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Measuring the scattering tensor of a disordered nonlinear medium

Nature Physics volume 19, pages 1709–1718 (2023)Cite this article

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Abstract

A complex scattering medium offers spatial mixing of the incoming waves via numerous randomly wired channels, making it act as a unique linear optical operator. However, its use as a nonlinear operator has been unexplored due to the difficulty in formulating the nonlinear wave–medium interaction. Here we present a theoretical framework and experimental proof that a third-order scattering tensor completely describes the input–output response of a nonlinear scattering medium made of second-harmonic-generation nanoparticles. The rank of the nonlinear scattering tensor is higher than that of a second-order scattering tensor describing a linear scattering medium, scaling with the number of the spatially orthogonal illumination channels. We implement the inverse of the nonlinear scattering tensor by tensor reshaping and minimization operation, which enables us to retrieve the original incident wave from the speckled nonlinear wave. Using the increased rank of the scattering tensor along with its inverse operation, we demonstrate that the disordered nonlinear medium can be used as a highly scalable nonlinear optical operator for optical encryptions, all-optical multichannel logic AND gates, and optical kernel methods in machine learning.

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Fig. 1: Wave–medium interaction of an SHG scattering medium.
Fig. 2: Experimental measurement of a scattering tensor.
Fig. 3: Solving the inverse of the scattering tensor.
Fig. 4: Experimental demonstration of all-optical multichannel logic gates.

Data availability

Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The MATLAB codes are available from the corresponding author upon reasonable request.

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Acknowledgements

We acknowledge competent discussions with K. Lee and D. Kim. This work was supported by the Institute for Basic Science (IBS-R023-D1, J.M., Y.-C.C., S.K. and W.C.) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2021R1C1C1011307, M.J.)

Author information

Authors and Affiliations

  1. Center for Molecular Spectroscopy and Dynamics, Institute for Basic Science, Seoul, Korea

    Jungho Moon, Ye-Chan Cho, Sungsam Kang & Wonshik Choi

  2. Department of Physics, Korea University, Seoul, Korea

    Jungho Moon, Ye-Chan Cho, Sungsam Kang & Wonshik Choi

  3. Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea

    Mooseok Jang

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Contributions

W.C. and J.M. developed the theory and designed the experiment. J.M. and Y.-C.C. constructed the experimental set-up and prepared the nonlinear scattering samples. J.M. and Y.-C.C. conducted data acquisition and data analysis along with S.K. and M.J. W.C. and J.M. wrote the paper. All authors discussed the results and commented on the paper.

Corresponding author

Correspondence to Wonshik Choi.

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The authors declare no competing interests.

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Peer review information

Nature Physics thanks Alexandre Aubry, Sushil Mujumdar 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 SHG interferometric microscope system for the scattering tensor measurement.

OL1 and OL2: objective lenses, SLM: spatial light modulator, A: aperture diaphragm for controlling both the amplitude and phase of the incident wave, SM: scanning mirrors for controlling the path length of the reference beam, BBO: Beta Barium Borate crystal, BP1 and BP2: bandpass filters, SP: spatial filter, HWP: half-wave plate, PBS: polarizing beam splitter, BS: beam splitter, and FM: flip mirror. Input and output planes of the nonlinear scattering medium are conjugate to SLM1, SLM2, and Camera1, respectively. Camera2 is in the Fourier plane of the SLM2. The set-up in the dashed rectangular box is used for the experimental demonstration of multichannel optical logic gates.

Supplementary information

Supplementary Information

Supplementary Figs. 1–8 and Text.

Source data

Source Data Fig. 1

Source data for Fig. 1g.

Source Data Fig. 2

Source data for Fig. 2f.

Source Data Fig. 3

Source data for Fig. 3b,f.

Source Data Fig. 4

Source data for Fig.4h.

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Moon, J., Cho, YC., Kang, S. et al. Measuring the scattering tensor of a disordered nonlinear medium. Nat. Phys. 19, 1709–1718 (2023). https://doi.org/10.1038/s41567-023-02163-8

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