Abstract
Optical anisotropy, which is an intrinsic property of many materials, originates from the structural arrangement of molecular structures, and to date, various polarizationsensitive imaging (PSI) methods have been developed to investigate the nature of anisotropic materials. In particular, the recently developed tomographic PSI technologies enable the investigation of anisotropic materials through volumetric mappings of the anisotropy distribution of these materials. However, these reported methods mostly operate on a single scattering model, and are thus not suitable for threedimensional (3D) PSI imaging of multiple scattering samples. Here, we present a novel referencefree 3D polarizationsensitive computational imaging technique—polarizationsensitive intensity diffraction tomography (PSIDT)—that enables the reconstruction of 3D anisotropy distribution of both weakly and multiple scattering specimens from multiple intensityonly measurements. A 3D anisotropic object is illuminated by circularly polarized plane waves at various illumination angles to encode the isotropic and anisotropic structural information into 2D intensity information. These information are then recorded separately through two orthogonal analyzer states, and a 3D Jones matrix is iteratively reconstructed based on the vectorial multislice beam propagation model and gradient descent method. We demonstrate the 3D anisotropy imaging capabilities of PSIDT by presenting 3D anisotropy maps of various samples, including potato starch granules and tardigrade.
Introduction
Over the past decade, significant efforts have been devoted to the development of a variety of computational imaging modalities for structural and molecular interpretation of microscale transparent specimens^{1}. Among the numerous achievements, computational quantitative phase imaging (QPI) techniques have emerged as innovative labelfree imaging methods that can not only reconstruct complex object fields, i.e., amplitude and phase delay or refractive indices of twodimensional^{2,3,4,5,6,7,8,9,10} (2D) and threedimensional^{11,12,13,14,15,16,17,18,19} (3D) transparent objects, but also overcome some of the limitations exhibited by the conventional imaging methods (e.g., limited spacebandwidth product, etc.). With the remarkable advances in phase retrieval algorithms and computing hardware, computational microscopy is now widely applied in biomedicine, chemistry, materials science, and various other disciplines^{5,20}. However, these imaging techniques are based on the scalar Helmholtz equations, and therefore, do not provide accurate information on the intrinsic properties of optically anisotropic materials.
Recently, diverse quantitative polarizationsensitive imaging (QPSI) systems have been developed to explore the nature of optically anisotropic materials. Various forms of digital holographybased QPSI techniques have been introduced based on offaxis interferometry^{21,22,23,24,25,26} and inline holography^{27,28}, and demonstrated their potentials in material characterization and disease diagnosis. Noninterferometric QPSI methods have also been developed based on, for examples, differential phase contrast microscopy, Fourier ptychography, ptychography, lensless imaging, and singlepixel imaging with their unique advantages^{29,30,31,32,33,34,35,36,37}. However, these QPSI methods provide only the 2D projection information that conceal the 3D anisotropy distributions. The recently developed interferometric and noninterferometric configurations of polarizationsensitive (PS) tomography have demonstrated the reconstruction of 3D anisotropy distributions of various specimens^{38,39,40,41}. However, interferometric approaches^{38,41} have inherent drawbacks, such as phase instability, speckle noise, and complex alignment, whereas the noninterferometric approaches^{39,40} require mechanical sample/objective scanning along the zdirection to reconstruct the 3D anisotropy information. In addition, the reconstruction model of the reported 3DQPSI methods relies on the weak scattering approximation, and thus, relatively complex structures, such as multicellular organisms, which are optically transparent but exhibit multiple scattering, may not be analyzed by these techniques.
Here, we introduce a novel computational 3D imaging technology, namely polarizationsensitive intensity diffraction tomography (PSIDT), that enables 3D Jones matrix imaging of multiplescattering objects using intensityonly measurements based on vectorial multislice beam propagation (MSBP) model. For the 2D intensity measurements, an object is illuminated at multiple angles and recorded with four different polarizer/analyzer configurations to encode the 3D structure and anisotropy information. To reconstruct the 3D anisotropy map, we introduce a novel reconstruction framework that accounts for the vectorial nature of polarization by reformulating the scalar MSBP model^{13,42} and gradient descent method in the vectorial form. PSIDT enables 3DQPSI imaging with only a simple modification of conventional polarization light microscope, i.e., replacement of its light source with a ring lightemitting diode (LED) array. We demonstrate the 3DQPSI capability of PSIDT by presenting the 3D anisotropy images of a digital phantom and various birefringent samples, including potato starch granules and tardigrade.
Results
Vectorial forward model and inverse problem formulation of PSIDT
First, we introduce the theoretical formulation of PSIDT to understand the propagation of polarized light through an anisotropic object, and the resultant image formation. The light propagation inside the object is modeled with a vectorial MSBP forward model, and based on the forward model, an inverse problem for the 3D Jones matrix is derived.
