Solution to the Fresnel Zone Plate Problem

The diffraction problem for the Fresnel half-period zone plate is solved. The zone plate behaves as a lens with marked chromatism which can be taken advantage of for use in a spectrometer. The multiplicity of foci associated with the zone plate are examined in detail, and it is shown that the spectral range that can be focused without overlap is $3λmin≥λmax$⁠. The wavelength resolution is $λ/Δλ ≈ (m+1)/0.445$⁠, where $m$ is the number of transparent half-period zones. For $m = 300$⁠, this corresponds to a $34$-wave number resolution at 20 microns. The intensity at the focus is given approximately by $4I (m+1)2$⁠, where $I$ is the intensity at the zone plate. The transmitted power that falls within the airy disk is proportional to the area of the zone plate.