Open Access
Add to BibSonomy
Abstract
Many important microscopy samples, such as liquid crystals, biological tissue, or starches, are birefringent in nature. They scatter light differently depending on the polarization of the light and the orientation of the molecules. The complete characterization of a birefringent sample is a challenging task because its 3 × 3 dielectric tensor must be reconstructed at every three-dimensional position. Moreover, obtaining a birefringent tomogram is more arduous for thick samples, where multiple light scattering should also be considered. In this study, we developed a new dielectric tensor tomography algorithm that enables full characterization of highly scattering birefringent samples by solving the vectoral inverse scattering problem while accounting for multiple light scattering. We proposed a discrete image-processing theory to compute the error backpropagation of vectorially diffracting light. Finally, our theory was experimentally demonstrated using both synthetic and biologically birefringent samples.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Birefringence results from polarization-dependent scattering of light by optically anisotropic materials. Samples with heterogeneous birefringence frequently emerge in soft matter physics and biophysics when molecules locally align their orientations because of complex many-body interactions [1–7]. Quantitative imaging of three-dimensional (3D) birefringence reveals the underlying structural order of these molecules, which may comprise molecules or crystals in biological tissues. However, these birefringent materials are mostly translucent and exhibit low contrast under conventional microscopic methods, which results in inaccurate characterization when using conventional bright-field microscopes [8–12].
Quantitative phase imaging (QPI) exploits the refractive index (RI) as an intrinsic quantitative imaging contrast and has been successfully deployed in many applications [13–15] where translucent samples should be imaged. Previously, 2D polarization sensitive QPI techniques have been reported and utilized for measuring polarization sensitive phase retardance or Jones matrix information [8–11]. Optical diffraction tomography (ODT) reconstructs the 3D refractive index (RI) distribution of a sample [16–19]. Extending from the ODT principle, recent progress has enabled direct 3D imaging of sample birefringence [20–22] and even direct measurement of the full 3D dielectric tensor [23,24]. However, these methods were all based on the weak scattering approximation in order to obtain an analytical solution for the inverse scattering program. This weak scattering approximation limits the general applications to complex materials with strong birefringence.
This study solved the inverse scattering problem beyond the weak scattering regime to reconstruct the dielectric tensor distribution of highly scattering 3D birefringent samples [Fig. 1(a)]. We specifically extended our previous discrete model of error backpropagation from scalar wave theory to a vectoral one [25]. The advantages of the proposed theory were experimentally demonstrated via the reconstruction of the 3D dielectric tensor distribution of synthetic liquid crystal droplets and biological starch grains. The dielectric tensor can be decomposed using singular value decomposition into three orthogonal polarization directions and three refractive indices for each of the corresponding polarizations. Consequently, the director and amount of birefringence can also be quantified [Fig. 1(a)]. The polarization direction corresponding to the highest refractive index corresponds to the underlying molecular alignment and is referred to as the director. Contrary to previous methods [25–28], the proposed model does not require strict assumptions, such as weak scattering or scalar diffraction. Moreover, it inversely solves Maxwell’s equations to fully characterize birefringent dielectric materials [Fig. 1(b)].
Fig. 1. Schematic of the proposed reconstruction method. (a) Dielectric tensor tomography of a radially distributed liquid crystal droplet, which is reconstructed from the measured multiple vectoral field images. The dielectric tensor can be decomposed using singular value decomposition into three orthogonal polarization directions and three principal refractive indices for each of the corresponding polarizations. Then the dielectric tensor can be converted into the refractive index, director, and birefringence for visualization. (b) A representative sample (potato starch) reconstructed using both the single and multiple scattering models. Birefringence direction is consistent with the growth ring structure of starches.
2. Results
2.1 Vectoral diffraction theory in discrete space
Understanding the light scattering by a birefringent object is essential for the development of advanced dielectric tensor imaging techniques. Consider a sample with dielectric tensor distribution, $\overleftrightarrow \varepsilon (\overrightarrow r )$, which scatters the incident polarized light field, ${\overrightarrow E _{in}}(\overrightarrow r )$. Here the double arrow indicates and tensor while the single arrow indicates vectors. For a dielectric material illuminated with monochromatic light, Maxwell’s equations are reduced to the inhomogeneous Helmholtz equation (derivation available in Supplement 1)
To perform numerical simulations, this equation is discretized on a voxel grid
The matrix inversion on the right-hand side of the equation is a challenge to overcome when computing the scattered field. For a large system, direct inversion of the matrix is not feasible and must be evaluated using iterative algorithms. Here, we used the convergent Born series [30–32] to perform the inversion; however, other approaches are also possible [26,33].
