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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.

References

  1. Vellekoop, I. M. & Mosk, A. P. Focusing coherent light through opaque strongly scattering media. Opt. Lett. 32, 2309–2311 (2007).

    ADS  Google Scholar 

  2. Popoff, S. M. et al. Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media. Phys. Rev. Lett. 104, 100601 (2010).

    ADS  Google Scholar 

  3. Mosk, A. P., Lagendijk, A., Lerosey, G. & Fink, M. Controlling waves in space and time for imaging and focusing in complex media. Nat. Photon. 6, 283–292 (2012).

    ADS  Google Scholar 

  4. Lerosey, G., De Rosny, J., Tourin, A. & Fink, M. Focusing beyond the diffraction limit with far-field time reversal. Science 315, 1120–1122 (2007).

    ADS  Google Scholar 

  5. Vellekoop, I. M., Lagendijk, A. & Mosk, A. P. Exploiting disorder for perfect focusing. Nat. Photon. 4, 320–322 (2010).

    Google Scholar 

  6. Choi, Y. et al. Overcoming the diffraction limit using multiple light scattering in a highly disordered medium. Phys. Rev. Lett. 107, 023902 (2011).

    ADS  Google Scholar 

  7. Park, J. H. et al. Subwavelength light focusing using random nanoparticles. Nat. Photon. 7, 455–459 (2013).

    ADS  Google Scholar 

  8. Kim, D. & Englund, D. R. Quantum reference beacon-guided superresolution optical focusing in complex media. Science 363, 528–531 (2019).

    ADS  Google Scholar 

  9. Pappu, R., Recht, R., Taylor, J. & Gershenfeld, N. Physical one-way functions. Science 297, 2026–2030 (2002).

    ADS  Google Scholar 

  10. Goorden, S. A., Horstmann, M., Mosk, A. P., Skoric, B. & Pinkse, P. W. H. Quantum-secure authentication of a physical unclonable key. Optica 1, 421–424 (2014).

    ADS  Google Scholar 

  11. Redding, B., Liew, S. F., Sarma, R. & Cao, H. Compact spectrometer based on a disordered photonic chip. Nat. Photon. 7, 746–751 (2013).

    ADS  Google Scholar 

  12. Boyd, R. W. Nonlinear Optics (Elsevier Science, 2003).

  13. Teğin, U., Yıldırım, M., Oğuz, İ., Moser, C. & Psaltis, D. Scalable optical learning operator. Nat. Comput. Sci. 1, 542–549 (2021).

    Google Scholar 

  14. Samanta, R., Pierrat, R., Carminati, R. & Mujumdar, S. Speckle decorrelation in fundamental and second-harmonic light scattered from nonlinear disorder. Phys. Rev. A 18, 054047 (2022).

    Google Scholar 

  15. Hou, J. & Situ, G. Image encryption using spatial nonlinear optics. eLight 2, 3 (2022).

    Google Scholar 

  16. Katz, O., Small, E., Guan, Y. F. & Silberberg, Y. Noninvasive nonlinear focusing and imaging through strongly scattering turbid layers. Optica 1, 170–174 (2014).

    ADS  Google Scholar 

  17. Katz, O., Small, E., Bromberg, Y. & Silberberg, Y. Focusing and compression of ultrashort pulses through scattering media. Nat. Photon. 5, 372–377 (2011).

    ADS  Google Scholar 

  18. de Aguiar, H. B., Gigan, S. & Brasselet, S. Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping. Phys. Rev. A 94, 043830 (2016).

    ADS  Google Scholar 

  19. Qiao, Y. Q., Peng, Y. J., Zheng, Y. L., Ye, F. W. & Chen, X. F. Second-harmonic focusing by a nonlinear turbid medium via feedback-based wavefront shaping. Opt. Lett. 42, 1895–1898 (2017).

    ADS  Google Scholar 

  20. Hsieh, C. L., Pu, Y., Grange, R. & Psaltis, D. Digital phase conjugation of second harmonic radiation emitted by nanoparticles in turbid media. Opt. Express 18, 12283–12290 (2010).

    ADS  Google Scholar 

  21. Hsieh, C. L., Pu, Y., Grange, R., Laporte, G. & Psaltis, D. Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle. Opt. Express 18, 20723–20731 (2010).

    ADS  Google Scholar 

  22. Hsieh, C. L., Pu, Y., Grange, R. & Psaltis, D. Second harmonic generation from nanocrystals under linearly and circularly polarized excitations. Opt. Express 18, 11917–11932 (2010).

    ADS  Google Scholar 

  23. Mouthaan, R., Christopher, P. J., Gordon, G. S. D., Wilkinson, T. D. & Euser, T. G. Robust correction of interferometer phase drift in transmission matrix measurements. Appl. Opt. 61, 4315–4321 (2022).

