Waveguides and Resonators

Abstract

The preceding chapter introduced general properties of Maxwell’s equations. It identified power flow and energy density and derived the uniqueness theorem and the reciprocity theorem. This background is necessary for the analysis of metallic waveguides and resonators as used in microwave structures. In this chapter, we analyze the modes of waveguides with perfectly conducting cylindrical enclosures. We determine the mode patterns and the dispersion relations, i.e. the phase velocity as a function of frequency. We derive the velocity of energy propagation and show that it is equal to the group velocity, i.e. the velocity of propagation of a wavepacket formed from a superposition of sinusoidal excitations within a narrow band of frequencies. Then we study the modes in an enclosure, a so-called cavity resonator. We determine the orthogonality properties of the modes. Next, resonators coupled to the exterior via “ports of access” are analyzed. Their impedance matrix description is obtained and the reciprocity theorem is applied to the impedance matrix. This analysis is in preparation for the study of noise in multiports, which begins in Chap. 5. Finally, we look at resonators in a general context. The analysis is based solely on the concept of energy conservation and time reversal. The derivation is applicable to any type of resonator, be it microwave, optical, acoustic, or other. Most of the results obtained here are contained in the literature [21,27–30]. The concepts of the waveguide mode and of resonant modes are necessary for the quantization of electromagnetic systems. Even though the analysis in this chapter concentrates on waveguides and resonators in perfectly conducting enclosures, the generic approach to resonance is independent of the details of the electromagnetic mode and is based solely on the concept of losslessness and time reversibility. This is the approach used in the analysis and quantization of the modes of optical resonators.