Nicolas Verrier
email: nicolas.verrier@uha.fr

| Asemare

Mengistie Taddese

Riadh Abbessi

Matthieu Debailleul

Olivier Haeberlé

3D Differential Interference Contrast microscopy using polarisation sensitive Tomographic Diffraction Microscopy

Keywords: Microscopy, Tomography, Holography 1

Tomographic Diffraction Microscopy (TDM) is a generalisation of Digital Holographic Microscopy (DHM), for which the illumination angle onto the sample is fully controlled, which has become a tool of choice for 3-D, high-resolution imaging of unlabelled samples. TDM makes it possible to obtain the optical field in both amplitude and phase for each illumination angle.

Proper information reallocation eventually allows for 3D reconstruction of the complex refractive index map. On the other hand, Polarisation Array Sensors (PAS) paves new way for TDM, as vectorial information assessment about the investigated sample.

In this contribution, we show an alternative use of this polarisation information based on the phase sensitive nature of TDM.

Here, we demonstrated that TDM coupled with PAS can lead to a 3D Differential Interference Contrast (DIC) microscope with almost no experimental configuration modification.

| INTRODUCTION

As far as imaging resolution and feature selectivity are concerned, fluorescence microscopy/nanoscopy techniques are a gold standard [START_REF] Hell | Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy[END_REF]2,[START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)[END_REF]4]. However, phenomena like photobleaching, likely to induce phototoxicity [START_REF] Icha | Phototoxicity in live fluorescence microscopy, and how to avoid it[END_REF], limits acquisition duration and sample durability. Moreover, in certain specific conditions, markers can perturb measurements [START_REF] Kumari | Structural insights into actin filament recognition by commonly used cellular actin markers[END_REF][START_REF] Pospich | Structural effects and functional implications of phalloidin and jasplakinolide binding to actin filaments[END_REF].

Interferometric maker-less imaging techniques can be seen as a solution to these limitations. As a matter of fact, these allow for quantitative phase imaging (QPI) of biomedical samples [START_REF] Park | Quantitative phase imaging in biomedicine[END_REF]. Among all the existing implementation of QPI, we here focus on Digital Holographic Microscopy (DHM). DHM consists in a coupling of interferometric acquisition together with back-propagation algorithms so as to retrieve 3D information about the investigated scene over a large field of view [START_REF] Schnars | Direct recording of holograms by a CCD target and numerical reconstruction[END_REF][START_REF] Verrier | Off-axis digital hologram reconstruction: some practical considerations[END_REF][START_REF] Verrier | High numerical aperture holographic microscopy reconstruction with extended z range[END_REF].

However, as the sample is illuminated parallel to the optical axis, resolution along this light propagation direction is almost lost.

Moreover, the phase information that one can assess in QPI techniques mixes both physical length and refractive index into an optical path length information only, which limits their application to specific samples. Tomographic Diffraction Microscopy (TDM) can circumvent these limitations. TDM is a generalisation of DHM for which the illumination angle of the sample is accurately known, and adjustable. This angle can be adjusted either by rotating the object within the illumination beam [START_REF] Kozacki | Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system[END_REF], or by sweeping the illumination beam over the investigated object [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF]. Combination of both approaches can also be envisaged [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF][START_REF] Vinoth | Integrated dual-tomography for refractive index analysis of free-floating single living cell with isotropic superresolution[END_REF][START_REF] Lee | Isotropically resolved label-free tomographic imaging based on tomographic moulds for optical trapping[END_REF]. Using the angular variety of the acquisitions allows, under adequate assumptions [START_REF] Wolf | Three-dimensional structure determination of semi-transparent objects from holographic data[END_REF], to retrieve a 3D quantitative information about the complex refractive index map (refraction and absorption contributions) of the investigated sample, with improved resolution compared to conventional intensity-only transmission microscopy [START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF][START_REF] Sarmis | High resolution reflection tomographic diffractive microscopy[END_REF][START_REF] Cotte | Marker-free phase nanoscopy[END_REF]. Commercial microscopes have been made available on the market [21,22]. These allow for out of the box, high-throughput, high-resolution imaging of transparent samples. Add-ons are available for correlative fluorescent imaging.

At present time, polarisation imaging is not included, and probably could not easily be added by the end user, requiring a redesign by the manufacturers.

Recently, polarisation information about the light passing through the sample have been beneficially used, and proven to allow for sample's 3D birefringence characterisation [START_REF] Saba | Polarization-sensitive optical diffraction tomography[END_REF][START_REF] Shin | Tomographic measurement of dielectric tensors at optical frequency[END_REF]. Extraction of this information is made possible considering, either the addition of polarisation sensitive elements coupled with multi-step acquisitions, or the use of a Polarisation Array Sensor (PAS). It should be noted that when a non-birefringent sample is considered, PAS indeed behaves as a phase-shifting device [START_REF] Nomura | Phase-shifting digital holography with a phase difference between orthogonal polarizations[END_REF].

In this note, we propose to beneficially use the phase-shifting capabilities of such a PAS to transform a TDM experiment into a 3D Differential Interference Contrast (DIC) microscope. After a quick discussion about the TDM acquisition and reconstruction framwork, feasibility of the approach will be discussed through imaging of olive tree pollens, leading to a cost-effective DIC configuration.

| TOMOGRAPHIC DIFFRACTION MICROSCOPY WITH A PAS

Transmission TDM is a generalisation of DHM with full control of the illumination angle of the sample [START_REF] Haeberlé | Tomographic diffractive microscopy: basics, techniques and perspectives[END_REF][START_REF] Jin | Tomographic phase microscopy: principles and applications in bioimaging[END_REF][START_REF] Balasubramani | Holographic tomography: techniques and biomedical applications (Invited)[END_REF]. A scheme of our implementation is proposed Fig. 1. Here, light emerging for a He-Ne laser (𝜆 0 = 632.8 nm) is split in two beams: a reference beam, depicted in red in Fig. 1, and the sample illumination beam, depicted in blue in Fig. 1. The sample, positioned between the condenser and a microscope objective, is illuminated with a circularly polarised plane wave (circular polarisation being brought by the Quarter Wave Plate (QWP)), which angle of incidence onto the sample is driven by the tip/tilt mirror (Newport FSM300 Fast Steering Mirror). It should be here noted that both condenser and objective consist in oil immersion (𝑛 imm = 1.518), NA = 1.3 100× Olympus UPLFLN100XO2-2 objectives. Light diffracted by the investigated sample (depicted in green Fig. 1) is collected and a magnified image of the object is recorded at the Lucid Phoenix PHX050S-P/Q PAS plane (optional sampling lenses can be added for diffraction pattern sampling considerations). Illumination beam, diffracted beam and reference beam are brought to interfere onto the PAS using a tilted cube beam combiner. This angular tilt aims at reconstructing both amplitude and phase of the optical field from a conventional Fourier filtering of the acquired information [START_REF] Cuche | Spatial filtering for zero-order and twin-image elimination in digital off-axis holography[END_REF]. Considering the first Born approximation [START_REF] Wolf | Three-dimensional structure determination of semi-transparent objects from holographic data[END_REF], acquired holograms can be combined considering a synthetic aperture process [START_REF] Taddese | Optimizing sample illumination scanning in transmission tomographic diffractive microscopy[END_REF], leading to a 3D extraction of the complex refractive index of the imaged sample (i.e. refraction, but also absorption, which often brings complementary information [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[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]).

