4.4. RECEIVER NOISE 159
Shot-Noise Limit
Consider the opposite limit in which the receiver performance is dominated by shot
noise (σs^2 σT^2 ). Sinceσs^2 increases linearly withPin, the shot-noise limit can be
achieved by making the incident power large. The dark currentIdcan be neglected in
that situation. Equation (4.4.12) then provides the following expression for SNR:
SNR=
RPin
2 q∆f=
ηPin
2 hν∆f. (4.4.15)
The SNR increases linearly withPinin the shot-noise limit and depends only on the
quantum efficiencyη, the bandwidth∆f, and the photon energyhν. It can be writ-
ten in terms of the number of photonsNpcontained in the “1” bit. If we useEp=
Pin
∫∞
−∞hp(t)dt=Pin/Bfor the pulse energy of a bit of duration 1/B, whereBis the
bit rate, and note thatEp=Nphν, we can writePinasPin=NphνB. By choosing
∆f=B/2 (a typical value for the bandwidth), the SNR is simply given byηNp.In
the shot-noise limit, a SNR of 20 dB can be realized ifNp≈100. By contrast, several
thousand photons are required to obtain SNR=20 dB when thermal noise dominates
the receiver. As a reference, for a 1.55-μm receiver operating at 10 Gb/s,Np= 100
whenPin≈130 nW.
4.4.3 APD Receivers
Optical receivers that employ an APD generally provide a higher SNR for the same
incident optical power. The improvement is due to the internal gain that increases the
photocurrent by a multiplication factorMso that
Ip=MRPin=RAPDPin, (4.4.16)whereRAPDis the APD responsivity, enhanced by a factor ofMcompared with that of
p–i–nphotodiodes (RAPD=MR). The SNR should improve by a factor ofM^2 if the
receiver noise were unaffected by the internal gain mechanism of APDs. Unfortunately,
this is not the case, and the SNR improvement is considerably reduced.
Shot-Noise Enhancement
Thermal noise remains the same for APD receivers, as it originates in the electrical
components that are not part of the APD. This is not the case for shot noise. The APD
gain results from generation of secondary electron–hole pairs through the process of
impact ionization. Since such pairs are generated at random times, an additional con-
tribution is added to the shot noise associated with the generation of primary electron–
hole pairs. In effect, the multiplication factor itself is a random variable, andMappear-
ing in Eq. (4.4.16) represents the average APD gain. Total shot noise can be calculated
by using Eqs. (4.2.3) and (4.2.4) and treatingieandihas random variables [86]. The
result is
σs^2 = 2 qM^2 FA(RPin+Id)∆f. (4.4.17)
whereFAis theexcess noise factorof the APD and is given by [86]
FA(M)=kAM+( 1 −kA)( 2 − 1 /M). (4.4.18)