Page 484 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 465

Next consider the correlation between linear and degenerate IFWM,
∑ ∑
<I I IFWM,d > = Re[u ]u lin,n ′ < b b ′ >
lin
n n
lln
l+m−n=0,l=m n ′ ,n ′ ≠0
( )
2 2
√ ∑ −n T s
= P 0 Re[u ] exp 2 . (10.298)
lln
l+m−n=0,l=m,n≠0 T 0
The variance of I IFWM,nd is
∑ ∑
2
2
<I > = 4 Re[u ]
IFWM,nd lmn
′
′
′
l+m−n=0 l +m −n =0
′
′
′ ′
l<m,l≠m≠n l <m ,l ≠m ≠n ′
Re[u ′ m ′ n ′] < b b b b ′b ′b ′ >. (10.299)
l l m n l m n
Since < b b b b ′b ′b ′ >= ′ mm ′ nn ′, Eq. (10.299) is simplified as
l m n l m n ll
∑ 2
2
2
<I >= 4 (Re{u }) . (10.300)
IFWM,nd lmn
l+m−n=0,l<m,l≠m≠n
Using Eqs. (10.289) and (10.290) in Eq. (10.287), it is easy to show that
<I >= 0, (10.301)
2
2 = <I >=<I 2 > + <I 2 > + <I 2 > +2 <I I >
PSK lin IFWM,d IFWM,nd lin IFWM,d
+2 <I I IFWM,nd > +2 <I IFWM,nd I IFWM,d >. (10.302)
lin
It can be shown that the correlation between degenerate IFWM and non-degenerate IFWM is zero and that
between linear ISI and non-degenerate IFWM is also zero. Hence, Eq. (10.302) becomes
2 =<I 2 > + <I 2 > + <I 2 > +2 <I I >. (10.303)
PSK lin IFWM,d IFWM,nd lin IFWM,d
Next, let us consider a direct detection OOK system. The photocurrent is
2
I ∝ P = |u(T, L )| . (10.304)
tot
Setting the constant of proportionality to be unity and using Eq. (10.283),
2
0 tot 1 tot
I(T = 0)= |u (T = 0, L )+ u (T = 0, L )|
2
2
2
∗
= |u (0, L )| + 2Re{u (0, L )u (0, L )} + |u (0, L )| . (10.305)
0 tot 0 tot 1 tot 1 tot
In Eq. (10.305), the first, second, and last terms on the right-hand side represent the currents due to the lin-
ear transmission, signal–nonlinear distortion beating, and nonlinear distortion–nonlinear distortion beating.
When the nonlinear distortion is small, the last term can be ignored. Eq. (10.305) can be written as
I = I + I + I , (10.306)
0 lin nl
where
I = P , (10.307)
0
0