2.2 Dielectric tensor tomography with weak scattering approximation
Previously, our group demonstrated the use of dielectric tensor tomography (DTT). a polarization-sensitive holographic microscopy technique for the reconstruction of 3D dielectric tensor [Fig. 2; Supplement 1 for details]. Based on the polarization-resolved holograms of a weakly birefringent sample, DTT reconstructed its 3D dielectric tensor distribution using linear reconstruction theory [23]. This theory assumed the Rytov approximation, wherein the optical path difference of the sample slowly varies in the transverse direction. While the method allows for a complete measurement of the dielectric tensor, reconstruction results are often compromised when measuring samples with large refractive index contrasts. Thus, more advanced algorithms that consider multiple light scattering are necessary.
2.3 Reconstruction algorithm of highly scattering dielectric tensor
Based on the complete understanding of the vectoral diffraction process, the main goal of our study can be addressed, that is, imaging highly scattering birefringent samples. This is referred to as the inverse problem, and it aims to retrieve the dielectric tensor from a known illumination $\overrightarrow {{E_{in}}} $ and scattered field $\overrightarrow E $. We developed a multiple scattering algorithm to find the dielectric tensor $\overleftrightarrow \varepsilon $ wherein the simulated scattered fields most closely matched the imaged fields [Fig. 3(b)]. This goal can be mathematically expressed as the following minimization problem:
This is the square norm of the difference between the measured and simulated scattered fields $\overrightarrow y $ and $\overrightarrow {y^{\prime}} ({\overleftrightarrow {\varepsilon^{\prime}}} )$, respectively. To perform the minimization, the gradient of the minimized function must be computed to enable the use of a gradient descent algorithm.
Fig. 3. Reconstruction algorithm principle. (a) Optical setup. DMD: digital micromirror device, BS: beam splitter, PBS: polarized beam splitter. (b) Dielectric tensor reconstruction algorithm (i) simulation of the scattering through the sample (ii) backpropagation of the error (iii) update of the refractive index. (c) Different terms in the gradient equation.
To compute the gradient, we used the scattering potential ($\overleftrightarrow F$) instead of the dielectric tensor ($\overleftrightarrow \varepsilon $) to simplify the derivations. While the measured field $\overrightarrow y $ is obtained experimentally, the simulated scattered field can be expressed using Eq. (1).
A comparison of Eqs. (1) and (3) indicates that the gradient can be computed using two sequences of vectoral forward scattering computations. The left term corresponds to the scattering simulation by the dielectric tensor estimate of the sample, and the right term corresponds to the scattering simulation of the phase conjugate of the difference between the measured and scattered fields [Fig. 2(c)]. This last step can be understood as error backpropagation or as the time reversal of the error owing to the use of phase conjugation on the error field [34,35]. The similarity between error backpropagation in neural networks and scattering stems from the fact that scattering can be modeled using convolution operations. Consequently, this gives rise to analogous properties during the computation of the gradient of the error in neural networks [27].
2.4 Optical setup
In order to measure vectoral scattered field we used a polarization-sensitive quantitative phase imaging setup based on a Michelson interferometer [Fig. 3].
A 532 nm continuous wave laser (Cobolt AB) was split into a sample and a reference arm using a beam splitter. In the former, a digital micromirror device (DLi 4130, Digital Light Innovations) and a liquid-crystal retarder (LCC1223-A, Thorlabs) were used to control both the illumination angle and polarization states of the incident light. A water-immersion condenser (UPLSAPO60XW, NA = 1.2, Olympus) and oil-immersed objective lens (UPLSAPO60XO, NA = 1.42, Olympus) were used to illuminate the sample and collect the diffracted light field. Finally, the diffracted light field was split using a polarizing beam splitter and interfered with the reference beam on two cameras (Lt425R, Lumenera) to form two polarization-sensitive off-axis holograms.
The pseudocode of the reconstruction algorithm is presented in Fig. 3(c). During the gradient descent step, we improved the quality of the reconstructed tomograms by imposing constraints related to prior knowledge of the samples. In this study, we used a total variation regularization algorithm combined with a fast-iterative shrinkage-thresholding algorithm (TV-FISTA) [36,37]. In our previous study, we imposed the algorithm on individual tensorial elements and circumvented the missing cone problem [38]. However, the method of independent regularization caused errors in the reconstructed directors in the presence of multiple scattering. To mitigate this problem, we improved the regularization algorithm by also imposing a similarity between the principal RIs by constraining the total birefringence (TB). Details on the implementation of the constraints are presented in Supplement 1.