Consider a 3D anisotropy object illuminated by a polarized plane wave at an incidence angle, i.e., \(\vec E_{m,0}^\ell \left( {{{\boldsymbol{r}}}} \right)\), where r denotes the 2D spatial coordinates, and m and \(\ell\) represent the indices for the input polarization state and illumination angle, respectively (Fig. 1a). In the vectorial MSBP, a 3D object is considered as a series of N equallyspaced opticallyanisotropic thin layers separated by a constant distance Δz, and the light propagation through the sample is modeled by considering a sequential layertolayer propagation of the electric field (Fig. 1b). Using the Jones formalism^{43}, an electric field vector of the nth layer, \(\vec E_n\), is described as:
where \(\bar O_n\left( {{{\boldsymbol{r}}}} \right)\) is a 2 × 2 Jones matrix of the nth layer, defined as \(\bar O_n\left( {{{\boldsymbol{r}}}} \right) = \left[ {\begin{array}{*{20}{c}} {O_{n,xx}\left( {{{\boldsymbol{r}}}} \right)} & {O_{n,xy}\left( {{{\boldsymbol{r}}}} \right)} \\ {O_{n,yx}\left( {{{\boldsymbol{r}}}} \right)} & {O_{n,yy}\left( {{{\boldsymbol{r}}}} \right)} \end{array}} \right]\). Note that each layer is considered as a thin nondepolarizing transparent layer, which is a widely used assumption in QPSI^{30,36,44,45,46}. The propagation operation matrix with distance ∆z, \(\bar H_{{\Delta}z}\), is expressed as \(\bar H_{{\Delta}z} = \bar F^{\dagger} h_{{\Delta}z}\left( {{{\boldsymbol{k}}}} \right)\bar F\), where \(\bar F\) and \(\bar F^{\dagger}\) are the 2D discrete Fourier and inverse Fourier transformation operator matrices, respectively, and k denotes the 2D spatial frequency coordinates. Here, the superscript † denotes the complexconjugate transpose, and h_{Δz} represents the 2D angular spectrum propagator expressed as \(h_{{\varDelta}z}\left( {{{\boldsymbol{k}}}} \right) = \exp \left\{ {j{\varDelta}z\sqrt {\left( {{\textstyle{{2\pi } \over \lambda }}n_{media}} \right)^2  \left {{{\boldsymbol{k}}}} \right^2} } \right\}\), where n_{media}, λ, and j are the refractive index of the surrounding medium, wavelength of light, and imaginary unit, respectively. Then, the electric field vector at the last layer can be compactly written as:
The exit field vector from the 3D object, \(\vec E_{m,N}^\ell \left( {{{\boldsymbol{r}}}} \right)\), is collected by the objective lens, and the image is formed on the image sensor passing through the tube lens (Fig. 1c). The final electric field vector at the image plane can then be written as:
where \(\bar P\left( {{{\boldsymbol{k}}}} \right)\) is the aberrationfree polarizationindependent pupil matrix expressed as \(\bar P\left( {{{\boldsymbol{k}}}} \right) = \left[ {\begin{array}{*{20}{c}} {P\left( {{{\boldsymbol{k}}}} \right)} & 0 \\ 0 & {P\left( {{{\boldsymbol{k}}}} \right)} \end{array}} \right]\). Note that in Eq. (3), the operator \(\bar H_{  N{\Delta}z/2}\) is applied by assuming that the focal plane is located at the center of the 3D object. The intensity vector of the measurement is \(\vec I_m^\ell \left( {{{\boldsymbol{r}}}} \right) = \vec E_m^{\ell ,^\ast }\left( {{{\boldsymbol{r}}}} \right) \odot \vec E_m^\ell \left( {{{\boldsymbol{r}}}} \right)\), where \(\odot\) is the Hadamard product, and the superscript * denotes the complex conjugate operator. Each element in the intensity vector is acquired with two orthogonal analyzer states, and the detailed acquisition procedure is described in Methods subsection “Data acquisition”.
With given input field and the measured intensity vectors, the 3D sample Jones matrix is recovered through an optimization scheme. The reconstruction task is formulated as the minimization problem given by:
where \({\boldsymbol{\mathcal{D}}}\) is the data fidelity term, \({\boldsymbol{\mathcal{R}}}\) is the regularization term, and τ is the regularization parameter, which controls the amount of regularization. The data fidelity term is formulated as the \(\ell _2\)norm between the square root of the measured intensity vectors and the estimated amplitude vectors through the forward model computed over the entire illumination conditions:
where L and M denote the number of illumination angles and input polarization states, respectively; r_{3D} is a 3D spatial coordinate, and \({\boldsymbol{\mathcal{G}}}_m^\ell \left\{ \cdot \right\}\) denotes the \({\mathbb{C}}^3 \to {\mathbb{C}}^2\) nonlinear operator of the vectorial MSBP forward model with the \(\ell {{\rm{th}}}\) illumination angle and mth input polarization state. Note that modulus in Eq. 5 is elementwise operation. For the regularization term, we implement the 3D total variation^{47}:
where ∇_{x}, ∇_{y}, and ∇_{y} denote the finite difference operations along the x, y, and z directions, respectively.
