Tilt-invariant theory of rough-surface scattering: I

Abstract

The small-slope approximation (SSA) in rough-surface scattering theory uses the surface slope as a small parameter of expansion. But, from the physical point of view, the slope may not be a restrictive parameter because we can change the slope of a surface simply by tilting the coordinate system. We present the theory of rough-surface scattering in a coordinate-invariant form. The new method, tilt-invariant approximation (TIA), leads to a different expansion that does not require that the slope of a surface be small. For a small Rayleigh parameter this approximate solution provides the correct perturbation theory, for a large Rayleigh parameter it provides the Kirchhoff approximation with several correcting terms.     

A reciprocal phase-perturbation theory for rough-surface scattering

We study the scattering of a scalar plane wave from a two-dimensional, randomly rough surface, on which the Dirichlet boundary condition is satisfied. The scattering amplitude is obtained in the form of the Fourier transform of an exponential, in which the exponent is written as an expansion in powers of the surface profile function. It is shown that the latter expansion can be written in such a way that the corresponding scattering matrix is manifestly reciprocal. Numerical results for the reflectivity, and for the contribution to the mean differential reflection coefficient from the incoherent component of the scattered field, are obtained and compared with the predictions of small-amplitude perturbation theory and the Kirchhoff approximation. As the wavelength of the incident wave is varied continuously the results of the phase-perturbation theory change continuously from those of the small-amplitude perturbation theory to those of the Kirchhoff approximation.     

A reciprocal phase-perturbation theory for rough-surface scattering

Abstract

We study the scattering of a scalar plane wave from a two-dimensional, randomly rough surface, on which the Dirichlet boundary condition is satisfied. The scattering amplitude is obtained in the form of the Fourier transform of an exponential, in which the exponent is written as an expansion in powers of the surface profile function. It is shown that the latter expansion can be written in such a way that the corresponding scattering matrix is manifestly reciprocal. Numerical results for the reflectivity, and for the contribution to the mean differential reflection coefficient from the incoherent component of the scattered field, are obtained and compared with the predictions of small-amplitude perturbation theory and the Kirchhoff approximation. As the wavelength of the incident wave is varied continuously the results of the phase-perturbation theory change continuously from those of the small-amplitude perturbation theory to those of the Kirchhoff approximation.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

Synthetic spectra from rough-surface scattering

We present a method for designing a one-dimensional, deterministic, perfectly conducting rough surface that scatters light at a fixed scattering angle with an intensity whose dependence on the frequency of a plane wave incident normally upon it reproduces the infrared spectrum of a known substance within a specified region of frequencies. Such a surface can therefore be used in a correlation spectrometer for the identification of unknown substances.     

High-order solutions of three-dimensional rough-surface scattering problems at high frequencies. I: the scalar case

We present a new high-order numerical method for the solution of high-frequency scattering problems from rough surfaces in three dimensions. The method is based on the asymptotic solution of appropriate integral equations in the high-frequency regime, in a manner that bypasses the need to resolve the fields on the scale of the wavelength of radiation. Indeed, inspired by prior work in two dimensions, we seek a solution of the integral equation in the form of a slow modulation of the incoming radiation, and we choose a series expansion in inverse powers of the wavenumber to represent the unknown slowly varying envelope. As we show, this framework can be made to yield an efficiently computable recursion for the terms in the series to any arbitrary order. The resulting algorithms generally provide a very significant improvement over classical (e.g. Kirchhoff) approximations in both accuracy and applicability and they can, in fact, effectively produce results with full double-precision accuracy for configurations of practical interest and up to the resonance regime.     

Convergence of solutions to the rough-surface scattering problem

Abstract

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.     

The IEM2M rough-surface scattering model for complex-permittivity scattering media

The integral equation model (IEM) was developed in the late 1980s and arguably became the most cited and implemented rough-surface scattering model in the field of radar remote sensing for Earth observation. It was derived by applying a second-order iteration in the incident electromagnetic field to the integral equations of the surface fields as given by Poggio and Miller. It is thus an extension of the first-order, Born approximation of these equations that produce the classical Kirchhoff approximation. The IEM has been subject to numerous amendments and variations over the last 20 years due to the imperfect introduction and handling of the Weyl representation of the spherical wave in its first version. The work presented here is a further development of the contribution made by the same author in 2001 (IEM2M), which was the first version of IEM able to blend analytically both the Kirchhoff and the small-perturbation approximations for the bistatic case. The improvement reported in this article is concerned with the inclusion of evanescent waves in the formulation of the model and the extension of the range of applicability of the second-order scattering terms to interfaces with complex-permittivity scattering media.     

