TheInfoList

Diffraction refers to various phenomena that occur when a
wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

encounters an obstacle or opening. It is defined as the bending of waves around the corners of an obstacle or through an
aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of ray (optics), rays that come to a focus (optics), focus ...

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 Francesco Maria Grimaldi (2 April 1618 – 28 December 1663) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Itali ...

coined the word ''diffraction'' and was the first to record accurate observations of the phenomenon in 1660. 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 Netherlands, Dutch physicist Christiaan Huygens and France, French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation both in the Far-field diffraction p ...
that treats each point in a propagating
wavefront In physics, the wavefront of a time-varying field is the set () of all where the wave has the same of the sinusoid. The term is generally meaningful only for fields that, at each point, vary in time with a single temporal frequency (otherwise ...

as a collection of individual spherical
wavelet A wavelet is a wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion a ...

s. The characteristic bending pattern is most pronounced when a wave from a coherent 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 (waves), phase on the wave, such as two adja ...

, 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 extran ...
, 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, (e.g., a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...

), a complex pattern of varying intensity can result. These effects also occur when a light wave travels through a medium with a varying
refractive index In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes how fast EM radiation, light travels through the material. It is defined as :n = \frac ...

, or when a
sound wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

travels through a medium with varying
acoustic impedance Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. The SI unit The International System of Units, known ...
– all waves diffract, including
gravitational wave Gravitational waves are disturbances in the curvature of spacetime In , spacetime is any which fuses the and the one of into a single . can be used to visualize effects, such as why different observers perceive differently where and wh ...
s,
water waves In , a wind wave, or wind-generated wave, is a water that occurs on the of . Wind waves result from the blowing over a fluid surface, where the contact distance in the direction of the wind is known as the '. Waves in the oceans can travel ...
, and other
electromagnetic waves In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

such as
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency ...

s and
radio waves Radio waves are a type of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space a ...

. Furthermore,
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
also demonstrates that matter possesses wave-like properties, and hence, undergoes diffraction (which is measurable at subatomic to molecular levels).

# History

The effects of diffraction of light were first carefully observed and characterized by
Francesco Maria Grimaldi Francesco Maria Grimaldi (2 April 1618 – 28 December 1663) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Itali ...

, who also coined the term ''diffraction'', from the Latin ''diffringere'', 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in 1665.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

studied these effects and attributed them to ''inflexion'' of light rays. James Gregory (1638–1675) observed the diffraction patterns caused by a bird feather, which was effectively the first
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...

to be discovered. performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves.
Augustin-Jean Fresnel Augustin-Jean Fresnel ( or ; ; 10 May 1788 – 14 July 1827) was a French civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engineering, professional engi ...
did more definitive studies and calculations of diffraction, made public in 1816 and 1818, and thereby gave great support to the wave theory of light that had been advanced by
Christiaan Huygens Christiaan Huygens ( , also , ; la, Hugenius; 14 April 1629 – 8 July 1695), also spelled Huyghens, was a Dutch mathematician, physicist, astronomer and inventor, who is regarded as one of the greatest scientists of all time and a major fig ...

and reinvigorated by Young, against Newton's particle theory.

# Mechanism

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 ...
diffraction arises because of the way in which waves propagate; this 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) is a method of analysis applied to problems of wave propagation both in the Far-field diffraction p ...
and the principle of superposition of waves. The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a point source for a secondary
spherical wave The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. I ...
. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima. In the modern quantum mechanical understanding of light propagation through a slit (or slits) every photon has what is known as a
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex number, complex-valued probability amplitude, and the probabilities for the possible results of me ...
. The wavefunction is determined by the physical surroundings such as slit geometry, screen distance and initial conditions when the photon is created. In important experiments (A low-intensity double-slit experiment was first performed by G. I. Taylor in 1909, see
double-slit experiment In modern physics Modern physics is a branch of physics either developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum physics, special r ...
) the existence of the photon's wavefunction was demonstrated. In the quantum approach the diffraction pattern is created by the probability distribution, the observation of light and dark bands is the presence or absence of photons in these areas, where these particles were more or less likely to be detected. The quantum approach has some striking similarities to the Huygens-Fresnel principle; based on that principle, as light travels through slits and boundaries, secondary, point light sources are created near or along these obstacles, and the resulting diffraction pattern is going to be the intensity profile based on the collective interference of all these lights sources that have different optical paths. That is similar to considering the limited regions around the slits and boundaries where photons are more likely to originate from, in the quantum formalism, and calculating the probability distribution. This distribution is directly proportional to the intensity, in the classical formalism. There are various analytical models which allow the diffracted field to be calculated, including the which is derived from the
wave equation The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and . Historically, the problem of a such as that of a was studied by , , , and ...
, the
Fraunhofer diffraction 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 (in the far-field region), and also when it is viewed at the focal plan ...
approximation of the Kirchhoff equation which applies to the
far field and Fresnel diffraction Augustin-Jean Fresnel ( or ; ; 10 May 1788 – 14 July 1827) was a French civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engi ...
and the
Fresnel diffraction Augustin-Jean Fresnel ( or ; ; 10 May 1788 – 14 July 1827) was a French civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engineering, professional engi ...

