Capacity of electromagnetic communication modes in a noise-limited optical system

We present capacity bounds of an optical system that communicates using electromagnetic waves between a transmitter and a receiver. The bounds are investigated in conjunction with a rigorous theory of degrees of freedom (DOF) in the presence of noise. By taking into account the different signal-to-noise ratio (SNR) levels, an optimal number of DOF that provides the maximum capacity is defined. We find that for moderate noise levels, the DOF estimate of the number of active modes is approximately equal to the optimum number of channels obtained by a more rigorous water-filling procedure. On the other hand, for very low- or high-SNR regions, the maximum capacity can be obtained using less or more channels compared to the number of communicating modes given by the DOF theory. In general, the capacity is shown to increase with increasing size of the transmitting and receiving volumes, whereas it decreases with an increase in the separation between volumes. Under the practical channel constraints of noise and finite available power, the capacity upper bound can be estimated by the well-known iterative water-filling solution to determine the optimal power allocation into the subchannels corresponding to the set of singular values when channel state information is known at the transmitter.