Asemare Mengistie

Bushra Canaan

Mohamad Al Assaad

Sonia Ouali

Yacine Belguermi

Gandorj Darambazar

Sophie Kohler

Stéphane Bazeille

Clément Roume

HIGH-RESOLUTION TOMOGRAPHIC DIFFRACTIVE MICROSCOPY: SAMPLE SCANNING OPTIMIZATION AND 3-D QUANTITATIVE POLARIMETRIC IMAGING

Keywords:

The discussions I had with Patrick Ferrand (who is also my former professor during my M.Sc. study

Label-free imaging techniques are highly demanded and are recently emerging for several applications in biology. Tomographic Diffractive Microscopy (TDM) is one of the well-established label-free Quantitative Phase Imaging (QPI) techniques. In TDM, the sample is illuminated by a coherent laser beam at various angular directions, and the scattered field's amplitude and phase is collected, sequentially. Numerical inversion models that characterize the object-light interaction are then used to reconstruct and quantify the weakly scattering sample's 3-D refractive index (RI) distribution.

One limitation of TDM is its sequential data acquisition. Acquiring few holograms leads to faster imaging, but, with a degraded reconstructed image quality due to unrecorded object frequency components. The use of either advanced reconstruction method with limited holograms, or a simplified reconstruction with many holograms leads to high-resolution images, however, with a slower imaging. Inspired by this problem, the first part of this thesis focuses on optimizing the angular sample scanning scheme to obtain high quality images using few holograms. For that, several classes of scanning patterns are studied and the corresponding frequency support is analyzed. Image simulation considering transmission, reflection and 4p TDM configurations as well as experimental validation using a transmission TDM reveals that the optimized scheme better fills the Fourier space, leading to better optical sectioning along the optical axis.

Moreover, TDM uses a scalar wave inverse scattering theory that allows to simplify data acquisition and image reconstruction process. This can still provide 3-D highresolution quantitative RI maps and works well for samples exhibiting isotropic property, however, fails to quantify samples with anisotropic nature. The second part of this work shows the advantage and necessity of having a simplified polarization sensitive TDM (PS-TDM). Starting from formulating a simplified vectorial expression by considering a weak polarization response along the optical axis, image simulations based on beam propagation forward model are done to validate the theory. A simplified PS-TDM is then built by upgrading the scalar TDM. Finally, its ability to image samples with anisotropic molecular structures is studied considering variety of samples.

Furthermore, besides alleviating the background phase problem, the use of circularly polarized beam with a polarization analyzer sensor for PS-TDM allows to mimic a differential interference contrast microscopy (DIC). Experimental demonstration of using PS-TDM as a 3-D DIC is also made using a non-birefringent sample.

Five years ago, when I first got a master's scholarship to join one of the well know international master's programs in optics and photonics (Europhotonics), I still remember my brother's (Alemneh's) reaction. He was even more excited than I was. In fact, he takes the lion's share in all my academic milestones. I couldn't thank him enough for perfectly playing his elder brother's role. After finishing my master's study, I was hesitating to decide what kind of path I should follow. From all the discussions I had with several people (friends, family and former professors), it was Awoke A. Negash who convinced me to go for a PhD by saying "you should know that doing a PhD is a lifetime diploma". After 3 years, I'm still convinced that it is true and I'm grateful for his advice.

I would like to express my gratitude to my primary supervisor, Olivier Haeberlé, who has guided me throughout this thesis. I started to acclaim him when I first arrived at Mulhouse. He picked me up from the train station, helped me to do all administrative stuff and grocery. Once I started my thesis, his help and guidance during COVID19 period was one of a kind. He has provided his help from the smallest personal issues (administrative, French language translation, lunch time call...) up to the strong academic challenges (theoretical and practical). He has been the guiding light throughout the project by being always available and directing me to the right path. I am lucky to work with such a humble human being, rich with scientific rigor and personal foresight.

I would also like to thank my co-supervisor, Nicolas Verrier, for all the support and time he has given me and for always being available to provide any help. He has never been annoyed by many of my questions. He is one of the best researchers I ever met, who has always many amazing ideas and is enthusiastic about everything we do. He has helped me to think outside the box and guided me to analyze thoroughly any kind of experimental or numerical result. Thank you for helping me to be a better scientist.

My warmest gratitude goes to Matthieu Debailleul, the senior engineer, who was always there to help me and allow me to proceed by myself at the same time. He has guided me a lot and taught me many things by showing me how to work smartly. I am always grateful for all the talks we had together, including the scientific and nonscientific discussions.

I would also like to thank Jean-Baptiste Courbot for his guidance in many aspects, especially in the early times of my thesis. He is always positive thinking and has been asking me about my progress and always provided me with some wisdom. My gratitude goes to Bruno Colicchio for all his support and positive attitude towards any academic problem. I'm always grateful for all the discussions we had and for all his practical insights.

I also wish to warmly thank the examiners, Denis Lebrun and Virginie Chamard, for reviewing this manuscript, as well as the jury members, Bertrand Simon and Pascal Picart. I would like to thank Hugues Giovannini and Corinne Fournier for being the Jury on my mid thesis. Their comments on my progress have helped me in many aspects. Besides being my former professor, Hugues Giovannini has been my guidance and reference, from my master's study up to my PhD. I am really grateful to have known such a person with an agreeable and delightful personality.

I owe a warm thanks to two of Institut Fresnel COMiX team memebres, Virginie Chamard and Patrick Ferrand. They were happy to welcome me to their lab (for short collaboration work) and to show me their system as well as for allowing me to work with their setup using my samples and theirs. Not only for reviewing my thesis but also as a collaborator Virginie Chamard is one of the nicest people I have worked with.

OVERVIEW

Introduction

T he optical microscope has been a crucial tool in basic scientific research, and many more applied domains, since its invention in the 16 th century [1]. It has revolutionized science, by allowing human beings to uncover the unseen world where the disease-causing agents become visible, studied and prevented. Thanks to its discovery and advancements in light sources, detectors sensitivity, images restoration mechanisms and improvements in optical components design, we have now the luxury to choose from a huge assortment of imaging methods to visualize various small structures (such as individual cells and their intracellular compartments, cytoskeletal proteins, or membranes) in microscopic or nanoscopic scale. Its applications are not limited to biology, but include chemistry, electronics study, geology and material science, among others.

Biological samples in general as well as cells and their internal structures in particular are transparent, hence lack sufficient contrast for direct investigation. The most prominent approach to improve image contrast is to use specific dyes to stain these structures. This necessitates labeling the sample with fluorescent dyes that can attach to a specific structure of interest, leading to fluorescence-based techniques [2]. The invention of fluorescent microscopy has catapulted biomedical research to new heights, allowing researchers to go beyond the diffraction limit of light for optical imaging, which was set by Abbe in 1873 [3], hence enabling super-resolution imaging. A variety of super-resolution fluorescence microscopy techniques are currently available, including Stimulated Emission Depletion (STED) [4], Stochastic Optical Reconstruction Microscopy (STORM) [5], Photo Activated Localization Microscopy (PALM) [6] and many others [2].

However, fluorescent-based microscopy has some limitations. First, staining can add artifacts, which come from structures that are added during sample preparation and do not constitute the inherent nature of the specimen. Even with careful preparations, such approaches allow to view specific structures of an object, and could fail to image the whole sample structures. Second, labeling could give wrong insight by creating undesired changes on the sample as the result of photobleaching and phototoxicity [7]. Hence, for long time lapse studies such as cell division that takes hours or days, the fluorophores may disappear through time, which make them challenging to observe.

In such applications, as well as in material characterization when knowledge about the sample is limited, the use of label-free imaging is highly significant. That is why much interest is given to label-free imaging techniques. This covers several transmission microscopy approaches, including Second Harmonic Generation microscopy (SHG) and Coherent Anti Stokes Raman Spectroscopy microscopy (CARS) [8].

In conventional wide-field microscopy (bright field, oblique illumination, dark field, dispersion staining, phase contrast, differential interference contrast, etc) the image contrast is produced by the scattering of light by the structural distribution of the sample structures. The weak contrast limits its application for biological imaging. One way of obtaining better contrasted image is by upgrading the bright field system with add-ons, to detect and interpret the phase shift of light caused by the density variations of the sample structures (related to refractive index variations) into image contrast.

Despite the fact that contrast improvement mechanisms have been identified and used, the refractive index of the material cannot be quantitatively determined from the observed intensity of light. In fact, for quantitative imaging, one needs to record both amplitude and phase alterations. With the development of digital holography [9] in 1948 by Gabor (Nobel Prize in physics in 1971) and advances in electronics and computer processing, which paved the way for the discovery of Digital Holographic Microscopy (DHM), it is now possible to retrieve quantitatively the absorption and refractive index distribution of an object.

This PhD work is devoted to Tomographic Diffractive Microscopy (TDM). Alternatively known as digital holographic tomography, tomographic phase microscopy or optical diffraction tomography, TDM is an extension of DHM for 3-D imaging. It is an emerging and well-known label-free 3-D high-resolution quantitative imaging technique, which was first proposed by E. Wolf in 1969 [10]. Then, in 1982, a solution in the image space was defined by Devaney [11] for its experimental implementation to image semi-transparent samples. Currently, TDM is implemented by several groups [12][13][14][15][16][17] and even commercially available. The basic idea is to illuminate the sample with a coherent laser beam from various angular directions, and collecting the scattered field's amplitude and phase, holographically. Then, numerical inversion models, that characterize the sample-light interaction, are used to reconstruct and quantify the sample's 3-D refractive index map.

This thesis addresses mainly two issues. First, it focuses on optimizing TDM angular sample illumination scheme to improve data acquisition and image resolution. Then, aiming at quantifying birefringence of anisotropic objects in 3-D, development of a simplified high-resolution polarization sensitive imaging system for several applications will be provided.

Outline of the document

This thesis contains 4 chapters:

• In Chapter 1, I will describe the state of the art of tomographic diffractive microscopy along with a review of some recent works in the field. First, I will give a brief introduction to holographic microscopy. This is followed by broad discussion on the principles of TDM using Born and Rytov approximation models of scalar description of electromagnetic fields. Several possible configurations of TDM will be explored including transmission, reflection and 4Pi TDM configurations. TDM resolution improvement approaches, along with their numerical data recombination processes, will also be recalled from the literature and will be discussed in detail. Furthermore, I will also highlight TDM data acquisition simplification and quantitative 3-D polarimetric imaging techniques.

• Chapter 2 focuses on optimizing angular sample illumination scheme for reflection, transmission and 4Pi TDM configurations. The Fourier space filling as the result of these scanning schemes will be validated and analyzed with numerical simulations and experiments.

Most research groups that are developing TDM consider light as a scalar electromagnetic field, leading to scalar approximations. This is a simplified and relatively easier approach for isotropic and homogeneously weakly scattering objects. For anisotropic samples, considering the vectorial nature of light (even using paraxial field approximation) provides better contrasted and more accurate quantitative RI maps.

• Chapter 3 is devoted to high-resolution polarization sensitive TDM. Firstly, the principles of TDM for polarimetric imaging will be detailed using Born and Rytov approximation models of vectorial description of electromagnetic fields. Then, the numerical validation of implementing a polarization sensitive TDM considering birefringent samples will be presented. Next, the experimental set-up development will be discussed in detail. Experimental validation of the reconstruction algorithm will be done by considering several anisotropic sample's and reconstructing their 3-D RI tensor as well as other polarimetric information, including birefringence, average refractive index and fast-axis orientation maps. Inspired by the limitations of conventional differential interference contrast microscopy, a mechanism to use the already built polarization sensitive TDM system as a 3-D differential interference contrast microscope to image non-birefringent samples will also be presented with experimental validations.

• Finally, in Chapter 4, I will conclude this thesis's main milestones together with future perspectives.

Introduction

In conventional wide field optical microscopy techniques, an incoherent light's complex interaction and propagation through the object is recorded by a camera as an intensity variation image. However, such approaches only allow to image strongly absorbing specimens. Hence, transparent samples (such as biological samples) lack image contrast. In such conditions, capturing the phase variation of the diffracted field due to the optical index difference of the specimen highly improves the visibility of sample's inner structures. Several phase imaging techniques can be mentioned. Proposed by Zernike and Nomarski, phase contrast microscopy [18] and differential interference contrast (DIC) [19] microscopy, allows to qualitatively reveal the phase alterations. Also, quantitative methods such as the transport of intensity equation, Fourier ptychographic microscopy [20] and Shack Hartman wave-front detection approach [21] can be used to quantify the optical path difference.

Chapter 1. STATE OF THE ART

In 1948, Gabor [9] introduced the concept of holography, and showed a direct approach to retrieve the phase and amplitude change of the interacting light with a scattering object. Relying on interferometric techniques, holography has shown its invaluable importance, especially in biological imaging [16,22]. Inspired by this, E. Wolf proposed a 3-D imaging approach, known as Tomographic Diffractive microscopy (TDM) in 1969 by combining holography with a synthetic aperture process [10], which has emerged through time related to advancements in sensor technology and computation efficiency. This chapter will give a brief overview of Digital Holographic Microscopy (DHM) and principles of TDM, which will be followed by detailed discussion on the various TDM techniques utilized by several research laboratories to extend the 3-D image resolution. Finally, it will introduce and highlight TDM's general numerical reconstruction process, and its future perspectives.

Digital Holographic Microscopy

Holography allows to record both the amplitude and phase of light scattered by a sample. It is a coherent imaging technique, which requires measuring a hologram as a result of interference between the field of interest (diffracted field by the object) u d and a known reference field u r .

Figure 1.1 is a schematic illustration to understand the general holography process, where the object is illuminated by an incident beam u i . The interference between the diffracted field and the reference field is then recorded using a CCD or CMOS camera.

Incident beam Object

Reference beam Diffracted Experimentally, the camera captures the intensity of the hologram I h , which can be represented mathematically as:

I h = |u d + u r | 2 = |u d | 2 + |u r | 2 | {z } 0 order + u ⇤ d u r |{z} -1 order + u d u ⇤ r |{z} +1 order
.

(1.1)

The zero order in equation (1.1) contains both the DC components and auto-correlation of the diffracted field amplitude. The 1 and +1 order terms contain information regarding the virtual and real images of the object, respectively. Recreation of the original object field from the hologram is done by illuminating it with a beam identical to the reference beam.

To compute the phase and amplitude contained in the interferogram, various holography approaches are used, which differ by the imaging system configuration. Different configurations of holography exist depending on the type of light source, the position of the object and the way of capturing the interferogram. The main types of configurations used in digital holography includes: Gabor holography (in-line), off-axis holography and phase-shifting holography.

In-Line Holography

Gabor, or in-line holography was first developed for electron microscopy [9], and then adapted to optical holography [23]. As depicted in Fig. 1.2, the coherent beam generated by a point source illuminates the object. A portion of the illumination beam is diffracted by the sample and the remaining light, which does not interact with the object, acts as a reference. The interference fringes resulting from these 2 beams can be detected by the camera and demodulated numerically to get the phase and amplitude information of the object. Its relative tolerance to variations in experimental circumstances, such as experimental setup movement or perturbations, as well as its relatively simpler configuration piqued interest to use it for variety of applications [16]. However, since the reference and diffracted beams share the same path, acting on one of them is more difficult, and digital filtering to suppress the conjugate image is required [24].

Off-axis Holography

As previously explained, in Gabor holography, the beam associated with reference and diffracted beams share the same path, hence, any of the images could suffer from an out-of-focus contributions. To avoid an overlap between the diffraction orders, the illumination wave and the reference wave (background beam) are tilted by an angle from each other. This was first done by Leith and Upatnieks in 1962 [25] for analog holography, and subsequently adapted for digital holography [26,[START_REF] Briones | Holographic microscopy[END_REF]. In this case, only a single recorded hologram is required to reconstruct the image of an object.

The experimental setup of off-axis holographic microscopy is based on a Mach-Zehnder interferometer, as depicted in Figure 1.3, for transmission case here. A coherent light source is split into a reference and illumination beam by a beam splitter (BS). After illuminating the sample by the incident beam, the diffracted field is collected by a microscope objective and recombined with the reference beam using a recombining cube (RC). The basic idea here is to interfere the diffracted beam with the reference beam, which is tilted at an angle by controlling the axes of the RC. This introduces a mechanism to modulate the hologram so that the various diffraction orders can be retrieved. Recalling Equation (1.1), and now illuminating the sample using a plane wave, we can re-write the intensity at the detector as:

I h (x, y)=|u d | 2 + |u r | 2 | {z } 0 order + u ⇤ d u r exp[i(k cx x + k cy y)] | {z } -1 order + u d u ⇤ r exp[ i(k cx x + k cy y)] | {z } +1 order , (1.2)
where (k cx , k cy )=( k c cos q c , k c sin q c ) are the coordinates of the illumination wavevector in the Fourier space, q c and k c being the angle introduced by the RC and a constant, respectively. The two first terms of Equation (1.2) are zero order of diffraction, while the third and the fourth terms are twin and real images, respectively. The three diffraction orders correspond to signals centered at the spatial frequencies 0,

(k cx ,k cy ) 2p
and

( k cx , k cy ) 2p
. Figure 1.4 depicts the general off-axis holography demodulation process. One advantage of using off-axis holography is that the object field (phase and amplitude) can be easily retrieved from the recorded hologram, since it allows to demodulate both (twin and real images) diffraction orders from a single acquisition. The recorded interferogram, as depicted in Fig 1.4 (a), clearly shows interference fringes on the object. Reconstruction of the object in off-axis configuration is done in the spatial Fourier space. Figure 1.4 (b) depicts the computed 2-D Fourier transform F of the recorded interferogram acquired in the off-axis holography. Mathematically, this can be expressed as:

F (I h (x, y)) = F{|u d | 2 + |u r | 2 + u d u ⇤ r exp[ i(k cx x + k cy y)] + cc}, (1.3)
where cc is the complex conjugate of u d u ⇤ r exp[ i(k cx x + k cy y)]. u r (x, y) and u d (x, y) are the complex amplitudes of the reference and diffracted field, respectively. Thanks to the tilt angle introduced by the RC, we observe a clear separation of the several diffraction orders. Note that the tilt angle has to be high enough to forbid the overlap between the diffraction orders.

Note that since the reference field is constant, the +1 order is proportional to the diffracted field but shifted in the Fourier space. Hence, in order to select only the object, we filter the +1 order of the object spectrum and cut out the 0 and 1 orders. The selected order is then centered at zero frequency

F [I obj (x, y)] = F [u d u ⇤ r ]. (1.4)
We finally compute the 2-D inverse Fourier transform,

I obj (x, y)=u d u ⇤ r .
(1.5)

I obj (x, y) contains the object amplitude and phase information of the object, which can easily be computed by normalizing it with the reference, Fig 1.4 (c). Note that the phase computed from the filtered field may be wrapped, hence a phase unwrapping procedure is performed [START_REF] Vyacheslav | Deterministic phase unwrapping in the presence of noise[END_REF].

Phase-Shifting Holography

In off-axis holography, due to the need for carrier fringes, we do not effectively make use of the pixel number of the camera. Also, the demodulated image size is limited, which comes from the reconstruction process in the Fourier space. Phase shifting holography was proposed and implemented by Yamaguchi and Zhang in 1997 [START_REF] Yamaguchi | Phase-shifting digital holography[END_REF][START_REF] Yamaguchi | Phase-shifting digital holography[END_REF][START_REF] Tahara | Parallel phase-shifting digital holographic microscopy[END_REF]. It necessitates to record a series of holograms by changing the phase of the reference wave, using phase shifting devices such as a piezoelectric transducer mirror or an electro-optic modulator (see Fig. 1.5). This makes demodulation possible, which allows to extract the phase and amplitude information of the object. Let us now consider the reference field:

u r (x, y)=u r (x, y) exp(if r (x, y) J), (1.6)
and the object wave

u o (x, y)=u o (x, y) exp(if o (x, y)) = u o (x, y)[cos(f o (x, y)+isin(f o (x, y))]. (1.7)
Here u r (x, y) and u o (x, y) are the complex amplitudes, f r (x, y) and f o (x, y) are the phases, and J is the phase change introduced to the reference field by the phase shifting device. Now, the interference between these fields gives:

I h = |u o (x, y)+u r (x, y)| 2 = |u r (x, y)| 2 + |u o (x, y)| 2 + 2 u o (x, y) u r (x, y) cos[f o (x, y) f r (x, y)+J] = I 1 (x, y)+I 2 (x, y) cos[f(x, y)+J], (1.8)
where

I 1 (x, y)=|u r (x, y)| 2 + |u o (x, y)| 2 , I 2 (x, y)=2u o (x, y)u r (x, y) and f(x, y)= f o (x, y) f r (x, y).
Three unknowns u r , u o and f exist in Equation (1.8). Mathematically, it is necessary to record at least three holograms to solve the holographic equation, and hence to compute the phase and amplitude of the diffracted field. However, it is demonstrated that extracting the phase and amplitude could be possible by using a three step or four-step phase shifting holographic approaches [16]. Note that, a two step phase shifting method is possible by two phase-shift acquisitions (plus separate measurement of the reference and object intensities) [START_REF] Liu | Two-step-only quadrature phase-shifting digital holography[END_REF].

In practice, implementing a four step phase-shifting holography has the advantage of minimizing the numerical reconstruction error [16,[START_REF] Creath | Phase-shifting speckle interferometry[END_REF]. Let us consider 4 holograms that are recorded with the following phase shifts, J = 0, p/2, p and 3p/2,

I 0 (x, y)=I 1 (x, y)+I 2 (x, y) cos(f(x, y)), I p/2 (x, y)=I 1 (x, y)+I 2 (x, y) cos[f(x, y)+p/2], I p (x, y)=I 1 (x, y)+I 2 (x, y) cos[f(x, y)+p], I 3p/2 (x, y)=I 1 (x, y)+I 2 (x, y) cos[f(x, y)+3p/2].
(1.9)

Combining them numerically allows us to compute the amplitude and phase information: .11) Note that arc-tangent function gives the phase in the interval [ p/2, p/2]. One can also use an arctan2 (which computes the angle between the given vector to the x-axis) rather than arc-tangent function, in order to compute the phase in the interval [ p, p]. The first TDM system built at IRIMAS is based on phase shifting interferometry with electro-optic modulator [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF]. One limitation of phase-shifting holography is that it requires sequential recording of multiple holograms, which slows down the speed of data acquisition. One way to solve this could be by using several cameras, or with a parallel acquisition approach [14].

|u o (x, y)| = ⇣ [I 0 I p ] 2 +[I 3p/2 I p ] 2 ⌘ 1/2 4|u r (x, y)| ; (1.10) f(x, y)=arctan  I 3p/2 I p/2 I 0 I p . ( 1

Limitations

In previous sections, we have seen that DHM has been utilized to measure phase and amplitude of diffracted field from an object. Practically, the spatial object frequency detected in DHM is directly limited by the numerical aperture (NA) of the microscope objective. Indeed, the objective acts as a low pass-filter on the spatial spectrum, in which the frequency components that continue to propagate are those that pass through the aperture of the lens.

Note that coherent light imaging brings better frequency transmission compared to conventional wide-field microscopy system. As depicted in Fig. 1.6 (c), holographic imaging has better frequency transmission compared to wide field imaging as it uses a coherent beam.

One of the limitation of holographic microscopy is its poor resolution when imaging 3-D transparent specimens, which is more noticeable along the optical axis. This can be seen in Fig. 1.6 (a,b), where the axial sectioning of the pollen structures are relatively limited. This is mainly due to acquisition of limited object frequency, since single object illumination is used for the reconstruction. In fact, for better 3D imaging, it is necessary to acquire more information from specimen, which helps to extend and fill the spectral support using an approach known as Fourier space synthetic aperture process. This will be explained in the following section.

Tomographic Diffractive Microscopy

Although it was proposed in 1969 by E. Wolf [10], TDM has gained much attention in the beginning of 21 st century. The first optical setup (as well as the first reconstruction (holographic) and incoherent imaging system, adapted from [START_REF] Bailleul | Microscopie tomographique diffractif á resolution isotrope[END_REF].

algorithm and reconstructed images) in our laboratory, Institut de Recherche en Informatique, Mathematiques, Automatique et Signal (IRIMAS), was successfully built [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF][START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF] in 2006-2007 during the thesis of B. Simon.

Principles

In the following, I will recall Wolf's [10] approach to compute the electric field E t (r) propagating in a weakly scattering medium. Starting from the simplified Maxwell's equation (by omitting the time dependence exp( iwt) term), r⇥r⇥E t (r)=k 2 0 e r (r)E t (r), (1.12) one can solve the time independent inhomogeneous Helmholtz equation,

(r 2 + k(r) 2 )E t (r)=0. (1.13)
Here r is the vector differential operator. r 2 is the Laplacian operator and vector r = (x, y, z) is a position in 3-D space. The modulus of the wave vector k(r) is given as:

k(r)= 2p n(r) l , (1.14)
where n(r)=n 0 + ∆n(r) is the complex refractive index distribution. n 0 , ∆n(r) and l are the immersion medium refractive index, local refractive index variation of the scattering medium and the wavelength of the incident plane wave in vacuum, respectively. The 3-D coordinate system for our computation is given in Fig. 1.7.

In Equation (1.13), E t (r) is the total propagating electric field detected at position r, and regarded as the sum of scattered E s (r) and incident E i (r) electric fields: The incident field is what we measure in the absence of the object, and can be given by the homogeneous Helmholtz equation:

E t (r)=E s (r)+E i (r).
(r 2 + k 2 0 )E i (r)=0, (1.16)
with k 0 = 2pn 0 l being the magnitude of the incident field's wave vector in the background medium. The solution for such kind of equation is a plane wave. Now, recalling Equation (1.13) and substituting Equation (1.16) leads to:

(r 2 + k 2 0 )E s (r)=V(r)E t (r), (1.17)
where V(r) is the scattering potential of the object at position r,

V(r)=k 2 0 " 1 n(r) 2 n 2 0 # . (1.18)
Such kind of differential equation can not be solved analytically. The standard approach to solve it is by using the Green's function, G(r):

(r 2 + k 2 0 )G(r)=d(r), (1.19)
with d(r) being the Dirac delta function. The Green's function enables one to find the outgoing electric field from a given input source, which is given as [START_REF] Jelena | On the derivation of the Green's function for the Helmholtz equation using generalized functions[END_REF]:

G(r, r')= exp(ik 0 ||r r'||) 4p||r r'|| . (1.20)
It describes the outgoing wave generated at position r by the source from position r', given that the wave is not diffracted during its propagation from r' to r. Equation (1.17) can now be transformed into an integral form using the Green's function:

E s (r)= Z G(r, r')V(r')E t (r')dr', (1.21)
where the total field is represented using the Lippmann-Schwinger equations:

E t (r)=E i (r)+ Z G(r, r')V(r')E t (r')dr'. (1.22)
Next, we will discuss on how to solve the above equation.

Approximated Solutions

TDM uses inverse propagation concept for its reconstruction algorithms with some approximated assumptions. The inverse Radon transform used in X-ray, CT-Scan and other similar techniques assume that the electromagnetic wave (EM) travels straight line when it propagate through the object to be imaged [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF][START_REF] Müller | The theory of diffraction tomography[END_REF]. The Fourier slice theorem can also be another approach as it has been used in classical tomography, which allows to reconstruct an image by means of the Fast Fourier Transform (FFT) [START_REF] Tc Wedberg | Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent, birefringent fibres[END_REF]. These are good assumptions and work very well in those techniques since the wavelength of the EM waves are comparatively smaller than the structure to be observed. Conversely in optical imaging, the wavelength of light is closer to the size of the object structures (for instance in biological tissues), hence diffraction becomes prominent and straight line assumptions can not be used anymore.

Considering visible light's wave nature, analytical or numerical methods to solve Maxwell's equation could be exact for modeling light propagation in a medium. However these approaches are complex and considering them leads to high computational cost. Hence it makes them almost impossible to be applicable in TDM reconstruction algorithms. Rayleigh-Sommerfeld diffraction theory could also be an accurate method if the object structures are relatively smaller than the wavelength of light. Geometrical optics concepts such as reflection and/or transmission could be also used if the object size is much larger than the wavelength. None of these cases are true for our purpose and hence the wave nature of light should be considered. In the next sections, we will see the formulation of the two basic assumptions in TDM: Born and Rytov approximations [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF].

The solution given by E. Wolf in 1969 was a scalar solution based on Born approximation [10]. In fact, the scalar approximation is the most implemented approaches in TDM reconstruction algorithms [13][14][15][16][17].

In Equation (1.22), the presence of E t in the left and in the integral makes it difficult to find the solution analytically. This is known as Fredholm equation of the second kind [START_REF] Kendall E Atkinson | The numerical solution of Fredholm integral equations of the second kind[END_REF], which is usually solved using iterative methods. To solve this problem, one can assume Born first order approximation (commonly known as Born approximation [START_REF] Born | Principles of Optics, 7th (expanded) edition[END_REF]), in which the scattered field inside the object is negligible and solely assumed as the incident field, i.e: E s ⌧ E i [10, 38]. .23) This leads to the Born field E B (r):

E t (r)=E i (r)+E s (r) ' E i (r)+ Z G(r, r 0 )V(r 0 )E i (r 0 )dr 0 . ( 1
E B (r)= Z G(r, r 0 )V(r 0 )E i (r 0 )dr 0 . (1.24)
Similarly using an other method, known as Rytov approximation [11], it is also possible to compute the Rytov field E R and complex phase f R (see the full derivation of

3. Tomographic Diffractive Microscopy 15 Rytov approximation in Appendix A). E R (r)=ln (E R )+f R (r) = ln  E t (r) E i (r) + i[f t (r) f i (r)], (1.25)
f t (r) , f i (r), E(r) are the complex phase of measured field, incident field, and the com- plex amplitudes, respectively. The validity of the Born approximation states that it is necessary for the phase change between the incident field and the wave propagating through the object to be less than 2p. In general, the validity is given in the literature [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF] as:

∆n ⌧ l 2pa , (1.26)
where a is the size of the object under consideration (for example, a = radius, if the object considered is a cylinder or sphere). Conversely, the Rytov approximation is comparatively less restrictive in terms of phase variations. It considers the change in the scattered phase f s rather than the total phase variations, [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF][START_REF] Baños | Dipole radiation in the presence of a conducting halfspace[END_REF]. The Rytov approximation is then valid for objects with low refractive index contrast [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF]:

∆n  ∆f s (r)l 2p 2 .
For now, let us find the solution for the scalar scattering potential using the Born method. If we consider that the sample is illuminated by a plane wave ;

E i = E exp[ik i .r],
with E being the complex amplitude, it is possible to further simplify equation (1.24):

E B (r)= Z G(r, r 0 )V(r 0 )E exp(ik i .r)dr 0 .
(1.27)

This leads to:

E B (r)=  exp( ik 0 r) 4pr Ã(k k i ), (1.28) with Ã(k k i )=k 2 0 E Z V(r 0 ) exp( ik.r) exp(ik i .r)dr 0 = k 2 0 E Z V(r 0 ) exp( i(k k i )r)dr 0 .
(1.29)

The above equation corresponds to a 3-D Fourier transform. Considering an arbitrary field ỹ(k) and its inverse Fourier transform, y(r), is given as:

ỹ(k)= 1 8p 3 Z y(r) exp( ik.r)d 3 r, (1.30) y(r)=8p 3 Z y(k) exp(ik.r)d 3 k. (1.31)
Similarly, multiplication of the field by a phase p in the direct space leads to a shift in the Fourier space, which can be easily recalled from Fourier transform properties:

F [y(r) exp( ip.r)] = ỹ(k p). ( 1
V(r), Ã(k k i )=8p 3 k 2 0 E Ṽ(k k i ). (1.33)
This provides a linear relationship between the diffracted far-field amplitude and the object's scattering potential. Finally, considering far-field assumption, the scattered field can be solved in the Fourier space at a given plane z, which leads to the mathematical expression given by Wolf [10]:

Ṽ(k k i )= ik dz 2p exp(ik dz .z) ẼB (k dx , k dy ; z), (1.34)
where

k dz = q k 2 0 k 2 dx k 2 dy , (1.35)
and

k k i =(k dx k ix , k dy k iy , k dz k iz ). (1.36)
Hence, from the above relation, we can realize that each measurement of E B gives a particular value of V(r). This allows us to obtain the object wave vector sets k o from the diffracted wave vector sets k d :

k o = k d k i = k d k 0 êi = 2 4 k dx k dy k dz 3 5 k 0 2 4 êix êiy êiz 3 5 , (1.37)
êi being a unit vector in the direction of the incident beam.

3-D Frequency Support Computation

We consider the elastic conditions provided by the Born approximation:

||k d || = k 0 = ||k i || = q k 2 ix + k 2 iy + k 2 iz . (1.38)
As a consequence, each measured 2-D diffracted wave vector corresponds to parts of the 3-D object wave vectors in the Fourier space. Thanks to the the elastic relationship, we can now compute the longitudinal components of the diffracted (k dz ) and incident wave vector (k iz ).

After the sample is illuminated by an incident beam, the diffracted field maps into the so called 'Ewald sphere' of radius k 0 . However, due to the limited numerical aperture of the objective lens, we collect only a cap of Ewald sphere as shown in Figure 1.8(a), see also Figure 1.8(b) for simplified 2-D representation in the (k x , k z ) plane. To obtain the 3-D frequency support of the object, it is necessary to shift the diffracted wave vector sets (cap of sphere) by the incident wave vectors. Figure 1.8(c) and (d) illustrates the computation process for single incident (equivalent to DHM) for reflection and transmission configurations, respectively. Considering many angular sample illuminations, it is possible to obtain an extended object frequency support or Optical Transfer Function (OTF) of the object, which will be detailed in section 3.2.

Resolution Improvement

As explained earlier in this chapter (section 3.1.2), the lateral extension of the frequency support for holographic technique is relatively limited, and the longitudinal extension is negligible, inducing poor resolution along the optical axis. Hence, the aim of TDM is to extend the set of recorded frequencies by merging several measurements so that improved resolution in both lateral and longitudinal direction is achieved. The possibilities include: illumination rotation, object rotation as well as varying the probing beam wavelength.

Illumination Rotation TDM

One of the most common configuration in TDM is Illumination Rotation TDM (IR-TDM), which can be done by keeping the sample static while changing the illumination beam direction. It is mostly favored and widely used for biological imaging due to its fast data acquisition and stability [12][13][14]17], as well as by commercial TDM systems (see for instance Nanolive1 and Tomocube2 systems). The diffracted fields from the sample at various illuminations are collected and recombined in Fourier space, using a so-called Fourier space "synthetic aperture" process.

Chapter 1. STATE OF THE ART At a given incidence the 2-D complex diffracted field from the object can be extracted from the holograms as seen in previous section. It should be noted that in such configuration, the illumination beam is tilted, hence a complimentary information is recorded for each oblique incidence.

The data recombination process for a transmission and reflection based IR-TDM is depicted in Figure 1.9. At each incidence, a set of diffracted field (in red) is translated by the incident wave (black doted line arrow) to give the object waves (in green). Considering many illuminations, the effective Optical Transfer Function (OTF) for reflection is a fully filled cap of sphere, allowing to collect higher spatial frequency components. Conversely, for transmission configuration a horn torus looking like OTF in 3-D is computed. One can observe that transmission based IR-TDM allows to collect lower spatial frequency of the object. In fact, it can be observed that there is a missing frequency component along the optical axis, known as "missing-cone problem" [13,[START_REF] Di | Tomographic phase microscopy: principles and applications in bioimaging[END_REF].

object k y k z k y k z k x k x k z k z

Sample Rotation TDM

IR-TDM has a good transversal resolution, however leads to poor axial resolution due to the missing frequency spectrum along the optical axis. Another alternative is to implement sample rotation TDM (SR-TDM) scheme. SR-TDM is realized by rotating the sample under the microscope objective while keeping the probing beam fixed. Note that rotation in the direct space with angle a corresponds to rotation in the Fourier space with the same rotation angle. In this sense, the set of captured frequency components are acquired by rotating the cap of Ewald sphere. As shown in Figure 1.10 (a), the specimen is rotated sequentially in the y axis, which corresponds to rotation of the object spectrum in the k y axis. The effective OTF at the end of many sample rotations is depicted in Figure 1.10 (b-d). The sample rotation allows to obtain an extended frequency support along the optical axis compared to IR-TDM. The effective frequency support is enlarged by a factor of p 2 in the k x , k z plane for a hypothetical objective of maximum collection angle (a max = p/2). One can still notice that the OTF is not really isotropic, rather a quasi-isotropic, due to a set of inaccessible frequencies along the sample rotation axis, known as missing apple core (shown in green, Fig 1 .10(c)). To acquire a completely isotropic OTF, another sample rotation in the other axis (y-axis for this case) can be done as well [START_REF] Foucault | Microscope tomographique diffractif: simplification des acquisitions et reconstructions numeriques associées[END_REF]. Another limitation is, however, the OTF extension along the optical axis is smaller than the x, y transverse direction [START_REF] Vertu | Improved and isotropic resolution in tomographic diffractive microscopy combining sample and illumination rotation[END_REF]. SR-TDM is relatively simple and inexpensive, especially if implemented for imaging non-biological samples, such as optical fibers [START_REF] Gorski | Tomographic imaging of photonic crystal fibers[END_REF]. In this case, the sample rotation can be done using a motorized rotation stage. This may create mechanical instability, and/or makes it hard to implement for bio-imaging applications. Moreover, to achieve optimal image quality with such system, a large number of acquisitions are necessary [14,[START_REF] Vertu | Improved and isotropic resolution in tomographic diffractive microscopy combining sample and illumination rotation[END_REF][START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF].

