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 (see Section 7.3 of LT1) that increases the photocurrent by a multiplication factor M so that

where /?apd = MRj is the APD responsivity, enhanced by a factor of M compared with that of p-i-n photodiodes. The SNR would improve by a factor of M2 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 less than M2.

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. As discussed in Section 7.3 of LT1, APD gain results from the generation of secondary electron-hole pairs through the process of impact ionization. Since such pairs are generated at random times, an additional contribution 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, and M appearing in Eq. (5.2.7) represents the average APD gain. The shot noise for APDs can be written in the form [15]

where FA is called the excess noise factor and is given by

The dimensionless parameter kA = Gth/ae if ah < ae but is defined as kA = ae/ai, when a^ > ae, where ae and ah represent the impact ionization coefficients for electrons and holes, respectively. By definition, 0 < kA < 1. In general, FA increases with M monotonically. However, although FA is at most 2 for kA = 0 and increases with M sublinearly for small values of kA, it continues increasing linearly (FA = M) when kA = 1. Clearly, the ratio kA should be as small as possible for a low-noise APD [16].

If the avalanche-gain process were noise-free (FA = 1), both I and cts would increase by the same factor M, and the SNR would be unaffected, as far as the shot-noise contribution is concerned. It is the dominance of thermal noise in practical receivers that makes APDs attractive. By adding the contributions of both the shot and thermal noises, the SNR of an APD receiver can be written as

2qM2FA(RdPm+Id)Af + 4(kBT/RL)FnAf, where Eqs. (5.1.10), (5.2.7), and (5.2.8) were used. Figure 5.2 shows the dependence of SNR on received power P,n for three values of APD gain M with Oj = 1 pA using the same receiver parameters used in Figure 5.1 and assuming that kA = 0.7 for the APD. This value of kA is realistic for InGaAs APDs designed to operate in the spectral region near 1.55 jum.

Several points are noteworthy from Figure 5.2. Noting that M = 1 case corresponds to the use of a p-i-n photodiode, it is evident that the SNR is in fact degraded for an APD receiver when input powers are relatively large. Some improvement in SNR occurs only for low input power levels below -20 dBm. The reason behind this behavior is related to the enhancement of shot noise in APD receivers. At low power levels, thermal noise dominates over shot noise, and the APD gain helps. However, as the APD gain increases, shot noise begins to dominate over thermal noise, and APD performs worse than a p-i-n photodiode under the same operating conditions. To make this point clear, we consider the two limits separately.

In the thermal-noise limit (as <c cry), the SNR becomes

and is improved, as expected, by a factor of M2 compared with that of p-i-n receivers [see Eq. (5.2.3)]. By contrast, in the shot-noise limit (<7S Cfj), the SNR is given by

Received Power (dBm)

Figure 5.2: Increase in SNR with received power Pm for three values of APD gain M for a receiver with a bandwidth of 30 GHz. The M = 1 case corresponds to a p-i-n photodiode.

Received Power (dBm)

Figure 5.2: Increase in SNR with received power Pm for three values of APD gain M for a receiver with a bandwidth of 30 GHz. The M = 1 case corresponds to a p-i-n photodiode.

and is reduced by the excess noise factor Fa compared with that of p-i-n receivers.

Equation (5.2.10) and Figure 5.2 indicate that for a given l]„, the SNR of APD receivers is maximum for an optimum value Mopt of the APD gain M. It is easy to show that the SNR is maximum when Mopt satisfies the following cubic polynomial:

4 knTF

The optimum value Mopt depends on a large number of the receiver parameters such as the dark current, the responsivity Rj, and the ionization-coefficient ratio kA. However, it is independent of receiver bandwidth. The most notable feature of Eq. (5.2.13) is that Aiopt decreases with an increase in Pm. Figure 5.3 shows the variation of Mopt with Pin for several values of kA using typical parameter values for a 1,55-/J m InGaAs receiver: Rl = 1 kQ, Fn = 2, Rd = 1 AAV, and ld = 2 nA.

The optimum APD gain is quite sensitive to the ionization-coefficient ratio kA ■ For kA — 0, Mopt decreases inversely with Pjn, as inferred readily from Eq. (5.2.13) after noting that the contribution of is negligible in practice. By contrast, Mopl varies as — 1 /3

Pin for kA ~ 1, and this form of dependence appears to hold even for as small as 0.01 as long as Mopt > 10. In fact, by neglecting the second term in Eq. (5.2.13), Mopt

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