Parity-Time Symmetry and Leaky Wave Antennas: A Generalized Dispersion Equation

A generalized dispersion equation is derived featuring coupled mode theory, parity-time symmetry, and leaky wave antennas of arbitrary periodic modulation. It can be specialized to each of these cases individually or can describe a structure containing all three electromagnetic phenomena simultaneously in a single antenna. This very general dispersion equation is derived using both mode matching and the transverse resonance method, the latter lacking the ability to provide the field descriptions and wave impedances with the advantage of computational simplicity. The dispersion equation is first used to design a parity time symmetric waveguide consisting of conjugate impedance sheets coupled in close proximity. The example shows both the eigenvalues (wavenumbers) and eigenvectors (modes) coalescing at a single point in the parameter space known as the exceptional point. In another example, the same dispersion equation is again used to model a sinusoidally modulated reactive sheet (SMRS) supported by an active impedance sheet backed dielectric spacer. The active impedance sheet is designed to compensate for the SMRS radiative leakage loss when coupled in close proximity. Hence, each spatial harmonic is described by a purely real wavenumber despite the radiative losses to the open far field channel. Plots of the spatial harmonics show a constant amplitude envelope and hence leaky wave radiation is generated from spatial harmonics which do not decay as they leak. Full wave simulations corroborate our results.