Achromatic combinations of a diffractive phase Fresnel lens and a refractive correcting element have been proposed for x-ray and gamma-ray astronomy and for microlithography, but considerations of absorption often dictate that the refractive component be given a stepped profile, resulting in a double Fresnel lens. The imaging performance of corrected Fresnel lenses, with and without stepping, is investigated, and the trade-off between resolution and useful bandwidth in different circumstances is discussed. Provided that the focal ratio is large, correction lenses made from low atomic number materials can be used with x rays in the range of approximately 10-100 keV without stepping. The use of stepping extends the possibility of correction to higher-aperture systems, to energies as low as a few kilo electron volts, and to gamma rays of mega electron volt energy.
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