Figure 2 presents a comparative analysis of our algorithm with preceding works in the field of optical diffraction tomography. When juxtaposed with the reconstruction of the scalar refractive index, one can anticipate a threefold surge in computation time and memory requirements. This increase primarily arises from the majority of computations now engaging light fields with three polarization components.
2.5 Experimental validation using synthetic liquid crystals
We experimentally validated the proposed reconstruction method by imaging a synthetic liquid crystal with a known structure. The experimental setup used to retrieve vectoral field is shown in [Fig. 3(a)] and detailed in Supplement 1. We chose to use a liquid-crystalline microsphere made of sodium dodecyl sulfate (LCSDS) in a 1.52 ultraviolet-cured medium. The LCSDS microsphere exhibits a radial arrangement of directors and was used as a reference sample. The amount of birefringence was defined as the $\frac{{\sqrt {{{(n_1^2 - n_2^2)}^2} + {{(n_1^2 - n_3^2)}^2} + {{(n_3^2 - n_2^2)}^2}} }}{{\sqrt 8 \cdot {n_m}}}$.
We compared the reconstruction results of three different methods: (i) slight tilting method (Dielectric tensor tomography); (ii) TV-FISTA with single-scattering approximation (see Supplement 1 for details); and (iii) our proposed method [Fig. 4]. All reconstruction methods showed consistent imaging results, with similar RI directions and birefringence. However, we observed that the proposed method outperformed the other methods, particularly by providing the most homogeneous birefringence limited to the spatial bandwidth and expected spherical shape in the vertical direction. The other reconstruction methods suffered from elongation in the axial direction. This suggests that inaccurate scattering models suffer from both the missing cone problem and multiple scattering artifacts even after using a regularization algorithm. Thus, our inverse model can significantly improve the reconstruction quality in DTT even in the case of small RI contrast.
Fig. 4. Reconstruction results of an LCSDS microsphere using (a) slight tilting, (b) single scattering DTT, and (c) multiple scattering DTT method.
2.6 3D reconstruction of the highly scattering and birefringent biostructure
A promising application of DTT is in the analysis of molecular alignment in biological specimens. However, most birefringent biological specimens are too optically heterogeneous for imaging. A highly scattering starch sample with a birefringent RI contrast exceeding 0.1 was prepared in a mounting medium with a refractive index of 1.43.
The results indicated a significant improvement in reconstruction quality when using the proposed method [Fig. 5].
Fig. 5. Reconstruction results of multiple starches. (a),(b) Refractive index, birefringence, and directors of the sample using (a) the single scattering and (b) multiples scattering DTT method. (c),(d) Corresponding three-dimensional renderings of principle refractive index values and directors.
The single-scattering model exhibited noisy artifacts in the 3D retrieved birefringence and director maps. In contrast, the proposed method clarified that the directors were aligned perpendicularly to the surface of the starches while measuring homogeneous birefringence. Remarkably, the perpendicular alignment of the directors was also visible in the axial cross-section, even when the starch was non-spherical. The improved result is consistent with the “growth ring” structure observed by X-ray microscopy, where the starch grows by alternating amorphous and crystalline layers, wherein the principal axis is perpendicular to the surface [39]. Additionally, the hilum, the singular center of the largest starch, was more clearly visualized using the multiple scattering model.
3. Discussion
We developed a reconstruction method to image the 3D dielectric tensors of highly scattering samples. The original DTT reconstruction method although computationally fast, assumed weakly scattering approximations and required angularly differential measurements, which limited its applications. Although our method is slower than the previous algorithm, our technique improved reconstruction accuracy by considering multiple light scattering. This was experimentally validated by reconstructing synthetic liquid crystal droplets and biological starch samples.
The developed theory was numerically implemented using the convergent Born series, which is among the most efficient forward solvers for electromagnetic waves. However, our method is compatible with other electromagnetic solvers and can be tested in immediate follow-up studies. In addition, the proposed technique can be further improved by reducing the computational burden and developing methods to enable convergence to the global minimum [40,41]. Finally, a synergistic approach between optical theory and computational modeling may extend our method to the high-throughput analysis of biological tissues and multicellular interactions.
The present approach will provide precise reconstruction of dielectric tensor tomography, and enable various new investigations, including addressing collagen fibers in volumetric thick tissue slides [42], birefringence of microtubules in 3D live cell cultures such as organoids.
