166 CHAPTER 4. OPTICAL RECEIVERS
Equation (4.5.17) shows howP ̄recdepends on various receiver parameters and how
it can be optimized. Consider first the case of ap–i–nreceiver by settingM=1. Since
thermal noiseσTgenerally dominates for such a receiver,P ̄recis given by the simple
expression
(P ̄rec)pin≈QσT/R. (4.5.18)
From Eq. (4.5.15),σT^2 depends not only on receiver parameters such asRLandFnbut
also on the bit rate through the receiver bandwidth∆f(typically,∆f=B/2). Thus,
P ̄recincreases as
√
Bin the thermal-noise limit. As an example, consider a 1.55-μm
p–i–nreceiver withR=1A/W.IfweuseσT=100 nA as a typical value andQ= 6
corresponding to a BER of 10−^9 , the receiver sensitivity is given byP ̄rec= 0. 6 μWor
− 32 .2 dBm.
Equation (4.5.17) shows how receiver sensitivity improves with the use of APD
receivers. If thermal noise remains dominant,P ̄recis reduced by a factor ofM, and
the received sensitivity is improved by the same factor. However, shot noise increases
considerably for APD, and Eq. (4.5.17) should be used in the general case in which
shot-noise and thermal-noise contributions are comparable. Similar to the case of SNR
discussed in Section 4.4.3, the receiver sensitivity can be optimized by adjusting the
APD gainM. By usingFAfrom Eq. (4.4.18) in Eq. (4.5.17), it is easy to verify thatP ̄rec
is minimum for an optimum value ofMgiven by [3]
Mopt=k
− 1 / 2
A(
σT
Qq∆f
+kA− 1) 1 / 2
≈
(
σT
kAQq∆f) 1 / 2
, (4.5.19)
and the minimum value is given by
(P ̄rec)APD=( 2 q∆f/R)Q^2 (kAMopt+ 1 −kA). (4.5.20)The improvement in receiver sensitivity obtained by the use of an APD can be esti-
mated by comparing Eqs. (4.5.18) and (4.5.20). It depends on the ionization-coefficient
ratiokAand is larger for APDs with a smaller value ofkA. For InGaAs APD receivers,
the sensitivity is typically improved by 6–8 dB; such an improvement is sometimes
called the APD advantage. Note thatP ̄recfor APD receivers increases linearly with the
bit rateB(∆f≈B/2), in contrast with its
√
Bdependence forp–i–nreceivers. The
linear dependence ofP ̄reconBis a general feature of shot-noise-limited receivers. For
an ideal receiver for whichσT=0, the receiver sensitivity is obtained by settingM= 1
in Eq. (4.5.17) and is given by
(P ̄rec)ideal=(q∆f/R)Q^2. (4.5.21)A comparison of Eqs. (4.5.20) and (4.5.21) shows sensitivity degradation caused by
the excess-noise factor in APD receivers.
Alternative measures of receiver sensitivity are sometimes used. For example, the
BER can be related to the SNR and to the average number of photonsNpcontained
within the “1” bit. In the thermal-noise limitσ 0 ≈σ 1. By usingI 0 =0, Eq. (4.5.11)
providesQ=I 1 / 2 σ 1 .AsSNR=I^21 /σ 12 , it is related toQby the simple relation SNR=
4 Q^2. SinceQ=6 for a BER of 10−^9 , the SNR must be at least 144 or 21.6 dB for
achieving BER≤ 10 −^9. The required value of SNR changes in the shot-noise limit. In