Approximate analytical expression for the spectrum of polychromatic Gaussian beams passing through a multi-apertured ABCD optical system

The calculation for the spectrum of beams passing through a multi-apertured ABCD optical system has not been finished in any personal computer when the aperture number is more than a certain value. By means of the expansion of the window function of the hard-edge aperture into a finite sum of complex Gaussian functions, an approximate analytical expression for the spectrum of polychromatic Gaussian beams passing through a multi-apertured ABCD optical system is derived and illustrated with numerical examples. The numerical results find an agreement with those by straightforward integral of the cross-spectral density function, but the computing time is strongly reduced.