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Abstract
Tomographic diffractive microscopy (TDM) based on scalar light-field approximation is widely implemented. Samples exhibiting anisotropic structures, however, necessitate accounting for the vectorial nature of light, leading to 3-D quantitative polarimetric imaging. In this work, we have developed a high-numerical aperture (at both illumination and detection) Jones TDM system, with detection multiplexing via a polarized array sensor (PAS), for imaging optically birefringent samples at high resolution. The method is first studied through image simulations. To validate our setup, an experiment using a sample containing both birefringent and non-birefringent objects is performed. Araneus diadematus spider silk fiber and Pinna nobilis oyster shell crystals are finally studied, allowing us to assess both birefringence and fast-axis orientation maps.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization imaging is a promising technique in several fields, and hence gaining much attention, especially due to its capability of identifying structures that are unseen by conventional imaging. Applications of polarization imaging range from macroscopic to microscopic scales, and from the simplest and robust to unique and comprehensive techniques [1]. Single cell study and tissue polarization imaging [2], as well as clinical applications of such imaging technologies, mainly for qualitative analyzing of biological samples in large Field of View (FOV) [3], are also of great interest.
In recent years, several research groups have shown interest towards quantitative 2-D Jones matrix polarization sensitive holography [4–10]. For instance, such an approach has been used to study anisotropic samples in 2-D holographic microscopy for live cell imaging [5], and more recently, synthetic aperture holographic tomography has been developed for noninvasive imaging of cancer cells, with better sensitivity compared to standard microholography [10].
For 3-D imaging, polarization sensitive optical coherence tomography [11], 3-D polarized light imaging [12] and fluorescence confocal microscopy [13] are the well established techniques. While these approaches have shown their invaluable importance for clinical application along with their sub micrometer resolutions, quantification of birefringence in 3-D is difficult, and only qualitative anisotropic contrast is revealed. Recently, large scale polarization contrast TDM in transmission configuration has been implemented for improving image contrast of anisotropic specimens such as zebrafish embryos [14].
Three-dimensional and truly quantitative polarimetric imaging modalities are less common. Proposed by E. Wolf in 1969 [15], TDM under scalar approximation has been used in several research areas [16–22]. However, such configurations fail at quantifying samples with anisotropic nature. Although V. Lauer reformulated Wolf’s work in 2002, providing full vectorial formulation by accounting for polarization of light [23], it is only very recently that polarimetric TDM regained attention.
Advances towards polarization sensitive TDM have confirmed the importance of quantitative 3-D polarimetric approaches. Polarization was first taken into account in reflection TDM by Zhang in 2013, [24], and recently polarization sensitive transmission TDM configurations have also been implemented [25,26].
In general, experimental implementations and image reconstruction processes of polarization sensitive TDM to image weakly birefringent specimens, such as biological sample (cells, cell structure, and tissue), with high-resolution and good sensitivity will need to be well explored and understood before becoming a routine technique. In this work, we have developed a high-resolution polarization sensitive TDM (PS-TDM) based on a Polarized Array Sensor (PAS) to image weakly birefringent samples. The use of a PAS paves the way for implementing a simplified Jones and/or Mueller matrix PS-TDM by multiplexing data acquisition. Image simulations of a synthetic sample as well as experiments using a sample containing both birefringent and non-birefringent objects is performed to validate the system.
2. Theory
2.1 Vectorial Born approximation
The vectorial Helmholtz equation expresses a monochromatic light-wave propagating through a non-magnetic and weakly scattering anisotropic object embedded in a homogeneous and isotropic background medium, and writes as:
The total field vector is given as the sum of scattered $\textbf {E}_s(\textbf {r})$ and incident field $\textbf {E}_i(\textbf {r})$. The simplified solution for Eq. (1) writes as [23,25]:
Equation (3) is further simplified by considering a weakly scattering birefringent sample, the total field vector inside this sample can be regarded as the incident field vector (which is the consequence of first order Born approximation [27]), i.e. $\textbf {E}_s(\textbf {r}) \ll \textbf {E}_i(\textbf {r})$ $=>$ $\textbf {E}_t(\textbf {r}) \approx \textbf {E}_i(\textbf {r})$.
