We introduce a method to analyze the diffraction integral for evaluating the point-spread function. Our method is based on the use of higher-order Airy functions along with Zernike and Taylor expansions. Our approach is applicable when we are considering a finite, arbitrary number of aberrations and arbitrarily large defocus simultaneously. We present an upper bound for the complexity and the convergence rate of this method. We also compare the cost and accuracy of this method with those of traditional ones and show the efficiency of our method through these comparisons. In particular, we rigorously show that this method is constructed in a way that the complexity of the analysis (i.e., the number of terms needed for expressing the light disturbance) does not increase as either defocus or resolution of interest increases. This has applications in several fields such as biological microscopy, lithography, and multidomain optimization in optical systems.
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