Compared to our previous work [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[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], the configuration presented Fig. 1, relies on the use of a PAS. Here, a micro-polariser array is positioned directly on the sensor chip. Therefore, each pixel is associated with a linear polarisation angle as illustrated in the inset presented Fig. 1. One of the interesting features of these sensors is that they allow for extracting information about light polarisation along four linear states, namely 0°, 45°, 90°, and 135°. Extraction of polarisation images for each analysis direction eventually relies on the use of a demosaicing algorithm, similarly to what is commonly done for colour Bayer filter analysis [START_REF] Mihoubi | Survey of Demosaicking Methods for Polarization Filter Array Images[END_REF]. A less sophisticated way to perform demosaicing is to directly extract the desired pixels to build under-sampled sub-images as depicted Fig. 2. In this configuration, each polarisation image will be shifted by one pixel along 𝑥 direction, 𝑦 direction, or both (Δ𝑥, Δ𝑦 shifts on Fig. 2).

Considering an anisotropic/polarising sample, the PAS will allow, depending on the acquisition configuration, for estimating key parameters such as Stokes parameters of light. These can then be used to estimate the diattenuation, retardance and depolarisation of the light going through the sample [START_REF] Lu | Interpretation of Mueller matrices based on polar decomposition[END_REF]. If we now consider an isotropic sample (i.e. a sample that does not modify the polarisation of light), the four sub-images should be identical. However, for illumination of the sample with a circular polarisation, it can be proven that images analysed through orthogonal direction (e.g. image analysed at 0°and 90°) are in phase-quadrature. This effect as been beneficially used to build a two-step phase-shifting holography configuration [START_REF] Nomura | Phase-shifting digital holography with a phase difference between orthogonal polarizations[END_REF].

Two effects can therefore be expected when using a PAS to image an isotropic sample: a spatial shift effect, and a phaseshifting effect. In the next section, we show that combination of these two effects allows for mimicking the behaviour of a conventional DIC microscope. Moreover, as TDM indeed delivers 3D data reconstructions, the so-build TDM-DIC microscope will give 3D images.

| FROM PAS IMAGES TO 3D-DIC IMAGES

The discussion bellow is only be valid when considering an isotropic sample illuminated under circular polarisation. Going back to the experimental configuration proposed Fig. 1, the QWP fast-axis is adjusted to 45°in the illumination arm. The angle of the QWP in the reference beam is adjusted to maximise the intensity contrast in the PAS plane. This allow to make sure that the phase-shift between the 0°hologram  0 (𝑥, 𝑦) and the 90°hologram  90 (𝑥, 𝑦) is 𝜋∕2 [START_REF] Nomura | Phase-shifting digital holography with a phase difference between orthogonal polarizations[END_REF]. Moreover, considering Fig. 2 we can show that information from  90 (𝑥, 𝑦) is shifted by a Δ𝑥, Δ𝑦 quantity to give the  0 (𝑥, 𝑦). Therefore, we can write:

 0 (𝑥, 𝑦) =  90 (𝑥 + Δ𝑥, 𝑦 + Δ𝑦) exp ( 𝑖 𝜋 2 ) . ( 1
)
If we now estimate the difference between  90 (𝑥, 𝑦), and  0 (𝑥, 𝑦), we obtain:

 90 (𝑥, 𝑦) - 0 (𝑥, 𝑦) =  90 (𝑥, 𝑦) - 90 (𝑥 + Δ𝑥, 𝑦 + Δ𝑦) exp ( 𝑖 𝜋 2 ) . ( 2
)
One can notice that Eq. ( 2) is nothing else that the phase-shifted gradient of  90 (𝑥, 𝑦), which is the formal equivalent of a DIC image [START_REF] Nomarski | Microinterféromètre différentiel à ondes polarisées[END_REF][START_REF] Kim | Three-dimensional differential interference contrast microscopy using synthetic aperture imaging[END_REF][START_REF] Vishnyakov | Differential interference contrast tomography[END_REF][START_REF] Abbessi | Multimodal image reconstruction from tomographic diffraction microscopy data[END_REF].

Experimental proof of concept is proposed Fig. 3. Here, a slice of the TDM reconstructed data is proposed. Holograms of an olive tree pollen have been acquired according to the configuration depicted Fig. 1. Demosaicing procedure proposed Fig. 2 have been applied on the acquired holograms. TDM reconstruction at 90°(a) and 0°(b) illustrate the refraction contribution of the pollen (real part of the refractive index). One can clearly see the inner structure of the pollen. A computed, intensity-only gradient image (c) is given for illustration purposes. It would have been obtained if no QWP where added in the illumination path of Fig. 1, leading to a zero phase-shift in Eq. ( 2). Addition of a QWP allows for calculation of the DIC image (d) using Eq. ( 2).

Both gradient and phase-shift effects, which are the main feature of a DIC microscope, are clearly visible. 3D rendering of Fig. 3 is given in the TDM2DIC.avi file in supplemental data.

In the final section of this note, influence of the experimental parameter on the DIC results is discussed.

| DIC PARAMETERS TUNING AND LIMITATIONS

Looking at a conventional DIC microscope or considering Eq. ( 2), one can modify two parameters: the phase-shift (also known as retardance) and the spatial-shift (also known as bias). Until now, both parameters were fixed by our experimental arrangement: spatial-shift is fixed to one pixel in each direction, and the associated phase-shift is of 𝜋∕2 if we want to use images with orthogonal polarisation [START_REF] Nomura | Phase-shifting digital holography with a phase difference between orthogonal polarizations[END_REF]. Spatial shift is here limited by the pixelated structure of the PAS. In fact, looking at fig. 2, discrete nature of the sensor limits the shearing to (𝑛Δ𝑥, 0), (𝑛Δ𝑥, 𝑛Δ𝑦), or (0, 𝑛Δ𝑦), 𝑛 being a multiplicative factor for the pixel-shift. To explore negative gradients, when can consider using pixel with 0°polarisation as a reference instead of using pixel with 90°103 polarisation. As far as phase-shift is concerned, rotating the QWP fast axis modify the phase-shift in between two orthogonally polarised pixels. It should, however, be noted that the only configuration for which the phase-shift if accurately known is the configuration associated with results proposed Fig. 3, i.e. a pixel-shift of (𝑛Δ𝑥, 𝑛Δ𝑦) associated with a QWP whose fast axis is at 45°. Nevertheless, if only a qualitative information about the phase-shift is needed, the other configurations can be considered. This is clearly one of the limitations of the proposed approach. As a matter of fact, the digital approach presented in Ref. [START_REF] Kim | Three-dimensional differential interference contrast microscopy using synthetic aperture imaging[END_REF] is more versatile and allows for fully controllable tuning of both phase-shift and lateral shearing between the views.