It should be noted that, under scalar approximation, it is possible to easily adapt these developments to Rytov or Born approximation, as studied in literature. The Rytov complex field is given by $\textbf {E}_r\left (\textbf {r}\right ) = \ln (\textbf {E}_B(\textbf {r}))$. Imaginary and real parts are computed as $\Im \left \{\textbf {E}_r\left (\textbf {r}\right )\right \}=\arg \left [\textbf {E}_t/\textbf {E}_i\right ]$, and $\Re \left \{\textbf {E}_r\left (\textbf {r}\right )\right \}=\ln (|\textbf {E}_t|/|\textbf {E}_i|)$, respectively. In that case, the phase difference $\phi _t-\phi _i = \arg \left [\textbf {E}_t/\textbf {E}_i\right ]$ has to be unwrapped.
2.2 Jones formalism
Quantitative polarimetric imaging techniques relying on a coherent light commonly utilize the Jones formalism to relate the input and output fields [5,10,28,29]. Aiming at retrieving the scattered field amplitude, we start by developing approximated mathematical expression for PS-TDM imaging using Jones calculus. The Jones tensor $\textbf {J}_\textrm{obj}$ of the birefringent sample under examination relates the scattered field vector $\textbf {E}_s(\textbf {r})$ and the incident field vector $\textbf {E}_i(\textbf {r})$ as:
Recalling Born approximation given in Eq. (6), one gets:
Following vectorial first order Born approximation, in the same way as for scalar approximation [15], one gets a linear relationship between the Jones tensor components and the sample’s permittivity tensor of the anisotropic sample:
From now on, we approximate the RI tensor by neglecting its third component, as in [25]. Such approximation is valid for anisotropic objects exhibiting weak polarization response along the optical axis. Doing so makes it possible to compute all the 9 components of $\tilde {\textbf {V}}(\textbf {k} - \textbf {k}_i)^{[3\times 3]}$ with only two independent polarization components, as one is able to measure only the x and y components of the scattered field using a PS-TDM. As a consequence, Eq. (10) now reduces to a second order tensor:
Jones matrix analysis of birefringent samples necessitates illumination along two independent incident polarization states ($x$ and $y$), as well as analysis of the scattered light along these two axes. Hence, for full Jones matrix retrieval, four measurement are usually required [14,25,26], which reduces speed, and can also decrease sensitivity due to measurement artifacts.
A first advantage of using a PAS as detector, providing mosaicked images in a single acquisition for various polarization states, is that data acquisition can be simplified, by multiplexing detection.
Also, for interferometric systems, linearly cross-polarized configurations lead to zero background intensity, making phase estimation of the reference beam ambiguous. This results in a random background phase, which makes phase unwrapping challenging [25,30]. Conversely, the use of circularly polarized beam greatly alleviates this problem, since it results in a non-zero background intensity, allowing for easier phase unwrapping. In such configuration, the sample is sequentially illuminated with right-, then left-circular polarization. We therefore now extend the formalism proposed in [25] to the case of circular polarization illumination.
Let us symbolize the output field at each micro-analyzer’s orientation:
- • $E^o_{1x}$: right-circular illumination and detection at $0^\circ$ polarization
- • $E^o_{1y}$: right-circular illumination and detection at $90^\circ$ polarization
- • $E^o_{2x}$: left-circular illumination and detection at $0^\circ$ polarization
- • $E^o_{2y}$: left-circular illumination and detection at $90^\circ$ polarization
Fig. 1. Schematics representing the incident field’s polarization for an oblique illumination of wave-vector $\textbf {k}_{i}$: $z-axis$ is the normal incident beam direction. $\textbf {E}^{xy}_{i}$ is the incident field $\textbf {E}_{i}$ projected at $xy$ plane. $\theta$ and $\phi$ are respectively the polar and azimuthal angle.
