Advanced concepts in laser beam characterization

In the paraxial approximation a partially coherent beam can be characterized by its intensity moments. In the most general case a three-dimensional beam has 10 second order moments, which describe the beam radii, far field divergences, radii of curvature, orientations in the near field and the far field, etc.. The 10 second order moments can be written in a 4×4 symmetric matrix, called the variance matrix. In first order optical systems the variance matrix obeys a simple propagation law. The unknown parameters of the second order moments are the twist parameters, which describe the rotation of the beam during propagation. The twist is directly related to the z-component of the intrinsic angular momentum flux of the field. The ten second order moments can be experimentally determined by measuring the intensity of the beam in a reasonable number of positions around the focal region and measuring the beam twist.