262 CHAPTER 6. OPTICAL AMPLIFIERS
the component of ASE that is copolarized with the signal as the orthogonally polarized
component cannot beat with the signal.
The current noise∆Iconsists of fluctuations originating from the shot noise, ther-
mal noise, and ASE noise. The ASE-induced current noise has its origin in the beating
ofEswithEspand the beating ofEspwith itself. To understand this beating phe-
nomenon more clearly, notice that the ASE fieldEspis broadband and can be written
in the form
Esp=
∫√
Sspexp(φn−iωnt)dωn, (6.5.4)whereφnis the phase of the noise-spectral component at the frequencyωn, and the
integral extends over the entire bandwidth of the amplifier (or optical filter). Using
Es=
√
Psexp(φs−iωst), the interference term in Eq. (6.5.1) consists of two parts and
leads to current fluctuations of the form
isig−sp= 2 R∫
(GPsSsp)^1 /^2 cosθ 1 dωn, isp−sp= 2 R∫∫
Sspcosθ 2 dωndωn′, (6.5.5)whereθ 1 =(ωs−ωn)t+φn−φsandθ 2 =(ωn−ωn′)t+φn′−φnare two rapidly varying
random phases. These two contributions to current noise are due to the beating ofEs
withEspand the beating ofEspwith itself, respectively. Averaging over the random
phases, the total varianceσ^2 =〈(∆I)^2 〉of current fluctuations can be written as [5]
σ^2 =σT^2 +σs^2 +σsig^2 −sp+σsp^2 −sp, (6.5.6)whereσT^2 is the thermal noise and the remaining three terms are [113]
σs^2 = 2 q[R(GPs+Psp)]∆f, (6.5.7)
σsig^2 −sp= 4 R^2 GPsSsp∆f, (6.5.8)
σsp^2 −sp= 4 R^2 S^2 sp∆νopt∆f, (6.5.9)where∆νoptis the bandwidth of the optical filter and∆fis the electrical noise band-
width of the receiver. The shot-noise termσs^2 is the same as in Section 4.4.1 except
thatPsphas been added toGPsto account for the shot noise generated by spontaneous
emission.
The BER can be obtained by following the analysis of Section 4.5.1. As before, it
is given by
BER=^12 erfc(Q/
√
2 ), (6.5.10)
with theQparameter
Q=I 1 −I 0
σ 1 +σ 0=
RG( 2 P ̄rec)
σ 1 +σ 0. (6.5.11)
Equation (6.5.11) is obtained by assuming zero extinction ratio (I 0 =0) so thatI 1 =
RGP 1 =RG( 2 P ̄rec), whereP ̄recis the receiver sensitivity for a given value of BER
(Q=6 for BER= 10 −^9 ). The RMS noise currentsσ 1 andσ 0 are obtained from Eqs.
(6.5.6)–(6.5.9) by settingPs=P 1 = 2 P ̄recandPs=0, respectively.
The analysis can be simplified considerably by comparing the magnitude of various
terms in Eqs. (6.5.6). For this purpose it is useful to substituteSspfrom Eq. (6.1.15),