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 $\mathcal{N}\hspace{0.17em}\text{of}$ degrees of freedom and an effective number N of uncorrelated random variables characterizing the source. Relations $\mathcal{N}⩽N\equiv \text{Tr}\hspace{0.17em}W/{\mathrm{\lambda }}_0⩽{(\text{Tr}\hspace{0.17em}W/\parallel W\parallel )}^2=V_e/V_{ce}$ 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; V_e is an effective volume of the source; and V_ce is its effective coherence volume. The main results are illustrated by a Gaussian Schell-model source.