Reconstruction of 3D anisotropic structures
To validate the 3D polarization imaging capability of PSIDT, we first numerically reconstructed the 3D anisotropy maps of a digital phantom. We considered a multiplescattering ellipsoidal anisotropic cell with an internal structure compartmentalized with a large nucleus and multiple granules of different sizes and refractive indices. The body of the phantom was set to be optically isotropic with refractive index of 1.34. The nucleic structure was optically anisotropic with refractive indices of the fast and slow axes of 1.36 and 1.38, respectively, and its opticaxis orientation varies in both lateral and axial directions. Ten isotropic and ten anisotropic vesicles were then randomly distributed inside the isotropic body and anisotropic nucleus. The refractive index of the isotropic vesicles was 1.42, and those of the fast and slow axes of the anisotropic vesicles were 1.42 and 1.45, respectively. The surrounding medium had a refractive index of 1.33. Notably, we generated the multiple scattering effect of the sample by designing a relatively larger refractive index difference. The operating conditions for the simulation were identical to that of the experimental setup.
Figures 2a, c show the groundtruth lateral and axial crosssectional maps of the mean phase and anisotropy of the digital phantom, and Figs. 2b, d are the corresponding images reconstructed with PSIDT. The origin (x = 0 μm, y = 0 μm, and z = 0 μm) of the coordinate system is defined at the center of the reconstructed volume. The images in the first row in Fig. 2a–d show the information in the xyplane (z = 0 μm) at the center of the 3D phantom, and the second and third rows correspond to the yz (x = 1.75 μm) and xz (y = 0.65 μm) crosssectional images through the red and orange dashed lines in Fig. 2a1. The reconstructed mean phase and anisotropy distributions agree well with the groundtruth information, while the features are blurred in the reconstructed images because of the limited resolution (NA = 0.6). It is clearly seen that the PSIDT successfully reconstructed not only the complex anisotropic nucleic structure but also the particles randomly distributed in both isotropic body and anisotropic nucleus. Note that the unwanted signal, indicated by the yellow arrows in Fig. 2b1, d1, comes from particles in adjacent layers due to the relatively low axial resolution of the angular scanning diffraction tomography technique. The 3Drendered mean phase and anisotropy tomograms of the phantom are shown in Fig. 2e, f, demonstrating the 3D PSI capability of the PSIDT technique in the multiplescattering regime. This 3Drendered view was obtained using the Amira imaging software (Amira3D pro). A video clip of the 3D rendering results is provided in Supplementary Video 1. We also numerically performed the PSIDT imaging of liquid crystal sphere as a weakly scattering anisotropy object, and the results are described in section 2 of the Supplementary Information and Supplementary Video 2.
Next, we performed the PSIDT imaging of a 5 μm polystyrene bead and aggregated potato starch granules as isotropic and multiplescattering anisotropic objects, respectively. For the experiment, the polystyrene bead and potato starch granules of various sizes were suspended in an indexmatching oil (\(n_{oil} = 1.516\)) and placed between glass coverslips. The sample was illuminated by plane waves from 24 different angles, and the polarization states of the illuminated plane waves were sequentially modulated to be left and righthanded circularly polarized. The images were obtained through two orthogonal analyzer states, and then used to produce the 3D Jones matrix map. We set Δz = 1 μm over a depth range of −30 μm to 30 μm. The regularization parameter of 1 × 10^{−4} was used for the image reconstruction.
In the case of the polystyrene beads (Fig. S2), one can note that the isotropic information (i.e., mean phase) was well recovered, whereas the anisotropic information was obtained only at the edge of the bead due to edge birefringence^{48}. Details of the PSIDT reconstruction of the polystyrene bead are provided in Supplementary Information (Section 3). In contrast, a potato starch granule features an alternating layered shell structure with amorphous and crystalline regions composed of polymer chains^{49}. Thus, this granule exhibits a strong birefringence because of the high degree of radial orientation of the amylopectin crystallites^{50,51}. Figure 3a–i shows the Jones matrix and the corresponding anisotropy maps of the potato starch granules obtained at different depths. The measured offdiagonal terms (O_{xy}) at each depth indicate the measured anisotropy properties of the potato starch granules. However, the anisotropic properties, such as linear retardance and opticaxis orientation, are coupled to the elements of the object Jones matrix, and thus, a quantitative interpretation of these properties from the Jones matrix images may be elusive. Therefore, we performed eigenvalue decomposition of the Jones matrix to obtain the linear retardance and identify the opticaxis orientation of the potato starch granules (see “Methods”). Figure 3g–i presents the visualization results of the anisotropic properties (i.e., linear retardance and azimuth opticaxis orientation) using pseudo colormaps. It is clearly seen that the PSIDT could map out the wellknown radially oriented opticaxis distribution and anisotropy information of the aggregated starches, but, due to the limited axial resolution and the use of paraxial BPM model, the starch particles could not be clearly resolved for the particles separated by smaller than 3 μm in axial direction. Recently, splitstep nonparaxial (SSNP) method^{52,53} has been introduced as the forward model for multiplescattering objects. The implementation of the SSNP model in the PTIDT platform is expected to improve the reconstruction performance for complicated multiple scattering samples. The rendered 3D volumetric anisotropy map of the potato starch granules is shown in Fig. 3j and Supplementary Video 3, demonstrating that the PSIDT technique can map the 3D anisotropy information from the intensityonly measurements.
