32 CHAPTER 2. OPTICAL FIBERS
whereA,A′,C, andC′are constants andJm,Ym,Km, andImare different kinds of Bessel
functions [32]. The parameterspandqare defined by
p^2 =n^21 k^20 −β^2 , (2.2.25)
q^2 =β^2 −n^22 k^20. (2.2.26)Considerable simplification occurs when we use the boundary condition that the optical
field for a guided mode should be finite atρ=0 and decay to zero atρ=∞. Since
Ym(pρ)has a singularity atρ=0,F( 0 )can remain finite only ifA′=0. Similarly
F(ρ)vanishes at infinity only ifC′=0. The general solution of Eq. (2.2.18) is thus of
the form
Ez={
AJm(pρ)exp(imφ)exp(iβz); ρ≤a,
CKm(qρ)exp(imφ)exp(iβz); ρ>a.(2.2.27)
The same method can be used to obtainHzwhich also satisfies Eq. (2.2.18). Indeed,
the solution is the same but with different constantsBandD, that is,
Hz={
BJm(pρ)exp(imφ)exp(iβz); ρ≤a,
DKm(qρ)exp(imφ)exp(iβz); ρ>a.(2.2.28)
The other four componentsEρ,Eφ,Hρ, andHφcan be expressed in terms ofEzandHz
by using Maxwell’s equations. In the core region, we obtain
Eρ=
i
p^2(
β
∂Ez
∂ρ+μ 0
ω
ρ∂Hz
∂φ)
, (2.2.29)
Eφ=i
p^2(
β
ρ∂Ez
∂φ−μ 0 ω∂Hz
∂ρ)
, (2.2.30)
Hρ=i
p^2(
β∂Hz
∂ρ−ε 0 n^2ω
ρ∂Ez
∂φ)
, (2.2.31)
Hφ=i
p^2(
β
ρ∂Hz
∂φ+ε 0 n^2 ω∂Ez
∂ρ)
. (2.2.32)
These equations can be used in the cladding region after replacingp^2 by−q^2.
Equations (2.2.27)–(2.2.32) express the electromagnetic field in the core and clad-
ding regions of an optical fiber in terms of four constantsA,B,C, andD. These
constants are determined by applying the boundary condition that the tangential com-
ponents ofEandHbe continuous across the core–cladding interface. By requiring
the continuity ofEz,Hz,Eφ, andHφatρ=a, we obtain a set of four homogeneous
equations satisfied byA,B,C, andD[19]. These equations have a nontrivial solution
only if the determinant of the coefficient matrix vanishes. After considerable algebraic
details, this condition leads us to the following eigenvalue equation [19]–[21]:
[
J′m(pa)
pJm(pa)+
Km′(qa)
qKm(qa)][
Jm′(pa)
pJm(pa)+
n^22
n^21Km′(qa)
qKm(qa)]
=
m^2
a^2(
1
p^2+
1
q^2)(
1
p^2+
n^22
n^211
q^2