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 I^0(i) chosen to be orthogonal, to the transverse wave equation, (?^?2t + ?320)I^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, ?32k, with associated eigenmodes, I^?o, which can be expanded (in lowest order) in terms of the N unperturbed eigenmodes: I^?o = ?LiaiI^(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
?L(±) = Bz = B0J1 (?30p) ??N?N? (± iN?) exp(ikz – iwt)
The eigenvalue parameter is ?30 = 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, ?321 = ?320 (1 = ??^?R/R) and ?322 = ?320 (1 – ??^?R/R). Determine the numerical value of ?? and find the eigenmodes as linear combinations of I^(±) Explain physically why the eigenmodes turn out as theydo.