Beam quality M² factor matrix for non-circular symmetric laser beams

It is standard to use $M_x^2$ and $M_y^2$ to characterize the beam quality of a non-circular symmetrical beam on its x-axis and y-axis orientation. However, we knew that the values of $M_x^2$ and $M_y^2$ are inconsistent if one selects a different coordinate system or measures beam quality with different experimental conditionals, even when analyzing the same beam. To overcome this, a new beam quality characterization method, the M ² factor matrix, is developed. It not only contains the beam quality terms, $M_x^2$ and $M_y^2$, to characterize the beam quality along x-axis and y-axis orientation for the non-symmetric beam, but also introduces two additional cross terms, M _xy and M _yx, which are used to characterize the location relationship between the principal axis of the test beam and coordinate system in experiment. Moreover, M ² factor matrix can be measured with a similar procedure to the traditional M ² factor whose measurement instructions are described in ISO11146 by adding some additional image and signal processing procedure. The measurement principle and method is present and the experiment system for beam quality M ² factor matrix is built to demonstrate the performance of M ² factor matrix with real experiments. (paper)