- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Dynamics of the transverse optical flux in random media

Yuchen Ke, Nandini Bhattacharya, and Fabian Maucher*

  • Department of Precision and Microsystems Engineering, Faculty of Mechanical Engineering, Delft University of Technology, 2628 CD Delft, The Netherlands

  • *Contact author: f.maucher@tudelft.nl
PDF

Phys. Rev. Research 8, 013113 – Published 30 January, 2026

DOI: https://doi.org/10.1103/5rww-88k5

Abstract

We study the evolution of the kinetic energy (or gradient norm) of an incident linearly polarized monochromatic wave propagating in correlated random media. We explore the optical flux transverse to the mean Poynting flux at the paraxial-nonparaxial (vectorial) transition along with vortex counting. Here, by paraxial-nonparaxial transition we mean a gradual loss of validity of the paraxial approximation such that it is necessary to solve Maxwell-consistently employing the dyadic Green’s function. The vortex number appears to increase approximately with a cubic root of the propagation distance for sufficiently small correlation length. Furthermore, a kink appears in nucleation rate at the position of maximum scintillation upon increasing correlation length. A driven steady state is reached due to the filtering of evanescent waves upon propagation. Finally, we present the spectrum of the incompressible kinetic energy and how it evolves from the paraxial case to that of a (nonparaxial) random field.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (50)

