The Fast Fourier Transform (FFT) algorithm has been widely applied to frequency analysis of many electrical and optical systems. Conceptually, the FFT decomposes a spatial distribution into a frequency domain distribution. An excellent discussion of Fourier optics can be found in reference [1] at the end of this article. There is also a summary of diffraction theory in the chapter "Physical Optics Propagation" in the ZEMAX manual, reference [2]. Both of these references describe Fresnel and Fraunhofer diffraction theory.

Most optical imaging systems meet the simplifying assumptions necessary for the Fraunhofer diffraction theory used by the FFT PSF algorithm. The main assumptions are:

• The F/# is large enough so that scalar diffraction theory applies
• The region over which the diffraction PSF has significant energy is small compared to the distance from the exit pupil of the optical system to the image surface.
• The exit pupil is not significantly distorted with respect to the entrance pupil. This means a uniform distribution of rays on the entrance pupil remains reasonably uniform on the exit pupil.
• The sampling is set high enough to accurately model the PSF
• The chief ray incident upon the image surface is close to normal incidence

The FFT PSF of an optical system is computed as follows. A grid of rays is traced from the source point to the exit pupil. For each ray, the amplitude and optical path difference is used to compute the complex amplitude of a point on the wavefront grid at the exit pupil. The FFT of this grid, appropriately scaled, is then squared to yield the real valued PSF. If the computation is polychromatic, the PSF's are summed incoherently.

To compute an FFT PSF for a sequential system, choose Analysis > PSF > FFT PSF from the main ZEMAX menu. Below is shown a sample FFT PSF for the on-axis field point of the Newtonian telescope sample file. The settings have been modified from the default settings, and this will discussed shortly.



Note the familiar Airy Disk shape. This is the expected result for the Newtonian, which is aberration free for the on axis field point.

To generate the picture above, the FFT PSF settings dialog should look like this:



The sampling refers to the grid of rays traced to the entrance pupil. Internally, ZEMAX doubles the size of the grid, filling the region outside of the entance pupil with zero valued data. Because of this doubling, the output PSF is always on a grid with twice as many points as the sampling grid. If the aberrations are reasonably small, the region of interest is concentrated near the center of the plot. Rather than plot all of these near-zero amplitude points, the display grid may be selected to be smaller than the total grid computed.

There are many ways to display the same underlying PSF data. Try the settings shown below:



Note Display is 128 x 128, Field is 2, Type is Logarithmic, and the Show As is set to False Color. Here is the resulting PSF: