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In physics, the wavefront of a time-varying ''
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
field'' is the set (
locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...
) of all
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points ...
having the same '' phase''. The term is generally meaningful only for fields that, at each point, vary
sinusoidally A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
in time with a single temporal frequency (otherwise the phase is not well defined). Wavefronts usually move with time. For waves propagating in a unidimensional medium, the wavefronts are usually single points; they are
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s in a two dimensional medium, and
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s in a three-dimensional one. For a
sinusoidal plane wave In physics, a sinusoidal (or monochromatic) plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. For any position \vec x in space and any time t, the val ...
, the wavefronts are planes perpendicular to the direction of propagation, that move in that direction together with the wave. For a sinusoidal spherical wave, the wavefronts are spherical surfaces that expand with it. If the speed of propagation is different at different points of a wavefront, the shape and/or orientation of the wavefronts may change by
refraction In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomen ...
. In particular, lenses can change the shape of optical wavefronts from planar to spherical, or vice versa. In
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, the diffraction phenomenon is described by the
Huygens–Fresnel principle The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating ...
that treats each point in a propagating wavefront as a collection of individual spherical
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s. The characteristic bending pattern is most pronounced when a wave from a
coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deriv ...
source (such as a laser) encounters a slit/aperture that is comparable in size to its
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, as shown in the inserted image. This is due to the addition, or interference, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface. If there are multiple, closely spaced openings (e.g., a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structur ...
), a complex pattern of varying intensity can result.


Simple wavefronts and propagation

Optical systems can be described with
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
, and linear propagating waves such as sound or electron beams have similar wave equations. However, given the above simplifications, Huygens' principle provides a quick method to predict the propagation of a wavefront through, for example,
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
. The construction is as follows: Let every point on the wavefront be considered a new point source. By calculating the total effect from every point source, the resulting field at new points can be computed. Computational algorithms are often based on this approach. Specific cases for simple wavefronts can be computed directly. For example, a spherical wavefront will remain spherical as the energy of the wave is carried away equally in all directions. Such directions of energy flow, which are always perpendicular to the wavefront, are called ''
rays Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gra ...
'' creating multiple wavefronts. The simplest form of a wavefront is the
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...
, where the rays are
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
to one another. The light from this type of wave is referred to as ''
collimated A collimated beam of light or other electromagnetic radiation has parallel rays, and therefore will spread minimally as it propagates. A perfectly collimated light beam, with no divergence, would not disperse with distance. However, diffraction p ...
'' light. The plane wavefront is a good model for a surface-section of a very large spherical wavefront; for instance, sunlight strikes the earth with a spherical wavefront that has a radius of about 150 million kilometers (1 AU). For many purposes, such a wavefront can be considered planar over distances of the diameter of Earth. Wavefronts travel with the speed of light in all directions in an isotropic medium.


Wavefront aberrations

Methods utilizing wavefront measurements or predictions can be considered an advanced approach to lens optics, where a single focal distance may not exist due to lens thickness or imperfections. For manufacturing reasons, a perfect lens has a spherical (or toroidal) surface shape though, theoretically, the ideal surface would be ''aspheric''. Shortcomings such as these in an optical system cause what are called
optical aberrations In optics, aberration is a property of optical systems, such as lenses, that causes light to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with th ...
. The best-known aberrations include
spherical aberration In optics, spherical aberration (SA) is a type of aberration found in optical systems that have elements with spherical surfaces. Lenses and curved mirrors are prime examples, because this shape is easier to manufacture. Light rays that strik ...
and
coma A coma is a deep state of prolonged unconsciousness in which a person cannot be awakened, fails to respond normally to painful stimuli, light, or sound, lacks a normal wake-sleep cycle and does not initiate voluntary actions. Coma patients exhi ...
.''Encyclopaedia of Physics (2nd Edition)'', R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3 However there may be more complex sources of aberrations such as in a large telescope due to spatial variations in the
index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of the atmosphere. The deviation of a wavefront in an optical system from a desired perfect planar wavefront is called the ''wavefront aberration''. Wavefront aberrations are usually described as either a sampled image or a collection of two-dimensional polynomial terms. Minimization of these aberrations is considered desirable for many applications in optical systems.