Practically, for rotating the sample several alternatives can be used. One approach is by embedding the sample within a rotating micro-capillary [START_REF] Lin | Sectional imaging of spatially refractive index distribution using coaxial rotation digital holographic microtomography[END_REF][START_REF] Kujawi Ńska | Problems and solutions in 3-D analysis of phase biological objects by optical diffraction tomography[END_REF][START_REF] Kuś | Tomographic phase microscopy of living three-dimensional cell cultures[END_REF]. Figure 1.11 shows an SR-TDM system taken from [START_REF] Kuś | Tomographic phase microscopy of living three-dimensional cell cultures[END_REF] where the sample is put in a micro-capillary for Chapter 1. STATE OF THE ART rotation purpose. This enables it to be applicable for biological imaging, such as single cell study. One of the limitation of such a system is that it should be implemented using a long working distance, hence a lower NA microscope objective. This limits the maximum achievable frequency support, which consequently reduces the image resolution. Alternatives to rotate the sample for SR-TDM could be: using optical tweezers [START_REF] Habaza | Tomographic phase microscopy with 180 rotation of live cells in suspension by holographic optical tweezers[END_REF] or considering cells' natural flow [START_REF] Merola | Tomographic flow cytometry by digital holography[END_REF], bearing in mind that care should be taken in the sample manipulation and reconstruction process.

TDM with Varying Wavelength

The use of multiple wavelengths could also be another alternative to increase the object frequency bandwidth of a holographic technique, as proposed by R. Dandliker and K. Weiss in 1970 [START_REF] Dändliker | Reconstruction of the three-dimensional refractive index from scattered waves[END_REF] and as experimentally implemented by [START_REF] Yu | Wavelength-scanning digital interference holography for tomographic three-dimensional imaging by use of the angular spectrum method[END_REF]. Previously, we saw that the radius of the cap of sphere consisting the collected frequency at each illumination is directly related to the wave vector of the illumination beam, k 0 = 2p l , in vacuum. Hence, It is straightforward that when changing the illumination light's wavelength, the curvature radius of the cap of sphere which consisted the collected frequency can be varied. Figure 1.12 describes schematically how the effective frequency support can be extended by introducing varying wavelength in the illumination. Fig 1 .12(b) depicts the frequency support collected with 3 illuminating light's wavelengths (blue, green and red), which translates into varying cap of spheres' curvature. The final frequency support is shown in Fig 1 .12(c) when a continuum of wavelength is used.

This indeed gives an advantage to implement such kind of TDM system, for example in a wide field, confocal, or phase contrast microscope, without requiring any moving part (such as a mirror as in IR-TDM) or sample (as SR-TDM). However, the frequency support collected as a result of this TDM is limited comparatively to previous approaches. This limits its experimental implementations. Nevertheless, some publications concerning such approach for reflection TDM configuration can be found in [START_REF] Yu | Wavelength-scanning digital interference holography for tomographic three-dimensional imaging by use of the angular spectrum method[END_REF][START_REF] Kühn | Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection[END_REF][START_REF] Yu | Digital holographic tomography based on spectral interferometry[END_REF].

Moreover, white-light diffraction tomography is proposed and implemented by Kim et al. [START_REF] Kim | White-light diffraction tomography of unlabelled live cells[END_REF] by making use of a white light source and a phase mask, which is possible by upgrading the phase contrast microscope. (c) for a continuum of wavelengths

Experimental Implementations

Depending on the ease of use, application and the kind of object under study, TDM can work either in transmission and/or reflection configurations. Schematic representation of a transmission TDM setup is illustrated in Fig. First the coherent laser is split by a beam-splitting device BS (such as beam splitting fiber) into a reference and an illumination beam. The illumination beam, which is collimated as plane wave is reflected-off the fast tip/tilt mirror (FSM-300) and is focused at the back focal plane of the condenser objective. By controlling the angle of the plane wave using the FSM, the sample is illuminated from various directions by the condenser objective. Light scattered by the sample is then collected by the microscope objective. After passing through the two imaging lenses (used to control the image sampling), the diffracted field is recombined with the reference beam with a recombining cube. The interferogram is finally detected by the camera.

For reflection TDM, the technique is similar, except that the same objective is used as both condenser and objective, to illuminate and collect the diffracted field from the sample, respectively [START_REF] Sarmis | High resolution reflection tomographic diffractive microscopy[END_REF].

Comparing IR-TDM and Digital Holographic Microscopy

The extended frequency support of IR-TDM leads to a relatively better lateral and longitudinal sectioning than DHM. Conversely, the missing cone along the axial direction translates into poor resolution along the optical axis. In general, as depicted in Fig. 1.15 (e), TDM has identical frequency bandwidth as wide field imaging thanks to the synthetic aperture process. But, contrary to wide field imaging, TDM (theoretically) allows to fully transmit all the frequencies without attenuation.

Comparing TDM and Other Modalities

Transmission IR-TDM configuration has been developed and implemented by several laboratories and commercial systems, in which the application is more focused on biological imaging such as for studying neurons, blood cells, cancerous cells, pollen grains and micro-algae [START_REF] Simon | High-resolution tomographic diffractive microscopy of biological samples[END_REF][START_REF] Choi | Tomographic phase microscopy[END_REF][START_REF] Park | Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum[END_REF][START_REF] Sung | Optical diffraction tomography for high resolution live cell imaging[END_REF][START_REF] Kim | High-resolution three-dimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography[END_REF][START_REF] Lee | Measurements of morphology and refractive indexes on human downy hairs using three-dimensional quantitative phase imaging[END_REF][START_REF] Jung | Label-free non-invasive quantitative measurement of lipid contents in individual microalgal cells using refractive index tomography[END_REF][START_REF] Chowdhury | Refractive index tomography with structured illumination[END_REF]. Fig 1.16 shows the better resolving capability of TDM compared with wide field and confocal fluorescent microscopy when imaging a carbon mesh [14]. Here, images taken by scanning electron microscope are used as a reference for comparison. The reflection IR-TDM configuration is relatively less implemented. However, it has an advantage of imaging reflective surfaces at high resolution along the optical axis [START_REF] Maire | Phase imaging and synthetic aperture super-resolution via total internal reflection microscopy[END_REF][START_REF] Sergey | Synthetic aperture Fourier holographic optical microscopy[END_REF], since it allows to collect high frequency components of the sample. Figure 1.17 shows the images of a microscopic USAF target taken by holographic, confocal reflection microscope and reflection TDM [START_REF] Sarmis | High resolution reflection tomographic diffractive microscopy[END_REF]. Images taken by reflection TDM (Fig. 1.17 (c,f)) exhibit improved signal-to-noise ratio relative to holography (Fig. 1.17 (a,d)), due to the synthetic aperture process of several acquisitions. Reflection confocal microscope image are characterized by better contrast and low noise, as can be seen in Fig. 1.17 (b,e).

Combined Approaches

Until this point, we have seen TDM techniques, which can extend the bandwidth of the holography system to gain better 3-D optical sectioning and improved resolutions. It is also possible to combine some of the earlier explained approaches to further extend the bandwidth, which will be discussed in the next sections.

Combining Illumination Rotation and Multi-wavelength TDM

One way to extend the frequency bandwidth of the TDM system is possible by combining IR-TDM and by varying the illumination beam wavelength. Zhang et al. [START_REF] Zhang | Multi-wavelength multi-angle reflection tomography[END_REF] have implemented such a configuration in the reflection mode TDM using 5 wavelengths.

The use of multiple wavelengths in the IR-TDM provides significant improvement in the reconstruction and consequently optical sectioning, especially along the optical axis as can be seen in Fig. 1.18. One advantage of such configuration is that no modification of the system is needed from the classical illumination rotation TDM, apart from replacing the light source. Another unique advantage of performing such combined approach is that it allows for multidimensional imaging [START_REF] Jung | Hyperspectral optical diffraction tomography[END_REF]. Transmission IR-TDM along with hyperspectral laser (400-700 nm wavelength) scanning allows to obtain four-dimensional images. This was done by Sung et al. [START_REF] Sung | Spectroscopic microtomography in the visible wavelength range[END_REF] considering absorbing chromophores at a given wavelength, where they directly estimate the absorption-coefficient and refractive index spectra of such specimens.

Another way is by making use of colored LED matrix for large scale TDM [START_REF] Zuo | Lensless phase microscopy and diffraction tomography with multi-angle and multi-wavelength illuminations using a LED matrix[END_REF]. They have implemented multi-wavelength phase retrieval and IR-TDM. However, up to now, they provide images with lower resolution (⇡ 3.72 µm lateral and ⇡ 5 µm axial resolution) compared to what is achieved by an IR-TDM using a coherent laser.

Combining Illumination Rotation and Sample Rotation TDM

TDM with illumination rotation has a unique advantage for biological applications since it provides high-resolution 3-D images of such samples by only rotating the probing beam, which allows to increase the spatial frequency bandwidth through Fourier space synthetic aperture process. This improves the axial and lateral resolution compared to DHM. However, the missing frequency (non-captured) components along the optical axis translates into relatively poor axial resolution compared to lateral one. Hence, the anisotropic OTF limits the optical sectioning capability of IR-TDM.

Conversely, considering SR-TDM enables to capture an almost isotropic OTF, except some missing frequencies along the rotation axis, the so called missing apple core problem. Also, to fully gain the benefits of SR-TDM, a large number of sample rotations are required, so that the OTF is effectively filled.

In order to gain the advantages of both approaches, it is possible to simultaneously combine both approaches in a single experiment. This permits to gain isotropic and extended OTF, which highly improves the resolution. This was first proposed and studied numerically by Vertu et al. [START_REF] Vertu | Improved and isotropic resolution in tomographic diffractive microscopy combining sample and illumination rotation[END_REF] in 2011. In 2018 Vinoth et al. have implemented combined approach of full-angle SR-TDM with those of IR-TDM [START_REF] Vinoth | Integrated dual-tomography for refractive index analysis of freefloating single living cell with isotropic superresolution[END_REF]. Doing so, they achieved an optical transfer function which they call it, 'UFO' (unidentified flying object), as can be observed in Figs. 1.19 (a-c). However, the OTF is not yet isotropic since the lateral and longitudinal extension of the IR-TDM and SR-TDM is not the same. In general, such approach still leads to improved resolution.

Another alternative to gain isotropic OTF is to combine many IR-TDM and few sample rotations. The experimental work was initiated during the thesis of J. Bailleul [START_REF] Bailleul | Microscopie tomographique diffractif á resolution isotrope[END_REF] and finally realized experimentally by Simon et al. in 2017 [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF]. The experiment is done in such a way that many holograms as the result of illumination rotations are acquired, after each sample rotations. Figures 1.19 (d), (e) and (f) show the 3D OTF as the result of such process, when only illumination rotation is realized, illumination rotations with two sample rotations, and 4 sample rotations, respectively.

The above-mentioned approaches require careful image registrations to reduce reconstruction artifact. Conversely, for axis-symmetric objects, such as fibers, it is possible to make use of their geometrical property as an advantage to simulate physical sample rotations TDM. For such samples, Foucault et al. have proposed and implemented a numerical sample rotation [START_REF] Foucault | Simplified tomographic diffractive microscopy for axisymmetric samples[END_REF], to achieve high resolution (if not super-resolution) refractive index distributions.

Figure 1.20 shows the reconstructed image of an optical fiber tip, first using classical illumination rotation ( Fig. 1.20(a-c)), then by adding 4 sample rotations (0 , 54 , 90 , and 126 ), see Fig. 1.20(d-f) [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF]. The experiment is done using 475 nm laser with NA = 1.4 objective in oil immersion media, for which 200 nm isotropic resolution is achieved. Theoretically, the spatial resolution is 85 nm, and with this system, an axial resolution of 95 nm is achieved. Furthermore, since the optical fiber is an axis-symmetric object, Foucault et al. [START_REF] Foucault | Simplified tomographic diffractive microscopy for axisymmetric samples[END_REF] have used the same sample to compare the applicability of numerical rotations. Reconstructed images are shown in Fig. 1.20(g-i) for the same 4 sample rotations (but here numerically), and 20 numerical object rotations Fig. 1.20(j-l). In terms of refractive index quantification, numerical rotations give better quality images compared to physical sample rotations. However, one drawbacks of such reconstruction is the multiplicability of noise, since artifacts coming from the reference arm are coherently added for each sample rotations. This is indicated with red double arrow in Fig. 1.20(g).

Combining Transmission and Reflection IR-TDM

Although transmission TDM configurations have been commonly used in many application, reflection mode is also a choice by some research groups [START_REF] Sarmis | High resolution reflection tomographic diffractive microscopy[END_REF][START_REF] Maire | Phase imaging and synthetic aperture super-resolution via total internal reflection microscopy[END_REF][START_REF] Sergey | Synthetic aperture Fourier holographic optical microscopy[END_REF][START_REF] Zhang | Full-polarized tomographic diffraction microscopy achieves a resolution about one-fourth of the wavelength[END_REF]. Both transmission and reflection TDM configurations can highly extend the bandwidth and enhance the resolution. In section 3.2.1, the OTFs as the result of both (reflection and transmission) TDM configurations are synthesized for illumination rotation TDM. Recalling the OTF, reflection TDM scheme allows to collect higher frequency components of the sample, which helps to distinguish sharp structures along the optical axis. Hence, such configuration is implemented for imaging samples with strong reflectivity. Conversely, transmission TDM helps to collect lower frequency components of the weakly scattering object. Hence, to further extend the bandwidth and improve the resolution of TDM, one can effectively combine the two configurations.

After TDM had been proposed by Wolf [10] in 1969 and reformulated by Dandliker and Weiss in 1970 [START_REF] Dändliker | Reconstruction of the three-dimensional refractive index from scattered waves[END_REF], V. Lauer studied such a system for experimental applications in 2002 [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF]. He proposed a 4p system similar to fluorescence microscopy, using a two-facing-objective configurations [2,[START_REF] Hell | Properties of a 4Pi confocal fluorescence microscope[END_REF]. He illustrated the synthesized OTF for a 4p TDM system using two high resolution objective lenses, which allows to acquire a quasi-isotropic resolution.

Figure 1.21 shows a slice of the 3-D frequency support as the result of combined reflected and transmitted fields to make 4p TDM. The middle OTF ("doughnut" in 3-D) in Fig. 1.21 is acquired when illuminating and detecting diffracted fields with two facing Chapter 1. STATE OF THE ART objectives, sequentially. Note that unlike to the one proposed by V. Lauer, the final frequency support is illustrated with non-overlapping region between the two OTFs, as a consequence of using 1.4NA objectives.

Another simplified alternative, called mirror-assisted TDM, has been proposed and implemented numerically by Mudry et al. in 2010 [78]. For that, the sample is prepared and placed in the vicinity of a highly reflecting mirror. Such a system was also implemented during Foucault's thesis [START_REF] Foucault | Microscope tomographique diffractif: simplification des acquisitions et reconstructions numeriques associées[END_REF] to simplify acquisition for transmission TDM. More discussion on 4p TDM will be given in Chapter 2.

Simplifying TDM Data Acquisition

TDM data acquisition is commonly based on Mach-Zehnder interferometry system, comprised of an object and a reference arm. Such system could suffer from phase noise, coming from measurement system. To alleviate this issues, common-path interferometry TDM system has been investigated [START_REF] Kim | Common-path diffraction optical tomography for investigation of three-dimensional structures and dynamics of biological cells[END_REF][START_REF] Popescu | Diffraction phase microscopy for quantifying cell structure and dynamics[END_REF]. For that, a phase grating placed at the image plane is used to generate several diffraction orders. The zeroth and first diffraction orders are then separated by spatial filter located at the back-focal plane of the imaging lens. Finally, the interference between the zeroth order (playing the role of the reference beam) and first order diffraction (containing the diffracted field) generates a hologram to be recorded by the CCD.

Alternatively, to alleviate the sequential data acquisition, snapshot TDM scheme was studied by Sung et al. [START_REF] Sung | Snapshot Holographic Optical Tomography[END_REF]. 32 focused beams at the back focal plane of the condenser lens are generated by the help of micro-lens arrays to simultaneously illuminate the sample from several directions and acquire spatially multiplexed holograms, which alleviates the classical sequential IR-TDM system, however, at the cost of reduced field of view and resolution.

Recent work on scan-free TDM approach by Mirsky et al. [START_REF] Simcha K Mirsky | Dynamic Tomographic Phase Microscopy by Double Six-Pack Holography[END_REF][START_REF] Simcha | First experimental realization of sixpack holography and its application to dynamic synthetic aperture superresolution[END_REF], whereby simultaneously illuminating the sample at six angular directions enables to acquire the corresponding six holograms (they call it six-pack TDM) in a single camera exposure. Lately, their setup has improved to acquire twelve holograms (double six-pack) [START_REF] Simcha K Mirsky | Dynamic Tomographic Phase Microscopy by Double Six-Pack Holography[END_REF]. This is done by successfully tuning the polarization state of these illumination and reference beams.

Another simplified TDM data acquisition is possible by using wavefront analyzers for direct amplitude and phase measurement [START_REF] Kim | Diffraction optical tomography using a quantitative phase imaging unit[END_REF]. This could help to reduce the complexity of currently available TDM setup, by fitting them as an add-on module into the conventional wide-field microscope. Furthermore, it helps to implement constrained inversion algorithm to correctly recover the sample refractive index by even using a partially coherent illumination beam [START_REF] Ruan | Tomographic diffractive microscopy with a wavefront sensor[END_REF], or allows for multiplexed wavefront acquisition for rapid reconstruction [START_REF] Natan | Off-axis digital holographic multiplexing for rapid wavefront acquisition and processing[END_REF].

Until this point, I have explained TDM relying on coherence nature of an illuminating beam. However, using partially coherent (or low coherence) interferometry can also be helpful to increase the speed of TDM reconstruction process [START_REF] Langehanenberg | Application of partially coherent light in live cell imaging with digital holographic microscopy[END_REF][START_REF] Juan M Soto | Label-free quantitative 3D tomographic imaging for partially coherent light microscopy[END_REF][START_REF] Chen | Wolf phase tomography (WPT) of transparent structures using partially coherent illumination[END_REF]. Such technique has relatively low cost system (which can be built as extension on the conventional microscope) and speckle noise free images, but needs some more measurements or a priori information to recover both refractive index and absorption information about the sample [START_REF] Juan M Soto | Optical diffraction tomography with fully and partially coherent illumination in high numerical aperture label-free microscopy[END_REF].

Fourier Ptychography Diffraction Tomography

One active research of interest, linked to phase imaging, is Fourier ptychography [20,[START_REF] Chandra | Fourier ptychography: current applications and future promises[END_REF]. It is a computational imaging method where the sample is variably illuminated at several directions and the corresponding intensity images are detected to iteratively retrieve the absorption and phase properties.

The notion of ptychography was initially introduced by Hegerl and Hoppe for electron diffraction measurements in 1969 [START_REF] Hoppe | Diffraction in inhomogeneous primary wave fields. 1. Principle of phase determination from electron diffraction interference[END_REF]. However, it has only recently becomes widely used, after several decades, after Rodenburg and colleagues have extended it for X-ray and optical imaging setups [20]. Fourier ptychography in optical imaging has shown its capability to provide phase information of thin samples [START_REF] Zheng | Wide-field, high-resolution Fourier ptychographic microscopy[END_REF].

Recently, Fourier ptychography has been extended for 3-D phase imaging, from 2D intensity measurements, with Born [START_REF] Horstmeyer | Diffraction tomography with Fourier ptychography[END_REF] and Rytov approximations [START_REF] Zuo | Wide-field high-resolution 3d microscopy with fourier ptychographic diffraction tomography[END_REF] using appropriate phase retrieval computation algorithm without phase measurements. Figure 1.22 depicts the synthetic aperture process of Fourier ptychography diffraction tomography. The 3-D OTF computation is similar to TDM synthetic aperture process. What is unique in Fourier ptychography TDM (unlike holographic based TDM techniques) is that the phase and amplitude information of the sample are computed iteratively. Afterwards, the 2-D Fourier spectrum is computed and mapped into the Ewald sphere (considering the elastic conditions) to get the 3-D OTF. In general, the main advantages of implementing Fourier ptychography tomography is that it uses low-resolution images acquired using low NA objective and allows to retrieve wide FOV and high-resolution complex refractive index (RI) image across large volumes, which (unlike TDM), does not rely on holographic phase measurement. However, as long as 1 st order approximations are used, the missing frequency along the optical axis always exist and provides poor resolution along the optical axis.

To improve the reconstructed image quality, one way could be to combine it with other imaging techniques, such as with CARS microscopy [START_REF] Heuke | 3D-Coherent anti-Stokes Raman scattering Fourier ptychography tomography (CARS-FPT)[END_REF] to reconstruct the complex non-linear susceptibility of a sample in 3-D. They proposed and demonstrate numerically on solving the missing cone problem. To do so, they assumed that the RI is unchanged with the wavelengths. Then by keeping the Stokes beam along the optical axis and varying the angles of the pump+probe beam (or vise versa), it could allow to access sample spatial frequencies along the optical axis.

Chapter 1. STATE OF THE ART

Polarization Sensitive TDM

Until this point, light has been regarded as a scalar quantity. In general, light as an electromagnetic field should be treated as a vectorial field, which has several applications in biology [START_REF] Valery V Tuchin | Tissue optics[END_REF]. For anisotropic samples, such consideration provides better contrasted and quantitative RI map [START_REF] Wang | Jones phase microscopy of transparent and anisotropic samples[END_REF]. In recent years, quantitative 2-D Jones matrix polarization sensitive techniques, such as vectorial ptychography [START_REF] Ferrand | Ptychography in anisotropic media[END_REF], as well as polarization sensitive 2-D holographic microscopy under paraxial approximation [START_REF] Todorov | Polarization holography. 1: A new high-efficiency organic material with reversible photoinduced birefringence[END_REF][START_REF] Daniel | Single-shot digital holographic microscopy for quantifying a spatially-resolved Jones matrix of biological specimens[END_REF][START_REF] Park | Jones Matrix Microscopy for Living Eukaryotic Cells[END_REF] have gained much attention for several applications.

However, polarization imaging in TDM has gained relatively less attention. For example, the work from Zhang et al. [START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF], based on full-vectorial field along with the non-approximated rigorous reconstruction algorithm helps to provide isotropic resolution for a reflection TDM. However, its application is limited to materials with known permittivity as well as highly reflective nature.

Lately, some works on polarization sensitive TDM have emerged. Recent work on large scale polarization contrast TDM in transmission configuration, has shown its potential to improve the image contrast of anisotropic specimen [START_REF] Van Rooij | Polarization contrast optical diffraction tomography[END_REF]. Figures. 1. 23(a-c) show results of zebra fish muscle structure's 3-D image obtained with phase and polarization contrast approaches. 3-D quantitative RI tensor maps and phase retardance of anisotropic objects, especially for microscopic biological samples exhibiting strong anisotropicity (such as muscle fibers) [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF] have also been investigated. Furthermore, Shin et al. investigated its application for material characterizations [START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF]. Figure 1.24 shows the experimental setup they have implemented to retrieve all the three-dimensional dielectric tensor components of anisotropic samples. They use a digital micro-mirror device (DMD) and liquid-crystal retarder plus half waveplate to control the angle of the illumination beam and polarization, respectively. Then with the help of 2 cameras, polarization-sensitive holograms are recorded, which are further processed to reconstruct the 9 dielectric tensor components by solving the inhomogenous Maxwell's equation for a non-magnetic material based on first order approximation.

In general, experimental realization and reconstruction process of polarization sensitive TDM to image low birefringent samples, such as biological samples (cells and cell structures) in high-resolution and good sensitivity needs to be well understood and explored. This will be detailed in chapter 3. to compute the object's 3-D refractive index map. In the early times, filtered backpropagation method was used to compute the 3-D refractive index map [START_REF] Tc Wedberg | Comparison of the filtered backpropagation and the filtered backprojection algorithms for quantitative tomography[END_REF]. It considers straight ray approximation, where the transmitted light's phase is the line integral of the RI along the propagation direction. The most popular and well implemented approach is E. Wolf's single scattering approximation (under Born or Rytov approximations), which allows to use the Fourier diffraction theorem and reconstruct the 3-D RI maps from the phase and amplitude information [10,[START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF]. Kim et al. have provided a comparative study and has given better practical insight about the two algorithms [START_REF] Kim | High-resolution three-dimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography[END_REF],

showing the validity of the two methods. Figure 1.25 depicts the general reconstruction process considering a typical illumination rotation TDM system. The inverse 2-D Fourier transform of the hologram taken at various angular directions are computed. As discussed in off-axis holography in Fig. 1.4, re-centering at the origin plus normalization by the specular spot is done on the spectrum before back-propagation. Then, back-propagation to merge each acquired information in the Fourier space to obtain the extended 3-D frequency support of the object can be done, following the Born approximation.

As the name indicates, the back-propagation process based on Fourier diffraction theorem takes into account diffraction of the incident wave by the specimen. As explained in section 3.1.2, the momentum conservation as the results of first order Born approximation states that each measured 2-D diffracted spectrum is part of the 3-D object spectrum, contained in a cap of sphere. Hence, we translate each cap of Ewald sphere by the incident wave vector to compute the final extended 3-D OTF as shown in Fig. 1.25.

Note that when projecting each object spectrum, an overlap between the various cap of spheres may happen. Hence, accounting the redundancy information due to the overlap helps to reconstruct accurate 3-D refractive index map as explained in Foucault's thesis [START_REF] Foucault | Microscope tomographique diffractif: simplification des acquisitions et reconstructions numeriques associées[END_REF] (in chapter 1, section 2) with schematic illustration.

The direct linear inversion method has been used by various researchers, because of its simplicity and fast reconstruction [14,25,[START_REF] Sung | Optical diffraction tomography for high resolution live cell imaging[END_REF][START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF]. Reconstruction algorithm using Fourier diffraction method with direct linear inversion algorithm considering single scattering is only valid to reconstruct RI of a weakly diffracting objects, which limits its application to such kind of samples. In the previous sections, we have seen mechanisms to improve the image resolution and refractive index estimation using experimental approaches. It is also possible to improve the image reconstruction quality by implementing more sophisticated non-linear inversion algorithms, instead of Born or Rytov approximations, or a priori information along with regularization approaches with more accurate forward models [15,17].

One technique to improve image contrast and to reduce noise could be by performing 3-D image deconvolution. Y. Cotte et al. [12] have used a sub-nanometer aperture to calibrate the system, in which they showed that the classical deconvolution free TDM reconstruction is outperformed compared to the non-deconvolved image. This approach was validated by using a 75 nanometer diameter nano-apertures, which could be resolved.

Iterative reconstruction schemes to retrieve the missing frequency of the sample as the results of IR-TDM experiments [START_REF] Zheng | Wide-field, high-resolution Fourier ptychographic microscopy[END_REF][START_REF] Lim | Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography[END_REF][START_REF] Krauze | Generalized total variation iterative constraint strategy in limited angle optical diffraction tomography[END_REF][START_REF] Ulugbek | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF][START_REF] Tian | 3D intensity and phase imaging from light field measurements in an LED array microscope[END_REF] are also of interest. One of the most common approach to address this missing information is to use Gerchberg-Papoulis-Saxton (GPS) algorithm [START_REF] Rw Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF][START_REF] Rw Gerchberg | Super-resolution through error energy reduction[END_REF] by iteratively computing the 3-D Fourier transform of the sample's complex RI and the 3-D inverse Fourier transform of the sample's spatial spectrum. This is done by incorporating prior information about the sample in both direct and inverse domain. For instance, imposing of simple non-negativity constraint on the refractive index [START_REF] Lim | Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography[END_REF][START_REF] James R Fienup | Phase retrieval algorithms: a comparison[END_REF], or adapting discrete algebraic reconstruction approach that employs known information about sample's uniform structures [START_REF] Lee | High-resolution Refractive Index Tomography Using Discrete Algebraic Reconstruction Technique[END_REF] helps to better estimate the RI values. An image simulation example for missing information retrieval using the GPS algorithm is given in Appendix A.

Lately, multimodal imaging techniques combining TDM and confocal fluorescence microscopy have emerged. A prior information about the sample acquired by spinning disc confocal scanning fluorescence microscopy can be used as an input for the GPS based TDM to accurately estimate the refractive index profile [START_REF] Guo | Limited-angle tomographic phase microscopy utilizing confocal scanning fluorescence microscopy[END_REF]. It is even reported that super-resolution can be achieved by combining 2-D fluorescence Hessian structured illumination microscopy and TDM [START_REF] Dong | Super-resolution fluorescence-assisted diffraction computational tomography reveals the three-dimensional landscape of the cellular organelle interactome[END_REF].

Reconstruction approaches based on multiple scattering are also of great interest,

Conclusion and Outlook

33

which give higher quality results when applied to samples exhibiting such property. For example, extended-depth-of-focus filtered back-propagation method with forward and backward propagation of the incident and measured field (based on Beam Propagation Method(BPM)) along with error minimization [START_REF] Shengli | Optical fiber refractive index profiling by iterative optical diffraction tomography[END_REF][START_REF] Shengli | Iterative optical diffraction tomography for illumination scanning configuration[END_REF] allows to iteratively retrieve the missing information for both SR-TDM and IR-TDM. BPM with edge preserving 3-D total variation regularization is also utilized to retrieve multiple scattering information [START_REF] Ulugbek | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF]. Recently proposed technique, known as Multi-layer Born, slices the object into a sequence of small layers or slabs to propagate the field sequentially [START_REF] Chen | Multi-layer Born multiple-scattering model for 3D phase microscopy[END_REF], similar to beam propagation approaches. One advantage of such approach over the above mentioned methods is that it enables to retrieve the backward scattered field. In general, as such methods are iterative, computation takes relatively more time and are not suitable for fast imaging but could result in better quality images if performed correctly.

The implementation of a non-linear inversion approach based on solving the vectorial wave equation plus a priori knowledge of the sample's permittivity contrast could result in fast and improved reconstruction. For instance, authors in [START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF] (at Intitut Fresnel) have implemented such approach, and reported a resolution of l/10 in reflection TDM.

Advancements in computational efficiency has triggered a tremendous demand for deep learning approaches (for example, Convolutional Neural Networks, CNNs) in several fields. Such techniques could be useful for TDM, especially in terms of enhancing the speed and quality of reconstruction [22]. Deep learning algorithms can be applied at various levels of TDM reconstruction process. At the pre-processing level, learning algorithms can be used to correct error during phase unwrapping [START_REF] Wang | One-step robust deep learning phase unwrapping[END_REF] to replace the classical approach [START_REF] Vyacheslav | Deterministic phase unwrapping in the presence of noise[END_REF]. Through residual learning, they can also be used to reduce phase noise coming from the imaging system, which highly enhance the reconstructed image quality [START_REF] Balasubramani | Influence of noise-reduction techniques in sparsedata sample rotation tomographic imaging[END_REF]. The use of CNNs in the reconstruction algorithm is also another growing research area [START_REF] Ulugbek | Learning approach to optical tomography[END_REF][START_REF] Ryu | Deep learning-based optical field screening for robust optical diffraction tomography[END_REF][START_REF] Yang | Deep-learning projector for optical diffraction tomography[END_REF][START_REF] Pierré | Deep learning framework applied to optical diffraction tomography (ODT)[END_REF]. It has some promises to decode the RI map from the measured phase and amplitude information. Learning TDM based on iterative reconstruction scheme that uses split-step non paraxial method as the forward model has been proposed to gain insight about the multiple scattering layers of sample structures, which relays less on sample's a priori information [START_REF] Lim | High-fidelity optical diffraction tomography of multiple scattering samples[END_REF]. Another promising application of learning approach could be in the post-processing of TDM images. It can be used to false color several cell compartments [START_REF] Javidi | Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography[END_REF] as well as to classify cells, tissues and microorganisms [START_REF] Sheneman | Deep learning classification of lipid droplets in quantitative phase images[END_REF].

Overall deep learning has been growing and advancing to play an important role in pre-processing step to reduce artifacts and error, to speed up reconstruction, and in postprocessing to digitally label and classify various structures of biological samples [17].

Conclusion and Outlook

In this chapter, I have described the notion of holography, and presented the three main types of digital holographic microscopy techniques: in-line (Gabor holography), offaxis and phase shifting holography. Aiming at extending the frequency support of digital holographic microscopy for 3-D imaging, I have introduced TDM and detailed its principles. Several modes of TDM: reflection TDM configurations, transmission TDM configurations and 4Pi TDM configuration, sample rotation TDM, illumination rotation TDM, multi-wavelength TDM, as well as their respective data recombination process in the Fourier space (synthetic aperture process) have been detailed. Inspired by the anisotropic 3-D OTF of these several methods and to further improve the 3-D image resolution, I have also introduced various combined TDM approaches.

Chapter 1. STATE OF THE ART

One limitation of TDM is that, as long as sequential data acquisition is used, the dilemma between speed of image reconstruction, and image quality is always there. As discussed in this chapter, one way of resolving such a problem could be by simplifying the TDM data acquisition scheme. Techniques, such as: snapshot TDM and doublesix-pack TDM could allow to increase the speed of TDM data acquisition. In fact, an elegant approach can be by optimizing the angular sample illumination schemes so that it could be possible to acquire enough data from the sample for high-resolution quantitative images with limited number of holograms. This is the main objective of Chapter 2.

At IRIMAS, we have been implementing the scalar TDM approach, which is widely utilized by several research groups. In addition to reducing experimental complexities, such scalar assumption allows for a simplified link between the measured diffracted fields and the sample scattering potential. However, for anisotropic samples, light as an electromagnetic wave should be treated as a vectorial field, which leads to polarimetric imaging. Polarization imaging, in general, and polarization sensitive TDM, in specific, is recently getting much attention. Its application to image some classes of objects, such as those made with anisotropic molecular structures, needed the recording and analyzing of the polarimetric properties of light. This will be discussed in Chapter 3.

ANGULAR SAMPLE ILLUMINATION OPTIMIZATION

Introduction

Illumination rotation TDM, either in reflection or transmission configuration, provides the 3-D complex RI distribution of the sample by combining 2-D holograms acquired from multiple illumination angles. In practice, one needs to acquire tens [START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF] to hundreds [12,[START_REF] Sung | Optical diffraction tomography for high resolution live cell imaging[END_REF][START_REF] Simon | Tomographic diffractive microscopy of transparent samples[END_REF] of holograms, depending on the required image quality, and the reconstruction algorithms to be used. These holograms at several angular illuminations are captured sequentially. As a general rule of thumb, using few interferograms results in a fast acquisition and reconstruction. However, such approach results in a degraded reconstructed image quality, due to the lack of frequency components in the Fourier space, mainly when using Born and Rytov 1 st order approximations.

On the contrary, to achieve high-resolution imaging, the Fourier space has to be filled. While increasing the number of illumination angles is the straightforward approach to achieve a high-quality image, doing so would slow the speed of TDM acquisition/reconstruction/display. Hence, a trade-off between image quality and acquisition/reconstruction speed has been the main concerns in the field. To improve image quality with limited number of holograms, advanced reconstruction algorithms can also be implemented which are mostly slower.

One way to address this trade-off is the implementation of a snapshot scheme as proposed by Sung et al. [START_REF] Sung | Snapshot Holographic Optical Tomography[END_REF]. In this case, 32 focused beams at the focal plane of the condenser lens are generated by the help of micro-lens arrays, which are converted into plane waves to simultaneously illuminate the sample from several directions. Holograms are multiplexed and spatially dispersed to be detected by the CCD at the image plane. However, because of the multiplexing in the acquisition, this approach divides the pixels of the camera to record holograms corresponding to several illuminations. As a result, the Field Of View (FOV) of such system is reduced as a function of the number of illumination angles. Moreover, the complexity of image reconstruction increases as the function of holograms.

Inspired by such limitations, Mirsky et al. [START_REF] Simcha K Mirsky | Dynamic Tomographic Phase Microscopy by Double Six-Pack Holography[END_REF] have proposed a scan-free (they call it double six-pack) TDM approach, whereby simultaneously illuminating the sample at twelve angular directions enables to acquire the corresponding twelve holograms in a single camera exposure. It allows us to image dynamic samples without sacrificing the FOV by utilizing two cameras and by successfully tuning the polarization state of these illumination beams. The latter makes it unsuitable for polarimetric imaging. Furthermore, the camera's dynamic range between various multiplexed interferograms may lead to reduction in the signal to noise ratio of the reconstructed images.

In fact, an elegant way to address this issue is by optimizing the angular sample scanning scheme so that with limited number angles, the spatial Fourier space is well filled and enough information for high-resolution imaging is collected from the sample. In this work, an optimization approach for data acquisition is proposed. For such purpose, various scanning schemes from simple 2-D scanning patterns up to the more elaborate 3-D ones are investigated. Simulations and experimental results are used to quantify and evaluate these acquisition schemes.

TDM Synthetic Aperture Process

Data recombination in Fourier space is a crucial part of tomographic diffractive microscopy. For weakly scattering samples, reconstruction can be done under Born or Rytov approximations to retrieve the 3D RI maps [10,11,[START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF]. In such cases, for each illumination, object wavevectors k o , diffracted wavevectors k d and the incident wavevector k i are linked by the elastic condition:

k o = k d k i .
(2.1) So for N acquisitions, one has N k i vectors, and N k d and k o vector sets, to be properly relocated in Fourier space. Hence, the object waves are computed from the diffracted waves through illumination wave vector translations, i.e k d are translated by

k i . (c) (a) (b) k o -k i k d k z k x k i k o -k i k d k z k x k i
Reflec�on TDM Transmission TDM 4Pi TDM As explained previously in chapter 1, one way to improve the image quality is by collecting more Fourier components of the sample. Specifically in IR TDM, this can successfully be done by illuminating and recording the corresponding scattered fields from the sample at various directions. These collected scattered field components need to be merged effectively using Equation (2.1). Recombining the various frequency components in several directions, through the synthetic aperture process, allows one to access spatial frequency components beyond the limit of the microscope's objective lens aperture.

