Outline of a variational formulation of zone-plate theory

Transmission zone plates composed of materials describable by the complex index of refraction n^[n^=(1-δ)+ik] modulate both the amplitude and the phase of the light passing through them. In this paper, a general variational approach to the design of zone-plate profiles that maximize system transfer functions, such as I(m)/IIN or I(m)/IOUT (where IIN and IOUT are the total input and output powers and I(m) is the mth-order diffracted power), is presented in terms of the real and imaginary components of n^. The variational problem, as formulated here, is shown to generate some important classical zone-plate configurations as special solutions and to lead naturally to the derivation of the most general zone-plate configurations possible. Some specific optimum solutions are derived for gold in the soft-x-ray range and are tabulated for convenient reference.