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In physics, the wavefront of a time-varying ''
wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
field'' is the set ( locus) of all
points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
having the same ''
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
''. The term is generally meaningful only for fields that, at each point, vary sinusoidally 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, 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 transmission medium, 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 commo ...
. In particular,
lenses A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
can change the shape of optical wavefronts from planar to spherical, or vice versa. In
classical physics Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
, the diffraction phenomenon is described by the
Huygens–Fresnel principle The Huygens–Fresnel principle (named after Netherlands, Dutch physicist Christiaan Huygens and France, French physicist Augustin-Jean Fresnel) states that every point on a wavefront is itself the source of spherical wavelets, and the secondary w ...
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 n ...
s. The characteristic bending pattern is most pronounced when a wave from a
coherent Coherence is, in general, a state or situation in which all the parts or ideas fit together well so that they form a united whole. More specifically, coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics ...
source (such as a laser) encounters a slit/aperture that is comparable in size to its
wavelength 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 (waves ...
, as shown in the inserted image. This is due to the addition, or
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...
, 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 grating with a periodic structure that diffraction, diffracts light, or another type of electromagnetic radiation, into several beams traveling in different directions (i.e., different diffractio ...
), 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, Electrical network, electr ...
, 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 (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
. The construction is as follows: Let every point on the wavefront be considered a new
point source A point source is a single identifiable ''localized'' source of something. A point source has a negligible extent, distinguishing it from other source geometries. Sources are called point sources because, in mathematical modeling, these sources ...
. 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'' creating multiple wavefronts. The simplest form of a wavefront is the
plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
, where the rays are parallel 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 laser beam is an archetypical example. A perfectly collimated light beam, with no divergence, would not disp ...
'' 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. In an isotropic medium wavefronts travel with the same speed in all directions.


Wavefront aberrations

Methods using 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 and mirrors, that causes the ''image'' created by the optical system to not be a faithful reproduction of the ''object'' being observed. Aberrations cause the image formed by ...
. 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. This phenomenon commonly affects lenses and curved mirrors, as these components are often shaped in a spherical ...
and
coma A coma is a deep state of prolonged unconsciousness in which a person cannot be awakened, fails to Nociception, respond normally to Pain, painful stimuli, light, or sound, lacks a normal Circadian rhythm, sleep-wake cycle and does not initiate ...
. 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 the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
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 is a device which measures the wavefront aberration in a coherent signal to describe the optical quality or lack thereof in an optical system. There are many applications that include
adaptive optics Adaptive optics (AO) is a technique of precisely deforming a mirror in order to compensate for light distortion. It is used in Astronomy, astronomical telescopes and laser communication systems to remove the effects of Astronomical seeing, atmo ...
, optical metrology and even the measurement of the aberrations in the
eye An eye is a sensory organ that allows an organism to perceive visual information. It detects light and converts it into electro-chemical impulses in neurons (neurones). It is part of an organism's visual system. In higher organisms, the ey ...
itself. In this approach, a weak laser source is directed into the eye and the reflection off the
retina The retina (; or retinas) is the innermost, photosensitivity, light-sensitive layer of tissue (biology), tissue of the eye of most vertebrates and some Mollusca, molluscs. The optics of the eye create a focus (optics), focused two-dimensional ...
is sampled and processed. Another application of software reconstruction of the phase is the control of telescopes through the use of adaptive optics. 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 suffer from linearity issues and so are much less robust than the original SHWFS, in term of phase measurement. There are several types of wavefront sensors, including: * Shack–Hartmann wavefront sensor: a very common method using a Shack–Hartmann lenslet array. * Phase-shifting Schlieren technique * Wavefront curvature sensor: also called the Roddier test. It yields good correction but needs an already good system as a starting point. * Pyramid wavefront sensor *
Common-path interferometer A common-path interferometer is a class of interferometers in which the reference beam and sample beams travel along the same path. Examples include the Sagnac interferometer, Zernike phase-contrast interferometer, and the point diffraction interfer ...
* Foucault knife-edge test * Multilateral shearing interferometer *
Ronchi test In Optics, optical testing a Ronchi test is a method of determining the surface shape (Figuring, figure) of a mirror used in Optical telescope, telescopes and other Optics, optical devices. Description In 1923 Italian people, Italian physicist Vasc ...
er * Shearing interferometer Although an amplitude splitting
interferometer Interferometry is a technique which uses the '' interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber opt ...
such as the
Michelson interferometer The Michelson interferometer is a common configuration for optical interferometry and was invented by the American physicist Albert Abraham Michelson in 1887. Using a beam splitter, a light source is split into two arms. Each of those light be ...
could be called a wavefront sensor, the term is normally applied to instruments that do not require an unaberrated reference beam to interfere with.


See also

* Huygens-Fresnel principle * Wavefront sensor *
Adaptive optics Adaptive optics (AO) is a technique of precisely deforming a mirror in order to compensate for light distortion. It is used in Astronomy, astronomical telescopes and laser communication systems to remove the effects of Astronomical seeing, atmo ...
* Deformable mirror *
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 from elemen ...
*
Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...


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

* * * *
Wavefront tip/tilt estimation from defocused images


External links


LightPipes
– Free
Unix Unix (, ; trademarked as UNIX) is a family of multitasking, multi-user 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, a ...
wavefront propagation software
AO Tutorial: Wave-front Sensors
Research groups and companies with interests in wavefront sensing and adaptive optics. {{Authority control Optics Waves