We investigate the achievable rate and capacity of a non-perfect photon-counting receiver assuming that the shot and thermal noise is negligible, and that the sampling interval is not shorter than the dead time. For long and fixed symbol duration, the achievable rate under on-off keying modulation is investigated based on Kullback-Leibler divergence and Chernoff divergence. We prove the tightness of the derived bounds for large peak power with zero background radiation at an exponential convergence rates, and for low peak power at an order-two convergence rates. Moreover, we propose an approximation on the achievable rate, which is more accurate compared with the derived bounds in the medium signal to noise ratio (SNR) regime. We also verify the accuracy of the proposed model on the achievable rate analysis via the comparison with the model with non-negligible shot and thermal noise. As for the capacity analysis, we assume that the symbol duration can be arbitrarily short, and demonstrate that the capacity approaches that of the continuous-time Poisson channel as both the sampling interval and the symbol duration approach zero, and the sampling interval equals the symbol duration. For large peak power, the capacity with a non-perfect receiver converges, while that of continuous Poisson capacity channel linearly increases.