To further demonstrate the 3D anisotropy imaging capability of PSIDT for multiplescattering samples, we performed PSIDT imaging of a tardigrade. Tardigrades, also known as water bears, with lengths and heights of less than 0.5 and 0.1 mm, respectively, have various organs, such as cuticles, brains, muscles, claws, guts, and stylets^{54}. Due to the optically thick and relatively complex organ distribution, tardigrades are transparent but exhibit multiple scattering characteristics, making it difficult to visualize their structures clearly with a conventional microscope. In addition, it is wellknown that some organs, such as the stylet in the buccal apparatus and granules in the midguts, exhibit birefingence^{55,56}. The stylet is composed of calcite with a hard structure to penetrate food^{57}, and the granules containing birefringent minerals are produced in the midgut as digestive products through the tryptophan catabolism^{55,58}. Because of these optical characteristics, a tardigrade was chosen as a suitable sample for demonstrating the imaging capabilities of PSIDT for transparent multiplescattering anisotropy objects.
Because a tardigrade is larger than the fieldofview (FOV) of the PSIDT, the whole tardigrade image was obtained through 3D stitching using standard 3D rigidbody registration algorithms^{59} after performing the PSIDT reconstruction on each region of interest (ROI) (80 μm × 80 μm). Figures 4a1–c1 and a2–c2 show the representative intensity measurements and the corresponding Fourier magnitudes of the tardigrade at three different ROIs. As evident from Fig. 4a2–c2, the pupil shifts (yellow dashed line) due to oblique illumination. However, unlike the spectra commonly observed in weakly scattering samples, in this case, we obtain the spatial frequency information outside the pupil (white arrow). These images validate the multiplescattering characteristics of tardigrades^{13}.
Figure 4d–f shows the PSIDT mean phase images of the whole tardigrade at depths of −10, −3, and 4 μm, respectively. Because the anisotropy information is distributed only in small regions such as the mouse and midgut, most of the information of the tardigrade structure can be visualized in the mean phase term. It can be seen that the 3D structures of many organs, such as claws (Fig. 4d), stylet (Fig. 4e), and gut (Fig. 4e, f), are clearly visualized over the entire volume. The PSIDT images of various other major structures of a tardigrade are provided in Supplementary section 4. Figures 4e1–2 and f1–2 show the enlarged mean phase and anisotropy images that correspond to the regions marked with orange and green square boxes in Fig. 4e, f, respectively. One can see that the optically anisotropic stylet (Fig. 4e2) and granules (Fig. 4f2) are clearly visualized via the anisotropybased information, while the overall structure of the tardigrade can be observed from the mean phase information. The 3D mean phase delay and anisotropy tomograms of the whole tardigrade are shown in Fig. 4g, h, demonstrating the 3D anisotropy imaging capability of PSIDT for optically transparent multiplescattering samples. The depth scanning results and 3D rendering movie clips of the reconstructed 3D mean phase and anisotropy are provided in Supplementary Videos 4 and 5. In particular, we could selectively visualize the 3D structure of the stylet in polarizationbased contrast (Supplementary Video 5), which is in good agreement with the structural information observed in scanning electron microscopy^{57}. These results clearly indicate that PSIDT is capable of recovering the 3D anisotropy/isotropic information by taking into account the multiplescattering characteristic of the sample, even when the anisotropy information is hidden inside the complex isotropic structures of the sample.
Discussion
In summary, we presented a novel form of computational 3D polarization microscopy that can quantitatively reconstruct the Jones matrix of a 3D anisotropic object based on the vectorial MSBP method and gradientdescentbased optimization strategy. The proposed PSIDT prototype features referencefree and inertiafree operation, with a lateral resolution of 0.54 μm and an axial resolution of 2.89 μm across an FOV of 80 μm × 80 μm. We developed the vectorial extension of the MSBP forward model to recover 3D Jones matrix of multiplescattering anisotropic object, and successfully reconstructed its 3D anisotropy information, overcoming the challenges of other 3D PSI techniques based on the singlescattering model (i.e., Rytov and Born)^{60}. Our prototype can reconstruct a 3D Jones matrix map from only 96 images with a numerical aperture (NA)matched annular illumination, and can be implemented simply by adding an LED array and polarization optics in a conventional optical microscope.
