Modeling Propagation in Negative Index Media Using Causal Complex Dispersion Relations

Starting from the causality of the permittivity and permeability of a medium, we investigate the causality of the propagation constant. We show that a reduced dispersion relation, obtained from the frequency dependence of the propagation constant by neglecting a linear frequency dependent term, obeys causality. The propagation constant is identical to the reduced propagation constant under appropriate limiting values of the physical parameters. We illustrate the causality of the reduced propagation constant through examples of (a) a nonmagnetic material where the permittivity is given by the Lorentz model, (b) a material where the permittivity and permeability are both Lorentz-type, and (c) an effective medium comprising a nonmagnetic material with Lorentz-type permittivity in a dispersionless host medium, where the effective permittivity is given by the Maxwell–Garnett rule. Causality of the propagation constant enables the use of simple operator formalisms to derive the underlying partial differential equations for baseband and envelope wave propagation, as demonstrated through an illustrative example of a negative index medium with gain