Figure 2.1 shows the 2D data recombination process for various TDM configurations. One can get the OTF in Figure 2.1(a) and (b) when using reflection and transmission TDM configurations, respectively. Considering the combination of these two TDM modes to make 4Pi TDM as proposed by [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF][START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF], two acquisitions are made using both reflection and transmission. One can get the OTF in Fig. 2.1(c, left-up), when illuminating through first objective, and detecting reflected field through that objective and transmission through the facing objective, which are a "doughnut" and a "cap of Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION ball" in 3D. When scanning the sample through the second objective, one records again the "doughnut", but in a different way, plus the complementary "cap of ball" for the reflection, Fig. 2.1(c, left-below). Recombining the four subsets of data leads to the final 4Pi OTF, Fig. 2.1(c, right). Note that the data acquired in transmission mode from both directions of illumination are not identical but combined to obtain a better filling of the central "doughnut".

Moreover, considering a linear image formation model (such as TDM), the measured 3-D scattering potential of the object, V m (r), is actually the convolution between the actual object scattering potential V(r) and the Point Spread Function of the system (PSF). We can equivalently say also, the collected object spectrum is the product of the actual object spectrum by the OTF of the microscope, which is given as a 3-D mask with ones inside and zeros outside the effective bandwidth. Mathematically, we can write it as:

V m (r)=V(r) ⇤ PSF, (2.2) or Ṽm (k)=F 3D [V m (r)] = Ṽ(k) • OTF, (2.3)
where F 3D is the Fourier transform in 3D. The ⇤ and • symbols denote the spatial convolution and the element-wise product, respectively. At this point, it is important to understand that various TDM configuration (reflection, transmission or combination of both as 4Pi) allows one to access different features of the object, since the OTF is directly related to the implemented system.

In either case, the use of few angular illuminations result in a degraded reconstructed image quality. It is obvious that collecting more Fourier components of the object highly improves the image quality, which could be possible by utilizing several angular illuminations of the sample. However, for fast imaging this seems challenging to do since the number of holograms is directly related to the speed of acquisition and reconstruction. An implementation of advanced technologies (for instance by performing a GPU computation) for fast imaging using 100s of holograms by J. Bailleul et al. [START_REF] Bailleul | Tomographic diffractive microscopy: towards high-resolution 3-D real-time data acquisition, image reconstruction and display of unlabeled samples[END_REF] is one recent advancement towards real time imaging. Nevertheless, due to measurement redundancy, increasing the number of angular sample illumination may not be always the best solution.

Figure 2.2 depicts the spectrum of a 2 µm spherical bead simulated with 200 and 100 illuminations by considering transmission, (Fig. 2.2(b,c)) and 4Pi TDM (Fig. 2.2(d,e)), respectively. Generally, the OTF with 200 illuminations allows one to collect more Fourier components of the object compared to the 100 illuminations. However, increasing the number of angles also increases the overlap between these caps of spheres, hence creating a redundant information in Fourier space. In fact, reallocation of each cap of spheres in the inverse space is linked with the illumination schemes. Hence the choice of an appropriate scanning pattern is also critical for optimizing the number of angles to be used in TDM experiments, which is the main goal of this study.

Scanning Patterns and Devices

In TDM data acquisition, the application of 1-D angular scanning along a line improves the speed, however, it does not permit isotropic gain in resolution [START_REF] Sung | Optical diffraction tomography for high resolution live cell imaging[END_REF][START_REF] Shan | Image formation in holographic tomography: high-aperture imaging conditions[END_REF]. Conversely, considering 2-D angular scanning of the incident beam has the advantage to enhance the spatial resolution in all directions [START_REF] Sung | Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy[END_REF]. For easy implementations, one can use LED arrays manufactured with different patterns as in Li et al. [START_REF] Li | Optical diffraction tomography microscopy with transport of intensity equation using a light-emitting diode array[END_REF][START_REF] Li | High-speed in vitro intensity diffraction tomography[END_REF]. Figure 2.3 shows the results taken from [START_REF] Li | Optical diffraction tomography microscopy with transport of intensity equation using a light-emitting diode array[END_REF]. They selected LEDs to illuminate the specimen within the condenser's back focal plane limit and to gain isotropic transversal resolution plus to The main limitations of using LEDs is that it result in non-tunable illumination patterns. For flexible tuning of the sample scanning scheme, one can use scanning devices, Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION which gives wide possibilities of the illumination pattern. However, controlling the precise illumination direction is not always straightforward and is highly dependent on the available technology. Several scanning technologies have been implemented in TDM.

At IRIMAS tip/tilt mirrors were used [START_REF] Simon | Tomographic diffractive microscopy of transparent samples[END_REF] at first. However, they have slow mechanical movements, and are prone to parasitic vibrations. Since interferometric systems are very sensitive to mechanical movements and parasitic vibrations, which could be magnified for fast acquisitions, we later replace them by Fast Steering Mirrors (FSM-300) [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF]. The FSM that we use in our lab provides two-axis, fast rotation with submicroradian (⇡ 1 µrad) resolution, up to ±1.5 voice coil actuators.

Several groups are also implementing various scanning devices. Rotating Pechan prisms are used by [START_REF] Noda | Three-dimensional phasecontrast imaging by a computed-tomography microscope[END_REF], where the polar angle is adjusted by the prism and azimuthal angle by rotating the prism using a motor. Spatial light modulators (SLM) are also used by [START_REF] Lim | High-fidelity optical diffraction tomography of multiple scattering samples[END_REF][START_REF] Kuś | Active limitedangle tomographic phase microscope[END_REF]. SLM helps to modulate the intensity, phase and polarization of incident light beam. Controlling the average voltage imposed on the Liquid crystal (LC) cell induces variable tilt of the LC molecules. They are employed in various fields with their advantages of providing a high mechanical stability [START_REF] Lazarev | LCOS spatial light modulators: trends and applications[END_REF].

Digital Micro mirror Devices (DMD) are other choice of use in TDM [START_REF] Lee | Low-coherent optical diffraction tomography by angle-scanning illumination[END_REF][START_REF] Shin | Optical diffraction tomography using a digital micromirror device for stable measurements of 4D refractive index tomography of cells[END_REF][START_REF] Di | Dynamic spatial filtering using a digital micromirror device for high-speed optical diffraction tomography[END_REF] for their higher spatiotemporal flexibility when modulating optical beams. They are manufactured with several thousands of microscopic switchable mirrors arranged in rectangular grid array that can individually be turned on and off, which acts as a binary light switch. However, failure caused by an internal contamination due to seal failure corroding the mirror supports is one their limitations, especially for those applications that require rapidly varying light pattern [START_REF] Bansal | Digital micromirror devices: principles and applications in imaging[END_REF]. Fast dual-axis galvanometer-driven scanning mirror in conjunction with a highspeed camera has also been implemented by Kim et al. [START_REF] Kim | High-speed synthetic aperture microscopy for live cell imaging[END_REF]. The two axes of the galvanomirror allow one to vary both the polar and azimuthal angles. This way, various illumination schemes can be achieved.

(d) (e)
Figure 2.4 shows several possible illumination patterns of the focused illuminating beam at the back focal plane of the condenser. Figure 2.4 (a) shows double spiral pattern along with the resulting OTF for low and high NA, NA eff = 0.5 and 0.8, respectively [START_REF] Lim | Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography[END_REF]. The later allows to collect relatively more extended OTF. Annular scanning at maximum NA, Figure 2.4 (e), [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF] and single spiral illuminations, Figure 2.4 (b), [START_REF] Kim | High-speed synthetic aperture microscopy for live cell imaging[END_REF] are also generated by using the dual-axis galvanomirror. Note that the annular scanning at maximum NA shows both the theoretical (denoted by circles) and experimental (denoted by dots) positions of the specular spot at the back focal plane of the condenser, in which the measured angular positions show a slight drift from the desired ones.

Overall, the use of these scanning devices gives a wide flexibility to chose from various possible sample illumination schemes. In the next sections, I will discuss several 2D and 3D angular scanning schemes for optimal illuminations of the sample.

Star Pattern

The star pattern is realized by scanning the sample along various axes as shown in scanning directions are considered to study such constraint. Please note that 1D linear scanning corresponds to the special case of a star scanning with only 1 axis. However, such peculiar scanning cannot provide isotropic (x y) resolution [START_REF] Sung | Optical diffraction tomography for high resolution live cell imaging[END_REF].

(a) (b) (c) k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.

Grid Pattern

Grid patterns, as depicted in Fig 2 .5 (c), are evenly spaced illuminations similar to the LEDs matrix located in the back focal plane of the the illumination setup as used in Li et al. [START_REF] Li | Optical diffraction tomography microscopy with transport of intensity equation using a light-emitting diode array[END_REF], but here with a tunable angular spacing, as the function of number of illuminations. The total number of illuminations is therefore simply tuned by adjusting the grid spacing, and keeping only those angular positions that fit within the illumination NA.

Annular Pattern

Annular sample scanning is achieved by using concentric circles within the limit of the back focal plane of the condenser as implemented by [START_REF] Lim | High-fidelity optical diffraction tomography of multiple scattering samples[END_REF][START_REF] Shin | Optical diffraction tomography using a digital micromirror device for stable measurements of 4D refractive index tomography of cells[END_REF]. Hence, the number of concentric circles and the distance between points are the determining variables. Here, two to four concentric circles have been considered, with same number of illuminations for each circle or with same azimuthal spacing, as depicted in Figure 2.6.

To achieve even azimuthal spacing, I use the circumference of each concentric circle. Hence the number of points in each circle p i , can be computed from total number of Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION points N, as:

p i = r i r t N ,
where r i is the radius of i th circle, and r t is the sum of the radius of all concentric circles. This is depicted in Figure 2.6(d-f). Single annular illumination at maximum NA has been introduced in [START_REF] Fiolka | Simplified approach to diffraction tomography in optical microscopy[END_REF] and used in [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF][START_REF] Park | Generalized quantification of three-dimensional resolution in optical diffraction tomography using the projection of maximal spatial bandwidths[END_REF] for simplifying data acquisitions. It was also considered in Foucault's thesis [START_REF] Foucault | Microscope tomographique diffractif: simplification des acquisitions et reconstructions numeriques associées[END_REF] that if properly used, such a scheme allows for a simplified data acquisition for mirror-assisted TDM [START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF], since it transforms a reflection TDM setup into a transmission one by using special mirrored slides. This case is illustrated by outermost circles in Fig 2 .4. This configuration records the highest amount of lateral high-frequencies, but also leads to the most asymmetric OTF, only capturing the upper half of the 'doughnut', which will be discussed in the later section in detail.

(a) (b) (c) k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 (d) k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 (e) (f)
k x /k -1 0 1 k y /k -1 -0.

Spiral Pattern

Spiral angular illumination schemes are generated using the general equation:

x = q cos(q) and y = q sin(q), q 2 [0, 2pt],

where t is the number of turns and q the polar angle. Single or double spirals with varying number of turns, and even angular spacing or along the curvilinear abscissa can be generated.

For an even sampling along the curvilinear abscissa, first the total length of the curve, L, is computed:

L = Z 2pt 0 p 1 + q 2 dq.
Then, instead of using a constant ∆q as for classical spiral pattern, one computes ∆q for each points:

∆q = L N 1 p 1 + q 2 .
Figure 2.7 illustrates the forms of the spiral illumination schemes considered for the study. I simulate and implement single and double spirals from 2 to 6 turns, with both even and non-even sampling along the curvilinear abscissa are implemented. Note that, double spirals are simply obtained by applying central symmetry to single spirals, as shown in Figure 2.7 (c,d), the ones with red dots are computed from black dots using central symmetry.

(a) (b) (e) (c) (d) k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.

Fermat's Spiral Pattern

The Fermat's spiral is a uniform 2D pattern, (see Fig. 2.7 (e)), which was proposed by H. Vogel in 1979 [START_REF] Vogel | A better way to construct the sunflower head[END_REF] to represent a sunflower head. It is generated using the equation:

x = p q cos(g q), y = p q sin(g q), q > 0 when g is the golden angle, g = p(3 p 5) rad. The golden angle allows for the points to traverse equal annuli in equal turns, and hence to implement 2-D uniform angular sample scanning.

Rose Pattern

Flower kind of sample scanning helps to have a wide variety of illumination schemes. The generalized equation for generating rose scanning patterns is given by [START_REF] Weisstein | Rose, from MathWorld-A Wolfram web resource[END_REF]:

x = cos(kq) and y = sin(kq), where k is the ratio of two integers k = c/d. Considering these main parameter (c, d), it is possible to change the nature of the flower. When c and d are odd numbers with periods pd, we found a rose pattern with c petals. Similarly for even c and d, the rose pattern has a period of 2pd with 2c petals. The simulated rose patterns are depicted in Fig. 2.8, which are generated using c 2 [2, 7] and d 2 [1,9]. One can realize that rose illumination patterns generated when c/d = 1, plus some other conditions such as when the simplified ratio, c/d, gives asymmetric scanning, hence, are not considered in this study.

1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 k y /k -1 -0.

Uniform 3-D Sample Scanning Pattern

Scanning schemes discussed in the previous subsections are 2-D ones, such as required when running an experiment by scanning a focused illumination spot throughout the condenser back-focal plane. These has to be transformed by the illumination setup into a plane wave of appropriate inclination. Considering a direct 3-D angular distributions could be crucial. For instance, one could obtain illumination schemes looking like LEDs distributed onto a cap of sphere, as performed in ptychography experiments [START_REF] Sen | Fourier ptychographic microscopy using an infrared-emitting hemispherical digital condenser[END_REF][START_REF] Pan | Subwavelength resolution Fourier ptychography with hemispherical digital condensers[END_REF].

In fact, to optimize angular sample illuminations, it is important to consider a 3-D equivalent of Fermat's spiral. E. B. Saff et al. [START_REF] Edward | Distributing many points on a sphere[END_REF] and J. Cui et al. [START_REF] Cui | Equidistribution on the sphere[END_REF] both in 1997 have demonstrated how to distribute two points on a sphere with constant minimal distance between each other. Since there is no analytical solution for such a problem, they used a commonly known "Tammes problem". It is a problem to pack circles on a sphere surface such that the minimum distance between those circles is maximized. Their work was inspired by electrons at equilibrium and their Dirichlet cells as shown in Fig. 2.9 (a). Both algorithms [START_REF] Edward | Distributing many points on a sphere[END_REF][START_REF] Cui | Equidistribution on the sphere[END_REF] have very good agreement to each other for less than 12,000 points. Fig. 2.9 (b) shows distributing 6 points on a sphere using such algorithm [START_REF] Edward | Distributing many points on a sphere[END_REF]. After distributing many points on the sphere, projecting them on disc is shown in Fig. 2.9 (c) [START_REF] Cui | Equidistribution on the sphere[END_REF]. 3-D UDHS pattern is obtained by evenly distributing N points over a hemisphere. The angular inclinations are computed using the method explained in [START_REF] Edward | Distributing many points on a sphere[END_REF], limited to the hemisphere. Then, these points are back-projected onto a 2-D disk. This process allows to obtain the positions, which would have to be addressed in the back focal plane of the condenser to actually perform such an illumination in a TDM experiment (Fig. 2.9 (e)).

(d) (e) k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 k x /k -1 0 1 k y /k -1 -0.5 0 0.5 1 (a) (c) (b)

3D-Uniform Distribution on a Cap of Sphere

3-D UDCS pattern is obtained by evenly distributing N points over a cap of sphere with angular inclinations (max. angle, a max ) limited by the considered objective's NA. Then, these points are back-projected onto a 2-D disk, similarly to the 3-D UDHS pattern. Note that the radius of the so-obtained disk being smaller than 1, it is enlarged to 1, for consistency with patterns previously described, see Fig. 2.9 (d). Note that for a hypothetical objective with 2p sr aperture, the two scanning patterns (3-D UDCS and 3-D UDHS) would be identical, but they are not for real objectives.

Evaluation Criteria

Filling Factor

Aiming at optimizing the efficiency of the various proposed angular sample scanning schemes, I use an evaluation method to compare the OTF as a result of each patterns. Thus, after computing the corresponding OTF of each scanning patterns using the synthetic aperture process for transmission, reflection and the combined 4p TDM configurations, I quantify the OTF "filling factor" FF(%) in the Fourier space. V s being the volume of the OTF effectively filled by the cap of spheres using the various scanning schemes, the FF(%) can be defined as:

FF(%)= V s V T ⇥ 100 . (2.4)
V T is the theoretical OTF volume, generated using parameters such as the NA of the objective and condenser lenses, the immersion medium refractive index as well as pixel size of the camera sensor [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF][START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF]. This reference OTF is a totally filled mask with 'doughnut' shape in 3-D for transmission TDM. Note that here, for the simulations and experimental results, a symmetric system with the same NA of condenser and objective lenses is used, as in [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF].

Error Estimation

The filling factor computation gives an indication at how efficient a given scanning pattern is to fill the Fourier space, but does not quantify the reconstructed image quality. So, in addition to the FF(%) computations, image simulations using the various scanning patterns is performed. The results are then compared using the computed Root Mean Square Error (RMSE) compared to the test objects:

RMSE(n t ,n m )= v u u t 1 I ⇥ J ⇥ K I ∑ i=1 J ∑ j=1 K ∑ k=1 h n t ijk n m ijk i 2 , (2.5)
where I, J, K are the dimensions of the object under consideration: n t ijk is the index of refraction at voxel (i, j, k) and n m ijk is its reconstructed value.

Transmission TDM

OTFs construction Results

To better understand the frequency distributions, I simulate the OTFs resulting from the sample scanning patterns presented in the previous sections. The simulation is done by considering high-resolution TDM experiments [12,[START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF][START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF]; 100⇥ oil immersion (n imm = 1.515) objective and condenser of same numerical aperture, NA = 1.4, camera with 5.5 µm sensor size, and an illumination wavelength of 633 nm. The computation is done in 512⇥512⇥512 3-D volume in pixels, with pixel size of 55 nm. The OTFs are obtained by filling the Fourier space with caps of spheres, whose locations are derived from the scanning positions under mechanism described in the synthetic aperture process. Figure 2.10 shows the k x k y and k z k y cross-sections of the 3D OTF at the central plane. For each scanning scheme described in the previous section, 600 illumination angles [12,[START_REF] Sung | Optical diffraction tomography for high resolution live cell imaging[END_REF][START_REF] Simon | Tomographic diffractive microscopy of transparent samples[END_REF] are considered. Fig. 2.10 (a) recalls the theoretical transmission OTF, obtained if 100% filling could be achieved.

From Figs. 2.10 (b-l), one can qualitatively conclude that:

• Scanning with 3-axes star pattern, Fig. 2.10 (b), and 8 overlapping petals of flower scheme, Fig. 2.10 (j), allows for good (k x -k y ) frequency coverage, but privileged directions are observed due to the nature of the corresponding scanning schemes. Flower patterns of 3 non-overlapping petals in Fig. 2.10 (i), shows non-symmetric frequency distributions. Note that the star pattern can be considered as a special case of flower pattern, with zero-width petals. Along the privileged directions, an accumulation of low-frequency components are observed, seen on the k x -k z crosssection;

• Using a grid illumination Fig. 2.10 (c) allows for better acquisition of high lateral frequencies, but empty regions appear in the OTF, which reflect the periodic structure of the illumination pattern.

• Thanks to their circular nature, some schemes induce accumulation of low frequency components without exhibiting privileged directions, such as annular scanning (Fig. 2.10 (d) and (f), spiral scanning (Fig. 2.10 (f) and (g)). Spiral illuminations however produce non rotationally-symmetric OTFs. This may be detrimental when imaging samples that exhibit such symmetries, such as beads; natural or artificial fibers, e.g. textile-or optical fibers [START_REF] Foucault | Simplified tomographic diffractive microscopy for axisymmetric samples[END_REF], or SNOM tips obtained from tapered optical fibers [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF].

• Scanning with Fermat's spiral, Fig. 2.10 (h), 3-D UDCS, and 3-D UDHS patterns, Fig. 2.10 (k-l), allow for more regular filling without presenting privileged directions, nor peculiar structures, as can be seen on both k x -k y and k x -k z cuts.

Filling Factor Results

The previous section qualitatively demonstrated that the OTFs as the result of several sample scanning schemes vary from one another. To deeply understand the frequency distributions, I compute the filling factors of the OTFs. Figure 2.11 shows the filling factor as a function of the number of illuminations for the considered scanning patterns. Figure 2.11 shows the computed FF(%) plot from 1 to 600 illumination angles. For few number of illuminations, FF(%) values are similar for each scanning, but curves start to further apart when increasing the number of illumination angles. This tells that some scanning patterns are able to fill the Fourier space without inducing much redundancy in the captured object frequency when registering the cap of spheres, as explained in the synthetic aperture process. Conversely, some scanning patterns exhibit redundancy, as can already be observed in Figs. 2.10(b-j). In an actual experiment, redundancy can help to increase signal to noise ratio by averaging multiply recorded frequencies. One can observe that 3-D scanning schemes (3-D UDCS and 3-D UDHS) always give the best FF(%). Annular, flower, Fermat's and grid scanning deliver very similar results in terms of Fourier space filling, while spiral and star scanning appear less efficient.

Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION (a) k x k y k z k y (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
Benchmark plots are common methods of choice to evaluate efficiency of certain approaches in research. They can be an empirical tool to analyze data sets and to compare them to an optimal method. In Figs. 2.12, I provide the benchmark plot using the FF(%) values of 3D-UDHS pattern as a reference, which helps to compare the efficiency of other schemes to it. Note that, the benchmark plot as a function of number of illumination angles shows somehow similar trend, except, some curves which shows an abrupt jump. Since it is a relative FF(%) plot, the shape of the curves is highly dependent on 3-D UDHS. In general, the normalized Fourier space filling plots oscillate between 0.85 and 1.

Star, spiral, annular and flower scanning are families of patterns depending on various parameters as discussed in earlier sections. Hence, in Fig. 2.13, I plot error bars around the mean FF(%) value of each family, showing the maximum and minimum values for all the studied members of these families. Also, I displayed the FF results computed for 3D-UDHS to compare with the results of each family. Note that, since not much deviation is observed for low number of angular illuminations the curve starts at 50 angles. One can note that: minimum is obtained using a 2-turn non-regularly sampled single spiral and the maximum being for a 6-turn regularly sampled double spiral.

• For star patterns, a narrow difference is found between the maximum filling obtained for a 3-axes star, and the minimum filling obtained with 4-axes star.

• Considering the so-called flower patterns family, an increase in the number of petals translates into a better distribution of illumination angles, which results in 5. Transmission TDM 51 better filling of Fourier space with reduced redundant information. However, the arrangement of petals as non-overlapping as well as overlapping, with more than 8 petals, showed almost no difference in the filling factor. Considering the same number of petals, odd number of petals results in higher redundancy compared to even number of petals. The maximum and minimum filling are achieved using 8 and 3 petals, respectively.

• Annular scanning for various number of concentric circles, as well as the regularity of sampling angular illumination induces a change in the filling factor. The maximum and minimum FF(%) values achieved at 4 evenly sampled and 2 nonevenly sampled concentric circles, respectively.

Summarizing, in terms of spatial Fourier space filling, 3-D UDHS scanning pattern achieves optimal results.

Image Reconstruction

It should be clear by now that the influence of angular sample scanning schemes in the collected spatial frequency of the object is important. However, the quantification of the OTF filling factor alone could not be enough. Hence the influence of the scanning pattern on image reconstruction is simulated using a synthetic object. For this purpose, only three scanning patterns, 3-D UDHS (with best filling), Fermat (with average filling) and star scheme (with low filling) are considered.

The simulation is made with a synthetic microsphere of 2 µm diameter with complex RI n 1 = n 0 0 + in 0 " = 1.46 + 0i suspended in a background medium of RI n 0 = 1.49 + 0i, in which the refractive index variation ∆n = 0.03 between the bead and the background medium. The bead consists in two absorptive inclusions of 0.8 µm diameter, with RI n 2 = n 3 = 1.46 + 0.02i, and located along the optical axis, at z = ±0.5 µm, respectively.

Figures 2.14(a,e,i) depicts the (x y) and (y z) cross sections of the synthetic object's real and imaginary parts of the 3-D complex refractive index. The reconstructed images are simulated under 1 st Born approximation. This is done by filtering the object's spectrum by the OTFs generated as the result of the three scanning patterns using 60 illuminations. Figures 2.14(b-d) and Figs. 2.14(f-h) depicts reconstructed images cross sections in the (x y) and (y z) planes of the real and imaginary parts.

Simulated image using star illumination is of lower overall quality (also exhibiting star-shaped artefacts on the index image) compared to Fermat's spiral and 3-D UDHS illumination patterns. This is expected mainly due to the lower filling factor. Note that the three considered scanning patterns permit detection of the absorbing inclusions along the optical axis, depicted in Figs. 2.14(f-h). In terms of separating the absorbing inclusions, 3-D UDHS gives better resolution compared to both Fermat's and star patterns.

Figures 2. 15 show profiles through the center of the object, laterally and longitudinally for the index of refraction n 0 . For n 0 (Fig. 2. 15(a-c)) the best estimation of the bead's index is obtained using 3-D UDHS scanning, Fermat's spiral giving the smoothest profile, star scanning delivering a lower index estimation, while also exhibiting stronger residual oscillations. Fig. 2. 15(d) shows the profile through the simulated beads longitudinally for the imaginary part n". The results show that 3-D UDHS is able to clearly distinguish between the two inclusions, which have less than 200 nm separation between them. Conversely, star pattern detects both the inclusions almost as a single structures. This comes mainly from the privileged directions of the star scanning along with the missing cone problems. Note also that a residual of the 2 µm non-absorbing bead is visible around the absorbing inclusions on the images of the star and 3-D UDHS scanning, but not on the Fermat's scanning. This effect is however very weak. This is attributed to the OTF's shape, which is more asymmetric with respect to the optical axis (k z ) for star and To evaluate the efficiency of these scanning patterns, I also compute the RMSE (Equation (2.5)) in a 25 ⇥ 25 ⇥ 25 µm 3 cubic volume around the object, given in table 2.1 for the refractive index and for the absorption. 3-D UDHS scanning gives the lowest RMSE for these simulations.

Experimental Investigations

For experimental investigations, I adapted and used our previously developed transmission TDM system, which was implemented during the thesis of J. Bailleul [START_REF] Bailleul | Microscopie tomographique diffractif á resolution isotrope[END_REF] and L. Foucault [START_REF] Foucault | Microscope tomographique diffractif: simplification des acquisitions et reconstructions numeriques associées[END_REF]. It is built from off-the-shelf opto-mechanical elements, and based on a Mach-Zehnder interferometer. A He-Ne laser at l = 633 nm is used to illuminate the sample. Two 100⇥ NA = 1.3 oil immersion objectives are installed, as condenser and objective lens, in a symmetric configuration. These are slightly lower NA objectives than used in previous experiments (NA = 1.4) [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF]. But they are chosen to allow for a larger working distance while still keeping a high NA experiment, which simplify handling of samples sandwiched between two high-magnification objectives. Detailed explanation of the transmission TDM is given in chapter 1.

The angular scanning is based on a Newport FSM 300 fast steering mirror. The polar and azimuthal angles are changed by applying varying voltage on the mirror. For such purpose, I have modified the previous FSM300 programming to easily switch between each scanning schemes. Acquisitions have been performed sequentially, by ensuring that no drift of the object occurs. For such purpose, instead of reconstructing the RI corresponding to each scanning right-away, all holograms are acquired first and images are post-acquisition reconstructed. This guaranties that same voxel in each image corresponds to the same point of the object under examination.

Silica Beads

Beads with well-defined diameter are used as test samples (Sigma Aldrich 904465-2G monodisperse non-porous 2 µm diameter silica beads 1 ). Note that while monodisperse beads are (by definition) well defined in diameter, depending on their fabrication method (from Stöber-Fink-Bohn process), the index of refraction of the silica constituting them can greatly vary, from n bead = 1.38 to 1.45, while n bead = 1.46 for pure fused silica beads [START_REF] Hu | The use of index-matched beads in optical particle counters[END_REF]. Beads are prepared between two 170 µm thickness cover-glasses, embedded in Eukitt from Sigma Aldrich2 with index of refraction close to that of the cover-glasses: n Eukitt = 1.49 to 1.50. The absorption of silica and Eukitt at l = 633 nm is negligible.

For validations, I mimic fast acquisition by considering only 60 illumination angles using star, Fermat and 3-D UDHS scanning patterns. But one has to ensure that such fast imaging would lead to images with adequate quality. Figure 2.16 shows images of a silica bead, as well as RI profiles along lateral and longitudinal directions. Furthermore, the corresponding OTF for each illuminations is shown for k x k y and k y k z slices, Fig. 2.16 (top row). The 3-D UDHS and Fermat's spiral scanning give slightly better contrast images, mainly attributed to the acquisition of more frequency components, as they present less redundancy than star scanning. Star pattern scanning leads to a slightly smoother image (less noise) as this redundancy translates into averaging those redundant frequency components. In this transverse plane, these three scanning methods however achieve similar lateral resolution, as can be concluded from the index of refraction profiles. Since the beads are spherical, these profiles have been obtained by radial averaging in the (x y) plane.

Along the z axis, the situation is slightly different: images are noticeably affected by the "missing cone", hence, the beads are significantly elongated along the optical axis. That means, radial averaging cannot be used to plot the profile anymore. I rather averaged 4 different beads for each scanning patterns. One can note that star scanning delivers slightly more elongated, less-contrasted images than Fermat or 3-D UDHS scanning. One should therefore favor such more-elaborated scanning patterns for TDM experiments.

Helianthus Tuberosus Pollen Grain

Beads are simple test samples that are suitable to validate proposed sample illumination schemes. It is also interesting to investigate more complex and elaborate samples, such as Helianthus tuberosus pollen grains. Such pollen grains constitute interesting structures at various scales, including pores and external ornamentations (large spikes of several micrometers and small perforations of size ranging down to the 100 nm), inner structures like double walls, the outer exine (a layered structure) and the inner intine (exhibiting pillars between its footlayer and tectum) [START_REF] Halbritter | Pollen Morphology and Ultrastructure[END_REF]. Also it contains structures that are often colored, which present absorption. In that case, a simplifying hypothesis or reconstruction algorithms neglecting absorption cannot be used. Nevertheless, absorption in itself represent an interesting contrast mechanism in TDM [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF][START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF][START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF][START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF][START_REF] Simon | Tomographic diffractive microscopy of transparent samples[END_REF][START_REF] Wedberg | Tomographic reconstruction of the crosssectional refractive index distribution in semi-transparent, birefringent fibres[END_REF][START_REF] Kim | High-resolution three-dimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography[END_REF][START_REF] Sung | Spectroscopic Microtomography in the Visible Wavelength Range[END_REF], but often neglected. Note that since these are thicker samples, Rytov reconstruction is used, since it provides, in general, better results compared to the Born approximation.

Figures 2.17 show the real and imaginary parts of a Helianthus tuberosus pollen grain cross sections along the same z plane. Note several structures of the pollen on both images. The double-wall structure appears, with intine (arrow) visible as a filled layer, while columellae (circle) are visible in the exine (arrow). A cavea in the exine is also visible (arrow), more easily distinguished on the index image than on the imaginary image. The wall structure, which is almost homogeneous in index, appears on the contrary with very marked borders on the imaginary component image (rectangle). This suggests that the absorptive components (at l = 633 nm) are concentrated in the outer region of the exine and in the intine layers. This is different to what is observed in other types of pollen grains [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF][START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF], while in [START_REF] Park | Three-dimensional Refractive-index Distributions of Individual Angiosperm Pollen Grains[END_REF][START_REF] Kim | Three-dimensional label-free imaging and analysis of Pinus pollen grains using optical diffraction tomography[END_REF] pollen membranes and innerstructures characterization in terms of index of refraction was reported, but not about absorption. The "bubble" which is attached to the pollen seems to correspond to an ejection of inner material through the pore. Note the small, non-absorbing particles only visible on the index of refraction image. Taking Fig. 2.18(a) as a ground truth, the 60-angle 3D-UDHS image is marginally more detailed/contrasted (double-arrows) than 60-angle star and Fermat scanning images. This is confirmed by plotting a z-direction profile through one of the echinate (lower row, right). For fast scanning images, the 3D-UDHS image indeed depicts better contrast/resolution than Fermat's and star images, which confirms the previous results obtained using beads.

C2C12 Myoblast cells

Samples containing calibrated beads or those containing complex structures, such as pollen grains, have shown their advantage to validate the proposed sample scanning schemes. To further investigate the effect of such schemes, I have also considered fixed C2C12 cells, which are provided by biologists as part of collaboration with IS2M, CNRS lab.

C2C12 are immortalized mouse myoblast cells discovered by Yaffe and Saxel [START_REF] Yaffe | Serial passaging and differentiation of myogenic cells isolated from dystrophic mouse muscle[END_REF] in 1977, which are now being used as a model for in vitro understanding of several mechanisms in biomedical research and for exploring various biochemical pathways, including: glucose metabolism, insulin signalling and resistance mechanisms, glucose transporters at cellular and molecular levels [START_REF] Chun Y Wong | C2C12 cell model: its role in understanding of insulin resistance at the molecular level and pharmaceutical development at the preclinical stage[END_REF].

After taken from normal mouse myoblasts, C2C12 cells can be cultured and differentiated into mononucleated myoblasts, which afterwards can form multinucleated myotubes networks (which is done by reducing the serum content or by starvation). After a few days this may lead to formation of sarcomeres and Z-lines muscle cells in the process of myogenesis [START_REF] Chun Y Wong | C2C12 cell model: its role in understanding of insulin resistance at the molecular level and pharmaceutical development at the preclinical stage[END_REF]. The full protocol of preparing such fixed sample is given in Appendix C. Fig. 2.19 shows reconstructed image cross sections of mononucleated C2C12 myoblasts cells acquired by transmission TDM using star, Fermat's and 3-D UDHS illumination schemes. The image reconstructed using 600 illumination angles of 3-D UDHS in Fig. 2.19(a,e) is used as a reference to compare the other images reconstructed using Fermat's spiral sample scanning patterns with the same number of holograms, as observed in Fig. 2.19(f) and (g), respectively. One can also realize that the RI estimation of the nucleus (as observed in the zoom in area) acquired using 3-D UDHS in Fig. 2.19(d) is closer to the reference than acquired by star and Fermat's spiral schemes.

Other considerations

Varying Numerical Aperture

Until this point, I have only considered an illumination rotation TDM system where the condenser and objective lenses have the same numerical aperture. However, this is not always the case, and in such conditions the OTF is not symmetric. This was studied numerically by Kou et al. [START_REF] Shan | Image formation in holographic tomography: high-aperture imaging conditions[END_REF] in 2009, shown in Figure 2.20(a). Kim et al. [START_REF] Kim | High-resolution three-dimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography[END_REF] have also implemented such a system in 2013, where the sample is illuminated angularly (600 angular illuminations) and the diffracted fields from the object are collected using a 1.4 NA objective. It should be clear by now that asymmetric OTF (as the result of system asymmetry) leads to wrong estimation of the complex RI estimations, mainly by intermixing the real and imaginary parts of the spatial frequencies of the object. Hence researchers have used a constraint in their reconstruction algorithm to compensate for the nonsymmetries of their OTF. For instance, using the known immersion media and considering that the media has greater RI than the sample under investigation helps to retrieve better estimation of the RI as used by Kim, et al. [START_REF] Kim | High-resolution three-dimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography[END_REF].

Annular Sample Illumination at Maximum NA

Annular sample illumination scanning using a single concentric circle, with maximum polar angle and varying azimutal angle (from 0 to 2p), has been used by several groups, including the commercial TDM manufacturers, Nanolive 1 and Tomocube 2 as shown in Fig. 2.4(d). It was first introduced by Fiolka et al. [START_REF] Fiolka | Simplified approach to diffraction tomography in optical microscopy[END_REF] for simplifying data acquisitions. It has been also considered in the thesis of L. Foucault [START_REF] Foucault | Microscope tomographique diffractif: simplification des acquisitions et reconstructions numeriques associées[END_REF], where he showed that, if properly performed, such a peculiar scheme permits a simplified approach for mirrorassisted tomography [START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF]. This allows for a versatile transmission/reflection approach, as it transforms a reflection TDM setup into a transmission one, if special mirrored slides are used. Moreover, Park et al. have implemented such a scheme to attain origin symmetric OTF that could result in a real-valued point spread function (PSF) [START_REF] Park | Generalized quantification of three-dimensional resolution in optical diffraction tomography using the projection of maximal spatial bandwidths[END_REF].

The OTF construction of such scheme is illustrated in Fig. 2.21 for transmission TDM. Fig. 2.21(a) is when the polar angle is lower than the maximum angle achieved by the condenser lens NA and Fig. 2.21(b) is when collecting diffracted fields by the full aperture of the objective. One can realize that such scheme allows to obtain maximum possible object frequency in the half space, k z > 0. In general, it will lead to asymmetric frequency support which leads to wrong estimations of the RI.