Several improvements can be considered in the reported platform. Dynamic imaging of optically anisotropic specimens is significant in various applications, such as cell dynamics^{61}, contractile activity analysis of beating cardiomyocytes^{62}, and liquid crystal monitoring^{63}. However, the PSIDT system developed in this study requires a relatively long acquisition time because of the use of a divergent LED light source and polarization optics. The LEDs with a 120° divergence angle were positioned at ~47 mm to generate a quasiplane wave illumination, and thus the amount of light intensity at the sample plane was very small. A domeshaped LED with builtin collimators^{13} or a combination of lasers and beam scanners (e.g., galvanometric scanners) can be considered for improving the light throughput, although the system may become bulky and introduce coherent speckle noise for lasers. In terms of the light loss due to polarized optics, polarization detection modules consisting of a polarizing beam splitter^{62} and multiple cameras can be employed. Another method to improve the imaging speed is to reduce the number of measurements. This approach reduces the total acquisition time as well as enables an efficient use of the memory and processing resources. In ref. ^{15}, the tomographic scalar refractive index map of dynamic C. elegans were successfully obtained using only four LED annular illuminations, although it compromised the reconstruction image quality. Integration with more sophisticated reconstruction strategies, such as deep learning and deep image priors, may also be explored to realize highspeed 3D PSI without compromising the image quality. Indeed, several recent studies have demonstrated a highly effective deeplearningaided tomographic reconstruction with limited angle illuminations^{64,65}.
In the forward model of this demonstration, we assumed the PSIDT prototype is an aberrationfree system. However, practical polarizationsensitive imaging system may have optical aberrations including polarizationindependent (e.g., defocus, astigmatism) and polarizationdependent aberrations induced by the birefringence of optics and interfaces^{66,67}. These optical aberrations can degrade the reconstruction performance by breaking the consistency between the measured and estimated information. Recently, Dai et al., demonstrated vectorial Fourier ptychography reconstruction along with aberrated pupil matrix recovery by extending scalar aberration correction (embedded pupil recovery) without additional measurements^{31}. Applying this approach, aberrationcorrected PSIDT reconstruction can be achieved by introducing an additional ptychographic update process into backpropagation to obtain both object spectrum and aberration pupil matrix in the Fourier plane.
The presented PSIDT produces 3D anisotropy map of an object using vectorial MSBP forward model derived with Jones matrix in paraxial approximation, which considers only the inplane optic axis in each layer. For biological applications involving flat or thin specimens (e.g., sectioned tissue slides or cell layers), this model may provide an efficient and accurate measurement. However, for applications that require a complete analysis of 3D anisotropic information, including the outofplane opticaxis orientation, our method must be refined to describe the complete dielectric tensor. Moreover, for high resolution (or highNA) imaging of multiple scattering objects, our method should be refined to consider nonparaxial multiple scattering model. Recently, SSNP model was introduced as the forward model in scalar diffraction tomography^{52,53}, and demonstrated its improved reconstruction performance for a nonparaxial multiplescattering model. Shin et al. also demonstrated a holographybased tomographic 3 × 3 dielectric tensor imaging scheme for biaxial materials^{41}. To realize the 3 × 3 tomographic measurement of dielectric tensors, additional measurements with a slightly tilted illumination were performed. The integration of these experimental and algorithmic strategies into the PSIDT platform would enable a highly viable QPSI imaging platform for high resolution 3 × 3 tensor reconstruction for both weakly and multiple scattering objects.
The MSBPbased reconstruction framework requires a high computational power to compute the propagation operator at every layer in the forward and backward directions. In addition, PSIDT, which handles four elements of the Jones matrix, performs numerically four times more computations than the scalar MSBPbased reconstruction method. This limitation can be addressed by leveraging the recent advances in parallel processing, achieved via GPU acceleration or cloud computing, to reduce the reconstruction time significantly.
Finally, we believe that PSIDT can serve as a versatile, multimodal imaging platform to provide a variety of structural and functional information about a 3D object. For example, combining this technique with computational fluorescence microscopy can provide the 3D distribution of labeled species along with the isotropic/anisotropic information^{14}, and can also be utilized for fluorescence polarization analysis to reveal the protein orientations^{68} and structures of the cellular organelles^{69}. In addition, 3D spectral anisotropy analysis can be performed by using either multicolor LEDs or hyperspectral image sensors, which will allow us to conduct various quantitative studies of, for example, 3D mapping and localization of plasmonic nanostructures based on their distinctive spectral and polarization signatures^{70,71,72}.