    ADS  Google Scholar 

  24. Liu, S., Guo, C. L. & Sheridan, J. T. A review of optical image encryption techniques. Opt. Laser Technol. 57, 327–342 (2014).

    ADS  Google Scholar 

  25. Hazer, A. & Yildirim, R. A review of single and multiple optical image encryption techniques. J. Opt. 23, 113501 (2021).

    ADS  Google Scholar 

  26. Javidi, B. et al. Roadmap on optical security. J. Opt. 18, 083001 (2016).

    ADS  Google Scholar 

  27. Ruan, H. W., Xu, J. & Yang, C. H. E. Optical information transmission through complex scattering media with optical-channel-based intensity streaming. Nat. Commun. 12, 2411 (2021).

    ADS  Google Scholar 

  28. Peng, X., Wei, H. Z. & Zhang, P. Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain. Opt. Lett. 31, 3261–3263 (2006).

    ADS  Google Scholar 

  29. Carnicer, A., Montes-Usategui, M., Arcos, S. & Juvells, I. Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys. Opt. Lett. 30, 1644–1646 (2005).

    ADS  Google Scholar 

  30. Liao, M., He, W., Lu, D. & Peng, X. Ciphertext-only attack on optical cryptosystem with spatially incoherent illumination: from the view of imaging through scattering medium. Sci. Rep. 7, 41789 (2017).

    ADS  Google Scholar 

  31. Popoff, S., Lerosey, G., Fink, M., Boccara, A. C. & Gigan, S. Image transmission through an opaque material. Nat. Commun. 1, 81 (2010).

    ADS  Google Scholar 

  32. Kim, M., Choi, W., Choi, Y., Yoon, C. & Choi, W. Transmission matrix of a scattering medium and its applications in biophotonics. Opt. Express 23, 12648–12668 (2015).

    ADS  Google Scholar 

  33. Yu, H. et al. Measuring large optical transmission matrices of disordered media. Phys. Rev. Lett. 111, 153902 (2013).

    ADS  Google Scholar 

  34. Liu, Y. F., Yu, P. P., Li, Y. M. & Gong, L. Exploiting light field imaging through scattering media for optical encryption. OSA Contin. 3, 2968–2975 (2020).

    Google Scholar 

  35. Bai, X. et al. Coherent imaging of objects through thin-layer highly scattering medium based on optical encryption. Opt. Commun. 506, 127558 (2022).

    Google Scholar 

  36. Minzioni, P. et al. Roadmap on all-optical processing. J. Opt. 21, 063001 (2019).

    ADS  Google Scholar 

  37. Hardy, J. & Shamir, J. Optics inspired logic architecture. Opt. Express 15, 150–165 (2007).

    ADS  Google Scholar 

  38. Singh, J. J., Dhawan, D. & Gupta, N. All-optical photonic crystal logic gates for optical computing: an extensive review. Opt. Eng. 59, 110901 (2020).

    Google Scholar 

  39. Hussein, H. M. E., Ali, T. A. & Rafat, N. H. A review on the techniques for building all-optical photonic crystal logic gates. Opt. Laser Technol. 106, 385–397 (2018).

    ADS  Google Scholar 

  40. Caulfield, H. J. & Dolev, S. Why future supercomputing requires optics. Nat. Photon. 4, 261–263 (2010).

    Google Scholar 

  41. Wei, H., Wang, Z. X., Tian, X. R., Kall, M. & Xu, H. X. Cascaded logic gates in nanophotonic plasmon networks. Nat. Commun. 2, 387 (2011).

    ADS  Google Scholar 

  42. Fu, Y. L. et al. All-optical logic gates based on nanoscale plasmonic slot waveguides. Nano Lett. 12, 5784–5790 (2012).

    ADS  Google Scholar 

  43. McCutcheon, M. W. et al. All-optical conditional logic with a nonlinear photonic crystal nanocavity. Appl. Phys. Lett. 95, 221102 (2009).

    ADS  Google Scholar 

  44. Rachana, M., Swarnakar, S., Krishna, S. V. & Kumar, S. Design and analysis of an optical three-input AND gate using a photonic crystal fiber. Appl. Opt. 61, 77–83 (2022).

    ADS  Google Scholar 

  45. Kildishev, A. V., Boltasseva, A. & Shalaev, V. M. Planar photonics with metasurfaces. Science 339, 1232009 (2013).

    Google Scholar 

  46. Almeida, E., Bitton, O. & Prior, Y. Nonlinear metamaterials for holography. Nat. Commun. 7, 12533 (2016).

    ADS  Google Scholar 

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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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