Observation of branched flow of light

Abstract

When waves propagate through a weak disordered potential with correlation length larger than the wavelength, they form channels (branches) of enhanced intensity that keep dividing as the waves propagate1. This fundamental wave phenomenon is known as branched flow. It was first observed for electrons1,2,3,4,5,6 and for microwave cavities7,8, and it is generally expected for waves with vastly different wavelengths, for example, branched flow has been suggested as a focusing mechanism for ocean waves9,10,11, and was suggested to occur also in sound waves12 and ultrarelativistic electrons in graphene13. Branched flow may act as a trigger for the formation of extreme nonlinear events14,15,16,17 and as a channel through which energy is transmitted in a scattering medium18. Here we present the experimental observation of the branched flow of light. We show that, as light propagates inside a thin soap membrane, smooth thickness variations in the film act as a correlated disordered potential, focusing the light into filaments that display the features of branched flow: scaling of the distance to the first branching point and the probability distribution of the intensity. We find that, counterintuitively, despite the random variations in the medium and the linear nature of the effect, the filaments remain collimated throughout their paths. Bringing branched flow to the field of optics, with its full arsenal of tools, opens the door to the investigation of a plethora of new ideas such as branched flow in nonlinear media, in curved space or in active systems with gain. Furthermore, the labile nature of soap films leads to a regime in which the branched flow of light interacts and affects the underlying disorder through radiation pressure and gradient force.

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Fig. 1: Thin liquid membranes as a platform for observing branched flow of light.
Fig. 2: Observation of branched flow of light for an input beam generated by a single mode fibre.
Fig. 3: Observation of the branched flow of light for a plane wave input beam.
Fig. 4: Statistical properties of experimentally observed optical branched flow for a narrow input beam.

Data availability

The data are available upon request.

References

  1. 1.

    Topinka, M. A. et al. Coherent branched flow in a two-dimensional electron gas. Nature 410, 183–186 (2001).

    CAS  Article  ADS  Google Scholar 

  2. 2.

    Shaw, S. E. J. Propagation in smooth random potentials. PhD thesis, Harvard (2002); https://search.proquest.com/docview/305536276.

  3. 3.

    Aidala, K. E. et al. Imaging magnetic focusing of coherent electron waves. Nat. Phys. 3, 464–468 (2007).

    CAS  Article  Google Scholar 

  4. 4.

    Jura, M. P. et al. Unexpected features of branched flow through high-mobility two-dimensional electron gases. Nat. Phys. 3, 841–845 (2007).

    CAS  Article  ADS  Google Scholar 

  5. 5.

    Maryenko, D. et al. How branching can change the conductance of ballistic semiconductor devices. Phys. Rev. B 85, 195329 (2012).

    Article  ADS  Google Scholar 

  6. 6.

    Liu, B. & Heller, E. J. Stability of branched flow from a quantum point contact. Phys. Rev. Lett. 111, 236804 (2013).

    Article  ADS  Google Scholar 

  7. 7.

    Höhmann, R., Kuhl, U., Stöckmann, H.-J., Kaplan, L. & Heller, E. J. Freak waves in the linear regime: a microwave study. Phys. Rev. Lett. 104, 093901 (2010).

    Article  ADS  Google Scholar 

  8. 8.

    Barkhofen, S., Metzger, J. J., Fleischmann, R., Kuhl, U. & Stöckmann, H.-J. Experimental observation of a fundamental length scale of waves in random media. Phys. Rev. Lett. 111, 183902 (2013).

    CAS  Article  ADS  Google Scholar 

  9. 9.

    Heller, E. J., Kaplan, L. & Dahlen, A. Refraction of a Gaussian seaway. J. Geophys. Res. Oceans 113, https://doi.org/10.1029/2008JC004748 (2008).

  10. 10.

    Ying, L. H., Zhuang, Z., Heller, E. J. & Kaplan, L. Linear and nonlinear rogue wave statistics in the presence of random currents. Nonlinearity 24, R67 (2011).

    MathSciNet  Article  ADS  Google Scholar 

  11. 11.

    Degueldre, H., Metzger, J. J., Geisel, T. & Fleischmann, R. Random focusing of tsunami waves. Nat. Phys. 12, 259–262 (2016).

    CAS  Article  Google Scholar 

  12. 12.

    Wolfson, M. A. & Tomsovic, S. On the stability of long-range sound propagation through a structured ocean. J. Acoust. Soc. Am. 109, 2693–2703 (2001).

    CAS  Article  ADS  Google Scholar 

  13. 13.

    Mattheakis, M., Tsironis, G. P. & Kaxiras, E. Emergence and dynamical properties of stochastic branching in the electronic flows of disordered Dirac solids. EPL 122, 27003 (2018).

    Article  ADS  Google Scholar 

  14. 14.

    Mattheakis, M. & Tsironis, G. P. Quodons in mica. In Extreme Waves and Branched Flows in Optical Media 425–454 (Springer International Publishing, 2015).

  15. 15.

    Mattheakis, M., Pitsios, I. J., Tsironis, G. P. & Tzortzakis, S. Extreme events in complex linear and nonlinear photonic media. Chaos Solitons Fractals 84, 73–80 (2016).

    Article  ADS  Google Scholar 

  16. 16.

    Dudley, J. M., Dias, F., Erkintalo, M. & Genty, G. Instabilities, breathers and rogue waves in optics. Nat. Photon. 8, 755–764 (2014).

    CAS  Article  ADS  Google Scholar 

  17. 17.

    Akhmediev, N., Soto-Crespo, J. M. & Ankiewicz, A. Extreme waves that appear from nowhere: on the nature of rogue waves. Phys. Lett. A 373, 2137–2145 (2009).

