaps with one of the fluorescence maxima in A to extract the fluorescence signal. (C) Magnified view of the red square in B. The background shows the simulated emission PSF (FWHM 266 nm) of a fluorophore, with the emission wavelength at 650 nm. The black grid illustrates the pixels of the camera, and the white grid shows the mask.

Figure 2 .

 2 38 and Figure 2.40).

Figure 2 .

 2 Figure2.33: "Cross-talk" between two "detectors". A) Two detectors are separated with a distance of 290 nm (period of the optical lattice). The background shows the emission PSF of a fluorophore centred at "detector" 1; B) Evolution of the relative amount of fluorescence detected by "detector" 2 compared to that detected by "detector" 1, as a function of the size of the "detectors".

Figure 2 . 34 :

 234 Figure 2.34: Effective excitation PSF of the three-beam Lattice-microscope with a homogeneous excitation. (A) Effective excitation PSF measured by scanning a fluorescence bead across the focal plane of the objective and recording its fluorescence simultaneously. The blue circle (diameter of 2.3 µm) is the region where depletion is efficient enough for the Lattice-STED microscope. Fluorescence is normalized to the maximal value. (B) Intensity distribution of the three-beam optical lattice for the angle of incidence of 60 • , normalized relative to the maximal value. (C) Intensity profiles along the blue and green lines in B.

Figure 2 .Figure 2 . 35 :

 2235 Figure 2.36) on these raw images to extract the fluorescence signal originating from each intensity minima of the optical lattice. The CMOS camera together with the digital

Figure 2 . 36 :

 236 Figure 2.36: Binary mask for the three-beam Lattice-STED microscope. (A) Effective excitation PSF. (B) Binary mask. Black and white parts are respectively opaque and transparent. Each white mask comprises 21 camera pixels.

Figure 2 .Figure 2 . 37 :

 2237 Figure 2.37: Three-beam Lattice-STED imaging of a sample containing 20 nm fluorescent beads. (A) Diffraction limited image; (B) super-resolved image; (C) intensity profiles along the dashed lines in A (blue) and B (red). P exc =2 mW, P ST ED = 400 mW. Acquisition time: 72 ms per super-resolution image.

Figure 2 . 38 :

 238 Figure 2.38: Four-beam Lattice-STED imaging of a sample containing 20 nm fluorescent beads. (A) and (B) are respectively the diffraction limited and the super-resolved images without and with the depletion optical lattice. (C) Intensity profiles of a bead without and with the depletion optical lattice. I exc = 2.5 kW/cm 2 , I max = 2.4 MW/cm 2 , acquisition speed: 80 ms per super-resolution image.

Figure 2 Figure 2 . 39 :Figure 2 .

 22392 Figure 2.39B.

Figure 2 . 41 :

 241 Figure 2.41: Simulations of the intensity distributions of the excitation (A) and STED (B) optical lattices for Lattice-STED. The optical lattices are complementary, i.e. A displays maxima where B has minima, and vice versa. The dashed boxes show the unit cells of the two optical lattices.

Figure 2 .

 2 Figure 2.42 shows the experimental setup of the Lattice-STED microscope with structured excitation. The setup is similar to Figure 2.22, except that both the excitation and STED beams are divided into four beams, respectively with a SLM (Pluto-Vis, Holoeye) and two Wollaston prisms (W1 and W2), to generate complementary optical lattices.

Figure 2 . 42 :

 242 Figure 2.42: Experimental setup of the Lattice-STED microscope with structured excitation and depletion patterns. As before, the depletion optical lattice is obtained with two Wollaston prisms. The excitation one is generated with a spatial light modulator (SLM), which permits the control of the relative phases between the interfering excitation beams and the incident angle of the beams (affecting the period), in order to position the excitation optical lattice over the depletion one. Abbreviations: NPBS : Non-Polarising cube BeamSplitter; HWP: Half-Wave Plate; FI: Faraday Isolator; SMPM Fiber: Single-mode Polarisation-Maintaining Fiber; Triplet: Triplet Fiber Collimator; PBS: Polarizing Beamsplitter; W: Wollaston Prism; L: Lens; SLM: Spatial Light Modulator; DM: Dichroic Mirror.

