Radio Science

Covariance operators, Green functions, and canonical stochastic electromagnetic fields

Authors


Abstract

[1] In this paper, we show that for a large class of lossless reciprocal configurations, a “canonical” stochastic electromagnetic field can be defined such that its spatial covariance is proportional to the real part of the Green tensor function of the configuration. This result generalizes some relations between stochastic plane waves and free space Green functions which are well known in theoretical physics. A special application of the ideas presented here is elaborated for the configuration of a semi-infinite, short-circuited waveguide. It is shown that the canonical stochastic electromagnetic field in that case is a pseudoisotropic plane wave corresponding to a very special stochastic linear combination of propagating modes only. This has important consequences for the electromagnetic theory of so-called mode-stirred chambers.

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].

2. Basic Relations for Time-Harmonic Electromagnetic Fields

[13] We shall base our theoretical development on the frequency domain Maxwell equations for a configuration containing a distribution of lossless, reciprocal, dielectrics and perfect conductors. The electromagnetic field, corresponding to a given distribution of time-harmonic electric current density, with radian frequency ω, satisfies the Maxwell partial differential equations

equation image
equation image

Here, the constitutive coefficients, ɛ and μ are real, symmetric, tensor functions and the electromagnetic field, E and H, and electric current density, J, are complex vector valued functions on an open domain Ω equation imageequation image3.

[14] In order to make the solution of the partial differential equations unique, we require the usual continuity and boundary conditions for the electric and magnetic field components near smooth surfaces of discontinuity in the constitutive parameters. In addition, we require the standard outgoing radiation conditions

equation image

componentwise for some finite constant vector M, and

equation image

Here Z0 = equation image is the wave impedance of vacuum and θ = x/∥x∥. A triple {E, H, J}, satisfying the above Maxwell equations and additional conditions, will be called an electromagnetic state in the configuration.

2.1. Integral Representations of the Field

[15] The relation between the electromagnetic field solution of the partial differential equations and the current distribution can be given in the form of an integral representation. Let {fk; k = 1, 2, 3} be a fixed orthonormal frame, then

equation image

where Gkl[eh]j(x, y) are the coefficients of the various Green tensor functions applying to the fields generated by electric current sources in the configuration (hence the second superscript j). In fact, we have a quite intuitive interpretation of equation imagefkGklej(x, y) as the electric field of an elementary electric dipole located in y, with polarization along fl. Similarly, for the superscript hj, we have the magnetic field of the same elementary dipole. Thus this pair of fields corresponds to an electromagnetic state, satisfying (1) and (2) and boundary/radiation conditions, with J = δyfl, where δy is the Dirac measure with support in y.

[16] The tensor-valued functions G[eh]j : Ω × Ω → equation image3equation image (equation image3)′, defined by G[eh]j(x, y) = equation imageG[eh]jkl(x, y)fkflt, will be referred to as the Green tensor functions of the configuration. Similar expressions are defined for magnetic sources, but we shall not elaborate this here. The dual here is with respect to the standard Euclidean inner product. In matrix algebra terms, this means that if elements of equation image3 are represented by column matrices, the elements of (equation image3)′ are simply represented by row matrices. For two vectors a, bequation image3, we shall write the inner product either as a · b or as atb, with at ∈ (equation image3)′, whenever this “dotless” notation is advantageous. For a given x ∈ Ω, we shall also write Gxej(y) = Gej(x, y) and

equation image

where the integration over the support of J is understood. All our integration domains are defined as two- or three-dimensional submanifolds of equation image3, with the Euclidean metric, and hence they get their surface or volume elements unambiguously defined for any parameterization. As in most cases the parameterization does not matter in our theoretical development, we use an abbreviated notation just indicating over which surface or volume the integral extends.

