J Pernot

E Bustarret

M Rudziński

P R Hageman

P K Larsen

Strain relaxation in GaN grown on vicinal 4H-SiC(0001) substrates

Keywords: numbers: 62.20.Fe, 78.66.Fd, 61.72.Hh. I

The strain of GaN layers grown by Metal Organic Chemical Vapor Deposition (MOCVD) on three vicinal 4H-SiC substrates (0, 3.4 and 8 offcut from [0001] towards [11-20] axis) is investigated by X-ray Diffraction (XRD), Raman Scattering and Cathodoluminescence (CL). The strain relaxation mechanisms are analyzed for each miscut angle. At a microscopic scale, the GaN layer grown on on-axis substrate has a slight and homogeneous tensile in-plane stress due to a uniform distribution of threading dislocations over the whole surface. The GaN layers grown on miscut substrates presented cracks, separating areas which have a stronger tensile in-plane stress but a more elastic strain. The plastic relaxation mechanisms involved in these layers are attributed to the step flow growth on misoriented surfaces (dislocations and stacking faults) and to the macroscopical plastic release of additional thermoelastic stress upon cooling down (cracks).

Introduction

GaN is an attractive semiconductor because of its intrinsic properties for electronic (high power, high frequency and high temperature) and optoelectronic (UV) applications. However, up to now, no substrate combining large size and high quality is commercially available. GaN-based devices are therefore developed using heterostructures on a host substrate having interesting structural, thermal and electrical properties. Among such templates, silicon carbide is a good candidate with a high thermal conductivity and a relatively large substrate sizes available (conducting and semi-insulating substrates). Its lattice parameters (a 4H-SiC = 3.08051 Å and c 4H-SiC = 2.52120 Å for 4H 1 ) are close to those of GaN (a GaN = 3.1890 Å and c GaN = 2.5932 Å for hexagonal GaN 2 ). Combined with a larger thermal expansion coefficient for GaN (5.59×10 -6 ) than for SiC (3.7×10 -6 for 4H), this yields GaN epilayers generally under tensile in-plane strain 3 and leads to the formation of a high density of linear and planar defects such as dislocations, stacking faults or cracks. At present, one of the most important challenges is to determine the best growth conditions to obtain a thin GaN epilayer fully unstrained with a high crystal quality. A promising technique is the vicinal surface epitaxy using a misoriented substrate 4 . This technique has the advantage to grow the GaN epilayer with a step flow mechanism limiting the generation of threading dislocations 5 . Recently, Huang et al. proposed to control the strain relaxation with a new mechanism, using an optimized value of the miscut angle of the SiC substrate 6 . In this work, we will study the influence of the miscut angle of 4H-SiC substrate on strain relaxation in the GaN epilayer. First, the experimental details will be described.

Then, the results of X-Ray diffraction as well as microRaman and cathodoluminescence spectroscopies are used in order to evaluate the strain state of the layers grown on differently oriented 4H-SiC. Finally, the strain relaxation mechanisms are discussed before a brief conclusion.

II. Experimental details

We focus on three GaN/AlGaN/4H-SiC samples grown under identical growth conditions (side by side in the low pressure MOCVD reactor). Trimethylgallium (TMGa), trimethylaluminium (TMAl), and ammonia (NH 3 ) were used as precursors and H 2 as a carrier gas. A nucleation layer consisting of AlGaN with a low Al content (≅ 1 at.%) and with a thickness of 120 nm was employed. The non-intentionally doped GaN layer was grown at identical conditions as the nucleation layer, 1170 o C at 50 mbar total reactor pressure, and the growth time was set to yield about 1.8 µm thick GaN layers for all samples discussed in this paper. For the present experiments, we used nominally (0001)-oriented (on-axis) 4H-SiC substrates and substrates that were 3.4 o , and 8 o misoriented from (0001) towards ] 0 2 11 [ . To check the structural quality of epilayers grown on these substrates, the samples have been examined by X-ray using the rocking curve mode using a Bruker D8 Discovery X-ray diffractometer with a Cu target (λ=1.54060 Å) and a 4-bounce Ge (022) monochromator.

Micro-Raman measurements were performed using a Jobin-Yvon Labram Infinity spectrometer fitted with an Olympus BX40 microscope and a liquid nitrogencooled CCD detector. A 2400 gr/mm grating led to a pixel spectral separation of 0.42 cm -1 . The 633 nm line of a HeNe laser served as excitation source. To achieve a spatial resolution of 2 µm 3 , a ×100 objective was used with a confocal pinhole of 100 µm. The backscattering configuration was employed with both the incident and scattered light propagating along the z direction (parallel to the [0001] direction of h-GaN) and analyzed for polarization along an arbitrary direction x, perpendicular to the [0001] direction of h-GaN, and undetermined in this plane. For samples with 3. Optical properties of the samples were assessed by cathodoluminescence (CL). The CL measurements were performed in a Scanning Electron Microscope (SEM) equipped with a He cooling stage at beam currents and energies varying respectively from 10 pA to 80 nA and from 20 to 30 keV. The temperature of the sample was between 12 K and 20 K. The emission spectra were analyzed using a 0.4m monochromator equipped with a UV-enhanced liquid nitrogen-cooled CCD camera. The spectral resolution was estimated to be 0.5 Å, which, in the energy range of interest, is equivalent to 0.5 meV. Depending on the resolution requirements, a 600gr/mm grating blazed at 300 nm and a 1800 gr/mm grating blazed at 500 nm were employed.

