162 CHAPTER 4. OPTICAL RECEIVERS
approximated by
Mopt≈[
4 kBTFn
kAqRL(RPin+Id)] 1 / 3
(4.4.23)
forkAin the range 0.01–1. This expression shows the critical role played by the
ionization-coefficient ratiokA. For Si APDs, for whichkA1,Moptcan be as large
as 100. By contrast,Moptis in the neighborhood of 10 for InGaAs receivers, since
kA≈ 0 .7. InGaAs APD receivers are nonetheless useful for optical communication
systems simply because of their higher sensitivity. Receiver sensitivity is an important
issue in the design of lightwave systems and is discussed next.
4.5 Receiver Sensitivity
Among a group of optical receivers, a receiver is said to be more sensitive if it achieves
the same performance with less optical power incident on it. The performance criterion
for digital receivers is governed by thebit-error rate(BER), defined as the probability
of incorrect identification of a bit by the decision circuit of the receiver. Hence, a
BER of 2× 10 −^6 corresponds to on average 2 errors per million bits. A commonly
used criterion for digital optical receivers requires the BER to be below 1× 10 −^9. The
receiver sensitivity is then defined as the minimum average received powerP ̄recrequired
by the receiver to operate at a BER of 10−^9. SinceP ̄recdepends on the BER, let us begin
by calculating the BER.
4.5.1 Bit-Error Rate
Figure 4.18(a) shows schematically the fluctuating signal received by the decision cir-
cuit, which samples it at the decision instanttDdetermined through clock recovery.
The sampled valueIfluctuates from bit to bit around an average valueI 1 orI 0 , depend-
ing on whether the bit corresponds to 1 or 0 in the bit stream. The decision circuit
compares the sampled value with a threshold valueIDand calls it bit 1 ifI>IDor bit
0ifI<ID. An error occurs ifI<IDfor bit 1 because of receiver noise. An error also
occurs ifI>IDfor bit 0. Both sources of errors can be included by defining theerror
probabilityas
BER=p( 1 )P( 0 / 1 )+p( 0 )P( 1 / 0 ), (4.5.1)
wherep( 1 )andp( 0 )are the probabilities of receiving bits 1 and 0, respectively,P( 0 / 1 )
is the probability of deciding 0 when 1 is received, andP( 1 / 0 )is the probability of
deciding 1 when 0 is received. Since 1 and 0 bits are equally likely to occur,p( 1 )=
p( 0 )= 1 /2, and the BER becomes
BER=^12 [P( 0 / 1 )+P( 1 / 0 )]. (4.5.2)Figure 4.18(b) shows howP( 0 / 1 )andP( 1 / 0 )depend on the probability density
functionp(I)of the sampled valueI. The functional form ofp(I)depends on the
statistics of noise sources responsible for current fluctuations. Thermal noiseiTin Eq.
(4.4.6) is well described by Gaussian statistics with zero mean and varianceσT^2 .The