An analytical exact time domain normal mode solution of a lossy resonator

Abstract

In this study, an analytical exact time domain solution for electromagnetic waves generated in a rectangular resonator is found in three-dimensional Cartesian coordinates. The resonator is fully filled with a lossy medium. Therefore, an inhomogeneous lossy wave equation for a component of a vector potential is solved by a Time Domain Normal Mode (TDNM) method. In the TDNM method, a space domain and a time domain eigenvalue differential equation are formulated by using separation of variables and Sturm-Liouville techniques. Then, an exact solution is found by combining the solutions of the two eigenvalue differential equations. Specially, the TDNM solutions are obtained for a modulated rectangular pulse and an unmodulated rectangular pulse type point source, separately. The analytical TDNM solutions of the lossless resonator are also extracted and validated as a limit case where a loss-free condition is satisfied.
The calculated analytical TDNM results in the sense of the field distributions, the time signatures and the frequency responses of the lossy resonator and the lossless resonator are validated by comparing them with numerical Finite Difference Time Domain (FDTD) solutions. The excellent agreements are observed between the fully analytical TDNM and the fully numerical FDTD results. In the future, it is planned to extend this study for an analytical exact TDNM solution of a resonator filled with a dispersive material.