III. Results and discussion

A. X-Ray

The X-ray rocking curves of our samples have been presented in previous works 7 . The structural quality of the GaN epilayer is decreasing at larger off-cut angles, in particular the Full width at Half Maximum (FWHM) of the (105) peak increases dramatically. The lattice parameters a and c are calculated using the standard formula taking into account the refraction index 8 . The normal (ε zz ) and in plane (ε xx ) components of the strain tensor are deduced using the strain free lattice parameters of GaN 2 : a 0 =3.1890 Å and c 0 =5.1864 Å. The ratio R B = ε zz / ε xx is evaluated for comparison with the value predicted by the elastic theory in the case of a pure biaxial stress R B = -0.53±10, according to the experimental data given by Polian et al. 9 . Results are summarized in Table 1. The in-plane strain is tensile for the three samples and increases with the misorientation angle of the substrate. Along c-axis direction, the normal strain is compressive, yielding a negative ratio R B in agreement with the elastic strain rules. The R B value of the 8° off axis sample is very close to the one corresponding to an ideal elastic relaxation of the stress. At the opposite, the ratio R B deviates from this value as the miscut angle decreases. These internal strain measurements have been confirmed by microRaman spectroscopy on the same layers.

ε xx (%) ε zz (%) R B E 2 (cm -1 ) σ B (GPa) Cracks #1 0 1.

B. Micro-Raman spectroscopy

Micro-Raman scattering is a useful non destructive probe of the symmetry of a crystal and of the strain in epilayers at the micrometer scale. Hexagonal GaN grows in the wurtzite structure and belongs to the C 6v (6mm) point group. Near k=0, group theory predicts four Raman active modes: one A 1 , one E 1 , and two E 2 . The strong E 2 mode at 568 cm -1 is allowed for the two backscattering configurations (parallel

z x x z ) , ( and crossed z y x z ) , (
). Its nonpolar character and its elevated scattering cross section make it the best phonon mode to evaluate the stress in GaN 3 . The figures 1 and 2 show the E 2 Raman mode spectra for the three samples. The presence of the TO mode of c-GaN at 554.5 cm -1 in the 8 o off-axis sample, indicated in Fig. 1 by an arrow, confirms the presence of cubic inclusions in the 8° miscut angle sample as already observed by HRTEM 10 . The shift of the line position and the broadening of the E 2 mode in Fig. 2 indicate a tensile stress which increases with the miscut angle of the substrates. The Raman spectra have been fitted using Lorentzian lineshapes. The energy positions are reported in Table 1 together with the X-Ray data and in Fig. 3 11 and used here with their error bars (a= -850±177 cm -1 and b= -963±220 cm -1 ). The dependence of the shift ∆ω versus ε xx is not linear in our case since R B is not a constant. Then, the biaxial stress σ B for each layer has been evaluated using the Raman biaxial pressure coefficient K B and the E 2 phonon shift measured for each sample 11 . It is important to notice that we have systematically located the laser spot far away from the cracks, observed in the 3.4 and 8 off axis samples. The present GaN layer grown on the on axis substrate is less strained than the thin GaN epilayers grown on 6H-SiC (0001) substrates studied in Ref. 12.

C. Cathodoluminescence spectroscopy

The low-temperature CL spectra of the hexagonal GaN layers grown on the differently oriented 4H-SiC substrates (0°, 3.4° and 8° off-axis) are shown in Fig. 4. The spectra have some common features in two spectral ranges. First, between 3.44 and 3.475 eV the donor bound exciton (DBE) emission lines are seen for all three substrate orientations, as well as their LO phonon replica red-shifted by 91-92 meV. Secondly, the donor-acceptor pair band (DAP) of h-GaN at 3.26 eV and its first (LO) and second (2LO) phonon replica, as assigned in Figs. 4. Extra features, indicated by arrows and star in Fig. 4, appear in the CL spectra of the 8° misoriented sample: i) a broad peak at ~3.39 eV. Recently, Liu et al. 13 presented the results of CL and Transmission Electron Microscopy (TEM) performed on the exactly same regions of thinned GaN/sapphire foils and gave clear evidence that basal plane stacking faults (BPSF) are responsible for an emission line at 3.41 eV, which is quite close to the peak observed at 3.39 eV for our 8° off-axis sample. ii) A prominent peak at 3.21 eV and its LO phonon replica at 3.12 eV and 3.03 eV. Very recently, Bai et al. 14 combined PL and TEM measurements and suggested that prismatic stacking faults and/or the associated stair-rod dislocations may be responsible for the PL features in the ~3.21 eV range. iii) A narrow feature at 3.28 eV. The best candidate for this peak is the exciton transition of the c-GaN inclusions observed by micro-Raman and TEM.

Figure 5 shows the CL spectra in the energy range of the donor bound exciton. As expected, this transition line decreases linearly with the biaxial stress. Using the coefficient dE A /dσ B of the A exciton of Ref. 11 , the DBE line shift permits to extrapolate the donor bound exciton transition energy for unstrained GaN at 3.47 eV, in good agreement with experimental data for free standing material. This confirms the tensile strain measured by X-Ray and Raman (away from the cracks).

IV. Discussion

The strain values measured in the three GaN layers grown during the same run on (0001) 4H-SiC substrates with different miscut angles (0°, 3.4° and 8° off [0001] towards [11-20] axis) show that the strain relaxation mechanisms involved are dependent of the substrate miscut angle. At a microscopic scale, the strain distribution of the on axis sample is homogeneous over the whole surface of the sample. For this sample, threading dislocations are formed upon coalescence of misoriented islands during the first step of the growth 15 . These isolated threading dislocations relaxes the strain through a plastic deformation yielding an uniformly and slightly stressed layer over the whole surface with a R B value far from the ideal elastic deformation case. The samples grown off axis are more favorable to a step flow growth, limiting the island formation and so decreasing the isolated threading dislocation density 5 . On the other hand, Rudziński et al. observed the formation of walls of dislocation along the [11-20] direction 10 for the 8° miscut angle. The observation of cubic inclusions 10 in sample #3 is certainly due to the anisotropic growth with triangularly shaped islands (2D nucleation growth mode) 5 . For the miscut samples, others relaxation mechanisms occur. Recently, Huang et al. 6 found that unpaired geometrical partial misfit dislocations (GPMDs) are formed along the step of the AlN/6H-SiC interface to correct the stacking sequence between 2H and 6H structures for misoriented substrates. This strain relaxation mechanism must be present at the AlGaN/SiC interface with a 2H/4H stacking structure in this case. For this system, the proportion of steps corresponding to GPMDs is 1/2 (1/3 for 2H with 6H [HuangPRL05]) since the number of cubic sites is 1/2 in the ABAC stacking sequence (one cubic site is defined as the B bilayer in the ABC sequence and one hexagonal site is defined as the B layer in the ABA sequence). This mechanism takes place during the step-flow growth, giving GPMDs lines along [1-100] axis and so, relaxing the [11-20] strains. Huang et al. 6 , on the basis of geometrical considerations (miscut angle and stacking sequence) argue that the number of GPMDs can be controlled, relaxing more or less the [11-20] strains. This explains the difference between the samples #2 and #3. Indeed, the critical miscut angle for 2H-GaN/4H-SiC 16 is 6.5° which is closer for sample #3 than #2. During the growth and prior to cooling, more GPMDs are formed along the [1-100] axis for sample #3 than sample #2, limiting more efficiently the [11-20] strain. Then, during the cooling step of the growth, the thermal expansion coefficient difference between GaN and SiC puts the GaN epilayer under tensile inplane strain, and then creates some cracks 17 . So, during the cooling step, the thermal expansion coefficient difference induces a high tensile strain along the [1-100] direction for the two miscut angles. This explains the formation of cracks along the [11-20] direction perpendicular to the [1-100] direction. The [11-20] strain is highly relaxed by the GPMDs for the sample #3 and not so much for sample #2, explaining the formation of additional cracks in the [2-1-10] and [-12-10] directions (not perpendicular to the [1-100] axis) for this sample. FIG. 6. Schematic draws of the crack and geometrical partial misfit dislocations (at the substrate epilayer interface) distribution in GaN grown on 4H-SiC substrates (3.4° and 8° off-axis). This is schematically represented in Fig. 6 for samples #2 and #3. For the off cut samples (#2 and #3), the strain relaxation seems to be mainly governed by two successive steps: i) Creation of GPMDs along the [1-100] axis 6 , misfit dislocation walls along the [11-20] axis 10 and basal plane stacking faults, prismatic stacking faults and stair rod dislocation 18 during the step flow growth due to the misorientation of the substrate.

ii) Creation of cracks along the [11-20] direction (and also the [2-1-10] and [-12-10] for the sample #2) during the cooling process due to the difference of the thermal expansion coefficients and in particular directions to minimize the residual stress.

The resulting layers are heterogeneously in-plane strained with a strong plastic relaxation near the cracks and in between, an elastic deformation close to the ideal case with a lower threading isolated dislocation density than in the on axis sample.

V. Conclusion

The strain of GaN layers grown by MOCVD on three vicinal 4H-SiC substrates (0, 3.4 and 8° offcut towards the [11-20] axis) have been measured and analyzed. For on axis samples, the threading dislocations due to the coalescence of misoriented islands during the first step of the growth relaxed the strain, resulting in a slightly but homogeneously tensile in-plane strained layer. For the miscut substrate, step flow growth occurs and other relaxation mechanisms appear during growth, such as the creation of various types of dislocations and stacking faults. During the cooling down step, cracks are formed in particular directions in order to limit the residual stress due to the difference of thermal expansion coefficients and to defect formation. Away from the cracks, the GaN areas are under a tensile in-plane stress with an almost pure elastic biaxial deformation.

  and scattered light are equally misoriented with respect to the [0001] direction.

  as a function of ε xx . The experiments are within the expected frequency range represented by the grey area deduced from the usual relation ∆ω(E 2 ) = 2 a ε xx + b ε zz , where a and b are the phonon deformation potentials. The phonon deformation potential values were recently determined by Demangeot et al.

FIG. 1 .

 1 FIG. 1. Room temperature micro-Raman spectra of h-GaN epilayers deposited on 4H-SiC substrates with misorientation of 0 o , 3.4° and 8° toward ] 0 2 11 [ . Arrows show signature of c-GaN inclusions in 8° off-axis sample.

FIG. 4 .

 4 FIG. 4. Cathodoluminescence spectra measured at 12-20 K with E EXC =20 keV of GaN epilayers deposited on misoriented 4H-SiC substrates (0°, 3.4° and 8° off-axis).

Table 1 :

 1 Samples characteristics and strain data deduced from X-Ray and Raman measurements.

			80	3.1930 ± 0.0005	5.18085 ± 0.0003	+0.1270 ± 0.016	-0.1070 ± 0.004	-0.843	567.1	-0.3745 ± 0.12	No
	#2	3.4	1.65	3.1980 ± 0.001	5.1776 ± 0.0003	+0.2822 ± 0.029	-0.1697 ± 0.007	-0.601	565.9	-0.8642 ± 0.28	along the [0112], [0211] and [0121] axis.
	#3	8	1.65	3.2005 ± 0.001	5.17675 ± 0.0003	+0.3606 ± 0.029	-0.1861 ± 0.006	-0.516	564.9	-1.2757 ± 0.42	parallel along [1120] axis

