Global Minimality in Constrained Inverse Source Problems for Complex Electromagnetic Media

Abstract

Global minimality, boundedness, and uniqueness are established for a general, physically motivated class of inverse source problems in non-homogeneous electromagnetic media with generalized constitutive parameters. The existence of a solution was addressed earlier. The radiating source, represented by the current density, was reconstructed earlier by minimizing its $L^{2}$-norm constrained to produce a prescribed radiated field while ensuring vanishing reactive power. Using the $L^{2}$-norm allows for an analytically tractable measure of the physical resources of the source, while the reactive power constraint maximizes transmitted power. Numerical study suggests that sources within active metamaterial substrates can have remarkable tuning behaviors. Tuning stability can be achieved along specific permittivity and permeability curves on zero-reactive power plots. Each permittivity or permeability value can correspond to a discrete set of dual parameter values that enable effective tuning. The tuning characteristics observed suggest that double-positive (DPS) and double-negative (DNG) substrates are more favorable for tuning than single-negative (SNG) materials, possibly due to interactions dominant in DPS and DNG media.