Fiolka et al. [START_REF] Fiolka | Simplified approach to diffraction tomography in optical microscopy[END_REF] used water immersion microscope objective and condenser lenses of NA = 1.2, which leads to maximum collecting angle of 64 . In Fig. 2.22 (I) the ef- fect of changing the NA of condenser lens NA cond when using annular illumination scheme is highlighted [START_REF] Fiolka | Simplified approach to diffraction tomography in optical microscopy[END_REF], in which symmetric OTF along x axis is obtained when NA cond = NA obj . They showed that decreasing only NA cond leads to less extended lateral bandwidth in Fourier space and more asymmetric OTF. After studying these constraints they preferred to use NA cond that provides a maximum incident angle (a = 30 ). There are also special configurations for which the OTF filling can be improved. In the next sections, I'll explain on how the peculiar properties of either the observed sample as well as the microscopic setup can improve Fourier space filling.

Axi-Symmetric Sample

Some objects such as spherical beads, rods and fibers present morphological symmetries. This includes top-down symmetry with respect to the focal plane or rotational symmetry. In either cases, tomographic data can be replicated in Fourier space to reflect the object's symmetry. In the extreme case of axi-symmetric objects such as beads a single holographic data set recorded with inclined illumination can even be sufficient to efficiently reconstruct the sample as demonstrated in the thesis of L. Foucault [START_REF] Foucault | Microscope tomographique diffractif: simplification des acquisitions et reconstructions numeriques associées[END_REF].

He has demonstrated that rotating the experimentally acquired data numerically in the Fourier space allows to reconstruct the 3-D refractive index distribution of axial symmetric samples. When starting the numerical rotation process from holographic acquisition, it leads to better reconstructed images, which he called it FINER: Fourier Image NumErical Rotation method. Reconstruction using such approach can successfully help to mimic sample rotation experiments, which is possible by initiating the algorithm using an illumination rotation TDM, TINER: TDM Image NumErical Rotation. Fig. 2.23 shows numerical image simulations of an optical fiber reconstructed using holography, TDM, FINER and TINER approaches [START_REF] Foucault | Simplified tomographic diffractive microscopy for axisymmetric samples[END_REF]. The Fourier spectra for holography, FINER, TDM and 2 angular rotation-TINER is shown in Fig. 2. 23 (a), (b), (c) and(d) respectively along with their respective reconstructed image in the lower row.

Non-absorbing Sample

A purely transparent and non-absorptive sample has a real refractive index. That means the object spectrum satisfies the following relation:

Ṽ(k)= Ṽ⇤ ( k), (2.6)
with Ṽ⇤ (k) being the complex conjugate of the Fourier transform of the object's scattering potential Ṽ(k) . Recalling Equation (2.3) between the measured scattering potential and true object potential:

V(r) m = F 1 3D [ Ṽ(k)OTF(k)],
(2.7) noting F 1 3D as inverse 3-D Fourier transform, the real R and imaginary part I of the scattering potential can be related using the consequences of equation 2.6: R[

(I) (II)
Ṽ(k)] = R[ Ṽ(-k)] and I[ Ṽ(k)] = I[ Ṽ(-k)], which leads to: R[V(r) m ]=R Z Ṽ(k)OTF(k) exp(ik.r).dk = R Z Ṽ(k) 1 2
[OTF(k)+OTF(-k)] exp(ik.r).dk.

(2.8)

Hence, in case the scanning induces an asymmetry in the effective OTF, it can be compensated by forcing Hermitian symmetry on the object's spectrum [START_REF] Park | Generalized quantification of three-dimensional resolution in optical diffraction tomography using the projection of maximal spatial bandwidths[END_REF][START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF], considering an effective OTF:

OTF eff (k)= 1 2 [OTF(k)+OTF(-k)].
(2.9)

The use of Hermitian symmetry on a transparent objects results in an extended OTF, Fourier spectra corresponding to the images below. (e),(j) Simulated cylinder. Reconstruction using (f),(k) DHM, (g),(i) FINER method, (h),(m) illumination rotation TDM (600 angles), (i),(n) F-TINER approach with two rotation angles (0°and 90°). Taken from [START_REF] Foucault | Simplified tomographic diffractive microscopy for axisymmetric samples[END_REF]. Scale bar is 2.5 µm.

shown in Fig. 2.24(a) and (b) before and after symmetrization of the OTF [START_REF] Park | Generalized quantification of three-dimensional resolution in optical diffraction tomography using the projection of maximal spatial bandwidths[END_REF]. This is of peculiar interest for annular illumination scheme at maximum NA, which doubles the filling factor by forcing Hermitian symmetry.

Mirror-Assisted Transmission TDM

Mirror-assisted tomography was first introduced by Mudry et al. [START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF] to simplify image acquisition and extend the frequency support. In such experiments, the sample is placed on the slide in the vicinity of a mirror for reflecting back the primarily diffracted light from the sample. Hence, the use of special mirrored slide induces a symmetry in the illumination system as well as in the detection setup.

As I have discussed in the previous section, such a device acts in fact as folded 4p system, originally introduced in the framework of diffractive tomography by V.

Lauer [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF]. Two transmission OTFs, as well as two reflection OTFs can indeed be recorded using a single objective.

L. Foucault has also shown another application of such system by considering the sample's property. He discussed that for mostly transparent and very weakly backdiffracting samples, one annular illumination is sufficient to acquire the entire transmission OTF [START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF]. Hence, properly taking into account both the acquisition system symmetries and the specific sample properties, a supplemental simplification is possible.

Figure 2.25(a) shows the numerical demodulation process after acquisition of the data using annular illumination at maximum NA [START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF]. The preliminary experimental results using polystyrene bead is also shown in Fig. 2. 25(b). But this comes with some experimental limitations, such as: the object should not be in contact with the mirror but at optimal distance from it, and should not be too large or too small, rather to be as flat as possible if first Born or Rytov approximation is used for the reconstruction.

OTF Reconstruction and Filling

The OTF for transmission TDM when considering annular scanning at maximum polar angle is shown in Fig. 2.26(a,b). Such scheme records the highest possible lateral frequencies in transmission TDM configurations. For reflection, it relatively allows to collect the lower lateral object frequencies. In both cases, such a scanning leads to the most asymmetric OTF, since we only capture the upper half of the 'doughnut' and lower parts of the cap of ball [START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF].

When the polar angle is lower than the maximum angle achieved by the condenser lens's NA (a = 0.8a m ), the computed OTF has a lower size as shown in Fig. 2. 26(c,d). For such cases, the OTF is not only in k z 0 but also in k z  0, however, it is asymmetric.

For both conditions, considering constraints on the OTF helps to get rather better filled and symmetric OTF. system, where the central 'doughnut' is now fully filled. Also, note that the OTF reconstructed using lower condenser NA shows more redundancy. Point symmetry helps to act only on the OTF, however, it may not be possible to implement it in an experimental procedure. This is due to the reason that the OTF and the scattering potential of the sample are measured at once, and hence entangled, see Equation (2.3).

Filling Factor

In terms of Fourier space filling, it is straightforward that, scanning along a single circle at maximum NA delivers the lowest values. In fact, it only fills the upper half of the OTF (0  k z  k max ) leading to a theoretical maximum FF(%) of 50 %.

.27 shows the filling factor results of transmission TDM for the previous considered illumination schemes by forcing Hermitian symmetry on the object's spectrum. This results in a better filled spectrum, which is equivalent to considering an instrument having a larger or better filled OTF. This is of peculiar interest for annular scanning at maximum NA, with doubled filling factor. But applying symmetry is then also valid to improve filling factor of the other scanning patterns, with a lower gain, but starting from higher values. The most remarkable feature is that annular scanning at maximum NA is now as efficient as 3D-UDHS, which justifies using this simple scanning method when Hermitian symmetry holds.

Benchmark plots provided in Fig. 2.28, which are computed using the FF(%) values of 3D-UDHS pattern as a reference, helps to compare the other schemes to it. Note that, the benchmark plot as a function number of illumination angles shows somehow similar trend except circular grid pattern. The normalized Fourier space filling plots oscillate between 0.82 and 1.

Image Reconstruction

Figure 2.29 depicts reconstructed image cross-sections of 2 µm microsphere with absorbing inclusions using the same parameters and 60 angular illuminations as earlier simulations. Without regard for the sample, annular scanning at maximum NA yields the lowest reconstructed image quality. This is attributed to its strongly asymmetric OTF (the so-called theoretical half 'doughnut'), which translates into a mixing of real and imaginary parts when performing Fourier transforms. The intermixing is more magnified along the optical axis as can be observed in Fig. 2.29 (f,g), where residual of the absorption inclusions superimpose into the index of refraction image and vise-versa. Imposing point symmetry on the OTF results in better reconstructed image (see Fig. 2.29 (c,g,h)), as it helps to double the filling factor (see Figs. 2.27 and 2.26). Note that in this case, the point symmetric method is only applied to the OTF of the imaging system (instrument), not to be confused with the Hermitian symmetry applied to the sample's spatial spectrum when a non-absorbing object is considered. Implementing a 4Pi scheme with annular illumination at maximum aperture, on the other hand, aids in the collection of the sample's higher frequencies, which enhances the reconstructed image but not drastically (Fig. 2.29(d,h,l)). Annular scanning may be an efficient scanning in transmission TDM, but should therefore be avoided for efficient 4Pi scanning. Considering a point symmetry in the OTF for transmission TDM setup and adapting annular illumination at the maximum angle attainable by the condenser NA could lead to a better reconstructed image. However, in order to use such a scheme experimentally, it is necessary to have a stable and well-aligned instrumentation. To reduce such experimental artifacts, it is sometimes necessary to consider an illumination angle lower than the maximum angle, such as a = 0.8a m , as in [145, 146]. Fig. 2.30 shows the comparison when using both approaches. One can observe clearly that limiting the illumination angle allows to collect more frequency information around the central plane, which leads to a better x y resolution and RI estimations compared to using the maximum NA (a = a m ), see Fig. 2.30(d) and (g). However, since using the maximum NA helps to collect higher object frequencies, the elongation along the optical axis is relatively reduced, see Fig. 2.30(e) and (h). One can also realize that the two absorbing inclusions are well resolved when using the maximum NA but well estimated using the symmetrized OTF achieved with a lower angle.

(i) (j) (k) (l) (b) (d) (f) (g) (h) (c)
However, such considerations are used when the OTF of the system and the object information is known in advance. In such cases, various advanced regularization and reconstruction approaches could be implemented, hence its application is limited to image simulation purposes. One can, rather, consider the exact experimental condition where the OTF and the object potential is measured as a product with the imaging system. Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION

Experimental Results: C2C12 Myoblast cells

Multinucleated C2C12 myotube networks are used to experimentally validate the annular sample illumination scheme at maximum collecting angle. Reconstructed RI cross sections of such sample acquired using annular illumination of limited polar angle (a = 0.8a max ) and at maximum polar angle (a = a max ) are shown in Fig. 2.32. As ex- plained in the previous subsection, using limited condenser angle provides slightly low intermixing of the real and imaginary parts of the 3-D RI compared to the results that are obtained at maximum polar angle in the illumination, as can be seen in Fig. 2.32 (d) and (b), respectively. Also, looking closely on the absorption images, residuals of the nuclear compartments and some structures are more visible in Fig. 2.32 (e,f) than in Fig. 2.32 (g,h). The main purpose of using annular illumination scheme at maximum angle is in fact to maximize the Fourier space filling by reducing the number of illumination angles by half. This is done by symmetrizing the OTFs, which could be done using the fact that biological samples are mostly non-absorbing. Fig. 2.33 shows those results when symmetrizing the spectrum by considering Hermitian symmetry, which forces the absorption to be zero as can be observed in Fig. 2.33 (e-h). In general, Imposing Hermitian symmetry of the object spatial frequency support allows to obtain relatively better image quality. y z cross-sections of the reconstructed RI maps acquired using annular illumination at maximum polar angle a = a max (Fig. 2.33 (b)) provides slightly better optical sectioning along the optical axis compared to the images obtained by limited angle illumination a = 0.8a max in Fig. 2.33 (d).

Reflection TDM

Reflection TDM configuration enables to capture higher frequency components, which allows to provide better lateral resolution and optical sectioning capability along the optical axis. Its application up to now is limited to reflecting and quasi-flat objects, such as for surface studies in semiconductor inspection [START_REF] Sarmis | High resolution reflection tomographic diffractive microscopy[END_REF][START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF]. However, such configuration has limitations. First, it lacks collecting low frequency components relative to transmission TDM [START_REF] Haeberlé | Tomographic diffractive microscopy: basics, techniques and perspectives[END_REF]. Hence if first order Born or Rytov approximations are used for reconstruction, it fails to properly quantify the smoother refractive index variations. Furthermore, the asymmetric Optical Transfer Function of reflection TDM could lead to wrong estimation by intermixing the real and imaginary components of the complex refractive index [START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF] unless a constraint is put into the reconstruction algorithm.

OTFs Construction

Unlike transmission OTF, which takes the shape of an angularly limited horn torus (so called "doughnut"), the synthesized OTF of reflection TDM looks the shape of a "cap of ball", shifted toward high longitudinal frequencies. Simulations of the corresponding OTFs when using the considered illumination schemes are shown in fig. 2.34(b)-(l), using 600 illumination angles in each case.

Similarly to transmission TDM, Fig. 2. [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF] shows OTFs corresponding to scanning patterns with constraints; such as annular, spiral and rose patterns with the optimal and worse case scenarios. One can qualitatively realize that: • Figure 2.34(d),(f) and (i), annular pattern with 2 concentric circles, spirals with 2 turns and 3 petal flower schemes, respectively, results in non-symmetric and redundancy in the spatial frequency coverage.

Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION k y k z (a) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
• Fig. 2.34(b) and (j) 3-axes star pattern and 8 non-overlapping flower petals, respectively, show poor frequency coverage along k y -k z directions, leaving large gaps in the reflection OTFs. Such large gaps are not observed in the corresponding transmission OTF.

• Fig. 2.34(c), (g) and (h) of grid, double spiral with even sampling, and Fermat's spiral illuminations, respectively, provides relatively a more even spatial frequency coverage.

• Fig. 2.34(b), (c), (g), (h), (k) and (l) of star, grid, spiral, Fermat's spiral, 3-D UDCS, and 3-D UDHS scanning schemes lead to OTFs characterized by string of holes along the optical axis. Also Fig. 2.34(d,e) of annular scanning results in few, but large holes along the k z direction.

Filling Factor results

The Filling Factor FF(%), computed as a function of the number of illuminations, from 1 to 600 illuminations, is plotted in Fig. 2 As for transmission TDM, for simplicity, I only display the curves obtained for the patterns giving the best results for Fourier space filling in their respective families: star pattern with 3 axes, flower pattern with 8 petals, regularly sampled double spiral along the curvilinear abscissa with 6 turns, and regular azimuthal spacing annular pattern with 3 concentric circles (in that case, different from transmission for which the optimum was attained with 4 concentric circles).

Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION

Note that for low numbers of illuminations, FF(%) is very similar for each scanning pattern, but curves plot further apart when increasing the number of illumination angles. These curves follow the same trend as for transmission TDM , but with two main differences. First, the annular pattern now gives similar results as 3D-UDHS scanning and spiral pattern give much higher filling compared to star illumination pattern. Secondly, the dispersion of the curves is higher for reflection TDM. Flower, Fermats's spiral and grid scanning deliver very similar results in terms of Fourier space filling, while star scanning appears to be the least efficient. Fig. 2.36 provides the benchmark plots that are computed using the FF(%) values of 3D-UDHS pattern as a reference, which helps to compare the other schemes to 3D-UDHS. Note that, the benchmark plot as a function of number of illumination angles shows somehow similar trend. One can realize that the curve as the result of annular pattern is monotonically increasing (even surpasses 1 at 500 illumination angles), while the benchmark plot of star pattern is monotonically decreasing. In general, the curves oscillate between 0.65 and 1.

Since star, spiral, annular and flower scanning are families of patterns that their OTFs exact shape depends on various parameters. In Figure 2.37, therefore, I provide the mean value plot of each family, with error bars showing the maximum and minimum values obtained when computing FF(%) for each family: • The spiral pattern is characterized by the largest variations as a function of its parameters: number of turns, simple or double spiral, even azimuthal or angular sampling. As for transmission TDM, 2-turn non-regularly sampled single spiral gives the lowest FF(%), while 6-turn regularly sampled double spiral delivers the highest.

• Star patterns are characterized by low dispersion of the filling factor, since only two of them (3 and 4 axis star) are used.

• For flower patterns family, increasing the number of petals results into more evenly distributed illumination angles, which in turn results in a better filling factor, thanks to reduced redundant information. Maximum and minimum filling are obtained with 8 and 3 petals, respectively, as for transmission TDM.

• Annular scanning with concentric circles gives best results with 3 circles, with even azimuthal spacing.

The interesting remark here is that, in terms of Fourier space filling, 3-D UDHS and annular scanning patterns deliver the best overall results. 3-D UDHS giving the best filling factor for both transmission and reflection has interesting implications in view of practical implementation, especially if one wants to combine both configurations, which will be discussed in the next section.

Image Reconstruction

To validate with image simulations, I consider the same phantom as for the transmission TDM, a bead with absorbing inclusions. But here I used 200 illumination angles, contrary to 60 angular illuminations that was used in transmission TDM. This is because the phantom under consideration has weak refractive index variations (∆n = 0.03), which exhibit weakly back-diffracted field compared to the transmitted forward-diffracting field. Figure 2.38 shows cross section of reconstructed images using a reflection TDM under first order Rytov/Born approximations considering star, Fermat's spiral and 3D UDHS. In general, the result emphasizes the rather poor optical sectioning capability of reflection TDM compared to transmission one in x y directions. As discussed earlier, reflection TDM helps only to collect higher spatial frequency components, and hence ignores the smooth regions of the object [START_REF] Sarmis | High resolution reflection tomographic diffractive microscopy[END_REF], hence only structures with high refractive index transitions are only observed. Moreover, since the OTF of the reflection TDM is not axis-symmetric, i.e. only in the k z 0, intermixing between imaginary and real Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION parts of the index is observed, Figure 2.38(f-h) and (j-l) qualitatively shows similar structures.

One can also notice rebound artifacts along the optical axis. This comes from the simplified image reconstruction process performed in Fourier space, which can be interpreted as Gibbs oscillations. Overall, refractive index estimation of weakly reflecting objects utilizing only reflection TDM system and 1 st Born approximation for inversion algorithm gives poor results.

Combining Transmission and Reflection TDM

Transmission TDM allows to collect lower spatial frequencies of the sample, so that smooth surfaces of the object are reconstructed. However, the missing frequency at the final 3-D frequency support limits its resolving power along the optical axis. Conversely, as discussed in the previous section, reflection TDM configuration enables to detect high spatial frequencies so that sharp edges or abrupt changes along the object contour can be reconstructed.

To fully use these advantages, V. Lauer proposed an approach in 2002 to combine both configurations to make a 4Pi tomography, as illustrated in Fig. 2.39 [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF]. Detecting reflected and transmitted fields using only one objective gives an asymmetric OTF as shown in Fig. 2.39(a). In Fig. 2.39(b), he uses high NA objectives for illumination and detection to get a complete OTF, in which the three sub-parts of the OTF overlap. But that happens only for aperture angles above 70. An alternative configuration is mirror-assisted tomography, introduced by Mudry et al. [START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF] in 2010, which has the advantage to implement 4Pi tomography, from reflection system, easily. In its full configuration, mirror-assisted TDM is equivalent to a 4Pi setup, and for mostly forward-diffracting samples, can be simplified to constitute an alternative to a conventional transmission system [START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF].

OTFs Construction

The OTFs as the result of combining transmission and reflection TDM is shown in Fig. 2.40. The combined OTFs allow one to collect the low frequency (from transmis- sion TDM) and high frequencies (from reflection TDM) of the specimen. However, as it is observed from the k y , k z cross-sections, the OTFs exhibit non-symmetricity along the optical axis, which introduces wrong estimation of the the complex refractive index. The 3D rendered OTF is provided in Fig. 2.40(j) for visualization. For more general specimens and better reconstruction of the complex refractive index, one can consider 4Pi tomography. Figure 2.41 shows the OTFs as the result of combining transmission and reflection TDM to make a 4Pi configuration. One can realize that the central "doughnut" in Figure 2.41 is axis-symmetric and better filled compared to the OTFs in Fig. 2.40. Also two reflection OTFs, in both k z  0 and k z 0, are observed, to give symmetric frequency support. These are due to the two successive illuminations and detection of the diffracted fields via both objectives.

(b) k y k z (a) (c) (d) (e) (f) (g) (h) (i) (j) k y k z k x

Filling Factor results

Figure 2.42 shows the FF(%) results for full 4Pi configuration. One can realize that the curves exhibit same trend as for reflection, Figure 2.35, and transmission, Figure 2.11, FF(%) results. There are however some noticeable differences.

First, the filling factor systematically increases, compared to separate transmission and reflection cases, reaching a maximum close to 80% in the best case. This is a benefit of the symmetric 4Pi configuration, which, with same number of illumination angles, allows for recording two transmission OTFs, resulting in a better filling factor of that part of the final 4Pi OTF. Secondly, the combination of reflection and transmission results in families of curves, with 3D-UDHS, 3D-UDCS and annular scanning giving best results, and star patterns being noticeably less efficient than flower, Fermat, spiral and grid scanning, which deliver very similar results. In terms of Fourier space filling, 3-D UDHS patterns deliver the best overall results. 3-D UDHS giving best filling factor for both transmission and reflection has interesting implications in view of practically implementing 4Pi tomographic diffractive microscopy system.

In Fig. 2.43 the benchmark plots that are computed using the FF(%) values of 3D-UDHS pattern as a reference is provided, which helps to compare the other schemes to 3D-UDHS. Note that the benchmark plot as a function number of illumination angles shows somehow similar trend. One can realize that the curve as the result of annular pattern is monotonically increasing, but unlike the case in reflection TDM, it does not surpasses 1. The benchmark plot of star pattern is monotonically decreasing. In general, the curves oscillate between 0.70 and 1.

Image Reconstruction

Shepp-Logan phantom

For validating the 4Pi TDM, I first consider a Shepp-Logan phantom with contrast corresponding to pure index of refraction variations [START_REF] Shengli | Iterative optical diffraction tomography for illumination scanning configuration[END_REF]. Here the background medium is set as RI n 0 = n 0 0 + in 0 " = 1.51 + i0, with maximum index of refraction departure of 0.04, and the phantom is 5µm wide along the optical axis. Fig. 2.44(b) depicts the characteristic elongation along the optical axis of transmission microscopes. Note how the small inclusions are smeared out, pointed with singlehead arrow. Also, the two inclusions near the big structure along the z-axis are not distinguishable anymore, (see double-head arrow). In Fig. 2.44(c), only edges are visible, since a pure reflection TDM only captures sample's high spatial frequencies, and therefore acts as a high-pass filter. Combining both techniques into a 4Pi method gives better reconstruction and RI estimation, as shown in Fig. 2.44(d). Note however that oscillations along the z-direction is visible, which come from the reflection TDM. Morphological reconstruction and RI estimations are of better quality, even considering the simplest first order Born/Rytov approximation.

This modified 3-D Shepp-Logan phantom present complex features, which makes it ideal to test the influence of scanning patterns in 4Pi TDM. Figures 2.45(a-h) show cross-sections of the phantom, and reconstructions using star, Fermat's and 3-D UDHS patterns, now considering only 60 illuminations. The zoomed area emphasizes ability of the various scanning patterns to reconstruct smaller features, especially along the optical axis, which is the direction for which 4Pi TDM is expected to improve results with respect to transmission TDM only.

Combining Transmission and Reflection TDM

79

3D-UDHS illumination pattern gives better size and shape estimation of the 3 small structures (see the zoomed image below). Conversely, star scanning fails to reconstruct them, and is plagued by more pronounced artefacts. Fermat's spiral identifies these structures on the left as 3D-UDHS, but has lower performances along the z-axis, which is due to the nature of the illumination scheme.

Refracting bead with absorbing inclusion

The Shepp-Logan phantom is purely refracting but with some intricate structures. In addition to RI variations, TDM is capable to detect and quantify absorption [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF][START_REF] Tc Wedberg | Tomographic reconstruction of the cross-sectional refractive index distribution in semi-transparent, birefringent fibres[END_REF][START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF][START_REF] Kim | High-resolution three-dimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography[END_REF][START_REF] Sung | Spectroscopic Microtomography in the Visible Wavelength Range[END_REF][START_REF] Simon | High-resolution tomographic diffractive microscopy of biological samples[END_REF], albeit often neglected contrast mechanism. Moreover, absorptive and refractive components could mix, resulting in wrong estimation of the complex refractive index [START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF]. To demonstrate such measurements, I have used previously considered 2 µm diameter microsphere with two absorptive inclusions of 0.8 µm diameter. Figures 2.46 depict x y and y z cross sections of this synthetic object with the corresponding reconstructed images considering 3-D UDHS, Fermat, and star scanning of 4Pi TDM configuration. This highlights another advantage of the 4Pi configuration, since we observe no (or extremely low) intermixing of real and imaginary components, compared to Fig. 2.14.

For such a small number of 60 illuminations, the filling factor FF(%) is rather low and very similar for each scanning pattern. As a result, image reconstruction is somehow of similar quality, but with slightly better index estimation and separation of the inclusions with 3-D UDHS, than with Fermat's and star scanning.

This behaviour is confirmed by analysing lateral and longitudinal profiles, depicted in Fig. 2.46(m,n). For refractive index n 0 along z axis, the best profile estimations are obtained for 3-D UDHS scanning. Note how the two inclusions are better resolved using 4Pi TDM compared to transmission configuration. In particular, considering the small separation between the two inclusions (⇡ 200 nm), it is possible to look at how each scanning reconstructs them well. This gives an insight about the z-axis resolution: 3D-UDHS illumination pattern more clearly separates the inclusion than Fermat's spiral, and finally star patterns. In all cases, Fermat scanning results into smoother profiles, with less oscillations, a feature also noticeable in transmission TDM. Note that Fermat scanning also presents interesting advantages in Magnetic Resonance Imaging scanning [START_REF] Winkelmann | An Optimal Radial Profile Order Based on the Golden Ratio for Time-Resolved MRI[END_REF][START_REF] Fyrdahl | Generalization of three-dimensional golden-angle radial acquisition to reduce eddy current artifacts in bSSFP CMR imaging[END_REF].

(i) (j) (k) (l)
To evaluate accuracy of the reconstruction when using the three scanning schemes, I have also computed the RMSE (Eq. 2.5) in a 25 ⇥ 25 ⇥ 25 µm 3 cubic volume around the object. Table 2

Experimental Implementations

Although 4pi TDM was originally proposed by V. Lauer in 2002 [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF], it has not yet been implemented. 4Pi in fluorescence microscopy [START_REF] Hell | Properties of a 4Pi confocal fluorescence microscope[END_REF] is difficult to setup, mainly because of the short coherence length of fluorescence, which requires a very tight balancing of the 4Pi interferometer arms. Thanks to the long coherence length of the lasers (for example: He-Ne laser), this problem would be greatly leveraged in 4Pi TDM. Furthermore, unlike 4Pi fluorescence microscopy, the data recombination process in 4Pi TDM is performed numerically, after sequential detection of the diffracted field by both objectives. The proposed full 4Pi TDM system is illustrated in Fig. 2.47. As I have shown in previous sections (using OTF FF(%) results, image simulation and experimental validations), 3D-UDHS proves to be the optimal scanning scheme for both transmission and reflection TDM configurations. The use of such optimized scanning plus multiplexed acquisition of Mirsky et al. [START_REF] Simcha K Mirsky | Dynamic Tomographic Phase Microscopy by Double Six-Pack Holography[END_REF] help to speed up acquisitions. With two successive illuminations, two transmission OTFs and two reflection OTFs, can be recorded and registered to form a final 4Pi OTF (see Fig. 2.1 (c)). Some challenges are however foreseeable to implement such approach. One challenge will be to register each sub-part of the OTF in relation to the others. As the central "doughnut" and the two cap of balls corresponding to the transmission and reflection TDM, respectively, are recorded in separate experiments, their relative phase and/or amplitude may not be properly normalized, which is unfavourable for Fourier inversion.

An alternative to solve this problem could be to perform sample rotation TDM [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF][START_REF] Ziemczonok | 3D-printed biological cell phantom for testing 3D quantitative phase imaging systems[END_REF][START_REF] Lee | Isotropically resolved label-free tomographic imaging based on tomographic moulds for optical trapping[END_REF][START_REF] Vinoth | Integrated dual-tomography for refractive index analysis of freefloating single living cell with isotropic superresolution[END_REF], in the 4Pi TDM configuration. Since large overlap could exist between the recorded 4Pi OTFs, it would facilitate registering and completely filling the OTF with spherical shape of radius 2k 0 . This would also allows for isotropic Nyquist resolution of l/4n (n being the average index of refraction of the sample) [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF]. Such resolution Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION is higher than the combined sample/illumination rotation TDM, which is limited to l/4NA Nyquist resolution [START_REF] Vertu | Improved and isotropic resolution in tomographic diffractive microscopy combining sample and illumination rotation[END_REF][START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF]. The use of advanced constrained reconstructions approaches could also contribute to partially fill the remaining gaps in the 4Pi OTF and improve image reconstruction quality [110, 111, 124-126, 165, 173, 174]. The use of specific sample property (axis-symmetry and Hermitian symmetry) may help to reduce constraints on the sample scanning scheme [START_REF] Krauze | Total variation iterative constraint algorithm for limited-angle tomographic reconstruction of non-piecewise-constant structures[END_REF].

Another practical hurdle could be the difficulty to record the reflected field, as it may be very dim relative to the transmitted diffracted field. Note that the back-diffracted field is detected on a dark background, unlike the transmitted field that contains the initial illumination beam. This allows to detect the weak components of such field more easily. Also, a solution could be to implement different exposure times on the two cameras that are used to detect transmission and reflection components. In this case, the reflection camera acquisition time determines the speed of the acquisition process. A dynamic range approach, which necessitates to illumination with low intensity for transmitted (to not saturate the camera) and to illuminate with higher intensity for the reflection, could also be used. This could allow for higher dynamic detection of the back-scattered components of the diffracted field, but slows down the process (by another factor 2 roughly).

Conclusion and Perspectives

In this chapter, the influence of angular sample illumination schemes in transmission, reflection and 4Pi TDM configurations was studied. To do so, 8 main classes of scanning patterns in TDM experiments were investigated . The aim is to optimize Fourier space filling in the low-diffraction regime.

After performing OTF simulations, 3-D UDHS illumination pattern best fills the Fourier space for all TDM configurations. For transmission TDM, these results have been confirmed with image simulations using absorbing beads as well as experiments using glass beads, pollen grains with complex structures and fixed C2C12 myoblast cells. These are made by considering only 60 holograms, and the most basic and direct inversion algorithms (based on Rytov or Born approximations). The motivation behind such configuration is the implementation of a real-time 3-D acquisition/reconstruction /display system [START_REF] Bailleul | Tomographic diffractive microscopy: towards high-resolution 3-D real-time data acquisition, image reconstruction and display of unlabeled samples[END_REF]. By using such a low number of acquisitions, coupled to fast GPU reconstructions, one could indeed envisage a real-time combined acquisition, reconstruction and 3-D display of TDM images. Such a system would not deliver highquality images, but could be very useful for fast investigations of preparations in view of selecting samples of best interest, for which slower, but high-quality, high-resolution images could then be taken.

Image simulations on a weakly absorbing phantom using a reflection TDM can only reconstruct abrupt refractive index changes or sharp corners of a refractive index. Since the OTF is asymmetric along the optical axis, an intermixing of real and imaginary parts of the complex RI is also observed. A 4Pi TDM configuration (not experimentally implemented yet), which can be achieved by combining the transmission and reflection TDM [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF] or mirror assisted system [START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF], yields the advantages of both approaches. For such purpose, image simulations using a refractive only Shepp-Logan phantom and a bead phantom with absorptive inclusions is performed. It showed that the image reconstruction quality has improved and 3-D UDHS being superior from all illumination schemes. Also, 3-D UDHS being an optimal scanning for 4Pi TDM configurations helps to simplify and speed up image acquisition in an actual setup. Practical implementation hurdles, and possible solutions of the 4Pi TDM system is also discussed.

Overall, understanding and selecting a favorable sample scanning pattern in TDM experiments can greatly contribute in speeding up the image acquisition, by reducing the number of illumination angles needed to achieve required image quality. These investigations can complement hardware optimizations, in view of improving performances of TDM system and minimizing image reconstruction errors [START_REF] Kuś | Illumination-related errors in limited-angle optical diffraction tomography[END_REF].

This work has led to two publications in Applied Optics; A.M. Taddese, et al., "Optimizing sample illumination scanning in transmission tomographic diffractive microscopy", Applied optics, (2021) 1 and A.M. Taddese, et al., "Optimizing sample illumination scanning for reflection and 4Pi tomographic diffractive microscopy", Applied optics, (2021)2 . Parts of the work was also presented in various conference; two national conferences: OPTIQUE Dijon 2021 3 and Journées Imagerie Optique Non Conventionnelle 2021 4 , as well as on an international conference: Focus on microscopy 2021 5 .

Introduction

Up to now, the vectorial nature of electromagnetic waves, known as polarization state of light, has been discarded. For certain classes of samples, considering the light-field's polarization state helps to gain more insight about the exact signatures and anisotropic structures of the sample. Such an approach is used to study and visualize samples exhibiting birefringence, which is known as polarization imaging [START_REF] Valery V Tuchin | Tissue optics[END_REF].

Polarization imaging is a well-established technique in several fields, hence one of the evolving hot topics in TDM. It is gaining a lot of attention especially due to its capability to identify structures that are unseen by conventional imaging. For example, polarized light microscopy (PLM), is used to image various classes of anisotropic samples using a polarized light [START_REF] Allen | Polarized light microscopy[END_REF]. For example, a commonly known birefringent sample, potato starch granules, is imaged using a bright-field, PLM and PLM with introduction of a phase plate (at Institut Fresnel, COMiX research team), as shown in Fig. 3 In general, polarization imaging covers from macroscopic to microscopic scales, and from the simplest and robust to unique and comprehensive techniques, such as polarized fluorescence imaging, polarization gating imaging, polarization-speckle imaging technique, Mueller matrix polarimetric analysis, etc [START_REF] Valery V Tuchin | Tissue optics[END_REF]. Single cell polarization studies and nanoparticle enhanced cell and tissue polarization imaging [START_REF] Anthony | Single cell optical imaging and spectroscopy[END_REF], as well as clinical applications of different polarization imaging technologies, especially for qualitative analyzing of biological samples in large FOV [START_REF] Zeng | Linear polarization difference imaging and its potential applications[END_REF], are also of great interest.

In recent years, quantitative 2-D Jones matrix polarization sensitive techniques, such as vectorial ptychography [START_REF] Ferrand | Ptychography in anisotropic media[END_REF] and holography have gained attention for several applications, including photoinduced birefringence property analysis [START_REF] Todorov | Polarization holography. 1: A new high-efficiency organic material with reversible photoinduced birefringence[END_REF] and biological imaging [START_REF] Wang | Jones phase microscopy of transparent and anisotropic samples[END_REF][START_REF] Daniel | Single-shot digital holographic microscopy for quantifying a spatially-resolved Jones matrix of biological specimens[END_REF]. It has been used to study anisotropic samples in 2-D holographic microscopy under paraxial approximation, in which it has been implemented for biological applications [START_REF] Wang | Optical polarization in biomedical applications[END_REF]. Lately, it has been implemented for biomedicine, and biological applications, where along with synthetic aperture technique, it was used for noninvasive imaging of cancer cells with a better sensitivity [START_REF] Park | Jones Matrix Microscopy for Living Eukaryotic Cells[END_REF]. However, extending such a system for 3-D imaging is not straightforward and needs to be dealt with care.

For 3-D imaging, optical coherence tomography [START_REF] Boer | Polarization sensitive optical coherence tomography-a review[END_REF], three-dimensional polarized light imaging [START_REF] Menzel | A Jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue[END_REF] and fluorescence confocal microscopy [START_REF] Ivan I Smalyukh | Three-dimensional imaging of orientational order by fluorescence confocal polarizing microscopy[END_REF] are the well-known approaches. While these approaches have shown their invaluable importance in clinical applications along with their sub-micrometer resolutions, quantification of birefringence in 3-D is difficult. Instead, often qualitative anisotropic contrast is detected and revealed.

Quantitative birefringence imaging modalities in 3-D are less explored in this manner. TDM, proposed in 1969 by E. Wolf is based on scalar assumption of Born first order approximations. It came a long way in terms of resolution and contrast improvements, which become possible through experimental manipulation tricks and/or sophisticated numerical reconstructions, as well as post processing algorithms. TDM with the scalar approximation has been used at several research areas and even being implemented as commercial systems [10,11,14,16,[START_REF] Di | Tomographic phase microscopy: principles and applications in bioimaging[END_REF][START_REF] Park | Quantitative phase imaging in biomedicine[END_REF]. The scalar approximation, however, fails to quantify samples with anisotropic nature. Although V. Lauer [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF] reformulated Wolf's work in 2002 for TDM polarimetric imaging to give a full vectorial formulation by accounting polarization of light, polarization sensitive TDM has gained comparatively less attention until recent years.

