Angular Momentum in Circular Waveguides

1 Problem In view of recent interest in orbital angular momentum modes in optical fibers, 1 discuss angular momentum in the modes of a circular waveguide of radius a, with perfectly conducting walls and filled with a medium of index of refraction n. 2 Solution Circular-waveguide modes were first discussed in 1893 by J.J. Thomson in sec. 300 of [2] and then by Rayleigh in 1897 [3]. 2 Although these modes are " well known, " discussion of them seldom includes mention of angular momentum. We first recall the general formalism appropriate for discussion of waves in pipes along the z-axis. 3 The dependence on z and t of all field components is taken to have the form e i(kzz−ωt). Inside the waveguide these field components ψ obey the Helmholtz wave equation 0 = ∇ 2 + n 2 index ω 2 c 2 ψ = ∇ 2 ⊥ + ∂ 2 ∂z 2 + k 2 0 ψ = ∇ 2 ⊥ − k 2 z + k 2 0 ψ = ∇ 2 ⊥ + k 2 ψ, (1) where c is the speed of light in vacuum, n index = √ relative μ relative is the index of refraction at angular frequency ω of the (linear, isotropic) medium inside the guide, and k 0 = n index ω c = √ μ ω , k 2 = k 2 0 − k 2 z. (2) Note that k z and not k is propagation constant. The use of the decomposition ∇ 2 = ∇ 2 ⊥ + ∂ 2 /∂z 2 in the wave equation (1) suggests a use of a similar decomposition of the gradient operator, ∇ = ∇ ⊥ + ∇ z , such that two of Maxwell's equations can be written