Zone-bandwidth ratio as a design parameter for narrow-band X-ray microscopy

Finite spectral bandwidth fundamentally limits the performance of Fresnel zone plates (FZPs) in laboratory X-ray microscopy. Although the commonly cited monochromaticity condition Y=N/ζ≲1, where Δλ is defined as the full width at half maximum, provides a qualitative guideline, its quantitative implications for depth of focus, peak intensity, and resolution remain insufficiently characterised. Here, we use the zone-bandwidth ratio Y as a convenient design parameter for quantifying chromatic degradation under incoherent illumination. By modelling spectral effects as the convolution of the monochromatic axial response with the distribution of chromatic focal shifts, second-moment scaling relations are derived that predict bandwidth-limited broadening of DOF and reduction of peak intensity. Numerical simulations confirm these relations and demonstrate asymptotic limits determined by focal length and spectral width. In contrast, resolution behaviour derived from the modulation transfer function does not follow variance-based scaling. For fixed focal-length configurations, an optimal regime is observed, with the 10% cut-off frequency maximised near Y ≈ 1.4. These results illustrate how the spectral bandwidth and numerical aperture (NA) can be optimised and provide a physically interpretable framework for balancing resolution, modulation efficiency, and depth of focus in polychromatic FZP imaging.