Lately, advancements towards polarization sensitive TDM have shown the importance of quantitative 3-D polarimetric approaches. Works done on polarization sensitive reflection TDM with non-linear inversion algorithm [START_REF] Zhang | Full-polarized tomographic diffraction microscopy achieves a resolution about one-fourth of the wavelength[END_REF], as well as polarization sensitive transmission TDM configuration with 1 st order Born or Rytov approximations [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF][START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF] have been recent advancements in the field.

This chapter is devoted to high-resolution polarization sensitive TDM (PS-TDM) imaging, which has been developed during the late period of this thesis. First, I will detail the vectorial inversion procedure, which is based on first order approximations (Born and Rytov approximations). Image simulations using synthetic data, followed by experimental results using various classes of anisotropic samples will be discussed.

Principles of Polarization Sensitive TDM

Principles

Polarization sensitive TDM is based on solving the simplified Maxwell's equation (by omitting the time dependence exp( iwt) term), when a monochromatic light wave is propagating through a linear, non-magnetic, weakly scattering anisotropic object embedded in a homogeneous and isotropic background medium, described as: r⇥r⇥E t (r)=k 2 0 e r (r)E t (r).

(3.1) E t (r) is the total propagating vectorial electric field detected at position r =(x, y, z). e r (r), r⇥ and r are the relative electric permittivity, curl and gradient operators, respectively. The modulus of the wave vector is k 0 = 2pn 0 /l, with l and n 0 being the wavelength and immersion medium RI index, respectively. Equation (3.1) can be re-written as:

r⇥r⇥E t (r) k 2 0 E t (r)=k 2 0 [e r (r) 1]E t (r). (3.2)
The standard approach to solve this kind of wave equations is by using the Green's tensor of the background medium. (3.5)

r⇥r⇥G(r, r 0 ) k 2 0 G(r, r 0 )=Id(r r 0 ), (3.3
Substituting y(r) by G(r, r 0 ) and using Equation (3.3) leads to:

r 2 G(r, r 0 )+r[r•G(r, r 0 )] k 2 0 G(r, r 0 )=Id(r r 0 ), (3.6) [r 2 + k 2 0 ]G(r, r 0 )= rr • G(r, r 0 ) Id(r r 0 ). (3.7)
In the above equation, we approximate the rr • G(r, r 0 ) ⇡ 0, since the polarization coupling is negligible for a homogeneous and isotropic background medium, which allows to represent the vectorial Green's tensor as a diagonal tensor [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF]:

G(r, r 0 )=Ig(r, r 0 ). (3.8)
The total electric field vector is actually the sum of scattered E i (r) and incident E s (r) electric field vectors, E t (r)=E s (r)+E i (r), which can be simplified using the mathematical property of the tensor product in Equation (3.5) and neglecting the polarization coupling [START_REF] Ciattoni | Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation[END_REF],

[

r 2 + k 2 0 ]E t (r)=k 2 0 [e r (r) I]E t (r). (3.9)
Considering n as the refractive index tensor of the birefringent sample, which is immersed in a background media of refractive index n 0 , the relative electric permittivity tensor e r (r) can be given as, e r (r)=(n 0 I + dn(r)) 2 ⇡ n 2 0 I + 2n 0 dn(r).

(3.10)

The above expression is an approximation considering a weak birefringence of the sample so that |dn|⌧1. The RI tensor n(r) can be given as: dn 1 , dn 2 and dn 3 are the principal refractive index, with R m and R 1 m being the 3-D rotation matrix and its inverse, respectively. Full mathematical expression of the R m will be given later, see Equation (3.38).

n(r)=n 0
The incident field E i (r) is what we measure in the absence of the scatterer, for which we can similarly find differential expression as the consequence of Maxwell's equation:

(r 2 + k 2 0 )E i (r)=0 (3.12)
Substituting this into Equation (3.9) allows to express the scattered field as:

(r 2 + k 2 0 )E s (r)= V(r)E t (r) (3.13)
where the scattering potential tensor V(r) is:

V(r)=k 2 0 h n(r) 2 n 2 0 I i (3.14)
2. Principles of Polarization Sensitive TDM

89

We can realize from Equation (3.13) that V(r) ⇥ E t (r) is the source term, hence solving for E s (r) using the Green's function:

E s (r)= Z G(r, r 0 ) ⇥ V(r 0 ) ⇥ E t (r, r 0 )dr 0 (3.15)
The total field at position r 0 can now be computed as:

E t (r)=E i (r)+ Z G(r, r 0 ) ⇥ V(r 0 ) ⇥ E t (r, r 0 )dr 0 (3.16)
Solving such mathematical expression analytically is difficult due to the presence of E t in the left as well as in the right hand, which is usually solved using iterative methods [START_REF] Kendall E Atkinson | The numerical solution of Fredholm integral equations of the second kind[END_REF]. One needs physical approximations to obtain a simplified and direct solution.

Vectorial First Order Born Approximation

Before discussing about the experimental implementation PS-TDM, I will first detail the retrieval process for the vectorial scattering potential tensor (hence the refractive index tensor) components from the measured vectorial scattered field. The solution given by E. Wolf in 1969 was a scalar equation [10]. Then, in 2002, the complete equation for both scalar and vectorial wave along with the experimental implementation of scalar TDM was given by V. Lauer [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF]. A. Saba et al. [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF] have provided more practical solution and an experimental implementation of vectorial TDM in 2021. Inspired by this, I will modify Equation (3.15) into a simpler form to find a solution to retrieve the scattered field E s (r) from the recorded hologram.

As mentioned in the previous section, the linear, non-magnetic, weakly scattering anisotropic object under study is embedded in a homogeneous and isotropic background medium, and illuminated by a monochromatic and spatially slowly varying phase of light wave. Considering a weakly scattering birefringent sample, the total field vector inside it can be regarded as the incident field vector (which is the consequence of first order Born approximation [START_REF] Born | Principles of Optics, 7th (expanded) edition[END_REF]), i.e. E s (r)

⌧ E i (r)=> E t (r) ⇡ E i (r).
This helps us to re-formulate Equation (3.15) as:

E B (r)= Z G(r, r 0 ) ⇥ V(r 0 ) ⇥ E i (r, r 0 )dr 0 , (3.17)
where E B (r) is Born's scattered field.

Considering a plane wave illumination to the sample, E i (r)=E i (r) exp(ik i .r), we have:

E B (r)= Z G(r, r 0 ) ⇥ V(r 0 ) ⇥ E i exp(ik i .r 0 )dr 0 . (3. 18
)
E i is the illumination beam vectorial complex amplitude. The above approximated relation makes it possible to compute the Born's scattered field vector from the incident field vector. For a general birefringent sample, V(r) is a 3 ⇥ 3 tensor, which is also true for the dyadic Green's tensor G(r) and the incident field E i (r),

E i (r) [3⇥3] = 2 4 E ix1 E ix2 E ix3 E iy1 E iy2 E iy3 E iz1 E iz2 E iz3 3 5 exp(ik i .(r)) , (3.19) V(r) [3⇥3] = 2 4 V xx (r) V xy (r) V xz (r) V yx (r) V yy (r) V yz (r) V zx (r) V zy (r) V zz (r) 3 5 , (3.20) G(r) [3⇥3] = I  exp(ik 0 n 0 |r|) 4p|r| (3.21)
Hence three independent measurements are required to compute all the components. Next, computation of the scattering potential tensor of the sample from the measured vectorial scattered field will be given using the Jones formalism.

Jones Formalism

Quantitative polarimetric imaging techniques relying upon coherent light uses the Jones Matrix formalism to relate the input and output beams [START_REF] Wang | Jones phase microscopy of transparent and anisotropic samples[END_REF][START_REF] Ferrand | Ptychography in anisotropic media[END_REF][START_REF] Park | Jones Matrix Microscopy for Living Eukaryotic Cells[END_REF]. It is a matrix expression used to represent polarized light field components and polarizing components. Although it discards light depolarization effect and its application is limited to only fully polarized light, it has been widely used to understand interference phenomena or in other similar areas that needs to study light-field's amplitudes variation [START_REF] Jones | A new calculus for the treatment of optical systemsv. A more general formulation, and description of another calculus[END_REF].

Aiming at retrieving the scattering field amplitude, we start by developing approximated mathematical expression for PS-TDM imaging using Jones calculus, which links the measured data with the vectorial optical fields. For that, each optical element is represented by a Jones matrix and the polarized light by the Jones vector.

Once the sample is illuminated by multiple polarized light, it is possible to record several polarization sensitive holograms. The purpose in polarization sensitive TDM is then to extract the vectorial complex diffracted field from these measured holograms. The Jones matrix J obj of the sample under examination relates the scattered field vector E s (r) and the incident field vectors E i :

E s (r)=J obj (r)E i (r). (3.22)
Also by recalling the Born approximation given in Equation 3.18, we get:

E B (r)=J obj (r)E i (r) exp(ik i .r) = Z G(r, r 0 ) [3⇥3] ⇥ V(r 0 ) [3⇥3] ⇥ E [3⇥3] i exp(ik i .r 0 )dr 0 , (3.23)
where:

J obj = 2 4 J xx J xy J xz J yx J yy J yz J zx J zy J zz 3 5 . (3.24)
Now following the vectorial version of Born's first order approximation, which can be done similar to the scalar version in Equation 1.34, we get: .25) This leads to the relationship between the Jones matrix and the sample's permittivity tensor:

F  J obj E [3⇥3] i exp(ik i .r) = 2pi k z Ṽ(k k i ) [3⇥3] ⇥ E [3⇥3] i . ( 3
J[3⇥3] obj (k k i )= 2pi k z Ṽ(k k i ) [3⇥3] . (3.26)
The above expression gives a linear relation between the computed Jones matrix components (which are calculated from the measured light-field) and the scattering potential tensor of the anisotropic sample. From now on, we make use of the approximated version of the RI tensor by neglecting the third component, as used in [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF]. Such approximation is valid for anisotropic objects exhibiting weak polarization response along the optical axis. Otherwise, it would not be possible to compute all the 9 components of e r (k k i ) [3⇥3] with two independent polarization components, as we are able to measure only the x and y components of the scattered field using our polarization sensitive system. Note that there are some indirect ways of measuring the polarization effect along the optical axis, such as by placing specially manufactured polarizing components referred to as a stressed-engineered optic (SEO) made from a two fused silica and BK7 glass windows with spatially varying birefringence pattern that encodes polarization information even along the optical axis [START_REF] Curcio | Birefringent Fourier filtering for single molecule coordinate and height super-resolution imaging with dithering and orientation[END_REF]. However, first, such optical element is not commercialized. Second, to the limit of my knowledge, it has only been used in polarization fluorescence imaging, which could make TDM implementation challenging. Hence, the implementation of such approaches are beyond the the scope of this thesis.

As a consequence, Equation (3.26) is now reduced into second order tensor,

Japprox obj (k k i )= 2pi k z Ṽ(k k i ) [2⇥2] . (3.27)
In Jones formalism, when light propagates through the several optical elements, the variations of the electric field vector from beginning to the end can be expressed as the product of each associated Jones matrices or Jones vectors. Figure 3.2 shows the simplified schematic for Jones matrix analysis, in both reference and sample arms. In the sample arm, the output field O ij is expressed as cascaded Jones components:

O ij = J q anl J s E inc , (3.28)
where J q anl is the Jones matrix of analyzer before the detector, and J s is the anisotropic sample's complex Jones matrix. E inc is the incident field to illuminate the sample. Similarly, the reference arm is represented by the analyzer's Jones matrix and reference beam's Jones vector:

R ij = J q anl IE re f . (3.29)
Once the beam is split into reference and illumination beams, light in both arms is changed into circular polarized beam. In the sample arm, the sample is illuminated by right and left circularly polarized light (LC and RC), described by their Jones vector,

E inc = 1 p 2  1 ±i . ( 3

.30)

As far as interferometric systems are concerned, an interferogram generated by two perpendicularly polarized beams leads to the disappearance of interference fringes in the background. Generally in cross-polarized configuration, we record a zero background intensity that makes the phase of the reference beam ambiguous to estimate. This results in a random background phase and this makes phase unwrapping challenging [START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF][START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF]. Conversely, the use of circularly polarized beam greatly alleviates this problem, since it results in a non-zero background intensity.

As described in the earlier paragraph, since we are constrained by two polarization states in the illumination and detection (x and y polarization), we represent the birefringent sample under consideration by the approximated 2 ⇥ 2 Jones matrix, rather polarization analyzer sensor than the 3 ⇥ 3 matrix. This works very well for anisotropic samples exhibiting weak polarization response along the optical axis [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF], so we have:

J approx obj = J s =  J xx J xy J yx J yy . ( 3.31)
The polarization sensitive hologram that we record using the polarization sensitive camera is the interferogram between the output field O ij and the reference R ij beam,

I ij = |O ij + R ij | 2 = |O ij | 2 + |R ij | 2 + O ij R ⇤ ij + O ⇤ ij R ij . (3.32)
To estimate the full Jones matrix components, hence permittivity tensor of the sample, two independent incident polarization states (x and y) are required, which are measured along two orthogonal directions using an analyzer. Thanks to the polarization sensitive camera, as it provides mosaic images in a single acquisition for various polarization states, the later step is automated. For such purpose, the sample is illuminated at right or left circular polarization, which helps to use either of the orthogonal polarization for the estimation.

Let us symbolize the output field at each micro-analyzers orientation:

• E o 1x measured
O ij R ⇤ ij =  E o 1x E o 2x E o 1y E o 2y , (3.33)
2. Principles of Polarization Sensitive TDM 93 which allows us to find the relationship between the measured scattered field along specific analyzer orientation and the Jones matrix of the sample as:

 J xx J xy J yx J yy =  E o 1x + E o 2x i(E o 1x E o 2x ) i(E o 2y E o 1y ) E o 1y + E o 2y . (3.34)
The above expression provides a linear relationship between the measured complex scattered field and the birefringent sample's Jones tensor components.

Effect of Oblique Illuminations

In illumination rotation TDM configuration, the object is illuminated at various angular directions, where the angular illuminations are tuned by the tip/tilt mirror. Hence, decomposing the respective polarization contributions in the illumination and analysis end is crucial to compute the corresponding polarization states. Such consideration is vital to reduce the error in the reconstructed RI tensor quantification when using an imaging system equipped with high NA objective lenses. Generally, we are illuminating the sample with left and right circular polarized beams. Hence, if we consider normal incident light to the specimen, which is propagating in z direction with wave vector k i = êz , the electric field can be described as:

E i = 2 4 E ix E iy 0 3 5 , (3.35)
where E ix and E iy are complex fields, with amplitude and phase. Depending on the nature of the polarized light, either of the complex field components could vanish (linearly polarized light), can be equal to each other (for circularly polarized light, E ix = E iy ), or different from each other but non zero (elliptically polarized light). For example, for a linearly polarized light polarized along x axis, E ix 6 = 0 and E iy = 0. Conversely, for oblique illumination, which is the case for TDM, the incident beam is tilted by the illumination optics as depicted in Fig. 3.3 (a) and (b). At this point, since we are using a high NA condenser lens, we need to model the rotation of the illumination beam by such lens. The incident beam in the sample is represented by its wave vector,

k i = 2 4 cos(f) sin(q) sin(f) sin(q) cos(q) 3 5 , (3.36)
with f and q being the azimuthal and polar angles, respectively, as depicted in Fig. 3.3. Now considering a circularly polarized beam entering the condenser, the incidence beam can be computed from the input beam at any polar angle by using the rotation matrix:

E i (r)=R m 2 4 E ix E iy E iz 3 5 , (3.37)
where R m is the rotation matrix and for transverse electric field (TE) E iz is always zero for the normal incidence but non zero for illumination at an arbitrary angle. In general

R m is an orthogonal 3 ⇥ 3 matrix provided that R 1 m = R T m and R m R T m = R T m R m = I, R m = 2 4 cos 2 (f) cos(q)+sin 2 (f) sin(f) cos(f)[cos(q) 1] sin(q) cos(f) sin(f) cos(f)[cos(q) 1] sin 2 (f) cos(q)+cos 2 (f) sin(q) sin(f) sin(q) cos(f) sin(q) sin(f) cos(q) 3 5 . (3.38)
For instance, assuming x-polarized input beam entering the condenser lens, the incident electric field entering into the sample with unity amplitude can be regarded as [START_REF] Heuke | 3D-Coherent anti-Stokes Raman scattering Fourier ptychography tomography (CARS-FPT)[END_REF]:

E i (r)= 2 4 cos 2 (f) cos(q)+sin 2 (f) sin(f) cos(f)[cos(q) 1] sin(q) cos(f) 3 5 exp(ik i .r). (3.39)
Once the sample is illuminated by a polarized plane wave, the diffracted field passes through the high NA objective lens before it is measured by the polarization sensitive camera. The measured electric field, E m , is then computed from the scattered field just before the objective by implementing the same procedure as above,

E m = R T m 2 4 E sx E sy E sz 3 5 . ( 3.40)
Finally the polarization sensitive camera records only along two perpendicular polarization axes, which we can compute the projection along various micro-analyzers as:

E mx = E m . âx (3.41)
E my = E m . ây . (3.42)
âx and ây are the unit vectors along the horizontal and vertical microanalyzers relative to the illumination beam. This in fact allows one to illuminate the sample with two orthogonal polarizations and to collect the corresponding scattered field along each micro-analyzers axes, which permits to compute the refractive index tensors components by utilizing full Jones formalism approach.

Pre-Processing

The polarization sensitive camera is made from 4 polarization filters deposited on each photo-receptor, which are related by 45 . It provides sub-sampled images in a 2 ⇥ 2 neighborhood represented by 4 pixels, referred to as a "super pixel". Once polarization sensitive holograms are acquired for different illuminations and detection, pixel-wise demosaicking of the holograms is performed. This allows to decompose the hologram in the respective micro-analyzer axis. Schematic representation of the prepossessing work-flow is depicted in Fig. 3 One main limitation of polarization sensitive cameras is that due to the different instantaneous FOV of each sub-pixels of the super pixel, errors are introduced in the demosaicked holograms [START_REF] Gao | Bilinear and bicubic interpolation methods for division of focal plane polarimeters[END_REF][START_REF] Mihoubi | Survey of demosaicking methods for polarization filter array images[END_REF]. Hence, to re-generate accurate high-resolution polarization sensitive holograms, a 2-D image interpolation method is performed on the four demosaicked polarization sensitive holograms.

For that, bicubic interpolation method is used. Note that, bicubic interpolation utilizes 4 neighbouring pixels (16 pixels in total), which takes more computation time compared to bilinear or nearest-neighbor interpolation techniques, that only performs the computation using 4 pixels and requires less computation time [START_REF] Gao | Bilinear and bicubic interpolation methods for division of focal plane polarimeters[END_REF]. However, the former interpolation is often preferred over the later methods, mainly, since it provides smoother images with less interpolation artifacts [START_REF] Mihoubi | Survey of demosaicking methods for polarization filter array images[END_REF].

Bicubic interpolation is implemented by using the third order polynomial fitting to interpolate an area delimited by four corners, and to reconstruct missing values from adjacent pixels by using additional constraint that the first and second derivative at the interpolation points are continuous. Mathematically, we can express the 4 ⇥ 4 bicubic interpolation as:

f (x, y)= 3 ∑ 0 3 ∑ 0 a i,j x i y j , (3.43)
where a i,j is the coefficients of the polynomial function. The formula necessitates to determine the values of all the 16 coefficients of f (x, y). It uses each adjacent pixel values and sums over i and j to approximate the value of the pixels. Note that, starting with a hologram of size N ⇥ N pixels, the demosaicking process leads to N/2 ⇥ N/2 pixels. After interpolation, we arrive at N ⇥ N pixels, as shown in Fig. 3.4

Chapter 3. POLARIZATION SENSITIVE TDM

Once we decompose the polarization sensitive hologram by demosaicking and interpolation, the corresponding Jones matrix components can be computed to analyze the refractive index (RI) tensor components of the sample using the vectorial 1 st order Rytov approximation [10,11,[START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF][START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF].

Retardance, Orientation and Effective Refractive Index

One advantage of polarimetric imaging is that it helps to determine the principal refractive index of optically anisotropic object such as crystals, biological tissues, polymers, etc. Common way of measuring these principal refractive indicatrix can be by using a crystal compensator, which requires several intensity image acquisitions for computing it mathematically.

Recent quantitative method to estimate such polarimetric parameters of the specimen is, for example, 2-D quantitative retardance measurement method based on quadriwave lateral shearing interferometry, which was employed by Aknoun et al. [START_REF] Aknoun | Quantitative retardance imaging of biological samples using quadriwave lateral shearing interferometry[END_REF]. By using several linearly polarized light to illuminate the sample, they acquire sets of polarization dependent phase shifts using their wave front sensor to reveal sample's anisotropic structures. This permits to compute the Optical Path Difference (OPD) corresponding to various incident polarization direction. The measured intensity in 1-D can be related with the OPD as:

I m (x)=I 0 ⇢ 1 + cos  2p p ✓ x ∂OPD ∂x z ◆ , (3.44)
where I 0 , p and z are the incident beam intensity, the period of the diffraction grating and the distance between the grating and the sensor, respectively. ∂OPD ∂x is the partial derivative of OPD along x. We can represent the OPD at each pixel as a function of the incident polarization direction q, which can be given mathematically as a sinusoidal function [START_REF] Aknoun | Quantitative retardance imaging of biological samples using quadriwave lateral shearing interferometry[END_REF]:

OPD m (q)=  n 1 + n 2 2 n 0 l +  n 1 n 2 2 l cos [2(q q 0 )] . (3.45)
Here l is the thickness of the sample, q 0 is the direction of the birefringence optical axis, n 1 and n 2 corresponds to the ordinary and extraordinary refractive index, respectively. Note that, for isotropic materials, the retrieved OPD is identical for any polarization. Hence, only the first term is non-zero, which gives information about samples morphology. Figure 3.5 shows how multiple polarization sensitive OPD measurements help to provide retardance and orientation information. Note from Fig. 3.5 (a) that at least six OPD measurements (as the function of q) are required to retrieve both 2-D orientation and retardance maps of the sample. In practice, more acquisitions are required to reduce phase noise and experimental artifacts.

An alternative approach is 2-D vectorial ptychography that allows to retrieve the polarimetric information of anisotropic sample by using a linearly polarized light. Ferrand et al. [START_REF] Ferrand | Ptychography in anisotropic media[END_REF][START_REF] Ferrand | Quantitative imaging of anisotropic material properties with vectorial ptychography[END_REF] have proposed and implemented such a scheme. The experimental set-up used for such purpose is shown in Fig. 3.6(a). Their system works in transmission configuration. The linearly polarized laser beam (635 nm wavelength) is delivered through a polarization maintaining optical fiber and is then collimated before arriving the scanning device. The scanning stage allows to scan each position of the sample.

The incident light polarization is controlled by using a half-waveplate. In total, 3 different polarizations (0 , 45 and 90 ) are used for illumination. Once the transmitted light is collected by an objective lens (20⇥, NA=0.4), it is analyzed in the corresponding

.5: Graphical illustration to retrieve polarimetric information using quadri-wave lateral shearing interferometry of several OPD measurement. l: object thickness, n 1 : ordinary RI, n 2 : extraordinary RI and q: analyzer axis direction. Adapted from [START_REF] Aknoun | Quantitative retardance imaging of biological samples using quadriwave lateral shearing interferometry[END_REF] polarization by using a linear polarizer. Full polarization state control in the illumination and detection allows to record nine intensity measurements for each scanning position. Using Jones formalism, the complex Jones matrix components are then computed through iterative fitting strategies. What is interesting about this technique is that, it allows to retrieve several polarimetric information (in 2-D) from the computed complex Jones matrix components O(r) of the object. For instance, O(r) can be decomposed into eigenvalues and eigenvectors to get phase-retardance f(r) and fast axis orientation q, in which light traveling along this direction encounters a lower RI and travels faster.

O(r)=RW(f(r)) R 1 , (3.46)
where R is a rotation matrix, which is a 2-D form of R m defined in Equation (3.38). Note that the transpose R T and inverse R 1 of such rotation matrix is the same: R =  cos(q) sin(q) sin(q) cos(q) . (

In Equation (3.46), W(f(r)) is a diagonal tensor:

W(f(r)) =  exp(if fast (r)) 0 0 exp(if slow (r)) (3.48)
Hence, if one knows the thickness of the anisotropic sample l, it is possible to compute the birefringence ∆n = l h f fast f slow 2pl i , with l being the illumination beam wavelength in vacuum.

Unlike the 2-D polarimetric technique that was explained above, thanks to the holographic imaging system and the numerical reconstruction algorithm based on first order approximation, PS-TDM directly estimates the 3-D refractive index tensor n of a weakly scattering sample. This helps to estimate other polarimetric information by using further mathematical simplification. For instance, for a homogeneous anisotropic sample it is possible to implement dn pixel-by-pixel diagonalization of the RI tensor, to obtain two principle refractive indices, n 1 and n 2 . When the anisotropic sample's optical axes are orthogonal (such as collagen fibers), the subscripts can be replaced by the standard Cartesian coordinates n x and n y [START_REF] Park | Jones Matrix Microscopy for Living Eukaryotic Cells[END_REF]. This can be given as:

n(r)=  n xx n xy n yx n yy = R  n 1 0 0 n 2 R 1 (3.49) (3.50)

=

 n 1 cos 2 (q)+n 2 sin 2 (q)( n 2 n 1 ) cos(q) sin(q) (n 2 n 1 )cos(q) sin(q) n 1 sin 2 (q)+n 2 cos 2 (q) . (

From the above expression, it is possible to obtain several 3-D polarimetric information, such as birefringence ∆n, average RI n avg and fast-axis orientation q: ∆n = n 1 n 2 (3.52)

n avg = n 1 + n 2 2 (3.53) q = 1 2 arctan  n xy + n yx n xx n yy . ( 3.54)
Note that to get an orientation map, q 2 [ p, p], one needs to use arctan 2 function and the computed map would be prone to phase wrapping problem. Hence a phase unwrapping algorithm is implemented [START_REF] Vyacheslav | Deterministic phase unwrapping in the presence of noise[END_REF].

Image Simulations

The main interest of image simulation here is to validate the polarization sensitive TDM by mimicking the experimental condition. Even though the inversion algorithm that is used to reconstructed the refractive index tensor components from the vectorial scattered field is based on first order approximation, the use of an accurate forward model can lead to a more realistic reconstructed result. Unlike the scalar approximation, which was simulated by considering the elastic conditions, here, the amplitude and phase variation needs to be accurately simulated. However, for such purpose, the implementation of more sophisticated calculation methods is required. One possibility is to solve Maxwell's equations using the vectorial Lorenz-Mie model [START_REF] Slimani | Near-field Lorenz-Mie theory and its application to microholography[END_REF], which is applied when the scattering sample structures size are comparable to the incident beam's wavelength. It is performed using the analytical scattering solution of Mie scattering for some class of samples such as stratified spheres and cylinders with infinite height. This has been also used to generate holograms for our recent publication [START_REF] Denneulin | GSURE criterion for unsupervised regularized reconstruction in tomographic diffractive microscopy[END_REF]. However, it is limited to objects with known geometries when it is possible to write separate equations for both angular as well as radial dependence of solutions. One can also consider using a finite-difference time-domain method (FDTD) for near-field computation, and a conventional wave propagation method to numerically retrieve the far-field [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF][START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF]. However, the computation time to calculate the output field from the input field varies depending the sampling, the size of the object and the computation device, but usually takes longer time.

Beam Propagation Method (BPM) provides a compromise between propagation accuracy and computation time, compared to FDTD, hence can be another method of choice [START_REF] Ulugbek | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF][START_REF] Shengli | Reconstructing complex refractive-index of multiply-scattering media by use of iterative optical diffraction tomography[END_REF]. For example, scalar Wide-Angle Beam Propagation Method (WABPM) was used as a forward model to compute the scattered field, mainly chosen for its computation efficiency [START_REF] Shengli | Reconstructing complex refractive-index of multiply-scattering media by use of iterative optical diffraction tomography[END_REF]. However, when simulating synthetic samples exhibiting absorbing structures using WABPM, we may realize complex-valued artifacts resulting in an intermix between the real and imaginary parts of the complex RI values. One solution to reduce this problem could be by simulating a weakly absorbing sample (which is the case in TDM) so that the imaginary parts of the complex RI can be assumed negligible, which allows to force the reconstruction process. Also, the vectorial BPM (VBPM) to generate the scattered field after propagated through a birefringent sample has been successfully used and compared to FDTD results [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF].

For purely phase objects, such as spherical beads in a background medium, vectorial first order approximation can be used for the simulation. In this case, it is possible to simply take the vectorial object permittivity tensor and the OTF of the imaging system to compute the measured spectrum of the sample, which allows to compute the reconstructed image through 3-D inverse Fourier transform. This is similar to the scalar simulation given in Equation (2.3). The main difference is that, since the object under considered is birefringent, it requires to compute the Fourier and inverse Fourier transform three times for each principal refractive index components.

Vectorial Beam Propagation Method as a Forward Model

The Vectorial Beam Propagation Method (V-BPM) is used as a forward model to propagate the incident field through the phantom. It necessitates dividing the 3-D sample into finitely small thickness 2-D slices, which allows to propagate the incident light beam through these slices and to get the output diffracted field, as shown in Fig. 3.7(a). The V-BPM model considering a slowly varying electromagnetic amplitude (SVEA) wave is given as [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF]:

e(x, y, z + dz)=exp(i2k v dndz/ cos a) | {z } refraction F 1 " F n e(x, y, z) o I exp ⇢ i k 2 x + k 2 y k 0 + k z dz # | {z } propagation , (3.55
) where e(x, y, z + dz) is the vector field obtained from the initial vector field e(x, y, z) after propagating a small dz step-size. k x , k y and k z defines the propagation kernel in the 3-D spatial Fourier space. Note that dz is divided by cos(a) to account for the exact optical path length when propagating at oblique angle a. The first term exp(i2k v dn/dz) corresponds to the refraction through the sample, with k v = 2p/l being the modulus of the wavevector in vacuum. Detailed derivation of the scalar BPM and the above-used V-BPM expressions are given in Appendix B.

Simulation Parameters

The simulation parameters are similar to our experimental conditions. The sample is assumed to be illuminated by a plane wave, with unity amplitude. The detector (PAS) has a size of 6.9 µm. The simulation is done in a x ⇥ y ⇥ z = 28 ⇥ 28 ⇥ 28 µm 3 volume. The volume is sampled into 512 ⇥ 512 ⇥ 512 cubic voxels of dimension ∆x = ∆y = ∆z = 55 nm. In all simulation, the sample is assumed to be a homogeneous anisotropic sample, in which the RI tensor can be diagonalized into the principal refractive indices.

The sample is assumed to be illuminated by a monochromatic plane wave of wavelength l = 633 nm. For simplicity, the sample is illuminated by a linearly polarized beam at ±45 polarization. Using the V-BPM forward model, 100 holograms are generated for each polarization states using a rose scanning (8 overlapping petals) scheme within a cone of illumination whose polar angle is between a max =[ 67 , 67 ]. Such scheme is chosen due to its simplified analytical expression, which helps to easily generate the incident beam at several directions (see section 3.6).

Once the diffracted field vector are computed using V-BPM as forward model, the Jones matrix components are retrieved. Vectorial Rytov back-propagation (full derivation given in Appendix A) is then implemented to calculate the synthetic object's 3-D RI tensor components. These are diagonalized to give the principal refractive indexes.

Simulation Results

Before showing the numerical results, I have displayed the simulated scattered field's amplitude and phase computed using BPM in Figure 3.8. The images are taken in the central planes.

In Figure 3.8(a), I illustrate beam propagation at normal incidence through a 2 µm diameter bead immersed in a background medium (Eukit). The scattered field's amplitude results shows clearly the edge of the bead. Conversely, the phase of the simulated bead shows the internal composition of the bead, which tells the phantom is more a phase object. Figure 3.8(b) shows the same simulation but at oblique incidence (a = p/16). Note that the simulation results at oblique incidence show artifacts around the border, which are coming from the the sampling problem of the propagator. Since the accuracy may decrease for longer distance propagation, it makes it less suited for longer distance propagation. One way to reduce such numerical errors would be to use a wider calculation window, but at the cost of longer computation time. Furthermore, Amp. to show the efficiency of such numerical method, diffracted field simulated using BPM is compared with the results computed using Lorenz-Mie calculation (see Appendix B) [START_REF] Denneulin | GSURE criterion for unsupervised regularized reconstruction in tomographic diffractive microscopy[END_REF].

Chapter 3. POLARIZATION SENSITIVE TDM

In the first part of the simulation based on V-BPM, I consider the 2 µm diameter anisotropic microsphere immersed in a background medium (Eukitt from Sigma Aldrich) of RI n 0 = 1.49, with RI tensor given by:

n = n 0 I + R  n 1 0 0 n 2 R 1 = R  1.52 0 0 1.51 R 1 (3.56)
where the rotation matrix is given as:

R = 1 2  1 p 3 p 31 , (3.57)
by considering q = p/3.

Once the diffracted field at several angular directions are computed using V-BPM as a forward model, inversion algorithm based on Rytov approximation is used to compute the RI tensor component of the sample (see Appendix A for full derivation). Then, assuming transparent anisotropic sample, I calculate the principal RI components through pixel-by-pixel diagonalization of the RI tensor. Figure 3.9 shows the simulated RI cross-sections along x y and x z planes. One can notice that, the RI estimation of reconstructed values are lower than the original one. This is expected, since only 100 holograms are used. Even if we increase the number of holograms, employing V-Rytov approximation as inversion algorithm affects the image reconstruction efficiency. Also we can observe an elongation along the optical axis due to the known missing cone problem.

In the second part of the simulation, I consider a spherical bead containing more complex structures. The computed principal refractive index cross-section that are retrieved from the reconstructed RI tensor components are shown in Figure 3.10. One can observe blurring around those structures that are multiple scattering layers and around the edges, which results from the inversion algorithm based on Rytov approximation single-scattering assumption. Also, underestimation of the RI as well as residual of the elliptical structure from dn 1 is observed in dn 2 image.

Experimental Investigations

Experimental Set-up

In this section detailed description of the first experimental implementation of the polarization sensitive TDM at IRIMAS will be given. Although the general architecture of the PS-TDM is the modified version of the scalar system, most features had to be built from scratch. The main tasks include adding polarization optical components and optical setup realignment. I built the set up from off-the-shelf optical components, and it is based on the Mach-Zehnder interferometer as depicted in Fig. 3.11. The laser beam (He-Ne, wavelength l = 633 nm) is split into two beams, reference beam and input beam, using a beam splitting optical fiber. Note that to reduce the polarization fluctuation of the reference beam, it is Then, a coherent plane wave exhibiting circular polarization is generated by using polarizing components. The polarization generating components, mounted in both the illumination beam and the reference beam, consist of a linear polarizer and a quarter waveplate. The main purpose of using circularly polarized beam instead of linearly polarized beam is to get rid of the phase unwrapping problem. The later causes the background intensity to be turned off for cross-polarization between illumination and analysis. The circularly polarized illumination beam is then deflected by the tip/tilt mirror (Newport FSM300 Fast Steering Mirror), and scans the weakly scattering anisotropic object sample positioned between the condenser and a microscope objective (oil immersion n imm = 1.518, NA = 1.4 100⇥ Olympus UPLFLN100XO2-2) from several angular directions.

Once the sample is illuminated by the plane waves with varying angular direction and polarization, the diffracted field along with the non-diffracted background beam is collected by the microscope objective. Finally, the three beams (background or illumination, diffracted and reference beams) are recombined to make polarization sensitive interferograms, which are detected by the polarization sensitive camera (PS-camera) from LUCID Vision Labs (PHX050S-PC). The PS-camera consists in of polarization analyzer sensors that allows light to pass in 4 linearly polarization states, parallel, perpendicular, as well as at 45 and 135 polarization, which makes snapshot Stokes polarimetry possible without the need for mechanical rotation of the analyzer.

Data Acquisition and Reconstruction Process

The optimized angular sample scanning scheme, 3-D uniform distribution over a hemisphere (3-D UDHS) is used to acquire several polarization sensitive holograms, as discussed in chapter 2, section 3.7.1. Note that some recent works in this field used annular sample illumination scheme near maximum NA [START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF][START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF][START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF]. As discussed in chapter 2, such scheme uses a non-absorbing sample assumption to double the Fourier space filling factor. This simplifies and increase the speed of TDM data acquisition, which also make preprocessing steps easier (such as when considering the rotation matrix). However, such consideration may affect the accuracy of the inversion algorithm, especially if objects in fact contain absorbing structures.

Once a mosaicked polarization sensitive hologram is recorded by the PAS, pixel-bypixel demosaicking and image interpolation algorithms are performed. Note that by making use of the rotation matrix, each interpolated hologram is then decomposed into its respective polarization. The same procedure is done for the holograms that are acquired without the sample (blank acquisitions) but only with the background medium, which are used as references. Afterwards, each decomposed holograms are preprocessed and reconstructed separately, similar to the scalar TDM reconstruction process.

Experimental Results

In this section, validation of the PS-TDM experimental setup will be investigated. The image reconstruction algorithm to compute the refractive index tensor components from the pre-processed Jones matrix of the sample is based on the vectorial Rytov 1 st order approximation. Indeed, Rytov approximation is preferred than Born approximation, since I am investigating thicker samples with small refractive index variation, as constrained by its assumption [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF].

