Sommerfeld Integral Path for Layered Double Negative Metamaterials

A new Sommerfeld integral path is proposed for a dielectric slab with double negative metamaterials. In order to conceive the proper Sommerfeld integral path, the mode properties of a grounded dielectric slab with double negative metamaterials are first reviewed. Dispersion diagrams and power flows are investigated, followed by a summary of sign choices for possible solutions. The mode loci in the k_ρ-plane is studied, as it provides crucial information in proposing a proper Sommerfeld integral path. It is found that layered double negative materials have a different Sommerfeld integral path as compared to the conventional layered double positive media. The numerical difficulties associated with the Sommerfeld integral path are investigated, and an accelerating method that greatly speeds up convergence is suggested by properly transforming the integral domain from k_ρ-plane to k_z-plane. Finally, several numerical examples are provided to demonstrate the importance of choosing correct integral paths and to unveil the exotic properties and potential applications of double negative metamaterials.