An analytic continuation method for the analysis and design of dispersive materials

All materials, by nature, possess a frequency-dependent permittivity. This dispersion can be expressed in the form of the Kramers-Kronig relations by invoking the analytic consequences of causality in the upper half of the complex frequency plane. However, the Hilbert transform pair character of these relations makes them useful only when half of the answer is already known. In order to derive a more general form useful for both synthesis and analysis of arbitrary materials, it is necessary to analytically continue the permittivity function into the lower half plane. Requiring that the dielectric polarization be expressible in terms of equations of motion, in addition to obeying causality, conservation of energy and the second law of thermodynamics is sufficient to obtain the desired expression as a sum of special complex functions. In the appropriate limits, this sum reduces to the Debye relaxation and Lorentz resonance models of dielectrics, but it also contains phenomena not expressible in terms of those classical models. In particular, the classic problem of the existence of optical transparency in water is resolved.