2.2. Integral Relations Between Field States

[17] Let {E, H, J}a,b be two electromagnetic states in the configuration. Let the support of the two current distributions be contained in some bounded subdomain, D, of the configuration, D ⊂ Ω. Then we have the “field reciprocity relation”

equation image

where we used the symmetry of ɛ and μ. Using the fact that the two states satisfy the same, source-free, Maxwell equations and additional conditions outside D, one shows that the surface integral on the left-hand side vanishes,

equation image

[18] We obtain another integral relation between electromagnetic states in a configuration by considering the following equality:

equation image

Using the symmetry of the (real) functions ε and μ, we obtain the “power reciprocity relation”

equation image

It is important to distinguish between the result of taking complex conjugates of a valid electromagnetic state in the configuration, that is, making the substitution {E, H, J} → {equation image, equation image, equation image}, and the electromagnetic state corresponding to the complex conjugate current distribution equation image. This substitution does not yield a solution of the Maxwell equations. On the other hand, a substitution like {E, H, J} → {−equation image, equation image, equation image}, which does indeed provide a solution of the partial differential equations, does not satisfy the outgoing radiation condition.

[19] Let us label the electric field by the current distribution which generates it, that is, Ea ≡ , etc. The field reciprocity then shows that

equation image

If we subtract this relation from the power reciprocity relation, we get

equation image

Using EJ(x) = equation imageGxej(y)J(y), where Gej is the pertinent Green tensor function of the configuration, we find

equation image

and substitution into the previous relation leads to

equation image

where C is the real, symmetric operator with kernel distribution 2 Re[Gxej] and 〈,〉L(D)the standard inner product in the space of square integrable vector functions, that is, 〈f, gL(D) = equation image(equation imagetg) = equation image (equation image · g).

[20] Equation (6) represents a very general relation which can be elaborated in various different ways according to the choice of Ω and the boundary conditions of the fields in the configuration. Of course, one has to be aware under what conditions the relation has been derived. In particular, if we impose special boundary conditions on the fields, the Green tensor function is the one which accounts for those boundary conditions.

3. Natural Field Decompositions

[21] In this section, we sketch the proof that in a general configuration we can always define two systems of basis fields, one for regular fields in a bounded interior subdomain and a complementary one of fields radiating to infinity. Thereto, we introduce a decomposition of space equation image3 = equation image ∪ Ω+ where Ω is a bounded interior and Ω+ is an unbounded exterior (see Figure 1). For ease of presentation, we consider ɛ = ɛ0 and μ = μ0 throughout Ω+ and in an open neighborhood of ∂Ω. These constraints can be relaxed, but treating the most general case would only make the presentation more cumbersome and hide the basic ideas. The example discussed in section 6 shows how the ideas can be applied in slightly different configurations.

Figure 1.

Decomposition of space.

[22] Any solution of the Maxwell equations in equation image ∪ Ω+, can be uniquely decomposed into two constituents (see Appendix A),

equation image

where {E±, H±} satisfies the source-free Maxwell equations in Ω±, respectively. These fields will be considered parameterized by their boundary limits on ∂Ω (also referred to as their Cauchy data).

[23] There exist countable bases of tangential electric and magnetic fields on ∂Ω, allowing the expansion of the general traces {E±, H±}

equation image

with complex amplitudes Ap±. Each pair {ep±, hp±} has a “Maxwellian continuation” into the corresponding domain, Ω±, that is, an electromagnetic field having this pair as boundary values and satisfying the appropriate source free Maxwell equations and other conditions there.

[24] In Appendix A, it is shown that we can construct two related systems of basis fields, solutions of the source-free Maxwell equations in Ω+ and in Ω, respectively, such that

equation image

for any p and for any solution {E+, H+} of the Maxwell equations in Ω, with sources only in the bounded interior Ω. This relation will turn out to be the key for the construction of the specific stochastic field we are looking for.

4. Scalar Product Relations for Radiating Current Distributions

[25] In this section, we recall the basic relations between the radiation fields of given current distributions in Ω. Let J be an oscillating current distribution with bounded support in Ω, generating a time harmonic electromagnetic field, {E, H}. This field has an X+ expansion in Ω+, the amplitudes of which have the following alternative expressions,

equation image

or, using (7),

equation image

or, using (3) with jp = 0,

equation image

[26] Substituting the outgoing wave expansion into (6), we can also identify

equation image

and therefore (see (A1)),

equation image

This puts into direct correspondence the inner product of the outgoing wave expansion and the special inner product between the sources of the radiation field, introduced in section 2.