Potato Starch Granules and Isotropic Silica Beads

To validate our high-resolution polarization sensitive TDM system, I have prepared a sample made of potato starch granules, which is a well-investigated birefringent microobject [START_REF] Ortega-Ojeda | Interactions between amylose, native potato, hydrophobically modified potato and high amylopectin potato starches[END_REF], mixed with isotropic silica beads, suspended in an isotropic background medium (Eukitt from Sigma Aldrich) with refractive index n 0 = 1.49, between two cover glasses of thickness 170 µm. As explained when describing the experimental setup in the previous section, the high NA objective (NA = 1.4, 100⇥) is chosen to collect object frequency components required for high resolution imaging, at maximum angle a max ⇡ 67 . To compromise between the image acquisition/reconstruction speed and image quality, the sample is illuminated from 400 angles using the optimized angular sample scanning scheme, 3D-UDHS.

I first study the recorded hologram considering scalar approximation inversion algorithm. This is done by considering a polarized beam illumination and by using only the hologram along one of the analyzer axis, which can be understood as if it is recorded by a CCD camera. Note that, such acquisition may result in a slightly improved reconstructed image quality compared to the classical TDM approach, since it analyzes the acquired hologram along a specific analyzer axis. This is actually discussed in Zhang et al. [START_REF] Zhang | Full-polarized tomographic diffraction microscopy achieves a resolution about one-fourth of the wavelength[END_REF], where a full vectorial TDM approach provides lower elongation along the optical axis. Nevertheless, to mimic the scalar approach, I reconstruct the 3-D RI using scalar Rytov inversion after implementing demosaicking. Fig. 3.12 shows the reconstructed 3-D refractive map cross-sections along (x, y) and (x, z) directions. Note the elongation of the spherical structures along the optical axis, resulting from the missing frequency (missing cone problem) when using Rytov approximation. The reconstructed image with scalar approximation delivers high-resolution images of the provided sample, however, such approach does not provide any polarimetric information, hence fails to discriminate any anisotropic structure of the potato starch granules. Furthermore, the absorption maps of the potato starch granules in Fig. 3.12(b) seems contains structural information about the starch composition, wherein the edges of non-absorbing silica beads are clearly visible, probably being reconstruction artefacts. To investigate polarimetric imaging capabilities of our PS-TDM system, two acquisitions with right and left circular polarization illumination and various micro-analyzer orientations were made, which is enough to retrieve the full Jones matrix components of the specimen as explained in the previous sections. After acquisitions, image demosaicking to extract the hologram corresponding to 0 and 90 analyzer axes were made. Image interpolation were also implemented to improve the quality of the complex field corresponding to horizontal and vertical polarization. Note that the interpolation part is critical due to two reasons. First, after circularly polarized beam is generated by the quarter wave plate, the phase shift introduced between two perpendicularly polarized light-field would lead to a differential interference contrast looking like effect. Moreover, since we are using a polarization sensitive camera, a shear (spatial shift) between two neighboring pixels may affect the RI estimation, which leads to a differential interference contrast looking like image (this will be discussed in the later section).

After image demosaicking and interpolation on the polarization sensitive holograms, the complex Jones matrix components were computed. Figure 3.13 shows the processed 2-D phase and amplitude maps of the Jones matrix elements of potato starch granules (anisotropic object) and silica beads (isotropic object) in the same field of view for a normal incidence. Note that the isotropic beads are visible in both J xx and J yy , but disappear in the anti-diagonal Jones matrix components J xy and J yx , which show only the anisotropic structures of the potato starch granules.

In Fig 3 .14, the cross-section view of reconstructed refractive index map of the potato starch granules and silica beads is displayed. Compared to the scalar reconstruction, the RI tensor components seem to show slightly lower elongation along the optical axis. Serrels, et al. [START_REF] Ka Serrels | Nanoscale optical microscopy in the vectorial focusing regime[END_REF] have demonstrated such effect, where they showed the influence of different polarization of light on the spatial resolution of the imaging system. Hence, the improvement may come from the separate treatment of the polarization of light. In fact, investigating the effects of polarization has crucial impact on image quality. As long as single scattering is dominant (which is the case for such transparent samples), analyzing the scattered field at several polarization directions could affects the reconstructed image resolution [START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF].

As it can be noticed from the results, n xx and n yy in Fig. 3.14(a) RI tensor components do not provide any discrimination between isotropic and anisotropic objects. They show either the ordinary or extraordinary RI components of the refractive index. Conversely, n xy and n yx show only the potato starch granules apart from some residual of the beads, which are either diffraction artifacts or edge birefringence effect at the bead boundaries as studied in 1991 by [START_REF] Oldenbourg | Analysis of edge birefringence[END_REF] and using vectorial ptychography in [START_REF] Dai | Quantitative Jones matrix imaging using vectorial Fourier ptychography[END_REF]. The circularly varying crystalline structure is clearly visible along the off-diagonal RI tensor components. In fact, native potato starch consists in about 20% amylose, whose image in polarimetric imaging give birth to the so-called Maltese cross [START_REF] Ortega-Ojeda | Interactions between amylose, native potato, hydrophobically modified potato and high amylopectin potato starches[END_REF].

To further investigate several polarimetric information of such sample, I provide the birefringence ∆n (Fig. 3.14(c)), average RI n avg (Fig. 3.14(b)) and fast-axis orientation q (Fig. 3.14(d)) maps. The potato starch granules are visible in all those images. However, the isotropic silica beads are only visible in the n avg , since they do not contain any polarimetric information except the edge birefringence effect.

Araneus Diadematus Silk Fiber

Spider silk from Araneus Diadematus is a well known fiber abundant in nature. Attempts to manufacture synthetic textile fibres of desirable properties, such as high strength, texture, and with ability to bind a wide variety of chemical dyes, makes the silk made by the spider primary choice. That's why mass culturing of such insect remains a major industry focus in several countries and such silks are the only silks commercially used [START_REF] John M Gosline | The structure and properties of spider silk[END_REF]. These kind of spider silks allow a unique feature to study the molecular structure of fibrous biopolymers, since they are made up of amino acid chains dissolved in water based solution. Figure 3.15 shows the RI tensor components of single-stranded axial spider silk fiber. Due to the molecular nature of spider silk containing distinct axes, interaction between polarized light and the crystalline lattice orientation refracts light into 2 direction, that are oriented depending on the vibration directions of the molecules. The size (specifically the fiber's diameters) of many spider silk fibers is higher than the optical resolution limit [START_REF] John M Gosline | The structure and properties of spider silk[END_REF]. However, one difficulty to perfectly measure the size comes from the fact that the fiber exhibit birefringent nature, which makes diameter quantification uncertain [START_REF] John M Gosline | The structure and properties of spider silk[END_REF]. The use of non-polarized light such as in wide-field microscope results in higher order diffraction fringes. This could make the fiber to look like having larger diameter than it actually has. Conversely, polarimetric imaging approaches using polarized light alleviates such issues, by aligning the fiber's axis along the electric field vector component thereby resulting in a more accurate size measurement.

In terms of shape, one can qualitatively realize ( that it is actually a highly specialized silk spin fiber made by Araneus diadematus spider, which seems to consist two interwoven strands exhibiting asymmetry. This results somehow less elongation along the optical axis. Note that this information is not revealed using classical scalar imaging approach which disregards anisotropicity information.

Several polarimetric images in 3-D are also computed. The birefringence of the spider silk, which is determined as the difference between ordinary and extraordinary RIs, is shown in Figure 3.15(c). Looking at the zoom in insets of the image, one can observe the non-homogeneous structures of the spider silk. Note that negative values in this case do not have any meaning, rather shows the residual error during computation. This may be the result of approximations: reconstruction using Rytov approximation and/or approximation of the anisotropic sample under consideration to have weak polarization response along the optical axis. This effect is clearly visible (if not magnified) in the average refractive index map as shown in technique used to image thin samples, but its quality to retrieve several polarimetric information attracts us to use it as another validation method for our PS-TDM system.

For that, a sample containing only potato starch granule is prepared and raw intensity images are acquired by illuminating each position of the specimen in the given FOV using linearly polarized light (0 , 60 and 120 ) and analyzed in the corresponding polarization. Note that here a different polarization state is used (to optimize the iterative Jones matrix components retrieval algorithm) but the general approach is similar to what was used in [START_REF] Ferrand | Ptychography in anisotropic media[END_REF]. The Jones matrix retrieval algorithm takes 1500 iteration.

The same sample is then imaged with our PS-TDM system. Figure 3.17 shows the reconstructed RI tensor components of the single potato starch granule using our PS-TDM. As you can observe, the starch granule is composed of amorphous and semicrystalline (containing amylopectin) structures. The crystaline structure is clearly discriminated in the diagonal components (n xy and n yx ).

Figure 3.18 shows several polarimetric information of the same single potato starch granule retrieved using both PS-TDM and V-Ptychography. In general, our PS-TDM gives better resolved and detailed polarimetric information of the considered anisotropic sample, especially for such kind of thick potato starch granule. The resolution improvement of PS-TDM comes from angular sample scanning (that results in the Fourier synthetic aperture process). However, some qualities of V-Ptychography cannot be disregarded, such as its ability to observe large FOV and to retrieve much more polarimetric information (such as diattenuation), which lacks in PS-TDM.

The effective RI (in Fig. 3.18(a)) shows the amylopectin distribution in the granule, which is arranged in a radial manner. This can be confirmed from the phase-retardance measurement using V-Ptychography, as shown in Fig. 3.18(d). Since the object is thicker, the maximum phase variation between the object and the background medium (Eukitt) is about 11 radians, which gives retardance up to 3 radians. From the diattenuation map in Figure 3.18(e), lower parts of the granule is polluted by an off-focus spot. Furthermore, the fast axis orientation map computed in both PS-TDM in Figure 3.18(c) and that calculated by V-Ptychography in Figure 3.18(f) agrees to each other, wherein the former approach provides detailed information. At last, the 2-D images in Figure 3.18(g) that are taken using the commercial brightfield, PLM at cross polarization as well as PLM with the addition of a phase plate (at COMiX team) can be regarded as reference images. As expected, the bright-field image does not provide any birefringent information of the potato starch granule.

From Polarization Sensitive Holograms to Simplified 3-D Differential Interference Contrast Microscopy

Recording the phase variation of a propagating light-field through a transparent sample is of great interest, as it carries invaluable structural information about the specimen.

For such weakly scattering object, the phase variation carries more information than the amplitude so it allows an intrinsic image contrast mechanism. Proposed by Nomarski in 1955 [19] Differential Interference Contrast (DIC) microscopy (equivalently called Nomarski interference contrast, Nomarski microscopy...) is a powerful and commonly employed tool that reveal the inner structures of transparent samples (which initially have little or no contrast) in detail without exogenous labeling. The images produced using conventional DIC have a pseudo 3D-effect, making the technique ideal for electro-physiology experiments. It employs the implementation of polarization optical elements to retrieve an interference between the light field coming from the object and the laterally displaced (phase-shifted) image in the form of intensity to enhance both spatial and phase variation within the sample [19]. This is however made possible considering specialized optical components that cannot always be fitted to conventional microscope light-paths.

Conversely, interferometric techniques like DHM have proven their capability to extract quantitative phase information of the sample. TDM system extends the DHM's concept for 3D label-free RI distribution imaging of a semi-transparent sample, allowing to extract both light absorption and refraction occurring in the sample. Also, PAS under certain illumination conditions considering an isotropic sample could give rise to two spatially-and phase-shifted images, as necessitated by DIC microscopy.

Experimental Implementation

A schematics diagram of conventional DIC microscopy experimental setup is shown in Fig. 3.19. A polarized beam at 45 is generated by a linear polarizer. The linearly polarized light is split into two beams with perpendicular polarized beams to each other (depicted in red and blue arrows) using Nomarski-modified Wollaston prism. The two parallel and slightly shifted perpendicular beams are then propagate through the sample and pass through the adjacent area of the specimen that are separated by a shear. The beams experience a phase delay from one another, in which the phase shift is directly related to the optical path difference (difference in the sample RI or thickness). Once the phase-shifted light field is collected by the objective, the second prism is used to combine the two beams into single polarised light. If both beams experience the same dephasing, they are recombined by the prism into a linearly polarized one. If one of the beam arrives in advance, an elliptically polarized beam is formed. Finally, the beam passes through the analyzer and the imaging tube lens (eyepiece) to make an image onto the CCD. Note that when both beams pass through the analyser their wavefronts interfere with each other, resulting in a laterally duplicated image (a 3-D looking 2-D image).

Principles and Image Formation

Let us consider a coherent light field incident to an object:

E i = E i exp[ if i (r)], (3.58)
where the optical axis is along z direction, and f i (r) is the phase function. After the beam passes through the Nomarski prism, the generated two beams can be represented as:

E 1 i (r)=E i exp[ if i (r + r s )+∆f] (3.59) E 2 i (r)=E i exp[ if i (r r s )], (3.60)
with ∆f being the constant phase difference and 2r s being the shear (lateral displacement) introduced by the prism.

After passing through the object, we can have the object field E o (r):

E 1 o (r)=E i exp[ if o (r + r s )+∆f], (3.61)
E 2 o (r)=E i exp[ if o (r r s )], (3.62)
f o contains the phase variation due to the specimen. Now we can represent the field in the image space E img , which is the convolution between the imaging system's point spread function P (which describes the propagation from object plane to image plane) and the object fields:

E img (x, y)= ZZ P(x x img , y y img ) h E 1 o (x, y, z = z n ) E 2 o (x, y, z = z n ) i dxdy. (3.63)
Hence, the recorded intensity image is given by: I img (x, y) shows both amplitude and phase variations.

I img (x, y)=|E img (x, y)| 2 . ( 3

3-D Differential Interference Contrast Microscopy

Besides allowing to record polarimetric information about anisotropic sample, PAS acts as a phase shifting device, when implemented for imaging isotropic samples. Hence, we can benefit from the PS-TDM system built based on PAS to mimic a DIC microscope. When illuminating the sample using a circularly polarized light, the images analyzed along the orthogonal directions are in phase-quadrature. For our purpose, the sample is illuminated by a circularly polarized beam and does not exhibit any birefringence or depolarizing structures. In fact, the use of circularly polarized incident light to image an isotropic sample using PAS gives rise to two spatially-and phase-shifted images, as required in DIC microscopy.

If we adjust the the Quarter WavePlate (QWP) fast-axis at 45 with respect to the horizontal axis and the reference beam along with the fast axes, the phase difference between the 2 beams could be p/2. The analyzers for such acquisition are at parallel and perpendicular axes with the QWP. The holograms recorded at 45 and 45 polarization of the micro-analyzer are H 0 (x, y) and H 90 (x, y), respectively. This allows us to link the hologram as:

H 0 (x, y)=H 90 (x + ∆x, y + ∆x)exp[ip/2]. (3.65)
It can be shown that for non-birefringent sample, the information from H 90 (x, y) is shifted by ∆x, ∆y quantity and a phase shift to give the hologram H 0 (x, y). Now if we compute the difference between H 0 (x, y) and H 90 (x, y): H 0 (x, y) H 90 (x, y)=H 90 (x + ∆x, y + ∆y) exp(ip/2) H 90 (x, y).

(3.66)

Considering the above equation, one can modify two parameters: the phase-shift (also known as retardance) and the spatial-shift (also known as bias). When using PAS, both parameters are fixed by our experimental arrangement: spatial-shift is fixed to one pixel in each direction, and the associated phase-shift is of p 2 if we want to use images with orthogonal polarization [START_REF] Nomura | Phase-shifting digital holography with a phase difference between orthogonal polarizations[END_REF].

Schematics of hologram demosaicking is shown in Fig. 3.20. One can observe that, the spatial shift is constrained by the pixelated structure of the PAS, which limits the shearing to a multiplicative factor for the pixel-shift, (n∆x, 0), (n∆x,n ∆y) or (0, n∆y), with n being pixel-shift factor multiplicative.

Chapter 3. POLARIZATION SENSITIVE TDM

Now by rotating the QWP fast axis, one can modify the phase-shift between two neighbouring polarization. For our purpose, the QWP's fast axis is at 45 , since the phase-shift associated with such configuration is accurately known. However, one can use other fast axis orientations and analyze in orthogonal directions if qualitative information about the phase-shift is required only.

Experimental Results

Before investigating the resulting reconstructed images, I determined the slow and fast axes of the QWP. To do so, I have acquired blank holograms at multiple quarter waveplate orientations and by fixing the PAS micro-analyzers. Figure 3.21 shows the measured and computed normalized intensity plots of the transmitted light at vertical analyzer directions. The two minima are measured at 15 and 105 , which show the fast and slow axis of the QWP, respectively. For experimental validation of the proposed method, the single-shot acquired holograms by the PAS, after the non-birefringent sample is illuminated by a circularly polarized beam, are separated into four polarization sensitive holograms using pixel-by-pixel demosaicking. For our purpose, 2 holograms corresponding to horizontal and vertical micro-analyzer directions are needed at each illumination direction to compute the 3-D DIC image. In total, 400 holograms are acquired using 3-D UDHS illumination scheme (chapter 2, section 3.7.1). Once holograms are acquired and demosaicking is done for each hologram, the preprocessing and reconstruction steps based on Rytov approximation are implemented similarly to PS-TDM.

Figure 3.22 shows the reconstructed 3-D RI (real part of the refractive index) slices of a pollen from an olive tree. The insets clearly show the inner structure of the pollen grain. The TDM images retrieved by illuminating with a circularly polarized beam, and then analyzed by 0 polarisation in Fig. 3.22(a), and 90 polarization in Fig. 3.22(b) seem similar qualitatively. Subtracting the two images allows to have a Pseudo-DIC results. One attribute of having DIC images is its good contrast, as shown in Fig. 3.22(c). The structures of the pollen in the DIC image that appear from bright to dark color As explained earlier, the fast axis is at 45 , since the phase-shift associated with such configuration is accurately known. For qualitative information, we can also consider fast axis orientations at another angle. Fig. 3. 22(d-f) shows results for fast axis at 20 .

Chapter 3. POLARIZATION SENSITIVE TDM

One can observe, the shadow effect as the result of the QWP orientation at specific direction.

Conclusions and perspectives

Scalar approximation in TDM has some important features that help to simplify data acquisition and image reconstruction process, and hence, provide 3-D high-resolution quantitative RI maps of a semi-transparent specimen. Such considerations work well for samples exhibiting isotropic property, however, fail to quantify samples with an anisotropic nature. In this chapter, I have shown the advantage of upgrading such scalar TDM system into a polarization sensitive TDM.

Inspired by those mentioned limitations and recent advancements towards polarization sensitive TDM [START_REF] Zhang | Full-polarized tomographic diffraction microscopy achieves a resolution about one-fourth of the wavelength[END_REF][START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF][START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF], as well as understanding of the importance of implementing quantitative 3-D polarimetric imaging, I have built and validated a simplified high-resolution polarization sensitive transmission TDM system. This was done during the late period of my thesis. First, the method is validated using image simulations on a synthetic data. To do so, V-BPM is used as a forward model to generate scattered field from the sample at several angular illumination directions. The reconstruction algorithm based on vectorial 1 st order Rytov approximations is then used for the inversion algorithm. Following image simulations, experiments using a sample that contains mixture of well known birefringent (potato starch granules) and non-birefringent (synthetic silica beads) is performed to validate PS-TDM system. RI tensor components of such samples have clearly identified the limits of the V-Rytov reconstruction algorithm, where it discriminates the isotropic and anisotropic structures of the specimen. After validation, other experiments using spider silk fiber is conducted to show the potential application of quantitative 3-D polarimetric imaging method. Another line of validating the PS-TDM is also done by comparing our PS-TDM system with V-Ptychography technique.

Polarization imaging is an important tool in several fields, its broad applications touches from material analysis to biological and biomedical areas. One potential application of PS-TDM could be for studying the morphology and structure of potato starch to provide insight for its utilization in food industry [START_REF] Ortega-Ojeda | Interactions between amylose, native potato, hydrophobically modified potato and high amylopectin potato starches[END_REF]. As far as industrial and environmental issues are concerned, starch has renewable and biodegradable quality. Native potato starch's application is limited due to the lack of desired characteristics required for industry. But modification through engineering of starch granule (such as its morphology, amylose to amylopectin ratio, and lipid content) allows it to have wider range of applications either in food and non-food industries [START_REF] Ortega-Ojeda | Interactions between amylose, native potato, hydrophobically modified potato and high amylopectin potato starches[END_REF].

Besides, neurodegenerative disorders including Alzheimer's disease and prion diseases are know by abnormal conformational transformation of proteins to amyloid fibrils [START_REF] Jin | Imaging linear birefringence and dichroism in cerebral amyloid pathologies[END_REF]. Although the structural bases of fibrillogenesis in such kind of diseases are not yet determined, identification of amyloid presence in diseased tissues, structural characterization as well as categorization of their various forms helps to facilitate amyloid disorders diagnosis [START_REF] Jin | Imaging linear birefringence and dichroism in cerebral amyloid pathologies[END_REF]. The relevance of quantitative polarization imaging techniques (for example, to classify and quantify samples to assess tumour development by using the optical birefringence of collagen fiber) allows to reveal quantitatively collagen structures and organizations [START_REF] Aknoun | Quantitative retardance imaging of biological samples using quadriwave lateral shearing interferometry[END_REF].

Recent challenges in quantitative Jones matrix holography techniques lack phase sensitivity, which comes from the diffraction noise as the result of unwanted interference when implementing interferometric imaging systems using coherent light source. In this thesis, the development of a simplified semi-automated PS-TDM system reduces such problem to make it advantageous for biological samples exhibiting relatively low birefringence. Full automation of the PS-TDM system to make it suitable for medical applications, as it is done for cancer study [START_REF] Park | Jones Matrix Microscopy for Living Eukaryotic Cells[END_REF] in 2-D, could be another line of work. Extending the reconstruction algorithm based on non-approximated assumption to include the 3 rd anisotropic parameter could be done for the purpose of material analysis [START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF]. In the future, computation of multi-wavelength 3-D birefringence map (as used in 2-D [START_REF] Honda | Simple multi-wavelength imaging of birefringence: case study of silk[END_REF]) to get polarimetric information as a function of wavelength to have insight about absorbing structures (for spectroscopic analysis) of anisotropic sample could be considered.

Besides polarimetric imaging, it has also been demonstrated that the already developed PS-TDM system based on a PAS leads to a DIC microscope when used to image non-birefringent samples. After understanding the theoretical background, experimental validations considering a non-birefringent sample (pollen grain from an olive tree) was performed.

One advantage of a polarization sensitive camera is that it can be incorporated into an existing bright-field microscope system, leading to a cost effective simplified DIC technique. One limitation of the conventional DIC microscope is that it does not allow to image samples that are birefringent, since the information retrieved at orthogonal polarization could create undesired interference with the background as well as each other. Our PS-TDM allows to check if the sample is birefringent while implementing a 3-D DIC microscope, which lacks by conventional systems. Moreover, it provides polarimetric information of anisotropic samples leading to 3-D quantitative PS-TDM.

This work has led to two publications, in Journal of Microscopy; N. Verrier, A.M. Taddese, R. Abbessi, M. Debailleul and O. Haeberlé, "3D Differential Interference Contrast microscopy using polarisation sensitive Tomographic Diffraction Microscopy"(2022)1 , and in Optics Express: A.M. Taddese, M. Lo, N. Verrier, M. Debailleul and O. Haeberlé, "Jones Tomographic Diffractive Microscopy with a Polarized Array Sensor"(2023)2 . Also, two conference presentation were provided; "OPTIQUE Nice 2022"3 and Focus on microscopy 2021 4 .

CONCLUSION AND PERSPECTIVES

T omographic Diffractive Microscopy (TDM) is now an established label-free quan- titative imaging technique. The basic principle is to illuminate the sample with a coherent laser beam from various angular directions, and to collect the scattered field's amplitude and phase, holographically. Numerical inversion models that characterize the sample-light interaction are then used to reconstruct and quantify its 3-D refractive index.

The numerical reconstruction process in TDM uses Born and Rytov approximation. TDM has several configurations including transmission, reflection and 4Pi TDM as well as combined approaches for resolution improvement. One of the configuration, illumination rotation TDM, is commonly preferred and implemented for biological imaging, due to its stability. Chapter 2 focused on understanding such system and its Fourier space synthetic aperture process. At first, the influence of limiting the number of 2-D holograms acquired from several angular illumination directions is studied.

In practice, depending on the required image quality, one can record tens [START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF] to hundreds [12,[START_REF] Sung | Optical diffraction tomography for high resolution live cell imaging[END_REF][START_REF] Simon | Tomographic diffractive microscopy of transparent samples[END_REF] of holograms. As the holograms are acquired sequentially, recording fewer interferograms could lead to faster imaging, but, results in a degraded reconstructed image quality, mainly when using Born and Rytov approximations. Also, for high quality images with limited number of holograms, advanced reconstruction algorithms can be performed, which are usually slower. In general, to achieve highresolution imaging, the Fourier space has to be filled adequately. While increasing the number of illumination angles could be the straightforward approach to solve such issue, doing so would slow the speed of TDM acquisition/reconstruction/display. Overall, we have seen that a trade-off between image quality and acquisition/reconstruction speed has been the main concerns in the field.

This thesis sought to study the angular sample scanning scheme, so that enough data is acquired from the sample for high-resolution imaging with limited number of holograms. To do so, 8 main classes of scanning patterns (star, annular, circular grid, rose, spiral, Fermat's spiral, 3-D UDCS and 3-D UDHS) were considered for transmission, reflection and 4Pi TDM configurations. As the aim was to optimize the Fourier space filling in the low-diffraction regime, numerical study was performed to understand the corresponding OTFs of the above-mentioned schemes. I found that 3-D UDHS illumination pattern best fills the Fourier space for all TDM configurations. For transmission TDM, these simulations have been confirmed with image simulations using absorbing beads as well as experiments using glass beads, pollen grains with complex structures and fixed C2C12 myoblast cells. These were made by considering a low number of holograms (60 holograms), and the linear inversion algorithms, as it may lead to real-time 3-D acquisition/reconstruction /display system [START_REF] Bailleul | Tomographic diffractive microscopy: towards high-resolution 3-D real-time data acquisition, image reconstruction and display of unlabeled samples[END_REF].

Image simulations of a weakly absorbing phantom considering a reflection TDM allows to reconstruct only abrupt RI changes or sharp corners. Since the OTF is asymmetric along the optical axis, intermixing of real and imaginary parts of the complex RI was observed. Simulation of the OTF for 4Pi TDM configuration by combining the transmission and reflection TDM, as well as image simulations using more complex phantoms was performed. As this approach has the advantages of both reflection plus transmission TDM configurations, the image reconstruction quality is improved, 3-D UDHS scheme giving better RI estimation. Also, 3-D UDHS being an optimal scanning for 4Pi TDM configurations helps to simplify and speed up image acquisition in an actual setup. At last, practical experimental hurdles and possible solutions to implement 4Pi TDM system were also discussed as a perspective.

As a remark, the OTF and image simulations as well as experimental results showed that understanding and selecting a favorable sample scanning pattern in TDM experiments can greatly contribute to image acquisition speed by reducing the number of illumination angles needed to achieve required image quality. Future works could consider experimentally validating the optimized scheme for reflection TDM as well as implementing a 4Pi TDM either using the proposed setup in chapter 2 or a mirror-assisted TDM [START_REF] Mudry | Mirror-assisted tomographic diffractive microscopy with isotropic resolution[END_REF].

The optimization can complement hardware optimizations, in view of improving performances of TDM system and minimizing image reconstruction errors [START_REF] Kuś | Illumination-related errors in limited-angle optical diffraction tomography[END_REF]. Furthermore, it can be used to optimize LEDs distribution for Fourier ptychography tomography [START_REF] Horstmeyer | Diffraction tomography with Fourier ptychography[END_REF][START_REF] Zuo | Wide-field high-resolution 3d microscopy with fourier ptychographic diffraction tomography[END_REF][START_REF] Sen | Fourier ptychographic microscopy using an infrared-emitting hemispherical digital condenser[END_REF][START_REF] Pan | Subwavelength resolution Fourier ptychography with hemispherical digital condensers[END_REF]. Application of such optimization could even go beyond TDM, especially for some techniques that require sample scanning. Also, Fermat's spiral scheme could be used in one of super-resolution optical microscopy technique, 3D structured illumination microscopy. I believe that the currently performed random illumination microscopy could be replaced by Fermat's spiral.

In chapter 3, it was discussed that TDM based on scalar electromagnetic field assumption has several advantages. Besides reducing experimental complexities, such scalar approximation leads to a simplified link between the measured diffracted fields and the sample scattering potential. Thanks to its simplified data acquisition and image reconstruction process, it is preferred and currently being used by several groups and even commercially available. With such scalar inversion theory, it provides 3-D highresolution quantitative RI maps of a sample exhibiting isotropic property. However, it fails to quantify samples with anisotropic nature.

In fact, imaging of samples with anisotropic structures necessitates recording and analyzing the polarization properties of light. The main aim of chapter 3 was to upgrade the scalar TDM into a simplified high-resolution polarization sensitive TDM for 3-D quantitative polarimetric imaging. To do so, the commonly used polarization analyzing components (linear polarizer and a CCD camera), were replaced by a polarization sensitive camera. Thanks to the micro-analyzers, such camera allows to measure mosaicked holograms that leads to a semi-automated PS-TDM system and reduces the acquisition time by a factor of 2.

Furthermore, previous works in the field used a linearly polarized light to illuminate the sample at horizontal and vertical polarization. Since zero background intensity is generated by two cross-polarized beams, leading to background phase noise [START_REF] Zhang | Far-field diffraction microscopy at l/10 resolution[END_REF][START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF], previous works in the field record the corresponding scattered field along an intermediate polarizations (±45 ). However, such approach needs to generate illumination beam at exactly ±45 polarization, otherwise requiring a normalization procedure. Also, a link between the intermediate polarization states and the vertical/horizontal polarizations is required to compute the field along these directions. To alleviate such computational and normalization issue, I have proposed and used a left and right circularly polarized beam for illuminating the sample.

The proposed method was first validated using image simulations on a synthetic data. For that, the synthetic anisotropic object is divided into small slices. Then by using the vectorial incident beam at x and y polarizations, the diffracted light-field vectors at several angular illumination directions were computed through vectorial Beam Propagation Method (V-BPM) as a forward model. The inversion algorithm based on V-Rytov approximations was then performed to reconstruct the RI tensor components of the synthetic birefringent object. The simulation had been useful to validate the approximated vectorial linear inversion algorithm. Some discrepancy between the reconstructed and original RI values were observed, coming from the limited efficiency of the V-BPM forward-model.

Once the inversion algorithm was validated with the image simulation, experiments were conducted using the simplified PS-TDM system. For that a sample containing both birefringent (potato starch granules) and isotropic objects (synthetic silica beads) was prepared and used to validate the PS-TDM system. RI tensor components of such samples has clearly identified the advantages of using V-Rytov reconstruction algorithm, where it discriminates the isotropic and anisotropic structures of the specimen. The experimental validations were followed by imaging other anisotropic samples, including a spider silk fiber and oyster shells. This showed the potential application of quantitative 3-D polarimetric imaging approach for analyzing naturally abundant anisotropic materials. Moreover, another line of validation was also the comparison of our PS-TDM system with already implemented and calibrated polarimetric system, 2-D vectorialptychography technique [START_REF] Ferrand | Ptychography in anisotropic media[END_REF], in collaboration with Institut Fresnel's COMiX team.

In general, polarization imaging is a useful tool in several fields, from microscopic to macroscopic scales, applied in material analysis, clinical application and biological study. For example, our PS-TDM system could be useful to study the morphology of potato starch to provide an insight about its use in food industry to genetically modify and create a potato starch with a desired characteristics for food and non-food industries. Medically, understanding abnormal conformational transformation of proteins to amyloid fibrils could help to facilitate neurodegenerative disorders diagnosis [START_REF] Jin | Imaging linear birefringence and dichroism in cerebral amyloid pathologies[END_REF].

Besides, quantitative polarization imaging could also help to quantify structure and organizations of collagen fiber using its optical birefringence, which could be useful to assess tumour development [START_REF] Aknoun | Quantitative retardance imaging of biological samples using quadriwave lateral shearing interferometry[END_REF]. However, quantitative Jones matrix approaches developed using coherent light source are lacks phase sensitivity due to diffraction noise as the result of unwanted interference. In this thesis, I believe that development of a simplified semi-automated PS-TDM system will help to reduce such problem, making it advantageous for biological samples exhibiting relatively low birefringence. Full automation of the PS-TDM system to make it suitable for medical applications, such as for cancer study [START_REF] Park | Jones Matrix Microscopy for Living Eukaryotic Cells[END_REF] could be considered in the future.

For instance, one could use motorized rotation stages and DC motor controllers to control the quarter waveplate so that it can easily switch between left and right circular polarizations. This necessitates to modify the acquisition, so that polarization sensitive holograms corresponding to left and right circular polarizations are automatically saved in two different folders. Another alternative approach to achieve automated PS-TDM system could be to replace the quarter waveplates with liquid crystal retarders. However, the liquid crystal regulator required to control the liquid crystal retarders could be costly and not recommended if one aims at implementing a low budget PS-TDM system.

Our linear inversion algorithm (V-Rytov and Vectorial Born approximation) used in this work considers two assumptions for simplified mathematical expression. First, it ignores polarization coupling between the x and y polarization states. Second, the sample under consideration was assumed to have a weak polarization response along the optical axis. This allows to estimate only the two principal refractive indexes of the sample. In the future, extending the reconstruction algorithm based on non-approximated assumption to include the third principal RI could be done [START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF].

The perspectives of such extension could be seen in two ways. The first approach could be by implementing a tomographic ptychography approach with vectorial illumination and detection. This could be done by discussing and extending the Vptychography system [START_REF] Ferrand | Ptychography in anisotropic media[END_REF]. However, computation time due to the addition of the third dimension and measurement hurdles because of the need for additional mechanism to control the angles of illumination beam could be challenging.

The second approach includes some indirect ways of measuring the polarization effect along the optical axis. For instance, a polarization component referred to as a stressed-engineered optic that consists in spatially varying birefringence pattern to encode polarization information even along the optical axis [START_REF] Curcio | Birefringent Fourier filtering for single molecule coordinate and height super-resolution imaging with dithering and orientation[END_REF] could be considered. However, up to now it is only used for polarization fluorescence imaging, hence TDM implementation could be challenging. Another similar alternative is by generating and detecting an arbitrary 3-D polarized beam [START_REF] Li | Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam[END_REF]. The 3-D polarization controlling is made by superposing both radially polarized beam and an azimuthally polarized beam with weighting factors, as well as by modulating the vectorial beam's amplitude using an apodizer [START_REF] Li | Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam[END_REF].

The presented PS-TDM is based on Jones formalism, where a purely polarized beam is considered. However, limitation of Jones formalism is that it fails to measure depolarization of light by a specimen [START_REF] Park | Jones Matrix Microscopy for Living Eukaryotic Cells[END_REF]. For that, one need to perform a Mueller calculus. The Mueller Matrix M of a sample transforms the incidence stokes vector S in to an output stokes vector S out . Mathematically, this can be expressed as: The Mueller matrix allows to gain information about the diattenuation, retardance and degree of polarization. It can be retrieved through computation of the Stokes parameters at several directions. At least 16 independent measurements have to be made to compute the total Mueller matrix. The use of PAS will make the implementation easier as it allows a single shot measurement of the stokes components. The practical requirement is to add a quarter waveplate at the analyzing arm and perform several measurements. The computation however could take more space, since we need to do the conventional TDM data acquisition at least 16 times for full Mueller matrix calculation. One solution could be to reduce the number of holograms.

S out = MS in . ( 4
In the future, computation of multi-wavelength 3-D birefringence map (as used in 2-D [START_REF] Honda | Simple multi-wavelength imaging of birefringence: case study of silk[END_REF]) to get polarimetric information as a function of wavelength to have insight about absorbing structures of anisotropic sample could be considered. This is possible by using hyperspectral laser [START_REF] Zhang | Multi-wavelength multi-angle reflection tomography[END_REF][START_REF] Sung | Spectroscopic microtomography in the visible wavelength range[END_REF] similarly to the combined multiwavelengths and illumination rotation TDM.

Besides polarimetric imaging, it was also discussed that the already developed PS-TDM system based on a PAS leads to a DIC microscope when used to image nonbirefringent samples. This was supported with experimental validations considering a non-birefringent sample (pollen grain from an olive tree). Another important feature of polarization imaging is that it allows to distinguish multiply scattered photons from single scattered ones. This can be useful in several application, especially if we extend the PS-TDM for large field of view. Also, when it is used for non-birefringent samples, polarization sensitive TDM along with advanced reconstruction approaches could lead to super-resolution imaging [START_REF] Zhang | Full-polarized tomographic diffraction microscopy achieves a resolution about one-fourth of the wavelength[END_REF]. Combining such TDM approach with fluorescent based imaging techniques [2] could allow to break the Abbe resolution limit [3].

1. Fourier transform of an estimated object g n = n 0 + ∆n(r)+in 0 (r). Here the momentum conservation (as a result of Born/Rytov approximations) is used to get initial estimate of the 2 µm phantom.

2. The modulus of the computed object spectrum G n is replaced by the measured Fourier modulus to obtain an estimate of the Fourier transform G 0 In Chapter 3, the potential application of Beam Propagation Method (BPM) was briefly described. In this section, mathematical derivation of such model is discussed, which can be used as a forward model in the image simulation. It is usually favored for techniques that necessitate simulating optical field propagation in a material with multiple scattering layers. Next, both the scalar and vectorial BPM formulation will be derived by adapting prior works in the field [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF][START_REF] Ulugbek | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF][START_REF] Feit | Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams[END_REF].

