Reciprocity theorems for electromagnetic or acoustic one‐way wave fields in dissipative inhomogeneous media

Abstract

Abstract. Reciprocity theorems have proven their usefulness in the study of forward and inverse scattering problems. Most reciprocity theorems in the literature apply to the total wave field and are thus not compatible with one-way wave theory, which is often applied in situations in which there is a clear preferred direction of propagation, like in electromagnetic or acoustic wave guides and in seismic exploration. In this paper we review the theory for one-way wave fields (or bidirectional beams), and we extensively discuss the symmetry properties of the square root operator appearing in the one-way wave equation. Using these symmetry properties, it appears to be possible to derive reciprocity theorems of the convolution type and of the correlation type for electromagnetic or acoustic one-way wave fields in dissipative inhomogeneous media along the same lines as the usual derivation of the reciprocity theorems for the total wave field. The one-way reciprocity theorem of the convolution type provides a basis for representations of scattered one-way wave fields in terms of generalized Bremmer series expansions or generalized primaries. The one-way reciprocity theorem of the correlation type finds its application in reflection imaging based on inverse one-way wave field propagators. IntroductionReciprocity In many wave propagation problems one can define a "preferred direction of propagation." Electromagnetic and acoustic waveguides are obvious examples, but also in laterally unbounded media it is often advantageous to define a preferred propagation direction, such as in seismic exploration. In all those situations it is useful to decompose the wave equation into a system of coupled equations for oppositely propagating waves. In the literature on electromagnetic wave theory these oppositely propagating waves are known as "bidirectional beams" [Hoekstra, 1997; van Stralen, 1997]; in the acoustic literature one usually speaks of "one-way wave wave fields" [Claerbout, 1971; Fishman et al., 1987]. In this paper we adopt the latter terminology.The reciprocity theorems discussed above apply to the total electromagnetic or acoustic wave field. Obviously, these reciprocity theorems are not compatible with one-way wave theory. In a recent paper we derived reciprocity theorems for acoustic one-way wave fields in lossless inhomogeneous fluids [Wapenaar and Grimbergen, 1996]. With some minor modifications these theorems are also applicable to transverse electric or transverse magnetic one-way wave fields in lossless inhomogeneous media. The aim of the current paper is to extend those one-way reciprocity theorems to electromagnetic or acoustic wave 851

A single-sided representation for the homogeneous Green's function of a unified scalar wave equation

“…When the original medium is dissipative, the adjoint medium is effectual [22][23][24] (a wave propagating through an effectual medium gains energy; effectual media are usually associated with a computational state). The wave operator for the adjoint medium is defined as…”

“…24,[53][54][55][56] The square root of an operator is not uniquely defined. We take the square root such that the imaginary part of the eigenvalue spectrum of H 1 has the same sign as that of H 2 , i.e., positive for a dissipative medium and negative for an effectual medium (for details see Ref.…”

“…We take the square root such that the imaginary part of the eigenvalue spectrum of H 1 has the same sign as that of H 2 , i.e., positive for a dissipative medium and negative for an effectual medium (for details see Ref. 24, keeping in mind that the imaginary unit j in that paper has been replaced by -i in the present paper, to be consistent with the use of i in Schr€ odinger's equation). We combine Eq.…”

“…To simplify this result further, we first consider the transposed and adjoint versions of the scalar operators H 1 ; L 1 , and L 2 , which are given by 24,55 …”

“…24,[53][54][55][56] With the specific choice for the operators L 1 and L 2 , made in Eqs. (B19) and (B20), u þ and u -are so-called flux-normalised downgoing and upgoing wave fields.…”