Beam quality M2(ψ) factor, spot rotation angle, and angular speed of general laser beams

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. In this paper, we characterize the rotation of a general beam by observing the rotational behavior of the directions of the extreme spot sizes during propagation. Moreover, we introduce the beam quality $M^2\left(\psi \right)$ 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. Furthermore, this paper presents rapid calculation formulas for these parameters, utilizing the mode expansion method. By integrating the advanced fast mode decomposition techniques with the rapid calculation formulas derived in this paper, we can significantly enhance the real-time beam evaluation methods, thereby enabling the instantaneous acquisition of these beam properties. Subsequently, we demonstrate that four angular solutions for extreme spot sizes exist in a given detection plane, coalescing into two physically distinct states through ${180}^{\circ }$ rotational symmetry: one corresponding to the maximum spot size(maximum spot rotation angle)and the other to the minimum (minimum spot rotation angle). The angular separation between the maximum and minimum spot rotation angles is consistently ${90}^{\circ }$ during the propagation. We also prove the spot rotation angles converge as $z$ approaches 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.