## 1. Introduction

[2] An electromagnetic wave with wavelength *λ* much larger than a certain object does not really “see” that object. Consequently a wave propagating in a homogeneous substrate with small inclusions that are, let us say, ∼*λ*/10 will interact effectively not with the individual inclusions but with the system {substrate,inclusions} as a whole. The wave, thus, sees a new effective medium having electromagnetic properties different from those of its components. This suggests that effective media with properly chosen inclusions may exhibit electromagnetic responses not found in nature. Metamaterials are such media. They are engineered materials whose effective constitutive parameters can, in principle, have any value, even negative.

[3] The onset of the “metamaterials era” is frequently attributed to the landmark theoretical article by *Veselago* [1968], though the study of complex media has a long history (see, for instance, the review by *Ziolkowski and Engheta* [2006]). Yet, until just a few years ago, the study of these materials was largely ignored by scientists and engineers alike. Then, at the turn of the century, *Smith et al.* [2000] announced the fabrication of a metamaterial with negative index of refraction in the microwave regime. This discovery instigated several studies that resulted in the identification of several novel and exciting applications of metamaterials. For example, metamaterials have been suggested for the realization of perfect lenses [*Pendry*, 2000], subwavelength cavities [*Engheta*, 2002], highly resonant electrically small antennas [*Ziolkowski and Kipple*, 2003], subwavelength waveguides [*Alù and Engheta*, 2004], cloaking devices [*Alù and Engheta*, 2005a], and ultrathin laser cavities [*Ziolkowski*, 2006a].

[4] In this paper we present a comparative study of radiation enhancement due to metamaterial substrates. Antenna radiation enhancement due to metamaterial substrates is a very active field of research [see *Engheta et al.*, 2006, and references therein] and several leading groups have addressed several important aspects of the problem. Here, we wish to address the issue of “fairness” in antenna performance comparisons which has been the subject of some controversy [*Kildal*, 2006; *Ziolkowski*, 2006b]. By “fairness” we mean the suitability of the reference antennas with respect to which the comparisons are to be carried out.

[5] This study complements our investigation of radiation enhancement with metamaterials for the core-shell system [*Khodja and Marengo*, 2008] which is a generalization of the homogeneous substrate case [*Marengo and Khodja*, 2007, 2008]. However, in the work of *Khodja and Marengo* [2008] we considered only the effects that adding a shell would have on the performance of an existing antenna. The novelty of the present treatment resides in the fact that it explores the effects that different definitions of the reference antenna would have on enhancement estimates (in particular three classes of reference antennas shall be considered.) As in the work of *Khodja and Marengo* [2008], the underlying formulation is a non-antenna-specific, full-vector, inverse-theoretic formulation in substrate media that is a generalization of the scalar inverse source theory by *Devaney et al.* [2007].

[6] The configuration investigated in the following is that of a piecewise-constant, radially symmetric, lossy core-shell system immersed in free space (three-region system). The core, of radius *a*, has relative permittivity ɛ_{a} and relative permeability *μ*_{a}. This core is the smallest spherical volume *V* that circumscribes the largest physical dimension of the original antenna. This antenna is treated, under a suppressed time dependence *e*^{−iωt}, as an arbitrary primary current density **J**(**r**). The core is surrounded by a spherical shell, of inner radius *a* and outer radius *b*. The constitutive parameters of the shell are relative permittivity ɛ_{b} and relative permeability *μ*_{b}. The core and the shell are assumed to be generally lossy; their relative constitutive parameters are, in general, complex.

[7] We formulate an inverse source problem in substrate media, whose objective is to deduce an unknown impressed, or primary, current density **J**(**r**) that is contained, along with the substrate, in the spherical volume *V*, and that generates a prescribed exterior field for *r* > *b*. The solution of the inverse problem is nonunique due to the possible presence of nonradiating source within *V* [*Devaney and Wolf*, 1973; *Marengo and Khodja*, 2006]. The uniqueness of the solution is guaranteed if, for instance, one requires that the square of the *L*^{2} norm of the source (as defined by equation (2)) be minimum. This quantity is usually termed “the source energy.” It is widely used in inverse theory literature [*Porter and Devaney*, 1982a, 1982b; *Langenberg*, 1987; *Kaiser*, 2005; *Devaney et al.*, 2008]. A minimized source energy would indicate that the resources of the antenna generating the given field pattern are optimally used within the prescribed volume of the source.

[8] Comparison of the required minimum source energies for different substrate configurations enables quantification of the enhancement due to such structures. This substrate enhancement characterization is non-device-specific. In particular, one would, in this framework, be able to compare the optimal source for a given substrate with the optimal source for another (including the free-space case).