Periodic electromagnetic and quantum systems

Abstract

We develop a new analysis of periodic electromagnetic structures which is based on their resonant properties rather than their traveling wave properties. By expanding the fields of Maxwell's equations in a set of short-circuit modes, defined as the resonant modes within a unit cell of the structure bounded by electric shorting planes on the coupling surfaces and excited by tangential electric field on those surfaces, we obtain a determinantal equation for k 2 = ω 2 μ 0 ϵ 0 for a given periodic phase shift φ. Alternately we expand the fields in a set of open-circuit modes, defined as the resonant modes within a unit cell with magnetic shorting planes on the coupling surfaces and excited by tangential magnetic field on those surfaces. We obtain the same determinantal equation for k 2( φ). We prove these theorems for a general symmetric structure: (1) the irrotational modes with zero curl are not necessary for determining the passband curves, (2) the shape of a narrow passband is a simple (1 ± cos φ) curve between the cutoff frequencies, independent of the coupling between cavities, and (3) the sum of all the k 2( φ) associated with the passbands is the sum of all the k 2 lying on the (1 ± cos φ)-curves between the actual cutoff frequencies, independent of the nature of the coupling. The determinantal equation for k 2( φ) can be obtained from variational expressions for k 2, and these lead us by analogy to a variational expression for the energy of a periodic quantum system. We solve the Schrödinger equation for the first energy band of an infinite line of positive ions, in terms of the first “short-circuit” and “open-circuit” eigenfunctions to illustrate the analysis. Then we outline a cellular procedure for solving Schrödinger's equation variationally within a unit cell of a more complicated crystal. The procedure provides for continuity of both ĝY and ∇ĝY over the whole surface of a unit cell and should prove useful for many crystals with unit cells of arbitrary shapes, described by very general Hamiltonians.