A discussion of the properties of the Rayleigh perturbative solution in diffraction theory

doi:10.1016/j.wavemoti.2005.01.004

Wave Motion

Volume 42, Issue 2, August 2005, Pages 169-176

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Abstract

The problem of two-dimensional plane-wave scattering from an infinite periodic profile with a Dirichlet boundary condition is considered. It is shown that the Rayleigh theory yields the same perturbation series for the reflection coefficients as the extinction-theorem formalism. This identity holds up to any order if the boundary profile is infinitely continuously differentiable, and at least up to some finite order if the boundary profile is finitely continuously differentiable. The nature of the derivation proposed allows the extension of these results to all linear homogeneous media encountered in electromagnetism and elastodynamics, to all usual linear boundary conditions, and to the case of scattering from finite bodies. It is indicated that, for interfaces which are periodic finite linear combinations of sinusoids, the Rayleigh perturbative solution is identical to the perturbative solutions obtained with any other approach, and verifies the reciprocity relationships.

Introduction

Among the various approaches used to deal with reflection and transmission of waves at rough interfaces [1], [2], the Rayleigh methods are simple to implement. They are based on the hypothesis, postulated by Rayleigh [3], that the field scattered from an interface can be represented by the same sum of outgoing and evanescent waves in all of an half-space bounded by the interface; the amplitudes of those waves are numerically determined by projecting the boundary conditions on a given complete set of functions. The Rayleigh methods turn out to give good numerical results well beyond the limited domain of validity of the Rayleigh hypothesis [4], [5]. An explanation of this fact has been indicated for an interface separating isotropic solid media [6] and later given in detail for all homogeneous linear media encountered in electromagnetism and elastodynamics [7]. Other references on the rich and long-standing debate on the Rayleigh hypothesis can be found in Refs. [1], [2], [7] for instance.

Contrary to the Rayleigh method, the extinction-theorem method has no theoretical limitation and is therefore regarded as exact. In this approach, the null-field equations [1], [2] are solved for the unknown surface fields and the result is used to compute the reflected and transmitted fields. It is formally possible to evaluate the reflection and transmission coefficients by the application of a perturbative approach to either method; the unknown surface fields and the scattered amplitudes are then sought in the form of an integral-power series in the surface elevation [8].

The issue of the identity of these formal series has been investigated in the literature. Jackson et al. [8] have shown for the Dirichlet problem that the five initial terms of the Rayleigh perturbation series are identical to those found with the extinction-theorem perturbative approach. For the same problem, Roginsky [9] has shown that the terms of the perturbation series of the two theories are related through the reciprocity transformation to all orders. In the case of electromagnetic scattering at an analytic interface separating linear homogeneous isotropic media, Bruno and Reitich [10], [11] have shown that there exists a domain of sufficiently small surface elevation where the Rayleigh hypothesis holds; they have also shown that the scattered field is analytic in the surface elevation up to and beyond the surface. These results are of fundamental importance as they justify both the Rayleigh perturbative approach and perturbative approaches based on the preliminary evaluation of unknown surface fields; the integral-power series in the surface elevation of the reflection and transmission coefficients derived from these methods must therefore coincide and be invariant under reciprocity. The problem of scattering from finite bodies has recently been investigated by Skaropoulos and Chrissoulidis [12] in the case of plane-wave scattering in an isotropic medium from a soft circular surface with sinusoidal roughness; the scattered wave amplitudes given by the Rayleigh and extinction-theorem perturbative methods are numerically compared and found to be identical at any order considered.

In the present work, we propose a demonstration of the identity of the formal perturbative solutions of the Rayleigh and extinction-theorem formalisms in the case of two-dimensional plane-wave scattering from a periodic boundary with a Dirichlet condition. This identity holds up to any order if the boundary profile is infinitely continuously differentiable, and at least up to some finite order if the boundary profile is finitely continuously differentiable. In the discussion of these results, it will appear that the derivation proposed can be easily extended to all linear homogeneous media encountered in electromagnetism and elastodynamics, to all usual linear boundary conditions, and to finite bodies.

