Page 552 - Fiber Optic Communications Fund

Appendix B

From Maxwell’s equations, we have
2
2 D
∇ E = . (B.1)
0 2
t
From Eq. (10.24), we have
D = E + P. (B.2)
0
Let us consider the case of a single polarization:
E = E x,
x
P = P x. (B.3)
x
From Eq. (10.42), we have
P (r,t)= P (r,t)+ P (r,t), (B.4)
x L NL
where
(1)
P (r,t)= E (r,t), (B.5)
L
0
x
(3) 3
P (r,t)= E (r,t). (B.6)
NL 0 x
Here, we have ignored the subscripts ’xx’ and ’xxxx’. For a dispersive medium, the first-order susceptibil-
ity (1) is a function of frequency (see Eq. (10.22)). Since the product in the frequency domain becomes a
convolution in the time domain, for a dispersive medium, Eq. (B.5) should be modified as
(1)
P (r,t)= (r,t) ⊗ E (r,t) (B.7)
L
x
or
(1)
̃
̃
P (r,)= ̃ (r,)E (r,), (B.8)
L x
where ⊗ denotes convolution. An optical pulse propagating down the fiber has rapidly varying oscillations at
the carrier frequency and a slowly varying envelope corresponding to the pulse shape. Therefore, the electric
field may be written in the following form:
1
E (r, t)= [E (r, t) exp(−i t)+ c.c.], (B.9)
x 0 0
2
Fiber Optic Communications: Fundamentals and Applications, First Edition. Shiva Kumar and M. Jamal Deen.
© 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.