Acknowledgements

The financial support of M. Rudziński by Philips Semiconductors B.V. Nijmegen is gratefully acknowledged. J.P. and E.B. wish to thank F. Omn\`{e}s (Institut NEEL, CNRS, Grenoble, France) for fruitful discussions


Nicholas Ayache

Towards a Personalized Computational Patient

Keywords: Medical Imaging Informatics, Personalized Computational Patient, Computational Anatomy, Computational Anatomy, Computational Medicine

a fast evolving discipline at the crossing of Informatics, Computational Sciences, and Medicine that is profoundly changing medical practices, for the patients' benefit.

techniques, and from the introduction of new imaging modalities at various scales. It is supported by the regular increase of computer performances in terms of both computing power and memory storage. Finally it is boosted by the development of large databases of medical images made available through the Internet and by the introduction of large--scale machine learning methods. In 2014, I had the privilege to be elected to the Chair of "Informatics and Computational Sciences" at the Collège de France. The inaugural lecture entitled "From Medical Images to the Digital Patient", the course on "The personalized digital patient: Images, Medicine and Informatics", and the associated invited lectures of 26 prominent colleagues are available on the web site of the Collège de France 1 . These lectures reveal that the research in Medical Imaging Informatics is supported by a virtuous triangle. The first vertex represents a multidisciplinary academic research where informatics is associated with other sciences (mathematics, biology, physics, chemistry, anatomy, physiology, etc.). The second vertex represents a clinical partnership, essential to address relevant medical problems and validate the proposed solutions. The third vertex represents the industrial partnership required to transform the research prototypes into clinically approved products. These three worlds (academic, clinical and industrial) do not have rigid frontiers; actually their interactions are very stimulating and rich in terms of innovations. A young and active international community of researchers supports the research in Medical Imaging Informatics. These researchers share a common vision and have developed a scientific society in 2004 called MICCAI (for Medical Image Computing and Computer Assisted Intervention). Born in 1998 with a few hundreds of researchers, its conference currently gathers each year on a different continent about 1,300 participants from 46 countries. About 40% of the participants are PhD students, which is a good sign of the vitality of this research community. Most of these young doctors find a position after their thesis, either in the academic, clinical, or industrial environment. An increasing number of them participate to the creation or development of young start--up companies. Medical Imaging Informatics has become a fast evolving discipline at the crossing of Informatics, Computational Sciences, and Medicine that is profoundly changing medical practices, for the patients' benefit! Note : the final printed version of this article is available as a free PDF at this address: http://www.schattauer.de/de/magazine/uebersicht/zeitschriften--a--z/imia--yearbook/imia--yearbook-first/issue/special/manuscript/25863/show.html


F Momey

A Berdeu

T Bordy

J.-M Dinten

F Kermarrec Marcel

N Picollet-D'hahan

X Gidrol

C Allier

Lensfree diffractive tomography for the imaging of 3D cell cultures

New microscopes are needed to help realize the full potential of 3D organoid culture studies. In order to image large volume of 3D organoid cultures while preserving the ability to catch every single cell, we propose a new imaging platform based on lensfree microscopy. We have built a lensfree diffractive tomography setup performing multi-angle acquisitions of 3D organoid culture embedded in Matrigel ® and developed a dedicated 3D holographic reconstruction algorithm based on the Fourier diffraction theorem. With this new imaging platform, we have been able to reconstruct a 3D volume as large as 21.5 mm 3 of a 3D organoid culture of prostatic RWPE1 cells showing the ability of these cells to assemble in 3D intricate cellular network at the mesoscopic scale. Importantly, comparisons with 2D images show that it is possible to resolve single cells isolated from the main cellular structure with our lensfree diffractive tomography setup.

Introduction

The study of in vitro cell populations remains a challenging task if one needs to gather large quantitative and systematic data over extended period of time while preserving the integrity of the living sample. As discussed in [START_REF] Scherf | The smart and gentle microscope[END_REF], there is a need for a new microscopy technique that must be label-free and non-phototoxic to be as "gentle" as possible with the sample, and "smart" enough to observe the sample exhaustively at a variety of scales both in space and time. Lensfree video microscopy is addressing these needs in the context of 2D cell culture [START_REF] Vinjimore Kesavan | High-throughput monitoring of major cell functions by means of lensfree video microscopy[END_REF][START_REF] Pushkarsky | Automated single-cell motility analysis on a chip using lensfree microscopy[END_REF][START_REF] Zheng | The ePetri dish, an on-chip cell imaging platform based on subpixel perspective sweeping microscopy (SPSM)[END_REF].

As scientists better understand the benefit of growing organoids in 3D and routinely adopt 3D culture techniques, lensfree imaging must also be adapted to 3D cultures. Therefore, the new challenging task is to extend lensfree microscopy techniques to the acquisitions and 3D reconstructions of large organoids structures [START_REF] Dolega | Label-free analysis of prostate acini-like 3D structures by lensfree imaging[END_REF][START_REF] Isikman | Lens-free optical tomographic microscope with a large imaging volume on a chip[END_REF][START_REF] Isikman | Giga-pixel lensfree holographic microscopy and tomography using color image sensors[END_REF]. The adaptation of lensfree microscopy techniques to 3D organoid cultures imaging is the scope of the present paper.

We first describe an experimental bench dedicated to lensfree diffractive tomography of 3D biological samples. Next, we present a 3D reconstruction algorithm based on diffractive tomographic microscopy methods [8][9][10][11] but applied here to lensfree microscopy. And as a first proof of principle, we present the 3D reconstruction of a RWPE1 prostatic cell culture in 3D within an extra-cellular matrix (Matrigel ® ). The reconstructed volume is as large as 21.5 mm 3 which allows to observe simultaneously a very large cellular networks and isolated cells.

