Wigner matrix formalism for phase-modulated signals

International audienceLaser beam can carry multi-scales properties in space and time that impact the beam quality. The study of their evolution along complex optical sequences is of crucial interest, particularly in high intensity laser chains. For such analysis, results obtained with standard numerical calculus are strongly dependent of the sampling. In this paper, we develop an analytic model for a sinusoidal phase modulation inside a sequence of first order optics elements based on the Wigner matrix formalism. A Bessel decomposition of the Wigner function gives pseudo-Wigner functions that obey to the general ABCD matrix law transformation without approximations and sampling considerations. Applied to a Gaussian beam, explicit expressions are obtained for the projections of the Wigner function in the sub-spaces and gives a powerful tool for laser beam analysis. The formalism is established in the spatial and temporal domain and can be used to evaluate the impact of the phase noise to the beam properties and is not limited to small modulation depths. In a sake of illustration, the model is applied to the Talbot effect with the analysis of the propagation in the spatial and phase-space domain. A comparison with full numerical calculations evidences the high accuracy of the analytic model that retrieves all the features of the diffracted beam