The fields diffracted by planar one- or two-dimensional periodic objects, and in particular their Fourier and Fresnel self-images, can be computed with the aid of a ray-tracing technique based on the Fermat principle. This method (geometrical self-imaging) yields accurate results for any numerical aperture and image field. An analytical study of the image formation, carried out in the fourth-order approximation for the phase, leads to the definition of self-imaging aberrations. These aberrations are strongly dependent on spatial frequency and render the well-known relationships derived by Rayleigh for the location and magnification of self-images approximate at best. The aberrations can be described graphically by a phase diagram and a magnification diagram, which permit interpretation of the properties of high-aperture, large-field self-images and the prediction of optimal imaging conditions. In the case of large magnifications (100× and larger), we present a simple method to eliminate all fourth-order aberrations completely and even sixth-order ones partially. This method consists of introducing a compensating spherical aberration to the incident wave, e.g., by the insertion of a glass plate of appropriate index and thickness just before the object. Thus object spatial frequencies up to about 800 mm−1 can be imaged almost without aberration for several image periods.
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