A Lévy flight for light

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 diffusion. 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 invariant. Lévy flights have accordingly turned out to be applicable to a diverse range of fields, describing animal foraging patterns, the distribution of human travel and even some aspects of earthquake behaviour. Transport based on Lévy flights has been extensively studied numerically, but experimental work has been limited 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.

[1]  P. Levy.  Théorie de l'addition des variables aléatoires , 1955 .

[2]  B. Mandelbrot.  The Variation of Certain Speculative Prices , 1963 .

[3]  B. Mandelbrot.  Fractal Geometry of Nature , 1984 .

[4]  Geisel,et al.  Accelerated diffusion in Josephson junctions and related chaotic systems. , 1985, Physical review letters.

[5]  M. Toda,et al.  In: Statistical physics II , 1985 .

[6]  Ott,et al.  Anomalous diffusion in "living polymers": A genuine Levy flight? , 1990, Physical review letters.

[7]  D. Frenkel,et al.  Evidence for universal asymptotic decay of velocity fluctuations in Lorentz gases , 1992 .

[8]  Solomon,et al.  Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. , 1993, Physical review letters.

[9]  Michael F. Shlesinger,et al.  Strange kinetics , 1993, Nature.

[10]  Bouchaud,et al.  Subrecoil laser cooling and Lévy flights. , 1994, Physical review letters.

[11]  Stanley,et al.  Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. , 1994, Physical review letters.

[12]  Object imaging using diffuse light , 1994 .

[13]  P. Sheng,et al.  Introduction to Wave Scattering, Localization and Mesoscopic Phenomena. Second edition , 1995 .

[14]  R. Mantegna,et al.  Scaling behaviour in the dynamics of an economic index , 1995, Nature.

[15]  G. Rikken,et al.  Observation of magnetically induced transverse diffusion of light , 1996, Nature.

[16]  G. Maret,et al.  Universal Conductance Fluctuations of Light , 1998 .

[17]  P. A. Robinson,et al.  LEVY RANDOM WALKS IN FINITE SYSTEMS , 1998 .

[18]  Robert P. H. Chang,et al.  Random laser action in semiconductor powder , 1999 .

[19]  H E Stanley,et al.  Average time spent by Lévy flights and walks on an interval with absorbing boundaries. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Michael F. Shlesinger.  Physics in the noise , 2001, Nature.

[21]  Mher Ghulinyan,et al.  Optical analogue of electronic Bloch oscillations. , 2003, Physical review letters.

[22]  R. Kubo.  Statistical Physics II: Nonequilibrium Statistical Mechanics , 2003 .

[23]  Frederic Bartumeus,et al.  ANIMAL SEARCH STRATEGIES: A QUANTITATIVE RANDOM‐WALK ANALYSIS , 2005 .

[24]  LÉVY STATISTICAL FLUCTUATIONS FROM A RANDOM AMPLIFYING MEDIUM , 2005, physics/0503059.

[25]  Alvaro Corral.  Universal earthquake-occurrence jumps, correlations with time, and anomalous diffusion. , 2006, Physical review letters.

[26]  Vasilis Ntziachristos,et al.  From finite to infinite volumes: removal of boundaries in diffuse wave imaging. , 2006, Physical review letters.

[27]  T. Geisel,et al.  The scaling laws of human travel , 2006, Nature.

[28]  J. Klafter,et al.  Some fundamental aspects of Lévy flights , 2007 .

[29]  Diederik S. Wiersma,et al.  Statistical regimes of random laser fluctuations , 2007 .