Partially coherent nonparaxial Hermite–Gaussian beams and their propagation properties

The concept of partially coherent nonparaxial Hermite–Gaussian (HG) beams is proposed. By using the generalized Rayleigh–Sommerfeld diffraction integral, the closed-form propagation equations of partially coherent nonparaxial HG beams in free space are derived. The on-axis intensity, far-field and paraxial expressions, and specifically, the free-space propagation equations of partially coherent nonparaxial Gaussian Schell-model beams are given, and treated as special cases of our results. It is found that the f and fσ parameters play a crucial role in determining the nonparaxiality of partially coherent TEMmn-mode HG beams. Only if the f and fσ parameters are small enough, the paraxial approximation is applicable.