The above expression provides a linear relationship between the scattered field and the birefringent sample’s Jones tensor components. For light illuminating the specimen at normal incidence, and propagating in $z$ direction, the illumination beam has either $x$ and/or $y$ polarization components, and can be written as:
$\hat {e}_x$ and $\hat {e}_y$ being unit vectors along $x$ and $y$ axis. $E_{i_x}$ and $E_{i_y}$ are complex input fields, with amplitude and phase.But in illumination rotation TDM configuration, the object is illuminated with various angular directions. Hence, to correctly extract the polarization information for each illumination beam, one has to decompose it into its respective polarization contributions. To do so, we use the approach proposed in [25], introducing the rotation matrix $\textbf {R}_m$ as an orthogonal $3\times 3$ matrix satisfying $\textbf {R}_m^{-1} = \textbf {R}_m^T$ and $\textbf {R}_m \textbf {R}_m^T = \textbf {R}_m^T \textbf {R}_m = \mathbb {I}$:
For transverse electric field (TE) $E_{inp_z}$ is always zero for the normal incidence but non zero for illumination at an arbitrary angle. Once the sample is illuminated by a polarized plane wave, the diffracted field passes through the high NA objective lens before it is measured by the polarization array sensor. The measured electric field, $\textbf {E}_m$, is then computed from the scattered field just before the objective by implementing the same procedure as above,
Finally the information recorded by the polarization array sensor along various micro-analyzers axes can be computed as:
where $\hat {a}_x$ and $\hat {a}_y$ are unit vectors along $0^\circ$ and $90^\circ$ micro-analyzer axis.2.3 Hologram demosaicking and Interpolation
The polarization array sensor we use consists in a CMOS sensor with polarization filters deposited on each photo-receptor, in groups of 4, oriented at 0, 45$^\circ$, 90$^\circ$ and 135$^\circ$, respectively. It provides sub-sampled images in a $2\times 2$ neighborhood represented by 4 pixels, referred to as a "super pixel". Once polarization sensitive holograms are acquired for several illumination directions and polarizations, pixel-wise demosaicking of those holograms is performed. This allows for decomposing each hologram onto the respective micro-analyzer axis. Schematic representation of the pre-processing workflow is depicted in Fig. 2.
One main limitation of such polarization array sensor are the different instantaneous field of views seen by each sub-pixels of the super pixel, inducing errors in the demosaicked holograms [31]. To compensate for this drawback, and reconstruct accurate high-resolution polarization sensitive holograms, a 2-D image interpolation is performed on the four demosaicked polarization sensitive holograms, using bicubic interpolation. Although bicubic interpolation utilizes 4 neighboring pixels, therefore needing more computation time compared to bilinear or nearest-neighbor interpolation techniques [32], we selected this approach since it provides smoother images with less interpolation artifacts [31].
An important consideration for optimal image reconstruction is to fulfill Nyquist sampling requirement [23]. For example, compared to our setup as described in [33], sampling has to be tuned, when switching from a standard (polarization insensitive) CMOS sensor to a polarization sensitive camera, to satisfy Nyquist criterion for each sub-image. This is performed by adapting magnification in the detection arm of the interferometer (see also section 4). Doing so ensures that hologram demosaicking and interpolation, as previously described, delivers images containing maximal frequencies as allowed by the optical system.
2.4 Other polarimetric information
Unlike other 2-D polarimetric techniques that need insights about the thickness of the sample (since they only measure phase changes) [28,34], thanks to holographic imaging and numerical reconstruction based on first order Born approximation, PS-TDM can directly estimates the 3-D refractive index tensor $\overline{\overline{n}}$ of weakly scattering samples. This allows for estimating other polarimetric information by using further mathematical simplification. For instance, for a homogeneous anisotropic sample, it is possible to implement pixel-by-pixel diagonalization of the RI tensor to obtain two principal refractive indices, $n_1$ and $n_2$. This can be given as:
From the above expression, it is possible to obtain several 3-D polarimetric information, such as birefringence $\Delta \overline{\overline{n}}$, average RI $n_{avg}$, and fast-axis orientation $\theta$:
Note that to get an orientation map, $\theta \in [-\pi,\pi ]$, one needs to use $\arctan 2$ function and the computed map would be prone to phase wrapping problem. Under the Rytov approximation, the algorithm needs the phase as an input data, hence a phase unwrapping algorithm is implemented [35].
