Negative Refraction at a Lossy Interface and a Bold Hypothesis via Complex Frequency

Abstract

We investigate the negative refraction effect at a planar interface of a highly absorptive material, where the direct experimental verification is difficult because of the loss-induced skin depth effect. An apparent contradiction occurs when we try to determine the group velocity direction by the method of equifrequency contours (EFCs) in detail. This contradiction forbids any physical solution to be found for negative refraction. We conclude that this paradox is mainly caused by the definition of complex wavevector ~k which is conventionally adopted in the case of complex permittivity. The complex wavevector may result in ambiguously defined optical path, which limits the application of the classical Snell’s law. We propose a bold suggestion that the complex wavevector ~k should be replaced by a complex frequency ~ω. Therefore, the optical path can always be defined as real. The proposed hypothesis is capable of resolving the contradiction about the loss-induced negative refraction, and the obtained theoretical prediction fits well with the reported experimental results.