Page 482 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 463

For a single-span system with constant loss and dispersion coefficient , Eq. (10.281) reduces to [38, 43]
2
L 2
2
3∕2 3 exp [−Z −(2CT − d) ∕(4C(1 + i2 CZ)− C)]
u lmn (T, L)= iP 0 T 0 ∫ √ dZ, (10.282)
0 (1 + i2 CZ)(T − i Z)(T + Z )
2
2 2
4
2 0 2 0 2
where L is the fiber length.
10.9.1 Variance Calculations
Without loss of generality, we consider the nonlinear distortion on the pulse located at T = 0. The total field
at the end of the transmission line is

u(T = 0, L )= u (T = 0, L )+ u (T = 0, L )
0
1
tot
tot
tot
√ [ ∞ ( 2 2 )]
P T ∑ n T s
0 0
= b + b exp −
n
0
T (L ) 2T 2
1 tot n=−∞,n≠0 1
∞ ∞ ∞
∑ ∑ ∑
+ b b b u . (10.283)
l m n lmn
l=−∞ m=−∞ n=−∞
The second term on the right-hand side of Eq. (10.283) represents the ISI from the neighboring symbols
and the last term on the right-hand side represents the nonlinear distortion due to SPM, IXPM, and IFWM.
The nonlinear interaction between pulses centered at lT , mT , and nT results in an echo pulse centered
s s s
approximately at (l + m − n)T . Therefore, the dominant contributions to the nonlinear distortion at T = 0
s
come from the symbol slots that satisfy l + m − n = 0 and all the other triplets in Eq. (10.283) can be ignored.
When l = m = n = 0, u corresponds to SPM. When l = 0 and m = n, u corresponds to intrachannel
000 0mm
XPM (IXPM). All the other triplets satisfying l + m − n = 0 represent the echo pulses due to intrachannel
FWM (IFWM). Let us first calculate the variance of ‘1’ in BPSK systems. Let us assume that the bit in the
symbol slot is ‘1’, i.e., b = 1. Considering only the triplets that satisfy l + m − n = 0, Eq. (10.283) can be
0
written as
√ [ ∞ ( )]
2 2
P T ∑ n T s
0 0
u(T = 0, L )= 1 + b exp −
n
tot
T (L ) n=−∞,n≠0 2T 2 1
1
tot
⎡ ⎤
⎢ ⎥
∞
∑ ∑
⎢ ⎥
+ u + 2 u + u b b b . (10.284)
⎢ 000 0mm lmn l m n⎥
⎢ ⏟⏟⏟ m=−∞ l+m−n=0,l≠0,m≠0 ⎥
⎢ SPM ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⎥
⎣ IXPM IFWM ⎦
The last term in Eq. (10.284) excludes SPM and IXPM. As can be seen from Eq. (10.284), the contribution
from SPM and IXPM leads to deterministic amplitude and phase changes. At the receiver, the dispersion
is fully compensated either in the optical domain or using the DSP (see Chapter 11). So, we assume that
T (L )= T in Eq. (10.284). For BPSK systems, the photocurrent is proportional to the real part of u(0, L ).
tot
tot
0
1
Setting the constant proportionality to be unity, the current at T = 0 can be written as
I = I + I, (10.285)
0