Methods
Experimental setup
The PSIDT prototype was constructed by implementing an LED microscopy with polarization optics for 3D PSI (Fig. 1c). As a light source, an annular LED array (central wavelength λ = 520 nm, 1586, Adafruit, USA) was used for the annular plane wave illuminations at various azimuthal angles. Recently, several researchers have shown that annular illuminations with the incident angle matching the objective NA (\(NA_{obj}\)) is the optimal condition for tomographic imaging^{15,73,74,75}. This illumination scheme optimally encodes both low and highspatial frequency information across the entire 3D volume using a small number of intensity measurements. An annular LED array featuring a radius of 35 mm with 24 LEDs equally spaced in the azimuthal direction was used. To match the objective NA and illumination NA (\(NA_{ill}\)), the LED array was placed at a distance of 47 mm from the sample plane, and its center was carefully positioned on the optical axis. The distance (h), i.e., the separation between the LED array and the sample, was calculated by the following relation:
where r is the radius of the annular LED array.
A 3D object was illuminated with a unique illumination angle by sequentially operating the individual elements of the LED array. A rotating wheel (LCFW5, Thorlabs, USA) equipped with left and righthanded circular polarizers (CP42HE and CP42HER, Edmund Optics, UK) was placed between the sample plane and the annular LED array, and the circular polarizer was manually changed to modulate the input polarization state. After the interaction with the 3D anisotropic sample, the diffracted light was collected by a 0.6 NA, ×40 objective lens (CFI S Plan Fluor ELWD ADM 40XC, Nikon, Japan), and the tube lens (ACT508200A, Thorlabs, USA) formed the image on a scientific CMOS camera (pco.edge 4.2, PCO, USA). To acquire an intensity vector containing two orthogonally polarized intensity components, a motorized rotating linear polarizer (LPVISA100MP2 and PRM1Z8, Thorlabs, USA) was placed in front of the image sensor. The angle of the absorption axis of the linear polarizer was aligned in the horizontal direction of the observation coordinates. The PSIDT exhibited an axial resolution of 2.89 μm and a lateral resolution of 0.54 μm with an FOV of 80 μm × 80 μm.
Data acquisition
For each polarizer/analyzer configuration, 24 raw intensity images were obtained by sequentially illuminating the LED elements. The LED array was driven by Arduino Uno, and all the images were acquired by a scientific CMOS camera (pco.edge 4.2, PCO, USA), which was synchronized with the LED source through an external trigger channel. Under the four polarizer/analyzer configurations, a total of 96 images were acquired for the PSIDT reconstruction. The exposure time of each measurement was 1.5 s, resulting in a total acquisition time of approximately 144 s. Note that a relatively long exposure time was required because of the use of a divergent LED and polarization optics.
For calibrating the LED brightness, all the LED elements were sequentially turned on without the polarization optics (circular polarizer and linear polarizer in the illumination and detection paths, respectively) and sample, and 24 images were captured with an exposure time 0.3 s. After obtaining the brightness of all the LEDs by averaging the pixel intensities in the FOV, we used the information to normalize the intensity of the raw images in the subsequent measurements.
Reconstruction procedures of PSIDT
The reconstruction algorithm of PSIDT is based on an iterative approach that involves gradient descent and backpropagation, along with the vectorial MSBP model. The reconstruction procedure is summarized as below:

(1)
Initialize the Jones matrix at each voxel in the reconstruction volume with an arbitrary 2 × 2 matrix. In our implementation, an identity matrix I was set as the initial Jones matrix. Then, initialize the iteration index i and the input polarization state index m as 0.

(2)
Increment the iteration index, \(i \leftarrow i + 1\), and initialize the cost function, \({{{\mathrm{c}}}}\left( i \right) = 0\).

(3)
Increment the input polarization state, \(m \leftarrow m + 1\), and initialize the illumination angle index, \(\ell = 0\).

(4)
Increment the illumination angle index, \(\ell \leftarrow \ell + 1\), and compute the incident electric field vector \(\vec E_{m,0}^\ell \left( {{{\boldsymbol{r}}}} \right) = \exp \left( {j{{{\boldsymbol{k}}}}_0^\ell \cdot {{{\boldsymbol{r}}}}} \right)\left[ {\begin{array}{*{20}{c}} 1 & {\left( {  1} \right)^mj} \end{array}} \right]^T\). Note that an even or odd m corresponds to an illumination with left or righthanded circular polarization.

(5)
Using the incident electric field vector and estimated Jones matrix, evaluate the electric field vectors \(\left\{ {\vec E_{m,n}^\ell \left( {{{\boldsymbol{r}}}} \right)n = 1,2, \ldots ,N} \right\}\) at each layer of the reconstruction volume, and estimate the measurement amplitude \(\left( {\left {\boldsymbol{\mathcal{G}}_m^\ell \left\{ {\bar O\left( {{{{\boldsymbol{r}}}}_{3D}} \right)} \right\}} \right} \right)\) using the vectorial MSBP forward model.

