Page 63 - Fiber Optic Communications Fund

44 Fiber Optic Communications

Solution:
n − n 2
1
Δ= = 0.0136. (2.25)
n 1
For a step-index fiber, from Eq. (2.23), we find

2
n LΔ
ΔT = 1 = 67.58 ns. (2.26)
cn
2
For a parabolic-index fiber, from Eq. (2.24), we find
2
2
n Δ L
1
ΔT = = 0.1133 ns. (2.27)
8c
Thus, we see that the intermodal dispersion can be significantly reduced by using a parabolic-index fiber.

2.4 Modes of a Step-Index Optical Fiber ∗
To understand the electromagnetic field propagation in optical fibers, we should solve Maxwell’s equations
with the condition that the tangential components of electric and magnetic fields should be continuous at the
interface between core and cladding [6, 7]. When the refractive index difference between core and cladding
is small, a weakly guiding or scalar wave approximation can be made [8–11] and in this approximation, the
electromagnetic field is assumed to be nearly transverse as in the case of free-space propagation. Under this
approximation, the one set of modes consists of E and H components (x-polarized) and the other set of
y
x
modes consists of E and H components (y-polarized). These two sets of modes are independent and known
x
y
as linearly polarized (LP) modes.The x-or y- component of the electric field intensity satisfies the scalar
wave equation Eq. (1.125),
2
1
2
∇ − = 0, (2.28)
2
(r) t 2
where (r) is the speed of light given by
c
(r)= (2.29)
n(r)
with {
n for r < a
1
n(r)= , (2.30)
n for r ≥ a
2
where a = core radius. We assume that n > n , as shown in Fig. 2.13. In cylindrical coordinates, the Laplacian
1 2
2
operator ∇ can be written as
2
2
2
1 1
2
∇ = + + + . (2.31)
r 2 r r r 2 z 2
2
Suppose this fiber is excited with a laser oscillating at angular frequency . In a linear dielectric medium,
the frequency of the electromagnetic field should be the same as that of the source. Therefore, we look for a
solution of Eq. (2.28) in the form
Ψ(r,, z, t)= f(r,, z)e −it . (2.32)