However, in the case of anisotropic samples, using a PAS makes it now possible to highlight birefringence of the sample, as illustrated by imaging potato starches together with pollen sample, as in Fig. 4. Demosaicked images obtained with 90°111 polarisation (a), and 0°polarisation (b) are used to compute the difference image (c). As it can be noticed, for samples mixing both isotropic and anisotropic objects, the situation is a bit different. On Fig. 4(c), isotropic objects appear with a DIC contrast, while anisotropic objects exhibit a well-known Maltese cross structure. It should be noted that such a distinguishing effect cannot be obtained considering the fully digital approach proposed in Ref. [START_REF] Kim | Three-dimensional differential interference contrast microscopy using synthetic aperture imaging[END_REF], as it relies on a scalar reconstruction formalism.

| CONCLUSION

We discussed the opportunity of adding a PAS sensor to a TDM configuration. Doing so with a non-interpolated demosaicing of the acquired images, allows to build a 3D DIC microscope with almost no configuration modification. The proposed method is intended to operate for all non-polarising (isotropic) samples, while still maintaining the possibility to obtain polarisation sensitive images of anisotropic samples. Feasibility of the approach is demonstrated using a olive tree pollen, and influence of the experimental parameters have been discussed. The proposed approach is more limited in the choice of phase-shift and shearing of the DIC views, compared to pure numerical DIC reconstructions [START_REF] Kim | Three-dimensional differential interference contrast microscopy using synthetic aperture imaging[END_REF][START_REF] Abbessi | Multimodal image reconstruction from tomographic diffraction microscopy data[END_REF], but keeps polarisation sensitivity for anisotropic samples. It should be noted that the proposed configuration relies here on a modification of a TDM set-up for 3D imaging purposes. The same concept can however be applied to conventional 2D microscopes, leading to a cost effective implementation of the DIC technique. The proposed method is indeed a positive side-effect of the use of PAS for TDM imaging, which can work with both isotropic and ansitropic sample with very specific signatures: DIC effect for isotropic sample, Maltese cross like structure for polarising sample. This property could, for instance, be beneficially used for a rapid-, 2-classes, isotropic/anisotropic samples distinction, as for identification of shocked quartz [START_REF] Bohor | Shocked quartz in the Cretaceous-Tertiary boundary clays: Evidence for a global distribution[END_REF], or nanoplastics, which represent an increasing threat for the whole food chain, and even possibly human health [START_REF] Schwaferts | Methods for the analysis of submicrometer-and nanoplastic particles in the environment[END_REF].

1 3

 13 (TDM) set-up with a Polarisation Array Sensor (PAS). FD: Field Diaphragm, AD: Aperture Diaphragm, QWP: Quarter Wave Plate, TL1,2 : Tube Lenses, SL1,2: Optional Sampling Lenses. Inset: PAS layout with four analysers, from left to right and top to bottom 90°, 135°, 45°, 0°. . . . . . . . . . Tomographic Diffraction Microscopy (TDM) set-up with a Polarisation Array Sensor (PAS). FD: Field Diaphragm, AD: Aperture Diaphragm, QWP: Quarter Wave Plate, TL1,2 : Tube Lenses, SL1,2: Optional Sampling Lenses. Inset: PAS layout with four analysers, from left to right and top to bottom 90°, 135°, Reconstructed TDM image of an Olive tree pollen, at (a) 90°polarisation, (b) 0 °polarisation, (c) spatial gradient, (d) obtained DIC image. See TDM2DIC.avi file for 3D rendering.

4

 4 Reconstructed TDM image of an Olive tree pollen and potato starches, at (a) 90°polarisation, (b) 0 °polarisation, (c) Difference without interpolation.

The authors declare no conflict of interest.

R E F E R E N C E S

Funding information

This work is funded by Agence Nationale de la Recherche (ANR) under contracts ANR-18-CE45-0010 and ANR-19-CE42-0004 and by the Région Grand Est under contracts FRCR 18P-07855 and FRCR 19P-10656.


Riadh Abbessi

Nicolas Verrier
email: nicolas.verrier@uha.fr

Asemare Mengistie Taddese

Steve Laroche

Matthieu Debailleul

Mohamed Lo

Jean-Baptiste Courbot

Olivier Haeberlé

| Asemare

Mengistie Taddese

Multimodal image reconstruction from tomographic diffraction microscopy data

Keywords: Microscopy, Tomography, Holography 1

teaching and research institutions in France or abroad, or from public or private research centers.

| INTRODUCTION

Optical microscopy is essential in bio-imaging. Fluorescence microscopy has become the reference, thanks to its unique chemical, therefore functional, specificity, and its unsurpassed resolution, allowed by the development of fluorescence nanoscopy techniques [START_REF] Hell | Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy[END_REF][START_REF] Betzig | Imaging Intracellular Fluorescent Proteins at Nanometer Resolution[END_REF][START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)[END_REF][START_REF] Balzarotti | Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes[END_REF]. However, fluorescence can also present limitations like photobleaching of fluorophores, which handicaps long term measurements and can induce phototoxicity [START_REF] Icha | Phototoxicity in live fluorescence microscopy, and how to avoid it[END_REF], or difficulties in labelling in some cases, or because of legal requirements, in particular in biology or in food-industry, when one cannot label samples or use genetically modified organisms. Also, markers can sometimes interfere with the measurements [START_REF] Kumari | Structural insights into actin filament recognition by commonly used cellular actin markers[END_REF][START_REF] Pospich | Structural effects and functional implications of phalloidin and jasplakinolide binding to actin filaments[END_REF].

Interferometric microscopy techniques can be considered to tackle this issue. These methods allow for Quantitative Phase Imaging (QPI) in biomedicine [START_REF] Park | Quantitative phase imaging in biomedicine[END_REF]. Implementations based on short coherence interferometry [START_REF] Lee | Quantitative phase imaging unit[END_REF], Fourier ptychography [10], and Digital Holographic Microscopy (DHM) [11] have been demonstrated. In the remaining, we will focus on DHM implementations of QPI. DHM, in association with light-backpropagation algorithms [12,13], makes it possible for pseudo-3D volumetric reconstruction over a large depth of field [14]. However, resolution along the light propagation axis is lost, inhibiting DHM to be considered as a full 3D imaging technique [START_REF] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF][START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF].

Full 3D reconstruction of biological samples have been demonstrated with Tomographic Diffraction Microscopy (TDM) [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] Debailleul | Holographic microscopy and diffractive microtomography of transparent samples[END_REF][START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF][START_REF] Kim | Real-time visualization of 3-D dynamic microscopic objects using optical diffraction tomography[END_REF][START_REF] Kostencka | Holographic tomography with scanning of illumination: spacedomain reconstruction for spatially invariant accuracy[END_REF]. TDM is a generalization of DHM experiments with a full control of the investigated object illumination angle. This angle can be modified by rotating the object within the illumination beam [START_REF] Kozacki | Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system[END_REF], sweeping the illumination beam on the object [START_REF] Simon | High-resolution tomographic diffractive microscopy of biological samples[END_REF], or combining both approaches [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][START_REF] Vinoth | Integrated dual-tomography for refractive index analysis of free-floating single living cell with isotropic superresolution[END_REF][START_REF] Lee | Isotropically resolved label-free tomographic imaging based on tomographic moulds for optical trapping[END_REF]. From these acquisitions, it is possible to reconstruct, in 3D, both the real (refraction), and imaginary part (absorption) of the imaged sample's refractive index [START_REF] Wolf | Three-dimensional structure determination of semi-transparent objects from holographic data[END_REF][START_REF] Devaney | Inverse-scattering theory within the Rytov approximation[END_REF][START_REF] Kamilov | Optical Tomographic Image Reconstruction Based on Beam Propagation and Sparse Regularization[END_REF], with an improved lateral resolution, compared to conventional microscopes, [START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF], even down to the 100 nm resolution [START_REF] Cotte | Marker-free phase nanoscopy[END_REF]. 3D isotropic resolution imaging has also been recently demonstrated [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF][START_REF] Lee | Isotropically resolved label-free tomographic imaging based on tomographic moulds for optical trapping[END_REF].

