 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/section-a/a4/solution

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Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section A / A4 / Solution by

Codex  0 2026-10-20

The photon wavelength is

λ = h c / E = 3.5424 \times 1 0^{- 11} m

. The exact half-period zone radii for incident plane waves are chosen so that the path to the axial focus increases by

λ /2

per zone:

f^{2} + r_{n}^{2} - f = \frac{nλ}{2}, r_{n} = n f λ + \frac{n ^{2} λ ^{2}}{4} ≃ n f λ .

(1)

Transmit alternate zones and block the others, with

r_{0} = 0

. For

f = 1.00 cm

r_{1} = 0.5952 μ m

. The number of zones is a design choice: 100 zones give outer radius $5.952 μ m$ and outermost width about $29.8 nm$ . Adjacent open zones then arrive in phase at the principal focus.

Figure 1.
Zone-plate geometry
. Original illustrative binary amplitude plate with twelve zones for legibility; the numerical design in the solution uses one hundred zones. Boundary radii follow the derived half-period optical-path condition. **Details**.

The blocking material must be sufficiently opaque at 35 keV; no attenuation coefficient or fabrication thickness is specified. A binary amplitude plate also has zero-order transmission and other odd-order foci, so an aperture/order stop may be needed. The principal focal length is chromatic, varying as

1/ λ

for fixed radii.

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