(6)
Using the corresponding intensity measurement \(\vec I_m^\ell\), update the cost function for the current iteration as \({{{\mathrm{c}}}}\left( i \right) \leftarrow {{{\mathrm{c}}}}\left( i \right) + \mathop {\sum}\nolimits_{{{\boldsymbol{r}}}} {\left\ {\sqrt {\vec I_m^\ell \left( {{{\boldsymbol{r}}}} \right)}  \left {\boldsymbol{\mathcal{G}}_m^\ell \left\{ {\bar O\left( {{{{\boldsymbol{r}}}}_{3D}} \right)} \right\}} \right} \right\_2^2}\), and initialize a residual vector, \(\vec {\Psi}_m^\ell \left( {{{\boldsymbol{r}}}} \right),\) as
$$\vec {\Psi}_m^\ell \left( {{{\boldsymbol{r}}}} \right) = {\boldsymbol{\mathcal{G}}}_m^\ell \left\{ {\bar O\left( {{{{\boldsymbol{r}}}}_{3D}} \right)} \right\}  diag\left( {\frac{{\boldsymbol{\mathcal{G}}_m^\ell \left\{ {\bar O\left( {{{{\boldsymbol{r}}}}_{3D}} \right)} \right\}}}{{\left {\boldsymbol{\mathcal{G}}_m^\ell \left\{ {\bar O\left( {{{{\boldsymbol{r}}}}_{3D}} \right)} \right\}} \right}}} \right)\sqrt {\vec I_m^\ell \left( {{{\boldsymbol{r}}}} \right)}$$(7)where \(diag\left( {\vec x} \right)\) is a diagonal matrix with the elements of the vector \(\vec x\) as the diagonal elements. Note that matrix division, square root, and modulus are elementwise operations. Then, by substituting the residual vector into the inverse operation of Eq. 3, the backpropagation vector of the last layer of the reconstruction volume, \(\vec {\Psi}_{m,N}^\ell \left( {{{\boldsymbol{r}}}} \right)\), is obtained.

(7)
The sample Jones matrix for each layer is updated based on the gradient descent and backpropagation from the last layer to the first layer using
$$\vec O_n\left( {{{\boldsymbol{r}}}} \right) \leftarrow \vec O_n\left( {{{\boldsymbol{r}}}} \right) + diag\left( {\vec \alpha } \right)\vec E_{m,n}^{\ell ,{\dagger} }\left( {{{\boldsymbol{r}}}} \right)\bar H_{{\Delta}z}^{\dagger} \vec {\varPsi}_{m,n}^\ell \left( {{{\boldsymbol{r}}}} \right)$$(8)where \(\vec {\varPsi}_{m,n}^\ell \left( {{{\boldsymbol{r}}}} \right) = \bar O_k^{\dagger} \left( {{{\boldsymbol{r}}}} \right)\bar H_{{\Delta}z}^{\dagger} \vec {\varPsi}_{m,n + 1}^\ell \left( {{{\boldsymbol{r}}}} \right)\), and \(\vec \alpha\) is the stepsize vector. Note that \(\vec O_n\left( {{{\boldsymbol{r}}}} \right)\) is the vector representation of the sample Jones matrix \(\bar O_n\left( {{{\boldsymbol{r}}}} \right)\), i.e.,\(\vec O_n\left( {{{\boldsymbol{r}}}} \right) = \left[ {\begin{array}{*{20}{c}} {O_{n,xx}} & {O_{n,xy}} & {O_{n,yx}} & {O_{n,yy}} \end{array}} \right]^T\). The details of the gradient computation is presented in Supplementary Information (Section 1).

(8)
Repeat steps (4)–(7) for each illumination angle.

(9)
Repeat steps (3)–(8) for each input polarization state.

(10)
Implement a proximal operation of 3D total variation^{76} and fast iterative shrinkagethresholding algorithm^{77} on the reconstructed 3D Jones matrix to promote stability and increase the convergence speed.

(11)
Repeat (2)–(10) until a selfconsistent solution is achieved. After reaching convergence, anisotropy properties, such as the mean phase delay (\(\tilde \phi\)), retardance (δ), and azimuthal opticaxis orientation (φ), are extracted from the reconstructed 3D Jones matrix using matrix diagonalization (Fig. 1d). Details on the extraction of the isotropic and anisotropic properties are provided in the “Methods” section.
Calibration of illumination angles
PSIDT requires an accurate knowledge of the illumination angles for robust 3D Jones matrix reconstruction. However, due to manufacturing imperfection or experimental misalignment of the LED array, the illumination angles may differ from the predefined angles in the experiment, which may cause significant reconstruction artifacts^{78,79}. To mitigate the mismatch between the implementation and numerical model, we performed a selfcalibration procedure before the reconstruction. In the demonstration, we implemented the algorithmic selfcalibration method^{78} originally developed to correct the LED position in Fourier ptychography. This technique, which can quickly calculate the illumination angle from the raw data without additional measurements, has been utilized in various computational microscopy techinques^{13,30,60} as well as in FP. In Supplementary Information section 6, a comparison of the PSIDT reconstruction performance before and after the calibration is provided.
