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Mesoscopic light transport in nonlinear disordered media

Alfonso Nardi1, Andrea Morandi1, Romain Pierrat2, Arthur Goetschy2, Xuanchen Li1, Frank Scheffold3, and Rachel Grange1

Phys. Rev. Research 7, 023323 – Published 30 June, 2025

DOI: https://doi.org/10.1103/ztyj-rc63

Abstract

Nonlinear disordered media uniquely combine multiple scattering and second-harmonic generation. Here, we investigate the statistical properties of the nonlinear light generated within such media. We report super-Rayleigh statistics of the second-harmonic speckle intensity, and demonstrate that it is caused by the mesoscopic correlations arising in extreme scattering conditions. The conductance measured is exceptionally low for an isotropically scattering three-dimensional medium, enabling applications in broadband second-harmonic generation, wavefront shaping in nonlinear disordered media, and photonic computing.

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References (49)

  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed. (The International Society for Optical Engineering, Bellingham, 2020).
  2. R. F. Wagner, S. W. Smith, J. M. Sandrik, and H. Lopez, Statistics of speckle in ultrasound B-scans, IEEE Trans. Sonics Ultrason. 30, 156 (1983).
  3. R. G. Dall, S. S. Hodgman, A. G. Manning, M. T. Johnsson, K. G. H. Baldwin, and A. G. Truscott, Observation of atomic speckle and Hanbury Brown–Twiss correlations in guided matter waves, Nat. Commun. 2, 291 (2011).
  4. R. Carminati and J. C. Schotland, Principles of Scattering and Transport of Light (Cambridge University Press, Cambridge, U.K., 2021).
  5. Y. Bromberg and H. Cao, Generating non-Rayleigh speckles with tailored intensity statistics, Phys. Rev. Lett. 112, 213904 (2014).
  6. R. Liu, B. Qing, S. Zhao, P. Zhang, H. Gao, S. Chen, and F. Li, Generation of non-Rayleigh nondiffracting speckles, Phys. Rev. Lett. 127, 180601 (2021).
  7. S. Han, N. Bender, and H. Cao, Tailoring 3D speckle statistics, Phys. Rev. Lett. 130, 093802 (2023).
  8. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, Correlations and fluctuations of coherent wave transmission through disordered media, Phys. Rev. Lett. 61, 834 (1988).
  9. N. Garcia and A. Z. Genack, Crossover to strong intensity correlation for microwave radiation in random media, Phys. Rev. Lett. 63, 1678 (1989).
  10. M. P. van Albada, J. F. de Boer, and A. Lagendijk, Observation of long-range intensity correlation in the transport of coherent light through a random medium, Phys. Rev. Lett. 64, 2787 (1990).
  11. F. Scheffold and G. Maret, Universal conductance fluctuations of light, Phys. Rev. Lett. 81, 5800 (1998).
  12. S. Balog, P. Zakharov, F. Scheffold, and S. E. Skipetrov, Photocount statistics in mesoscopic optics, Phys. Rev. Lett. 97, 103901 (2006).
  13. O. N. Dorokhov, On the coexistence of localized and extended electronic states in the metallic phase, Solid State Commun. 51, 381 (1984).
  14. Y. Imry, Active transmission channels and universal conductance fluctuations, Europhys. Lett. 1, 249 (1986).
  15. T. M. Nieuwenhuizen and M. C. W. van Rossum, Intensity distributions of waves transmitted through a multiple scattering medium, Phys. Rev. Lett. 74, 2674 (1995).
  16. T. Strudley, T. Zehender, C. Blejean, E. P. A. M. Bakkers, and O. L. Muskens, Mesoscopic light transport by very strong collective multiple scattering in nanowire mats, Nat. Photon. 7, 413 (2013).
  17. T. Strudley, D. Akbulut, W. L. Vos, A. Lagendijk, A. P. Mosk, and O. L. Muskens, Observation of intensity statistics of light transmitted through 3D random media, Opt. Lett. 39, 6347 (2014).
  18. F. Timpu, J. Sendra, C. Renaut, L. Lang, M. Timofeeva, M. T. Buscaglia, V. Buscaglia, and R. Grange, Lithium niobate nanocubes as linear and nonlinear ultraviolet Mie resonators, ACS Photonics 6, 545 (2019).
  19. R. W. Boyd, Nonlinear Optics (Academic Press, Burlington, 2008).
  20. M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials, Nature (London) 432, 374 (2004).
  21. E. V. Makeev and S. E. Skipetrov, Second harmonic generation in suspensions of spherical particles, Opt. Commun. 224, 139 (2003).
  22. R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Broadband femtosecond frequency doubling in random media, Appl. Phys. Lett. 89, 191105 (2006).
  23. Y. Qiao, F. Ye, Y. Zheng, and X. Chen, Cavity-enhanced second-harmonic generation in strongly scattering nonlinear media, Phys. Rev. A 99, 043844 (2019).
  24. R. Savo, A. Morandi, J. S. Müller, F. Kaufmann, F. Timpu, M. Reig Escalé, M. Zanini, L. Isa, and R. Grange, Broadband Mie driven random quasi-phase-matching, Nat. Photon. 14, 740 (2020).
