A simple analytic form for the intensity point spread function is obtained in terms of the power spectral density function for the phase. Two fourier transforms are required to compute the psf from the psd (A third fourier transform is needed to give the fraction of the light in the core). The analytic form is an infinite sum of convolution integrals of increasing order in the psd function multiplied by the simple renormalization factor exp(-sigma 2), where sigma 2 is the two-dimensional integral of the psd in radians squared. Computationally, the psf is evaluated on a discrete grid in kx-ky space. This infinite sum can be evaluated at all pixels other than the zero frequency pixel by taking the two-dimensional complex fourier transform X of the psd, computing exp(X)-1, and then taking the inverse fourier transform. There is also a simple expression for the value at the zero spatial frequency pixel. Like the psd, the psf is smooth because the psf is an ensemble average over all realizations for the phase: Each realization of the phase gives an intensity speckle pattern in the focal plane. The psf is the ensemble average over all realizations. This transformation has been extensively tested for azimuthally symmetric phase psd functions by comparing the computed psf using the analytic transformation with the azimuthally averaged psf computed using a specific realization for the phase. The psd functions that were compared this way were all azimuthally symmetric, but the analytic transformation from psd to psf doesn't require this. The final result for the halo is equivalent to the result in Hardy when the pupil is infinite. The derivation in this paper is simple and direct.
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