Extraction of the isotropic and anisotropic properties
Using the reconstructed Jones matrix information, the mean phase, retardance, and opticaxis orientation can be obtained through eigen analysis. The 3 × 3 Jones matrix of a nondepolarizing uniaxial material can be expressed as:
where \(\bar R_\varphi\) and \(\bar R_\theta\) are the coordinate rotation matrices for the azimuthal (φ) and polar angles (θ), respectively, and ϕ_{s} and ϕ_{f} are the phase delays of the slow and fast axes, respectively. Because Eq. 9 is symmetric, the opticaxis orientations (φ and θ) and phase delays (ϕ_{s} and ϕ_{f}) can be easily obtained from the 3 × 3 Jones matrix by taking the eigenvalues and eigenvectors via diagonalization^{22,34,80,81}. Yet, the reconstructed 2 × 2 Jones matrix is symmetric, there are three parameters that determine the birefringence in each voxel in PSIDT. Therefore, the extraction of the anisotropic properties is an underdetermined problem. As a result, through the eigen analysis of the reconstructed Jones matrix, the fastaxis term (\(e^{j\phi _f}\)) and opticaxis orientation of the xyplane (φ) can be directly extracted, but the slowaxis term (\(e^{j\phi _s}\)) and polar angle (θ) are coupled as
where \(\phi _s^{Approx}\) is the slowaxis phase term measured using the reconstructed 2 × 2 Jones matrix. As shown in Eq. 10, when the polar angle of the object’s optic axis decreases, the error in the extracted slowaxis phase term cannot be ignored. We performed a numerical error analysis on the estimated phase delay in the slow axis as a function of the optic axis polar angle, and found that the error is smaller than 10% for a polar angle larger than 68°. In our analysis, the mean phase (\(\tilde \phi\)) and retardance (δ) were defined as \(\tilde \phi = {\textstyle{{\phi _s^{Approx} + \phi _f} \over 2}}\) and \(\delta = \phi _s^{Approx}  \phi _f\), respectively, and were used to present the anisotropic information.
Sample preparations
Potato starch granules slide
In this study, 1 mg of potato starch (100%) and 50 μL of the index matching oil (n_{oil} = 1.516, Cargille Laboratories, USA) were carefully mixed with a vortex mixer (model VM10, Witeg Labortechnik, Germany). Then, the mixed sample was mounted on a microscope slide and covered with a coverslip for the PSIDT imaging.
Tardigrades
Live tardigrades (Hypsibius exemplaris, 133960, Carolina Biological, USA) fed with Chlorella algae (152069, Carolina Biological, USA) were first carefully extracted with a micropipette and placed on microscope slides. The sample was then mounted in Hoyer’s medium, which is widely used for mounting tardigrades for microscope imaging^{82,83}. The sample was covered with a coverslip for imaging. We prepared Hoyer’s medium by following the recipe reported in ref. ^{83}. In brief, 200 g of chloral hydrate (302170, SigmaAldrich, USA) was added to 50 mL of distilled water (7732185, SigmaAldrich, USA), and heated to 60° until dissolved. Then, 30 g of gum arabic (9000015, SigmaAldrich, USA) was added to the solution, and covered, stirred, and heated overnight. Finally, 20 g of glycerin (56815, SigmaAldrich, USA) was added to the mixture, and the solution was filtered through a blotting paper. Typically, the refractive index of Hoyer’s medium is ~1.48, which is similar to that of tardigrade organs. After sealing and waiting for ~3 min, the tardigrades were imaged using PSIDT.
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Acknowledgements
This work was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1A2C201206113), Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (P0019784), Samsung Research Funding & Incubation Center of Samsung Electronics under Project Number SRFCIT200207, Korea Medical Device Development Fund grant funded by the Korea government (the Ministry of Science and ICT, the Ministry of Trade, Industry and Energy, the Ministry of Health & Welfare, the Ministry of Food and Drug Safety) (Project Number: RS2020KD000099), and the Ministry of Science and ICT (Project Number: 20232203000420) (NTIS, 1711179106) and Commercialization Promotion Agency for R&D Outcomes (COMPA).
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C.J. conceived the initial concept. S.S. developed the principles and reconstruction algorithm, performed the experiments and simulations, and analyzed the data. J.K., TM., and W.K. performed the experiments and simulations. B.S. performed the simulation and illustrated graphics C.Y. and J.C. prepared and analyzed the samples, and participated in discussions during the development of the paper. All the authors contributed to writing the manuscript.
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Song, S., Kim, J., Moon, T. et al. Polarizationsensitive intensity diffraction tomography. Light Sci Appl 12, 124 (2023). https://doi.org/10.1038/s41377023011510
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DOI: https://doi.org/10.1038/s41377023011510