Eikonal equation for partially coherent fields

Coherence is the study of the amplitude correlations of optical fields. Its physical features can be obtained from the cross-spectral density function W(x1,x2,γ) which satisfies two coupled Helmholtz equations. In this article, we describe the amplitude of the optical field using the angular spectrum model. With this representation we calculate the propagation of the correlation function emerging from a transmittance plane. We show that the cross-spectral density function, can be described by just one Helmholtz equation. The treatment permits us to associate directional features to the coherence phenomena. This implies the existence of extremal trajectories of correlation, which are characterized by an eikonal equation, and the existence of a function for media fluctuations, which we term the correlation refractive index. Experimental results are shown for the synthesis of partially coherent focusing regions, which are described by an ensemble of extreme correlation trajectories.