Mathematical Optics

where $w_x/\sqrt{\pi }$ is the waist width of the embedded Gaussian beam.

For beams that are not circularly symmetric, this factor can be different for the two directions orthogonal to the beam axis and to each other. The HG beam HG_m,n, related to a TEM_m,n resonator mode, has an $M_{x,y}^2$ factor proportional to (2m + 1) and an $M_y^2$ factor proportional to (2n + 1). The $M_{x,y}^2$ factor is also used to quantify the degree of paraxiality of a monochromatic laser beam (Vaveliuk 2009). We underline that the $M_{x,y}^2$ factor cannot be considered as a complete characterization of the beam quality, since it does not reveal the beam phase structure encoded in the mixed second-order moments, M_rp, which we will consider now.

The moments m_xu and m_yv allow determining the beam curvature ...