Matrix formulation of the beam quality of the Hermite-Gaussian beam

Based on the M ²-parameter of Hermite-Gaussian beam, we deduced the matrix formulation of the beam quality for nonsymmetric two-dimensional Hermite-Gaussian beam, which at an arbitrary azimyth angle α. And coupled beam radius square, coupled M ²-parameter also have been introduced. The beam radius versus azimuth angle α at different positions has been presented by numerical simulation; and the track of M ²-parameter versus azimuth angle α has been given. The M ²-parameter matrix is universal and available both in the theoretical analysis and the practical measurement of the beam quality for a arbitrary two-dimensional nonsymmetric higher-order Gaussian beam.