The possibility of extracting 3D information about the refractive index of the imaged sample is an appealing feature of TDM. However, selectivity of the content is lost, and structural details can be missed. Therefore, TDM images can be difficult to interpret compared to images extracted with more common microscope modalities.

In this work, we propose to take benefit of the capabilities of TDM to accurately measure the 3D optical field coming from an object, to reconstruct images that would have been obtained with other microscopic imaging modalities. This will allow to take benefits of the TDM capabilities, together with the content discrimination brought by other techniques. Starting from TDM acquisitions, and using the image formation models of classical optical microscopes, we are able to simulate the behavior of "routine" microscopes (e.g. bright-field or dark-field microscopes), phase contrast improved microscopes (e.g. Phase contrast or Differential Interference Contrast (DIC) microscopes), as well as some less common imaging modalities (e.g. oblique illumination, or Rheinberg illumination microscopes), leading to a kind of "universal digital microscope" from TDM acquisitions.

| TOMOGRAPHIC DIFFRACTION MICROSCOPY PRINCIPLES

QPI techniques rely on the use of interferometry to extract both amplitude and phase of the field diffracted by the imaged object. As far as DHM is concerned, this extraction is either performed by spatial [START_REF] Cuche | Spatial filtering for zero-order and twin-image elimination in digital off-axis holography[END_REF] or temporal demodulation [START_REF] Yamaguchi | Phase-shifting digital holography[END_REF][START_REF] Atlan | Accurate phase-shifting digital interferometry[END_REF]. TDM is a generalized version of DHM with a full control of the object illumination angle [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] Haeberlé | Tomographic diffractive microscopy: basics, techniques and perspectives[END_REF]. A sketch of the transmission TDM we developed is proposed Fig. 1. Light, emerging from a He-Ne laser (λ = 632.8nm), is split into a reference beam (depicted in red in Fig. 1), and an object beam (in blue Fig. 1). Both paths are brought to interfere on the CMOS sensor within an off-axis Mach-Zenhder configuration. The main difference between DHM and TDM consists in the use of a Tip/Tilt mirror (Newport™ FSM 300 fast steering mirror) for condenser back-focal plane scanning. Therefore, a plane wave with a controlled angle illuminates the sample. From these acquired holograms, 3D information can be reconstructed using a synthetic aperture process. Considering the first Born approximation, the object wavevectors k o can be reconstructed using the elasticity condition

k o = k d -k i , (1)
with k i the illumination wavevector (associated with the illumination angle), and k d the associated set of diffracted wavevectors for each hologram [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. The k d wavevectors map the Ewald sphere [START_REF] Wolf | Three-dimensional structure determination of semi-transparent objects from holographic data[END_REF]. However, as diffracted light is collected in transmission with a limited Numerical Aperture (NA), the measured k d vector set maps the Mc Cutchen pupil, which can be viewed as a cap of sphere, whose cord is limited by the collection objective NA [START_REF] Mccutchen | Generalized Aperture and the Three-Dimensional Diffraction Image[END_REF]. hundreds of holograms makes it possible to build the full 3D object potential as can be seen Fig. 1(iv) in transmission [START_REF] Taddese | Optimizing sample illumination scanning in transmission tomographic diffractive microscopy[END_REF] (similar constructions holds for reflection or 4Pi TDM [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] Taddese | Optimizing sample illumination scanning for reflection and 4Pi tomographic diffractive microscopy[END_REF]). Taking the 3D inverse Fourier transform of this 3D object potential, one can reconstruct the 3D complex refractive index distribution of the investigated sample.

An example of tomographic reconstruction is proposed Fig. 2. The (x, y ) slices of the 3D reconstructed data cube show both the refraction (a), and absorption (b) of a Helianthus tuberosus pollen grain for a given z position (3D rendering of both refraction and absorption is provided in Media TDM.avi). The associated TDM acquisition consisted in the sequential acquisition of 400 holograms. The illumination scanning pattern was chosen to be the 3D-Uniform Sampled scheme proposed in [START_REF] Taddese | Optimizing sample illumination scanning in transmission tomographic diffractive microscopy[END_REF][START_REF] Taddese | Optimizing sample illumination scanning for reflection and 4Pi tomographic diffractive microscopy[END_REF], as it has been shown to provide the best possible filling of the Fourier space.

It can also be noticed that both images reveal different information about the sample. For instance, looking at the absorption image (Fig. 2(b)), we can remark that this pollen only weakly absorbs light, except for some structures of its double-wall envelope [START_REF] Halbritter | Pollen Morphology and Ultrastructure[END_REF], a behavior already observed for Betula pollen grain [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF].

So, to reconstruct 3D index of refraction images, TDM requires precise measurement of the diffracted field. This represents the entirety of the physical information carried out by the light-sample interaction, contrary to conventional transmission microscopes, which record intensity-only images. This property can be beneficially used if one is interested in obtaining an image of the sample under another imaging modality. Knowing the image formation model of a specific microscope, it is indeed possible to simulate the behavior of classical intensity (bright-field, dark-field, Rheinberg, . . . ) and phase contrast (Zernike, Differential Interference Contrast, . . .) microscopes, as shown in the next sections.

| FROM TDM TO OPTICAL MICROSCOPY

As recalled in Sec. 2, a TDM acquisition consists in the sequential capture of holograms acquired with different illumination angles. But instead of performing a classical tomographic reconstruction, one can also directly use the optical fields extracted for each hologram, using either Born or Rytov approximations [START_REF] Devaney | Mathematical Foundations of Imaging, Tomography and Wavefield Inversion[END_REF]. Combination and proper filtering of the holograms will then allow for mimicking different microscopic imaging modalities. We split these methods in two main families: intensity techniques relying on the combination of intensity-only images, and phase-dependent modalities, for which a phase modulation, possibly combined with an amplitude modulation, is introduced. One can also differentiate 2D approaches, recombining information at focal plane of acquisition, from true 3D reconstructions, which make use of the angular spectrum approach described Fig. 1.

| Intensity-only imaging

| Bright-field imaging

Bright-field imaging corresponds to optimal use of the condenser to simultaneously illuminate the sample under many directions, using incoherent light (Köhler illumination): in bright-field imaging, intensity information for each illumination angle is simply multiplexed. So, from an information point of view, bright-field microscopy can be considered as a parallel information processing system, while TDM is a sequential one. The synoptic of the bright-field microscope emulation is therefore straightforward, as proposed Fig. 4. Starting from the wavefront extracted from each angular hologram, an intensity image is calculated. All the intensity images are simply added to obtain the final bright-field image. Applying this scheme makes it possible to obtain the image in Fig. 3(a). As it has been noticed from Fig. 2, this pollen grain is weakly absorbing. This point is also noticeable on the reconstructed bright-field image which presents a small contrast, except in the pollen wall structures.