Figure 2 . 43 :

 243 Figure 2.43: Fluorescence decays recorded for the same maximum excitation intensity in the case of structured (red) and homogeneous excitations (blue). An exponential fit gives a decay time of 44.1 s for the structured excitation, two times longer than that for the homogeneous case 20.7 s. I exc = 127 W/cm 2 , I max = 1.7 MW/cm 2 .

Figure 2 . 44 :

 244 Figure 2.44: Diffraction-limited (left side) and Lattice-STED (right side) successive images of 20 nm fluorescent beads in carbopol gel (concentration 2%). Images are taken at the rate of 80 ms per image with I exc = 2.2 kW/cm 2 , I max = 1.7 MW/cm 2 . The squares indicate the region where one bead is moving around another one.

T 1 .

 1 The molecular vibrational levels of the molecule and the phonon states of the host add on top of each electronic levels. Various transitions are possible. Radiative transitions are indicated by solid arrows and non-radiative transitions by dashed arrows. Excitation of the molecule can involve electronic, vibrational and phonon states. In most low-temperature experiments, molecules are excited from the lowest vibrational level of S 0 to the lowest vibrational level of S 1 . This transition is referred to as the 0-0 transition and is a purely electronic zerophonon transition. In well chosen host-guest systems at 2K, dephasing of the transition dipole owing to phonons is drastically reduced, and the zero-phonon linewidth of a single

Figure 3 . 1 :

 31 Figure 3.1: Jablonski diagram of a single molecule embedded in a solid. The thickest lines indicate the lowest energy levels of each state. Lines marked with ν are vibrational levels. The thinnest lines denote the phonon states of the host. Inter-system crossing (ISC) and decays rates (k 21 , k 23 and k 31 ) are shown. Solid arrows indicate the radiative transitions and dashed arrows show the non-radiative ones.

1 Figure 3 . 2 :

 132 Figure 3.2: Schematic absorption spectrum of a single molecule showing only two vibronic bands.

Figure 3 . 3 :

 33 Figure 3.3: Principle of the spectral selection of single molecules in solid hosts at cryogenic temperature. The inhomogeneous line is formed by the sum of all the homogeneous lines of the molecules. Single molecules have absorption lines with different frequencies due to their different local host environments, as depicted in the right top corner. The general structure illustrates the host matrix and the rectangle indicates the impurity molecules.

Figure 3 . 4 :

 34 Figure 3.4: Excitation spectrum of single terrylene molecules in the X 2 site of a pterphenyl crystal. Laser wavelength at 578.5 nm, excitation intensity 1 W/cm 2 .

  T 1 represents the lifetime of the exited level, which takes into account both the radiative and non-radiative relaxation processes. T * 2 represents the pure dephasing time of the transition dipole. Dephasing is caused by a coupling of the molecular electronic transition to low frequency matrix vibrational modes like phonons and librations. Excitation of these modes strongly depends on temperature, such that dephasing usually vanishes below 2K. The zero-phonon linewidth of a single molecule therefore reaches its fundamental lower bound determined by the excited state lifetime.

  α F C and α DW are the Franck-Condon and the Debye-Waller factors. cos 2 θ takes into account the angle θ between the transition dipole moment and the laser field polarization.The main contribution of the cross section comes from the factor 3λ 2 /(2π), which is 1.6 • 10 -9 cm 2 at λ = 589 nm. Taking θ = 0, the other factors contribute by about 0.1 at low temperature. Therefore typical values of the absorption cross-section of a single molecule at 2K are at the order of 10 -10 cm 2 . They are about 10 6 times larger than the absorption cross-sections of dye molecules at room temperature, which are about10 -16

. 4 )

 4 Note that from Equation 3.4, we can obtain the saturation intensity of a two-level system by setting k 23 /2k 31 = 0. The effects of optical saturation on the fluorescence emission rate R(I) and the absorption linewidth ∆ν hom (I) of the molecule are given by: R(I) = R ∞ I/I S 1 + I/I S , (3.5) ∆ν hom (I) = ∆ν hom (0) 1 + I/I S (3.6)

Figure 3 .