    MathSciNet  CAS  Article  ADS  Google Scholar 

  18. 18.

    Brandstötter, A., Girschik, A., Ambichl, P. & Rotter, S. Shaping the branched flow of light through disordered media. Proc. Natl Acad. Sci. USA 116, 13260–13265 (2019).

    Article  ADS  Google Scholar 

  19. 19.

    Berry, M. V. & Upstill, C. in Progress in Optics. IV. Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns (ed. Wolf, E.) Vol. 18, 257–346 (Elsevier, 1980).

  20. 20.

    Kaplan, L. Statistics of branched flow in a weak correlated random potential. Phys. Rev. Lett. 89, 184103 (2002).

    Article  ADS  Google Scholar 

  21. 21.

    Kravtsov, Y. A. & Orlov, Y. I. Caustics, Catastrophes and Wave Fields (Springer, 1993).

  22. 22.

    Patsyk, A., Bandres, M. A. & Segev, M. Interaction of light with thin liquid membranes. In Conference on Lasers and Electro-Optics FF3E2 (Optical Society of America, 2018); https://www.osapublishing.org/abstract.cfm?uri=CLEO_QELS-2019-FTu3D.1.

  23. 23.

    Metzger, J. J., Fleischmann, R. & Geisel, T. Universal statistics of branched flows. Phys. Rev. Lett. 105, 020601 (2010).

    Article  ADS  Google Scholar 

  24. 24.

    Degueldre, H., Metzger, J. J., Schultheis, E. & Fleischmann, R. Channeling of branched flow in weakly scattering anisotropic media. Phys. Rev. Lett. 118, 024301 (2017).

    Article  ADS  Google Scholar 

  25. 25.

    Metzger, J. J., Fleischmann, R. & Geisel, T. Statistics of extreme waves in random media. Phys. Rev. Lett. 112, 203903 (2014).

    Article  ADS  Google Scholar 

  26. 26.

    Metzger, J. J., Fleischmann, R. & Geisel, T. Intensity fluctuations of waves in random media: what is the semiclassical limit? Phys. Rev. Lett. 111, 013901 (2013).

    Article  ADS  Google Scholar 

  27. 27.

    Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007).

    CAS  Article  ADS  Google Scholar 

  28. 28.

    Ashkin, A., Dziedzic, J. M. & Smith, P. W. Continuous-wave self-focusing and self-trapping of light in artificial Kerr media. Opt. Lett. 7, 276–278 (1982).

    CAS  Article  ADS  Google Scholar 

  29. 29.

    Lamhot, Y. et al. Optical control of thermocapillary effects in complex nanofluids. Phys. Rev. Lett. 103, 264503 (2009).

    Article  ADS  Google Scholar 

  30. 30.

    Schley, R. et al. Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories. Nat. Commun. 5, 5189 (2014).

    CAS  Article  ADS  Google Scholar 

  31. 31.

    Heller, E. J., Fleischmann, R. & Kramer, T. Branched flow. Preprint at https://arxiv.org/abs/1910.07086 (2019).

  32. 32.

    Bekenstein, R., Nemirovsky, J., Kaminer, I. & Segev, M. Shape-preserving accelerating electromagnetic wave packets in curved space. Phys. Rev. X 4, 011038 (2014).

    Google Scholar 

  33. 33.

    Patsyk, A., Bandres, M. A., Bekenstein, R. & Segev, M. Observation of accelerating wave packets in curved space. Phys. Rev. X 8, 011001 (2018).

    Google Scholar 

  34. 34.

    Bekenstein, R., Schley, R., Mutzafi, M., Rotschild, C. & Segev, M. Optical simulations of gravitational effects in the Newton–Schrödinger system. Nat. Phys. 11, 872–878 (2015).

    CAS  Article  Google Scholar 

  35. 35.

    El-Ganainy, R., Makris, K. G., Christodoulides, D. N. & Musslimani, Z. H. Theory of coupled optical PT-symmetric structures. Opt. Lett. 32, 2632–2634 (2007).

    CAS  Article  ADS  Google Scholar 

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Acknowledgements

We are indebted to M. V. Berry, for suggesting to us that what we saw in the experiments is related to branched flow. This research was supported by the German-Israeli DIP project, and by the Israel Science Foundation.

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Correspondence to Mordechai Segev or Miguel A. Bandres.

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Supplementary Information

This file contains Supplementary Text and Data, including Supplementary Figures 1-13.

Video 1

Branched flow Typical patterns of branched flow. The light is launched into the soap film through an optical fiber. The optical beam propagates within the thin film, which acts as a slab waveguide with a locally varying thickness. The light scatters from the thickness variations and forms the pattern of branched flow.

Video 2

Branched flow and background potential visualization The video shows branched flow patterns superimposed on top of the landscape of the scattering potential. The system is similar to video #1, but with additional RGB illumination incident upon the film from the top. The RGB light is used to map the thickness fluctuations as a function of position in the plane of the film. The film acts as a Fabry-Perot interferometer for the RGB light, giving rise to the colourful pattern shown in the video. The colours provide the information about the thickness of film as a function of lateral position. From these colours, we calculate the potential landscape, as explained in the SI. The thickness variations in the film act as a weak slowly-varying potential for the laser beam guided within the membrane.

Video 3

Branched flow in curved space In this video, the soap film forms a (semi) spherical bubble, and the branched flow forms within a positively curved 2D shell. In this case, in addition to thickness variations of the film, there is a global curvature of the film that affects the propagation. The curvature acts as an additional focusing potential and makes the branches propagate in a parallel fashion.

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Patsyk, A., Sivan, U., Segev, M. et al. Observation of branched flow of light. Nature 583, 60–65 (2020). https://doi.org/10.1038/s41586-020-2376-8

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