For such weakly scattering samples, other imaging modalities, such as dark-field, oblique, Rheinberg microscopy have been developed, which can also easily be computed from TDM data, as developed in the next paragraphs.

| Dark-field imaging

In conventional dark-field operation, the investigated sample is illuminated using an annular illumination, at a higher angle that than of the maximal collecting angle of the objective. Doing so ensures that only the light diffracted by the sample is captured, so that its image appears on a black background, therefore the name of the technique, and its high sensitivity. To simulate dark-field microscopy, we therefore only keep the holograms distributed along an annulus associated to the maximal NA of the condenser, while limiting the NA of the collection through Fourier space pupil filtering. Taking the intensity of each filtered hologram and summing the results allows to obtain the dark-field image proposed Fig. 3(b).

Compared to the result obtained Fig. 3(b), structural details about the pollen's membrane can be clearly noticed.

This is an intrinsic property of dark-field microscopy that allow to enhance high spatial frequency content in the images.

Note here the peculiar advantage of this purely numerical approach, which is that one can perform same NA, or even higher NA, at detection than at illumination by filtering out the specular illumination information in the Fourier space, while a physical dark-field microscope setup imposes a lower NA at detection than at illumination. Other highpass filtering can also be considered to extract smaller details about cellular samples [START_REF] Bon | Fast Label-Free Cytoskeletal Network Imaging in Living Mammalian Cells[END_REF]. However, we here limited our discussion to the most commonly available techniques.

| Oblique illumination microscope

Methods based on the directional selection of the illumination angle make it possible to enhance contrast along a specific direction. This is for instance the case of oblique illumination microscopy [START_REF] Sanchez | Oblique illumination in microscopy: A quantitative evaluation[END_REF], for which part of the illumination is blocked, as in dark-field microscopy, but here not only high-angle illuminations are allowed, and detection can also be performed at high numerical aperture. To mimic this behavior the synoptic of Fig. 6 can be considered. Here, only half of the condenser pupil is selected from the acquired holograms. The remaining holograms are then processed according to the classical bright-field scheme, i.e. the reconstructed optical field's intensities are summed for each processed hologram.

Comparing Figs. 3(c,d) with the bright-field result (see Fig. 3(a)), it can be noticed that the contrast is improved for the features whose direction is perpendicular to the illumination filter. This point is specifically pointed out when considering two different orientations of the oblique filter. Results proposed in Fig 3(c) for an oblique filter rotation of 144°, and 3(d) with an oblique filter rotated at an angle of 312°. Different features, pointed out by red arrows in Fig. 3(c,d) can be extracted from the two oblique filter position images. Full rotation of the oblique filter is proposed in Media ObliqueStack.avi.

| Rheinberg contrast imaging

In some cases, both low and high spatial frequency content is of interest, and in such cases, other methods, like

Rheinberg contrast microscopy, also known as optical staining, can be of interest [START_REF] Nelson | TRANSACTIONS OF THE SOCIETY: On an Addition to the Methods of Microscopical Research, by a new way of Optically producing Colour-Contrast between an Object and its Background, or between Definite Parts of the Object itself[END_REF]. Rheinberg illumination is similar to dark-field imaging, but (in its most classical form) instead of simply blocking by an opaque screen the central part of the cone of light emerging from the condenser, a colored disk filter is used. For high angle incidence illumination rays, a different colored annular filter is used (note that many different, more complex shapes of Rheinberg filters do exist). As a consequence, the observed specimen appears with the color of the outer rays, on a background of the central filter color, which greatly increases contrast.

Adapting the Rheinberg filter can improve the image contrast, but such modification is not really convenient/rapid on an actual microscope, and also the set of available Rheinberg filters is usually limited. On the contrary, one can easily imagine an equivalent of the colored illumination filter in the numerical version of Rheinberg microscope, which could provide a much greater flexibility, as in [START_REF] Samson | Dynamic contrast enhancement in widefield microscopy using projector-generated illumination patterns[END_REF].

A holographic implementation of Rheinberg microscope has been recently proposed [START_REF] Fan | Label-free color staining of quantitative phase images of biological cells by simulated Rheinberg illumination[END_REF], and we will here focus on the hologram processing.

The simulation scheme for Rheinberg contrast microscopy is proposed Fig. 7. As in its classical implementation, the illumination is color-filtered according to its angle, i.e. holograms are associated to a color filter according to their angles of incidence. The spectrum of each hologram (equivalently the collection objective backfocal plane) is then computed and filters are applied accordingly, in a similar way as what is done with dark-field microscopy.

Here, we propose a triple-color Rheinberg filter. Three filters are applied to the holograms: the first one for the low illumination angles, in green Fig. 7), the second one in red for the mid illumination angles (in red Fig. 7), and the last one in blue for the highest illumination angles. Three images are computed accordingly, as in bright-field microscopy.

These three images are then fused to build a RGB-composite intensity image, giving the Rheinberg contrast image proposed Fig. 3(e). As expected, low spatial frequency content like the image background or slowly varying features appear in green, while high spatial frequency content associated with the pollen shell is red-and blue-colored.

Up to now, we described intensity based microscopic imaging techniques. However, techniques based on the phase information can also be considered, as presented in the next section.

| Phase-induced enhanced contrast

In the previous sections, we have drawn attention onto intensity-based microscopy techniques. But contrast can also be brought to images by acting on the phase of the light-field. This can conveniently be performed from TDM images as we are able to measure both amplitude and phase of the optical field scattered by the investigated object. Classical techniques, like phase-contrast imaging, and differential interference contrast microscopy (DIC) can therefore also be emulated.

| Zernike phase-contrast microscopy

Phase-contrast microscopy has been introduced by F. Zernike in 1935 [START_REF] Zernike | Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode[END_REF][START_REF] Zernike | Phase contrast, a new method for the microscopic observation of transparent objects[END_REF][START_REF] Zernike | Phase contrast, a new method for the microscopic observation of transparent objects part II[END_REF]. The main idea of the method is, qualitatively, the following: an annular illumination is used, and noticing that at microscopic scale, a ray traversing a cell experiences an about 1/4 wavelength delay, inducing a supplemental 1/4 wavelength phase-shift between the illumination wave and the wave scattered by the object results in an enhancement of the image contrast, as both waves interfere destructively.

To mimic this behavior using TDM data, the processing scheme presented Fig. 8 is considered. After proper selection of the illumination angle, a back-focal plane filter P k x , k y , defined as

P k x , k y = α k x , k y exp i Φ k x , k y (2)
allows for shifting the phase of the specular illumination spot, while also slightly attenuating its amplitude. In Eq. ( 2), α corresponds to the amount of attenuation of the specular beam (note that it also attenuates/shifts the corresponding portion of the diffracted rays).

α k x , k y =          α if k x -k ix 2 + k y -k iy 2 ≤ ρ pc 1 elsewhere , (3)
with k ix , and k iy the spectral coordinates of the specular illumination spot, and ρ pc the radius of the pupil filter. The introduced phase-shift Φ k x , k y can be similarly defined by

Φ k x , k y =          Φ if k x -k ix 2 + k y -k iy 2 ≤ ρ pc 0 elsewhere . ( 4
)
Reconstruction of the pollen grain with phase-contrast configuration is depicted Fig. 3 for both positive (f) and negative (g) phase-contrasts. Here, α = 0.7 and Φ = ±π/4 (see Eqs. [START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)[END_REF][START_REF] Balzarotti | Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes[END_REF]) for positive and negative phase contrast, respectively.

As expected, one can notice than the pollen images exhibit better contrasts than the bright-field image reconstructed Fig. 4, which is explained by the attenuation coupled with the introduced phase-shift, as in an actual phase contrast microscope.

| Differential interference contrast microscopy

Differential Interference Contrast (DIC) microscopy, proposed by Nomarski in 1955, is an other classical technique for characterization of weakly scattering samples [START_REF] Nomarski | Microinterféromètre différentiel à ondes polarisées[END_REF][START_REF] Pluta | Nomarski's DIC microscopy: a review. Phase Contrast and Differential Interference Contrast Imaging[END_REF]. Here, the investigated object is illuminated using two orthogonally-polarized light-fields (ordinary and extraordinary light-fields emerging from a Nomarski prism). Parallel and slightly shifted orthogonally-polarized rays passing through the observed samples are recombined, and then pass through an analyzer. This leads to two cases. If both rays experience the same dephasing, they are recombined by the Nomarski prism into a linearly polarized ray, blocked by the analyzer. If one or the other ray emerges in advance, an elliptically polarized beam emerges from the analyzer to form an image onto the detector. Properly adjusted, the imaged contrast corresponds to gradients in specimen optical path lengths, hence the name of Differential Interference Contrast.

Simulations of DIC images have been proposed under the framework of DHM acquisition [START_REF] Kemper | 3D Tracking and Multi-Wavelength Techniques for Digital Holographic Microscopy based Cell Analysis[END_REF][START_REF] Miccio | Dynamic DIC by digital holography microscopy for enhancing phase-contrast visualization[END_REF]. Moreover, an implementation has recently been proposed for generalization to tomographic experiment [START_REF] Vishnyakov | Differential interference contrast tomography[END_REF]. As our reconstruction algorithms are based on the scalar implementation of the Born formalism, polarization of the incident light is not accounted for. This issue can easily be overcome considering that two orthogonally-polarized beam are in phase quadrature. Thus, for each hologram H θ corresponding to an illumination angle θ, we will estimate the differential interference contrast image I DIC considering

I DIC = |H θ (x, y ) -H θ (x + ∆x, y + ∆y ) exp (-i Φ DIC ) | 2 , (5)
with Φ DIC , and ∆x, y , denoting the introduced phase-shift and the spatial shift between the two holograms, respectively.

The associated simulation scheme is proposed Fig. 9. For each hologram, a difference image is calculated according to Eq. ( 5). Results are then summed in intensity for all illumination angles, leading to the final DIC image.

The obtained image is presented Fig. 3(h). The gradient effect, which is a basic signature of DIC microscopy, is here clearly noticeable, and makes it possible to better point out the sample structure.

Classical microscopy set-ups have been successfully simulated from TDM acquisitions. Depending on hologram combination and filtering, we were able to obtain multimodal 2D information about the investigated sample. But as far as we are working with holographic data, we can also mimic the focusing/defocusing occurring when moving the sample holder along the microscope optical axis, as described in the next section.

| Numerical refocusing

Illustrations proposed up to now consisted in the simulation of 2D microscopic image modalities. In this situation, determining the best focalized image of a thick sample can be challenging [START_REF] Baczewska | Refractive Index Changes of Cells and Cellular Compartments Upon Paraformaldehyde Fixation Acquired by Tomographic Phase Microscopy[END_REF]. So for our simulation framework to be complete, we can also consider taking into account the focus adjustment. This can be performed using classical angular spectrum propagation algorithms [12,13]. Other approaches can also be envisaged [START_REF] Kemper | Investigation of living pancreas tumor cells by digital holographic microscopy[END_REF][START_REF] Kim | Interference techniques in digital holography[END_REF]. Considering the simulated intensity images I z =0 (x, y ) for a given imaging modality, the intensity after propagation over a distance ∆z is given by:

I ∆z (x, y ) = F -1 F [I z =0 (x, y ) ] × H ∆z k x , k y , (6)
with F and F -1 respectively denoting Fourier and inverse Fourier transforms. Here H ∆z k x , k y is the angular spectrum propagation term expressed as:

H ∆z k x , k y = exp i 2πn imm λ 1 -(λk x ) 2 -λk y 2 , (7)
with n imm the refractive index of the immersion medium. Illustration of numerical refocusing is available in supplementary materials (Media Zfocus.avi).

| EXTENSION TO 3D IMAGING

Reconstruction modalities presented so far were built by acting on the reconstruction of individual holograms, leading to 2D modalities (with possible refocusing), as often done in holographic imaging. However, TDM being intrinsically a true three-dimensional imaging approach, applying our schemes to the 3D object frequency support will allow a direct extension to the most common microscopy imaging modalities, delivering 3D images.

| Dark-field and Phase-Contrast 3D microscopy

The processing scheme for 3D generalization of both dark-field and Phase-Contrast microscopy from tomographic data is proposed Fig. 10. Here, as for 2D schemes, the acquired spectra are filtered in the low spatial frequency region [START_REF] Chang | Computational approach to dark-field optical diffraction tomography[END_REF]. In such case, the filter to apply is defined as:

P 3D k x , k y , k z = α 3D k x , k y , k z exp i Φ 3D k x , k y , k z , (8)
with:

α 3D k x , k y , k z =        α 3D if k 2 x + k 2 y + k 2 z ≤ ρ 3D 1 elsewhere , (9)
and:

Φ 3D k x , k y , k z =        Φ 3D if k 2 x + k 2 y + k 2 z ≤ ρ 3D 0 elsewhere . ( 10
)
Here k x,y ,z are the 3D frequency coordinates of the object's spectrum, and ρ 3D is the filtering sphere radius. Attenuation and filter phase-shift are respectively given by α 3D k x , k y , k z and Φ 3D k x , k y , k z . If one wants to simulate dark-field images, Eqs. ( 9) and ( 10) are simply modified so that:

α 3D k x , k y , k z =        0 if k 2 x + k 2 y + k 2 z ≤ ρ 3D 1 elsewhere , (11)
and

Φ 3D k x , k y , k z = 0. ( 12
)
Here appears one advantage of this numerical approach compared to an actual dark-field or phase-contrast microscope. In these systems, an annular illumination has to be used to increase contrast, but can be detrimental to image reconstruction quality, especially in terms of resolution, as the optical transfer function is not optimally filled in Fourier space, contrary to TDM, which can use, via synthetic aperture, the full numerical aperture of the condenser [START_REF] Simon | Tomographic diffractive microscopy with isotropic resolution[END_REF][START_REF] Haeberlé | Tomographic diffractive microscopy: basics, techniques and perspectives[END_REF][START_REF] Streibl | Three-dimensional imaging by a microscope[END_REF], the necessary phase/amplitude correction factor being applied after acquisition of the data. Results obtained for both dark-field and phase-contrast microscopy are presented Fig. 11 for qualitative comparison (see Media 3D Modalities.avi for z-slicing). Imaging properties of both modalities are here compared with an axial cut (a,b), and longitudinal cut (c,d) in the reconstructed volume.

| DIC microscopy

The 3D generalization of DIC microscopy is fairly straightforward. Remembering that the contrast in DIC is directly proportional to gradients in the optical specimen, instead of shifting and phase-shifting the holograms to build a DIC image as in section 3.2.2 (see Figs. 9), one can now apply this approach directly to the 3D reconstructed tomographic images.

