Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations

Abstract

The analysis of many systems in optical communications and metrology utilizing Gaussian beams, such as free-space propagation from single-mode fibers, point diffraction interferometers, and interference lithography, would benefit from an accurate analytical model of Gaussian beam propagation. We present a full vector analysis of Gaussian beam propagation by using the well-known method of the angular spectrum of plane waves. A Gaussian beam is assumed to traverse a charge-free, homogeneous, isotropic, linear, and nonmagnetic dielectric medium. The angular spectrum representation, in its vector form, is applied to a problem with a Gaussian intensity boundary condition. After some mathematical manipulation, each nonzero propagating electric field component is expressed in terms of a power-series expansion. Previous analytical work derived a power series for the transverse field, where the first term (zero order) in the expansion corresponds to the usual scalar paraxial approximation. We confirm this result and derive a corresponding longitudinal power series. We show that the leading longitudinal term is comparable in magnitude with the first transverse term above the scalar paraxial term, thus indicating that a full vector theory is required when going beyond the scalar paraxial approximation. In spite of the advantages of a compact analytical formalism, enabling rapid and accurate modeling of Gaussian beam systems, this approach has a notable drawback. The higher-order terms diverge at locations that are sufficiently far from the initial boundary, yielding unphysical results. Hence any meaningful use of the expansion approach calls for a careful study of its range of applicability. By considering the transition of a Gaussian wave from the paraxial to the spherical regime, we are able to derive a simple expression for the range within which the series produce numerically satisfying answers.

Analytical beam propagation model for clipped focused-Gaussian beams using vector diffraction theory

“…The input plane is typically a transmission limiting aperture where propagation of light through the input surface is spatially limited to a particular area and the rest of the surface is opaque. Beyond plane-wave input light fields, non-paraxial diffraction theory has been applied to converging spherical waves [9,10], and diverging Gaussian beam [11][12][13][14][15] light fields. One limitation of the models presented in Refs.…”

“…One limitation of the models presented in Refs. [11][12][13][14][15] is that the aperture plane is chosen to be coplanar with the beam waist of the Gaussian light field. Recently, we applied Hertz vector diffraction theory (HVDT) [16,17] to the diffraction of converging focused Gaussian light fields (GHVDT) [18] and investigated the effects of diffraction on the behavior and propagation of the electromagnetic fields.…”

Orbital Angular Momentum Density of an Elegant Laguerre-Gaussian Beam

“…With the fixed values of n and m, the second integral can be neglected for the certain range of f , which has been demonstrated in Refs. [35][36][37][38]. When n = m = 0, the omission of the second integral is allowed for the case of f ≤ 0.2.…”

Phase and polarization changes of pulsed Gaussian beams during focusing and propagation

“…Aberrationdistorted focusing can also be properly treated using vector diffraction theory [155,156], but this is not a topic of this work. The only relevant idea regarding the vector nature of light that the longitudinal components of the field can be neglected in the paraxial approximation, so the the propagation of the two transverse components (the x-and y-linear or left-and right-circular constituents) can be treated independently [157,158].…”

“…One has to be careful, however, as beams generally do not have purely transverse polarization (see introduction of Section 2.3). Still, as it was mentioned in Section 2.3.3 (or in [157,158]), using paraxial approximation has the advantage that Gaussian beams can still be considered as transverse waves without causing flaws in the conclusions drawn. So, using the same theory as in Section 2.2.3, the vector electric field can be described by two components, either the linearly polarized scalar components x and y, or the left-and right circularly polarized ones.…”