B.I Scalar Beam Propagation Method

Let us recall the scalar Helmholtz equation:

(4 2 + k 2 )E t (r)=0, (B.1)
where the Laplacian 4 = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 . In the following, the differential equation can be solved using non-paraxial approximation of the ligh-field, as done by Feit and Flack [START_REF] Feit | Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams[END_REF] and implemented by [START_REF] Ulugbek | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF].

∂ 2 E t (r) ∂x 2 + ∂ 2 E t (r) ∂y 2 + ∂ 2 E t (r) ∂z 2 + k 2 E t (r)=0 (B.2)
Here, we can consider a complex envelope paraxial field e(r) within the wavelength distance: At this point, we can introduce a slowly varying envelop approximation, which implies that the the envelope does not vary much with z over a wavelength distance. [START_REF] Ulugbek | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF]. It tells how the optical field propagates along z direction.

E t (
| ∂ 2 e(
The aim here is to propagate the optical field slice by slice. If we consider finite but small z step, i.e. z = dz, we are able to write Equation (B.5) as a complex exponential equation [START_REF] Ulugbek | Optical tomographic image reconstruction based on beam propagation and sparse regularization[END_REF][START_REF] Gilmore | Baker-Campbell-Hausdorff formulas[END_REF]: e(x, y, z + dz)=e(x, y, z) exp(Ddz) exp(Rdz), (B.6) where D = i 2k 0 4 x,y accounts the diffraction operator and R = ik 0 rn/n 0 is the refraction term. This allows to separately model the diffraction and refraction phenomena. The diffraction term can be expressed using the modified shifted angular spectrum method [START_REF] Ritter | Modified shifted angular spectrum method for numerical propagation at reduced spatial sampling rates[END_REF]. We have: where k v is the modulus of the wavevector in vacuum. The refined and more general scalar BPM expression to simulate BPM at wide angle can be found [START_REF] Shengli | Iterative optical diffraction tomography for illumination scanning configuration[END_REF][START_REF] Feit | Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams[END_REF].

e(x, y, z + dz)=F 1

B.II Vectorial Beam Propagation Method

Let us recall the vectorial Helmholtz equation:

(r 2 + k(r) 2 )E t (r)=0.

(B.10)

One way to simplify such expression is by expressing the total field using slowly varying envelope, e(r), i.e. E t (r)=e(r) exp(ik o z). Similarly to the scalar case, we can introduce a slowly varying electromagnetic amplitude (SVEA) approximation, | ∂ where dn 2 ⇡ 0 for weak birefringence variation. The aim here is also to propagate the optical field slice by slice. If we consider finite but small z step, i.e. z = dz, we are able to decompose the above equation as a complex exponential equation [START_REF] Gilmore | Baker-Campbell-Hausdorff formulas[END_REF]:

e(x, y, z + dz)=e(x, y, z) exp(Ddz) exp(Rdz), (B.12)

where D = i 2k 0 r 2 x,y accounts the diffraction operator and R = 2k 2 0 dn is the refraction term. This allows to separately model the diffraction and refraction. The diffraction B.III. Comparing BPM with Mie Calculation 135 term can be expressed by using the modified shifted angular spectrum method [START_REF] Ritter | Modified shifted angular spectrum method for numerical propagation at reduced spatial sampling rates[END_REF]. This expression allows to propagate the incident field through a known 3-D sample. As a forward model, such expression allows to compute the scattered field vector from the incident field vector. a is the oblique angle. As inversion algorithm, it could also be implemented using an iterative technique or a deep-learning approach [START_REF] Ulugbek | Learning approach to optical tomography[END_REF].

B.III Comparing BPM with Mie Calculation

Mie calculation provides an analytical solution to the Maxwell's equation. It allows to compute light scattering by a dielectric sphere, by expanding the incident plane wave and the scattering field into a radiating spherical harmonics as well as by using boundary condition. It allows to compute the scattered field coefficients by solving the spherical Bessel functions of order n. For that, it uses input parameters, such as: the RI of the sphere, radius of the sphere, modulus of light field's wave vector and wavelength in the surrounding medium. The Mie calculation model is similar to what is used in [START_REF] Denneulin | GSURE criterion for unsupervised regularized reconstruction in tomographic diffractive microscopy[END_REF]. As the results show, the computed phase by BPM is lower than the Mie calculation result. For the amplitude, only the edges of the synthetic bead is observed in the difference image.

C C2C12 Cells preparation

The cells are acquired from the 'Institut de Science des Matériaux de Mulhouse (IS2M)' laboratory, thanks to the collaboration created between our group and IS2M's Biomaterials-Biointerfaces research group. They are working on several areas, including understanding biomolecular-surface interaction mechanisms that occur in biology as well as development of new materials or processes mainly for biological purposes. The samples are made after discussing the sample requirements of our imaging system. Procedure For proliferation purpose, cells are grown about 50-70% confluency. Afterwards, the cells are split in 4-well plates (surface 1.9 cm 2 ) with glass cover-slips • with the density of 5000 cells/cm 2 or 10000 cells/well • Cells are incubated at 37 and 5% CO 2 ;

• The cells are fixed with 4% PFA and incubated for 15 minutes at room temperature;

• Then they are washed with PBS before being fixed with the Eukitt (background medium).

Finally we let the sample dry by overnight to make sure that it is well fixed. Once the preparation is finished, we prepared them between two thin glass-slides and bring it to our lab for observation. Results are provided in Fig. 2.19 and Fig. 2.33.

Introduction

Depuis son invention au 16 ième siècle, le microscope optique a été un outil crucial pour la recherche scientifique fondamentale et dans de nombreux autres domaines appliqués [1]. Il a révolutionné la science en permettant aux êtres humains de découvrir le monde invisible où les agents pathogènes deviennent visibles, étudiés et évités. Avec cette découverte et les progrès réalisés dans les domaines des sources lumineuses, de la sensibilité des détecteurs, de la correction des aberrations optiques, des mécanismes de restauration des images, de la conception des objectifs et de l'amélioration des capteurs, nous avons aujourd'hui le luxe de pouvoir choisir parmi un vaste assortiment de méthodes d'imagerie pour visualiser diverses petites structures (telles que des cellules individuelles et leurs compartiments intracellulaires, des protéines du cytosquelette ou des membranes) à l'échelle microscopique ou nanoscopique. Ses applications ne se limitent pas à la biologie, mais s'étendent à la chimie, à l'électronique, la géologie et aux sciences des matériaux.

Les échantillons biologiques en général, et en particulier les cellules et leurs structures internes, sont transparents et ne présentent donc pas un contraste suffisant pour permettre une étude adéquate. L'approche la plus courante pour améliorer le contraste consiste à utiliser des colorants spécifiques pour marquer ces structures. Cela nécessite de colorer l'échantillon avec des marqueurs fluorescentes qui peuvent s'attacher à une structure spécifique d'intérêt, ce qui conduit aux techniques de microscopie basées sur la fluorescence [2]. L'invention de la microscopie de fluorescence a propulsé la recherche biomédicale vers de nouveaux sommets, en permettant aux chercheurs de dépasser la limite de diffraction de la lumière pour l'imagerie optique, fixée par Abbe en 1873 [3], et permettant une imagerie de super-résolution. Diverses techniques de microscopie de fluorescence à super-résolution sont actuellement disponibles, notamment la microscopie à déplétion par émission stimulée (STimulated Emission Depletetion STED), les microscopies pointillistes, STOchastic Reconstruction Microscopy STORM), ou Photo Activated Localization Micrscopy (PALM) [2] et bien d'autres encore.

Cependant, la microscopie de fluorescence présente certaines limites. Tout d'abord, le marquage peut induire des artefacts, c'est-à-dire des structures ajoutées lors de la préparation de l'échantillon, qui ne constituent pas la nature inhérente du spécimen. Même avec une préparation minutieuse, ces approches qui permettent de visualiser des structures spécifiques d'un objet pourraient ne pas permettre d'obtenir une image de l'ensemble des structures de l'échantillon. Deuxièmement, le marquage peut donner un aperçu erroné en créant des changements indésirables sur l'échantillon, à la suite de phénomènes de photoblanchiment et de phototoxicité. Par conséquent, pour des études à long terme telles que celles sur la division cellulaire, qui peuvent prendre des heures voire des jours, les fluorophores peuvent disparaître avec le temps, ce qui rend l'imagerie difficile.

Pour de telles applications, en particulier dans la caractérisation des matériaux et lorsque la connaissance de l'échantillon est limitée, l'utilisation de l'imagerie sans marquage est très importante. C'est pourquoi les techniques sans marquage présentent un grand intérêt. Elles comprennent plusieurs techniques de microscopie à transmission, la microscopie à génération de seconde harmonique (Second Harmonic Generation SHG) et la spectroscopie Raman anti-Stokes cohérente (Coherent Anti-Stokes Raman Spectrocopy CARS).

En microscopie conventionnelle à champ large (champ clair, éclairage oblique, champ sombre, coloration par dispersion optique, contraste de phase, contraste interférentiel différentiel...), le contraste de l'image est produit par la diffraction de la lumière par les structures de l'échantillon. Le faible contraste obtenu limite son application à l'imagerie biologique. L'un des moyens d'obtenir une image mieux contrastée consiste à améliorer le système à champ clair avec des accessoires, afin de détecter et d'interpréter le déphasage de la lumière causé par les variations de densité des structures de l'échantillon (liées aux variations de l'indice de réfraction) dans le contraste de l'image.

Bien que ces procédures d'amélioration du contraste aient été identifiées et utilisées, l'indice de réfraction du matériau ne peut pas être déterminé quantitativement à partir uniquement de l'intensité lumineuse observée. En fait, pour obtenir une imagerie quantitative, il faut enregistrer à la fois les variations d'amplitude et de phase. Avec le développement de l'holographie numérique (couronnée par un prix Nobel de physique en 1971) et les progrès de l'électronique et de l'informatique, il est aujourd'hui possible de mesurer quantitativement la distribution de l'absorption et de l'indice de réfraction d'un objet.

La technique de microscopie tomographique diffractive MTD (Tomographic Diffractive Microscopy TDM) (également appelé tomographie holographique numérique, microscopie de phase tomographique, tomographie par diffraction optique) est une technique d'imagerie quantitative 3D à haute résolution, qui a été proposée pour la première fois par E. Wolf en 1969 [4], puis améliorée dans les années 80 par A. J. Devaney [5], afin d'obtenir des images d'échantillons semi-transparents. Il s'agit d'une technique d'imagerie 3D sans marquage, actuellement développée par divers groupes [6,7,8,9,10,11] et même disponible dans le commerce, mais dont la disponibilté est encore faible par rapport aux techniques de fluorescence, et pour laquelle il reste encore de nombreux développements à faire.

Ce résumé contient une brève description des principales réalisations de ma thèse de doctorat. Il se concentre en deux parties. Dans la première partie, l'optimisation des schémas d'illumination des échantillons en microscopie tomographique diffractive pour améliorer les acquisitions et les reconstructions des données sera discutée. La deuxième partie est consacrée au développement d'un système simplifié d'imagerie à haute résolution sensible à la polarisation pour quantifier la biréfringence d'objets microscopiques anisotropes.

Principes de base de l'acquisition des données et des reconstructions en MTD

En MTD, la carte 3D des indices optiques d'un objet est calculée par la recombinaison numérique d'hologrammes 2D acquis sous plusieurs angles d'illumination (nécessitants de quelques dizaines [12] à quelques centaines [13,6,14] d'angles). L'algorithme de reconstruction du MTD utilise l'approximation de Born ou de Rytov d'ordre 1 st pour des échantillons faiblement diffusants (voir par exemple : [4,10,15,16,17,18,19,5]). Par conséquent, compte tenu de la conservation de la quantité de mouvement, le support des vecteurs d'onde de l'objet k o recueillis par l'objectif est une calotte de sphère translatée, les vecteurs d'onde diffractés k d , par le vecteur d'onde incident k i : On peut également calculer l'OTF d'un MTD 4Pi (dans la configuration proposée par Lauer [20]), en combinant les fréquences mesurées en réflexion et en transmission, obtenues à l'aide d'une configuration à deux objectifs. Le processus de recombinaison généralisée est illustré dans les figures 1(e-g). On obtient deux OTF en réflexion à la suite de l'éclairage du spécimen par un objectif et de la détection à l'aide de ce même objectif. Pour la transmission, deux OTF sont également collectées en utilisant un objectif pour l'illumination et l'autre pour la détection. L'OTF finale est alors obtenue en recombinant ces quatre informations.

k o = k d k i . (1

Figures pour l'illumination angulaire

Compte tenu de leur applicabilité, en vue de l'optimisation du remplissage de l'espace de Fourier dans le régime de faible diffraction, j'ai mis en oeuvre plusieurs classes de modèles de balayage d'échantillons. Ils sont représentés sur la Fig. 2.

Les différents modèles d'illumination sont générés à l'aide des équations qui les régissent ou éventuellement en rééchantillonnant soigneusement les distributions de points. Les paramètres susceptibles d'influer sur le remplissage de l'espace de Fourier sont également étudiés pour chaque motif. Le nombre d'axes pour l'illumination en étoile, le nombre de cercles concentriques pour l'illumination annulaire, le nombre de tours pour l'illumination en spirale et le nombre de pétales pour l'illumination en forme de rose sont les paramètres considérés.

Reconstruction des OTF et analyse

Les fonctions de transfert optique (OTF) correspondant à ces balayages pour un TDM 4Pi (résultant de la combinaison et de la symétrisation de la transmission et de la réflexion) sont représentées dans les Figs. 3. On peut constater la différence dans les distributions de fréquences résultant de chaque schéma de balayage : certains schémas de balayage montrent une accumulation de composantes à basse fréquence (Figs. 3(d,e)), tandis que d'autres introduisent une périodicité (Figs. 3(c,d,h,i)). En fonction des conditions expérimentales et de la nature de l'échantillon, il est possible de privilégier un modèle d'illumination présentant une propriété spécifique.

Pour comparer l'efficacité de ces différents modèles d'illumination, je quantifie le "facteur de remplissage" de l'OTF FF(%) dans l'espace de Fourier. Il est défini comme le rapport entre le volume (en nombre de pixels) qui est effectivement rempli par la calotte de sphère en utilisant le balayage V s et le volume théorique de l'OTF V T : À l'IRIMAS et dans plusieurs groupes de recherche travaillant sur cette technique, l'illumination et la détection ont été jusqu'à présent considérées comme des champs électromagnétiques scalaires, ce qui a conduit à des approximations scalaires. Il s'agit d'une approche de reconstruction simplifiée et relativement facile, et valide pour les objets isotropes et homogènes à faible diffusion. En général, la lumière en tant que champ électromagnétique devrait être traitée comme un champ vectoriel, ce qui a de nombreuses applications en biologie [22]. Pour cela, le système TDM actuel doit être amélioré pour permettre une imagerie polarimétrique quantitative tridimensionnelle.

FF(%)= V s V T ⇥ 100 . (2

Montage expérimental

Tenant compte de certaines limites du système de MTD scalaire, et inspiré par les progrès récents en matière de MTD sensible à la polarisation [23,24,25], j'ai construit un système de MTD simplifié en transmission, sensible à la polarisation et à haute résolution, à partir de composants optiques disponibles commercialement. Ce système est basé sur un interféromètre de Mach-Zehnder, illustré

Résultats expérimentaux

La méthode est d'abord validée à l'aide de simulations, puis d'expériences utilisant plusieurs échantillons. Pour la reconstruction d'image, l'algorithme utilisé jusqu'ici à l'IRIMAS a été modifié pour calculer les composantes du tenseur de l'indice de réfraction à partir de la matrice de Jones prétraitée de l'échantillon, en utilisant l'approximation de Rytov vectorielle à l'ordre 1 st . Pour valider notre système TDM à haute résolution sensible à la polarisation, j'ai préparé un échantillon composé de granules d'amidon de pomme de terre, qui est un micro-objet biréfringent bien connu [26], et de billes de silice isotropes. Cet échantillon est placé entre deux lamelles de microscope d'épaisseur 170 µm, et fixé dans un milieu isotrope (Eukit de Sigma Aldrich) avec un indice de réfraction n 0 = 1, 49.

La figure 8(a) montre des coupes x y (première ligne) et x z (deuxième ligne) des composantes du tenseur RI reconstruites pour les granules d'amidon de pomme de terre (objet anisotrope) et les billes de silice de 5 µm (objet isotrope) dans le même champ de vision (FOV). Comme on peut le remarquer, les composantes du tenseur RI n xx et n yy ne permettent pas de distinguer les objets isotropes des objets anisotropes. Elles indiquent les composantes ordinaires ou extraordinaires de l'indice de réfraction. Inversement, n xy et n yx ne montrent que les granules de fécule de pomme de terre, à l'exception de quelques franges résiduelles pour les billes, provenant d'artefacts de diffraction sur les bords de celles-ci. D'autres informations polarimétriques sur l'échantillon peuvent également être calculées, notamment l'IR moyen (

Conclusion

Dans la première partie de mon travail de doctorat, j'ai montré la nécessité d'optimiser l'éclairage des échantillons dans les expériences de microscopie tomographique difffractive. Plusieurs familles de modèles d'illumination ont été étudiées et comparées à l'aide de différents critères d'évaluation. Le calcul du facteur de remplissage, des simulations d'images et une mise en oeuvre expérimentales ont montré que le balayage angulaire uniforme 3D donne les meilleurs résultats, améliorant la résolution transversale et longitudinale, ainsi que le contraste des images.

Inspiré par certaines limitations des approches scalaire en MTD et par les récentes avancées en MTD sensible à la polarisation [23,24,25], j'ai aussi construit et validé un système de MTD simplifié en transmission à haute résolution, sensible à la polarisation, en améliorant le prototype déjà utilisé à l'IRIMAS. Cela a été fait pendant la dernière période de ma thèse (après la période de blocage de Covid19). La méthode a té validée en utilisant des simulations d'images sur des données synthétiques, suivies d'expériences utilisant plusieurs échantillons biréfringents.

Parmi les perspectives prometteuses de ce travail, on peut mentionner :

• l'extension des modèles de reconstruction aux cas des spécimens fortement diffractants ;

• l'extension de l'analyse à la polarimétrie de Mueller, plus riche que celle de Jones ;

• le développement d'une MTD multispectrale, qui pourra apporter une sélectivité chimique. Enfin, ce travail a donné lieu à plusieurs articles et communications dans des conférences internationales et française, données ci-après.

xvii 2 .

 2 31 Image reconstruction of a synthetic object using annular illumination scheme considering Hermitian symmetry ........................ 2.32 Reconstructed C2C12 cells acquired using annular illumination at maximum NA without OTF symmetrization .................... 2.33 Reconstructed C2C12 cells acquired using annular illumination at maximum NA after OTF symmetrization ...................... 2.34 2-D k y -k z representations of OTFs for reflection TDM using 600 illumination angles. ..................................... 2.35 OTF filling factor plots for reflection TDM. .................. 2.36 Normalized OTF filling factor plots by 3D-UDHS for reflection TDM. ... 2.37 OTF mean filling factor FF(%) plots for reflection with maximum variation due to the parameters in the illumination scheme. ........... 2.38 Image reconstruction of a synthetic object using reflection TDM, using star, Fermat and 3-D UDHS scanning patterns ................ 2.39 OTF representation to make 4Pi TDM ..................... 2.40 2-D and 3-D of representation of simulated OTFs for combined transmission and reflection TDM using 600 illumination angles. ........... 2.41 2-D k y -k z representation of simulated OTFs for 4pi TDM using 600 illumination angles. .................................. 2.42 OTF filling factor plots for 4Pi TDM. ...................... 2.43 Normalized OTF filling factor FF(%) plots by 3D-UDHS for 4Pi TDM. .. 2.44 Image reconstruction results of a Shepp-Logan phantom using 600 angular illuminations: transmission, reflection and 4Pi TDM ........... 2.45 Image reconstruction results of a Shepp-Logan phantom using 4Pi TDM: considering star, Fermat's and 3-D UDHS scanning schemes ........ 2.46 Image reconstruction of an absorbing phantom using 4Pi TDM: star, Fermat's and 3-D UDHS scanning schemes and corresponding RI profile plots 2.47 Schematic illustration of proposed 4Pi TDM setup .............. 3.1 Potato starch granules imaged by bright-field, PLM and PLM with a phase plate ......................................... 3.2 Schematics of simplified setup for Jones matrix analysis ........... 3.3 Schematics to represent the incident and scattered field's polarization for an oblique illumination .............................. 3.4 Schematics to represent the pre-processing of polarization sensitive holograms ........................................ 3.5 Graphical illustration to retrieve polarimetric information by quadri-wave lateral shearing interferometry ......................... 3.6 Vectorial ptychography: experimental setup and reconstruction process .3.7 Schematic diagram of image simulation in PS-TDM: Forward propagation using V-BPM and inversion algorithm using V-Rytov first order approximation. ....................................100 3.8 Simulation of the diffracted field by a spherical bead using beam propagation method ...................................101 3.9 Simulation results of a spherical anisotropic bead ..............102 3.10 Simulation results of a 2 µm spherical bead with anisotropic structures. ..103 3.11 Experimental setup of the developed high resolution polarization sensitive TDM ......................................104 3.12 Reconstructed complex RI of potato starch granules, and silica beads considering scalar approximation ..........................106 3.13 Jones matrix components of potato starch granules and silica beads ....107 xviii 3.14 Reconstructed 3-D polarimetric information of potato starch granules and silica beads in the same field of view. ......................108 3.15 Reconstructed 3-D polarimetric information of spider silk. .........109 3.16 Reconstructed 3-D polarimetric information of Pinna nobilis oyster shell. . 111 3.17 Reconstructed 3-D RI tensor components of a single potato starch granule acquired by PS-TDM. ...............................112 3.18 Reconstructed polarimetric information of single potato starch granule using PS-TDM, V-Ptychography, bright-field and polarization light microscopy ......................................113 3.19 Schematic diagram of a typical differential interference contrast microscope.114 3.20 Schematics of hologram demosaicking .....................115 3.21 Determining the extinction angles for the quarter wave plate ........116 3.22 Reconstructed TDM and DIC image of an Olive tree pollen .........117 A.1 Gerchberg-Saxton-Papoulis algorithm .....................130 A.2 Gerchberg-Saxton-Papoulis to retrieve missing information from a 2 µm synthetic bead ...................................131 B.1 Comparison of the diffracted field by a 2 µm spherical bead using Mie and BPM model. ..................................136 1

FIGURE 1 . 1 :

 11 FIGURE 1.1: Schematic diagram of holography process; u d : diffracted field by the object, u r : reference field, u i : incident beam and I h : hologram detected by the detector

FIGURE 1 . 2 :

 12 FIGURE 1.2: Schematic diagram of in-line holography; u d : diffracted field by the object and u i : incident beam

FIGURE 1 . 3 :

 13 FIGURE 1.3: Schematic diagram of off-axis holography; u r : reference beam and u i : incident beam, RC: recombining cube. TL: tube lens. M 1 and M 2 : mirrors, BS: beam splitter.

FIGURE 1 . 4 :

 14 FIGURE 1.4: Off-axis holography demodulation process considering a synthetic object: (a) interference pattern to make hologram (b) spatial 2-D Fourier transform of the hologram (note the separated 0 order, virtual image and object information), (c) demodulated phase and amplitude information.

FIGURE 1 . 5 :

 15 FIGURE 1.5: Schematic diagram of phase-shifting holography; u r : reference beam, u i : incident beam, RC: recombining cube. LT: tube lens. M 1 , M 2 : mirror, BS: beam spliter

FIGURE 1 . 6 :

 16 FIGURE 1.6: (a,b) A Pollen reconstructed by digital holographic microscope, (a) x y and (b) x z cross-sections. (c) transfer function of coherent(holographic) and incoherent imaging system, adapted from[START_REF] Bailleul | Microscopie tomographique diffractif á resolution isotrope[END_REF].

FIGURE 1 . 7 :

 17 FIGURE 1.7: Schematic illustration of the 3-D coordinate system: l c and z c are the thickness of the object and the detector position, respectively.

FIGURE 1 . 8 :

 18 FIGURE 1.8: (a) 3-D and (b) 2-D representation of diffracted field collected by holography. Effective object frequency support computation process for: (c) reflection and (d) transmission TDM configurations at normal object incidence. k o : object wave vector sets, k d : diffracted wave vector sets and k i : incident wave vector

FIGURE 1 . 9 :

 19 FIGURE 1.9: Synthetic aperture process for IR-TDM: (right) transmission TDM, (left) reflection TDM. The diffracted wavevectors k d are translated by incident wavevector k i to give the object wavevectors k o . The 2-D illustration (depicted in gray) as well as the rendered 3-D OTF results when considering finitely many angular sample illuminations.

FIGURE 1 .

 1 FIGURE 1.10: (a) Synthetic aperture process for SR-TDM when the sample is rotated with constant interval angle a. Effective frequency support along: (b) k x k y and (c) k y k z cross-sections. (d) Effective 3-D OTF

FIGURE 1 .

 1 FIGURE 1.11: (left) A SR-TDM by embedding the sample within a rotating fiber, (right) relative RI of a U937 cell. Taken from [51]. CL: condensor lens, PD: Petri dish, MO: Microscope objective, FC: Fiber capillary

FIGURE 1 .

 1 FIGURE 1.12: 2-D schematic representation for transmission TDM with wavelength variation of the illumination light: computation of the object frequency support (a) for a single wavelength, (b) for 3 wavelengths, and (c) for a continuum of wavelengths

  1.13. A photo of such experimental setup implemented in our laboratory is shown in figure1.14.

FIGURE 1 . 13 :

 113 FIGURE 1.13: Schematic diagram of transmission TDM system, adapted from [35]. BS: beam splitter, RC: recombination cube.

FIGURE 1 . 14 :

 114 FIGURE 1.14: Photo of transmission TDM set-up. FSM: Fast tip/tilt mirror, LP: linear polarizer, HWP: half-waveplates, and CMOS: camera

  Figures 1.15 (b,d) shows images of a pollen using a typical transmission TDM compared with images taken by DHM, Figs 1.15 (a,c). The frequency support recorded by DHM systems (off-axis holography) is highly limited by the aperture of the microscope objective, which reduces the axial resolution. Moreover, the transversal resolution is so poor that optical sectioning along the optical axis is almost impossible. Comparatively, one can witnessed that TDM helps to achieve better lateral x, y and longitudinal x, z resolution, as shown in Fig.1.15 (b,d).

FIGURE 1 . 15 :

 115 FIGURE 1.15: Comparing pollen grain images taken by (a,c) holography and (b,d) transmission TDM. (e) Transfer function for coherent (holography and TDM) and incoherent imaging.

FIGURE 1 . 16 :

 116 FIGURE 1.16: Comparing images of lacey carbon membrane acquired using several microscopy techniques; wide field (WFM), scanning electron (SEM) and confocal fluorescent (LSCM) microscopy, taken from [14].

FIGURE 1 . 17 :

 117 FIGURE 1.17: Comparing images of a microscopic USAF target taken by (a,d) holographic reflection microscopy, (b,e) reflection confocal microscopy, and (c,f) reflection TDM, taken from [59].

FIGURE 1 .

 1 FIGURE 1.18: OTF reconstruction for: (a) IR-TDM and monochromatic light, (b) IR-TDM and and multichromatic illumination light (red, green and blue light). Taken from [70].

FIGURE 1 . 19 :

 119 FIGURE 1.19: Effective OTF for: (a) full angle sample rotation TDM, (b) beam rotation TDM and (c) UFO looking like OTF obtained by combining (a) and (b), taken from [74]; (d) illumination rotation TDM, (e) illumination rotation TDM and 2 sample rotation, (f) illumination rotation TDM and 4 angular sample rotations along x axis. Taken from [48].

FIGURE 1 . 20 :

 120 FIGURE 1.20: Reconstruction of an optical fiber tip using: (a-c) conventional Rytov IR-TDM, (d-e) IR and 4 physical sample rotations in 0 , 54 , 90 , and 126 , taken from [48], (g-i) same 4 numerical sample rotations and IR-TDM and (j-l) 20 numerical sample rotations and IR, taken from [75]. Scale bar is 3 µm

FIGURE 1 . 21 :

 121 FIGURE 1.21: Effective OTF of a 4Pi TDM, resulted from effective combination of reflection TDM and transmission TDM.

FIGURE 1 . 22 :

 122 FIGURE 1.22: Object frequency support computation process for Fourier Ptychography diffraction tomography.(a, b) single illumination, (c,d) sample illumination along 3 angular directions. k 0 is the wave vector in the background medium.

FIGURE 1 . 23 :

 123 FIGURE 1.23: Zebra fish muscle structure imaged with (a) phase contrast, (b) polarization contrast TDM and (c) combined method, taken from [104]. Scale bar is 200 µm.

FIGURE 1 . 24 :

 124 FIGURE 1.24: Experimental setup for polarization sensitive TDM, taken from [106]. PBS: polarizing beam splitter; M: mirror; DMD: digital micromirror device; LCR: liquid-crystal retarder; L: lenses; C: condenser; O: objective lens; p: polarizer; Cam1, Cam2: Camera and HWP: half-wave plate.

FIGURE 1 . 25 :

 125 FIGURE 1.25: Schematic diagram of TDM reconstruction process. F 2D : 2-D Fourier transom, F 1 3D : inverse 3-D Fourier transom.

2 Filling

 2 ................................. 2 TDM Synthetic Aperture Process ...................... 3 Scanning Patterns and Devices ........................ 3.1 Star Pattern ............................... 3.2 Grid Pattern ............................... 3.3 Annular Pattern ............................. 3.4 Spiral Pattern .............................. 3.5 Fermat's Spiral Pattern ......................... 3.6 Rose Pattern ............................... 3.7 Uniform 3-D Sample Scanning Pattern ................ 4 Evaluation Criteria ............................... 4.1 Filling Factor .............................. 4.2 Error Estimation ............................ 5 Transmission TDM .............................. 5.1 OTFs construction Results ....................... 5.2 Filling Factor Results .......................... 5.3 Image Reconstruction ......................... 5.4 Experimental Investigations ...................... 5.5 Other considerations .......................... 6 Reflection TDM ................................. 6.1 OTFs Construction ........................... 6.Factor results .......................... 6.3 Image Reconstruction ......................... 7 Combining Transmission and Reflection TDM .............. 7.1 OTFs Construction ........................... 7.2 Filling Factor results .......................... 7.3 Image Reconstruction ......................... 7.4 Experimental Implementations ....................

FIGURE 2 . 1 :

 21 FIGURE 2.1: Synthetic aperture process to compute the effective OTF in 2D for: (a) reflection, (b) transmission and (c) 4Pi IR-TDM configurations. k o : object wavevectors, k d : diffracted wavevectors and k i : incident wavevector

FIGURE 2 . 2 :

 22 FIGURE 2.2: (top row) simulated 3-D 2µm radius bead and its spectrum; (a) collected object spectrum by using: (b,c) transmission and (d,e) 4Pi TDM; cross-sections of the spectrum for (b,d) 200 illuminations and (c,e) 100 illuminations.

FIGURE 2 . 3 :

 23 FIGURE 2.3: (a) An LED array used to illuminate a lung cancer cell sample, (b) computed object spectrum, and (c) cross-sections of reconstructed lungcancer cells using Rytov approximations, taken from[START_REF] Li | Optical diffraction tomography microscopy with transport of intensity equation using a light-emitting diode array[END_REF]

FIGURE 2 . 4 :

 24 FIGURE 2.4: Examples of sample illuminations angular scanning patterns: (a) spiral scanning, (b,c) cross-section of the 3-D OTF for NA eff = 0.5 (red) and 0.8 (green), from[START_REF] Lim | Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography[END_REF], (d) single spiral illumination scheme, from[START_REF] Kim | High-speed synthetic aperture microscopy for live cell imaging[END_REF], (e) annular illumination pattern, from[START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF]

Fig 2 . 5

 25 (a-b). Note that the points correspond to the specular position of each illumination angles, and 200 sample illuminations are considered for the illustration. The number of illumination is fixed by regularly sampling along each axis. The number of axes is the main constraint, hence star patterns of three (Fig 2.5 (a)) and four (Fig 2.5 (b))

1 FIGURE 2 . 5 :

 125 FIGURE 2.5: Scanning schemes: (a) 3 axes and (b) 4 axes star. (c) Circular grid.

1 FIGURE 2 . 6 :

 126 FIGURE 2.6: Annular scanning schemes: (a-c) non-even and (d-f) even azimuthal spacing of points; with (a,d) 2, (b,e) 3 and (c,f) 4 concentric circles.

1 FIGURE 2 . 7 :

 127 FIGURE 2.7: Spiral scanning schemes: single spiral with (a) non-even, (b) even sampling, double spiral with (c) non-even, and (d) even sampling of points along the curvilinear abscissa, (e) Fermat's spiral.

FIGURE 2 . 8 :

 28 FIGURE 2.8: Rose scanning schemes: c 2 [2, 7] and d 2 [1, 9]. The arrangement of the petals is dependent on the values of c and d as well as their ratio.

FIGURE 2 . 9 :

 29 FIGURE 2.9: (a) Equally spaced points inspired from electrons at equilibrium, (b) spiral points on a sphere, and (c) N equally spaced points on sphere projected to 2-D disc, taken from [151]. Uniform 3-D sample scanning schemes: (d) 3D UDCS and (e) 3D UDHS. In TDM the maximum inclination is limited by objective lens's aperture. Hence, I adapted the algorithm for such purpose. So, two possible cases of 3-D angular scanning are considered: 3D-Uniform Distribution on a Cap of Sphere (3-D UDCS), and 3-D Uniform Distribution on a Hemisphere (3-D UDHS).

Chapter 2 .

 2 ANGULAR SAMPLE ILLUMINATION OPTIMIZATION 3.7.1 3-D Uniform Distribution on a Hemisphere

FIGURE 2 .

 2 FIGURE 2.10: 2-D (k x -k y ) and (k x -k z ) representations of OTFs and the corresponding illumination scheme used. (a) Theoretically, fully filled OTF; (b)-(l) OTFs corresponding to 600 illumination angles: (b) star; (c) circular grid; (d,e) annular patterns with 2 and 4 concentric circles; (f) single spiral and (g) double spiral; (h) Fermat's spiral; (i,j) flower patterns with 3 nonoverlapping and 8 overlapping petals; (k) 3-D UDCS, and (l) 3-D UDHS.

FIGURE 2 .

 2 FIGURE 2.11: OTF filling factor plots for transmission TDM. (left) zoom in view of FF for low number of angles.

FIGURE 2 . 14 :

 214 FIGURE 2.14: Image reconstruction of a synthetic object using 60 angles of star, Fermat and 3-D UDHS scanning in transmission TDM. (b-d) x y and (f-h) x z cross sections. (a-h) real part n 0 and (i-l) imaginary part n" of the index of refraction. (a,e) original bead; reconstructions using: (b,f,j) 3-axes star, (c,g,k) Fermat's, and (d,h,l) 3-D UDHS scanning. Scale bar is 1 µm.

FIGURE 2 . 15 :

 215 FIGURE 2.15: Refractive index profile plots of the bead in Fig. 2.14: for the RI along (a) x, (b) y and (c) z axis; (d) for the absorption in the z axis.

FIGURE 2 . 16 :

 216 FIGURE 2.16: Computed object frequencies (top row), reconstructed RI maps (middle row), and profiles (bottom row) of silica bead. Reconstructed images: (a,b) star, (c,d) Fermat's spiral, and (e,f) 3D-UDHS; RI profile: (g) mean profile in x, y cross-section, and (h) along z. Green: 3-D UDHS. Black: Fermat's spiral. Blue: star scanning patterns.

  (a,b)

Figures 2. 18

 18

  (a-d) show 4 outer views of this pollen grain.

Figure 2 .

 2 18(a) is a highresolution image obtained with 600 angular illuminations using 3-D UDHS scanning. Note that the high-resolution image Fig.2.18(a) clearly depicts large echinus (E), smaller perforations (P) at the basis of these echinus, as well as the smallest (here seen as 200-250 nm structures) pollen ornementation (O) on the exine surface[START_REF] Buchner | Helianthus tuberosus. In: PalDat -A palynological database[END_REF]. Figs. 2.18(b-d) are results using 60 angles star, Fermat, and 3-D UDHS scanning, respectively, which has less resolving power compared to the 600 angular illuminations.

FIGURE 2 .

 2 FIGURE 2.17: x y plane cross sections of Helianthus tuberosus pollen grain TDM image. (a): index of refraction n 0 . (b): imaginary part n".

FIGURE 2 . 58 Chapter 2 FIGURE 2 . 19 :

 2582219 FIGURE 2.18: (a-h) Reconstructed RI maps, and (i) profile curves of Helianthus tuberosus pollen grain in Eukitt: using (a,e) 600 holograms of 3D-UDHS (red), and 60 holograms of (b,f) star (blue line), (c,g) Fermat's spiral doted black line) and (d,h) 3D-UDHS scheme (green line). Scale bar is 10 µm.

Figure 2 .

 2 20(b) shows the spectrum of the object slice along the x y plane for 5 angular illuminations Figure 2.20(b: c1,c3) and 600 Figure 2.20(b: c2,c4) illuminations.

FIGURE 2 .

 2 FIGURE 2.20: (a) Cross-sections of the OTF obtained at various ratio between NA of condenser and objective lens, taken from [133], (b) Spatial object spectrum obtained using (c1,c3) 5 angular illuminations and (c2,c4) 600 illuminations, from [64].

