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Truncation strategy for the series expressions in the advanced ENZ-theory of diffraction integrals

The advanced ENZ-theory of diffraction integrals, as published recently in J. Europ. Opt. Soc. Rap. Public. 8, 13044 (2013), presents the diffraction integrals per Zernike term in the form of doubly infinite series. These double series involve, aside from an overall azimuthal factor, the products of Jinc functions for the radial dependence and structural quantities $c_t$ that depend on the optical parameters of the optical system (such as NA and refractive indices) and the defocus value. The products in the double series have coefficients that are related to Clebsch-Gordan coefficients and that depend on the order of the Jinc function and the index $t$ of the structural quantity, as well as on the azimuthal order and degree of the involved Zernike term. In addition, the structural quantities themselves are also given in the form of doubly infinite series. In this paper, we give truncation rules for the various infinite series depending on specified required accuracy.