As pointed out by one of the anonymous reviewers, a point of attention is the following: as light propagates through a birefringent sample, polarization rotates proportionally to the path-length. This lead to a phase jump, forbidding detection to discern polarization states that are rotated by an angle $\phi$ from those rotated by an angle $\phi$+$n2\pi$ (n being an integer). To avoid this ambiguity, we here restrict investigations to small and low birefringent samples.
3. Image reconstruction simulations
3.1 Forward model
Before experimental investigations, image simulations are conducted to validate the PS-TDM method by mimicking experimental conditions. Thus, Vectorial Beam Propagation Method (V-BPM) is implemented as a forward model to propagate the incident field through the considered phantoms. It provides a compromise between propagation accuracy and computation time, compared to finite-difference time-domain method (FDTD) and Born approximation [25,26,36]. Afterwards, the inversion algorithm that is used to reconstructed the refractive index tensor components from the vectorial scattered field is based on first order approximation. V-BPM necessitates dividing the 3-D sample into finitely small thickness 2-D slices, which allows to propagate the incident light beam through these slices.
The V-BPM model considering a slowly varying electromagnetic amplitude (SVEA) wave is given as [25]:
3.2 Simulation parameters
The sample is assumed to be illuminated by a polarized plane wave of wavelength $\lambda = 633\,nm$, and with unit amplitude. Simulations are performed in a $x\times y\times z= 28\times 28 \times 28\,\mu \rm {m} ^3$ volume. The volume is sampled into $512 \times 512 \times 512$ voxels of dimensions $\Delta x= \Delta y = \Delta z = 55\,\rm {nm}$. In all simulations, the sample is assumed to be a homogeneous anisotropic sample, in which the RI tensor can be diagonalized into the principal refractive indices.
Using the V-BPM forward model, 100 holograms are generated for each polarization states using a rose scanning (8 overlapping petals, as given in [33,37]) scheme within a cone of illumination whose polar angle is between $\alpha _ {max} = [ -67^\circ, 67^\circ ]$, corresponding to our experimental setup. Such illumination scheme is chosen due to its simplified analytical expression, which helps to easily generate the incident beam at several directions. Once the diffracted field vectors are computed using V-BPM as forward model, the Jones matrix components are retrieved. Vectorial Rytov approximation is then implemented to calculate the synthetic object’s 3-D RI tensor components, which are processed to give the principal refractive indexes.
3.3 Simulation results
Image simulations have been performed using a home-developed Python code (Python 3.10.0). To do so, a spherical bead containing an ellipsoidal anisotropic structures is considered. The (central slices) of the computed refractive index distributions that are retrieved from the reconstructed RI tensor components are shown in Fig. 3. One can observe blurring around those structures that are multiple scattering layers and around the edges, which results from the inversion algorithm based on Rytov approximation single-scattering assumption. Also, underestimation of the refractive index is observed, as can be confirmed from the RI profile plot along the $y$-axis. This is expected, since only 100 holograms are used. Even if we increase the number of holograms, employing vectorial Rytov approximation as inversion algorithm affects the image reconstruction efficiency. Such behavior is often observed when using simpler reconstruction methods such as Born or Rytov linear inversions [33,37], and clearly constitutes a limit, but those methods have the advantage of being very fast, potentially allowing for real-time acquisition-reconstruction-display of the images, which would be interesting for high-throughput screening [38]. One can also observe an elongation along the optical axis due to the known missing-cone problem, characteristic of all (non-confocal) transmission microscopes [39]. Finally, note that due to intermixing of information, residual of the ellipsoidal structure from $\delta n_1$ is observed in the $\delta n_2$ image.
