Beam quality $M^2(\psi)$ factor, spot rotation angle, and angular speed in general laser beams

TL;DR

This paper introduces a comprehensive framework for characterizing the rotation, quality, and angular properties of general laser beams, including those with orbital angular momentum, using a unified approach and rapid calculation formulas.

Contribution

It defines the beam quality $M^2(cpsi)$ for all directions, introduces new parameters for beam characterization, and provides efficient formulas to analyze beam rotation and spot sizes.

Findings

Only two extreme spot sizes exist in a detection plane.

The angle between maximum and minimum spot sizes is always 90b0.

Spot rotation angles converge at large propagation distances.

Abstract

A unified definition for the rotation angle and rotation angular speed of general beams, including those with orbital angular momentum (OAM), has been lacking until now. The rotation of a general beam is characterized by observing the rotational behavior of the directions of the extreme spot sizes during propagation. We introduce the beam quality $M^2(\psi)$ factor to characterize the unique beam quality of a general beam across all directions, not limited to the $x$- or $y$-axes. Besides that, we present the beam center $s_{\psi}(\psi,z)$, spot size $w_{\psi}(\psi,z)$, waist position, waist radius, and divergence angle along the direction that forms an angle $\psi$ with the $x$-axis in the plane perpendicular to the $z$-axis for the general beam. Furthermore, this paper presents rapid calculation formulas for these parameters, utilizing the mode expansion method (MEM). Subsequently, we prove that only two extreme spot sizes exist in a given detection plane and the angle between the maximum and minimum spot angles is consistently $90^{\circ}$ during the propagation. We also prove the spot rotation angles converge as $z$ approaches either positive or negative infinity. We first show the extreme spot sizes, spot rotation angle, and angular speed for the vortex beam. Our formulas efficiently differentiate between vortex OAM beams and asymmetry OAM beams.