Global Minimality in Constrained Inverse Source Problems for Metamaterials

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 L2-norm constrained to produce a prescribed radiated field while ensuring vanishing reactive power. Using the L2-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. These results fill an analytical gap in the solution of a problem that is both intriguing and challenging due to its general formulation, which requires minimizing an objective functional with nonconvex functional constraints on an unbounded domain. They also offer numerical insights that may have implications for the design and optimization of sources in complex media, which is a topic of significant current interest.