Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics

The propagation of any general astigmatic laser beam through any paraxial optical system can be characterized to second order by a 4 × 4 matrix of second moments in the transverse spatial and angular coordinates, with this matrix having at most ten independent elements. We show how to interpret the two existing beam invariants that contain these elements in terms of an overall beam quality factor Meff4 and an intrinsic antigmatism factor a. We describe a general procedure for determining all ten parameters for any arbitrary beam by measuring only second-order moments of the beam irradiance in a single transversal plane after the beam has passed through at most four simple astigmatic optical systems. A number of such systems containing, in particular, only one or two rotating cylindrical lenses and a CCD camera are proposed and analyzed in more detail.