We employ a combination of asymptotic methods to speed up the computation of fields in the focal region of a diffractive lens (DL). The DL is treated locally as a linear grating with a slowly varying period and groove orientation. We employ rigorous electromagnetic diffraction theory locally to obtain the field just behind the DL. A simple diffracted-ray formula is derived for the field in the focal region of the DL at observation points that are not in the immediate vicinity of the optical axis. A careful study of the range of validity of this formula is made. For observation points that are not in the immediate vicinity of the optical axis the new algorithm is 3×10~(5) times faster than the application of numerical integration to the double integrals involved and approximately 1000-1200 times faster than a recently published algorithm based on using asymptotic theory to replace the double integral with a single integral.
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