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For uniform coherent plane-wave illumination, an ideal binary amplitude zone plate transmits alternate Fresnel half-period zones. If $N_o$ complete odd zones are open, every one contributes approximately twice the unobstructed axial complex amplitude, in the same phase, and $I_f/I_0\simeq 4N_o^2$. This is a local light intensity gain relative to removing the plate and outer stop. It differs from comparison with a finite clear aperture: an aperture containing an odd integer number of half-period zones has axial intensity $4I_0$, giving plate-to-aperture ratio $N_o^2$. Neither gain is a total power efficiency.

David Attwood studies the use of diffraction for X-ray microscopy. With Janos Kirz, he authored the X-Ray Data Booklet's account of zone plates: concentric rings select contributions that interfere at a focus. Their treatment connects outer-zone width to spatial resolution and explains why an ideal opaque/transparent zone plate sends only $1/{\pi }^2$ of incident power into its first diffraction order. Resolving a small structure and collecting a useful fraction of the incident X-rays are therefore separate design constraints.^[ref]

For a plane wave incident on a flat zone plate, the path difference from radius $r_n$ to axial focal distance $f$ is $\sqrt{f^2+r_n^2}-f$. Setting it equal to $n\lambda /2$ gives the title relation. The paraxial limit is $r_n\simeq \sqrt{nf\lambda }$, and the fine outer-zone width is approximately $f\lambda /(2r_n)$. Alternately removing half-period annuli makes the remaining contributions reinforce at the principal focus. Choosing the number of zones sets the aperture and fabrication scale, not the principal focal distance independently of these radii.

Janos Kirz developed X-ray imaging methods using zone plates. His joint technical account with David Attwood explains how diffraction replaces ordinary refraction in a small focusing optic. Replacing absorbing zones by zones that shift the wave phase can raise the focusing efficiency. Their account also connects finite illumination bandwidth to chromatic blurring: many zones improve focusing only if the incident X-rays occupy a sufficiently narrow range of wavelengths.^[ref]

For a circular structure, cylindrical symmetry reduces the scalar Fresnel-Kirchhoff diffraction integral to concentric annuli. With plane-wave illumination, radius $a$ and distance $z$, the relevant Fresnel number is $N_F=a^2/(\lambda z)$. Successive Fresnel half-period zones have radii $r_j^2\simeq j\lambda z$: their areas are approximately equal while their phases alternate by $\pi$, causing neighbouring-zone cancellation.

For Fresnel diffraction by a circular aperture, the on-axis field follows directly from the radial integral,Odd zone counts reinforce on axis; even counts cancel in this ideal scalar model. An annular opening gives the difference of the corresponding two edge exponentials. A zone plate alternately blocks or phase-shifts zones so that otherwise cancelling contributions reinforce at a focus, with $r_j^2\simeq j\lambda f$ for its principal focal distance.

Babinet's principle gives $U_{\mathrm{disk}}=U_{\mathrm{free}}-U_{\mathrm{ap}}=U_{\mathrm{free}}{e}^{i\pi N_F}$ for Fresnel diffraction by an opaque disk. Thus its ideal on-axis intensity equals the unobstructed intensity, the Arago spot, even within the geometric shadow. This concerns a narrow axial feature, not the whole shadow. Finite source size, imperfect circularity, bandwidth and alignment reduce its visibility. When $N_F\ll 1$, the circular-aperture far-field pattern becomes the Airy disk intensity pattern $[2J_1(u)/u{]}^2$, with $u=ka\sin \theta$ and first minimum near $\theta =1.22\lambda /(2a)$.

Take the aperture in $z=0$ and the observation point $P=(0,0,b)$, with $b>0$. Under scalar diffraction theory, the Huygens-Fresnel principle sums complex amplitudes from area elements, rather than summing their intensities. Apart from a common normalization,Here $h$ is the complex aperture transmission, $b$ is the axial distance, and $\theta$ is the angle of the secondary ray to the aperture normal. The obliquity factor $K$ describes the direction dependence of the secondary contribution; one Kirchhoff convention is $K=(1+\cos \theta )/2$ for normal illumination. Other scalar boundary approximations have slightly different angular weights, all tending to unity near the axis.

Put $r^2=x^2+y^2$. The paraxial approximation gives $R\simeq b+r^2/(2b)$ in the phase and $1/R\simeq 1/b$ in the amplitude. Removing common $e^{ikb}/b$ yieldsAssumptions are uniform normally incident plane-wave illumination, coherent monochromatic propagation, aperture dimensions large enough for the scalar description, $r/b\ll 1$, and phase accuracy $k{r}^4/(8b^3)\ll 1$ over the relevant aperture. The last condition justifies dropping the quartic distance term; geometric smallness alone does not guarantee phase accuracy.

For a circular aperture, $h=1$ for $r\le a$ and zero elsewhere. Rotational symmetry and $s=r^2$ give $dxdy=rdrd\phi$, $rdr=ds/2$, andThe constant $\pi$ can be absorbed into the proportionality factor, producing the required one-dimensional integral.

At a focus a distance $f$ away, half-period-zone boundaries obeyWith slowly varying obliquity and illumination, each zone has almost the same area and its integrated phasor has the opposite sign to its neighbour. A zone plate blocks alternate zones, so the remaining phasors add instead of nearly cancelling. Choosing an open central zone, the first three open regions have outer boundaries $n=1,3,5$:These are outer radii, not zone widths; the second open region is the annulus between $r_2$ and $r_3$.

For the specified outer radius, $R^2/(\lambda f)=49$: there are approximately 49 half-period zones, of which 25 odd zones are open. Normalize to the field ${\psi }_0$ of the unobstructed plane wave. The normalized Fresnel aperture integral isFor any odd open annulus this gives $2$, so the whole plate gives $\psi /{\psi }_0\simeq 50$ andThe axial intensity gain of an amplitude zone plate grows as the square of the number of open zones. Treating the number of zones continuously gives $49^2\simeq 2.40\times 10^3$, the same accuracy as an estimate. This large local intensity gain is consistent with conservation of energy because light is concentrated into a small focal spot. It is not a power efficiency; the ideal first-order power efficiency of a binary amplitude zone plate is only $1/{\pi }^2$.^[ref]

“Without the zone plate” has here meant removing the finite plate and its outer stop, giving the unobstructed incident wave. If instead a clear circular aperture of radius $R$ is retained, its axial field is ${\psi }_{\mathrm{clear}}/{\psi }_0=1-e^{i49\pi }=2$, so the ratio to that finite-aperture reference is $2500/4=625$. The two reference experiments differ; neither ratio should be confused with total transmitted power.

By Janos Kirz and David Attwood. Section 4.4 of the X-Ray Data Booklet, Lawrence Berkeley National Laboratory.

Official text: xdb.lbl.gov/Section4/Sec_4-4.html

The section specifies half-period-zone geometry and distinguishes focal resolution from first-order diffraction efficiency. Its ideal binary amplitude zone plate has first-order efficiency $1/{\pi }^2$.