  25. J. S. Müller, A. Morandi, R. Grange, and R. Savo, Modeling of random quasi-phase-matching in birefringent disordered media, Phys. Rev. Appl. 15, 064070 (2021).
  26. A. Morandi, R. Savo, J. S. Müller, S. Reichen, and R. Grange, Multiple scattering and random quasi-phase-matching in disordered assemblies of LiNbO3 nanocubes, ACS Photonics 9, 1882 (2022).
  27. V. M. Agranovich and V. E. Kravtsov, Effects of weak localization of photons in nonlinear optics: Second harmonic generation, Phys. Lett. A 131, 378 (1988).
  28. K. M. Yoo, S. Lee, Y. Takiguchi, and R. R. Alfano, Search for the effect of weak photon localization in second-harmonic waves generated in a disordered anisotropic nonlinear medium, Opt. Lett. 14, 800 (1989).
  29. S. Faez, P. M. Johnson, D. A. Mazurenko, and A. Lagendijk, Experimental observation of second-harmonic generation and diffusion inside random media, J. Opt. Soc. Am. B 26, 235 (2009).
  30. C. I. Valencia and E. R. Méndez, Weak localization effects in the second-harmonic light scattered by random systems of particles, Opt. Commun. 282, 1706 (2009).
  31. R. Samanta and S. Mujumdar, Intensity-dependent speckle contrast of second harmonic light in a nonlinear disordered medium, Appl. Opt. 59, 11266 (2020).
  32. R. Samanta, R. Pierrat, R. Carminati, and S. Mujumdar, Speckle decorrelation in fundamental and second-harmonic light scattered from nonlinear disorder, Phys. Rev. Appl. 18, 054047 (2022).
  33. J. Moon, Y.-C. Cho, S. Kang, M. Jang, and W. Choi, Measuring the scattering tensor of a disordered nonlinear medium, Nat. Phys. 19, 1709 (2023).
  34. F. Ni, H. Liu, Y. Zheng, and X. Chen, Nonlinear harmonic wave manipulation in nonlinear scattering medium via scattering-matrix method, Adv. Photonics 5, 046010 (2023).
  35. H. Wang, J. Hu, A. Morandi, A. Nardi, F. Xia, X. Li, R. Savo, Q. Liu, R. Grange, and S. Gigan, Large-scale photonic computing with nonlinear disordered media, Nat. Comput. Sci. 4, 429 (2024).
  36. C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, Correlation-enhanced control of wave focusing in disordered media, Nat. Phys. 13, 497 (2017).
  37. N. Bender, A. Yamilov, A. Goetschy, H. Yılmaz, C. W. Hsu, and H. Cao, Depth-targeted energy delivery deep inside scattering media, Nat. Phys. 18, 309 (2022).
  38. S. Yan, Y. Sun, F. Ni, Z. Liu, H. Liu, and X. Chen, Image reconstruction through a nonlinear scattering medium via deep learning, Photon. Res. 12, 2047 (2024).
  39. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ztyj-rc63 for the fabrication procedure of the nonlinear disordered sample, for the intensity fluctuations of fundamental light at different distances Δz, for the details of the data analysis, for the theoretical expressions of the mesoscopic distributions and the corresponding fit procedure, for the characterization of the transport mean free path and the minimum conductance, and for the derivation of the equations that govern the propagation of the linear and SH mean intensities through the disordered slab, which includes Refs. [44, 46, 47, 48, 49].
  40. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Statistical signatures of photon localization, Nature (London) 404, 850 (2000).
  41. F. Scheffold, W. Härtl, G. Maret, and E. Matijević, Observation of long-range correlations in temporal intensity fluctuations of light, Phys. Rev. B 56, 10942 (1997).
  42. C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69, 731 (1997).
  43. M. C. W. van Rossum and T. M. Nieuwenhuizen, Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion, Rev. Mod. Phys. 71, 313 (1999).
  44. A. García-Martín, F. Scheffold, M. Nieto-Vesperinas, and J. J. Sáenz, Finite-size effects on intensity correlations in random media, Phys. Rev. Lett. 88, 143901 (2002).
  45. A. Nardi, Dataset: Mesoscopic light transport in nonlinear disordered media (2025), https://doi.org/10.3929/ethz-b-000736656.
  46. G. Parry, Speckle patterns in partially coherent light, Laser Speckle and Related Phenomena (Springer, Berlin, 1975), pp. 77–121.
  47. N. Garcia, A. Z. Genack, and A. A. Lisyansky, Measurement of the transport mean free path of diffusing photons, Phys. Rev. B 46, 14475 (1992) 
  48. G. J. Edwards and M. Lawrence, A temperature-dependent dispersion equation for congruently grown lithium niobate, Opt. Quantum Electron. 16, 373 (1984).
  49. J. X. Zhu, D. J. Pine, and D. A. Weitz, Internal reflection of diffusive light in random media, Phys. Rev. A 44, 3948 (1991).

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