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A Lévy flight for light

Abstract

A random walk is a stochastic process in which particles or waves travel along random trajectories. The first application of a random walk was in the description of particle motion in a fluid (brownian motion); now it is a central concept in statistical physics, describing transport phenomena such as heat, sound and light diffusion1. Lévy flights are a particular class of generalized random walk in which the step lengths during the walk are described by a ‘heavy-tailed’ probability distribution. They can describe all stochastic processes that are scale invariant2,3. Lévy flights have accordingly turned out to be applicable to a diverse range of fields, describing animal foraging patterns4, the distribution of human travel5 and even some aspects of earthquake behaviour6. Transport based on Lévy flights has been extensively studied numerically7,8,9, but experimental work has been limited10,11 and, to date, it has not seemed possible to observe and study Lévy transport in actual materials. For example, experimental work on heat, sound, and light diffusion is generally limited to normal, brownian, diffusion. Here we show that it is possible to engineer an optical material in which light waves perform a Lévy flight. The key parameters that determine the transport behaviour can be easily tuned, making this an ideal experimental system in which to study Lévy flights in a controlled way. The development of a material in which the diffusive transport of light is governed by Lévy statistics might even permit the development of new optical functionalities that go beyond normal light diffusion.

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Figure 1: Random walk trajectories.
Figure 2: Thickness dependence of the total transmission.
Figure 3: Spatial dependence of the transmission on the output surface.
Figure 4: Average transmission on the output surface versus radial distance from the centre.
Figure 5: Lévy walk in an inhomogeneous medium.

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Acknowledgements

We wish to thank A. Lagendijk for discussions and for reading the manuscript. Also we thank R. Righini and M. Colocci for continuous support, the entire Optics of Complex Systems group at LENS for discussions. This project has been financed by the ATLAS program of the European Commission, as well as the European Network of Excellence PHOREMOST.

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Correspondence to Diederik S. Wiersma.

Supplementary information

Supplementary information

The file contains Supplementary Notes with Supplementary Figures 1-2.. Mathematical model of the relation between the alpha-parameter and the sphere diameter distribution. Details of the sample fabrication. Control measurement of the transmission profile, for a sample containing only one category of spheres. (PDF 319 kb)

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Barthelemy, P., Bertolotti, J. & Wiersma, D. A Lévy flight for light. Nature 453, 495–498 (2008). https://doi.org/10.1038/nature06948

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