Materials and methods

Cell lines and 3D organoid culture

RWPE1 (ATCC CRL-11609) cells were derived from non-neoplastic human prostate epithelial cells immortalized with human papillomavirus. Cells were maintained in KSFM (Life Technologies, Carlsbad, CA, ref. 17005-075) supplemented with 5 ng/mL epidermal growth factor (EGF) and 50 µg/mL bovine pituitary extract. The cells were maintained in culture until approximately 70% confluency. For passaging, cells were washed with Dulbeccos Ca2+/Mg-free PBS (D-PBS, Life Technologies, ref. according to a no top-coat protocol. Briefly, Matrigel ® was thawed overnight and poured into 4-well (160 µL of Matrigel ® ) or 8-well Labtek ® (90 µL of Matrigel ® ) plates on ice. For polymerization, Matrigel ® was incubated for 30 minutes at 37 • C. Cells were seeded in half the final volume (500 µL or 250 µL of culture media in 4-well or 8-well Labtek ® respectively) and allowed to adhere for approximately 45 minutes. Then culture media was slowly poured over the attached cells and was changed every other day. All cells were routinely cultured in a humidified atmosphere with 5% CO 2 at 37 • C.

Experimental bench

Unlike 2D lensfree imaging, where only one image is required for retrieving the 2D object, the reconstruction of a 3D object from lensfree acquisitions requires to multiply the viewing angles. For this purpose, we have developed an experimental bench, illustrated on 1 and2) and thus the illumination direction is tilted relative to the sensor plane. This configuration is the most adapted to 3D organoid culture in standard containers such as Petri dish or well plates.

Fig. 1. Left-hand side -Experimental bench dedicated to lensfree diffractive tomography. Right-hand side -Optical schema of the system. The semi-coherent incident plane wave U inc passes through the sample volume. Each element of the volume diffracts the incident plane, behaving like secondary spherical sources, creating a diffracted wave U di f . The sensor records the intensity of their summation:

I tot = |U tot | 2 = U inc +U di f 2 2.

Reconstruction methods

In the case of 3D lensfree imaging, the modelling of the wave U di f diffracted by the sample (see Fig. 1), defined in each point r = (x, y, z) of the space, is given by [8]: Fig. 2. Schema of three different acquisitions at the illumination angles ϕ j 1 , ϕ j 2 and ϕ j 3 , with ( j 1 , j 2 , j 3 ) ∈ 1, N 3 , where N is the total number of acquisitions.

U di f ( r) = -1 4π Ob ject f r U inc r exp ik 0 r -r r -r d 3 r = - 1 4π ( f .U inc ) * h (1)
where k 0 = 2πn 0 λ 0 is the wave number in an ambient medium of refractive index n 0 and h ( r) = 1 r exp (ik 0 r ) is the convolution kernel characteristic of spherical wave propagation. The scattering potential f ( r) is the 3D object to be reconstructed. It is related to the complex refractive index of the object n ( r) by the formula:

f ( r) = -k 2 0 n ( r) n 0 2 -1 (2)
These equations are valid when the overall phase delay introduced by the sample is weak (see [9]). In the case of 3D cell culture, this requirement is generally not met since these samples feature objects with dimensions larger than l = 20 µm. In water, for a typical cell refractive index of 1.35, the difference of refractive index is δ n = 0.02, introducing a phase delay of 2πlδ n/λ . In visible light (λ 550 nm), this phase delay is greater than π and therefore cannot be supposed to be weak. In the present work, we decided to use these equations nonetheless to determine if some information, interesting on the biological point of view such as positions, dimensions and shapes of the objects could be estimated. As a drawback of the methodology, it is clear that quantitative results such as refractive indexes or optical lengths cannot be obtained.

Algorithm

Our reconstruction algorithm recovers f based on N multiple acquisitions at different angles (ϕ j ) j∈ 1,N (see Fig. 2), and is based on the Fourier diffraction theorem, widely used in the field of diffractive tomographic microscopy [9][10][11]. This latter links Ûdi f , the 2D Fourier transform of U di f to f , the 3D Fourier transform of the object f . At a given illumination angle ϕ j on the sensor plane z = z s , Eq. ( 1) becomes (see [8,12]):

f α j , β j , γ j = 4iπw • exp (-2iπwz s ) H { k 0 j ;λ 0 } zs Û j di f (u, v; z s ) (3)
with,

         α j = u - n 0 .p j 0 λ 0 β j = v - n 0 .q j 0 λ 0 γ j = w - n 0 .m j 0 λ 0 and w = n 2 0 λ 2 0 -u 2 -v 2
, where (u, v) and (α, β , γ) correspond respectively to the Fourier space coordinates of the sensor and the object. p j 0 , q j 0 , m j 0 is the orientation of the ϕ j illumination, i.e. the orientation of the wave vector k 0 j :      p j 0 = 0 q j 0 = sin ϕ j m j 0 = cos ϕ j Notice that the Fourier transform is defined for a given function g as:

ĝ (u) = ∞ -∞ g(x) exp (-2iπux)dx (4)
Figure 3 illustrates the Fourier diffraction theorem: a geometrical description is that the 2D Fourier transform Û j di f of a given diffracted wave U j di f is mapped on a spherical cap (2D surface, also called Ewald's sphere) depending on k 0 j ; λ 0 inside the 3D frequency space of the object of interest f . To actually map the uniform 3D discrete Fourier space of f , the discrete Fourier transform of each projection is computed. The position of each calculated 2D Fourier coefficient on its spherical cap is then determined and its value is associated to the nearest neighbour into this uniform frequency volume. Once all the 2D Fourier transforms of projections are mapped, it results to gaps within the 3D frequency space, as it can be seen in the medallion of Fig. 3. In this paper, it was decided not to apply any interpolation method to generate a fully filled 3D frequency support. Finally a 3D inverse discrete Fourier transform retrieves the object f .

