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Tomographic measurement of dielectric tensors at optical frequency

Abstract

The dielectric tensor is a physical descriptor of fundamental light–matter interactions, characterizing anisotropic materials with principal refractive indices and optic axes. Despite its importance in scientific and industrial applications ranging from material science to soft matter physics, the direct measurement of the three-dimensional dielectric tensor has been limited by the vectorial and inhomogeneous nature of light scattering from anisotropic materials. Here, we present a dielectric tensor tomographic approach to directly measure dielectric tensors of anisotropic structures including the spatial variations of principal refractive indices and directors. The anisotropic structure is illuminated with a polarized plane wave with various angles and polarization states. Then, the scattered fields are holographically measured and converted into vectorial diffracted field components. Finally, by inversely solving a vectorial wave equation, the three-dimensional dielectric tensor is reconstructed. Using this approach, we demonstrate quantitative tomographic measurements of various nematic liquid-crystal structures and their fast three-dimensional non-equilibrium dynamics.

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Fig. 1: Direct 3D anisotropy-imaging modality.
Fig. 2: Principle of DTT operation.
Fig. 3: Validation of DTT by numerical simulations and experimental demonstrations.
Fig. 4: Quantitative assessments of an axially twisted, inclined and inhomogeneous LC structure and its non-equilibrium dynamics.
Fig. 5: Experimental reconstruction and direct observation of 3D topological point defects in the LCN.

Data availability

DTT datasets are available at https://drive.google.com/drive/folders/1r3X5rJPKskW8ceElz009v1CIgZFULZO3?usp=sharing. All other data supporting the findings of this study are available from the corresponding author on reasonable request.

Code availability

DTT source code and example data are available at https://github.com/BMOLKAIST/Dielectric-Tensor-Tomography.

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Acknowledgements

This work (S.S., H.H. and Y.P.) was supported by the Korea Advanced Institute of Science and Technology UP programme, BK21+ programme, Tomocube and National Research Foundation of Korea (2015R1A3A2066550) and by the Institute of Information & Communications Technology Planning & Evaluation grant funded by the Korean government (Ministry of Science and ICT; 2021-0-00745). S.S. and Y.P. acknowledge funding from the Korea Advanced Institute of Science and Technology Institute of Technology Value Creation, Industry Liaison Center (G-CORE Project) grant funded by the Ministry of Science and ICT (N11210014). J.E. and J.J. thank the National Research Foundation of Korea (NRF-2021R1A2C1011163) for support. This work (S.S.L.) was supported by the Korea Institute of Science and Technology Institution Program. C.L. and D.K.Y. thank the Ministry of Science and ICT (2021M3H4A3A01050378) for support. S.-H.K. acknowledges funding from the National Research Foundation of Korea (NRF-2020R1A2C2003245).

Author information

Authors and Affiliations

Authors

Contributions

Y.P. conceived the initial idea. S.S. developed the reconstruction algorithm, performed the experiments and simulations and analysed the data. J.E., S.S.L, C.L., D.K.Y., S.-H.K. and J.J. provided the samples and contributed analytic tools. H.H. contributed analytic tools. All authors wrote and revised the manuscript.

Corresponding author

Correspondence to YongKeun Park.

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Nature Materials thanks Jeroen Kalkman, Oleg Lavrentovich and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Experimental setup based on Mach-Zehnder interferometry equipped with a DMD.

PBS, polarizing beam splitter; M, mirror; DMD, digital micromirror device (DLi 4130, Digital Light Innovations Inc.); LCR, liquid-crystal retarder (LCC1223-A, Thorlabs Inc.); L1-L2, tube lenses (f = 300 mm); C, condenser (UPLSAPO 60XW, 60×, NA = 1.2, Olympus Inc.); O, objective lens (UPLSAPO, 60×, oil immersion, NA = 1.42, Olympus Inc.); BS, beam splitter; P1-P2, polarizer (LPVISE100-A, Thorlabs Inc.); Cam1-Cam2, image sensor (Lt425R, Lumenera Inc.); HWP, half-wave plate.

Extended Data Fig. 2 Experimental determination of the proper amount of slight tilt.

