3.1.2 The probability of correct detection with an APD detector

A low noise APD enhances the detection of weak optical signals by amplifying the electrical current generated by absorbed photons. This is illustrated in Fig. 2, in which the diode symbol represents the more complicated solid state components of the APD itself, and some of the APD parameters are shown in block diagram form.

Figure 2: The soft APD demodulator.

Unfortunately, in addition to amplifying the signal, the APD transforms the simple Poisson distribution of absorbed photons into a much more complicated probability density function at the APD output. This pdf is known [Con72,McI72], but extremely complex to evaluate numerically. This Conradi-McIntyre distribution has been accurately approximated in a simpler formulation by Webb [WMC74]. In particular, the probability that $m$ secondary electrons are emitted from the APD in response to the absorption of, on average, $\bar n$ primary photons in a slot, is approximately

$$_w(m \vert \bar n) = {\exp\left[-{(m-G\bar n)^2 \over 2\bar nG^2F... ...sqrt{2\pi\bar n G^2F} \left[1 + {m-G\bar n \over \bar nGF/(F-1)}\right]^{3/2}},$$ (2)

where $G$ is the average APD gain, $F$ is the excess noise factor given by

$$F = k_{\mbox{\small\em eff\/}}G + \left(2-{1 \over G}\right)(1-k_{\mbox{\small\em eff\/}}),$$

and $k_{\mbox{\small\em eff\/}}$ is the ionization ratio. For values of $m$ close to its mean $G\bar n$, Eq. (2) can be approximated by a Gaussian pdf; however, Pr$_w(m \vert \bar n)$ departs greatly from a Gaussian pdf at both tails, which form the main contribution to error events in decoders [DS88].

The detector output $x$ is the sum of the charge due to the approximately Webb-distributed secondary electron emissions, a contribution from the APD surface leakage current, and Gaussian distributed amplifier thermal noise, as shown in Fig. 2. Because of the thermal noise, the slot statistic $x$ is not necessarily an integer, and may even be negative. The pdf of the sum charge is given by the convolution

$$p(x\vert\bar n) = \sum_{m=0}^\infty \phi(x,\mu_m,\sigma^2) _w(m\vert\bar n),$$ (3)

where $\phi(x,\mu_m,\sigma^2)$ is a Gaussian pdf with mean $\mu_m = me_- + I_sT_s$ and variance $\sigma^2 = (2e_-I_s + (4\kappa T/R)) BT_s^2$, $e_-$ is the electron charge, $\kappa$ is Boltzmann's constant, $T$ is the noise temperature, $B$ is the single-sided noise bandwidth, and $I_s$ is the APD surface leakage current. Note that Pr$_w(m \vert \bar n)$ and $p(x\vert\bar n)$ are conditioned on the mean number of photons effectively absorbed by the detector, not incident the detector. The relationship between incident and absorbed photons is governed by the quantum efficiency $\eta$ of the detector, as shown in Fig. 2.

The average number of absorbed photons $\bar n$ depends on whether the slot contains the signal. In a signaling slot, $\bar n = \eta\bar n_s + \eta\bar n_b + I_b/e_-$; in a nonsignaling slot, $\bar n = {\eta\bar n_s\over \alpha_{er}} + \eta\bar n_b + I_b/e_-$. The $I_b/e_-$ term represents the additional effective absorbed photons resulting from the APD bulk leakage current. The $\eta\bar n_s/\alpha_{er}$ term represents the photons absorbed when the laser is not sending a pulse. For practical purposes, the extinction ratio $\alpha_{er}$ is often inconsequential, being as high or higher than $10^6$.

The probability of correct detection $p$ is given by

$$p = \int_{-\infty}^\infty p(x\vert\eta\bar n_s + \eta\bar n_b + I... ... p(y\vert\eta\bar n_b + \eta\bar n_s/\alpha_{er} + I_b/e_-) dy\right]^{M-1} dx,$$ (4)

where $p(x\vert\bar n)$ is the conditional pdf of the detector slot statistic given that an average of $\bar n$ photons are absorbed by the detector, using Eq. (3). By plugging Eq. (4) into Eq. (1), the capacity is determined. In cases where Eq .(4) is too cumbersome to numerically evaluate we may use a simpler expression as a bound and approximation. Using Jensen's inequality, $p$ can be bounded by [SV98]

$$p \ge \left[1-\int_{-\infty}^\infty p(x\vert\eta\bar n_s + \eta\b... ... p(y\vert\eta\bar n_b + \eta\bar n_s/\alpha_{er} + I_b/e_-) dy dx\right]^{M-1},$$ (5)

which will give a lower bound on capacity when plugged into Eq. (1). This bound is always tighter than the union bound [Hug92], which implies that as the probability of error gets small, the ratio of the bound to the true value tends to one.