An iterative solution of the rough-surface scattering problem

Abstract

An iterative solution to the problem of scattering from a one-dimensional rough surface is obtained for the Dirichlet boundary condition. The advantages of this method are that bounds for convergence of the solution can be established and that the solution may readily be iterated to sufficiently high order in the interaction to examine the rate at which it converges. Absolute convergence of the iterative solution is also a sufficient condition for the convergence of the operator expansion method for surfaces on which the slope is everywhere less than unity. A numerical example of scattering from an echelette grating is considered, and bounds for convergence established. It is found that for scattering from such surfaces the rate at which the iterative solution converges decreases as the surface slope is increased. Corresponding results are found for the operator expansion method.     

Convergence of solutions to the rough-surface scattering problem

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.     

Formulation of rough-surface scattering theory in terms of phase factors and approximate solutions based on this formulation

Abstract

This paper presents the formulation of rough-surface scattering theory in which the bounded phase shift factors, ζ(r, α) ζ exp[iαζ(r)], replace the elevation, ζ(r). Both the Dirichlet and the Neumann problems are considered. The integral equations for secondary surface sources are obtained that contain only this phase function in their kernels.

The Neumann (iterative) series for the solutions of the integral equations thus derived are functional Taylor series in powers of L(r, α), not in powers of ζ. If we expand L(r, α) in these series in powers of ζ(r), we obtain the standard perturbation theory series. Thus, the new formulation corresponds to the partial summation of the perturbation series.

Using the Neumann series, we obtain several uniform (with respect to αζ) approximate solutions that contain, as limiting cases, Bragg scattering, the Kirchhoff approximation, and most known advanced approximations.

In the case of random surface z = ζ(r), these new expansions contain the function ζ(r) only in the exponents, and, therefore, the result of averaging can be expressed only in terms of the characteristic functions of the multivariate probability distribution of elevations.     

On the theory of multiple scattering I

A systematic theory of multiple scattering is given for a nonpolar fluid of point dipoles. Local-field correction factors are consistently accounted for to all orders. To first order (single scattering) Einstein's result is obtained; the theory yields, however, automatically to this order the attenuation of the incoming beam before and of the scattered light after scattering. In the usual theories these effects are hidden in secular (shadow) contributions to multiple scattering. The single-triple and double-scattering intensities are briefly discussed.     

A phenomenological theory of light scattering by surfaces: I. General theory

A phenomenological theory of light scattering by surfaces is presented. The differential intensity of the scattered light is found in terms of autocorrelation functions characteristic for the distribution of matter in the surface layer. Local field factors are found.     

High-order solutions of three-dimensional rough-surface scattering problems at high-frequencies. Part II: The vector electromagnetic case

We introduce a new numerical scheme for three-dimensional electromagnetic rough-surface scattering simulations with the capability of delivering very accurate results from low to high frequencies at a cost that is independent of the wavelength of radiation. The method is an extension of the ideas and techniques introduced in the first paper of this series (Waves in Random and Complex Media, 15 (2005), pp. 1-16) to the vector electromagnetic case, and it is based on the solution of an integral equation formulation of the scattering problem. As in the scalar case, the solution of the integral equation (i.e. the current) is expressed as a slow modulation of an oscillatory exponential of known phase, and explicit recursive formulae are derived for the successive terms in a series expansion of the slow envelope in inverse powers of the wavenumber. As we show, and in spite of the considerably more involved nature of the derivations and resulting formulae, the performance of the method retains the favourable characteristics that were demonstrated in the treatment of acoustic scattering problems. In particular, results with full double-precision accuracy are presented for various geometries, incidences and polarizations.     

Non-unitary scattering and capture. I. Hilbert space theory

By carrying out a general analysis of properties of the wave operators for the non-unitary scattering theory which arises in connection with the use of complex optical potentials in nuclear scattering and elsewhere, we clarify some puzzling differences between two recent approaches to this subject.     

The effect of diatom-diatom collisions on depolarized light scattering linewidths : I. General theory

With the use of an appropriately generalized Waldmann-Snider collision (super-) operator, a unified kinetic theory treatment of depolarized light scattering (both depolarized Rayleigh and rotational Raman) and their related linewidths is presented for gases of rigid rotor molecules. Explicit expressions are given, both exactly and within a distorted wave Born approximation, for the various state-to-state (i.e. rotational quantum number dependent) collision processes which can contribute to any observed linewidths due to diatom-diatom collisions. The results of this paper are employed in the calculation of the depolarized linewidths for the hydrogen isotopes in the following article.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.