Obtained results are proposed Fig. 12(a,c) (see Media 3D Modalities.avi for z-slicing). Here, a 3 pixel spatial shift along the x-axis, corresponding to a 165 nm physical shift (accounting for imaging system magnification) and a 0.01 rad phase-shift is applied to the tomographic volume. As for the 2D case, the gradient effect can be clearly noticed. The advantage of this numerical approach is that bias and retardation can easily be tuned to highlight different features of interest. One could even apply gradient operation in out-of-plane directions (i.e. not only in the x-y plane), something which is not feasible in an actual DIC microscope. An example is given in the Discussion below. Such a feature remains to be fully explored, as it adds a new parameter to DIC imaging, which is promising but may also complicate interpretation of the observed contrast.

| Rheinberg illumination contrast

Rheinberg illumination contrast (see Figs. 7) can also be generalized for 3D operation. To this aim, each acquired hologram is associated to a filter color according to its corresponding illumination angle. For each color channel, a separate TDM reconstruction is performed. Reconstructed intensities are finally merged to build a composite RGB reconstruction of the object as depicted Fig. 12(b,d). Here, green channel is associated with small illumination angle holograms, red channel with "intermediate" illumination angles, and blue channel with the highest illumination angles.

Size and position of these illumination filters can easily be tuned depending on the sample's feature one wants to highlight.

Note that mimicking the Rheinberg illumination gives a different colored spectrum than simply assigning colors to low-, intermediary-, and high frequencies, as could also be done in Fourier filtering. The peculiarities of 2D/3D image formation in transmission microscopy/TDM [START_REF] Streibl | Three-dimensional imaging by a microscope[END_REF][START_REF] Haeberlé | Tomographic diffractive microscopy: basics, techniques and perspectives[END_REF]] make that, for example, high illumination angles, here assigned blue color, also contribute to the low frequency spectrum of the observed object,as can be seen Fig. 12(e,f) (see Media 3D Modalities.avi for z-slicing).

| DISCUSSION

This study is mainly driven by the fact that tomographic microscopes become more and more common, with commercial TDM devices being even now available [START_REF]Nanolive webpage[END_REF]59]. While this image contrast presents several advantages, it can in some cases remain difficult to interpret, due to the lack of selectivity of the reconstructed data. For this reason, manufacturers are now proposing coupled fluorescence/TDM devices. Another possible way to add selectivity in data reconstruction is to re-explore conventional imaging contrasts, in order to provide experimentalists, with images they are more familiar with and used to interpret. This is made possible by the QPI nature of a TDM configuration. In other words, this article is a tutorial on how to obtain different imaging contrasts, from TDM data (holograms), when available. Considering this, some remarks can be formulated considering the obtained results.

When using a DHM configuration with coherent illumination of the sample, coupled with simple/rapid Fourier inversions, noise can be noticed in reconstructed images Fig. 3. This noise structure is coming from a constructive interference phenomenon occurring at the sensor's protection window plane. This effect can be limited if one consider a short coherence light source, or using an image sensor without protection windows. The immediate consequence is that the simulated results are of a lower overall quality than that possibly obtained using a dedicated Phase-Contrast, DIC... configuration. However, all the simulated results share the same field of view, which is not that trivial for multimodal imaging if one has to use multiple microscopes, as such modalities can exclude each other: a physical dark-field microscope for example cannot use the same high numerical aperture objectives than a DIC .

As 3D microscopy techniques are concerned (see Figs. [START_REF] Betzig | Imaging Intracellular Fluorescent Proteins at Nanometer Resolution[END_REF]11,12,14), reconstructions can also be prone to Gibbs artifact at the border of the obtained field of view. These are coming from the Fourier space processing of acquired data. Moreover, looking at the associated supplementary media files, a strong noise structure can be noticed in the central imaging plane. This plane corresponds to the coherent summation of the reference beam for all the acquired holograms. It can be reduced considering either a blank acquisition (i.e. a sample less acquisition) or using a sensor without protection window. It should be noted that as 3D techniques are concerned, there is, to the best of our knowledge no commercial equivalent to our simulation protocol.

This work therefore does not aim at replacing routine microscopes. Instead, we propose a simple way to possibly add visualization modalities to existing TDM systems. As a matter of fact, superposing conventional imaging results to TDM results can bring back some of the contrast selectivity, which is lost when considering a marker-less technique.

| CONCLUSION AND OUTLOOKS

The sequential nature of data acquisition in TDM can be a drawback in terms of speed, but, contrary to classic optical transmission microscopes, allows for proper identification of the contribution of each illuminating ray, in both amplitude and phase. This makes it possible, considering adequate data processing, to emulate conventional 2D intensity and phase contrast microscopes, even taking into account possible defocus, similar to what occurs on a bench-top microscope. We have therefore studied reconstructing conventional microscopy imaging modalities from TDM acquisitions. Furthermore, considering the 3D reconstruction possibilities of TDM, we extended this work to full 3D imaging.

Note that other imaging modalities could also be similarly emulated. For example, Hoffman modulation contrast [START_REF] Hoffman | Modulation contrast microscope[END_REF] is based on an amplitude spatial filter, inserted at the back focal plane of the objective, therefore performing Fourier filtering, combined with a (usually offset) slit illumination, also equivalent to properly selecting illumination angles in TDM.

This study also paves the way for imaging new microscopy modalities that are not easily feasible in practice, for example out-of-focal-plane DIC, as suggested in previous section, and as illustrated Fig. 13. Here, effect of both lateral and axial shearing in either x, y or x, z cuts is illustrated. If lateral shearing is routinely performed with a DIC microscope, axis shearing cannot be performed. It, however seems to improve axial detection of our 3D digital DIC microscope. Similarly, confocal transmission microscopy should be feasible, as already done in reflection TDM [START_REF] Rasedujjaman | Three-dimensional imaging with reflection synthetic confocal microscopy[END_REF].

