Page 501 - Fiber Optic Communications Fund

482 Fiber Optic Communications

The Taylor series expansion of P(T − s, Z) around T is
2
2
P s P
P(T − s, Z)= P(T, Z)− s + +… (10.428)
T 2 T 2
2
If the spectral width of the signal is sufficiently small, terms that are proportional to s and beyond may be
ignored. Under this condition, Eq. (10.424) becomes
∞ [ 2 ]
2
2
2
|q(T, Z)| → (1 − )|q(T, Z)| + |q(T, Z)| − s |q(T, Z)| h(s)ds. (10.429)
∫
−∞ T
Using Eq. (10.425), Eq. (10.429) is rewritten as
2
2 2 |q(T, Z)|
|q(T, Z)| → |q(T, Z)| − , (10.430)
T
where
∞
= sh(s)ds. (10.431)
∫
−∞
Substituting Eq. (10.430) in Eq. (10.81), we obtain
2
q q 2 q |q(T, Z)| 2
2
i − + |q| q + i = q. (10.432)
Z 2 T 2 2 T
Eq. (10.432) is the modified nonlinear Schrodinger equation and the term on the right-hand side denotes
the Raman contributions. The energy exchange between the pump and the Stokes waves can be understood
from Eq. (10.432) by considering the pump and Stokes waves as CW:
q = q + q , (10.433)
p
s
q = A e −iΩ p T , (10.434)
p
p
q = A e −iΩ s T , (10.435)
s
s
where A and A denote the complex amplitudes of pump and Stokes’s waves, respectively, and Ω and Ω
p s p s
are the corresponding angular frequency offset from the reference. Let us first consider
∗
2
∗ −iΩT
2
2
|q| = |A | + |A | + A A e + A A e iΩT , (10.436)
p s p s p s
2
|q| 2 −iΩ p T 2 −iΩ s T
q =(−iΩ)A |A | e +(iΩ)A |A | e + terms at 2Ω −Ω s and 2Ω −Ω , (10.437)
s
p
p
p
s
p
s
T
where
Ω=Ω −Ω . (10.438)
p s
Substituting Eq. (10.437) in Eq. (10.432) and collecting the terms that are proportional to e −iΩ p T and e −iΩ s T ,
we find
dA p 2
2
2
2
2
i + Ω A + {|A | + 2|A | }A =−iΩ|A | A − A , (10.439)
p
p
p
s
p
s
p p
dZ 2
dA s 2
2
2
2
2
i + Ω A + {|A | + 2|A | }A = iΩ|A | A − A . (10.440)
s
s
s
s
s
p
s
p
dZ 2