OFFPRINT Space-frequency correlation of classical waves in disordered media: High-frequency and small-scale asymptotics

Two-frequency radiative transfer (2f-RT) theory is developed for geometrical optics in random media. The space-frequency correlation is described by the two-frequency Wigner distribution (2f-WD) which satisfies a closed-form equation, the two-frequency Wigner-Moyal equation. In the RT regime it is proved rigorously that 2f-WD satisfies a Fokker-Planck–like equation with complex-valued coefficients. By dimensional analysis 2f-RT equation yields the scaling behavior of three physical parameters: the spatial spread, the coherence length and the coherence bandwidth. The sub-transport mean-free-path behavior is obtained in a closed form by analytically solving a paraxial 2f-RT equation. Copyright c © EPLA, 2007 Introduction. – Correlation functions of fields arise naturally in the description of fluctuations and are ubiquitous objects in statistical physics. The most basic of those are the second-order correlations in the spacetime or space-frequency domain; the two are equivalent to each other via the Fourier transform. When the field fluctuations can be described as a Gaussian stochastic process, all the correlation functions of the field can then be expressed in term of the second-order ones, by the use of the moment theorem for Gaussian processes. The second-order space-frequency correlation then emerges as an indispensable tool for studying fluctuations of fields and is equivalent to the mutual coherence function describing the field correlation at two space-time points [1]. Spatial and temporal structures of ultrawide-band highfrequency fields can be appreciably affected by small random changes of the medium parameters characteristic of almost all astroand geophysical environments. An important step toward the analytical understanding of pulse propagation in multiply scattering media is then to derive the equation for the space-frequency correlation, obtain the qualitative information about its behavior and, if possible, find its (asymptotic) solutions. This problem has been extensively studied in the literature, see, e.g., [2–6]. The main distinction of our approach from previous works is that our approach to space-frequency correlation is carried out in terms of the two-frequency Wigner distribution (2f-WD) for which we will derive rigorously equations of relatively simple form in the radiative transfer (RT) regime and obtain an exact solution for the small-scale behavior below the transport mean-freepath [1,7]. The standard (equal-time or -frequency) Wigner distribution (WD) is a quasi-probability density function in phase space and was first introduced by Wigner [8] in connection to quantum thermodynamics and later found wide-ranging applications in classical [9,10], as well as in quantum optics [1,11]. In classical optics, a main use of the Wigner distribution is connected to high-frequency asymptotic and radiative transfer, both of which can be most naturally worked out from the first principle in phase space (see the reviews [12,13] and references therein). The main advantage of 2f-RT over the traditional equaltime radiative transfer theory is that it describes not just the energetic transport but also the two space-time point mutual coherence in the following way. Let the scalar wave field Uj , j = 1, 2, of wave number kj , j = 1, 2 be governed by the reduced wave equation ∆Uj(r)+ k 2 j (ν+V (r))Uj(r) = 0, r∈R, j = 1, 2, (1) where ν and V are, respectively, the mean and fluctuation of the refractive index associated are assumed to be realvalued, corresponding to a lossless medium. For simplicity, we restrict our attention to dispersionless media (see [14] for discussion on dispersive media). Here and below the