Influence of the spectral composition of radiation on the spatial coherence in the far-field pattern

A theoretical analysis is made of the propagation of a space-time correlation function in a homogeneous nondispersive medium. The solution of a six-dimensional wave equation, describing the propagation of the coherence function, is found by the angular spectrum method. The relationships obtained make it possible to analyze fully the influence of the spectral composition of radiation on the spatial coherence. For example, in the case of a spatially noncoherent source there is a critical distance $z_{cr}$, such that when $z\ll z_{cr}$ , the polychromaticity of the field makes a negligible contribution to the spatial coherence; however, if $z\gtrsim z_{cr}$, the linear behavior of the coherence radius, $R_c\sim z$, typical of the van Cittert–Zernike theorem, gives way to the dependence $R_c\sim\sqrt z$. Similar conclusions apply also to a spatially coherent source.