60 Chapter 1 FIGURE 2 . 21 :

 601221 FIGURE 2.21: Schematics for OTF reconstruction using annular scanning at maximum NA. (a) NA cond = 0.8NA obj and (b) NA cond = NA obj

FIGURE 2 .

 2 FIGURE 2.22: (I) Simulated frequency support for illumination rotation TDM under different incident angles. (II) Experimental results of human cheek cell; Fourier transforms (kx-kz cross-sections): Fourier spectrum for (a) axial illumination and (b) rotating illumination; reconstructed image using rotating illumination (c) in x, y and (d) y, z cross-sections, taken from [145].

FIGURE 2 . 23 :

 223 FIGURE 2.23: Image reconstruction of an optical fiber in TDM. (a)-(d)Fourier spectra corresponding to the images below. (e),(j) Simulated cylinder. Reconstruction using (f),(k) DHM, (g),(i) FINER method, (h),(m) illumination rotation TDM (600 angles), (i),(n) F-TINER approach with two rotation angles (0°and 90°). Taken from[START_REF] Foucault | Simplified tomographic diffractive microscopy for axisymmetric samples[END_REF]. Scale bar is 2.5 µm.

FIGURE 2 . 24 :

 224 FIGURE 2.24: Extending the frequency support O ef f by Hermitian symmetry. taken from [146].

FIGURE 2 .

 2 FIGURE 2.25: (a) Demodulation of the images obtained from simplified mirror-assisted TDM, to cancel the mirroring effect and properly reassign information relative to the real (index) and imaginary (absorption) contributions, (b) experimental results using polystyrene bead, (top) before and(bottom) after properly reassigning the information. taken from[START_REF] Foucault | Versatile transmission/reflection tomographic diffractive microscopy approach[END_REF]

Fig. 2 .FIGURE 2 . 26 :

 2226 FIGURE 2.26: OTF slices for scanning along a single circle: (top row) considering NA cond = NA obj and (bottom row) considering NA cond = 0.8NA obj

FIGURE 2 . 27 :

 227 FIGURE 2.27: OTF filling factor plots considering non-absorbing sample for transmission TDM

FIGURE 2 . 29 :

 229 FIGURE 2.29: Image reconstruction of a synthetic object with annular illumination scheme at maximum NA: (a-h): real part ∆n 0 of the index of refraction. (i-l): imaginary part n". (a,e,i): original bead; reconstructions using (b,f,j): transmission, (c,g,k): transmission and point symmetric, and (d,h,l): 4pi configuration. Scale bar is 1 µm

FIGURE 2 . 30 :

 230 FIGURE 2.30: Image reconstruction of a synthetic object using annular illumination scheme considering point symmetric OTF. (a-b, d-e, g-h) real, and (c,f,i) imaginary part of the complex RI. (a-c) original object; reconstructions using (d-f): annular illumination scheme at max. NA, and (g-i) 0.8 of max.NA. Scale bar is 1 µm.

Fig. 2 .FIGURE 2 . 31 :

 2231 FIGURE 2.31: Image reconstruction of a synthetic object using annular illumination scheme considering Hermitian symmetry. (a-b, d-e, g-h) real, and (c,f,i) imaginary part of the complex RI. (a-c) original object; reconstructions using (d-f): annular illumination scheme at max. NA, and (d-f): annular illumination scheme at max. NA and (g-i) 0.8 of max. NA. Scale bar is 1 µm.

FIGURE 2 . 32 :

 232 FIGURE 2.32: Reconstructed C2C12 cells acquired using annular illumination at maximum NA without OTF symmetrization. (top row): OTFs corresponding to lower row; reconstructed (a,d) RI and (e,h) absorption maps.Scale bar is 10 µm.

FIGURE 2 . 33 :

 233 FIGURE 2.33: Reconstructed C2C12 cells acquired using annular illumination at maximum NA after OTF symmetrization. (top row): OTFs corresponding to lower row; reconstructed (a,d) RI and (e,h) absorption maps.Scale bar is 10 µm.

FIGURE 2 .

 2 FIGURE 2.34: 2-D (k y -k z ) representations of OTFs for reflection TDM. (a) Theoretically, fully filled OTF; (b)-(l) OTFs corresponding to 600 illumination angles: (b) 3-axes star; (c) circular grid; annular pattern (d) of 2 concentric circles with non-even sampling and (e) 4 concentric circles with even sampling; (f) single spiral with 2 turns and non-regular spacing; (g) double spiral with 6 turns and regular spacing; (h) Fermat's spiral; (i) 3-petal and (j) 8-petal flower pattern; (k) 3-D UDCS, and (l) 3-D UDHS.

FIGURE 2 . 37 :

 237 FIGURE 2.37: OTF mean filling factor FF(%) plots for reflection with maximum variation due to the parameters in the illumination scheme.

FIGURE 2 . 38 :

 238 FIGURE 2.38: Image reconstruction of a synthetic object using reflection TDM: (a,e,i) original bead, and reconstructions using (b,f,j) star, (c,g,h) Fermat and (d,h,l) 3-D UDHS scanning. (a-h) real part n 0 of the index of refraction, (i-l) imaginary part n". Scale bar is 1µm

  5 , or NA=1.43 in oil (oil immersion objectives with NA=1.45 exist), or NA=1.25 in water (water immersion objectives with NA=1.27 exist).

FIGURE 2 . 39 :

 239 FIGURE 2.39: OTF representation to make 4Pi TDM by combinig reflection and transmission TDM, taken from [36].

FIGURE 2 .

 2 FIGURE 2.40: 2-D and 3-D representations of OTFs for combined transmission and reflection TDM. (a) Theoretically, fully filled OTF; (b)-(l) OTFs corresponding to 600 illumination angles: (b) 3-axes star; (c) circular grid; (d) annular pattern of 4 concentric circles with even sampling; (e) double spiral with 6 turns and regular spacing; (f) Fermat's spiral; (g) 8-petal flower pattern; (h) 3-D UDCS, and (i) 3-D UDHS. (j) 3-D representation

FIGURE 2 .

 2 FIGURE 2.41: 2-D (k y -k z ) representations of OTFs for 4pi TDM. (a) Theoretically, fully filled OTF; (b)-(l) OTFs corresponding to 600 illumination angles: (b) 3-axes star; (c) Circular grid; (d) annular pattern of 4 concentric circles with even sampling; (e) double spiral with 6 turns and regular spacing; (f) Fermat's spiral; (g) 8-petal flower pattern; (h) 3-D UDCS, and (i) 3-D UDHS.

Figures 2 .

 2 44(b-d) show y z views of the original phantom, and the reconstructed images with transmission, reflection, and 4Pi TDM, respectively. 3-D UDHS of 600 illumination angles, at l = 633 nm and using NA=1.4 oil immersion objectives (n oil = Chapter 2. ANGULAR SAMPLE ILLUMINATION OPTIMIZATION 1.515), is considered. As from the discussion, 4Pi TDM is expected to deliver improved results along longitudinal sections, while the lateral resolution is somehow similar to transmission TDM.

FIGURE 2 . 44 :

 244 FIGURE 2.44: Image reconstruction results of a Shepp-Logan phantom: (a) original phantom; reconstructed using 600 angular illuminations of: (b) transmission, (c) reflection, and (d) 4Pi TDM. Scale bar is 1 µm

FIGURE 2 . 45 :

 245 FIGURE 2.45: Image reconstruction results of a Shepp-Logan phantom using 4Pi TDM. (a,e) original phantom; reconstructed using: (b,f) star, (c,g) Fermat's, and (d,h) 3-D UDHS scanning scheme. Scale bar is 1µm

FIGURE 2 . 46 :

 246 FIGURE 2.46: Image reconstruction of an absorbing phantom using 4Pi TDM. (a,e) phantom refractive bead , and (i) absorptive inclusions; (a-l) corresponding reconstructed images: using (b,f,j) star, (c,g,k) Fermat's, (d,h,l) 3D-UDHS. (m,n) profile plots along z axis. Scale bar is 1 µm.

FIGURE 2 . 47 :

 247 FIGURE 2.47: Schematic illustration of proposed 4Pi TDM setup. BS: beam splitter, FM: flip mirror, TTM: tip-tilt mirror, RC: recombination cube.

) d(r r 0 )

 0 being the Dirac delta function that satisfies outgoing boundary condition. I is the diagonal identity tensor. The dyadic Green's function allows to solve the outgoing electric field vector from a given input source. The scalar Green's function in the homogeneous background medium, which is the response due to a point source d(r r 0 ), given in Equation (1.20), as it was solved by considering the observation point r' to be at far, i.e. |r| |r 0 | and |r| |r 0 | 2 /l, g(r, r 0 )= exp(ik 0 n 0 |r r 0 |) 4p|r r 0 | . (3.4) Chapter 3. POLARIZATION SENSITIVE TDM To simplify the vectorial Green's tensor, we use the mathematical property, considering an arbitrary 3-D vector field y(r): r⇥r⇥y(r)= r 2 y(r)+r[r•y(r)].

FIGURE 3 . 2 :

 32 FIGURE 3.2: Schematics of simplified setup for Jones matrix analysis. PAS: polarization analyzer sensor

  at right-circular illumination and detection at 0 polarization • E o 1y measured at right-circular illumination and detection at 90 polarization • E o 2x measured at left-circular illumination and detection at 0 polarization • E o 2y measured at left-circular illumination and detection at 90 polarization Now, recalling Equation (3.32) and extracting the first diffraction order leads to:

FIGURE 3 . 3 :

 33 FIGURE 3.3: Schematics to represent the incident and scattered field's polarization for an oblique illumination of wave-vector k i : z axis is the normal incident beam direction. E xy inc is the incident field E inc projected at xy plane.

  .

FIGURE 3 . 4 :

 34 FIGURE 3.4: Schematics to represent the pre-processing of polarization sensitive holograms. Demosaicking of the acquired hologram leads to 4 holograms corresponding to 0 , 45 , 90 and 135 polarizations. Interpolation solves spatial shear and phase shift problem in the demosaicked holograms.

FIGURE 3 . 6 :

 36 FIGURE 3.6: Vectorial ptychography: (a) Experimental set-up. PM optical fiber: Polarization maintaining fiber, HWP: Half-waveplate, S s scanning stage, LP: linear polarizer, L T : tube lens. (b) computed Jones maps, and (c) transmittance, optical path difference, retardance and orientation maps,taken from[START_REF] Baroni | Joint estimation of object and probes in vectorial ptychography[END_REF]

FIGURE 3 . 7 :

 37 FIGURE 3.7: Schematic diagram of image simulation in PS-TDM. (a) Forward propagation using V-BPM and (b) inversion algorithm using V-Rytov first order approximation. n(x, y, z): object RI tensor, E(x, y, a i , e p ): polarization sensitive hologram for specific illumination angle a i and incident beam E i polarization e p .

FIGURE 3 . 8 :

 38 FIGURE 3.8: Simulation of the diffracted field by a spherical bead using beam propagation method: diffracted light-field's phase and amplitude computed at (a) normal incidence a = 0, and (b) oblique incidence, a = p/16

FIGURE 3 . 9 :

 39 FIGURE 3.9: Simulation results of a spherical 2 µm anisotropic bead: (left) the original phantom and (right) reconstructed principal RI cross-sections. Underestimation of the RI values is observed on the reconstructed results

FIGURE 3 . 10 :

 310 FIGURE 3.10: Simulation results of a 2 µm spherical anisotropic bead with anisotropic structures. The two principal RIs (dn 1 and dn 2 ) contain different structures. Intermixing between the two principal RI is clearly visible, especially in y z cross-sections.

Chapter 3 .FIGURE 3 . 11 :

 3311 FIGURE 3.11: Experimental setup of the developed high resolution polarization sensitive TDM; CL: collimating lens, M 1,2 : Mirrors, PC 1,2 : Polarizing components, L 1,2,3,4 : lenses, BS: beam spliter, PAS: polarization analyzer sensor or polarization sensitive camera, and RC: recombination Cube

FIGURE 3 . 12 :

 312 FIGURE 3.12: Reconstructed complex RI of potato starch granules, and silica beads considering scalar approximation: (a) refraction and (b) absorption; scale bar is 10 µm

FIGURE 3 . 13 :

 313 FIGURE 3.13: Computed 2-D Jones tensor maps: (left) amplitude, and (right) phase values of potato starch granules (large spherical objects, visible in all components), and silica beads (not visible in the diagonal components). Scale bar is 10 µm.

FIGURE 3 . 14 :

 314 FIGURE 3.14: Reconstructed 3-D polarimetric information of potato starch granules and silica beads in the same field of view. (a) RI tensor components, (b) average RI, (c) retardance and (d) fast-axis orientation. scale bar is 10 µm

Figure 3 .FIGURE 3 . 15 :

 3315 FIGURE 3.15: Reconstructed 3-D polarimetric information of spider silk. (a) RI tensor components, (b) average RI, (c) birefringence and (d) fast-axis orientation.

  Figure 3.15(b). The 3-D map of the fast axis orientation ranging in [0, p] displayed in Figure 3.15(d), also give an insight about how light could be affected when propagating through the sample.

FIGURE 3 . 16 :

 316 FIGURE 3.16: Reconstructed 3-D polarimetric information of Pinna nobilis oyster shell. (a) RI tensor components, (b) average RI, (c) birefringence and (d) fast-axis orientation. Scale bar is 20 µm.

FIGURE 3 . 17 :

 317 FIGURE 3.17: Reconstructed 3-D RI tensor components of a single potato starch granule acquired by PS-TDM. Scale bar is 20 µm.

5 .FIGURE 3 . 18 :

 5318 FIGURE 3.18: Reconstructed polarimetric information of single potato starch granule. (a-c) acquired using PS-TDM: (a) average RI, (b) birefringence, (c) fast-axis orientation; (d-f) acquired using V-Ptychography: (d) phase-retardance, (e) diattenuation and (f) fast-axis orientation; (g, left) Bright-field, (g, middle) PLM and (g, right) PLM plus a phase plate. Scale bar is 20 µm.

FIGURE 3 . 19 :

 319 FIGURE 3.19: Schematic diagram of a typical differential interference contrast microscope.

.64) 5 .

 5 From Polarization Sensitive Holograms to Simplified 3-D Differential Interference Contrast Microscopy 115

FIGURE 3 . 20 :

 320 FIGURE 3.20: Schematics of hologram demosaicking: spatial shear is observed in each demosaicked holograms. +∆x and +∆y represents relative spatial shear along x and y axis, respectively.

FIGURE 3 . 21 :

 321 FIGURE 3.21: Determining the extinction angles for the QWP, minima and maxima showing the fast and slow axis of the QWP.

5 . 117 (

 5117 From Polarization Sensitive Holograms to Simplified 3-D Differential Interference Contrast Microscopy displayed on a grey background) clearly show both the gradient and phase-shift attributes. The missing cone problem results in elongation along the optical axis, which seems more magnified in the 0 and 90 polarization TDM images than the DIC image. This may come from the summation of the polarization effects at the edges of the pollen.

FIGURE 3 . 22 :

 322 FIGURE 3.22: Reconstructed TDM image of an Olive tree pollen, fast axis orientation at (a-c) 45 and (d-f) 20 . Images corresponding to (a,d) 0 polarization, (b,e) 90 polarization and (c,f) retrieved DIC image. Scale bar is 20 µm.

n 3 .n 5 .FIGURE A. 2 :

 352 FIGURE A.2: Gerchberg-Saxton-Papoulis to retrieve missing information from a 2 µm synthetic bead

  r)=e(r) exp(ik 0 z),(B.3) where the envelop e(r) is the amplitude modulation and the field is propagating along the z direction in the background medium of refractive index n 0 and wavevectork 0 = 2p l n 0 . ∂ 2 E t (r) ∂z 2 =[

7 )

 7 e(x, y, z + dz)=F1 Similarly the refraction term is modeled as the phase modulation: e(x, y, z + dz)=e(x, y, z) exp(ik v rndz), (B.8)

I

  is the identity matrix. Similarly the refraction term is modeled as the phase modulation:e(x, y, z + dz)=e(x, y, z) exp(i2k v dndz/ cos a).(B.14)This leads to:e(x, y, z + dz)=exp(i2k v dndz/ cos a)

  Figure B.1 shows the simulated scattered field by a 2 µm spherical bead, and compares the results of Mie calculation and BPM model.

  ) Ainsi, pour N illuminations successives, on obtient N k i vecteurs, et N k d et k o ensembles de vecteurs, à repositionner correctement dans l'espace de Fourier, en suivant les principes du processus de synthèse d'ouverture. Ce processus de synthèse d'ouverture dans l'espace de Fourier pour diverses configurations de MTD (transmission, réflexion et configuration 4Pi est représenté sur le figure 1 en 2D pour des raisons de simplicité. La forme finale du support de fréquence étendue en 3D pour 600 illuminations est également montrée pour aider à la visualisation. Pour la configuration en réflexion, la Fonction de Transfert Optique (Optical Transfer Function OTF) 2D est représentée par un arc de cercle en 2D et comme le sommet d'une sphère en 3D, Figs. 1(a)-(b), qui sont décalées vers des fréquences plus élevées par le processus de synthèse d'ouverture. Pour la MTD en transmission, l'OTF 2D prend la forme dite d'un "papillon" en 2D (Fig. 1(c)), et une forme de tore limitée angulairement en 3-D (appelée "doughnut"), comme le montre la Fig. 1(c).

FIGURE 1 -

 1 FIGURE 1 -Processus de synthèse d'ouverture pour les configuration MTD en réflexion, transmission et 4Pi : (a) processus de synthèse d'ouverture 2D dans le plan (k y k z ) et (b) son OTF effective étendue en 3D pour la MTD en réflexion ; (c) processus de synthèse d'ouverture 2D dans le plan (k y k z ) et (d) l'OTF effective étendue correspondante en 3D pour la MTD en transmission ; (e-g) représentation 2D du calcul de l'OTF effective pour une configuration 4Pi : (e) détection par l'objectif 1, (f) par l'objectif 2 et (g) recombinaison des 2 acquisitions.

FIGURE 2 -

 2 FIGURE 2 -Modèles de balayage étudiés pour la MTD : (a) étoile à 3 axes ; (b) grille ; balayage annulaire en 4 cercles concentriques : (c) points non régulièrement et (d) points régulièrement espacés. Le cercle en bleu le plus à l'extérieur représente également le cas particulier du balayage annulaire avec un seul cercle à l'ouverture numérique maximale ; spirale simple avec 4 tours : (e) points non régulièrement, et (f) points régulièrement espacés ; double spirale avec (g) points non régulièrement, et (h) points régulièrement espacés : (g) points non régulièrement et (h) points régulièrement espacés ; (i) rose à 8 pétales avec k=4 (c = 4, d = 1) ; (j) spirale de Fermat ; (k) distribution uniforme en 3D sur une calotte de sphère complète, et (l) sur la moitié de la sphère.

FIGURE 4 -

 4 FIGURE 4 -Facteurs de remplissage des OTF dans la configuration MTD 4Pi pour les différents balayages considérés.

FIGURE 5 -

 5 FIGURE 5 -Résultats des simulations en considérant une configuration MTD 4Pi. Coupes du fantôme original pour (a) l'indice de réfraction et (f) l'absorption ; (b-d) et (g-h) images reconstruites correspondantes en utilisant 60 angles (b,f) balayage en étoile, (c,g) balayage de Fermat, et (d,h) balayage 3D-UDHS ; (i) profil de l'indice de réfraction le long de l'axe y et (j) profil d'absorption le long de l'axe z. La taille de la barre d'échelle est de 1µm

Figure 7 .

 7 Le faisceau laser (He-Ne, longueur d'onde λ = 633 nm) est divisé en deux, le faisceau de référence et le faisceau d'entrée, à l'aide d'une fibre optique de séparation de faisceau.

FIGURE 7 -

 7 FIGURE 7 -Experimental setup of high resolution polarization sensitive TDM ; CL : collimating lens, M 1,2 : Mirrors, PC 1,2 : Polarizing components, BS : beam spliter, PS-camera : polarization sensitive Camera, and RC : recombination component Ensuite, une onde plane cohérente présentant une polarisation circulaire est générée à l'aide d'un polariseur linéaire suivi d'une lame quart d'onde. Le faisceau d'éclairage à polarisation circulaire est ensuite dévié par le miroir de balayage rapide (Newport FSM300 Fast Steering Mirror) et balaye angulairement l'échantillon, composé d'objets anisotropes et faiblement diffusants, et placé entre le condenseur et un objectif de microscope (à immersion à l'huile n imm = 1,518, NA = 1,4 100⇥ Olympus

FIGURE 8 -

 8 FIGURE 8 -Images polarimétriques 3D de granules d'amidon de pomme de terre et de billes de silica dans le même champ de vue. (a) Composantes du tenseur RI, (b) RI moyen, (c) retardance et (d) orientation de l'axe rapide. Barre d'échelle : 10 µm.

  Fig 8(b)), la retardance (Fig 8(c)) et l'orientation de l'axe rapide (Fig 8(d)).

  Absorption 1.21 ⇥ 10 4 1.18 ⇥ 10 4 1.17 ⇥ 10 4

	RI	Star 2.1 ⇥ 10 4	Fermat 2.0 ⇥ 10 4	3-D UDHS 1.9 ⇥ 10 4

3-D UDHS scanning than for Fermat's scanning, therefore closer to what is needed to not intermix real and imaginary parts in Fourier imaging.

TABLE 2 .

 2 1: RMSE of the reconstructed images using transmission TDM, for both refractive index and absorption, in a 25 ⇥ 25 ⇥ 25 µm 3 cubic volume around the 2 µm bead.

  .35.

				70
				60
			Filling Factor (%)	30 40 50
				20
						star		Annular	3D-UDCS
				10		Grid		Fermat	3D-UDHS
						Spiral	Flower
				0
				0	100	200	300	400	500	600
					Number of Illumination Angles
		1.05
				star	Grid	Spiral	Annular	Fermat	Flower	3D-UDCS	3D-UDHS
		1
	Factor(%)	0.9 0.95
	Filling	0.85
	Normalized	0.7 0.75 0.8
		0.65	0	100		200		300	400	500	600
						Number of Illumination Angles
		FIGURE 2.36: Normalized OTF filling factor plots by 3D-UDHS for reflec-
							tion TDM.

FIGURE 2.35: OTF filling factor plots for reflection TDM.

  .2 displays RMSE results for the RI and absorption. 3-D UDHS scanning gives the overall (sum of RI and absorption RMSE) lowest RMSE, with star scanning very close and Fermat scanning slightly higher.

	Star 1.80⇥10 4 1.70⇥10 4 1.50⇥10 4 Fermat 3D-UDHS Absorption 0.68⇥10 4 0.96⇥10 4 0.88⇥10 4 RI Overall 2.48⇥10 4 2.66⇥10 4 2.38⇥10 4

TABLE 2 .

 2

2: RMSE of reconstructed images using 4Pi TDM, for refractive index and absorption in a 25 ⇥ 25 ⇥ 25 µm 3 volume around the 2 µm bead.

  .1.

	x	Bright-field	PLM	PLM + phase

y FIGURE 3.1: Potato starch granules imaged by bright-field, PLM and PLM with a phase plate (the sample is made at IRIMAS, Mulhouse and images are taken at Institut Fresnel COMiX group, Marseille). PLM: Polarization Light Microscope

  m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44

				.1)
	M is a 4 ⇥ 4 matrix given as:
		2	3
	M =	6 6 4	7 7 5 .	(4.2)

  + ik 0 rn/n 0 ]e(r), (B.5) where 4 x,y = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 . Equation (B.5) is the paraxial Helmholtz equation

	r) ∂z 2 |⌧|2ik 0	∂e(r) ∂z |. This helps to simplify Equation (B.2):
		∂e(r) ∂z	=[	i 2k 0	4 x,y

https://www.nanolive.ch/

https://www.tomocube.com/

Reconstruction ProcessOnce the diffracted field from the sample and the reference beam are collected and recorded by the detector as holograms, a numerical reconstruction algorithm is used

Conclusion and Perspectives ......................... 82

https://www.sigmaaldrich.com/FR/fr/product/aldrich/904465

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https://doi.org/10.1364/AO.417061

https://doi.org/10.1364/AO.435721

https://www.sfoptique.org/pages/congres-optique/optique-dijon-2021/

http://imabio-cnrs.fr/event/17ieme-journees-dimagerie-optique-non-conventionnelle/

https://focusonmicroscopy.org/past/2021/index.html

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https://doi.org/10.1111/jmi.13160

https://doi.org/10.1364/OE.483050

https://www.sfoptique.org/pages/sfo/congres-optique/optique-nice-2022/

https://focusonmicroscopy.org/past/2022/index.html

Acknowledgements 1 Partie I : Optimisation du Balayage en Illumination en MTD 1.1 Motivation

Dans la technique MTD la plus classique, l'objet est scanné en variant les directions d'illumination. Les différents champs diffractés par l'échantillon sont ensuite enregistrés sous forme d'interférogrammes (hologrammes), qui sont finalement recombinés numériquement pour reconstruire des images en trois dimensions. C'est ce que nous appelons la TDM par rotation de l'illumination (IR-TDM en anglais). Le balayage angulaire dans l'IR-TDM permet de remplir le support de fréquences spatiales dans l'espace de Fourier, ce qui conduit à une approche dite de synthèse d'ouverture. Le processus de synthèse d'ouverture permet de combiner les champs diffractés acquis à partir de l'échantillon pour obtenir une image finale à haute résolution.

L'une des principales limites des expériences de MTD est la nature séquentielle de l'acquisition des données. Par conséquent, si l'on souhaite obtenir une imagerie rapide, il faut limiter le nombre d'illuminations angulaires (et donc d'hologrammes) à utiliser, car le nombre d'angles de balayage est directement lié au temps de mesure. L'utilisation d'un petit nombre d'interférogrammes permet des acquisitions et des reconstructions rapides, mais la qualité des images reconstruites est dégradée. Bien que l'augmentation du nombre d'angles d'illumination soit l'approche la plus simple pour obtenir une image de haute qualité, cela ralentirait la vitesse d'acquisition/reconstruction/affichage des données. Il faut donc trouver un compromis entre la qualité de l'image finale et la vitesse d'acquisition/reconstruction.

Inspirée par ce problème, une partie de ma thèse a été consacrée à l'optimisation du schéma de balayage des échantillons, afin d'obtenir la meilleure image possible, tout en utilisant un nombre limité d'hologrammes. J'ai donc proposé et étudié plusieurs classes de modèles d'illumination, dont les performances ont été comparées.

funding organization ANR for recruiting and awarding me with a scholarship to pursue this PhD.

Chapter 3. POLARIZATION SENSITIVE TDM

In general, birefringence is highly dependent on the wavelength of the light-field used for illumination. Hence, absolute birefringence computation necessitates measurement of phase retardance at multiple wavelengths. Authors in [START_REF] Honda | Simple multi-wavelength imaging of birefringence: case study of silk[END_REF] have developed a system for such purpose (but a 2-D imaging technique) to measure the retardance of silk based on multi-wavelength measurement (contrary to those techniques that are done merely at single wavelength) to obtain range of retardance values. To do so, white light with interference filters is used to select 4 wavelengths at 10 nm bandwidth, in which the exact birefringence is obtained by averaging over a region of interest. This allows to get the average birefringence over a spectral range (425 nm -625 nm). For example, they found ∆n ⇡ (1.63 ± 0.05) ⇥ 10 2 by assuming the spider silk fiber to have a cylindrical shape.

Pinna Nobilis Oyster Shell

To further demonstrate the performance of our PS-TDM and to show that it can qualitatively image wide variety of complex specimens, Pinna nobilis oyster shells are used. The samples are acquired are prepared from COMiX research group at Institut Fresnel as part of collaboration. Oyster shells are one example of a mesocrystalline material that are made from calcium carbonate. Oysters create calcium carbonate structures cemented together by extracellularly secreting proteins and minerals from their mantle. Their shell layers are enlarged when they grow. Structurally, the shell is made from single crystal calcite prisms each surrounded by organic walls (thin layer of amorphous calcium carbonate), which is widely studied to hypothesize realistic biomineralization scenarios [START_REF] Ferrand | Quantitative imaging of anisotropic material properties with vectorial ptychography[END_REF][START_REF] Cuif | Biominerals and fossils through time[END_REF]. Such sample is well suited for observation using 2-D microscopy (for example for vectorial ptychography [START_REF] Ferrand | Quantitative imaging of anisotropic material properties with vectorial ptychography[END_REF]) as it has relatively adequate flatness and moderate thickness. Since it is also a semi-transparent anisotropic material it is well suited to be imaged by our PS-TDM.

For our purpose, the outer surface of the Pinna nobilis shell is taken to prepare the specimen for observation. Once immersed in the background medium (Eukitt), the samples are transported back to our lab and the results are shown in Fig. 3.16. The reconstructed RI tensor components of the shell in Fig. 3.16 (a) clearly show the micro structure of the shell relative to the background medium. We observe an inhomogeneous structures of the prisms, and the RI tensor component of one prism varies from the other. Note also the cross-section along the optical axis, where n xy and n yx RIs tensor components show the ordinary or extraordinary indexes (see the white rectangle).

Several optical parameters are also computed to gain more insight about the polarimetric properties of such sample. The average refractive index map in Fig. 3.16 (b) reveals an optically inhomogeneous structures of the various prisms. We also observe smooth transition between adjacent prisms. These results are also confirmed by the computed retardance map, as shown in Fig. 3.16 (c), as well as the fast-axis-orientation in Fig. 3.16 (d).

Comparing PS -TDM and Vectorial Ptychography

In order to demonstrate advantages of our PS-TDM system for 3-D quantitative polarimetric information retrieval, it is better to compare it with other polarimetric methods. In this section, I will compare the several polarimetric information of anisotropic sample that are acquired by PS -TDM system, with that of vectorial ptychography (V-ptychography) system. The V-ptychography system is at Institut Fresnel (COMiX research team), in which I had the chance to work with. The system is described in Ferrand et al.'s 2015 paper [START_REF] Ferrand | Ptychography in anisotropic media[END_REF][START_REF] Ferrand | Quantitative imaging of anisotropic material properties with vectorial ptychography[END_REF]. As it is well understood, V-ptychography is a 2-D A Rytov Approximation

A.I Scalar Rytov Approximation

The Rytov approximation is a well known method used in TDM, especially for reconstructing larger objects such as biological samples [11,[START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF]. Its main approach is representing the diffracted and incident fielding using a complex phase. The derivation starts by relating the total, incident and scattered field to the phase:

(A.1)

where f t (r)=f i (r)+f d (r). Now taking the relationship E t (r)=E i (r)+E s (r), we can write the extended equation for E s (r):

We can also write the homogeneous Helmholtz equation for the incident field as:

k = 2pn 0 /l is the wave vector amplitude in the surrounding medium of refractive index n 0 . Substituting the incident field expression and further manipulation leads to a simplified expression for the complex phase f i (r):

Now recalling the inhomogeneous Helmholtz equation (Equation (1.17)):

where V(r) is the scattering potential of the sample. Then, substituting E t (r) gives:

Further simplification and computing the Laplacian operator yields:

Now, using the relationship between the phases of total, scattered and incident field and inserting in the last expression of equation (A.7):

Using the expression from equation (A.4), we find a differential equation for f s (r) as:

Multiplying by E i (r) gives:

Making use of the relation:

Then re-arranging it and comparing it with equation (A.8) give the final formulation.

[

Now, we can make use of the Rytov approximation assumption, (i.e. rf s (r) << V(r)) and get the Rytov expression, [START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF]:

With the above expression, one can compute the Rytov phase f R (r) using the Green's function [START_REF] Jelena | On the derivation of the Green's function for the Helmholtz equation using generalized functions[END_REF].

This expression is similar to the the Born approximation. Indeed, we can relate f R (r) and Born field E B :

Hence to obtain the Rytov phase it is necessary to compute the ln of the complex fields, E i (r) and E t (r), in which we get:

This allows to compute the amplitude and phase of the Rytov field from the total field measured by the camera, which leads to the computation of the object scattering potential.

A.II Vectorial Rytov Approximation

Inspired by polarization sensitive TDM, derivation of the vectorial Rytov approximation is given in this section. This method is used in TDM especially for reconstructing thick samples [11,[START_REF] Slaney | Limitations of imaging with first-order diffraction tomography[END_REF]. We start by considering the spatially slowly varying phase of light, which is monochromatic propagating through anisotropic weakly scattering object. Recalling the vectorial Helmholtz equation:

which can be re-written as:

We start by expressing the electric field using the exponential function as in the scalar case:

which allows to simplify Equation (A.17):

[r 2 y(r)+k 2 0 +(ry(r)

Now by using the Rytov series for small phase variations, y = y (0) + y (1) + y (2) ..., and considering the Taylor expansion of the exponential function, the above equation gives the zeroth order, which is unperturbed one:

The first order perturbation is considered where the sample is illuminated by a polarized plane wave E i (r)=exp[ik i • r] êp , with wavevector k i , and polarization êp . The first order Rytov approximation can also expressed as:

[r 2 y (1) + 2ry (1) ry (0) = V (1) (r) exp[y (1)

Now let us express the total electric field by considering the first order Rytov expansion,

Also the scattered field can be written as:

If we substitute this into Equation (A.17), and by considering a plane wave illumination (from the solution of the unperturbed zero order field) to compute the scattered field:

[

Note that, for samples with weak birefringence response, the scattering potential tensor is considered to be similar to the overall scattering potential tensor (see [START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF] for more). At this point we can compute the analytical solution of the above expression, since each vectorial differential equations has similar mathematical expression to the Appendix A. Rytov Approximation first-order scalar Born field,

where logm is the logarithm of a matrix. This allows to relate the Rytov phase tensor and the vectorial Rytov field,

r).

(A.27)

The Born field vector and the 3-D object permittivity tensor are linked:

This permits to compute the object permittivity tensor from the measured scattered field vector.

A.III Rytov Approximation and Gerchberg-Saxton-Papoulis Algorithm

Image reconstruction using Gerchberg-Saxton-Papoulis (GPS) algorithm considering simple non-negativity constraint is given below. The image reconstruction takes a primarily reconstructed image using the linear inversion algorithm provided by first order approximations, and iteratively go back and forth in the Fourier and direct space to retrieve the missing information using constraints about the sample [START_REF] Rw Gerchberg | Super-resolution through error energy reduction[END_REF]. For this simulation, the constrains are:

• refractive index contrast, ∆n(r) 0,

• absorption, n 0 (r)=0,

The C++ GPS script was written by Bruno Colicchio (Dr.). Here it is adapted to be used in MATLAB. The algorithm is given below. V T est calculé à l'aide de paramètres tels que l'ouverture numérique de l'objectif et du condenseur, l'indice de réfraction du milieu d'immersion ainsi que la taille des pixels du capteur de la caméra [20,21]. Cette OTF de référence est un masque totalement rempli en forme de "beignet" (transmission) et une calotte de sphère partiellement remplie (réflexion) en 3-D (Fig. 3(a)).

J'ai constaté que le motif 3D-UDHS était celui qui remplissait le mieux l'espace de Fourier, tandis que les motifs en étoile et en spirale de Fermat remplissent respectivement le moins et moyennement l'espace de Fourier. Les résultats obtenus pour le facteur de remplissage FF dans les configurations en transmission et réflexion suivent globalement la même tendance.

Résultats des simulations

Pour approfondir ces résultats, j'ai aussi effectué des simulations en considérant un fantôme constitué d'une bille avec des inclusions absorbantes. Les résultats de ces simulations ont en effet montré que la qualité de l'image reconstruite dépend du schéma de balayage considéré. Les figures 5 montrent les résultats de ces simulations lors de la reconstruction du fantôme considéré à l'aide d'une configuration MTD 4Pi.

Le calcul de l'erreur (Root Mean Square Error RMSE) et le tracé du profil le long de différentes directions montrent que le balayage en spirale de Fermat fournit des résultats moyens, mais plus lisses, tandis que les motifs uniformes 3D et le balayage en étoile donnent respectivement les meilleures et les plus mauvaises estimations de l'absorption et de l'indice de réfraction.

Résultats expérimentaux

J'ai utilisé, pour valider mes résultats de manière expérimentale, un système de MTD en transmission développé précédemment, construit à partir d'éléments opto-mécaniques disponibles commer- Les Figures 6(a-d) montrent des vues d'un grain de pollen d'Helianthus tuberosus reconstruites avec (a) : 600 hologrammes avec balayage uniforme en 3D, servant de "vérité de base", et (b-d) : avec 60 hologrammes acquis respectivement par balayage en étoile (remplissage le plus faible de l'espace de Fourier), balayage en spirale de Fermat (remplissage moyen) et balayage 3D-uniforme (meilleur remplissage de l'esapce de Fourier).

Les résultats montrent que le balayage uniforme 3D donne effectivement l'image la plus contrastée et la plus détaillée, même avec un faible nombre d'acquisitions (voir les flèches). Cela prouve que la sélection d'un modèle de balayage adapté peut effectivement aider à atteindre la qualité d'image requise avec un nombre réduit d'hologrammes.

Enfin, une bille microscopique a été utilisée pour étayer et valider les résultats des simulations des Figures 6(e-f). Les tracés des profils confirment les simulations selon lesquelles une OTF mieux remplie permet une meilleure estimation de la taille et de l'indice de réfraction du spécimen observé.

Conférences Nationales : Posters :

• Mifobio2021 : "Optimizing Sample Illumination Scanning for Reflection and 4Pi Tomographic Diffractive Microscopy"