Nonreciprocal TM-mode thin-film phase shifters

TM modes guided by thin-film waveguides with one or more magnetooptic and anisotropic layers (i.e., substrate, film, and cover layers) have been studied with emphasis on its nonreciprocal characteristics. Our analysis shows that the nonreciprocity of TM modes is the result of nonreciprocal Goos-Haenchen phase shifts at the cover–film and/or film–substrate interfaces. Contributions to the nonreciprocity defined as (N+ − N−) from various regions are identified. Here N± are the effective indices of refraction of two counterpropagating modes. An accurate algebraic approximation for (N+ − N−) has been obtained to provide physical insight into the problem and facilitate numerical calculations. Guidelines for designing nonreciprocal TM mode phase shifters are also listed.