Page 553 - Fiber Optic Communications Fund

534 Fiber Optic Communications

where E (r, t) is the slowly varying function of time and c.c stands for complex conjugate. Substituting
0
Eq. (B.9) in Eq. (B.6), we find
(3)
0
3
2
P (r, t)= [3|E(r, t)| E(r, t) exp(−i t)+ E (r, t) exp(−i3 t)] + c.c. (B.10)
0
0
NL
8
The first term in the square bracket corresponds to oscillations at and the second term corresponds to third
0
harmonic frequency 3 . The efficiency of third harmonic generation in fibers is very small unless special
0
phase-matching techniques are used. Therefore, ignoring the second term and substituting Eqs. (B.7)–(B.10)
in Eq. (B.1), we obtain
2
1 Ψ(r, t) 1 2 (1) 3 (3) 2 2
2
∇ Ψ(r, t)− = [ (r,t) ⊗ Ψ(r, t)] + [|Ψ(r, t)| Ψ(r, t)], (B.11)
2
c 2 t 2 c t 2 4c 2 t 2
where
Ψ(r, t)= E (r, t) exp(−i t),
0
0
1
2
c = .

0 0
The electric field intensity in a single-mode fiber may be written as (see Chapter 2)
Ψ(r, t)= q(z, t)(x, y)e −i( 0 t− 0 z) , (B.12)
where = ( ) is the propagation constant, (x, y) is the transverse field distribution, and q(z, t) is the
0
0
0
field envelope which is a slowly varying function of t and z. Substituting Eq. (B.12) in Eq. (B.11) and taking
the Fourier transform, we obtain
[ 2 ]
̃q(z, Ω) ̃q(z, Ω) 2
+ 2i − ̃q(z, Ω)
0
0
z 2 z
{ 2 2 2 }
[ (1) ]
+ + + 1 + ̃ (r,) ̃ q(z, Ω)
x 2 y 2 c 2
2
3 (3) 3 ∗
=− { (x, y)[̃q(z, Ω) ⊗̃q (z, −Ω) ⊗̃q(z, Ω)]}, (B.13)
4c 2
where Ω= − . To obtain Eq. (B.13), we have used the Fourier transform relations
0
( 2 )
A
 =− A() (B.14)
2 ̃
t 2
and
̃
̃
[A(t)B(t)] = A() ⊗ B()
( ′ ) ( ′ )
̃
= A B ̃ − d . (B.15)
∫ 2 2 2
Under the slowly varying envelope approximation, the first term in Eq. (B.13) can be ignored, which is a good
approximation for pulse widths that are much longer than the period 2∕ . From Eq. (10.29), we have
0
(1)
2
1 + ̃ (r,)= n (r,), (B.16)