To treat perturbations if there is a degeneracy of modes in guides or cavities under ideal conditions one must use degenerate-state perturbation theory. Consider the two-dimensional (waveguide) situation in which there is an N-fold degeneracy in the ideal circumstances (of perfect conductivity or chosen shape of cross section), with no other nearby modes. There are N linearly independent solutions ?₀⁽ⁱ⁾ chosen to be orthogonal, to the transverse wave equation, (?²_t + ?²₀)?₀^{(i )?}= 0, i = 1, 2,... N. In response to the perturbation, the degeneracy is in general lifted. There is a set of perturbed eigenvalues, ?²_k, with associated eigenmodes, ?_?, which can be expanded (in lowest order) in terms of the N unperturbed eigenmodes: ?_? = ?_ia_i?⁽ⁱ⁾₀.

(a) Show that the generalization of (8.68) for finite conductivity (and the corresponding expression in Problem 8.12 on distortion of the shape of a wave-guide) is the set of algebraic equations,

Where

For finite conductivity, and

for distortion of the boundary shape.

(b) The lowest mode in a circular guide of radius R is the twofold degenerate TE₁₁ mode, with fields given by

?^(?) = B_z = B₀J₁ (?₀p) ??? (? i?) exp(ikz ? iwt)

The eigenvalue parameter is ?₀ = 1.841/R,, corresponding to the first root of dJ₁(x) dx. Suppose that the circular waveguide is distorted along its length into an elliptical shape with semimajor and semiminor axes, a = R + ?R, b = R ? ?R, respectively. To first order in ?R/R, the area and circumference of the guide remain unchanged. Show that the degeneracy is lifted by the distortion and that to first order in ?R/R, ?²_1?=_??²₀ (1 = ??R/R) and ?²₂ = ?²_0?(1 - ??R/R). Determine the numerical value of ? and find the eigenmodes as linear combinations of ?^(?) Explain physically why the eigenmodes turn out as they do.

Transcribed Image Text:

E [( - v)N,ô, + A,la, = 0 (j = 1, 2, ..., N) %3D