Fig. 3. Simulation results of a spherical anisotropic bead, with two principal RIs ($\delta n_1$ and $\delta n_2$). The images are the central slices of the 3-D object. The RI profile plot is taken from $\delta n_1$ along the $y$-axis. Scale bar is $2\,\mu \rm {m}$.
4. Experimental setup
The experimental setup that we developed is an improvement of our previous systems [20,33,38], based on the Mach-Zehnder interferometer, as depicted in Fig. 4. The laser beam (He-Ne, wavelength $\lambda$ = 633 nm) is split into a reference beam and illumination beam, using a beam splitting optical fiber. In order to reduce polarization fluctuations of the reference beam, it is fed through a polarization maintaining optical fiber from Thorlabs (PM460-HP). Both reference and illumination beams are collimated.
Fig. 4. Experimental setup of high-resolution polarization sensitive TDM based on off-axis digital holographic microscope; CL: collimating lens, $\rm {M}_{1,2}$: Mirrors, $\rm {PC}_{1,2}$: Polarizing components, BS: beam spliter, PS-camera: polarization sensitive Camera, $\rm {L}_{1,2,3,4}$: Lenses, and RC: recombining component.
Then, a coherent plane wave exhibiting circular polarization is generated by using polarizing components. The polarization generating components, mounted in both illumination and reference arms, consist of a linear polarizer and a quarter waveplate. The circularly polarized illumination beam is then deflected by the tip/tilt mirror (Newport FSM300 Fast Steering Mirror), and angularly scans the sample, sandwiched between two microscope objectives (oil immersion $n_{imm}$ = 1.518, NA = 1.4 100$\times$ Olympus UPLFLN100XO2-2), the first serving as high-magnification condenser, the second as the detection objective.
Once the sample is illuminated by the plane waves with varying angular direction and polarization, the scattered field and reference beam are recombined by the recombining cube to make polarization sensitive holograms that are finally detected by the Polarized Array Sensor (PAS) from LUCID Vision Labs (PHX050S-PC). The PAS with micro-analyzers allows to record mosaicked holograms corresponding to 4 linearly polarization states, parallel, perpendicular, as well as at $45^\circ$ and $135^\circ$ polarization orientations, without the need for mechanical rotation of the analyzer. Lenses $\rm {L}_3 - \rm {L}_4$ are to be chosen so as to properly satisfy Nyquist sampling for each sub-image on the PAS.
Automation of the acquisitions have been realized using a homemade C++ acquisition software. This software uses the following libraries: eBUS SDK from Pleora Technologies [40] for managing hologram acquisitions via the PAS, and a Labjack (Labjack U3 DAC) [41] to control the FSM300 tip-tilt mirror, which delivers controlling signal to the FSM driver. This acquisition software also allows for quick visualization of the illumination scanning, as well as of the back focal plane of the detection objective, to ensure optimal scanning and detection. A GPU reconstruction code allows for a fast reconstruction and previsualization.
It should be noted acquisition multiplexing could also have been performed using a conventional sensor, considering two orthogonally polarized reference beams generated using a polarized beam splitter [42] or a Wollaston prism [43]. This approach makes it possible to maintain the image field of view, with an imaging resolution limited by the microscope objective numerical aperture. To obtain the same results with a PAS, acquired images should be twice as big as the one obtained in a Fourier multiplexing framework [43]. Nevertheless, the choice of a PAS opens new outlooks such as Mueller imaging, or DIC imaging [44] with no modification of the experimental configuration, thus improving the versatility of the proposed arrangement. It also translates into a simpler experimental arrangement, as only the sensor is modified compared to a regular TDM experiment.
5. Experimental results
5.1 Hologram pre-processing
Once mosaicked polarization sensitive holograms are recorded by the PSA, pixel-by-pixel demosaicking, and image interpolation are performed. Note that by making use of the rotation matrix, each interpolated hologram is then decomposed into its respective polarization. Same procedure applies for the holograms that are acquired without the sample (blank acquisitions) but only with the background medium, which are used as references to correct the holograms with the procedure presented in [45]. Afterwards, each decomposed hologram is processed independently, similar to the scalar TDM reconstruction process.
