Shimshon Frankenthal

Two-scale solutions for intensity fluctuations in strong scattering

An exact integrodifferential equation is derived for a certain transform of the four-point coherence function by using a two-scale embedding procedure. The solution is expanded in an asymptotic series in the inverse strong-scattering parameter, the terms of which are solutions of a hierarchy of simple first-order partial differential equations. Further approximations simplify the retransformation and display the four-point coherence in a useful, numerically convenient form. Results compare favorably with existing numerical computations of the scintillation index and the covariance in two-dimensional plane-wave propagation. Results for three-dimensional plane-wave propagation are also presented.

Scattering calculations using the characteristic rays of the coherence function

We use the two‐scale expansion developed previously to determine the intensity variation in the neighborhood of caustics, to show how we may calculate the scattering of acoustic radiation in a random medium with a variable speed of sound profile. Equations are derived which show how the coherence function varies along the characteristic rays (identical to the rays of geometrical acoustics in the parabolic approximation). Conditions are given for simplifying the equations for high‐frequency quasi‐isotropic scattering. The problems that arise when the medium is highly anisotropic are discussed and an approximate solution is given. It is also shown how to calculate the intensity variation due to scattering in the neighborhood of a caustic.

PROPAGATION IN RANDOM STRATIFIED WAVEGUIDES : A MODAL-SPECTRAL TREATMENT

The statistics of a forward-propagating wave is considered in a random, anisotropic, stratified three-dimensional waveguide, where modal analysis offers unique advantages. After extracting the vertical dependence in the usual way, the equations are formulated which govern the range evolution of the transverse horizontal spectra of the modal field coefficients (MTWS). A short-range perturbation solution is used to derive the difference equations governing the long-range behavior of the two lowest moments of the field spectra. The conditions under which these difference equations can be approximated as differential equations are given. These equations are not limited by the parabolic approximation, and are amenable to numerical treatment by marching techniques. They are used here to study the effect of scattering on the spectral redistribution of the modal power, and the related problem of the coherence of plane and cylindrical waves. It is shown that, as a result of scattering in the transverse horizontal ...

Volume scattering in a shallow channel

Using a generalization of the modal coherence equations previously developed by Sutton and McCoy [J. Math. Phys. 18, 1052 (1977)], the effect of volume scattering in a shallow channel is treated. The difference between the range behavior of the cross‐modal coherence functions and the previously studied self‐modal coherence functions is shown. In particular, calculations are made to evaluate the characteristic scales that govern the range decay of the cross‐modal coherence functions. A particular example is given to illustrate the effect of the choice of different parameters.

Multiple foci in the range dependence of the intensity fluctuations of a plane wave propagating in a random medium

The differential equation for the fourth-order statistical moment of the field of a plane wave propagating in a two dimensional random medium is solved numerically. Results are presented for the scintillation index when the index-of-refraction correlation function of the medium is Gaussian and gamma (ratio of diffraction length to scattering length) is 50. It is pointed out that the solution has multiple foci, and it is suggested that this characteristic will be more pronounced for larger values of gamma.

The mutual coherence function in a scattering channel— A two‐scale solution

By using the two‐scale embedding procedure, an approximate expression for the mutual coherence function in a refractive and scattering channel is derived. The solution reduces to the known limiting monochromatic solutions, and provides a good approximation for the bichromatic coherence in a homogeneous scattering medium. Simple expressions are derived for both the coherence bandwidth and coherence time (Doppler spreading) in a quadratic channel with a quadratic scatterer. The possibility of inversion is explored.

Scintillations due to multiscale phase screens

The statistics of multiscale phase screens encompasses three fluctuation regimes: an inertial power-law regime, a short-scale (quadratic) dissipative regime, and a long-scale energy-input regime. We introduce a simple, analytically tractable, and numerically convenient model for such a screen and employ this model to analyze, compute, and classify the range profile of the scintillation index associated with a plane wave modulated by the screen. The screen classification is based on the ratios of the short scale and the long scale to the inertial scale, which is an inverse measure of the magnitude of the random phase fluctuations injected by a purely inertial screen. A screen is designated weak or strong, according to whether both ratios are less than or greater than 1, and moderate in the intermediate case (one ratio greater than 1 and one ratio less than 1). The scintillation profile increases quadratically with the range in the perturbation region near the screen and approaches a constant saturation level far from the screen. Weak screens produce monotonically increasing profiles, which saturate at less than 1. Moderate and strong screens produce profiles that saturate practically at 1. However, moderate screens produce profiles with a weak maximum because of the interplay between the inertial and the energy-input mechanisms. Moreover, the magnitude of this maximum can vary only in a narrow range whose bounds are determined by the inertial power-law exponent. Strong screens produce profiles with a pronounced maximum because of the interplay between the dissipative and the inertial mechanisms. The magnitude of this maximum scales with the logarithm of the screen strength (or the wavenumber) and therefore is unbounded in that sense.

Propagation of radiation in time-dependent three-dimensional random media

In Ref. [1] (Appendix A) we derived equations governing the frequency and spatial spectrum of radiation propagating in three-dimensional time-dependent random media with randomly varying sound speed c ( x , t). From the spectral equations we determine equations for the energy flux in both the forward and backward directions. We consider media that are spatially homogeneous and isotropic and stationary in time. In order to allow an independence assumption the analysis is restricted to fluctuations that satisfy the conditions τμ ≫ L z /c 0 and τμ ≪ L FS/c 0 where τμ is the characteristic time of the fluctuations, k 0 is the mean radiation wavenumber, L z is the characteristic correlation length of the random fluctuations in the mean propagation direction and L FS is a mean scattering length. We consider various values of γ = (k 0 L z )2/2. When γ ≪ 1 we find the usual radiation transfer equations. When γ ≫ 1, but back-scattering can be neglected, we find the forward-scattering equations. We also consider γ ≫ 1, when back-scattering cannot be neglected. We consider as initial boundary conditions a plane wave and an infinite incoherent source. We present numerical solutions for γ ≪ 1, γ = O (1) and γ ≫ 1 using a simple Gaussian form for the fluctuation correlation function.

Backscattering in stratified time-dependent random media-CW and narrow-band pulse propagation

Abstract We consider backscattering in a random stratified medium where the wave-speed fluctuations depend on time and on the range coordinate, which is normal to the planes of stratification. For the limit where the correlation time is shorter than the mean scattering time, we first derive radiation-transport equations that govern the range evolution of the spectra of the ensemble-averaged forward-and back-propagating components of the field and their bichromatic coherence functions. The latter are governed by integro-differential equations that account for the broadening of the signal spectra due to the time dependence of the random fluctuations. When the correlation time is longer than both the period of the radiating signal and the travel time of a wavefront across a range correlation length, the spectrum of the radiation due to a monochromatic CW excitation remains confined to a narrow band over extensive ranges. This permits a quasi-monochromatic approximation, whereby the integro-differential equations produce ordinary differential equations that govern the quantities of interest. We use this approximation to track the power flux associated with the propagation of a narrow-band pulse.

Propagation in one-dimensionally stratified time-independent scattering media

We calculate the probability density distributions of the power reflection coefficient, and of the various fluxes of the components of an erstwhile plane wave that propagates in a one-dimensionally stratified slab of a time-independent scattering medium. We determine the second- and fourth-order statistics of the power-fluxes, discuss the relevance of this problem to the localization phenomenon, examine the distribution of the emerging power-flux and the limitations on the assumption that it possesses a lognormal distribution, and finally discuss and rationalize the differences between the above and the corresponding characteristics of the radiation propagating in a time-dependent variant of this problem.