In sum, the algorithm operates as follows:

f = F -1 3D S H j z s • F 2D M j 0 •U j di f { k j 0 | j=1→N} (5)
where f constitutes the 3D image of size N x × N y × N z to be reconstructed. F -1 3D and F 2D are respectively the inverse 3D and direct 2D discrete Fourier transforms. They are performed by the FFT algorithm. S stands for the mapper into the 3D frequency space of the N j spherical caps corresponding to N j projections U j di f to which a given illumination wave vector k j 0 is associated. The projections are 2D complex images of size N d x × N d y corresponding to the sensor's size, with a pixel pitch denoted ∆ d , which is the same in each direction (see Fig. 1). M j 0 is the modulation of U j di f in the spatial domain corresponding to the translation of the spherical cap from the sensor's coordinate space (u, v) to the object's coordinate space α j , β j , γ j in the frequency domain in Eq. (3) (see Fig. 3): In our case, only the intensity of the total transmitted wave I j tot = |U j tot | 2 is acquired by the sensor, where U tot = U inc +U di f (see Fig. 1 and2). Hence we have to approximate the diffracted wave U j di f from I j tot . To do so, we first normalize I j tot so that the background value, corresponding to the unperturbed incident wave, is equal to 1. We note this normalized projection Ī j tot . We multiply the square root of Ī j tot by a phase ramp exp (i k j 0 . r) to take into account the non-uniform phase shift induced by the tilted incidence of k j 0 relative to the sensor plane. Then we subtract the theoretical incident plane wave U j inc = exp (i k j 0 . r):

M j 0 (x, y) = exp -2iπ n 0 .p j 0 λ 0 x + n 0 .q j 0 λ 0 y (6)
U j di f Ī j tot exp (i k j 0 . r) -U j inc ( 7
)
This formula is schematically represented on Fig. 4. This is a needed step in the presented methods. Indeed, the phase is very important since, inter alia, it contains the inclination of the wavefront. Without this information, the mapping of the Fourier space with Eq. ( 5) cannot work because the computed 2D spectra will be shift in the Fourier space. Equation ( 7) allows to reintroduce the information of the illumination angle in the data even if the simulated phase is just an approximation of the real one.

This calculation does not totally compensate the lack of phase information at the sensor plane. In particular, the phase distortion introduced by the object is not taken into account and could be approximated by means of a phase retrieval algorithm in order to diminish the artefactual twin image [13].

Fig. 4. Illustration of the steps to get an approximation of U j di f for a given j.

Reconstruction of regions of interest

If we take 1.67 × 1.67 × 1.67 µm 3 voxels based on the pixel pitch of the sensor (see Fig. 1), the size of the 3D reconstructed image is very huge -up to several tens of gigabytes -if we want to maintain the same resolution in each direction (N x ≈ N d x , N y ≈ N d y and N z ≈ N d x or N d y ). This could be too expensive for conventional computers and could cause memory overflows. To address this problem, we can extract from the data a region of interest (ROI) and reconstruct it separately. The extraction process of a ROI is illustrated on Fig. 5. The ROI is defined on the projection corresponding to ϕ = 0 • , i.e. where the incidence of illumination is normal to the sensor. Knowing the altitude z s and the angle ϕ j , we can predict where the particular structures in the ROI have been shifted in the other projections, due to the tilted incidence. The shift is theoretically known and equals:

x j 0 = 0 y j 0 = z s tan ϕ j (8)
The centering of the ROI whatever the angle of view implies to modify the algorithm to indicate the absolute position of the ROI on each projection. As it is a translation in the spatial domain, this yields to a modulation of the Fourier transform of the projection as a function of (x j 0 , y j 0 ), giving from Eq. ( 5):

f = F -1 3D S H j z s • D j 0 • F 2D M j 0 •U j di f ROI { k j 0 ,x j 0 ,y j 0 | j=1→N j } (9)
where

D j 0 (u, v) = exp -2iπ(ux j 0 + vy j 0 ) and M j 0 (x, y) = exp -2iπ n 0 .p j 0 λ 0 (x + x j 0 ) + n 0 .q j 0 λ 0 (y + y j 0 ) .
As a first approximation, we consider that the emitted light propagates in a homogeneous medium. As a result, it follows a straight path until it is diffracted by the objects of interest. However, our setup induces several refractive indexes' changes (air, culture medium, glass) along the light path. This causes uncertainties on the effective shift (x j 0 , y j 0 ) as a function of the illumination angle ϕ j . Moreover, the altitude z s is not known absolutely and has to be estimated with a focusing algorithm using Fresnel backpropagation at incidence ϕ = 0 • . As a solution, we use a registration algorithm to correctly center the extracted ROIs so that the projections match the theoretical shift. We only consider rigid translations, ignoring the fact that the holograms are also stretched differently as a function of the light incidence.

Results

We have acquired tomographic data of a culture of prostatic cells RWPE1 in Matrigel ® , over 61 angles from -30 • to 30 • with an angular pitch of 1 • . Figure 5 shows 3 projections at ϕ = -30 • , ϕ = 0 • and ϕ = 30 • . We have reconstructed a large volume of 4 × 4 × 1.34 mm 3 showing that the cells interestingly tend to form a structured network.

To reconstruct such a large volume, we worked by pieces of 1.34 × 1.34 × 1.34 mm 3 , i.e. nine reconstructions. Thus we have divided the data into nine adjacent ROIs of size 1.34 × 1.34 mm 2 (800×800 pixels), as illustrated on Fig. 6. The registration algorithm for the centering of each projection was applied relative to the central ROI (dash red on Fig. 6), as well as the focusing algorithm to estimate z s , the distance between the sensor plane and the cell culture. The eight other ROIs remain adjacent to this one whatever the angle of view, to maintain the continuity in the 3D reconstruction. Each reconstructed 3D image had a size of 800 × 800 × 400 voxels with the sampling rate 1.67 × 1.67 × 3.34 µm 3 . Owing this decoupling task, a single ROI reconstruction has a size of two gigabytes in double precision which can be handled on a desktop computer. As a result the global 3D image has a size of 2400 × 2400 × 400 voxels with the same sampling rate, this whole image having a size close to 20 gigabytes.

The whole reconstructed volume f is shown on Fig. 7. As it is a complex image, we only visualize its imaginary part which gave us the most observable information. Three different Fig. 5. 3D lensfree data of a culture of prostatic cells RWPE1 in Matrigel ® . 61 angles of view were acquired from -30 • to 30 • with an angular pitch of 1 • around the x axis. The illumination wavelength is 520 nm. A region of interest (ROI) is extracted on each projection and centered by registration. This aims at performing a localized reconstruction to save computation from RAM overflows. The centering has to be taken into account in the algorithm (Eq. ( 5)).

sectional views -xy (center), xz (right) and yz (bottom) -and 3D views (see Visualizations 1 and 2) show that the network formed by the cells is well focused at precise altitudes in the volume. The network fits a surface which is not horizontal, which demonstrates that the reconstruction algorithm effectively operates in 3D: isolated objects are also focused apart from the network at independent altitudes. Some of them are pointed out by red arrows on Fig. 7.

The central ROI and the ROI in the top-right corner of the reconstructed volume shown on Fig. 7 (equivalently the central and bottom-right ROIs on the raw data on Fig. 6) and are further detailed on Fig. 8 and9. Each of these figures represents the three sectional views -xy (center), xz (right) and yz (bottom) -and a 3D view of the ROI and two specific patterns such as a cell or an organoid which are pointed by arrows in the ROIs. The rings around the focused objects on the xy views are artefactual: this is the well-known artefactual phenomenon of twin image, induced by the lack of absolute phase information in the data.

To analyse the performances of our setup and 3D reconstruction algorithm to recover unambiguously biological objects, a comparative study with 2D reconstructions from 2D lensfree data acquired at the normal illumination (ϕ = 0 • ) was conducted. The algorithm used for performing the 2D reconstructions has been presented in [START_REF] Vinjimore Kesavan | High-throughput monitoring of major cell functions by means of lensfree video microscopy[END_REF], and has already shown, as a 2D lensfree imaging application, its ability to recover single cells and organoids while removing the twin images [START_REF] Vinjimore Kesavan | High-throughput monitoring of major cell functions by means of lensfree video microscopy[END_REF][START_REF] Dolega | Label-free analysis of prostate acini-like 3D structures by lensfree imaging[END_REF]. Figure 10 shows the 2D reconstruction of a region of the studied field of view. We can observe the efficiency of the reconstruction algorithm to recover the biological scene showing branching networks, single organoids and cells. To perform the comparisons with the 3D reconstruction, three objects were selected from the objects highlighted Fig. 8 and9: two cells (see ROI 1 and ROI 2 on Fig. 10) and one organoid (see ROI 3 on Fig. 10). Two axial cut profiles respectively in x and y directions were taken for both the 2D and 3D reconstructions, and one cut profile in the z direction for the 3D reconstruction. These cut profiles are illustrated on the three graphs on Fig. 10. As a first observation, we can see that axial profiles are globally equivalent in 2D and 3D, allowing to measure approximatively cell sizes of ∼ 15 µm for ROI 2 and ∼ 30 µm for ROI 1 considering the width of the central lobe of the profiles (the secondary lobes are known to be twin artefacts), which is consistent with typical cell sizes. For the organoid, the profiles are more erratic but we can distinguish a kind of a plateau curve, the width of which is measured at ∼ 90 µm. As expected, the observations are different in the z direction and the resolution is clearly degraded: whereas the objects must have a global isotropic behaviour in terms of sizes in 3D, our measures give ∼ 80 µm for the cell in ROI 1, ∼ 115 µm for the cell in ROI 2 and ∼ 200 µm for the organoid in ROI 3. Moreover, the disparities of the measures for the cells tend to show that the apparent resolution highly depend on the position of the object in the field of view: the more we go away from the centre (from ROI 1 to ROI 2), the worse is the resolution in the z direction. Note that the term of "resolution" that we employ has not to be confused with the strict definition of optical resolution. These are empirical measures that allow us to conclude that single biological objects are effectively and unambiguously identified in the 3D reconstruction.

Uncertainties in the calibration of the setup and the projection model can explain some artefacts in the image and the previous quantifications of Fig. 10. For example we did not take into account the stretching of holograms in the registration of the projections which depends of the lighting direction. Thus some misalignment can be present after the registration step. Moreover uncertainties remain on these lighting directions, due to the multiple refractions in between the different propagating media (air, culture medium, sensor glass, etc.) which change the illumination orientations. The induced artefacts then get worse as the distance from the center increases since the mistakes on the objects positioning grow with this distance. That is why the artefactual bow shape around the single objects get worse on the side, as one can see comparing Fig. 8 and 9 and the profiles on Fig. 10.

These critical points highlighted by our demonstrative results will be addressed in future works to allow us a precise quantification of the performances of 3D lensfree imaging, e.g. the effective optical resolution in all directions.

