Zone plate

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Binary zone plate: The areas of each ring, both light and dark, are equal.

Sinusoidal zone plate: This type has a single focal point.

A zone plate is a device used to focus light or other things exhibiting wave character.^[1] Unlike lenses or curved mirrors however, zone plates use diffraction instead of refraction or reflection. Based on analysis by Augustin-Jean Fresnel, 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.

Design and manufacture

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

$r_{n}={\sqrt {n\lambda f+{\frac {n^{2}\lambda ^{2}}{4}}}}$

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

$r_{{n}}\simeq {\sqrt {nf\lambda }}$ .

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 r_N:

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

In the long focal length limit, the area of each zone is equal, because 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,

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

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 intensity maxima along the axis of the plate at odd fractions (f/3, f/5, f/7, etc.). Although these contain less energy (counts of the spot) than the principal focus (because it is wider), they have the same maximum intensity (counts/m^2).

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:

${\frac {1\pm \cos(kr^{2})}{2}}\,$

Binary zone plates use almost the same formula, however they depend only on the sign:

${\frac {1\pm \operatorname{sgn}(\cos(kr^{2}))}{2}}\,$

where r is the distance from the plate center and k determines the plate's scale.^[2]

Free parameter

It does not matter to the constructive interference what the absolute phase is, but only that it is the same from each ring. So an arbitrary length can be added to all the paths

$r_{n}={\sqrt {(n+\alpha )\lambda f+{\frac {(n+\alpha )^{2}\lambda ^{2}}{4}}}}$

This reference phase can be chosen to optimize secondary properties such as side lobes.^[1]

Relation to a Fresnel lens

Wave propagation can be calculated by summing over all possible ray paths. For short wavelengths, it is usually only where many paths sum coherently, meaning that they have close to the same phase (are on the same half of the complex plane), that propagation is significant, because contributions with different phases cancel each other. (For electromagnetic radiation, for example, opposite fields cancel.)

A zone plate works by blocking the paths with contributions of the opposite phase. If a phase shifter such as a refractive material is available, one can increase the collected light (or other wave) by replacing the absorbing rings with half wavelength phase delays.^[1] Side lobes can be reduced by sloping these dielectric rings thicker on the inner side than on the outer. With optimal slope of the dielectric or other phase shifter, (in the thin lens approximation) the contributions from all parts of the lens can be the same, making it an ideal lens for one wavelength. The jump between zones can be any integer number of wavelengths. If there is a range in wavelengths (the source is not monochromatic), then the zones are not coherent with each other, but still coherent within each zone. The diffraction limit of the resolution is then determined by the width of each zone, and the zones add incoherently (with random phase). This is called a Fresnel lens.^{[citation needed]} If there is only one such zone, it is a conventional lens.

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.

Other waves such as sound waves and, due to quantum mechanics, matter waves can be focused in the same way. Wave plates have been used to focus beams of neutrons and helium atoms.^[1]

Photography

Example of an image taken with zone plate optics.

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. Common f-numbers for a pinhole camera range from f/150 to f/200 or higher, whereas zone plates are frequently f/40 and lower. This makes hand held shots feasible at the higher ISO settings available with newer DSLR cameras.

Gunsights

Zone plates have been proposed as a cheap alternative to more expensive optical sights or targeting lasers.^[3]

Lenses

Zone plates may be used as imaging lenses with a single focus as long as the type of grating used is sinusoidal in nature.

Zone Plate Image

A bitmap representation of a zone plate image may be used for testing various image processing algorithms, such as:

Image interpolation and image resampling;^[4]
Image filtering.^[5]

An open-source zone-plate image generator is available.^[6]

References

↑ 1.0 1.1 1.2 1.3 G. W. Webb, I. V. Minin and O. V. Minin, “Variable Reference Phase in Diffractive Antennas”, IEEE Antennas and Propagation Magazine, vol. 53, no. 2, April. 2011, pp. 77-94.
↑ Joseph W. Goodman (2005). Introduction to Fourier Optics (3rd ed.). p. 125. ISBN 0-9747077-2-4.
↑ New INL gunsight technology should improve accuracy for target shooters, hunters, soldiers, Mike Wall, Idaho National Laboratory, 5 May 2010.
↑ http://web.archive.org/web/20060827184031/http://www.path.unimelb.edu.au/~dersch/interpolator/interpolator.html Testing Interpolator Quality
↑ http://blogs.mathworks.com/steve/2011/07/22/filtering-fun/ Filtering Fun - Matlab Central
↑ http://web.archive.org/web/20060913203139/http://www.worldserver.com/turk/opensource/#ZonePlate Zone Plate generator, c code.

External links

Magnetic Soft X-ray microscopy
Making a photographic zone plate
Examples of zone plate photographs
"Telescope could focus light without a mirror or lens". New Scientist. 1 May 2008.

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