5. Stochastic Incident Fields and Observables

[27] In this section, we succinctly develop a theory of observables on stochastic electromagnetic fields. In the domain of classical, linear electromagnetics, observables are quantities like port voltages (or currents) in electronic circuits or changes in the state of electrical machines. Such observables can be expressed as linear forms on electromagnetic fields. Here, we shall concentrate on an electronic system's port voltage, which we shall write as V. Therefore we simply write

equation image

where j is a (normalized) current distribution, with some bounded support in Ω, characteristic of the observable V and E is the field in absence of the system. We refer to the literature for a more detailed analysis of how j can be obtained such that for a given incident field E, the observable V is the voltage on a port which corresponds to measured voltages when a system is introduced in the incident field (see, for example, de Hoop [1974] for the plane wave case and Michielsen [1985] for the general case).

[28] If the field is a stochastic field, the observable is a stochastic variable. The purpose of stochastic interaction theory is to relate the properties of the stochastic incident field to the properties of the observable, which we suppose to be defined by a deterministic j. One might also consider the case where j is a stochastic distribution and the incident field fixed or where both j and E are stochastic fields. However, in this paper we want to develop a special property of stochastic incident fields only. This situation corresponds to fields generated by stochastic sources in a configuration with a fixed geometry and physical constitution. The same models will also apply to situations where the field is stochastic because of its interaction with a stochastic environment, but then the coupling between the stochastic part of the geometry and the observed system must be weak. Irrespective of the precise probability distribution on the values of V we can relate its two main statistics, average and variance, to the corresponding statistics of the incident field. Because of the linearity of the expression (13), the average is computed from

equation image

just as in the deterministic case. As we are mainly interested in fluctuations around the average, we can without loss of generality assume that the average is equal to zero. Then, because of linearity again, the variance is given by

equation image

where we introduced the covariance operator of the stochastic incident field, which we write as CE,

equation image

with the kernel function given by the complex conjugate of the spatial covariance of the incident field

equation image

This covariance operator is the principal object of our study.

[29] We define a stochastic electromagnetic incident field, {E, H}, as a superposition of fields with a stochastic vector of amplitudes A,

equation image

where {ep, hp} is an element of the basis in X discussed in section 3. We can study the properties of the covariance operator of this field by evaluating 〈CEJb, JaL on two arbitrary current distributions Ja and Jb. Substitution of the wave expansion in this expression using (14) and (15) gives

equation image

We now choose the amplitudes to be statistically independent and identically distributed (IID) stochastic variables with variance σ2. With that choice, we get

equation image

and, after substitution of Ap+;a,b = 〈equation image, Ja,bL (see (10)), remembering that these are deterministic radiation field amplitudes, we obtain

equation image

Comparison of this last equation, (18), with equation (12), which both hold for arbitrary current distributions, allows us to identify the canonical operator C with the spatial correlation operator of the specific stochastic incident field we defined above.

equation image

This identity justifies, in a certain way, the name canonical correlation operator for the operator C. On the other hand, we might also call the specific stochastic field for which this identity hold a “canonical stochastic incident field.” It is clear that the identity allows for efficient computations of the fluctuations of observables in such “canonical stochastic fields” when the Green functions of the configuration are known.

6. Application to a Waveguide Problem

[30] The relations found in the previous sections are valid in general three-dimensional configurations. They can be elaborated in various different ways; for example, one could use it then to define a canonical stochastic spherical wave expansion in free space and use the known real part of the free space Green tensor function to compute the variance of observables in that field. Another application would be to establish the spatial covariance operator of a canonical stochastic field in the presence of a reflecting half plane.

[31] In this section, however, we shall study an application to the configuration of a short-circuited semi-infinite waveguide. Such configurations are of practical interest in the theory of mode-stirred chambers, for example. Moreover, because waveguides are not truly open three-dimensional configurations, this example allows us to show how the analysis of the previous sections can be adapted to such cases. In particular, the radiation condition in waveguide configurations takes a particular form. One of the consequences of this is that the canonical stochastic fields in these configurations concern only fields built from the propagating modes. Hence they are finite stochastic linear combinations of regular fields in the interior domain Ω.

[32] The geometrical configuration we are interested in here is depicted in Figure 2. It consists of a semi-infinite waveguide of cross section A, so we have Ω = A × equation image+ and ∂Ω = ∂A × equation image+A × {0}. A point in x ∈ Ω will be written as (xT, z) with xTAequation imageequation image2 and zequation image+. The fixed cross section at axial coordinate z will be referred to as Az. The theoretical analysis applies to any cross section for which eigenfunctions of the Laplace operator can be constructed; however, at the end of this section, we shall elaborate the field structure only for the special case of a rectangular cross section. We assume the waveguide walls and the short-circuit plane, A0, to be perfectly conducting and the constitutive parameters are those of vacuum throughout Ω.

Figure 2.

Semi-infinite waveguide configuration.

[33] The interior domain, Ω, is the parallelepiped region, delimited by A0 and AL, of which five faces coincide with the perfectly conducting waveguide walls and the short-circuit plane, respectively. The exterior domain, Ω+, is a semi-infinite waveguide. As we shall consider only fields which have vanishing tangential electric fields on five of the six faces of ∂Ω, we can restrict our attention to the waveguide cross section A in z = L.

[34] Our purpose is to establish a canonical stochastic field in Ω, that is, a stochastic field such that a relation like equation (19) holds. Let us recall that the definition of a canonical stochastic field was based on comparison of equation (12) with the equation (18). The first of these equations appeared in the power additivity relation of two arbitrary current distributions in Ω and involves a canonical inner product. The second of these equations expresses the evaluation of the spatial covariance operator of a very specific stochastic field on two arbitrary current distributions in Ω. The definition of the stochastic field was, of course, carefully designed such that the right-hand sides of the two equations became proportional.

[35] In order to reproduce similar results in the semi-infinite waveguide configuration, we have to consider first the equivalent of (12). We recall very succinctly the essentials of waveguide mode theory, which we need here, and assume that the reader is familiar with presentations like, for example, Collin [1991].

[36] In the waveguide configuration defined above, we have the following field representations,

equation image
equation image

The vector potentials Ape, for the TM modes, and Aph, for the TE modes, are defined by (we suppress the index p here, for ease of notation)

equation image

where −ΔTϕe;h = kc2ϕe;h, with ϕeA = 0 respectively ∂nϕh∣∂A = 0, and γ = equation image, with k0 = ω/c0. The factor N is a normalization coefficient which we shall compute in order that the requirements of the field decomposition in section 3 are met.

[37] The phase factors ψe;h determine the axial dependency of the mode fields and γ is called the propagation coefficient. For the X+ field expansion we choose ψe(z) = ψh(z) = e−γz. For the X field expansion, we choose ψe(z) = cosh(−γz) and ψh(z) = sinh(−γz) such that the tangential electric field vanishes on the cavity face at z = 0. Note that with both choices we have ∂zψe = −γψh and ∂zψh = −γψe.

[38] In the configurations we consider, the kc2 are positive and real, such that the propagation coefficients are either purely real, when k02kc2, or purely imaginary, k02 > kc2. The former correspond to fields with exponential decay in Ω+ and the latter to fields propagating to infinity in Ω+.

[39] We introduce a single index ordered set of fields generated by one of the potentials Ape or Aph,

equation image

where pI, a linearly order label set I labeling both the polarization type, TE or TM, and the harmonic orders. We shall partition this index set into two components, I = PE, where P labels the propagating modes and E the evanescent modes. Accordingly, we can decompose the space of tangential electric fields on the surface A, into two components F = FPFE, where FP corresponds to fields with a propagating mode expansion and FE to fields with an evanescent mode expansion in Ω+.

[40] The existence of evanescent modes marks an important difference with the truly three dimensional case; that is, nontrivial fields exist in Ω+ which do vanish rapidly at infinity. In truly three-dimensional open domains, that is, domains such that in the exterior of some finite ball we have free space everywhere, every nonvanishing surface field corresponds to an electromagnetic field satisfying the outgoing radiation conditions and hence cannot tend to zero faster than “1/R” [Rellich, 1943]. In the semi-infinite waveguide, however, we have evanescent modes with exponential decay at infinity. As a consequence, we have for any qE

equation image

because there are no sources of {eq+, hq+} in Ω+. Then, because this evanescent mode field vanishes on A, we have

equation image

This implies that substitution of the modal expansion into

equation image

gives

equation image

Therefore, if we maintain the definition of a canonical stochastic field of the general theory, we should construct such a field as the superposition, with statistically independent and identically distributed coefficients, of regular fields in Ω adapted to the subspace of exterior fields spanned by the propagating modes only.

[41] The first step is to orthonormalize the X+ fields. From the definition of these fields given above, we can readily prove the orthogonality (within one polarization class, because the eigenfunctions of the Laplacian are orthogonal, and between polarization classes, because “gradients are orthogonal to curls”). As to the normalization, we have to obtain

equation image

For TE modes, we get

equation image

and for the TM modes,

equation image

[42] The X basis fields are already orthogonal with respect to the integral of (A3), we only have to adapt the normalization such that

equation image

For the TE modes we get

equation image

and for the TM modes,

equation image

Finally then, we conclude that the canonical incident field in Ω is the linear combination of X basis fields with IID coefficients, if the basis field normalizations are given by

equation image

[43] We shall next investigate the structure of this field in detail for the case of a waveguide with rectangular cross section, A = (0, Lx) × (0, Ly). For this simple geometry, the eigenfunctions of the Laplacian are explicitly known

equation image

with ∥ϕhL = equation image and ∥ϕeL = equation image and where ξ = mπ/Lx and η = nπ/Ly for any m, nequation image2. The eigenvalues are given by kc2 = ξ2 + η2. Observe that for ξ = 0 and/or η = 0, ϕe ≡ 0 defines a null field.

[44] We shall now show that this canonical stochastic field corresponds to a stochastic linear combination of plane waves which in a certain way is “as isotropic as possible” given the boundary conditions on the walls. We simply remark that any of the X basis fields is a superposition of eight plane waves. In a spherical coordinate system, we can give an identification of the mode amplitudes and the electric field polarization. More precisely, we have, for propagation directions in the principal octant,

equation image
equation image

where θmn is a unit vector, the propagation direction, given by

equation image

The electric field of this plane wave is given by

equation image
equation image

With these expressions, we find for the variances,

equation image

where Z0 = equation image, σ2 = var(Amnh) = var(Amne), by definition of the canonical field, and cos(θ) = equation image is the cosine of the angle between the plane wave propagation direction and the positive waveguide axis.

[45] Now, the density of spectral coefficients is asymptotically homogeneous in the real plane. The formula for the propagation direction (24) shows that this corresponds to a density of propagation directions on the unit sphere given by the uniform density multiplied by cos(θ) (so the density diminishes toward the transversal directions). This is a straightforward computation. Using the standard coordinates on the unit sphere (ϑ, ϕ) and the standard cartesian coordinates in the plane (x, y) = (sin ϑ cos ϕ, sin ϑ sin ϕ), the uniform density in the plane dxdy is found to be

equation image

The second factor in the last expression is the uniform density of the unit sphere. The expressions for the plane wave electric field polarizations show that the variances have this cos(θ) in the denominator. The lack of contributing waves in a certain solid angle close to the equator is thus “compensated” for by an increased variance. We might therefore call this a pseudoisotropic stochastic plane wave. In contradistinction to a completely isotropic plane wave, with uniform distribution of propagation directions on the unit sphere and IID electric polarization for any different propagation directions, the pseudoisotropic plane wave has a discrete distribution on the propagation directions and the electric polarizations are only statistically independent in the principal octant.

[46] This kind of stochastic fields has already been introduced [Fiachetti, 2002] as a convenient model for the field in the measurement zone in mode-stirred chambers (see Figure 3). In such a configuration, we can identify the measurement zone, ΩM, with the domain Ω of the semi-infinite waveguide depicted in Figure 2, such that the five perfectly conducting walls coincide in the two configurations. The complementary region in the MSC contains the feed and mode stirrer. The feed antenna has an aperture through which any energy entering the chamber also leaves the chamber (we consider perfectly conducting walls everywhere, such that there is no net energy flow into the MSC). If we consider the mode stirrer orientation, a stochastic variable the field in the MSC becomes a stochastic field. In particular, in the measurement zone, we have a waveguide mode expansion of this field with stochastic amplitudes. Fiachetti [2002] proposed a stochastic field model, which came as close as possible to the isotropic plane wave, while satisfying the boundary conditions on the walls touching the measurement zone. Surprisingly, the result is identical to the one derived as the canonical stochastic field above, where we did not put the requirement of isotropy on the foreground. We think that the correspondence with the canonical correlation product, established in this paper, adds a new argument in favor of the naturality of these pseudoisotropic stochastic plane waves in mode-stirred chamber analysis.

Figure 3.

Typical mode-stirred chamber, with a measurement zone, ΩM, and a complementary stirrer and feed zone, ΩS.

7. Conclusion

[47] In this paper a fundamental relation has been established between a scalar product, measuring the radiation power of electric current distributions in arbitrary reciprocal and lossless configurations, and the covariance operator of a stochastic incident field in such configurations. The universality of the former scalar product suggests the specific stochastic field to deserve the qualification “canonical.” In many configurations such canonical stochastic fields can be constructed explicitly and their spatial covariance operators are then known to be proportional to the real part of the configuration's Green tensor function. As an example, the particular case of a short-circuited semi-infinite waveguide has been elaborated in detail. It appeared that the “canonical” stochastic field in that case is a pseudoisotropic stochastic plane wave. In a certain way, this stochastic field is the best possible approximation to the well-known isotropic plane wave in free space, while satisfying the boundary conditions on the waveguide walls.

Appendix A:: Existence of Naturally Dual Bases

[48] In this appendix, we present the essential elements of the proof that in an arbitrary scattering configuration, we can always define two systems of basis fields, one for regular fields in a bounded interior subdomain and a complementary one of fields radiating to infinity, such that the special relation (7) holds. We shall refer to the decomposition of space introduced in section 3. Let F be the space of tangential electric fields on ∂Ω and F′ its dual space, in which we find the boundary limits n × H. The space of all possible fields on the boundary will be called X, so we have X = F × F′.

[49] Using the integral representations of the field in the configuration, we can define a projection operator

equation image

where X+ is the subspace of Cauchy data for exterior solutions.

equation image

where the special notation of the limit indicates that the point p on ∂Ω is approached by x through a path in Ω+. Note that in this integral representation, the Green tensor functions, G[eh]kx, applying to the radiation of magnetic current sources, “k = n × e,” appear. These are defined in an analogous way as the ones for electric current sources, defined in section 2. One verifies that C+C+ = C+ by considering the fact that exterior solutions are uniquely defined by their boundary limits, so substitution of the boundary limits of an exterior solution in the integral representation must reproduce the same field. By taking the boundary limit through Ω we obtain another projection operator, C, which, because of the jump relations of the integral representations, is the complement of C+, that is, C = equation imageC+. From the properties of C+, we derive CC = C and C+C = 0 = CC+. These operators are called the Calderón projection operators. Hence it follows that the subspaces X± = im(C±) only intersect on the trivial field, which is equivalent to stating that the only solution to the Maxwell equations, plus continuity, boundary, and radiation conditions, which has neither sources in Ω+ nor in Ω, is the trivial solution.

[50] Since ∂Ω is compact, there exist countable bases of tangential electric and magnetic fields on ∂Ω, allowing the expansion of the general traces {E±, H±}

equation image

with complex amplitudes Ap±. By definition, each pair {ep±, hp±} corresponds to an electromagnetic field having this pair as boundary values and satisfying the appropriate source-free Maxwell equations and other conditions, in Ω+ and Ω, respectively. Thus, from the uniqueness of exterior boundary value problems, we can identify n × hp+ = Y+ep+, where Y+ is the so-called exterior admittance operator.

[51] We cannot directly do the same for the pairs in X because, at the so-called interior resonance frequencies of Ω, X is not the graph of a bounded operator. We know that these frequencies form a discrete subset of the real axis, which means that we have a well-defined interior admittance operator Y almost everywhere on the real frequencies. It should be remarked, however, that these singularities are removable, because the surface integrals in question do not change if we deform them through a source-free, reciprocal medium, and the exact shape of this surface does not matter for our considerations. Therefore, for any given frequency we can choose the surface such that the frequency is not an internal resonance frequency and Y is well defined. In applications to specific cases, this simply means that we should be able to find a singularity-free formulation independent of the exact position of the separation surface ∂Ω. We shall therefore continue our analysis and simply suppose that the surface has been chosen appropriately.

[52] From the field reciprocity relation, applied to Ω±,

equation image

it follows immediately that both admittance operators are symmetrical with respect to the bilinear form 〈u, v〉 = equation image(u · v).

[53] In the construction of our stochastic field, we need the two systems of basis fields, introduced above, to be related in a certain way such that the theoretical formulations will be as simple as possible. In particular, we should like to have basis fields which are orthogonal with respect to integrals of the type we find in the field and power reciprocity relations.

[54] We already know, from the field reciprocity relation, that fields of the same X± type give a vanishing Lorentz integral. Therefore the following “orthogonality”

equation image

(for the upper and lower superscripts separately) depends only on the fact that the fields have a regular Maxwellian continuation into the exterior or the interior domain (assumed reciprocal).

[55] For power reciprocity type integrals and field reciprocity integrals over mixed type fields we have to do a bit more work. We shall first show that there exists a basis of real vector functions, {ep : pequation image}, in L2(∂Ω) such that

equation image

where n × hq = Y+eq and Y+ is the exterior admittance operator and δk is Kronecker's symbol δk = 1 if k = 0 and zero otherwise.

[56] For that purpose, consider

equation image

Using the symmetry of Y+, we get

equation image

which defines a real, symmetrical matrix. Therefore we want to show that this matrix can be diagonalized.

[57] From the absence of sources in the exterior domain, Ω+, we derive

equation image

with Y0 = equation image and SR2 is the unit sphere with radius R. Substitution of the definition of Y+ gives

equation image

From a theorem by Rellich [1943], we know that ∥ep2 cannot vanish faster than O(R−1), or the field vanishes identically. This implies that Re(Y+) is an injective, positive definite operator because for any ep the image Re(Y+)ep cannot be identically zero as its evaluation on at least ep itself does not vanish.

[58] This positivity result implies that we can apply a Gram-Schmidt orthonormalization procedure with respect to the metric Re(Y+) and hence diagonalize the RHS of equation (A2). We write the thus constructed basis as {ep+, hp+}, where n × hp+ = Y+ep+.

[59] The next result shows that we can adapt the X basis to this particular, orthogonalized, X+ basis to get a pair of basis fields such that

equation image

Substitution of the two admittance operators in the expression on the left-hand side gives (using the symmetry of Y),

equation image

where el+ is a given function from the positive basis. The uniqueness of solutions of boundary value problems in the entire domain Ω ∪ Ω+ makes the operator YY+ invertible. If it were not injective, there would be an electromagnetic field solution of the source-free Maxwell equations in Ω+ ∪ Ω satisfying the continuity conditions and the outgoing radiation condition, which contradicts the uniqueness result. On the other hand, if the operator were not surjective, there would be a current distribution on ∂Ω corresponding to a vanishing tangential electric field on ∂Ω. This can only happen at internal resonance frequencies, but we suppose that for each frequency the surface has been chosen such that this frequency is not an internal resonance. Therefore the images (YY+)el+ can serve to define a basis for the function space of surface currents. Then, by taking ek as the basis dual to the latter, we have the desired basis for the X fields.

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