Page 495 - Fiber Optic Communications Fund

476 Fiber Optic Communications

The total phase given by Eq. (10.392) can be separated into two parts:

= + , (10.393)
d
where is the deterministic nonlinear phase shift given by
d
= EN L ∕T (10.394)
d a eff b
and represents the phase noise,

√
′
n ′ 0i 2 En (N − m)L eff
a
0r
= √ + . (10.395)
E T b
The first and second terms in Eq. (10.395) represent the linear and nonlinear phase noise, respectively. As
′
′
can be seen, the in-phase component n and the quadrature component, n are responsible for nonlinear and
0r 0i
linear phase noise, respectively. From Eq. (10.388), it follows that
<>= 0. (10.396)
Squaring and averaging Eq. (10.395) and using Eqs. (10.389) and (10.390), we find the variance of the phase
noise as
[ ] 2
(N − m)L eff
a
2
= + 2E . (10.397)
m
2E T b
So far, we have ignored the impact of ASE due to other amplifiers. In the presence of ASE due to other
amplifiers, the expression for the optical field envelope at mL − given by Eq. (10.384) is inaccurate since it
a
ignores the noise field added by the amplifiers preceding the mth amplifier. However, when the signal power
2
2
is much larger than the noise power, second-order terms such as n and n can be ignored. At the end of the
0r 0i
transmission line, the dominant contribution would come from the linear terms n and n of each amplifier.
0i 0i
Since the noise fields of amplifiers are statistically independent, the total variance is the sum of the variance
due to each amplifier,
N a
∑ 2
2
= m
m=1
2
[ ] N a −1
N a L eff ∑ 2
= + 2E (N − m)
a
2E T
b m=1
2 2
N a (N − 1)N (2N − 1)E L eff
a
a
a
= + . (10.398)
2E 3T 2
b
Refs. [55–58] provide a more rigorous treatment of the nonlinear phase noise without ignoring the
higher-order noise terms. From Eq. (10.398), we see that the variance of the linear phase noise (the first
term on the right-hand side) increases linearly with the number of amplifiers, whereas the variance of the
nonlinear phase noise (the second term) increases cubically with the number of amplifiers when N is
a
large, indicating that nonlinear phase noise could be the dominant penalty for ultra-long-haul fiber-optic
transmission systems. In addition, the variance of linear phase noise is inversely proportional to the energy of
the pulse, whereas the variance of nonlinear phase noise is directly proportional to the energy. This implies