Propagation of Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation

Abstract

On the basis of the vectorial Rayleigh-Sommerfeld formulas and by means of the relation between Hermite and Laguerre polynomials, the analytical expressions for the propagation of the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams beyond the paraxial approximation are derived, with the corresponding far-field propagation expressions and that for the Gaussian beams being given as special cases of the results. Some detailed comparisons of our results with the expansion series and paraxial expressions are made, which show the advantages of our results over the expansion series. With the results obtained, some typical intensity patterns of nonparaxial HG and LG beams are shown.

Light scattering of Laguerre-Gaussian beams: near-field structures and symmetries

“…The problem studied in Refs. [27,29,30] deals with the exact propagation of the optical field in the half-space, z > 0, when its transverse components at the initial (source) plane, z = 0, are known. In Ref.…”

“…In Refs. [26][27][28][29][30] this strategy has been applied to the important case of Laguerre-Gaussian (LG) beams using different methods such as the vectorial Rayleigh-Sommerfeld formulas [27,30], the vector angular spectrum method [29], approximating LG beams by nonparaxial beams with (near) cylindrical symmetry [26,28].…”

Mie scattering of Laguerre-Gaussian beams: Photonic nanojets and near-field optical vortices

“…[33,35,36] deals with the exact propagation of the optical field in the half-space, z > 0, when its transverse components at the initial (source) plane, z = 0, are known. In Ref.…”

“…[63] and those obtained using the vectorial Rayleigh-Sommerfeld integrals [33,36]. The resulting expression for the far-field angular distribution can be written in the following form…”

“…In Refs. [32][33][34][35][36] this strategy has been applied to the important case of Laguerre-Gaussian (LG) beams using different methods such as the vectorial Rayleigh-Sommerfeld formulas [33,36], the vector angular spectrum method [35], approximating LG beams by nonparaxial beams with (near) cylindrical symmetry [32,34].…”