2.2. WAVE PROPAGATION 31
2.2.2 Fiber Modes
The concept of the mode is a general concept in optics occurring also, for example, in
the theory of lasers. Anoptical moderefers to a specific solution of the wave equation
(2.2.16) that satisfies the appropriate boundary conditions and has the property that its
spatial distribution does not change with propagation. The fiber modes can be classified
as guided modes, leaky modes, and radiation modes [14]. As one might expect, sig-
nal transmission in fiber-optic communication systems takes place through the guided
modes only. The following discussion focuses exclusively on the guided modes of a
step-index fiber.
To take advantage of the cylindrical symmetry, Eq. (2.2.16) is written in the cylin-
drical coordinatesρ,φ, andzas
∂^2 Ez
∂ρ^2+
1
ρ∂Ez
∂ρ+
1
ρ^2∂^2 Ez
∂φ^2+
∂^2 Ez
∂z^2+n^2 k^20 Ez= 0 , (2.2.18)where for a step-index fiber of core radiusa, the refractive indexnis of the form
n={
n 1 ; ρ≤a,
n 2 ; ρ>a.(2.2.19)
For simplicity of notation, the tilde overE ̃has been dropped and the frequency de-
pendence of all variables is implicitly understood. Equation (2.2.18) is written for the
axial componentEzof the electric field vector. Similar equations can be written for the
other five components ofEandH. However, it is not necessary to solve all six equa-
tions since only two components out of six are independent. It is customary to choose
EzandHzas the independent components and obtainEρ,Eφ,Hρ, andHφin terms of
them. Equation (2.2.18) is easily solved by using the method of separation of variables
and writingEzas
Ez(ρ,φ,z)=F(ρ)Φ(φ)Z(z). (2.2.20)
By using Eq. (2.2.20) in Eq. (2.2.18), we obtain the three ordinary differential equa-
tions:
d^2 Z/dz^2 +β^2 Z= 0 , (2.2.21)
d^2 Φ/dφ^2 +m^2 Φ= 0 , (2.2.22)
d^2 F
dρ^2+
1
ρdF
dρ+
(
n^2 k^20 −β^2 −
m^2
ρ^2)
F= 0. (2.2.23)
Equation (2.2.21) has a solution of the formZ=exp(iβz), whereβhas the physical
significance of the propagation constant. Similarly, Eq. (2.2.22) has a solutionΦ=
exp(imφ), but the constantmis restricted to take only integer values since the field
must be periodic inφwith a period of 2π.
Equation (2.2.23) is the well-known differential equation satisfied by the Bessel
functions [32]. Its general solution in the core and cladding regions can be written as
F(ρ)={
AJm(pρ)+A′Ym(pρ); ρ≤a,
CKm(qρ)+C′Im(qρ); ρ>a,