156 CHAPTER 4. OPTICAL RECEIVERS
4.4.1 Noise Mechanisms
The shot noise and thermal noise are the two fundamental noise mechanisms responsi-
ble for current fluctuations in all optical receivers even when the incident optical power
Pinis constant. Of course, additional noise is generated ifPinis itself fluctuating be-
cause of noise produced by optical amplifiers. This section considers only the noise
generated at the receiver; optical noise is discussed in Section 4.6.2.
Shot Noise
Shot noise is a manifestation of the fact that an electric current consists of a stream
of electrons that are generated at random times. It was first studied by Schottky [83]
in 1918 and has been thoroughly investigated since then [80]–[82]. The photodiode
current generated in response to a constant optical signal can be written as
I(t)=Ip+is(t), (4.4.1)whereIp=RPinis the average current andis(t)is a current fluctuation related to shot
noise. Mathematically,is(t)is a stationary random process withPoisson statistics(ap-
proximated often by Gaussian statistics). The autocorrelation function ofis(t)is related
to the spectral densitySs(f)by theWiener–Khinchin theorem[82]
〈is(t)is(t+τ)〉=∫∞−∞Ss(f)exp( 2 πifτ)df, (4.4.2)where angle brackets denote an ensemble average over fluctuations. The spectral den-
sity of shot noise is constant and is given bySs(f)=qIp(an example ofwhite noise).
Note thatSs(f)is thetwo-sidedspectral density, as negative frequencies are included
in Eq. (4.4.2). If only positive frequencies are considered by changing the lower limit
of integration to zero, theone-sidedspectral density becomes 2qIp.
The noise variance is obtained by settingτ=0 in Eq. (4.4.2), i.e.,
σs^2 =〈i^2 s(t)〉=∫∞−∞Ss(f)df= 2 qIp∆f, (4.4.3)where∆fis theeffective noise bandwidthof the receiver. The actual value of∆f
depends on receiver design. It corresponds to the intrinsic photodetector bandwidth if
fluctuations in the photocurrent are measured. In practice, a decision circuit may use
voltage or some other quantity (e.g., signal integrated over the bit slot). One then has
to consider the transfer functions of other receiver components such as the preamplifier
and the low-pass filter. It is common to consider current fluctuations and include the
total transfer functionHT(f)by modifying Eq. (4.4.3) as
σs^2 = 2 qIp∫∞0|HT(f)|^2 df= 2 qIp∆f, (4.4.4)where∆f=
∫∞
0 |HT(f)|(^2) dfandHT(f)is given by Eq. (4.3.7). Since the dark current
Idalso generates shot noise, its contribution is included in Eq. (4.4.4) by replacingIp
byIp+Id. The total shot noise is then given by
σs^2 = 2 q(Ip+Id)∆f. (4.4.5)