Conclusion

In this paper, we have presented a novel technique for the imaging of 3D organoid cultures: lensfree diffractive tomography. Based on lensfree imaging and digital holography principles, we have demonstrated that it is possible to reconstruct large volumes up to several mm 3 , by multiplying the angles of view around the object of interest. We have developed a reconstruction algorithm based on the direct inversion of the data in the frequency domain using the known Fourier diffraction theorem. We have reconstructed a large volume (4 × 4 × 1.34 mm 3 ) of a culture of RWPE1 prostatic cells in Matrigel ® . As a result we have been able to observe focused objects of a few tens of microns (cells and network branches).

These first results tend to demonstrate that it is possible to acquire knowledge on the posi- tions and shapes of elements present in 3D cell culture, e.g. spheroids, branching and single cells, on a large and thus relevant volume (several mm 3 ). Acquiring such information is of prior importance for the biologists in the realms of e.g. embryogenesis, angiogenesis and tumorigenesis. In particular the ability to detect isolated single cell distant from the main cell structure, since these cells may present a metastatic phenotype of interest.

  14190) and incubated with 1 mL trypsin-EDTA (Lonza, Basel, CH, ref. CC-5012, 0.25 mg/mL) for approximately 7 minutes. The trypsin was neutralized with 2 mL trypsin neutralizing solution (Lonza, ref. CC-5002), and the cells were recovered by centrifugation and counted using a Scepter 2.0 Handheld Automated Cell Counter (Millipore, Billerica, MA, ref. PHCC 20060). The RWPE1 cells were cultured in 3D with KSFM (Life Technologies, ref. 17005-075) supplemented with 50 ng/mL EGF and 2% fetal bovine serum (FBS). The 3D culture was grown in Matrigel ® (BD Biosciences, San Jose, CA, ref. 356231)

Fig. 1 .

 1 It is composed of a semi-coherent illumination source (LED CREE, λ = 520 nmref. XLamp MC-E RGBW MCE4CT) and CMOS sensor (IDS -29.4 mm 2 , 3840 × 2748 monochromatic pixels, pixel pitch 1.67 µm -ref. UI-1942LE-M). The object and the sensor are static, the object being placed on top of the sensor at a distance of 1 to 3 mm. The source is rotated with a rotation stage (Newport -ref. M-URM80APP, controlled by a Newport universal motion controller -ref. ESP300) about the ϕ-axis (see Fig.

Fig. 3 . 1 di f , U j 2 di f and U j 3

 3123 Fig. 3. Illustration of the Fourier diffraction theorem with the notations of Fig. 2. The 3D frequency space of the object of interest f is mapped with the 2D Fourier transform of the projections U j 1 di f , U j 2 di f and U j 3 di f . An example of a Fourier region which is actually filled by the algorithm is given in the medallion in the 3D frequency domain part of the figure with the parameters used in section 3 to reconstruct a given region of interest: a volume of 1.34 × 1.34 × 1.34 mm 3 with a voxel size of 1.67 × 1.67 × 3.34 µm 3 at λ = 520 nm with an illumination angle varying from -30 • (blue) to 30 • (green) with a step angle of 5 • . The red cap corresponds to the region map with the normal illumination.

Fig. 6 .

 6 Fig. 6. Extraction of 9 ROIs of size 1.34 × 1.34 mm 2 (800 × 800 pixels) for the piecewise 3D reconstruction of the 3D culture of prostatic cells RWPE1 presented on Fig. 5.

Fig. 7 .

 7 Fig. 7. Piecewise 3D reconstruction (imaginary part of f ) of a large volume (4 × 4 × 1.34 mm 3 ) of the culture of prostatic cells RWPE1 from tomographic data presented on Fig. 5. (a) 3 sectional views -xy (center), xz (right) and yz (bottom) -are shown, each one focused at a given distance d = z sz from the sensor. (b) 3D views of the reconstruction from different observation points (see Visualizations 1 and 2). The dash red curves indicate the tilt of the network relative to the horizontal plane. The red arrows point out isolated objects that are not focused at the same altitude as the network.

Fig. 8 .

 8 Fig. 8. ROI at the center of the piecewise 3D reconstruction of the volume of Fig. 7. (a) 3 sectional views -xy (center), xz (right) and yz (bottom) -and a 3D view of the ROI. (a) same representation rules on two specific organoids at the full resolution of the reconstruction (see Visualizations 3, 4 and 5). These are pointed by arrows on the global view. The dashedline indicates the network plane. The red arrow points toward a 33 × 38 × 42 µm 3 object and the blue arrow toward a 16 × 17 × 40 µm 3 object.

Fig. 9 .

 9 Fig. 9. ROI at the to right corner of the piecewise 3D reconstruction of the volume of Fig. 7. (a) 3 sectional views -xy (center), xz (right) and yz (bottom) -and a 3D view of the ROI. (b) same representation rules on two specific organoids at the full resolution of the reconstruction (see Visualizations 6, 7 and 8). These are pointed by arrows on the global view. The dashed-line indicates the network plane. The red arrow points toward a 12 × 43 × 96 µm 3 object and the blue arrow toward a 80 × 80 × 100 µm 3 object.

Fig. 10 .

 10 Fig. 10. 2D reconstruction from 2D lensfree data acquired on the 3D culture of prostatic cells RWPE1 presented on Fig. 5. Comparisons of two axial cut profiles respectively in x and y directions taken on both 2D and 3D reconstructions, and one cut profile in the z direction taken on the 3D reconstruction, for three biological objects: two cells, one identified on Fig. 8 (ROI 1 in red) and one identified on Fig. 9 (ROI 2 in blue) and one organoid identified on Fig. 9 (ROI 3 in green). The dashed and solid curves correspond respectively to the 2D and 3D reconstruction cut profiles.