A closed-form conversion from spherical-wave- to complex-point-source-expansion

Authors


Abstract

[1] A simple closed-form expression for a complex point source (CPS) beam expansion of an arbitrary electromagnetic field is derived. The expansion process consists of two steps: first, a particular form of the equivalence principle is applied to a sphere enclosing the real sources, and a continuous equivalent electric current distribution is obtained in terms of spherical waves (SW); then, the continuous current is extended to complex space and its SW components are properly filtered and sampled to generate the discrete set of CPS. The final result is a compact finite series representation suitable for arbitrary radiated fields, and particularly efficient when the source is highly directional and/or the observation domain is limited to a given angular sector. The robustness of the process is demonstrated by showing its connection with the singular value decomposition of the radiation operator from a complex sphere.

1. Introduction

[2] In many electromagnetic problems, beam expansions represent a useful means to efficiently analyze interactions since, thanks to their angular selectivity, only a limited number of beams significantly contribute to the field in a given angular observation region. Among the various kinds of beams, the ones generated by analytically continuing the location of a point source to the complex space (complex point source, CPS) have the advantage of being exact solution of the wave equation, that can therefore be analytically propagated throughout the space [Felsen, 1976; Heyman and Felsen, 2001].

[3] CPS beam expansions have been utilized to accelerate the iterative solutions of large method of moments problems [Tap et al., 2007]. Recently, CPS beams have also been employed in a domain decomposition approach based on a generalized scattering matrix formalism for the analysis of complex antenna and/or scattering problems [Martini et al., 2010]. In this approach, the analysis domain is decomposed into disjoint subdomains which are analyzed independently, and the CPS beams are used to describe inter-subdomain interactions. More specifically, the subdomain containing the primary source is characterized by the coefficients of the CPS beam expansion of the radiated field, while the subdomains containing a scatterer are characterized by a scattering matrix relating the expansion coefficients of the incident field to the ones of the corresponding scattered field. Different subdomains are connected through transmission matrices which relate the CPS beams outgoing from one subdomain to the CPS beams incoming to another subdomain. Finally, the overall problem is reduced to the solution of a linear system whose unknowns are the coefficients of the interacting CPS beams, which, thanks to the beam selectivity, are just a small fraction of the total number of beams.

[4] A key issue for the efficiency of these procedures is the capability of constructing a complete CPS beam expansion of a given electromagnetic field with moderate redundancy. A complex source representation based on complex Huygens' principle for only scalar fields was first proposed by Dezhong [1995]. In the work of Norris and Hansen [1997], an exact CPS expansion for arbitrary scalar fields was presented, where the beams are launched from a single point in space and their coefficients are determined on the basis of the field radiated on a sphere in real space. Complex source representations for scalar transient radiation have also been proposed [Heyman, 1989; Hansen and Norris, 1997]. More recently, alternative formulations for time-harmonic vector electromagnetic fields have been provided by Tap et al. [2011] where the beams are launched from a sphere enclosing the real sources. A first formulation uses both electric and magnetic equivalent sources, whose weights are related, through the equivalence theorem, to the analytic continuation in complex space of the field to be represented. A second formulation uses only electric CPS and determines their weights through a numerical procedure by matching the original field with its CPS expansion on a proper test surface. In this second formulation the number of CPS beams is minimized, but the computation of the expansion coefficients requires the solution of a linear system.

[5] This work proposes a new approach that combines the advantages of these two formulations by providing a closed-form expression for a compact CPS beam expansion. The starting point is the derivation of a continuous equivalent electric current distribution on a spherical surface enclosing the real sources. This is done by applying the equivalence principle with only electric-type sources as formulated by Martini et al. [2008], that yields a closed-form expression of the current distribution involving the spherical wave (SW) coefficients of the original field. Then, the equivalence surface is extended to complex space, to obtain a continuous equivalent distribution of CPS. The resulting expression of the current distribution has the remarkable property of being an expansion in a series of terms ordered with increasing spatial-frequency and decreasing radiation efficiency. This allows for an immediate identification of the high spatial-frequency components, which provide a negligible contribution to radiation. Hence, the current is easily filtered to only retain its slowly varying part, which is accurately discretized with a limited number of samples.

[6] An alternative method to derive the coefficients of a compact field expansion in terms of CPS beams consists in using the singular value decomposition (SVD) to solve the linear system obtained by field matching. In this paper, the relationship between the proposed expansion procedure and the SVD-based method will be clarified and it will be shown that the expansion coefficients provided by the two approaches tend to coincide for increasing numbers of CPS beams used in the expansion.

[7] The organization of the paper is the following. In section 2 the concept of degrees of freedom is briefly reviewed in connection with CPS beam expansions. Then, the proposed representation is presented in section 3 and its connection with the numerical approach for the determination of the expansion coefficients is highlighted in section 4. Guidelines for the choice of the CPS expansion parameters are provided in section 5. Finally, numerical results are presented in section 6 and conclusions are drawn in section 7. A brief summary of the spherical wave functions notation is reported in Appendix A.