  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 1998).
  2. K. Vynck, R. Pierrat, R. Carminati, L. S. Froufe-Pérez, F. Scheffold, R. Sapienza, S. Vignolini, and J. J. Sáenz, Light in correlated disordered media, Rev. Mod. Phys. 95, 045003 (2023).
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).
  4. M. C. W. van Rossum and T. M. Nieuwenhuizen, Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion, Rev. Mod. Phys. 71, 313 (1999).
  5. R. Carminati and J. C. Schotland, Principles of Scattering and Transport of Light (Cambridge University Press, Cambridge, 2021).
  6. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. Al-Habash, Theory of optical scintillation, J. Opt. Soc. Am. A 16, 1417 (1999).
  7. A. Patsyk, U. Sivan, M. Segev, and M. A. Bandres, Observation of branched flow of light, Nature (London) 583, 60 (2020).
  8. S.-S. Chang, K.-H. Wu, S.-J. Liu, Z.-K. Lin, J.-B. Wu, S.-J. Ge, L.-J. Chen, P. Chen, W. Hu, Y. Xu, H. Chen, D. He, D.-Q. Yang, J.-H. Jiang, Y.-Q. Lu, and J.-H. Chen, Electrical tuning of branched flow of light, Nat. Commun. 15, 197 (2024).
  9. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  10. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Localization of light in a disordered medium, Nature (London) 390, 671 (1997).
  11. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Transport and Anderson localization in disordered two-dimensional photonic lattices, Nature (London) 446, 52 (2007).
  12. A. Lagendijk, B. V. Tiggelen, and D. S. Wiersma, Fifty years of Anderson localization, Phys. Today 62(8), 24 (2009).
  13. A. Yamilov, S. E. Skipetrov, T. W. Hughes, M. Minkov, Z. Yu, and H. Cao, Anderson localization of electromagnetic waves in three dimensions, Nat. Phys. 19, 1308 (2023).
  14. O. Dorokhov, On the coexistence of localized and extended electronic states in the metallic phase, Solid State Commun. 51, 381 (1984).
  15. I. M. Vellekoop and A. Mosk, Focusing coherent light through opaque strongly scattering media, Opt. Lett. 32, 2309 (2007).
  16. J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, Non-invasive imaging through opaque scattering layers, Nature (London) 491, 232 (2012).
  17. H. Cao, Perspective on speckle spectrometers, J. Opt. 19, 060402 (2017).
  18. B. Redding, S. F. Liew, R. Sarma, and H. Cao, Compact spectrometer based on a disordered photonic chip, Nat. Photon. 7, 746 (2013).
  19. J. F. Nye and M. V. Berry, Dislocations in wave trains, Proc. R. Soc. Lond. A 336, 165 (1974).
  20. M. S. Soskin and M. V. Vasnetsov, Singular Optics, in Progress in Optics, edited by E. Wolf (Elsevier, 2001), Chap. 4, Vol. 42, pp. 219–276.
  21. M. R. Dennis, K. O’Holleran, and M. J. Padgett, Singular Optics: Optical Vortices and Polarization Singularities (Elsevier, 2009), Chap. 5, pp. 293–363.
  22. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Terabit free-space data transmission employing orbital angular momentum multiplexing, Nat. Photon. 6, 488 (2012).
  23. D. G. Grier, A revolution in optical manipulation, Nature (London) 424, 810 (2003).
  24. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature (London) 412, 313 (2001).
  25. C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, Quantized Vortex Dynamics and Superfluid Turbulence (Springer, Berlin, Heidelberg, 2001).
  26. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett. 83, 2498 (1999).
  27. E. A. L. Henn, J. A. Seman, G. Roati, K. M. F. Magalhães, and V. S. Bagnato, Emergence of turbulence in an oscillating Bose-Einstein condensate, Phys. Rev. Lett. 103, 045301 (2009).
  28. M. V. Berry, Disruption of wavefronts: Statistics of dislocations in incoherent Gaussian random waves, J. Phys. A: Math. Gen. 11, 27 (1978).
  29. I. Freund, N. Shvartsman, and V. Freilikher, Optical dislocation networks in highly random media, Opt. Commun. 101, 247 (1993).
  30. I. Freund, Optical vortices in Gaussian random wave fields: Statistical probability densities, J. Opt. Soc. Am. A 11, 1644 (1994).
  31. N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, Dislocations of the wave-front of a speckle inhomogeneous field, Zh. Eksp. Teor. Fiz. 33, 206 (1981) [JETP Lett. 33, 195 (1981)].
  32. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, Wave-front dislocations: Topological limitations for adaptive systems with phase conjugation, J. Opt. Soc. Am. 73, 525 (1983).
  33. M. V. Berry and M. R. Dennis, Phase singularities in isotropic random waves, Proc. R. Soc. Lond. A 456, 2059 (2000).
  34. S. Zhang, B. Hu, P. Sebbah, and A. Z. Genack, Speckle evolution of diffusive and localized waves, Phys. Rev. Lett. 99, 063902 (2007).
  35. S. Zhang and A. Z. Genack, Statistics of diffusive and localized fields in the vortex core, Phys. Rev. Lett. 99, 203901 (2007).
  36. K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, Fractality of light’s darkness, Phys. Rev. Lett. 100, 053902 (2008).
  37. K. Staliunas, A. Berzanskis, and V. Jarutis, Vortex statistics in optical speckle fields, Opt. Commun. 120, 23 (1995).
  38. A. S. Bradley and B. P. Anderson, Energy spectra of vortex distributions in two-dimensional quantum turbulence, Phys. Rev. X 2, 041001 (2012).
  39. M. Baker-Rasooli, W. Liu, T. Aladjidi, A. Bramati, and Q. Glorieux, Turbulent dynamics in a two-dimensional paraxial fluid of light, Phys. Rev. A 108, 063512 (2023).
  40. R. Fante, Electromagnetic beam propagation in turbulent media, Proc. IEEE 63, 1669 (1975).
  41. 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 (2016).
  42. G. Osnabrugge, M. Benedictus, and I. M. Vellekoop, Ultra-thin boundary layer for high-accuracy simulations of light propagation, Opt. Express 29, 1649 (2021).
  43. E. A. Novikov, Functionals and the random-force method in turbulence theory, J. Exptl. Theoret. Phys. (U.S.S.R.) 47, 1919 (1964) [Sov. Phys. JETP 20, 1290 (1965)].
  44. K. Furutsu, On the statistical theory of electromagnetic waves in a fluctuating medium (I), J. Res. Nat. Bur. Standards D 67, 303 (1963).
  45. F. Maucher, W. Krolikowski, and S. Skupin, Stability of solitary waves in random nonlocal nonlinear media, Phys. Rev. A 85, 063803 (2012).
  46. J. J. Metzger, R. Fleischmann, and T. Geisel, Universal statistics of branched flows, Phys. Rev. Lett. 105, 020601 (2010).
  47. J. W. Goodman, Statistical Optics (John Wiley & Sons, Hoboken, New Jersey Wiley, 2015).
  48. F. Maucher, S. Skupin, S. A. Gardiner, and I. G. Hughes, Creating complex optical longitudinal polarization structures, Phys. Rev. Lett. 120, 163903 (2018).
  49. G. S. Settles, Schlieren and Shadowgraph Techniques (Springer, Berlin, Heidelberg, 2001).
  50. A. Nardi, A. Morandi, R. Pierrat, A. Goetschy, X. Li, F. Scheffold, and R. Grange, Mesoscopic light transport in nonlinear disordered media, Phys. Rev. Res. 7, 023323 (2025).

Outline

Information

Phys. Rev. Research 8, 013113– Published 30 January, 2026
  • Received 17 July 2025
  • Accepted 13 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation