Matrix formulation of the Gaussian expansion of coherent multiple beams in arbitrary dimensions

Modeling the propagation of beams along laser beamlines is very challenging due to the multidimensional and multiscale configuration of the problem. Spatio-temporal couplings are particularly difficult to address with conventional numerical methods. Here we derive the Wigner function of a sum of Gaussian beams by calculating the multidimensional Fourier transform of the intercorrelation function of the fields. The matrix formulation allows for a simple propagation of the Wigner function in the framework of matrix optics. The relevancy of this approach is assessed by applying this model to one-dimensional and multidimensional configurations and by studying the influence of spatio-temporal couplings when considering propagation and dispersion by a diffraction grating.