1 Introduction

The capacity of a channel is the highest data rate it can reliably support. Whenever the data rate is less than the capacity of the channel, there exists an error-correcting code for the channel that has an output probability of error as small as desired, and conversely, whenever the data rate is more than the capacity the probability of error is bounded away from zero.

The capacity of the optical channel depends on many factors, including the modulation scheme, laser, transmission medium, photodetector, and preamplifier. Unlike the bandlimited additive white Gaussian noise (AWGN) channel in which all performance-influencing factors are relevant to the channel capacity only in how they affect the bandwidth and signal-to-noise ratio, there is not a method to simplify the formulation of the capacity of the optical channel to so few variables. For example, the capacity depends separately on the signal and background light levels, not simply their ratio. In this report, the functional dependence of the capacity is distilled to the following six major parameters: (1) the PPM order $M$, (2) the laser pulse width $T_s$, (3) the necessary dead time between pulses $T_d$, (4) the average number of signal photons per pulse incident on the detector $\bar n_s$, (5) the average number of background photons per slot incident on the detector $\bar n_b$, and (6) the detector itself. These parameters are represented by the vector $(M,\bar n_s,\bar n_b,T_s,T_d,$detector$)$, and we will write the capacity as $C = C(M,\bar n_s,\bar n_b,T_s,T_d,$detector$)$. For an APD detector, the parameters used are the quantum efficiency $\eta$, the ionization ratio $k_{\mbox{\small\em eff\/}}$, noise temperature $T$, load resistance $R$, noise equivalent one-sided bandwidth $B$, bulk leakage current $I_b$, and surface leakage current $I_s$. Not explicitly included in the functional description of the capacity is the modulation extinction ratio $\alpha_{er}$ of the laser, which we fix at $10^6$ throughout the report. A description of these parameters is contained in [DS88,Mec86].

Numerical results in the report are based on a system using components currently available and suggested by X2000 2nd delivery for a Mars-type mission. This includes a 1064nm pulsed Q-switched Neodymium-doped Yttrium Aluminum Garnet (Nd:YAG) laser, a super low $K$ (SLiK) APD detector made by EG&G, where $K$ is the ionization ratio, and a transimpedance pre-amplifier.

Future improvements made in lasers and detectors can be evaluated with the methods outlined in this report. The increase in capacity can be projected by re-evaluating the equations with new $(M,\bar n_s,\bar n_b,T_s,T_d,$   detector$)$ parameters. These results will be given in a future report.

In the following section, the optical channel is described and the notation used in this report is given. We also discuss the various units in which capacity may be expressed. Section 3 gives the analytic capacity results, including derivations of the capacity of PPM, the probability of uncoded symbol error for the APD and ideal photon counting detectors, and implications of the converse of Shannon's capacity theorem. In Section 4 we give the numerical capacity results, and in Section 5 we state conclusions and discuss future research needed in this area.