Interpretation of complex frequencies in propagation problems

Derived from Helmholtz wave equation, the dispersion relation describes the temporal and spatial behaviors of a wave in a given electromagnetic medium. In general, this relation can be calculated using either driven mode analysis or eigen mode analysis. In this paper, the two approaches are compared using an example of a periodic leaky-wave antenna. An interpretation for the complex frequencies that arise from the eigen mode approach is given based on the general concept of mapping between two complex planes, namely the complex frequency plane and the complex propagation constant plane. It is shown that the complex frequency dispersion relation gives information on the transient response of the electromagnetic structure under test. Moreover, this paper shows that the results of the two approaches are independent and provide complementary information about the structure under test.