Funding
Tomocube Inc.; KAIST Institute of Technology Value Creation, Industry Liaison Center (G-CORE Project) grant funded by MSIT (N11230131); Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (2021-0-00745); National Research Foundation of Korea (2015R1A3A2066550, 2022M3H4A1A02074314).
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.
Supplemental document
See Supplement 1 for supporting content.
References
1. H. Almohammadi, M. Bagnani, and R. Mezzenga, “Flow-induced order–order transitions in amyloid fibril liquid crystalline tactoids,” Nat. Commun. 11(1), 5416–5419 (2020). [CrossRef]
2. L. Gianaroli, M. C. Magli, A. P. Ferraretti, A. Crippa, M. Lappi, S. Capitani, and B. Baccetti, “Birefringence characteristics in sperm heads allow for the selection of reacted spermatozoa for intracytoplasmic sperm injection,” Fertil. Steril. 93(3), 807–813 (2010). [CrossRef]
3. D. Needleman and Z. Dogic, “Active matter at the interface between materials science and cell biology,” Nat. Rev. Mater. 2(9), 17048 (2017). [CrossRef]
4. E. Priestly, Introduction to Liquid Crystals (Springer Science & Business Media, 2012).
5. S. Inou, Collected Works of Shinya Inou: Microscopes, Living Cells, and Dynamic Molecules (World Scientific, 2008).
6. V. V. Tuchin, “Tissue optics” (Society of Photo-Optical Instrumentation Engineers (SPIE), 2015).
7. S. J. Woltman, G. D. Jay, and G. P. Crawford, “Liquid-crystal materials find a new order in biomedical applications,” Nat. Mater. 6(12), 929–938 (2007). [CrossRef]
8. Z. Wang, L. J. Millet, M. U. Gillette, and G. Popescu, “Jones phase microscopy of transparent and anisotropic samples,” Opt. Lett. 33(11), 1270–1272 (2008). [CrossRef]
9. Y. Kim, J. Jeong, J. Jang, M. W. Kim, and Y. Park, “Polarization holographic microscopy for extracting spatio-temporally resolved Jones matrix,” Opt. Express 20(9), 9948–9955 (2012). [CrossRef]
10. J. Park, H. Yu, J.-H. Park, and Y. Park, “LCD panel characterization by measuring full Jones matrix of individual pixels using polarization-sensitive digital holographic microscopy,” Opt. Express 22(20), 24304–24311 (2014). [CrossRef]
11. T. Colomb, F. Dürr, E. Cuche, P. Marquet, H. G. Limberger, R.-P. Salathé, and C. Depeursinge, “Polarization microscopy by use of digital holography: application to optical-fiber birefringence measurements,” Appl. Opt. 44(21), 4461–4469 (2005). [CrossRef]
12. X. Dai, S. Xu, X. Yang, K. C. Zhou, C. Glass, P. C. Konda, and R. Horstmeyer, “Quantitative Jones matrix imaging using vectorial Fourier ptychography,” Biomed. Opt. Express 13(3), 1457–1470 (2022). [CrossRef]
13. P. Ferraro, C. Del Core, L. Miccio, S. Grilli, S. De Nicola, A. Finizio, and G. Coppola, “Phase map retrieval in digital holography: avoiding the undersampling effect by a lateral shear approach,” Opt. Lett. 32(15), 2233–2235 (2007). [CrossRef]
14. C. Lee, Y. Baek, H. Hugonnet, and Y. Park, “Single-shot wide-field topography measurement using spectrally multiplexed reflection intensity holography via space-domain Kramers–Kronig relations,” Opt. Lett. 47(5), 1025–1028 (2022). [CrossRef]
15. Y. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018). [CrossRef]
16. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969). [CrossRef]
17. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205(2), 165–176 (2002). [CrossRef]
18. K. Kim, J. Yoon, S. Shin, S. Lee, S.-A. Yang, and Y. Park, “Optical diffraction tomography techniques for the study of cell pathophysiology,” J. Biomed. Photonics Eng. 2(2), 020201 (2016). [CrossRef]
19. R. Abbessi, N. Verrier, A. M. Taddese, S. Laroche, M. Debailleul, M. Lo, J. B. Courbot, and O. Haeberlé, “Multimodal image reconstruction from tomographic diffraction microscopy data,” J. Microsc. 288(3), 193–206 (2022). [CrossRef]
20. J. van Rooij and J. Kalkman, “Polarization contrast optical diffraction tomography,” Biomed. Opt. Express 11(4), 2109–2121 (2020). [CrossRef]
21. A. Saba, J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “Polarization-sensitive optical diffraction tomography,” Optica 8(3), 402–408 (2021). [CrossRef]
22. L.-H. Yeh, I. E. Ivanov, S.-M. Guo, B. B. Chhun, E. Hashemi, M. H. Han, and S. B. Mehta, “Upti: Uniaxial permittivity tensor imaging of intrinsic density and anisotropy,” in Biophotonics Congress (Optica Publishing Group, 2021), paper NM3C.4.
