Far-field analysis of spectral shifts in Gaussian Schell-model beams propagating through media with arbitrary refractive properties

We derive the formula for the frequency shift of both scalar and electromagnetic Gaussian Schell-model beams propagating in deterministic isotropic media characterized by the real refractive index n (no gain/absorption). Remarkably, the far-field spectral shift in the paraxial region does not depend either on the rms width of the degree of coherence of the source or on the rms width of its spectral density, or on the refractive index n. It is solely a function of the central frequency and the frequency spread of the source