Theoretical study of the implications of causality when inferring metamaterial properties

The usual procedures to extract effective refractive indices for negative-index metamaterials are complicated by branching ambiguities in the inverse cosine function. The existing methods to eliminate ambiguities either involve calculations with varying geometries which are inherently flawed, as metamaterial properties depend strongly on geometry, or are more subjective as one must inspect and select branches to yield effec- tive parameters as a continuous function of frequency. We propose that the Kramers-Kronig relations, which hold for negative-index materials, naturally guide the selection of the proper branches, and in fact predict negative refractive index where other extraction procedures alone might not. The experimental realization of high quality negative-index materials requires a robust and reliable index retrieval procedure; the combined extraction/Kramers-Kronig retrieval can be used to design optimal metama- terials in a simple, systemmatic, and general manner.