Page 280 - Fiber Optic Communications Fund

Optical Amplifiers 261

and the corresponding photocurrent and electrical signal power are
I = RGP , (6.97)
out in
2
S =(RGP ) R , (6.98)
out in L
respectively. The noise power delivered to a resistor R consists mainly of two components. They are due
L
to shot noise and signal–ASE beating noise. In this analysis, we ignore the ASE–ASE beating noise and
assume that the optical filter is absent. The total noise power can be obtained by adding the shot noise given
by Eq. (6.94) and the signal–ASE beating noise power given by Eq. (6.60),
N = N + N
out shot s−sp
2
= 2qRP R B + 4R P B R . (6.99)
out L e ASE out e L
Note that the power spectral density of the ASE noise in Eq. (6.99) is in single polarization. Although the
amplifier adds noise in both polarizations, the noise in the polarization orthogonal to the signal polarization
does not interfere with the signal to generate signal–ASE beat noise. The SNR at the output of PD can be
2
written as
2
S out (RGP ) R L
in
SNR out = =
N out [q + 2R ASE ]2RGP B R
in e L
RGP in
= . (6.100)
(q + 2R )2B
ASE e
Substituting Eqs. (6.95) and (6.100) in Eq. (6.90), we find
RP (q + 2R ASE )2B e
in
F =
n
2qB e RGP in
q + 2R ASE
= . (6.101)
Gq
Using Eq. (6.92), Eq. (6.101) can be written as
=(GF − 1)hf ∕2. (6.102)
ASE n 0
The power spectral density can also be expressed in terms of n (see Eq. (6.17)),
ASE sp
ASE = n (G − 1)hf . (6.103)
0
sp
Equating the right-hand sides of Eqs. (6.102) and (6.103), we find an expression that relates the amplifier
noise figure and spontaneous emission factor n ,
sp
2n (G − 1) 1
sp
F = + . (6.104)
n
G G
When G ≫ 1,
F ≅ 2n . (6.105)
n sp
Since the minimum value of n is 1, the lowest achievable noise figure is 2. The noise figure is expressed in
sp
dB units as
F (dB)= 10 log 10 n (6.106)
F .
n
When n = 1, F = 3 dB, which corresponds to an ideal amplifier with the lowest ASE noise.
n
sp