Effective number of degrees of freedom of partially coherent sources

Orthogonal expansion of a partially coherent source with a given cross-spectral density W is used to define an effective number N of degrees of freedom and an effective number N of uncorrelated random variables characterizing the source. Relations N⩽N≡Tr W/λ0⩽(Tr W/∥W∥)2=Ve/Vce are established and discussed. Here Tr W, ‖W‖, and λ0 are, respectively, the trace, the norm, and the largest eigenvalue of W used as the kernel of a homogeneous Fredholm equation; Ve is an effective volume of the source; and Vce is its effective coherence volume. The main results are illustrated by a Gaussian Schell-model source.