Wavefront sensor and reconstruction techniques

A
wavefront sensor In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
is a device which measures the wavefront aberration in a coherent signal to describe the optical quality or lack thereof in an optical system. A very common method is to use a Shack–Hartmann lenslet array. There are many applications that include
adaptive optics Adaptive optics (AO) is a technology used to improve the performance of optical systems by reducing the effect of incoming wavefront distortions by deforming a mirror in order to compensate for the distortion. It is used in astronomical tele ...
, optical metrology and even the measurement of the aberrations in the eye itself. In this approach, a weak laser source is directed into the eye and the reflection off the
retina The retina (from la, rete "net") is the innermost, light-sensitive layer of tissue of the eye of most vertebrates and some molluscs. The optics of the eye create a focused two-dimensional image of the visual world on the retina, which the ...
is sampled and processed. Alternative wavefront sensing techniques to the Shack–Hartmann system are emerging. Mathematical techniques like phase imaging or curvature sensing are also capable of providing wavefront estimations. These algorithms compute wavefront images from conventional brightfield images at different focal planes without the need for specialised wavefront optics. While Shack-Hartmann lenslet arrays are limited in lateral resolution to the size of the lenslet array, techniques such as these are only limited by the resolution of digital images used to compute the wavefront measurements. That said, those wavefront sensors are suffering from linearity issues and so are much less robust than the original SHWFS, in term of phase measurement. Another application of software reconstruction of the phase is the control of telescopes through the use of adaptive optics. A common method is the Roddier test, also called wavefront curvature sensing. It yields good correction, but needs an already good system as a starting point. Indeed because of those linearity issues described above. That's why people are coupling different types of WFS in those next Gen adaptive optics systems.


See also

* Huygens-Fresnel principle *
Wavefront sensor In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
*
Adaptive optics Adaptive optics (AO) is a technology used to improve the performance of optical systems by reducing the effect of incoming wavefront distortions by deforming a mirror in order to compensate for the distortion. It is used in astronomical tele ...
*
Deformable mirror Deformable mirrors (DM) are mirrors whose surface can be deformed, in order to achieve wavefront control and correction of optical aberrations. Deformable mirrors are used in combination with wavefront sensors and real-time control systems in ad ...
*
Wave field synthesis Wave field synthesis (WFS) is a spatial audio rendering technique, characterized by creation of virtual acoustic environments. It produces ''artificial'' wavefronts synthesized by a large number of individually driven loudspeakers. Such wavef ...


References


Further reading


Textbooks and books

* ''Concepts of Modern Physics'' (4th Edition), A. Beiser, Physics, McGraw-Hill (International), 1987, * ''Physics with Modern Applications'', L. H. Greenberg, Holt-Saunders International W. B. Saunders and Co, 1978, * ''Principles of Physics'', J. B. Marion, W. F. Hornyak, Holt-Saunders International Saunders College, 1984, * ''Introduction to Electrodynamics'' (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007, * ''Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers'', Y. B. Band, John Wiley & Sons, 2010, * ''The Light Fantastic – Introduction to Classic and Quantum Optics'', I. R. Kenyon, Oxford University Press, 2008, * ''McGraw Hill Encyclopaedia of Physics'' (2nd Edition), C. B. Parker, 1994, *


Journals

* * * * {{Cite journal , last=Shcherbak , first=O. P. , year=1988 , title=Волновые фронты и группы отражений , via=
Russian Mathematical Surveys ''Uspekhi Matematicheskikh Nauk'' (russian: Успехи математических наук) is a Russian mathematical journal, published by the Russian Academy of Sciences and Moscow Mathematical Society and translated into English as ''Russi ...
, 43:3 (1988), 149–194 , trans-title=Wavefronts and reflection groups , url=http://www.mathnet.ru/links/100d8a1eb447dd5b7afb5374163c5c3d/rm1892.pdf , journal=Успехи математических наук , language=Russian , volume=43 , issue=3(261) , pages=125–160 , doi=10.1070/RM1988v043n03ABEH001741
Wavefront tip/tilt estimation from defocused images


External links


LightPipes
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Unix Unix (; trademarked as UNIX) is a family of multitasking, multiuser computer operating systems that derive from the original AT&T Unix, whose development started in 1969 at the Bell Labs research center by Ken Thompson, Dennis Ritchie, ...
wavefront propagation software Optics Waves