Unique Retrieval of Complex Permittivity and Permeability of Dispersive Materials From Reflection and Transmitted Fields by Enforcing Causality

The electromagnetic properties of general classes of materials can be obtained by inverting the measured or numerically simulated reflection ( S ₁₁) and transmission ( S ₂₁) coefficients through a known thickness of a planar slab of the material. The major difficulty is the uncertainty in the change of phase of the transmitted field, if the phase change exceeds 2mpi, m=0,1,2,.... This happens for thick samples as well as for highly dispersive materials. An infinite number of solutions are generated for the imaginary part of the complex wavenumber gamma = alpha + jbeta for different choices of m. The choice of m can be made unique by noting that for a causal medium, gamma is an analytic function of frequency and this results in a Kramers-Kronig-type relation between alpha and beta. Since alpha is independent of the choice of m and, hence, uniquely determined, a Kramers-Kronig reconstruction of beta using alpha will provide a guideline for the correct choice of m. This is a "foolproof" approach even for totally unknown materials since no initial guess is required. Indeed, this approach will work in optics and ultrasonics, and should also be effective for determining the effective refractive index of photonic-bandgap structures that are also highly dispersive.