5.2 Potato starch granules and silica beads
To validate our PS-TDM system, we have prepared a sample made of potato starch granules, which are well-investigated birefringent micro-objects [46], mixed with $5\,\mu \textrm{m}$ isotropic silica beads, embedded in an isotropic background medium (Eukitt from Sigma Aldrich) with refractive index $n_0 = 1.49$, and sandwiched between two cover glasses of thickness $170\, \mu \textrm{m}$. To balance between image acquisition/reconstruction speed and image quality, the sample is illuminated with 400 angles using the optimized angular sample scanning scheme, 3D-UDHS from [33,37]. Image demosaicking, then interpolations were conducted to retrieve the complex field corresponding to $x$ and $y$ polarization. Note that interpolation is critical, since a PAS introduces a phase shift and spatial-shear between two orthogonal polarized light. This effect can compromise PS-TDM image reconstruction, but it can be on the contrary beneficial in some cases, leading to a differential interference contrast (DIC) looking-like effect (see [44]).
Figure 5 shows the processed 2-D phase and amplitude maps of the Jones matrix elements of potato starch granules (anisotropic object) and silica beads (isotropic object) in the same field of view. Note that the isotropic silica beads are not visible in the off-diagonal Jones components. Unlike the silica beads, the potato starch granules structures are visible in all Jones components.
Fig. 5. Computed 2-D Jones tensor maps: (left) amplitude, and (right) phase values of potato starch granules (large spherical objects, visible in all components), and silica beads (not visible in the diagonal components). Scale bar is 10 $\mu \textrm{m}$.
Figure 6 shows x,y (first row) and x,z (second row) slices of the reconstructed RI tensor components of potato starch granules and silica beads in the same field of view (FOV). Note that $n_{xx}$ and $n_{yy}$ RI tensor components do not provide discrimination between isotropic and anisotropic objects. They show either the ordinary or extraordinary components of the RI. Conversely, $n_{xy}$ and $n_{yx}$ only show the potato starch granules, apart from some residual of the beads, coming from diffraction artifacts at the bead boundaries. The circularly varying crystalline structures of the potato starches are clearly visible along the off-diagonal RI tensor components, as so-called Maltese crosses.
Fig. 6. Reconstructed RI tensor components of potato starch granules, and silica beads in the same field of view; scale bar is $10\mu m$. (See Visualization 1 for details)
To the best of our knowledge, such mixed-sample acquisitions have not yet been presented, while in fact they ensure that the observed (computed) birefringence is not a consequence of potential image reconstruction artifacts, thus validating both our experimental setup, and our image reconstruction method. This polarization approach can also contribute increasing image contrast by clearly separating birefringent and non-birefringent elements of a sample, besides estimating the 3-D RI tensor.
5.3 Araneus diadematus silk fiber
So far, we illustrated benefits of our approach considering potato starch granules mixed with non birefringent beads, proving the ability to separate isotropic from anisotropic sample. In this subsection, we analyze the polarimetric information of a spider silk fiber. We selected Araneus diadematus silk fiber from dragline (radial line) of the web [47]. Such a sample is much smaller ($1\mu m$ or less) than potato starches, and known for exhibiting anisotropic structure at micrometer scale [48], which should be revealed using polarization sensitive TDM. The sample is immersed in Eukitt, and imaged with the same experimental conditions as that of Figs. 5, and 6
Obtained results are shown in Fig. 7. Such sample is thin and small enough to comply with both Born and weakly birefringent hypotheses. The reconstructed RI tensor image, shown in Fig. 7(a), emphasizes the anisotropic nature of the spider silk, which is proven by the presence of signal in the off-diagonal RI tensor components ($n_{xy}$ and $n_{yx}$).