Isotropic RI tomograms of a 7-μm-diameter polystyrene microsphere immersed in oil (n = 1.565), reconstructed by (a) principles of the conventional ODT, (b) the slight tilt as 0.014 μm-1, and (c) the slight tilt as 0.042 μm-1 along the polar angle.

Extended Data Fig. 3 Illustration of the angular spectrum decomposition method for obtaining vector components of diffracted fields.

(a), The polarization state of an oblique plane wave passing through a polarizer \(\overrightarrow {p_\alpha }\) is determined by the incident wave vector \(\overrightarrow {k_i}\) and the direction of the polarizer \(\overrightarrow \alpha\). (b), Under a right circularly polarized plane wave illumination, a diffracted field from a sample is polarization sensitively measured using two different directions of a polarizer, α1 = −45° and α2 = 45°, respectively (top). The measured fields are decomposed into the multiple plane waves (bottom). (c), Each decomposed plane wave is vectorized (top), and inverse Fourier transform of the multiple vectorized plane waves gives the vectorial diffracted field (bottom). In the color circle at the right side, the symbols A and φ denote the normalized amplitude and phase of the fields in the image plane, respectively. For visualization purposes, the symbols A and φ denote the normalized logarithmic amplitude and phase of the fields in Fourier plane, respectively.

Extended Data Fig. 4 Validation of DTT by numerical simulations.

The DTT results for simulated anisotropic microspheres. (a and c) Cross-sectional slices of the reconstructed 3D dielectric tensor of the anisotropic microspheres in the x–y, y–z, and z–x planes. (b and d) Cross-sectional slices of the principal refractive indices no, ne overlaid with the directors \(\hat n\), and a 3D view of the directors \(\hat n\) with the isosurfaces from no as 1.547. The white and black dashed lines indicate the cross-section positions.

Extended Data Fig. 5 Comparison of 3D directors of the LCN film.

(a, c) Reconstructed directors in the LCN film by DTT, and (b, d) estimated directors in the LCN film by commercial simulation software (TechWiz LCD 3D). (e) A recorded PLM image of the LCN film shown in (a, c). (f) A simulated PLM image from the estimated directors in (b, d). As in the case of LCN film comprising mixed and photopolymerized materials, our approach serves an invaluable tool for characterizing anisotropic materials whose properties are inaccessible. Unlike LC simulations demanding accurate material properties, DTT is based on experimental measurements and does not require any material properties for the reconstruction, such as the principal RIs, elastic constants, or morphological information.

Supplementary information

Supplementary Information

Supplementary Discussion Sections 1–3 and Figs. 1 and 2.

Supplementary Video 1

LC particle with radial configuration. The 3D rendered view of the directors with the isosurfaces from no as 1.547. The radial configuration and the radial hedgehog are shown for various perspectives.

Supplementary Video 2

LC particle with bipolar configuration. The 3D rendered view of the directors with the isosurfaces from no as 1.547. Various perspectives reveal the characteristic orientational alignments of bipolar configurations including the surface topological defects, boojums.

Supplementary Video 3

LCLC droplet in equilibrium. The lemon-shaped morphology and twisted bipolar orientational configuration of a stable LCLC droplet are clearly shown in the 3D rendered view for various perspectives.

Supplementary Video 4

Annihilation of an LCLC droplet. Incoming thermal energy breaks the equilibrium of the nematic–isotropic coexistence phase. Time-lapse 3D views for perspective views visualize the shrinking area, height and volume of the nematic tactoid.

Supplementary Video 5

Nucleation of a LCLC droplet. Time-lapse 3D views of the nucleation and growing of a LCLC droplet. For various perspectives, increasing non-equilibrium orientational defects can be found during the growth of the droplet.

Supplementary Video 6

Merging LCLC droplets. By cooling, neighbouring LCLC droplets are growing and merging into a single droplet. Relaxation to the lemon shape at equilibrium is verified for various perspectives.

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Shin, S., Eun, J., Lee, S.S. et al. Tomographic measurement of dielectric tensors at optical frequency. Nat. Mater. 21, 317–324 (2022). https://doi.org/10.1038/s41563-022-01202-8

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