2. CPS Expansions and Degrees of Freedom

[8] The definition of a non redundant field representation is strictly related to the concept of degrees of freedom (DoF) [Bucci and Franceschetti, 1989]. The degrees of freedom of the field radiated by an arbitrary source contained in a surface S in a given observation domain can be defined as the minimum number of independent coefficients of any type of discrete field expansion that allows for an accurate field description in that observation domain. A general way to determine the DoF relies on the use of a singular value decomposition of the radiation operator which maps the currents on the source boundary S onto the field radiated in the observation domain. The SVD identifies two sets of orthonormal functions (also referred to as “modes”) with which to expand the radiated field and the current distribution, respectively, and the associated singular values. Current modes associated with vanishingly small singular values only contribute to the evanescent field in the neighboring of the sources and contribute negligibly to the radiated field. Hence, assuming that some noise is present, they are not observable at a certain distance from the sources. For this reason, the dimension of the set of the significant (in the sense explained above) current modes defines the number of degrees of freedom of the radiated field. There is in principle an ambiguity in the definition of the number of DoF, since it apparently depends upon an assumed dynamic range of the radiation efficiencies. However, this is not a practical issue, because, due to the fact that the radiation operator is compact, the singular values tend exponentially fast to zero when their order tends to infinity, making the identification of the DoF almost insensitive to the required precision [Bucci, 2005; Stupfel and Morel, 2008].

[9] As a general rule, the number of DoF is a function of the electrical size of the source distribution and, although strictly speaking it depends on the observation distance, its value stabilizes outside the reactive region of the sources. In the particular case when both the surface S enclosing the sources and the observation surface Σ are spherical, the singular functions of the radiation operator are spherical modes, and the relevant singular values can be calculated in closed form; outside the reactive region, the number of DoF turns out to be approximately equal to

equation image

where k is the wave number and r0 is the radius of S. In this case, the spherical harmonics represent the optimal basis for the field representation. It is noted that equation (1) provides the minimum number of wave objects for the representation of the radiated field in the absence of any a priori knowledge of the sources enclosed by S. If, instead, the actual location of the sources inside S is known, the number of modes required for the field representation may be significantly lower than the number in equation (1). For instance, for planar source distributions the number of degrees of freedom is given by the Landau-Pollak bound [Arnold, 1995] and ad hoc sets of functions can be defined for the field representation [Bogush and Elkins, 1986; Prakash and Mittra, 2003; Shlivinski et al., 2004; Matekovits et al., 2007; Casaletti et al., 2009]. When also the source distribution is known, the number of basis functions can be further reduced by a priori selecting a subset of the general basis [Shlivinski et al., 2004] or by using field-matched wave objects [Škokić et al., 2011]. However, in this latter case a different set of basis functions must be defined for each source distribution. A pre-defined basis, independent of the sources, provides generality at the price of some redundancy.

[10] In certain cases one is interested in the field radiated only in directions belonging to a limited angular sector. This is the case, for instance, when calculating the interactions between well separated groups of basis functions in the approach proposed by Tap et al. [2007] or the interactions between two sub-domains in the approach proposed by Martini et al. [2010]. When the observation domain is restricted to a limited solid angle Ω, the analysis of the singular values reveals that the number of DoF is reduced by a factor approximately equal to the ratio between Ω and 4π

equation image

Spherical harmonics no longer represent an optimal basis for the current representation, since in general all the first NDoF harmonics provide a non-negligible contribution in the angular sector Ω, due to their poor angular selectivity. The optimal basis is constituted by the NDoFΩ orthogonal modes associated with the predominant singular values, which are obtained as SVD-based linear combinations of the first NDoF spherical harmonics. Such a basis must be constructed numerically and depends on the given observation domain Ω. When elements from the same basis must be used to represent the field in different observation domains, like in the approaches of Tap et al. [2007] and Martini et al. [2010], a more convenient way to reduce the redundancy consists in using beams as basis elements and exploiting their angular selectivity to identify the important contributions for any given angular sector Ω. Among the various kinds of beams, the CPS beams have the advantage of being exact solution of the wave equation and of maintaining their simple functional form in the whole space [Felsen, 1976; Heyman and Felsen, 2001].

[11] CPS beams are generated by analytically continuing the location of a point source to the complex space. In particular, by assuming a time dependence exp(jωt), a CPS located at r′ = ajb, with a, bequation image3, creates a beam-type field with axis parallel to b, which reduces to the field of an electromagnetic Gaussian beam in the paraxial region [Felsen, 1976; Heyman and Felsen, 2001]. In the work of Tap et al. [2011], a field representation in terms of CPS beams radially emerging from a sphere enclosing the real sources is obtained by locating a set of CPS on a sphere of complex radius. Different algorithms for the determination of the CPS beam weights are also suggested. In a first formulation, the equivalence theorem [Harrington, 1961] is used to represent the radiated field in terms of equivalent electric and magnetic surface currents located on a sphere enclosing the sources; then, the sphere radius is analytically extended to a complex value, thus converting the radiation contributions of the electric and magnetic point sources to CPS beams radially emerging from the real sphere, whose weights are related to the analytic continuation of the fields. Finally, the discretization of the radiation integral gives rise to a finite summation of beams [Tap et al., 2011].

[12] A second formulation uses only electric CPS and determines their coefficients by solving a linear system. The latter is constructed by matching the field to be expanded with the CPS beam expansion at a proper set of points on a spherical test surface. This second approach allows for the minimization of the number of CPS beams, but it requires the solution of a linear system for the determination of the expansion coefficients. On the other hand, the first approach provides a closed-form expression of the beam weights, but it requires a larger number of beams due to the high sampling density needed to accurately discretize the radiation integral. Further redundancy is related to the use of both electric and magnetic type sources, since the two associated sets of beams are not independent. If the equivalence surface is taken close to the sources, the equivalent currents exhibit rapid variations related to the reactive part of the fields. The reactive fields are not observable at a few wavelengths distance from the sources; however, considering a larger equivalence surface would not resolve the problem, since it would lead to faster variations of the dyadic Green's function appearing in the integrand of the radiation integral.

[13] An expansion algorithm aiming at combining the efficiency of the numerical matching approach and the advantages of a closed-form expansion is proposed in the next section.

3. Analytic CPS Expansion

[14] Consider an arbitrary set of currents radiating in a homogeneous medium that, for simplicity, we assume to be free-space. We look for a closed-form expression of the coefficients of a compact CPS beam expansion of the field radiated by these currents at a certain distance.

[15] It is well known that the equivalence principle allows one to replace an arbitrary set of sources by equivalent currents distributed over a virtual closed surface S enclosing the real sources, and providing the same field outside S. In the most common (Love) formulation, both electric and magnetic currents are present, whose values are obtained by imposing a null field inside S [Harrington, 1961]. Recently, a formulation has been provided for equivalent currents of only electric (magnetic) type by imposing inside S an electromagnetic field that guarantees the continuity of the tangential electric (magnetic) field at the interface [Martini et al., 2008]. Although for a generic surface the calculation of these currents requires the solution of a boundary value or cavity problem, for the particular case of a spherical equivalence surface an analytic expression of the equivalent currents has been obtained after expanding the internal and external electromagnetic field in a set of spherical harmonics and applying mode matching. Consider the expansion of the field radiated outside the equivalence surface S in terms of outgoing spherical wave functions

equation image

where ζ is the free-space impedance, Fs,m,n(c) are spherical wave functions defined in Appendix A and Qs,m,n(c) are the expansion coefficients. Then, the corresponding expression for the equivalent current on S is

equation image

where r0 is the radius of the spherical equivalence surface, and the functions Rs,n(c) and Ts,m,n result from the radial and angular factorization of the tangential components of Fs,m,n(c) as defined in Appendix A. Here, we follow Hansen [1988] for the definition of the SW functions, except for the opposite convention for time dependence, that here is implicitly ejωt.

[16] It is noted that equation (4) provides a closed-form expression of the equivalent current distribution which only depends on the coefficients of the SW expansion of the electric field outside S. These coefficients may be directly available from spherical near field measurements, or they can be easily calculated from field samples measured or calculated on a spherical surface external to the minimum sphere.

[17] The expression in (4) still holds if the radius of the sphere is analytically continued to a complex value equation image0. With the choice equation image0 = r0jb, with b > 0, the equivalent current in (4) becomes a continuous distribution of complex point sources, producing directive beam fields radially emerging from the real spherical surface S. Accordingly, the radiation integral over the complex equivalence surface can be interpreted as a continuum of beam contributions

equation image

where equation imageee denotes the electric dyadic Green's function pertaining to electric current sources, r ≡ (R, equation image, equation image) is the observation point and equation image′ ≡ (equation image0, θ′, ϕ′) is the radial vector on the equivalence complex sphere of radius equation image0 = r0jb.

[18] In order to obtain a usable formula, the radiation integral in (5) must be properly discretized to yield a finite summation of CPS contributions. The number of samples required in this process is very large if all the significant terms in the current expansion in (4) are retained, due to the rapid angular variation exhibited by the spherical wave functions associated with the higher values of the indexes n and m. However, it is known that the currents associated with higher order SW functions do not significantly contribute to the field radiated at a certain distance from S. As a consequence, when the distance between the source and the observer is larger than a wavelength the summation over the index n can be truncated to the value N = kr0 + n1 without affecting the accuracy of the field representation, n1 being an integer depending on the position where the field is observed and on the precision required. As a general rule, n1 = 10 is a satisfactory value for most practical applications [Hansen, 1988].

[19] When the SW expansion is truncated at N, a slowly varying equivalent current distribution is obtained; as a consequence, the radiation integral can be discretized with a moderate number of samples by applying a quadrature rule, yielding

equation image

with

equation image

for j = 1, ., P/2, and

equation image

where rp ≡ (equation image0, θp, ϕp) and wp are the nodes and weights of the quadrature scheme, respectively, and the notation n < N indicates that the summation over index n is truncated to the value N = kr0 + n1. Notice that the current expression in (4) is exact. Hence, the accuracy of the CPS expansion in (6) only depends on the truncation index chosen for the SW summation and on the quadrature scheme used for the discretization of the radiation integral. A discussion on the number of nodes required to obtain an accurate field representation is presented in section 5.1. It is also pointed out that the expression in (6) is implemented by analytically extending to complex space the exact expression of the Green's dyad, inclusive of the reactive contributions; for this reason, the CPS fields are exact solution of the wave equation for any value of kb.

[20] The final result is an accurate field representation consisting of a compact summation of CPS beam contributions whose amplitudes are related to the samples of the filtered current distribution at the corresponding node and can be immediately obtained from the knowledge of the spherical wave coefficients of the radiated field. A key point in the derivation of this compact CPS expansion is the SW expansion of the equivalent currents in (4), which makes it possible to only consider the spatial frequency components that provide a significant contribution to the radiated fields. It is remarked that the proposed representation is extremely efficient when the field is only observed inside a certain angular region, since in that case the angular selectivity of the CPS beams allows one to reduce the number of wave objects needed for the field representation by a factor approximately equal to the ratio between the observed solid angle and the full solid angle 4π.

4. Relationship With the CPS Expansion Based on Numerical Field Matching

[21] The coefficients of the CPS beam expansion in (6) can also be obtained by a totally different process, that consists in numerically solving the linear system

equation image

where equation image is the vector of the P unknown complex point source coefficients, the vector equation image contains the Q tangential components of the electric field radiated at a number Q/2 of points on the test sphere, and equation image is a matrix of Green's functions relating the tangential electric field radiated at the test points to the strength of the complex point sources (see Tap et al. [2011] and Martini et al. [2010] for more details). In practice, the system matrix equation image, which is of order Q × P, with QP, results from the discretization of the radiation operator relating the source distribution on the equivalent surface S to the tangential electric field radiated on the test surface Σ. A convenient technique to solve the system in (9) relies on the use of the singular-value decomposition of the matrix equation image, which yields the following factorization

equation image

where the superscript H denotes Hermitian conjugation, equation image and equation image are unitary matrices of dimensions Q × Q and P × P, respectively, and equation image is a Q × P matrix whose entries are zero apart from the diagonal elements λj, which are nonnegative real numbers ordered in decreasing amplitude, and represent the singular values of equation image. The columns Uj and Vj of equation image and equation image, respectively, are the left and right singular vectors of equation image and represent the mutually orthogonal field and current modes, respectively. The j-th field mode Uj is related to only the j-th source mode Vj by the corresponding singular value λj, whose amplitude provides therefore a measure of the radiation efficiency of that current mode.

[22] The unknown CPS beam coefficients in (9) are obtained by multiplying the rectangular matrix equation image+, which is the Moore-Penrose pseudoinverse of equation image [Golub and Loan, 1983], by the vector equation image

equation image

In (11), equation image+ is the pseudoinverse of equation image, which is formed by replacing every nonzero entry by its reciprocal and transposing the resulting matrix. Expression (11) reveals that in the presence of small singular values λj the problem is ill-posed, since small perturbations in the entries of the vector equation image caused by noise would be amplified by the division of the small values of λj. A regularization technique that can be used to face this problem is the so-called “truncated singular value decomposition,” TSVD, which consists in ignoring the singular values smaller than a given threshold σ [Varah, 1973]. This corresponds to truncating the summation in (11) to the value Pσ such that equation image, thus, yielding

equation image

[23] The expression in (12) has a simple physical interpretation: the electric field on the test surface is first projected onto the Pσ field modes associated with the highest radiation efficiencies; then, the field modes are transformed into the corresponding current modes through the relevant singular values. Finally, the CPS beam coefficients are reconstructed as a summation of contributions from the single current modes.

[24] This approach holds for arbitrary shapes of the test and the equivalence surfaces and can be exploited to construct a non redundant set of bases for the currents expansion [Prakash and Mittra, 2003; Matekovits et al., 2007; Stupfel and Morel, 2008; Casaletti et al., 2009]. The objective here is to show that in the particular case of a spherical geometry the numerical determination of the CPS beam expansion coefficients via TSVD corresponds to the analytical procedure proposed in the previous sections. This is illustrated in the following, with reference to the geometry introduced in section 3 and for the two cases in which the radiation operator has infinite or finite dimension.

4.1. Infinite Dimensional Radiation Operator for Spherical Geometry

[25] In the limit of infinite density of test and source points, the singular vectors become continuous functions, and are called “singular modes” or “singular functions” [Smithies, 1958]. In that case, the radiation operator L from the surface of the complex sphere is infinite dimensional and takes the following integral form

equation image

where the subscript t denote the radially transverse component, and equation imageee is the dyadic Green's function introduced in (5). After defining the inner product between two current- or tangential-field-functions as

equation image

we recognize that an orthonormal basis for both the space of radiated fields on Σ and the space of electric currents on S is provided by the set of functions Ts,m,n(equation image, equation image). These functions are eigenfunctions of the radiation operator, since it can be easily shown, after exploiting the orthogonality properties of the SWs, that [Hansen, 1988]

equation image

where the transformation

equation image

is assumed to convert the index triple (s, m, n) to the single index j. The self-adjoint operators LLH and LHL, where LH is the adjoint of the operator L, are diagonalizable with positive eigenvalues

equation image

and eigenfunctions Tj. By definition, the singular values of the operator L are the square roots of the eigenvalues of the self-adjoint operators LLH and LHL and are therefore

equation image

while the singular modes Uj and Vj are defined by the relationships

equation image

A possible choice is therefore

equation image

where ϕj = equation image, and the phase factor equation image is needed to convert the eigenvalues in (15) into the real positive singular values λj. Hence, the Tj functions coincide, up to a phase factor, with the singular functions of the radiation operator. Notice that the singular functions are independent of the radius of the equivalence surface S, and are therefore the same for real and complex point sources. On the contrary, the singular values λj depend on equation image0.

[26] According to (10), the SVD representation of the generic pq element of the Green's functions matrix is

equation image

Notice the correspondence between the expression in (21) and the eigenfunction expansion of the radially transverse part of the dyadic Green's function [Tai, 1993]

equation image

The relationship between the SVD of the radiation operator in a spherical geometry and the spherical wave expansion has also been highlighted for acoustic radiation [Borgiotti, 1990; Photiadis, 1990; Nelson and Kahana, 2001].

[27] According to equation (11), in the numerical approach the current distribution on S radiating a given field E is obtained through the following steps:

[28] 1. The tangential field on the test sphere is projected onto the set of the efficiently radiating SW yielding

equation image

This step corresponds to the calculation of the SW expansion coefficients of the radiated field, Qs,m,n(3), in the proposed procedure.

[29] 2. Each projection is multiplied by the inverse of the corresponding singular value yielding

equation image

The resulting value is the weight of the j-th current mode;

[30] 3. The contributions from all the current modes are combined to obtain the current distribution

equation image

which coincides with the current expression in (4), thus demonstrating the strict connection between the CPS expansion approach based on numerical field matching and the approach described in section 3.

4.2. Finite Dimensional Radiation Operator for Spherical Geometry

[31] For a finite density of test and source points the identification of the singular functions with the SW is no more exact; however, the physical interpretation of the outcome of the singular value decomposition remains the same. Indeed, when the samples on the equivalence surface S and on the test surface Σ are sufficiently dense and are taken at the center of patches of equal area, the product between the Green's function matrix and the current vector can be interpreted as a discretization of the radiation integral in (13) after multiplication by the elemental area dA = 4πequation image02/(P/2). In this case, the significant singular values of the Green's matrix stabilize to the value

equation image

[32] This is illustrated in Figure 1, where the black curve represents on a logarithmic scale the magnitude of the singular values in (26) normalized to the largest for equation image0 = 2λ(1 − j) and R = 7λ. As it can be seen, the singular values are grouped in spherical harmonic order, with the first 6 ones corresponding to n = 1, the next 10 corresponding to n = 2 and so forth, and decrease exponentially for increasing indexes. In the same figure, the colored curves represent the numerical singular values of a square matrix of Green functions of different dimensions P. As expected, numerical and analytical values tend to coincide for P → ∞, while for any finite value of P the first P′ < P singular values are stabilized to the corresponding analytical value. For instance, with a number of beams equal to 2NSW the first NSW singular values are retrieved with a relative error less than 2%. The corresponding discrete singular vectors should in principle correspond to sampled spherical harmonics. However, since the 2n + 1 modes characterized by a given couple of indexes (s, n) have the same singular value (i.e. the modes are degenerate with respect to the index m) the j-th left and right singular vectors generated by the SVD will be linear combinations of all the sampled spherical harmonics characterized by the corresponding indexes (s, n).

Figure 1.

Singular values of the radiation operator for equation image0 = 2λ(1 − j) and R = 7λ.

5. Guidelines for the Choice of the CPS Expansion Parameters

5.1. Choice of the Number of CPS Beams

[33] In the proposed expansion approach, the total number of CPS beams is twice the number of nodes used for the discretization of the radiation integral in (5). For a given variation rate of the integrand function with equation image′ and equation image′, the number of nodes depends on the quadrature rule used. A simple approach consists in taking the nodes at the center of regions of equal area, and using equal weights for all the samples. A partition of the sphere into regions of equal area can be defined through the algorithm presented by Leopardi [2006]. An alternative approach is proposed by Heilpern et al. [2007] and by Timchenko et al. [2010] and consists in dividing the sphere into a mesh of triangles with common nodes and setting the weight of each node equal to the associated spherical angle. These solutions have the advantage of providing a set of beams whose axes are approximately uniformly distributed in space, which can be desirable for certain applications, like the ones in the works of Tap et al. [2007] and Martini et al. [2010]. However, in order to minimize the number of beams required to obtain a certain accuracy, it is more convenient to use proper quadrature formulas defined over the unit sphere which are exact for spherical harmonics of order less than or equal to a given value. In this case, the location of the nodes is defined by the quadrature formula, along with the relevant weights. In order to determine the required degree of the quadrature rule (corresponding to the number of nodes), it is sufficient to consider the contribution provided to the radiation integral by the highest order spherical harmonics of current, characterized by the indexes n = N, m = N, s, which is

equation image

where equation image′ = (r0jb)equation image′ is the complex location of the sources.

[34] We now introduce in (27) the SW expansion of the dyadic Green's function, thus obtaining

equation image

Since the amplitude of the terms of the summation decreases exponentially for large values of ν, in practice the summation can be truncated to a certain index N′. The value of N′ can be determined by imposing that the fraction of power Ptr which is excluded by truncating the SWE is smaller than a given threshold. Sample results are shown in Figure 2. In particular, Figure 2a shows the behavior of the truncated power fraction Ptr with the truncation index Ntr for the function equation imageee(r, equation image′) · equation imageθ with r0 = 4λ, R = 50λ, θ = 0 and ϕ = 0. There, different curves correspond to different values of the complex displacement b. A similar behavior is found for equation imageee(r, equation image′) · equation imageϕ and/or different values of the observation angles θ and ϕ. Figure 2b shows the value of N′ as a function of b, where N′ is defined as the minimum value of Ntr which provides Ptr < −40 dB. As it can be seen, the value of N′ is minimum for b ∈ [0.5 – 1]r0 and it increases monotonically for larger values of b.

Figure 2.

(a) Truncated power ratio of the function equation imageee(r, equation image′) · equation imageequation image with r0 = 4λ, R = 50λ, equation image = 0 and equation image = 0 as a function of the truncation index for different values of the complex displacement b. (b) Value of the truncation index Ntr which provides Ptr < −40 dB as a function of b.

[35] After truncating the summation in equation (28) and interchanging summation and integration we get

equation image

It is now clear that the number of nodes must be chosen so as to correctly calculate the integral in (29) for all the indexes νN′(b). By applying the Lebedev quadrature formulas [Lebedev, 1999], this is obtained with a quadrature scheme of order Np = N + N′(b), which requires a number of nodes approximately equal to equation image. For instance, for b = r0 it results from Figure 2b that N′ = 21. Hence, for N = 26, an integration formula of order 47, corresponding to 770 nodes, provides accurate results. If instead b is chosen equal to 3r0, it results N′ = 27, which means that a similar accuracy is obtained with a quadrature formula of order 53, which requires 974 nodes.

[36] The procedure described above can be used to determine the number of nodes for an arbitrary value of r0. As a general rule, for b = 0 it can be assumed N′ = ⌈kr0 + 1.6equation image⌉ while the dependency of N′ on b is similar to the one depicted in Figure 2b.

5.2. Choice of the Imaginary Displacement b

[37] It is now clear that the value of the imaginary displacement b affects the effectiveness of the CPS expansion, as also discussed by Heilpern et al. [2007]. The optimal value of b is the one requiring the minimum number of beams to obtain a given accuracy, or, equivalently, providing the maximum accuracy for a given number of beams. This value may actually depend on the field to be represented and on the observation region; however, it is possible to provide general guidelines for the choice of this parameter. In the following, we consider separately the cases where the field representation is required to be accurate over the full solid angle (section 5.2.1) and over a limited angular region (section 5.2.2).

5.2.1. Field Representation Over the Full Solid Angle

[38] A generic field radiated by sources contained inside a given minimum sphere can be accurately represented as a superimposition of spherical harmonics with maximum polar index related to the radius of the minimum sphere. As a consequence, the analysis of the accuracy in the representation of these individual spherical harmonics also provides information on the accuracy in the representation of an arbitrary radiated field. From this analysis, it turns out that for a given radius of the minimum sphere r0 and a given set of CPS located on the complex sphere of radius r0jb the minimum error is achieved by choosing b in the range [0.5 – 1]r0. Sample results are shown in Figure 3a for the case of an equivalence sphere of radius r0 = 4λ. Lebedev quadrature rule with 974 nodes has been employed for the computation of the radiation integral. This corresponds to P = 1948 CPS beams, accounting for the two orthogonal polarizations. Each curve in Figure 3a represents the error in the reconstruction of a single spherical harmonic. Different values of the polar index n are considered, and in all the cases the other indexes are m = n (corresponding to the fastest variation in equation image) and s = 1. The relative error has been defined on the observation sphere of radius R = 50λ as

equation image

where ECPS = E(R, θ, ϕ) is the field reconstructed through the CPS expansion and Eref = kequation imageFs,m,n(3)(R, θ, ϕ) is the reference field.

Figure 3.

Relative error in the CPS representation of a single spherical harmonic with m = n, s = 1 for (a) the analytic expansion procedure and (b) the numerical one.

[39] It is apparent that the choice b ∈ [0.5 – 1]r0 provides a low error for all the spherical harmonics of interest. The same trend has been found also for s = 2 and for different values of the radius of the equivalence and/or observation spheres. This behavior is consistent with the results presented in the previous section. Since the only source of error in the proposed CPS representation of a single spherical harmonics is the discretization of the radiation integral, the maximum accuracy for a given number of nodes is obtained by minimizing the bandwidth of the SW spectrum of the Green's function.

[40] For any given value of n, the redundancy in the CPS representation can be quantified by the ratio between the number of CPS beams and the number of spherical harmonics of polar index smaller or equal to n, which is NSW = 2n(n + 2). This “redundancy factor” is indicated in the caption of Figure 3a. It is noted that with a proper choice of the complex displacement b it is possible to obtain an accurate field representation also for n = 28, when the redundancy factor is very small. The number of points chosen in the Lebedev's quadrature formula actually corresponds to distances approximately equal to λ/2 between the nodes.

[41] The error of the CPS expansion obtained through the numerical matching procedure has also been evaluated, for the sake of comparison, and the results are shown in Figure 3b. As it can be seen, the order of magnitude of the error is comparable for the two procedures, which confirms the effectiveness of the proposed analytical expansion procedure.

5.2.2. Field Representation in a Limited Angular Region

[42] When the knowledge of the field is only required in a certain angular region, it can be preferable to use collimated CPS beams (kb ≫ 1) in order to exploit their spatial selectivity to truncate the CPS beam expansion [Heilpern et al., 2007]. This case is of practical importance when dealing with MoM type reaction integrals between large/entire domain basis functions. To illustrate this concept, we consider an observation region corresponding to the interior of a cone with apex at the origin, and half-aperture angle α (Figure 6). Let us call CΩ the angular region associated with this cone, where Ω = 2π(1 − cos α) denotes the corresponding solid angle. The number NDoFΩ of degrees of freedom of the field inside CΩ is related to the number NDoF of degrees of freedom associated with the full solid angle by the relationship NDoFΩ = NDoFΩ/(4π). The number of CPS would be reduced by approximately the same factor by retaining in the expansion only those CPS beams whose axis is contained inside the conical region CΩ. However, due to the finite beamwidth of the beams, this is not sufficient to obtain a good accuracy. It is thus convenient to define a second coaxial “truncation” cone with half-aperture angle α + δ (see Figure 6) and to retain in the expansion only the CPS beams whose axis is contained inside this truncation cone. The value of the excess angle δ is chosen so that the amplitude of the field of a CPS beam with axis lying on the boundary of the truncation cone is attenuated of approximately 20 dB at the boundary of CΩ. For observation distances much larger than b and collimated beams, this condition leads to the simple formula

equation image

The number of beams is in such a case proportional to Ω′ = 2π[1 − cos(α + δ)]. It is then clear that increasing b reduces the fraction of beams that must be retained in the expansion to represent the field in the given angular sector. However, as shown in the previous section, values of b much larger than r0 require a total number of beams (i.e. number of beams for the representation of the field over the full solid angle) which can be significantly larger than the one corresponding to br0. As a consequence, the optimal choice actually depends on the extent of the observation region and is a trade-off between two requirements: minimizing the number of beams for the representation of the field in the whole solid angle, leading to b ∈ [0.5 – 1]r0, and increasing the beam angular selectivity, leading to larger values of b. For large observation regions, a possible approach consists in using curves like the ones in Figure 3a to find the maximum value of b that can be used for a certain admissible error with a given number of nodes and a for a certain maximum polar index of the spherical harmonics.

5.3. Region of Validity of the CPS Beam Expansion

[43] The field of a CPS located at the complex point r0jb exhibits branch point singularities on a circumference equation image perpendicular to b, with center in r0 and radius ∣b∣. With the branch choice equation imagee[equation imagerrequation image] ≥ 0, the field is continuous everywhere in equation image3 except on the flat disk equation image contoured by equation image (branch cut) [Felsen, 1976].

[44] The integral relationship in (5) expresses the radiated field as a continuum of CPS beams with complex location equal to (r0jb)equation image. For this set of CPS, the branch point singularities lie on the sphere of radius equation image while the branch cuts are located in the volume between the spheres with radii r0 and equation image (see Figure 4). In spite of these singularities, the expansion does not abruptly become inaccurate when the observation point enters the region R < equation image from outside, but it exhibits discontinuities when the observation point approaches the surface R = r0. A typical behavior is shown in Figure 5, which compares the field provided by the CPS expansion with the reference solution (the spherical harmonic with s = 1, m = 25, n = 25) when the observation points enters the region r0 < R < equation image. Expansions corresponding to different numbers P of CPS beams are considered. In this example we have r0 = 4λ and b = 8λ, corresponding to equation image ≃ 9λ (dashed line in Figure 5). As it can be seen, the representation is quite precise for R > 6λ, but inaccuracies arise for smaller values of R.

Figure 4.

Pictorial representation of the location of the branch cuts of the CPS. (a) Three-dimensional view. (b) Cross sectional view.

Figure 5.

Comparison between the CPS beam expansion with equation image0 = 4λj8λ and the reference solution for varying observation distances R and observation angles equation image = 60°, ϕ = 0°. The dashed line indicates the value R = equation image.

[45] The branch cut singularities can be removed by replacing the CPS beams by their gaussian beam approximation, but the results will be only accurate in the paraxial region of the beam and for distances much smaller that kb2 [Heilpern et al., 2007].

6. Numerical Results

[46] Numerical analyses have been carried out to verify the accuracy of the proposed representation. In all the examples, the numerical integration has been performed by Lebedev's quadrature and two orthogonal CPS have been defined at each node.

[47] In the first example the CPS expansion of the field is only required to be accurate inside the cone with apex at the origin, axis pointing in the direction (θp = 100°, ϕp = 60°) and half-aperture angle α = 30°. The CPS beams providing a non negligible contribution to the field in the observation region have been identified as explained in section 5.2 (Figure 6).

Figure 6.

Pictorial representation of the observation region and of the truncation cone.

[48] The SW expansion of the field to be represented has the following coefficients

equation image

corresponding to a truncation index N = 19 for the summation over the polar index n. A total number of 590 source points (corresponding to a total of 1180 beams, taking into account the two orthogonal polarizations) are defined on a spherical equivalence surface of radius equal to equation image0 = (3 − 10j)λ, but only 88 (corresponding to 176 beams) are retained in the expansion. The number of CPS used is significantly smaller than the total number of spherical harmonics with polar index less than or equal to 19, which is 798, even if it is larger than the number of degrees of freedom of the field in the given observation domain, which is around 75, as revealed by the analysis of the singular values of the radiation operator.

[49] In Figure 7, the field provided by the proposed CPS beam expansion is compared with the reference solution in the cut ϕ = 60° and for Robs = 50λ. As it is apparent, the CPS expansion is very accurate inside the region of interest, whose boundaries are indicated by the continuous lines. The dashed lines represent the boundaries of the truncation cone.

Figure 7.

Comparison between the electric field reference solution and the corresponding CPS expansion for the electric field described by the SW expansion in (32) in the cut ϕ = 60° for Robs = 50λ. (a) θ-component. (b) ϕ-component. Continuous and dashed vertical lines indicate the boundaries of the observation and truncation cones, respectively.

[50] For this first test case, the CPS beam weights provided by the proposed procedure have been compared with the ones obtained through the solution of the linear system in (9) with the same number of CPS. The results, reported in Figure 8, show the strong similarity between the two solutions, which testifies the effectiveness of the proposed procedure.

Figure 8.

Comparison between the amplitude of the CPS given by equation (8) and the ones provided by the numerical matching, for the first test case.

[51] As a second example, the field radiated by a 12 × 12 array of Huygens source radiators lying on the xy plane has been considered. Each Huygens source radiator consists of an electric dipole directed along equation image and a magnetic dipole directed along equation image, with a ratio between the amplitude of the two dipoles equal to the free space impedance; the inter-element distance is equal to λ/2 along both x and y. In this case, the CPS representation is required to be valid over the full solid angle. Only the spherical harmonics with n smaller or equal to 27 (corresponding to ⌊kr0 + 2⌋) are assumed to be important for the representation of the radiated field. This corresponds to a number of degrees of freedom equal to 1566. For the CPS expansion, the imaginary displacement b is set to 3λ, and Lebedev quadrature formula with 974 nodes (corresponding to 1948 CPS beams) is used. This leads to a redundancy factor equal to approximately 1.24. The relative error over the full solid angle, calculated according to equation (30), is 2.4 · 10−4. Figure 9 shows the comparison between the reference solution and the CPS expansion in the cut ϕ = 0°, demonstrating the accuracy of the proposed representation.

Figure 9.

Comparison between the electric field reference solution and the corresponding CPS expansion for a 12 × 12 array of Huygens' source radiators with interelement spacing equal to λ \ 2. Cut ϕ = 0°.

7. Conclusions

[52] A compact expansion in terms of CPS beams with closed form coefficients has been derived for the electromagnetic field radiated by an arbitrary finite source region. The CPS beams emerge from a spherical surface enclosing the real sources, and their weights are analytically calculated starting from the SW coefficients of the radiated field. The resulting representation is particularly efficient when the field knowledge is only required in a limited angular region; in this case, the CPS basis presents some redundancy with respect to the basis provided by SVD, but it has the advantages of consisting of wave objects, which can be propagated analytically, and of being independent of the angular observation region. These characteristics are important when the field expansion is used for the analysis of the interactions of a given source with neighboring obstacles. Furthermore, the proposed analytical expansion procedure also provides a physical insight into the mechanism behind the numerical determination of the weights of the CPS expansion through field matching. In fact, it has been shown that the results of the two approaches tend to coincide for increasing densities of CPS beams.

Appendix A:: Spherical Wave Functions

[53] The general expressions for the spherical wave functions are [Hansen, 1988]:

equation image

where equation imagenm(·) is the normalized associated Legendre function, and

equation image
equation image
equation image

In (A1)–(A4) the index n refers to the radial dependence and the index m refers to the azimuthal angular wave number. The index c = 1, 2, 3, 4 denotes different typologies of r-dependent functions, namely spherical Bessel, Neumann, Hankel outgoing (second type) and Hankel incoming (first type) functions, respectively. The subscript s = 1, 2 denotes TE, TM behavior with respect to r when Fs,m,n(c) is used to represent the electric field.

[54] The components of the spherical wave functions which are transverse with respect to the radial coordinate can be factorized as the product of Ts,m,n(c)(θ, ϕ) and the scalar radial function Rs,n(c)(kr)

equation image

Acknowledgments

[55] The authors would like to thank the anonymous reviewer for useful comments and suggestions.