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# Zone plate

A zone plate is a device used to focus light. Unlike lenses however, zone plates use diffraction instead of refraction. Created by Augustin-Jean Fresnel [freɪ'nel], they are sometimes called Fresnel zone plates in his honor. The zone plate's focusing ability is an extension of the Arago spot phenomenon caused by diffraction from an opaque disc.

A zone plate consists of a set of radially symmetric rings, known as Fresnel zones, which alternate between opaque and transparent. Light hitting the zone plate will diffract around the opaque zones. The zones can be spaced so that the diffracted light constructively interferes at the desired focus, creating an image there. Zone plates produce equivalent diffraction patterns no matter whether the central disk is opaque or transparent, as long as the zones alternate in opacity.

## Design and manufacture

To get constructive interference at the focus, the zones should switch from opaque to transparent at radii where

$r_n = sqrt\left\{n lambda f + frac\left\{n^2lambda^2\right\}\left\{4\right\}\right\}$
where n is an integer, λ is the wavelength of the light the zone plate is meant to focus and f is the distance from the center of the zone plate to the focus. When the zone plate is small compared to the focal length, this can be approximated as
$r_\left\{n\right\} simeq sqrt\left\{n f lambda\right\}$.

For plates with many zones, you can calculate the distance to the focus if you only know the radius of the outermost zone, r N, and its width, Δ rN:

$f = frac\left\{2 r_\left\{N\right\} Delta r_\left\{N\right\}\right\}\left\{lambda\right\}$

In order to get complete constructive interference at the focus, the amplitude of the diffracted light waves from each zone in the zone plate must be the same. This means that for an evenly illuminated zone plate, the area of each zone is equal.

Because the area of each zone is equal, the width of the zones must decrease farther from the center. The maximum possible resolution of a zone plate depends on the smallest zone width,

$frac\left\{Delta l\right\}\left\{Delta r_\left\{N\right\}\right\} = 1.22$

Because of this, the smallest size object you can image, Δl, is limited by how small you can reliably make your zones.

Zone plates are frequently manufactured using lithography. As lithography technology improves and the size of features that can be manufactured decreases, the possible resolution of zone plates manufactured with this technique can improve.

Unlike a standard lens, a binary zone plate produces subsidiary intensity maxima along the axis of the plate at odd fractions (f/3, f/5, f/7, etc.), though these are less intense than the principal focus.

However, if the zone plate is constructed so that the opacity varies in a gradual, sinusoidal manner, the resulting diffraction causes only a single focal point to be formed. This type of zone plate pattern is the equivalent of a transmission hologram of a converging lens.

For a smooth zone plate, the opacity (or transparency) at a point can be given by:

$frac \left\{1 pm cos\left(kr^2\right)\right\}\left\{2\right\},$
Binary zone plates use almost the same formula, however they depend only on the sign:
$frac\left\{1 pm sgn\left(cos\left(kr^2\right)\right)\right\}\left\{2\right\},$
where r is the distance from the plate center and k determines the plate's scale.

## Applications

### Physics

There are many wavelengths of light outside of the visible area of the electromagnetic spectrum where traditional lens materials like glass are not transparent, and so lenses are more difficult to manufacture. Likewise, there are many wavelengths for which there are no materials with a refractive index significantly larger than one. X-rays, for example, are only weakly refracted by glass or other materials, and so require a different technique for focusing. Zone plates eliminate the need for finding transparent, refractive, easy-to-manufacture materials for every region of the spectrum. The same zone plate will focus light of many wavelengths to different foci, which means they can also be used to filter out unwanted wavelengths while focusing the light of interest.

### Photography

Zone plates are also used in photography in place of a lens or pinhole for a glowing, soft-focus image. One advantage over pinholes (aside from the unique, fuzzy look achieved with zone plates) is that the transparent area is larger than that of a comparable pinhole. The result is that the effective f-number of a zone plate is lower than for the corresponding pinhole and the exposure time can be decreased.
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