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Recently, a new class of diffractive optical element called a photon sieve, which consists of a great number of pinholes, was developed for the focusing and imaging of soft x rays. In terms of the closed-form formula for the far field of individual pinholes and the linear superposition principle, we present a simple yet accurate analytical model for the focusing of the pinhole photon sieve. This model is applicable to arbitrary paraxial illumination with arbitrary complex amplitude distribution at the photon sieve plane. We check the validity range of this model by comparing it with the exact Fresnel diffraction integral. Some special problems, such as the individual quasi-far-field correction for very large pinholes and the related phase shift induced by this correction, are also discussed.

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A 2-D array of 10 x 10 diffractive lenslets was fabricated and tested. Each lenslet has a rectangular aperture and a size of 1.5 mm x 1.5 mm. The focal length of each lenslet is 47 mm. The array was produced by depositing thin films of silicon monoxide on a quartz glass substrate and by using photolithographic techniques. The performance of the lenslets is based on the diffraction of light at a Fresnel zone plate (FZP). The FZP pattern was implemented as a phase structure with eight discrete levels. The diffraction efficiency was measured to be 91%.

Recently, a paraxially individual far-field model was presented for the focusing and imaging analysis of pinhole photon sieves. By use of a local Taylor expansion of the integrated function of the Rayleigh-Sommerfeld diffraction formula, the small-size property of the individual pinholes, and the linear superposition principle, we extend this model to the nonparaxial case of high-numerical-aperture photon sieves. Some related problems, such as the validity range of this nonparaxial model and the selection conditions for the individual pinholes, are also discussed in detail.