Although most optical imaging systems have a circular or an annular pupil, with or without Gaussian illumination, there are times when the wavefront or the interferogram is hexagonal. This is most notable for the primary mirrors of large telescopes, such as the Keck, the James Webb, or the CELT. Although these mirrors are circular, they are large enough that they are segmented into small hexagonal segments. Optical testing of a hexagonal segment yields a hexagonal wavefront or interferogram, thus requiring polynomials that are orthogonal over a hexagon. Even a large hexagonal primary mirror consisting of hexagonal segments has been proposed.
Smith and Marsh have discussed the PSF of a hexagonal pupil, but their equation for it is incorrect. Sabatke et al. desribe the complex amplitude for a trapezoid forming the upper half of a regular hexagon, but do not carry out the summation of the diffracted amplitudes of the two trapezoids of the hexagonal pupil. We give closed-form expressions for the six-fold symmetric aberration-free PSF and OTF. Similar expressions for the PSF have been given by others. The PSF and OTF are plotted along with the ensquared power, and compared with the corresponding quantities for a system with a circular pupil. The ensquared power and the OTF are shown to be lower than the corresponding values for a circular pupil.
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