= 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>,
$$
\sqrt{f^2+r_j^2}-f=\frac{j\lambda}{2},\qquad
r_j^2=j\lambda f+\frac{j^2\lambda^2}{4}
\simeq j\lambda f.
$$
The paraxial radii grow as $\sqrt j$, so the zones become narrower outwards.
Using the <photon energy> relation $\lambda=hc/E$ gives \b[$\lambda=4.133\times10^{-11}\ {\rm m}=0.04133\ {\rm nm}$]. For $f=10^{-2}\ {\rm m}$,
$$
\boxed{r_j\simeq0.6429\sqrt j\ \mu{\rm m}.}
$$
The first three boundaries are approximately $0.6429$, $0.9092$ and $1.1135\ \mu{\rm m}$. This fixes the zone pattern but \b[does not fix a unique outside diameter]: the number of zones is a design choice. For example, $N=100$ gives radius $6.429\ \mu{\rm m}$, diameter $12.86\ \mu{\rm m}$ and outermost zone width about $32.2\ {\rm nm}$. 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 $f\propto1/\lambda$, so a narrow <energy> band is needed for a sharp common focus.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2016/paper-2-interference.png]
{title=Original zone-plate radii, Michelson visibility envelope and Fabry–Pérot transmission calculations}
{description=The zone-plate panel shows the first twelve mathematical zones, not a specified complete fabricated device. The Michelson panel uses a declared illustrative <wavelength> pair. The etalon panels use the calculated transmission; the added third line is plotted with a chosen equal input intensity.}
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