In their article “Imaging with ambient noise” (PHYSICS TODAY, September 2010, page 44), Roel Snieder and Kees Wapenaar discuss a relation between cross-correlated noise in a medium and a signal, the Green function, propagating in that medium. The article is devoted to some comparatively recent developments in acoustics and seismology. That kind of relation has been known for a long time in optics or, more generally, in electrodynamics of continuous media.
A medium in thermal equilibrium generates fluctuating electromagnetic fields, due to the random motion of the particles of the medium. The fluctuation–dissipation theorem, supplemented by the requirement of time-reversal symmetry, enables one to express the two-point correlation function of the fluctuating fields in terms of the corresponding Green function.1,2 The Green function describes a signal propagating between the two points, one point being a source and the other a receiver. The existence of such a relation is not particularly surprising. After all, the whole essence of the fluctuation–dissipation theorem is to relate properties of the noise to the system’s response to an external source.
Similar relations, although of quite distinct physical origin, were established in optics of disordered media. In that field the thermal fluctuations are neglected; instead, random sources of radiation are provided by the inhomogeneities in the refraction index of the medium. Those sources, unlike the thermal ones, are not naturally excited but are activated by an externally injected wave, which scatters off the inhomogeneities. That multiple scattering leads to a random speckle pattern of light intensity.
It turns out that in a weakly disordered medium, the cross correlation of field, or intensity, in such a pattern is related to the Green function in the frequency domain.3 In deriving that relation, one cannot generally use the fluctuation–dissipation theorem because the system is out of equilibrium. Furthermore, for a coherent monochromatic external source, some averaging is required, either over different realizations of randomness or over many pairs of points with the same fixed separation (for the incoherent thermal source, the averaging occurs naturally). Thus by studying correlations in the speckle pattern—a disorder-induced effect!—one can retrieve the Green function of the underlying clean medium. In that way one can use disorder to obtain information about various properties of the medium, such as the effective refraction index4 or the spatial structure of a photonic crystal.5
The developments in acoustics and seismology, as reviewed by Snieder and Wapenaar, are exciting and useful; however, the basic physics relating Green functions to noise is similar to that in the earlier work on stochastic electrodynamics and optics.
