Page 490 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 471

10.10 Nonlinear Phase Noise
So far we have ignored the nonlinear interaction between the signal and the ASE of inline amplifiers.
Owing to ASE, the amplitude or power of the optical signal fluctuates randomly about a mean value.
Since the nonlinear phase shift due to the SPM is proportional to power, the phase of the signal fluctuates
randomly. This type of noise was first studied by Gordon and Mollenauer [51] and, hence, this noise is also
known as Gordon-Mollenauer phase noise. The nonlinear phase-noise leads to performance degradations
in phase-modulated systems such as DPSK or QPSK systems. The analysis of nonlinear phase noise in
phase-modulated fiber-optic transmission systems has drawn significant attention [51–76]. In the following
section, we consider the impact of ASE when the nonlinear effects are absent and in Section 10.10.2, an
expression for the variance of phase noise including the SPM is derived.

10.10.1 Linear Phase Noise
Consider the output of the optical transmitter, s (T), which is confined to the bit interval −T ∕2 < T < T ∕2.
b
b
in
Let
√
s (T)= a 0 Ep(T), (10.345)
in
where a is the symbol in the interval −T ∕2 < T < T ∕2, p(T) is the pulse shape, E is the energy of the
0 b b
pulse, and
∞
2
|p(T)| dT = 1. (10.346)
∫
−∞
For BPSK, a takes values 1 and −1 with equal probability. In this section, we ignore the fiber dispersion and
0
nonlinearity and include only fiber loss. To compensate for fiber loss, amplifiers are introduced periodically
along the transmission line with a spacing of L . The amplifier compensates for the loss exactly and introduces
a
ASE noise. Let us consider a single-span fiber-optic system with a single amplifier at the fiber output. Let the
amplifier compensate for the fiber loss exactly. The output of the amplifier may be written as
s (T)= s (T)+ n(T), (10.347)
in
out
where n(T) is the ASE noise which can be treated as white,
< n(T) >= 0, (10.348)
∗
′
′
< n(T)n (T ) >= (T − T ), (10.349)
′
< n(T)n(T ) >= 0, (10.350)
where is the ASE power spectral density per polarization given by (Eq. (6.17))
= n hf(G − 1). (10.351)
sp
Here, G is the gain of the amplifier, n is a spontaneous noise factor, h is Planck’s constant, and f is the mean
sp
optical carrier frequency.
A signal of bandwidth B and duration T has 2J = 2BT degrees of freedom (DOF). From the Nyquist
b
b
sampling theorem, it follows that if the highest frequency component of a signal is B∕2, the signal is described
completely by specifying the values of the signal at instants of time separated by 1∕B. Therefore, in the interval
T , there are BT complex samples which fully describe the signal. Equivalently, the signal can be described
b
b