A hollow metallic waveguide with a distortion in the form of a localized bend or increase in cross section can support nonpropagating (``bound state'') configurations of fields in the vicinity of the distortion. Consider a rectangular guide that has its distortion confined to a plane, as shown in the figure, and TE10 as its lowest propagating mode, with perpendicular electric field E⊥=ψ. On either side of the distortion the guide is straight and of width a. Without distortion, ψ=E0sin(πy/a)exp(±ikz), where k2=(ω/c)2−(π/a)2. The distortion is described by a curvature κ(s)=1/R(s) and a width w(s). Locally the element of area in the plane is dA=h(s,t)dsdt, where s is the length along the guide wall and t the transverse coordinate, as shown in the figure, and h(s,t)=1−κ(s)t. In terms of s and t the Laplacian is
∇2ψ=h1∂t∂(h∂t∂ψ)+h1∂s∂(h1∂s∂ψ)
If the distortions are very small and change slowly in s on the scale of the width a, an ansatz for the solution is
ψ(s,t)=h(s,t)u(s)⋅sin[w(s)πt]
(a) Show that substitution of the ansatz into the two-dimensional wave equation, (∇2+ω2/c2)ψ=0, leads to the equation for u(s),
ds2d2u+[k2−v(s)]u=0with v(s)=π2(w2(s)1−a21)−41κ2(s)
if small terms are neglected. Interpret v(s) in analogy with the Schr\"odinger equation in one dimension.
(b) If the distortion is in the form of a bend through an angle θ with constant radius of curvature R≫a, show that for θa/R≪1 there is a ``bound state'' at frequency ω0 where
ω02≈(aπc)2[1−(8πRθa)2]
References: J. Goldstone and R. L. Jaffe, Phys. Rev. B 45, 14100 (1992); J. P. Carini, J. T. Londergan, K. Mullen, and D. P. Murdock, Phys. Rev. B 48, 4503 (1993).