The paper is organized as follows. The two-dimensional quasi-periodic Dirichlet problem is presented in Section 2. The Rayleigh perturbative method is presented in Section 3. The Rayleigh perturbation series for the reflection coefficients are shown to be identical to the extinction-theorem perturbation series in Section 4. We discuss and generalize the above results in Section 5.

Section snippets

Problem description and fundamental relations

We consider a one-dimensional periodic profile ( $S$ ) defined by the relation $z = h f (x)$ , where x and z are the cartesian coordinates of a two-dimensional position vector r, and h is a positive real. f has a period D and is assumed to be T times continuously differentiable with T being some non-negative, possibly infinite, integer. A complex scalar field $u$ obeys the Dirichlet condition $u = 0$ on ( $S$ ) and the scalar Helmholtz equation $Δ u + k^{2} u = 0$ in the upper half-space defined by $z > h f (x)$ , with k being a

Rayleigh perturbative solution

In the Rayleigh perturbative formalism, the boundary condition is formally written with the Rayleigh hypothesis and solved after expansion in h. Therefore, if the reflection coefficients obtained with the Rayleigh method are designated by $t_{n, R}$ , the perturbation series $E^{(N)} (t_{n, R})$ at order N in h are obtained by writing that, for any $r_{S}$ on ( $S$ ) $E^{(N)} [ψ_{0}^{-} (r_{S})] + \sum_{n = - \infty}^{+ \infty} E^{(N)} [t_{n, R} ψ_{n}^{+} (r_{S})] = 0,$ and by projecting (13) on the Fourier basis.

At order $P + 1$ , Eq. (13) gives the following recursion formula: $\sum_{n = - \infty}^{+ \infty} t_{n, R}$

Comparison with the extinction-theorem perturbative solution

If Eq.(6) is applied to the case where $v (r_{0}) = u_{d} (r_{0})$ in the upper half-space and $v (r_{0}) = - ψ_{0}^{-} (r_{0})$ in the lower half-space, we find, with the boundary condition $ψ_{0}^{-} + u_{d} = 0$ on ( $S$ ) and the definition (1), $- ψ_{0}^{-} (r_{0}) = \int_{(C)} G (r_{0} - r_{S}) \frac{\partial u}{\partial n} (r_{S}) d l,$ at any $r_{0}$ in the lower half-space. This is the extinction theorem [1], [2]; it is used to calculate $\partial u / \partial n$ on ( $S$ ) in the extinction-theorem method. In the perturbative approach, (19) is written on a horizontal line $z = z_{0}$ located at a finite distance below ( $S$ ) and solved

Discussion

It has been shown, for the two-dimensional quasi-periodic Dirichlet problem with an infinitely continuously differentiable boundary, that the Rayleigh perturbation series for the reflection coefficients is identical through any order to the extinction-theorem perturbation series. If the boundary is only finitely continuously differentiable, the two perturbation series are identical at least up to some finite order. The perturbation series have been regarded as formal, and their applicability

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Wave Motion

A discussion of the properties of the Rayleigh perturbative solution in diffraction theory

Abstract

Introduction

Section snippets

Problem description and fundamental relations

Rayleigh perturbative solution

Comparison with the extinction-theorem perturbative solution

Discussion

Wave Scattering from Rough Surfaces

The Theory of Sound

Plane-wave expansions used to describe the field diffracted by a grating

J. Opt. Soc. Am.

Rayleigh method for scattering from random and deterministic interfaces

J. Acoust. Soc. Am.

Comparison of the Rayleigh–Fourier and extinction theorem methods applied to scattering and transmission at a rough solid-solid interface

J. Acoust. Soc. Am.

Rayleigh methods applied to electromagnetic scattering from gratings in general homogeneous media

Phys. Rev. E

Comparison of perturbation theories for rough-surface scattering

J. Acoust. Soc. Am.

A fast and high-order accurate surface perturbation method for nanoplasmonic simulations: Basic concepts, analytic continuation and applications