
of a perfectly reflecting hollow cylinde r have the simple relationship between the
frequency, !, and propagation constant, , of the modes:
n
2
!
2
c
2
=
2
þ
j
pq
R
2
; (15.28)
where n is the refractive index of the core material, R is the radius of the core, c
is the speed of light in a vacuum, and j
pq
is the qth zero of the pth-order Bessel
function of the first kind. It is convenient to recast Eqn (15.28) in terms of the
effective index of the mode, n
eff
c/!:
n
2
eff
= n
2
j
pq
c
!R
2
: (15.29)
Interestingly, the effective indices of these modes are less than the index of the
core material, n. For a hollow core, with n = 1, the mode effective index is less
than ...