23. S. Shin, J. Eun, S. S. Lee, C. Lee, H. Hugonnet, D. K. Yoon, S.-H. Kim, J. Jeong, and Y. Park, “Tomographic measurement of dielectric tensors at optical frequency,” Nat. Mater. 21(3), 317–324 (2022). [CrossRef]
24. E. Wolf and M. Born, eds., “Optics of crystals,” in Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), pp. 790–852.
25. M. Lee, H. Hugonnet, and Y. Park, “Inverse problem solver for multiple light scattering using modified Born series,” Optica 9(2), 177–182 (2022). [CrossRef]
26. T.-A. Pham, E. Soubies, A. Ayoub, J. Lim, D. Psaltis, and M. Unser, “Three-Dimensional Optical Diffraction Tomography With Lippmann-Schwinger Model,” IEEE Trans. Comput. Imaging 6, 727–738 (2020). [CrossRef]
27. U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2(6), 517–522 (2015). [CrossRef]
28. J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 82 (2019). [CrossRef]
29. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
30. G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016). [CrossRef]
31. T. Vettenburg, S. A. R. Horsley, and J. Bertolotti, “Calculating coherent light-wave propagation in large heterogeneous media,” Opt. Express 27(9), 11946–11967 (2019). [CrossRef]
32. G. Osnabrugge, M. Benedictus, and I. M. Vellekoop, “Ultra-thin boundary layer for high-accuracy simulations of light propagation,” Opt. Express 29(2), 1649–1658 (2021). [CrossRef]
33. H. DeVoe, “Optical Properties of Molecular Aggregates. I. Classical Model of Electronic Absorption and Refraction,” The J. Chem. Phys. 41(2), 393–400 (1964). [CrossRef]
34. D. M. Pepper, “Nonlinear optical phase conjugation,” in Laser Handbook (Elsevier, 1985), pp. 333–485.
35. K. Lee, J. Lee, J.-H. Park, J.-H. Park, and Y. Park, “One-Wave Optical Phase Conjugation Mirror by Actively Coupling Arbitrary Light Fields into a Single-Mode Reflector,” Phys. Rev. Lett. 115(15), 153902 (2015). [CrossRef]
36. T. A. Waigh, I. Hopkinson, A. M. Donald, M. F. Butler, F. Heidelbach, and C. Riekel, “Analysis of the Native Structure of Starch Granules with X-ray Microfocus Diffraction,” Macromolecules 30(13), 3813–3820 (1997). [CrossRef]
37. S. Fan, S. Smith-Dryden, G. Li, and B. Saleh, “Optimization-based optical diffraction tomography using iODT initialization,” J. Opt. Soc. Am. A 38(7), 947–953 (2021). [CrossRef]
38. O. Yasuhiko and K. Takeuchi, “In-silico clearing approach for deep refractive index tomography by partial reconstruction and wave-backpropagation,” Light: Sci. Appl. 12(1), 101 (2023). [CrossRef]
39. H. Hugonnet, Y. W. Kim, M. Lee, S. Shin, R. H. Hruban, S.-M. Hong, and Y. Park, “Multiscale label-free volumetric holographic histopathology of thick-tissue slides with subcellular resolution,” Adv. Photonics 3(02), 026004 (2021). [CrossRef]
40. A. J. Lee, Y. S. Kim, J.-G. Kim, H. Hugonnet, M. Lee, T. Ku, and Y. Park, “Volumetric refractive index measurement and quantitative density analysis of mouse brain tissue with sub-micrometer spatial resolution,” bioRxiv, 2023.01.16.524195 (2023). [CrossRef]
41. L. Junqueira, G. Montes, and E. Sanchez, “The influence of tissue section thickness on the study of collagen by the Picrosirius-polarization method,” Histochemistry 74(1), 153–156 (1982). [CrossRef]
42. R. Oldenbourg, E. Salmon, and P. Tran, “Birefringence of Single and Bundled Microtubules,” Biophys. J. 74(1), 645–654 (1998). [CrossRef]