Fig. 7. Reconstructed 3-D polarimetric information of Araneus diadematus spider silk. (a) RI tensor components, (b) birefringence, (c) average RI and (d) fast-axis orientation. Scale bar is $10\mu m$ (a) and $2\mu m$ (b-d).
Further analysis is brought by computing the average RI $n_{avg}$, the birefringence of the silk $\Delta \overline{\overline{n}}$, and the fast axis orientation $\theta$ that are given by Eqs. (24), (23), and (25), respectively. Obtained results are illustrated in Figs. 7(b-d). Here, it should be noted that the rapid RI oscillations around the silk (from 1.45 to 1.5) are associated with a commonly known Rytov reconstruction artifact. However, the average RI seems to be constant throughout the silk. Conversely, non-homogeneous structure is observed in the birefringence map of the silk fiber, as pointed out by a white arrow (see 7(c)).
We finally can mention the fast-axis orientation map (Fig. 7(d)) showing that orientation of the silk core is along the fiber axis, consistent with structural data [49], but seems to differ from the orientation of its shell (see Fig. 2 in [50]). Interestingly, such orientation image results from computations, which eliminate reconstruction artifacts as seen on RI images. This could be of interest to obtain better sample discrimination from the background, but also possibly better resolved images.
5.4 Pinna nobilis oyster shell
To further demonstrate the performance of our PS-TDM, Pinna nobilis oyster shells are investigated. Oyster shells are one example of a mesocrystalline material made out of calcium carbonate. Structurally, the shell is made from single crystal calcite prisms, each prism being surrounded by organic walls (thin layer of amorphous calcium carbonate). Such structures are widely studied to hypothesize realistic biomineralization scenarios [51,52]. Such crystals are well suited for observation using 2-D microscopy (for example by 2-D vectorial ptychography [51]), as they are of relatively adequate flatness and moderate thickness. Since they are also made of semi-transparent anisotropic material, they are well suited to be imaged by our PS-TDM.
The Pinna nobilis shell pieces were sampled in October 2013 from specimen living in the Mediterranean Sea (Giens, France). No animal was captured or killed. Small, millimeter-sized pieces were removed in situ from the spines covering the outer shell, in the vicinity of the shell growth margin, and preserved in a 70% ethanol/water solution (see [53] for detailed explanations).
The reconstructed RI tensor components of the shell depicted in Fig. 8(a) clearly show the microstructure of the shell relative to the background medium. We observe an inhomogeneous structure of the prisms, and the RI tensor component of one prism varies from another. Note also the Region of Interest (ROI) depicted with white rectangles along the $x-z$ cross-section, where we observe bright colors (showing a positive RI contrast) relative to the background medium in both $n_{xy}$ and $n_{yx}$ RI tensor components. The same ROI shows dark structures in $n_{xx}$ and $n_{yy}$ (showing a negative RI contrast). Such phenomena is also observed on potato starch granules, see Fig. 6.
Fig. 8. Reconstructed 3-D polarimetric information of Pinna nobilis oyster shell. (a) RI tensor components, (b) birefringence, (c) average RI and (d) fast-axis orientation. Scale bar is $20\,\mu m$.
Other 3-D optical parameters are also computed to gain more insight about the polarimetric properties of such sample. The birefringence map in Fig. 8(b) reveals an optically inhomogeneous structure for the various prisms. We also observe smooth transition between adjacent prisms. These results are also confirmed by the computed average RI map, as shown in Fig. 8(c), as well as the fast-axis-orientation depicted Fig. 8(d).
Similar results have been obtained using optical vectorial ptychography (see Fig. 5 in [53]), but, up-to-now, at lower magnification/resolution than with our system, which uses high-numerical aperture objectives, the drawback being a reduced field of view compared to [53].