One of the difficulty of transmission confocal microscopy is to keep confocalized both illumination and detection, which are performed through two different optical trains, while reflection-and fluorescence confocal microscopy are autoconfocal. A numerical approach could help leverage this difficulty, even if transmission confocal does not provide same optical sectioning as fluorescence confocal microscopy. Imaging modalities based on Fourier filtering using phase plates or Spatial Light Modulators could also be simulated [START_REF] Fürhapter | Spiral phase contrast imaging in microscopy[END_REF][START_REF] Ikeda | Hilbert phase microscopy for investigating fast dynamics in transparent systems[END_REF][START_REF] Maurer | What spatial light modulators can do for optical microscopy[END_REF]. Residual aberrations correction has already been numerically implemented [START_REF] Colomb | Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation[END_REF][START_REF] Liu | Tomographic diffractive microscopy and multiview profilometry with flexible aberration correction[END_REF][START_REF] Kujawińska | Problems and solutions in 3-D analysis of phase biological objects by optical diffraction tomography[END_REF]. Finally, some microscopy modalities combination not implementable in practice, can easily be computed, for example the combined Rheinberg-DIC microscopy, as illustrated Fig. 14, in which Rheinberg illumination is simulated in green (NA < 0.4) and red (0.4 ≤ NA ≤ 1) and DIC is computed for NA > 1. Axial (a) and longitudinal (b) cuts are proposed. 3D rendering is available in Media RheinbergDIC.avi.

In the given simulated modalities, predetermined intensity filters have been proposed to reconstruct dark-field, Rheinberg or oblique illumination imaging. Similarly, fixed phase changes (phase contrast) or bias and retardation (DIC) have been used. A promising approach could be to introduce dynamic variations of the Fourier filtering, in both amplitude and phase, which, coupled to an adapted image contrast optimization criteria, could lead to a dynamic, purely-numerical contrast optimization, as performed in intensity-only imaging in [START_REF] Samson | Dynamic contrast enhancement in widefield microscopy using projector-generated illumination patterns[END_REF][START_REF] Bimber | Closed-loop feedback illumination for optical inverse tone-mapping in light microscopy[END_REF], in order to efficiently highlight the structures of interest to be investigated.

A scalar-approach TDM microscope delivers all the optical information, in terms of amplitude and phase, resulting from the sample under investigation with controlled illumination, and has already clearly demonstrated improved resolution compared to conventional transmission microscopy [START_REF] Debailleul | High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples[END_REF][START_REF] Simon | High-resolution tomographic diffractive microscopy of biological samples[END_REF][START_REF] Atlan | Accurate phase-shifting digital interferometry[END_REF]. Note that extension towards polarized TDM has already been proposed [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] Van Rooij | Polarization contrast optical diffraction tomography[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 multispectral or hyperspectral tomography [START_REF] Jung | Hyperspectral optical diffraction tomography[END_REF][START_REF] Sung | Spectroscopic microtomography in the visible wavelength range[END_REF][START_REF] Zdańkowski | Common-path intrinsically achromatic optical diffraction tomography[END_REF].

So, at least in theory, a combined polarized, hyperspectral TDM system could, with proper data processing, emulate all kind of existing optical transmission microscopes, even the classical polarization microscope, which works in white light. Such an approach would be tremendously challenging in terms of instrumentation (speed of acquisition [START_REF] Kim | High-speed synthetic aperture microscopy for live cell imaging[END_REF][START_REF] Jin | Dynamic spatial filtering using a digital micromirror device for high-speed optical diffraction tomography[END_REF][START_REF] Sung | Snapshot holographic optical tomography[END_REF] and data processing [START_REF] Kim | Real-time visualization of 3-D dynamic microscopic objects using optical diffraction tomography[END_REF][START_REF] Dardikman | Video-rate processing in tomographic phase microscopy of biological cells using CUDA[END_REF][START_REF] Bailleul | Tomographic diffractive microscopy: towards high-resolution 3-D real-time data acquisition, image reconstruction and display of unlabeled samples[END_REF]), but could pave the way towards a kind of all-in-one, universal system for optical transmission microscopy.

Fig. 1 )

 1 Fig. 1). The Mc Cutchen pupil, denoted k NA d is shown in red. From the Eq. (1), information is reallocated by subtracting the illumination vector contribution. The case of perpendicular illumination, along the optical axis, corresponding to classical DHM, is illustrated Fig. 1(i). Case of illumination at maximal NA is proposed Figs. 1(ii,iii). Combining tens to

F

  I G U R E 1 (a) Sketch of a typical DHM-based TDM configuration. Red beam is associated with the reference wavefront, blue beam is the illumination wavefront, and green beam is the wavefront diffracted by the investigated sample. TL 1 , and TL 2 respectively denote the condenser and microscope objective tube lenses. The sampling doublet made of SL 1 , and SL 2 lenses ensures adequate sampling of the interferograms. FD, and AD are the Field Diaphragm, and the Aperture Diaphragm, respectively. (b) Building of the 3D object potential k o from sequential hologram acquisitions (illustrated in 2D for convenience). Hologram spectrum registering according to Eq. (1) for perpendicular illumination (i), at collection objective maximum NA (ii,iii), full 3D object potential (iv). F I G U R E 2 Reconstruction of a Helianthus tuberosus pollen grain acquired with our TDM configuration. (a) Refraction contribution. (b) Absorption contribution.

3

 3 Simulated multimodal 2D images of a Helianthus tuberosus pollen grain. (a) Bright-field microscopy, (b) Dark-field microscopy, (c,d) Oblique illumination microscopy with an oblique filter at 144°(c), and 312°, (e) Rheinberg illumination microscope, (f) positive phase-contrast microscope, (g) negative phase-contrast microscope, (h) DIC microscope. Scale bar is 10 µm. F I G U R E 4 Synoptic of bright-field imaging from TDM acquisition. F I G U R E 5 Synoptic of dark-field imaging from TDM acquisition. F I G U R E 6 Synoptic of oblique illumination microscope from TDM acquisition. F I G U R E 7 Synoptic of Rheinberg imaging from TDM acquisition. F I G U R E 8 Synoptic of phase-contrast imaging from TDM acquisition. F I G U R E 9 Synoptic of DIC microscopy from TDM acquisition.DarkfieldPhase-ContrastTDM Frequency supportF I G U R E 1 0 TDM data processing for 3D dark-field and phase-contrast imaging.

1

 1 Slices of the 3D reconstruction considering (a) dark-field, and (b) Phase-Contrast modalities. (c), (d) associated x z -plane cuts 10 µm F I G U R E 1 2 Slices of the 3D reconstruction considering (a) DIC, and (b) Rheinberg illumination contrast modalities. (c), (d) associated x z -plane cuts (e) k x , k y cut of the frequency support. (f) (k x , k z ) cut of the frequency support.

3

 3 Influence of the shearing direction on the simulated 3D DIC results. (a) x, y cut, (c) x, z cut with a lateral shearing along x direction. (b) x, y cut, (d) x, z cut with an axial shearing along z direction. Shearing direction is figured out by the arrows in red.

Funding information

This work is funded by Agence Nationale de la Recherche (ANR) under contracts ANR-18-CE45-0010 and ANR-19-CE42-0004 and by the Région Grand Est under contracts FRCR 18P-07855 and FRCR 19P-10656.

Conflict of interest

The authors declare no conflicts of interest.

List of Figures