## 1. Introduction

[2] In this paper, we study time-harmonic stochastic electromagnetic fields in the context of classical, macroscopic, electromagnetic theory. Stochastic fields are used in the probabilistic modeling of situations where deterministic information, about source distributions and/or scattering obstacles, is inaccessible or too complicated to handle [see, e.g., *Corona et al.*, 1996; *Lehman*, 1993]. The observables of classical field theory modeling are the voltages and currents appearing on electronic system ports or forces on movable parts of devices, for example. Such observables are defined by linear forms on the electromagnetic field, that is, by integrating currents against fields. Both the current, which represents the responsive system, and the field, which represents the forces on the responsive system, are characterized as solutions of electromagnetic boundary value problems. In the absence of deterministic field models, one tries to compute certain statistics of the observables directly from the properties of stochastic fields. The most obvious statistic is the average value, which can be obtained straightforwardly from the average of the stochastic field. Another important statistic is the variance, which, in a certain way, measures the intensity of the fluctuations around the average. The variance of an observable on a stochastic electromagnetic field can be computed from the evaluation of the spatial covariance operator of the field on the current distribution characterizing the observable. Explicit, a priori knowledge of the spatial covariance operator of the stochastic field is then very important. This is because applying more conventional statistical techniques requires, in most cases, the numerical solution of large series of boundary value problems. The computational complexity of even a single such boundary value problem may already be prohibitive (remember that this was one of the reasons to turn to nondeterministic methods in the first place).

[3] These observations show that it is very important to have stochastic fields available for which the covariance operators are explicitly known. In various branches of theoretical physics, relations between covariances of stochastic fields and Green functions have been established [e.g., *Glimm and Jaffe*, 1996]. In particular, in the so-called linear response theory, the fluctuation-dissipation theorem identifies the generalized susceptibility of a quantum system in an external force field to correlation functions of the equilibrium state [see, e.g., *Agarwal*, 1975]. In the context of classical electromagnetics, the spatial correlation functions of blackbody radiation fields have been established by E. Wolf et al. [see, e.g., *Mehta and Wolf*, 1967]. These correlation functions have later also been identified with a part of the free space Green function. This identification has been emphasized in recent work on the spatial correlation function of fields generated by random current distributions in free space and has led to the idea of the universality of such correlation functions [see *Gbur et al.*, 1999; *Setälä et al.*, 2003, and references therein].

[4] The identification of the spatial correlation of a statistically isotropic plane wave and (part of) the free space Green function, for the Maxwell equations, has also been remarked, and applied, in recent work on the interaction of fields with electronic systems in so-called mode-stirred chambers [see *Warne and Lee*, 2001; *Fiachetti*, 2002]. This particular relation has been very advantageous in the numerical computation of variances of certain observables on stochastic fields [see *Fiachetti and Michielsen*, 2003]. Therefore the question arises naturally, whether this identification between a field correlation function and a Green function can be generalized to geometrical and physical configurations other than free space.

[5] This paper shows that such is the case indeed. In any lossless and reciprocal configuration, we can identify a natural (or “canonical”) stochastic field of which the spatial covariance is indeed the real part of the Green function of the Maxwell equations and boundary conditions for that configuration.

[6] As explained in section 1, the principal aim of this paper is to show the existence of a stochastic time-harmonic electromagnetic field in an arbitrary, three-dimensional, configuration with lossless media, such that its spatial covariance function is proportional to the real part of the Green function of that configuration. We shall always refer to the Green function when we mean the Green (tensor) function of the Maxwell equations, possibly incorporating special boundary conditions and material inhomogeneities. Note that this definition leads to the electric and magnetic field expressed as integrals of these Green functions over current distributions. This is not identical to using the Green function of the Helmholtz equation where one has additional multipliers. This is the reason why certain authors, when studying free space problems, speak of relations between spatial covariances and the imaginary part of the Green function; these authors consider the Green function of the Helmholtz equation. This way of presenting the theory is not well suited for inhomogeneous media, which we wish to include in our considerations.

[7] The central idea in our theory is to establish a proportionality between two inner products of current distributions in arbitrary but lossless configurations: on the one hand, a “radiation power” inner product, based on the real part of the appropriate Green function, and, on the other hand, a “reception variance” inner product, based on the field covariance. The radiation power inner product follows naturally from basic integral relations in electromagnetic field theory. The sought for relation can then be established by adjusting the structure of the stochastic field. In order to be able to do that, we construct a basis of solutions of the source-free Maxwell equations in the configuration, which is related, in a very special way, to a basis of radiation fields. It will appear then, as is detailed in the subsequent sections, that the relation between these two bases is such that we are able to choose a well-identified stochastic linear combination of these basis fields, such that the desired spatial covariance is obtained.

[8] In section 2, we present the general configuration for which we shall derive the relation between spatial covariances and Green functions. This introductory section also allows us to define notations and recall some basic results from electromagnetic field theory. Our principal tools, besides, obviously, the Green formulas, are the reciprocity relations.

[9] In section 3, we consider a field decomposition relative to a surface separating an interior domain from an exterior domain. In the interior domain, we consider source-free incident fields. In the exterior domain, we consider source-free radiation or scattered fields. We show that there exist two sets of related basis fields, which allow us to link radiation field expansions to incident field expansions in general scattering configurations. The technical details of the proofs are deferred to Appendix A.

[10] In the two subsequent sections, we shall see that the study of radiation fields leads to inner products which one also encounters in the study of stochastic incident fields. Section 4 presents a first relation between, on the one hand, an inner product of radiation or scattered fields and, on the other hand, an inner product between the current distributions that generate them. The kernel distribution of the metric operator of the latter inner product is the real part of the Green function of the configuration.

[11] In section 5, we define a very specific stochastic incident field. In fact, this stochastic field is the simplest choice given the natural field decomposition in the configuration; it could therefore be called the “canonical stochastic incident field” in the configuration. It is then shown that the spatial covariance operator of this “canonical stochastic field” is proportional to the inner product operator of the preceding section. This identification of the spatial covariance of a stochastic incident field with the real part of the Green function is the principal theoretical result of this paper.

[12] In section 6, we show the physical relevance of the relations obtained for the case of a “canonical” stochastic waveguide mode expansion in a semi-infinite short-circuited waveguide. The result obtained in that section shows why the most appropriate model of mode-stirred chamber environments is a “pseudoisotropic” plane wave [see *Fiachetti*, 2002; *Fiachetti and Michielsen*, 2003] instead of the isotropic stochastic plane wave model, which is sometimes used [see *Lehman*, 1993; *Hill*, 1998].