Page 77 - Fiber Optic Communications Fund

58 Fiber Optic Communications

If an optical fiber is excited with multiple frequency components, the total field distribution is the superposition
of the fields due to each frequency component,
N
∑ −i n t+i r ( n )z
−z∕2
(x, y, z, t)=Φ(x, y)e A( )e . (2.96)
n
n=1
In Eq. (2.96), we have ignored the frequency dependency of the transverse field distribution Φ and also that
of the loss coefficient . This is valid if the frequency spread Δ = | − | is much smaller than the mean
N 1
frequency of the incident field. If the incident field envelope is a pulse, its frequency components are closely
spaced and we can replace the summation in Eq. (2.96) with an integral
(x, y, z, t)=Φ(x, y)F(t, z) (2.97)
where
e −z∕2 ∞ −i[t− r ()z]
̃
F(t, z)= A()e d, (2.98)
2 ∫ −∞
A( )
n
̃
A()= 2 lim . (2.99)
Δ n →0 Δ
n
From Eq. (2.98), we have
+∞
1 −it
̃
F(t, 0)= A()e d. (2.100)
2 ∫ −∞
̃
̃
Eq. (2.100) represents the inverse Fourier transform of A(). Therefore, the Fourier transform A() of the
incident pulse F(t, 0) is
+∞
it
̃
A()= F(t, 0)e dt. (2.101)
∫
−∞
̃
Thus, for the given incident pulse shape, we can calculate A() using Eq. (2.101) and the optical field distri-
bution at any z can be calculated using Eqs. (2.97) and (2.98). The impact of the fiber is characterized by ().
However, in practice, the dependence of the propagation constant on frequency for the commercially available
fibers is not known. Besides, from the fiber-optic system design point of view, it is desirable to characterize
the fiber using a few parameters. Therefore, we do the following approximation. The propagation constant
at any frequency can be written in terms of the propagation constant and its derivative at some reference
frequency (typically the carrier frequency) using Taylor series,
0
1 2
()= + ( − )+ ( − ) +··· , (2.102)
0
r
1
2
0
0
2
where
= ( ), (2.103)
r
0
0
d | 1
r
= | = , (2.104)
1 |
d |= 0 g
2
d |
= r | . (2.105)
2 2 |
d |
|= 0
is the inverse group velocity and is the second-order dispersion coefficient (see Section 1.10). If the
1 2
signal bandwidth is much smaller than the carrier frequency , we can truncate the Taylor series after the
0