A phase space model of two-dimensional (2D) Gaussian beam propagation is generalized for three-dimensional (3D) general astigmatic Gaussian beam passing through first-order optical system. The general astigmatic Gaussian beam is represented by a four-dimensional (4D) phase super-ellipsoid that defined by an associated 4*4 real matrix, then the transformation formula of the phase super-ellipsoid of the beam through first-order optical system is derived. In particular, in the phase space framework, the beam propagation factor M2 value is proved to be a ratio of phase area of real beam to ideal beam, and a novel approach for a qualitative examination of the properties of fractional Fourier transform (FRT) for the beam is also provided.
© 2006 Chinese Optics Letters
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(070.2590) Fourier optics and signal processing : ABCD transforms
(080.2730) Geometric optics : Matrix methods in paraxial optics
(140.3300) Lasers and laser optics : Laser beam shaping
Baoxin Chen, "Transformation of general astigmatic Gaussian beams in a four-dimensional phase space," Chin. Opt. Lett. 4, 255-258 (2006)