6. Conclusion
Experimental realization of quantitative 3-D polarimetric imaging approaches to image samples exhibiting low birefringence, such as biological samples, necessitates the development of highly-sensitive optical systems. We have developed a simplified semi-automated high-resolution PS-TDM, which, thanks to the use of a PAS detector, multiplexes acquisitions, gaining a factor 2 in data collection. Full automation of the PS-TDM system could be done either by using motorized rotation stages or liquid crystal retarders in the polarization generation. Validation of our inversion algorithm based on Rytov approximation is done using synthetic data. The ability of our PS-TDM experimental setup to image birefringent specimens is validated using first a test samples made from birefringent and non-birefringent objects, then investigating a much smaller object such a spider silk fiber, and then tested onto biomineralized microcrystals.
Further analysis of our PS-TDM performances will require the use of calibrated samples. Calibrated birefringent samples that are made for 3-D quantitative polarimetric imaging system, include liquid crystals [26] and birefringent resolution targets such as for example those provided by Thorlabs (e.g. R2L2S1B). However, such samples are comprised of strong birefringence structures and may not be suitable for our application as they do not comply with our assumption of approximated Jones formalism approach. A highly-scattering 3-D sample with known RI deposited on silicon substrate is used in full polarization reflection TDM [24,30]. In these works, the authors showed the advantage of full polarimetric measurements to improve the reconstructed image resolution, which has been done by combining scattered field information at multiple polarizations.
In general, fabricating such samples is not a simple task, as the phantoms would have to be weakly scattering, as well as weakly birefringent. For instance, 3-D printed phantoms that are fabricated using two-photon polymerization [54,55], used to characterize 3-D quantitative phase imaging techniques are presently limited to scalar system. It would be of great interest in the domain to have real 3-D birefringent test patterns to simultaneously be able to measure spatial resolution as well as RI sensitivity.
Our system could be of interest to study biological samples exhibiting birefringence at the cellular or sub-cellular level, such as spindle fibers, made of bundled microtubules, or actin distribution. Stress-induced birefringence at microscopic scale could also be investigated, such as in diatom or radiolarian shells, microcrystals, textile fibers, combustion residues. As possible extension, spectrally resolved, polarization-sensitive TDM should allow for a much-improved chemical sensitivity, which could be useful to identify micro- and nano-plastics, free-floating, and possibly even when internalized by cells.
Appendix
In Jones formalism, when light propagates through several optical elements, the variations of the electric field vector from beginning to the end can be expressed as the product of each associated Jones matrices or Jones vectors. Light in both reference and illumination arms is changed into circularly polarized beam, right and left circularly polarized light (LC and RC), described by their Jones vector,
In the sample arm, the output field $O_{ij}$ is expressed as cascaded Jones components:
where $J^\theta _{anl}$ is the Jones matrix of analyzer before the detector, and $J_s$ is the anisotropic sample’s complex Jones matrix:Now using the $0^\circ$ and $90^\circ$ polarization, $O_{ij}$ can be computed as:
Similarly, the reference beam, $R_{ij}$, can be represented by $J^\theta _{anl}$ and reference beam’s Jones vector $E_{ref}$:
It could also be possible to accounts for light source fluctuation or optical system inaccuracies in the reference arm. The best way to make sure that we are measuring only the sample contribution is to take a blank acquisition and subtract from both object and reference fields.
Now, the polarization sensitive hologram that we record using the polarization sensitive camera is an interferogram between $O_{ij}$ and $R_{ij}$. Recalling Eq. (13) and extracting the first diffraction order leads to: Extracting the $+1$ order:
So 2 acquisitions are enough to compute all 4 Jones matrix components since the PS-Camera has mosaic images, containing both 0 and 90 degree polarization in a single acquisition. Thus, it provides a linear relationship between the measured complex scattered field and the birefringent sample’s Jones tensor components.
Funding
Conseil régional du Grand Est (FRCR 18P-07855, FRCR 19P-10656); Agence Nationale de la Recherche (ANR-18-CE45-0010, ANR-19-CE42-0004).
Acknowledgments
Julien Duboisset and Olivier Fauvarque are warmly acknowledged for their help during sample preparation of the Pinna nobilis sample. We also would like to thank Virginie Chamard and Patrick Ferrand for their fruitful discussions on the theoretical background on light polarization and polarization imaging.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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