 3 [START_REF] Chao | Soft X-ray microscopy at a spatial resolution better than 15 nm[END_REF] shows fluorescence excitation spectra of a DBATT molecule with excitation intensities varying between 0.1 kW/cm 2 and 2.5 kW/cm 2 . Increasing the excitation intensity leads to both a line broadening and a raise of the fluorescence signal. Figure3.6A shows the experimental fit of the linewidths by Equation 3.6, yielding the homogeneous linewidth ∆ν hom (0) = 50 MHz and the saturation intensity I S = 3 W/cm 2 . We find the same saturation intensity by fitting the intensity dependence of the fluorescence signal with Equation3.5 (see Figure3.6B). The polarization of the excitation light is set circular in order to excite identically molecules with dipole moments spread in the sample plane. The measured I S for different molecules is between 1 W/cm 2 and 5 W/cm 2 . Such saturation intensities are about 10 5 times lower than those obtained at room temperature.

Figure 3 . 5 :

 35 Figure 3.5: Fluorescence excitation spectra of a DBATT molecule in octadecane, measured at different excitation intensities.

Figure 3 . 6 :

 36 Figure 3.6: Optical saturation of a single molecule. (A) Broadening of the linewidth. (B) Saturation of the fluorescence signal. The squares indicate the experimental data, solid lines show the fits with Equation 3.6 (A) and Equation 3.5 (B).

  I D is the maximal intensity of the doughnut beam and d the diameter (the distance between two opposing intensity maxima) of the doughnut beam. For λ = 589 nm and an objective with NA = 0.95, d = λ/NA = 620 nm. At low intensity, the fluorescence of the molecule is proportional to the excitation intensity. The image of the molecule thus reproduces the intensity profile of the doughnut beam (see Figure 3.7A). The resolution can be defined as the FWHM (290 nm) of the

Figure 3 . 7 :Figure 3 . 8 :Figure 3 .Figure 3 . 9 :Figure 3 . 10 :

 3738339310 Figure 3.7: (A) Simulated image of a molecule excited with a doughnut beam. (B) Crosssectional profile of the image along the dashed line in A, whose FWHM determines the resolution. I D = 0.1I S . The fluorescence signal is given in unit of R ∞ , the fluorescence signal given by one molecule at full saturation.

Figure 3 .

 3 Figure 3.10B shows the direct-ESSat image obtained with a doughnut beam at I D /I S = 10. We can clearly resolve molecules which are not resolvable in Figure 3.10A. However, when more than two molecules are present in the focal spot, the fluorescence signal of each molecule adds a background signal to the other one, which degrades the contrast of the central dip.

Figure 3 .Figure 3 . 11 :

 3311 Figure 3.11A shows the intensity profiles of the doughnut beam (blue) and the Gaussian beam (red). The doughnut beam has the maximal intensity much higher than I S and saturates the molecule except in its center. The Gaussian beam has a maximal intensity equal to I S . Figure 3.11B is an illustration of the principle of the modulated-ESSat microscopy. The blue and red curves are respectively the fluorescence profiles of the molecule, excited with only the doughnut beam and only the Gaussian beam (not modulated). In modulated-ESSat microscopy, both beams excite the molecule, so that the excitation intensity is I total (t) = I donut + I Gaussian (t). Since the Gaussian beam is

. 13 )Figure 3 . 12 :

 13312 Figure 3.12: Simulations of modulated-ESSat imaging. (A) and (B) represent respectively images of a molecule excited with a Gaussian beam and a doughnut beam; (C) the modulated-ESSat image. (D -F) are images of 50 identical molecules taken with respectively a Gaussian beam, direct-ESSat microscopy and modulated-ESSat microscopy. I G = 0.1I S , I D = 10I S . The fluorescence signal is given in the unit of R ∞ .

