  We investigate the achievable rate and capacity of a non-perfect
photon-counting receiver. For the case of long symbol duration, the achievable
rate under on-off keying modulation is investigated based on Kullback-Leibler
(KL) divergence and Chernoff $\alpha$-divergence. We prove the tightness of the
derived bounds for large peak power with zero background radiation with
exponential convergence rate, and for low peak power of order two convergence
rate. For large peak power with fixed background radiation and low background
radiation with fixed peak power, the proposed bound gap is a small positive
value for low background radiation and large peak power, respectively.
Moreover, we propose an approximation on the achievable rate in the low
background radiation and long symbol duration regime, which is more accurate
compared with the derived upper and lower bounds in the medium signal to noise
ratio (SNR) regime. For the symbol duration that can be sufficiently small, the
capacity and the optimal duty cycle are is investigated. We show that the
capacity approaches that of continuous Poisson capacity as $T_s=\tau\to0$. The
asymptotic capacity is analyzed for low and large peak power. Compared with the
continuous Poisson capacity, the capacity of a non-perfect receiver is almost
lossless and loss with attenuation for low peak power given zero background
radiation and nonzero background radiation, respectively. For large peak power,
the capacity with a non-perfect receiver converges, while that of continuous
Poisson capacity channel linearly increases. The above theoretical results are
extensively validated by numerical results.
