  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 the focal ratio is
large, correction lenses made of low atomic number materials can be used with
X-rays in the range 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 $\sim$ mega
electron volt energy.
