Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates

Abstract

A new geometrical model for the fundamental mode of a Gaussian beam is presented in the oblate spheroidal coordinate system. The model is an interpretation of a Gaussian amplitude wave function, which is an exact solution of the scalar Helmholtz equation. The model uses the skew-line generator of a hyperboloid of one sheet as a raylike element on a contour of constant amplitude. The geometrical characteristics of the skew line and the consequences of treating it as a ray are explored in depth. Finally, the skew line is used to build a nonorthogonal coordinate system that permits straight-line propagation of a Gaussian beam in three-dimensional space.

Skew line ray model of nonparaxial Gaussian beam

“…This is mainly induced by the approximation of Eq. (20). This treatment is the paraxial approximation of optical path length.…”

“…Ray-optics models of paraxial Gaussian beam have been studied for years, including the complex ray representation of Gaussian beam [18,19], the skew line ray (SLR) model of Gaussian beam [20,21], the Poynting vector model of Laguerre-Gauss beam and Bessel-Gauss beam [22], and the Poincaré sphere ray-optics model of structured Gaussian beams [23]. Notably, the SLR model regards the skew lines (straight generatrixes) of one-sheet hyperboloid as the trajectories of light rays.…”

“…The other is the Gouy phase shift of with rotation angle of l in terms of the initial point. A geometrical explanation of Gouy phase shift has been given by Landesman[20], as shown inFig. 3.…”

Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation; Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates: reply to comment

“…In fact, this arctangent factor has a simple but intriguing geometrical interpretation that will be the subject of a future paper. 8 The exponential amplitude exp[-kd(1 -77)] specifies the amplitude distribution on the oblate ellipse. In the paraxial limit, this term reduces to the traditional Gaussian amplitude distribution, as shown in detail below.…”