Point spread function of hexagonally segmented telescopes by new symmetrical formulation

A point spread function of hexagonally segmented telescopes is derived by a new symmetrical formulation. By introducing three variables on a pupil plane, the Fourier transform of pupil functions is derived by a 3D Fourier transform. The permutations of three variables correspond to those of a regular triangle's vertices on the pupil plane. The resultant diffraction amplitude can be written as a product of two functions of the three variables; the functions correspond to the sinc function and Dirichlet kernel used in the basic theory of diffraction gratings. The new expression makes it clear that hexagonally segmented telescopes are equivalent to diffraction gratings in terms of mathematical formulae.