. 16 )

 16 As in the case of direct-ESSat microscopy, the resolution mainly depends on the inverse of I D /I S and can be drastically reduced when I D I S . In principle, it also depends on the intensity of the Gaussian beam and is proportional to A(I G ). The evolution of A(I G ) is shown in Figure 3.13. At I G I S , A(I G ) = 1 and the resolution of modulated-ESSat microscopy is the same as that of direct-ESSat microscopy. Indeed, in this case, the amplitude of the modulated fluorescence signal is ∆R(r) ≈ R ∞ -R[I donut (r)], with R ∞ the maximal fluorescence signal. Obviously, ∆R(r) has the same FWHM as that of R[I donut (r)] (the fluorescence distribution of direct-ESSat). For lower I G , √

Figure 3

 3 Figure 3.13: √ A as a function of I G /I S .

Figure 3 . 14 :Figure 3 .Figure 3 . 15 :Figure 3 . 16 :

 3143315316 Figure 3.14: Chemical structures of the aromatic DBATT molecule and octadecane.

Figure 3 . 18 :

 318 Figure 3.18: Interface of the Labview program for the optimisation of the doughnut beam profile.

  )where I exp is the experimental image of the doughnut beam, reflected on a gold mirror placed at the focal plane of the objective. I cal is the corresponding desired diffractionlimited image of the doughnut beam, calculated on the basis of the experimental diameter of the doughnut beam profile. x, y are the pixel numbers in the image, and a i is the weight of the i th Zernike polynomial.The algorithm is based on the simulated annealing method, which minimizes the error function by randomly changing the weight of one of the Zernike polynomials. It combines a direct research by successive tries and a random acceptance of certain "harmful" modifications. A Maxwell-Boltzmann factor is introduced to decide the acceptance of the random changes, allowing to find a better minimum than the closest local minimum. A parameter called Temperature appears in the term exp(-∆Err/T ), where ∆Err = Err 1 -Err 0 is the difference between the error function before and after the change. The Temperature evolves with time to ensure a convergence with a reasonable number of iterations and a sufficient exploration of the error function space.

Figure 3 .Figure 3 . 19 :Figure 3 .

 33193 Figure 3.19 shows the results of the aberration corrections.Figure 3.19A, B, C are respectively the reflected doughnut image at room temperature, at 2 K before and after aberration corrections. Aberrations that are clearly present at 2 K can be well corrected. At low temperature, single molecules are both small (∼ 1 nm) and photostable, they can therefore be used as local nanoscopic probe of the light intensity distribution in the focal plane of the objective. The fluorescence signal of a molecule is proportional to the excitation intensity at low saturation. By scanning a molecule across the doughnut beam while recording its fluorescence with an APD, we can reproduce the doughnut intensity profile.Figure 3.20A shows an experimentally obtained fluorescence image of the doughnut beam after the aberration correction, with I D = 0.4 W/cm 2 (I S = 2 W/cm 2 ),

Figure 3 .Figure 3 . 21 :

 3321 Figure 3.22 shows the image of another molecule and its profile for I D = 13 kW/cm 2 , which corresponds to about 10 4 I S . The lowest value of the fluorescence intensity in the image is about 0.4 of the maximal value, due to the residual intensity in the doughnut center. The residual intensity can be therefore deduced with Equation 3.5 to be about 6•10 -5 of the maximal intensity. The achieved resolution is 4.4 nm, close to the value 3.9 nm predicted by Equation3.10. This resolution is less than a hundredth of the laser wavelength (589 nm) and approaches the size of the molecule. In principle, applying higher laser intensities can further improve the resolution. However, we observed displacements of the minimal fluorescence signal between successive scanning lines in images acquired at higher laser intensities. This suggests that the resolution is currently limited by the mechanical stability of our microscope setup. Two factors have been identified.

Figure 3 .Figure 3 . 24 :Figure 3 . 25 :Figure 3 . 26 :

 3324325326 Figure 3.24: Modulated-ESSat imaging. (A) and (B) represent respectively images of a molecule obtained with the Gaussian beam (I G = 0.7 W/cm 2 ) and with the doughnut beam (I D = 26 W/cm 2 ); (C) the modulated-ESSat image. I G = 5 W/cm 2 , I D = 26 W/cm 2 , pixel dwell time 10 ms, modulation frequency 1 kHZ.

Figure 3 .Figure 3 . 27 :

 3327 Figure 3.27A shows the fluorescence excitation spectrum of two molecules. We tune the laser frequency to the frequency of maximal overlap of the two molecular lines and acquire a conventional confocal image of the two molecules, shown in Figure 3.27B. The images of the two molecules overlap and they are not resolved. Modulated-ESSat clearly resolves the fluorescence spots from the two molecules, and unravels a spatial separation of 200 nm between them, as shown in Figure 3.27C.

  require low light level to be switched. Chmyrov et al. parallelized RESOLFT microscopy based on photoswitchable proteins and achieved a field of view of about 100 µm * 100 µm with an imaging speed of several second per super-resolution images [209]. However,

cos 2 α 2 Figure B. 1 :R 2 -y 2 √ R 2 -y 2 P√ R 2 -√R 2 -y 2 √ R 2

 22122222222 Figure B.1: "Cross-talk" between two "detectors", separated with a distance of 290 nm (period of the optical lattice). The background shows the emission PSF of a fluorophore centred at "detector" 1.

  a(F l-P + F l+P + F l+1 + F l-1 ) + b(F l-P -1 + F l-P +1 + F l+P -1 + F l+P +1 ) (B.4)with l = 1, 2, ..., P * Q. Equations.B.4 consist of P * Q equations and can be written as:M * F = F (B.5)M is the cross-talk matrix depending on a and b, F and F are respectively the fluorescence values of "detectors" without and with the cross-talk.

2.1.6 Generalization of STED principle

	STED microscopy can be generalized to other methods based on different fluorescence
	depletion processes. Reversible saturable optical fluorescence transitions (RESOLFT)

  .[START_REF] Chao | Real space soft x-ray imaging at 10 nm spatial resolution[END_REF] shows the timescales in pulsed-STED microscopy, where t p (200 ps) is the STED pulse duration, T p (12.5 ns) the laser repetition period and T int (∼ 1 ms) the

	integration time per camera frame. Three different regimes of rotational mobility should
	be considered :
	(i) fast rotating molecules, where τ rot t p .
	(ii) fixed molecules, where τ rot	T int .
	(iii) slowly rotating molecules, where t p	τ rot < T int . Slowly rotating molecules can
	be regarded as fixed during each STED pulse.
			T int
	T
	STED
	t p

p Figure 2.7: Timescales in pulsed-STED microscopy. t p , T p and T int are respectively STED pulse duration, laser repetition period and integration time per camera frame. t p = 200 ps, T p = 12.5 ns and T int ∼ 1 ms.

d'onde de lumière, NA = nsinα l'ouverture numérique de l'objectif, avec n l'indice de réfraction du milieu de microscope et α le demi-angle du cône maximum de lumière qui peut entrer ou sortir de l'objectif. Avec la lumière visible et un objectif à immersion dans l'huile (typiquement de NA ∼ 1,4), la meilleure résolution latérale est d'environ 200 nm et la résolution axiale est d'environ 1 µm. En dépassant la limite de diffraction et en offrant une résolution allant jusqu'à quelques nanomètres, la microscopie à fluorescence de super-résolution a révolutionné la microscopie optique en champ lointain au cours de la dernière décennie. L'utilisation des propriétés spectroscopiques des émetteurs fluorescents joue un rôle important dans la réalisation des techniques de super-résolution. À cause de la limite de diffraction, les images de fluorescence des émetteurs dans un volume limité par la diffraction se recouvrent et ne se distinguent pas. Exploiter les propriétés photophysiques et photo-chimiques des émetteurs permet d'activer les émetteurs séquentiellement et ainsi de les résoudre optiquement. Cette séquentialité de l'imagerie est réalisée par des moyens stochastiques ou déterministes. Les méthodes de super-résolution stochastiques sont basées sur la localisation d'émetteurs uniques isolés, tels que PALM (photo activated localization microscopy), STORM (stochastic optical reconstruction microscopy) et (u)PAINT ((universal) points accumulation for imaging in nanoscale topography). Les méthodes déterministes sont basées sur l'excitation (ou la déplétion) des émetteurs par des motifs optiques qui limitent l'émission de fluorescence à des positions pré-déterminées, telles que STED (stimulated emission depletion), GSD (ground state depletion) et (S)SIM ((saturated) structured illumination microscopy).La microscopie de super-résolution a eu beaucoup de succès dans différents domaines notamment dans la bio-imagerie. En 2014, le prix Nobel de chimie a été attribué à trois chercheurs "pour le développement de la microscopie à fluorescence de super-résolution".Néanmoins, la microscopie de super-résolution est encore en cours de développement rapide. Le défi pour les chercheurs consiste à en améliorer les performances, telles que

