Zone plate

Last updated
Binary zone plate: The areas of each ring, both light and dark, are equal. Zone plate.svg
Binary zone plate: The areas of each ring, both light and dark, are equal.
Sinusoidal zone plate: This type has a single focal point. Zonenplatte Cosinus.png
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, zone plates use diffraction instead of refraction or reflection. Based on analysis by French physicist 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. [2]

Contents

A zone plate consists of a set of concentric rings, known as Fresnel zones, which alternate between being 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 [3]

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

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

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,

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.

Continuous zone plates

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/m2).

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:

where is the distance from the plate center, and determines the plate's scale. [4]

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

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

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

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 differing from 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. Christmas with zone plate.jpg
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. [5]

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. A specifically designed Fresnel zone plate with blazed phase structures is sometimes called a kinoform. [6]

Reflection

A zone plate used as a reflector will allow radio waves to be focused as if by a parabolic reflector. This allows the reflector to be flat, and so easier to make. It also allows an appropriately patterned Fresnel reflector to be mounted flush to the side of a building, avoiding the wind loading that a paraboloid would be subject to.

Software testing

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

An open-source zone-plate image generator is available. [9]

See also

Related Research Articles

<span class="mw-page-title-main">Diffraction</span> Phenomenon of the motion of waves

Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.

<span class="mw-page-title-main">Optics</span> Branch of physics that studies light

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Light is a type of electromagnetic radiation, and other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

<span class="mw-page-title-main">Refractive index</span> Ratio of the speed of light in vacuum to that in the medium

In optics, the refractive index of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.

<span class="mw-page-title-main">Wavelength</span> Distance over which a waves shape repeats

In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term "wavelength" is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

<span class="mw-page-title-main">Snell's law</span> Formula for refraction angles

Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air. In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material. The law is also satisfied in meta-materials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.

<span class="mw-page-title-main">Arago spot</span> Bright point that appears at the center of a circular objects shadow due to Fresnel diffraction.

In optics, the Arago spot, Poisson spot, or Fresnel spot is a bright point that appears at the center of a circular object's shadow due to Fresnel diffraction. This spot played an important role in the discovery of the wave nature of light and is a common way to demonstrate that light behaves as a wave.

<span class="mw-page-title-main">Pinhole camera</span> Type of camera

A pinhole camera is a simple camera without a lens but with a tiny aperture —effectively a light-proof box with a small hole in one side. Light from a scene passes through the aperture and projects an inverted image on the opposite side of the box, which is known as the camera obscura effect. The size of the images depends on the distance between the object and the pinhole.

<span class="mw-page-title-main">Angular resolution</span> Ability of any image-forming device to distinguish small details of an object

Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. It is used in optics applied to light waves, in antenna theory applied to radio waves, and in acoustics applied to sound waves. The colloquial use of the term "resolution" sometimes causes confusion; when an optical system is said to have a high resolution or high angular resolution, it means that the perceived distance, or actual angular distance, between resolved neighboring objects is small. The value that quantifies this property, θ, which is given by the Rayleigh criterion, is low for a system with a high resolution. The closely related term spatial resolution refers to the precision of a measurement with respect to space, which is directly connected to angular resolution in imaging instruments. The Rayleigh criterion shows that the minimum angular spread that can be resolved by an image-forming system is limited by diffraction to the ratio of the wavelength of the waves to the aperture width. For this reason, high-resolution imaging systems such as astronomical telescopes, long distance telephoto camera lenses and radio telescopes have large apertures.

Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

<span class="mw-page-title-main">Diffraction-limited system</span> Optical system with resolution performance at the instruments theoretical limit

In optics, any optical instrument or system – a microscope, telescope, or camera – has a principal limit to its resolution due to the physics of diffraction. An optical instrument is said to be diffraction-limited if it has reached this limit of resolution performance. Other factors may affect an optical system's performance, such as lens imperfections or aberrations, but these are caused by errors in the manufacture or calculation of a lens, whereas the diffraction limit is the maximum resolution possible for a theoretically perfect, or ideal, optical system.

<span class="mw-page-title-main">Airy disk</span> Diffraction pattern in optics

In optics, the Airy disk and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The Airy disk is of importance in physics, optics, and astronomy.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance from the object, and also when it is viewed at the focal plane of an imaging lens. In contrast, the diffraction pattern created near the diffracting object and is given by the Fresnel diffraction equation.

<span class="mw-page-title-main">Fresnel diffraction</span> Diffraction

In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation.

<span class="mw-page-title-main">Acousto-optics</span> The study of sound and light interaction

Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound through an ultrasonic grating.

Free spectral range (FSR) is the spacing in optical frequency or wavelength between two successive reflected or transmitted optical intensity maxima or minima of an interferometer or diffractive optical element.

<span class="mw-page-title-main">Thin-film interference</span> Optical phenomenon

Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, increasing reflection at some wavelengths and decreasing it at others. When white light is incident on a thin film, this effect produces colorful reflections.

Kirchhoff's diffraction formula approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave is the incoming wave of a situation under consideration. This formula is derived by applying the Kirchhoff integral theorem, which uses the Green's second identity to derive the solution to the homogeneous scalar wave equation, to a spherical wave with some approximations.

Fresnel zone antennas are antennas that focus the signal by using the phase shifting property of the antenna surface or its shape. There are several types of Fresnel zone antennas, namely, Fresnel zone plate, offset Fresnel zone plate antennas, phase correcting reflective array or "Reflectarray" antennas and 3 Dimensional Fresnel antennas. They are a class of diffractive antennas and have been used from radio frequencies to X rays.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

<span class="mw-page-title-main">Scanning helium microscopy</span>

The scanning helium microscope (SHeM) is a novel form of microscopy that uses low-energy (5–100 meV) neutral helium atoms to image the surface of a sample without any damage to the sample caused by the imaging process. Since helium is inert and neutral, it can be used to study delicate and insulating surfaces. Images are formed by rastering a sample underneath an atom beam and monitoring the flux of atoms that are scattered into a detector at each point.

References

  1. 1 2 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.
  2. Wood, Robert Williams (1911), Physical Optics, New York: The MacMillan Company, pp. 37–39
  3. "Zone Plates". X-Ray Data Booklet. Center for X-ray Optics and Advanced Light Source, Lawrence Berkeley National Laboratory. Retrieved 13 January 2015.
  4. Joseph W. Goodman (2005). Introduction to Fourier Optics (3rd ed.). p. 125. ISBN   0-9747077-2-4.
  5. New INL gunsight technology should improve accuracy for target shooters, hunters, soldiers, Mike Wall, Idaho National Laboratory, 5 May 2010.
  6. Jordan, J. A.; Hirsch, P. M.; Lesem, L. B.; Van Rooy, D. L. (1970-08-01). "Kinoform Lenses". Applied Optics. 9 (8). Optica Publishing Group: 1883. doi:10.1364/ao.9.001883. ISSN   0003-6935.
  7. https://web.archive.org/web/20060827184031/http://www.path.unimelb.edu.au/~dersch/interpolator/interpolator.html Testing Interpolator Quality
  8. http://blogs.mathworks.com/steve/2011/07/22/filtering-fun/ Filtering Fun - Matlab Central
  9. https://web.archive.org/web/20200516104605/http://www.realitypixels.com/turk/opensource/#ZonePlate Zone Plate generator, C code. Archived from the original on 16 May 2020.