Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Wave dynamics in correlated weakly scattering random media


Figure 1: Illustration of a branched wave flow. The wave intensity (gray scale) of an initially plane wave propagating from top to bottom through a weakly scattering random medium.

Wave propagation in random media — this might sound abstract but is in fact very tangible and almost omnipresent in science and everyday life. A common example of such random waves in our scientific and everyday environment are surface water waves such as the wind driven ocean waves, but also light, sound, electrons, tsunamis and even earth quakes are waves that in a natural environment typically propagate through a complex medium. Due to this complexity, the medium is best described as random, with examples including the turbulent atmosphere, complex patterns of currents in the ocean, translucent biological tissue or a semiconductor crystal sprinkled with impurities.

In recent years it has become clear that even very small fluctuations in the random medium, if they are correlated, lead to strong fluctuations in the wave intensities that have pronounced branch-like spatial structures (an example is shown in Fig. 1). This branching has been reported for electron, light, sound, and water waves. For the latter it has been shown to be a possible mechanism for the formation of giant freak waves that are known to be a serious hazard for seafaring ships. However, branching leads to strong intensity fluctuations with a heavy tailed distribution in all the wave systems mentioned above (and not only those), of relevance as well e.g. for remote sensing or communications. It is thus of high theoretical and practical interest to find the intensity distributions of branched flows and their dependence on the parameters characterizing the random medium.

The basic mechanism of branching is the formation of random caustics in the corresponding trajectory or ray flow [12]. Random caustics are expected to cause power-law tails in the wave intensity distribution. These, however, will only be observable when the wavelength is many orders of magnitude smaller than the typical spatial scale of the disorder. For more realistic wavelengths the intensity distribution of the waves in the branching regime remains an open problem because of the multiple mechanisms influencing this distribution. Thus, while the correspondence of theory and experiment is well established qualitatively, a quantitative comparison allowing to confirm the underlying mechanisms and to predict wave intensity distributions in applications was still missing. We recently we were able to make several important steps to approach this goal.



Figure 2: The formation of extreme waves in a random medium. The scintillation index in a random medium as a function of the distance from the source (top panel). A high variance of the wave intensity corresponds to the formation of branches (center panel) and the occurrence of extreme events: (bottom panel)Two sections of the wave amplitudes at locations indicated by arrows in the central panel. The amplitude is normalized to the standard deviation of fluctuations in the saturated regime far away from the source (thick black dashed lines). Wave amplitudes of up to 10 times of this standard deviation can be observed. The probability of observing such an event in random waves following Rayleigh’s law would be smaller than 10-20.

One of the most fundamental predictions of theory is the typical scale along the propagation direction at which random caustics occur and its scaling with the disorder parameters. We could recently, both numerically and experimentally, establish that the ratio of the variance to the squared mean of the intensity fluctuations, i.e. the scintillation index (SI), is an adequate quantity to assess the spatial scale of branched wave flows [3]. The SI exhibits a pronounced peak at the onset of branching, as illustrated in Fig. 2. As shown in Fig. 3a the position of this peak exhibits the same scaling behavior with the fluctuation strength ε of the medium as the random caustic statistics (even for a wider scaling range). This shows that branching leads to its own fundamental length scale of wave propagation in random media.


Figure 3: Statistics of extreme waves. (a) The position of the SI peak scales with ε-2/3, defining the branching length scale [3]. (b) The distributions of the highest waves in different random media can be collapsed to a universal curve by appropriate scaling [4]. (c) The distribution of intensities in the saturated branching regime for various degrees of coherence shows the transition from Rayleigh to log-normal [5].


The peak of the SI defines the strong fluctuation regime, where the fluctuations are much stronger and their distribution much more heavy-tailed as in the saturated fluctuation regime, where the intensities follow the exponential Rayleigh’s law and the SI becomes 1. The bottom panel of Fig. 2 shows examples of rogue waves, two cuts along a branched wave flow that exhibit fluctuations up to approximately 10 times the mean intensity! While the derivation of the full distribution of intensities in the strong fluctuation regime is very difficult and still elusive, we were able to derive the probability distribution of the highest waves and their scaling with the parameters of the medium [4], as illustrated in Fig. 3b.

In the saturated regime, wave intensities are exponentially distributed. We have recently shown, that in branched flows this regime can be reached long before the mean free path [5], due to the superposition of a exponentially increasing number of branches. Ray flows were conjectured to show a log-normal intensity distribution in this regime and we could show that this conjecture is indeed correct. Wave flows, however, do not exhibit log-normal behavior, even for the smallest wave length, but we demonstrated that decoherence allows to explain the semiclassical limit and we derived the intensity distribution for arbitrary degree of coherence as shown in Fig. 3c.

[1]    J.J. Metzger, R. Fleischmann, and T. Geisel, Phys. Rev. Lett. 105, 020601 (2010).
[2]    D. Maryenko, F. Ospald, K. v. Klitzing, J. H. Smet, J. J. Metzger, R. Fleischmann, T. Geisel, and V. Umansky, Phys. Rev. B 85, 195329 (2012).
[3]    S. Barkhofen, J. J. Metzger, R. Fleischmann, U. Kuhl, and H.-J. Stöckmann, Phys. Rev. Lett. 111, 183902 (2013).
[4]     J. J. Metzger, R. Fleischmann, and T. Geisel, Phys. Rev. Lett. 112, 203903 (2014).
[5]    J.J. Metzger, R. Fleischmann, and T. Geisel, Phys. Rev. Lett. 111, 013901 (2013).

Contact:  Jakob J. Metzger 

Members working within this Project:

 Ragnar Fleischmann 
 Theo Geisel 

Former Members:

 Henri Degueldre 
 Jakob J. Metzger 

Selected Publications:

H. Degueldre, J.J. Metzger, E. Schultheis, and R. Fleischmann (2017).
Channeling of Branched Flow in Weakly Scattering Anisotropic Media
Phys. Rev. Lett. 118:024301 . download file

H. Degueldre, J.J. Metzger, T. Geisel, and R. Fleischmann (2016).
Random Focusing of Tsunami Waves
Nature Physics 12:259–262.

J.J. Metzger, R. Fleischmann, and T. Geisel (2014).
Statistics of Extreme Waves in Random Media
Phys. Rev. Lett. 112:203903.

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

S. Barkhofen, J.J. Metzger, R. Fleischmann, U. Kuhl, and H. Stöckmann (2013).
Experimental Observation of a Fundamental Length Scale of Waves in Random Media
Phys. Rev. Lett. 111(18):183902. download file

D. Maryenko, J.J. Metzger, R. Fleischmann, T. Geisel, V. Umansky, F. Ospald, K. v. Klitzing, and J. Smet (2012).
How branching can change the conductance of ballistic semiconductor devices
Phys. Rev. B 85:195329. download file

J.J. Metzger, R. Fleischmann, and T. Geisel (2010).
Universal Statistics of Branched Flows
Phys. Rev. Lett. 105(2):020601. download file