Page 335 - Fiber Optic Communications Fund

316 Fiber Optic Communications

Figure 7.13 A long-haul fiber-optic system consisting of a transmitter, a receiver, N fibers, and N amplifiers.

In this section, we ignore the fiber dispersion and consider only the fiber loss, amplifier gain, and ASE.
Fig. 7.13 shows a fiber-optic system consisting of transmission fibers and in-line amplifiers. Let H and G ,
j j
j = 1, … , N be the fiber loss and amplifier gain of the jth stage, respectively. The power spectral density of
ASE introduced by the jth amplifier per polarization is

ASE,j = n hf(G − 1). (7.102)
sp
j
Let us first consider the signal propagation in the absence of noise. Let P be the mean transmitter output
in
power. In this section, we assume that the transmitter output is CW. Later, in Sections 7.4.3 and 7.4.4, we
consider the OOK/PSK modulation formats. The received power is
N
∏
P = H G P . (7.103)
r j j in
j=1
Next, let us consider the propagation of ASE due to the first amplifier. Let the full bandwidth of the optical
filter be Δf. The noise power per polarization within the filter bandwidth immediately after the first amplifier
is n hf(G − 1)Δf. Therefore, the mean noise power at the receiver due to the first amplifier is
sp
N
∏
P = n hf(G − 1)Δf H G . (7.104)
1,ASE sp 1 j j
j=2
The mean noise power at the receiver due to the nth amplifier is

N
∏
P = n hf(G − 1)Δf H G . (7.105)
n,ASE sp n j j
j=n+1
The total mean noise power at the receiver due to all the amplifiers is
N N N
∑ ∑ ∏
P = P = n hfΔf (G − 1) H G . (7.106)
ASE n,ASE sp n j j
n=1 n=1 j=n+1
When an in-line amplifier fully compensates for the fiber loss, we have G = 1∕H . From now on, we assume
j j
that amplifiers (and fibers) are identical and G = 1∕H . Now, Eq. (7.106) can be simplified as
j j
P ASE = Nn hfΔf(G − 1), (7.107)
sp
where G = G , j = 1, 2, … , N. The cascade of in-line amplifiers and fibers is equivalent to a single amplifier
j
with unity gain and power spectral density of ASE,
eq
= Nn hf(G − 1). (7.108)
ASE sp