158 CHAPTER 4. OPTICAL RECEIVERS
with approximately Gaussian statistics, the total variance of current fluctuations,∆I=
I−Ip=is+iT, can be obtained simply by adding individual variances. The result is
σ^2 =〈(∆I)^2 〉=σs^2 +σT^2 = 2 q(Ip+Id)∆f+( 4 kBT/RL)Fn∆f. (4.4.10)Equation (4.4.10) can be used to calculate the SNR of the photocurrent.
4.4.2 p–i–nReceivers.........................
The performance of an optical receiver depends on the SNR. The SNR of a receiver
with ap–i–nphotodiode is considered here; APD receivers are discussed in the follow-
ing subsection. The SNR of any electrical signal is defined as
SNR=
average signal power
noise power=
I^2 p
σ^2, (4.4.11)
where we used the fact that electrical power varies as the square of the current. By
using Eq. (4.4.10) in Eq. (4.4.11) together withIp=RPin, the SNR is related to the
incident optical power as
SNR=
R^2 Pin^2
2 q(RPin+Id)∆f+ 4 (kBT/RL)Fn∆f, (4.4.12)
whereR=ηq/hνis the responsivity of thep–i–nphotodiode.
Thermal-Noise Limit
In most cases of practical interest, thermal noise dominates receiver performance (σT^2
σs^2 ). Neglecting the shot-noise term in Eq. (4.4.12), the SNR becomes
SNR=
RLR^2 Pin^2
4 kBTFn∆f. (4.4.13)
Thus, the SNR varies asPin^2 in the thermal-noise limit. It can also be improved by in-
creasing the load resistance. As discussed in Section 4.3.1, this is the reason why most
receivers use a high-impedance or transimpedance front end. The effect of thermal
noise is often quantified through a quantity called thenoise-equivalent power(NEP).
The NEP is defined as the minimum optical power per unit bandwidth required to pro-
duce SNR=1 and is given by
NEP=
Pin
√
∆f=
(
4 kBTFn
RLR^2) 1 / 2
=
hν
ηq(
4 kBTFn
RL) 1 / 2
. (4.4.14)
Another quantity, calleddetectivityand defined as (NEP)−^1 , is also used for this pur-
pose. The advantage of specifying NEP or the detectivity for ap–i–nreceiver is that it
can be used to estimate the optical power needed to obtain a specific value of SNR if
the bandwidth∆fis known. Typical values of NEP are in the range 1–10 pW/Hz^1 /^2.