Acknowledgement

This thesis was prepared in the Nanophotonics

C H A P I T E R 3

Cryogenic super-resolution microscopy by excited state saturation

This chapter presents a super-resolution microscopy based on excited state saturation (ESSat), performed on single organic molecules inserted in solid hosts (guest/host system) at liquid helium temperature. The motivations for performing super-resolution microscopy at cryogenic temperature are discussed in the first section. In the second section, we give a brief introduction to the photophysics of the guest/host system and to the optical saturation of single molecules. The principle of ESSat microscopy is described in the third section, together with analytic expressions for the resolution and with simulations of super-resolution images. The fourth section presents the experimental realization of ESSat microscopy. The experimental results are then shown in the next section. Sub-ten nanometer resolution has been achieved, mainly limited by the mechanical stability of the experimental setup. The last section gives a summary of this chapter and discusses the perspectives of ESSat microscopy.

Summary and outlook

During the course of this Doctorate work, we developed two super-resolution microscopies: Lattice-STED microscopy and cryogenic ESS microscopy.

Lattice-STED for the first time massively parallelized STED microscopy. Optical lattices generated by multi-beam interferences are used as depletion patterns together with a camera for detection. In order to obtain a super-resolution image, the sample only needs to be scanned over a unit cell of the optical lattice instead of the whole field of view. A 100-fold parallelization was achieved with moderate laser powers. Superresolution images of 70 nm resolution and a field view of 2.9 µm * 2.9 µm can be acquired at a rate of 12.5 images per second. This imaging speed is only limited by the readout time of the camera and can be therefore improved by using faster cameras.

The resolution performances of Lattice-STED is related to the rotational mobility of the molecules. For slowly-rotating or fixed molecules, the polarization of the field of the optical lattice matters. For three-beam Lattice-STED , the effective fluorescence PSF does not depend on the relative phases of the three beams, therefore the superresolution images can be deconvolved with this PSF. For four-beam Lattice-STED, one can use circularly polarized beams combined with light-sheet microscopy [START_REF] Santi | Light sheet fluorescence microscopy a review[END_REF][START_REF] Keller | Light sheet microscopy of living or cleared specimens[END_REF], so We now consider the wave vector. The incident wave vector can be written as k i = (0 0 k), which only has the component along the z-axis e z . Therefore we have

The transmitted wave vector follows a rotation along the axis e y and can be expressed in the basis B 1 as:

A.2 Period of the three-beam interference pattern

In this section, we discuss the period of the three-beam interference pattern at α = 30 • .

Following Equation 2.9 and Equation 2.10, the periodicity of the optical lattices is linked to phase terms Φ p,q ., which writes as:

r and ϕ are the polar coordinates at the interference plane, see Fig

The phase terms can be thus expressed as:

Note that Φ 1,2 +Φ 2,3 +Φ 3,1 =0, we can treat simple cases by setting one of the phase terms equal to 0. For example, by setting Φ 1,2 =0 (ϕ = 90 • ), we have Φ 2,3 =-Φ 3,1 . The interference pattern has the expression:

APPENDIX B

Cross-talk between neighbour "detectors" in Lattice-STED

In Lattice-STED, the binary mask and the camera acts together as an array of "point detectors", with each "point detector" corresponding to one of the intensity minimum of the optical lattice. While a fluophore is at one intensity minima, it can emit fluorescence, which will be detected by the corresponding "point detector", named "detector" 1. Due to the short distance between the "detectors" (290 nm) compared to the emission PSF, "detectors" neighbouring to "detector" 1 can also detect a small amount of fluorescence emitted by the fluorophore (see Figure B.1). This causes cross-talk between "detectors" and leads to artefacts in the super-resolution images. The artefact is called "ghost", meaning that a fluophore has a brighter image at the right coordinates and darker images appearing in adjacent sub-images. Therefore a linear unmixing is applied to the "detectors" to eliminate the cross-talk, as explained below.

We define the cross-talk as the ratio between the fluorescence signals detected by "detector" 1 and 2. a and b are respectively the cross-talks between horizontal (or vertical) and diagonal neighbour "detectors", which are given by: