## 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.