Vectorial Nonparaxial Beams Research Articles

Partially coherent non-paraxial vectorial Hermite–Gaussian beams and their far-field properities

The far-field theory of partially coherent vectorial non-paraxial beams is expanded and their analytical propagation expressions of the Wigner distribution function matrices and the cross-spectral-density matrices in free space are derived using the far-field approximation. Some interesting cases, in particular the vectorial non-paraxial partially coherent Hermite–Gaussian (HG) beams, are discussed and treated as special cases of our general expressions. It is shown that the f σ and f parameters play an important role in determining vector and non-paraxiality of partially coherent HG beams. When two parameters are small enough, scalar and paraxial vectorial approximation is allowed; otherwise, non-paraxial vectorial approximation is applied. But the decided parameters additionally affect their far-field divergence angles.

Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams

The Wigner distribution function (WDF) matrix is introduced, and its analytical propagation expressions in free space and the corresponding far-field forms are derived for the first time. It is shown that the propagation expressions for the WDF matrix have general applicable advantages and reduce to those for the WDF of partially coherent scalar paraxial beams within the paraxial regime. The application of the WDF matrix to the partially coherent vectorial nonparaxial beams is illustrated with vectorial nonparaxial Gaussian Schell-model beams.

Partially coherent vectorial nonparaxial beams

Generalized vectorial Rayleigh-Sommerfeld diffraction integrals are developed for the cross-spectral-density matrices of spatially partially coherent beams. Using the Gaussian Schell-model (GSM) beam as an example, we derive the expressions for the propagation of cross-spectral-density matrices and intensity of partially coherent vectorial nonparaxial beams, and the corresponding far-field asymptotic forms, beyond the paraxial approximation. The propagation of the vectorial nonparaxial GSM beams are evaluated and analyzed. It is shown that a 3 x 3 cross-spectral-density matrix or a vector theory is required for the exact description of nonparaxial GSM beams.

A comparative study of the propagation of vectorial nonparaxial beams in the use of different approaches

Starting from the vectorial Rayleigh diffraction integral formula and using a simple expansion of the nucleus in the Rayleigh formula, an analytical propagation equation of vectorial nonparaxial Gaussian beams in free space is derived, which permits us to perform numerical calculations in comparison with the expression derived by Ciattoni et al. and with the direct numerical integration of the Rayleigh formula. It is found that as usual the use of expansion of vectorial Rayleigh diffraction integral is sufficient to provide satisfactory numerical results as compared with the direct integration of the Rayleigh formula. The above two analytical expressions are valid under certain conditions, however both are applicable in the far field.