Past exam of the physics A course of the University of Cambridge / 2016 / Paper 2 / Section A / 2 / Solution

A Fresnel zone plate replaces a conventional refracting surface by concentric alternating transmitting and absorbing annuli. Successive annuli span half a wavelength of optical-path excess to the chosen focus; blocking alternate zones prevents their approximately opposite wave phases from cancelling. For a normally incident plane wave,The paraxial radii grow as $\sqrt{j}$, so the zones become narrower outwards.

Using the photon energy relation $\lambda =hc/E$ gives $\lambda =4.133\times 10^{-11}\mathrm{m}=0.04133\mathrm{nm}$. For $f=10^{-2}\mathrm{m}$,The first three boundaries are approximately $0.6429$, $0.9092$ and $1.1135\mu \mathrm{m}$. This fixes the zone pattern but does not fix a unique outside diameter: the number of zones is a design choice. For example, $N=100$ gives radius $6.429\mu \mathrm{m}$, diameter $12.86\mu \mathrm{m}$ and outermost zone width about $32.2\mathrm{nm}$. A larger usable diameter requires finer outer zones.

One could pattern absorbing metal annuli on a thin X-ray-transparent support, choosing absorber thickness from its attenuation at the selected photon energy; that thickness cannot be determined from energy and focal length alone. The design is chromatic: for fixed radii $f\propto 1/\lambda$, so a narrow energy band is needed for a sharp common focus.