approximation which applies to the near field. Most configurations cannot be solved analytically, but can yield numerical solutions through
finite element The finite element method (FEM) is a widely used method for numerically solving differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
and boundary element methods. It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out. The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For water waves, this is already the case; water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three-dimensional nature of the problem. File:Square diffraction.jpg, Computer generated intensity pattern formed on a screen by diffraction from a square aperture. File:Two-Slit Diffraction.png, Generation of an interference pattern from two-slit diffraction. File:Doubleslit.gif, Computational model of an interference pattern from two-slit diffraction. File:Optical diffraction pattern ( laser), (analogous to X-ray crystallography).JPG, Optical diffraction pattern ( laser), (analogous to X-ray crystallography) File:Diffraction pattern in spiderweb.JPG, Colors seen in a
spider web A spider web, spiderweb, spider's web, or cobweb (from the archaic word '' coppe'', meaning "spider") is a structure created by a spider Spiders ( order Araneae) are air-breathing arthropod An arthropod (, (gen. ποδός)) is an inver ...

are partially due to diffraction, according to some analyses.

# Examples

The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD act as a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...

to form the familiar rainbow pattern seen when looking at a disc. This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the
hologram Holography is the science and practice of making holograms. A hologram, also known as a holograph, (from the Greek for "whole description" or "whole picture") is a recording of an interference pattern which uses diffraction Diffraction r ...

on a credit card is an example. Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The
speckle pattern A speckle pattern is produced by the mutual interference of a set of coherent wavefronts. Although this phenomenon has been investigated by scientists since the time of Newton, speckles have come into prominence since the invention of the laser. ...

which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When
deli meat Deli may refer to: * Delicatessen A delicatessen or deli is a retail establishment that sells a selection of fine, exotic, or foreign prepared foods. Delicatessen originated in Germany (original: ''Delikatessen'') during the 18th century and spr ...
appears to be
iridescent Iridescence (also known as goniochromism) is the phenomenon of certain surfaces that appear to Gradient, gradually change color as the angle of view or the angle of illumination changes. Examples of iridescence include soap bubbles, feathers, ...
, that is diffraction off the meat fibers. All these effects are a consequence of the fact that light propagates as a
wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

. Diffraction can occur with any kind of wave. Ocean waves diffract around
jetties File:Seebrücke Swakopmund, Namibia.jpg, Aerial view of a jetty at Swakopmund, Namibia (2017) A jetty is a structure that projects from land out into water. It may also refer more specifically to a walkway accessing the centre of an enclosed wa ...

and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope. Other examples of diffraction are considered below.

## Single-slit diffraction

A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity, in accordance with
Huygens–Fresnel principle The Huygens–Fresnel principle (named after Netherlands, Dutch physicist Christiaan Huygens and France, French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation both in the Far-field diffraction p ...
. A slit that is wider than a wavelength produces interference effects in the space downstream of the slit. These can be explained by assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is coherent, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by 2π or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit. We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to ''λ''/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is approximately $\frac$ so that the minimum intensity occurs at an angle ''θ''min given by :$d\,\sin\theta_\text = \lambda$ where * ''d'' is the width of the slit, * $\theta_\text$ is the angle of incidence at which the minimum intensity occurs, and * $\lambda$ is the wavelength of the light A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles ''θ''''n'' given by :$d\,\sin\theta_ = n\lambda$ where * ''n'' is an integer other than zero. There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the
Fraunhofer diffraction 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 (in the far-field region), and also when it is viewed at the focal plan ...
equation as :$I\left(\theta\right) = I_0 \,\operatorname^2 \left\left( \frac \sin\theta \right\right)$ where * $I\left(\theta\right)$ is the intensity at a given angle, * $I_0$ is the intensity at the central maximum ($\theta=0$), which is also a normalization factor of the intensity profile that can be determined by an integration from $\theta=-\frac$ to $\theta=\frac$ and conservation of energy. *$\operatorname \left(x\right) = \begin \frac,&x\neq 0\\ 1,&x=0 \end$ is the unnormalized sinc function. This analysis applies only to the
far field and Fresnel diffraction Augustin-Jean Fresnel ( or ; ; 10 May 1788 – 14 July 1827) was a French civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engi ...
(
Fraunhofer diffraction 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 (in the far-field region), and also when it is viewed at the focal plan ...
), that is, at a distance much larger than the width of the slit. From the intensity profile above, if $d \ll \lambda$, the intensity will have little dependency on $\theta$, hence the wavefront emerging from the slit would resemble a cylindrical wave with azimuthal symmetry; If $d \gg \lambda$, only $\theta \approx 0$ would have appreciable intensity, hence the wavefront emerging from the slit would resemble that of
geometrical optics Geometrical optics, or ray optics, is a model of optics Optics is the branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural sci ...
. When the incident angle $\theta_\text$ of the light onto the slit is non-zero (which causes a change in the path length), the intensity profile in the Fraunhofer regime (i.e. far field) becomes: :

## Diffraction grating

A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation :$d \left\left( \sin \pm \sin \right\right) = m \lambda.$ where * θi is the angle at which the light is incident, * ''d'' is the separation of grating elements, and * ''m'' is an integer which can be positive or negative. The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a
convolution In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of diffraction and interference patterns. The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.

## Circular aperture

The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the
Airy Disk A computer-generated Airy disk from diffracted white light ( D65 spectrum). Note that the red component is diffracted more than the blue, so that the center appears slightly bluish. In optics Optics is the branch of physics Physics (f ...
. The
variation Variation or Variations may refer to: Science and mathematics * Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon * Genetic variation thumb File:Genetic Variation and Inhe ...
in intensity with angle is given by :$I\left(\theta\right) = I_0 \left \left( \frac \right \right)^2$, where ''a'' is the radius of the circular aperture, ''k'' is equal to 2π/λ and J1 is a
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematici ...
. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.

## General aperture

The wave that emerges from a point source has amplitude $\psi$ at location r that is given by the solution of the
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (information theory), signals with respect to frequency, rather than time. Put simply, a time-dom ...
wave equation The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and . Historically, the problem of a such as that of a was studied by , , , and ...
for a point source (the
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Hermann von Helmholtz, Helmholtz equation. It corresponds to the linear partial differential equation: \nabla^2 f = -k^2 f where is the Laplace operator (or "Laplac ...
), :$\nabla^2 \psi + k^2 \psi = \delta\left(\mathbf r\right)$ where $\delta\left(\mathbf r\right)$ is the 3-dimensional delta function. The delta function has only radial dependence, so the
Laplace operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(a.k.a. scalar Laplacian) in the
spherical coordinate system File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is oft ...

simplifies to (see
del in cylindrical and spherical coordinatesThis is a list of some vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculu ...
) :$\nabla ^2\psi= \frac \frac \left(r \psi\right)$ By direct substitution, the solution to this equation can be readily shown to be the scalar
Green's function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which in the
spherical coordinate system File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is oft ...

(and using the physics time convention $e^$) is: :$\psi\left(r\right) = \frac$ This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector $\mathbf r\text{'}$ and the field point is located at the point $\mathbf r$, then we may represent the scalar
Green's function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(for arbitrary source location) as: :$\psi\left(\mathbf r , \mathbf r\text{'}\right) = \frac$ Therefore, if an electric field, Einc(''x'',''y'') is incident on the aperture, the field produced by this aperture distribution is given by the
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

: :$\Psi\left(r\right)\propto \iint\limits_\mathrm E_\mathrm\left(x\text{'},y\text{'}\right)~ \frac \,dx\text{'}\, dy\text{'},$ where the source point in the aperture is given by the vector :$\mathbf\text{'} = x\text{'} \mathbf + y\text{'} \mathbf$ In the far field, wherein the parallel rays approximation can be employed, the Green's function, :$\psi\left(\mathbf r , \mathbf r\text{'}\right) = \frac$ simplifies to :$\psi\left(\mathbf , \mathbf\text{'}\right) = \frac e^$ as can be seen in the figure to the right (click to enlarge). The expression for the far-zone (Fraunhofer region) field becomes :$\Psi\left(r\right)\propto \frac \iint\limits_\mathrm E_\mathrm\left(x\text{'},y\text{'}\right) e^ \, dx\text{'} \,dy\text{'},$ Now, since :$\mathbf\text{'} = x\text{'} \mathbf + y\text{'} \mathbf$ and :$\mathbf = \sin \theta \cos \phi \mathbf + \sin \theta ~ \sin \phi ~ \mathbf + \cos \theta \mathbf$ the expression for the Fraunhofer region field from a planar aperture now becomes, :$\Psi\left(r\right)\propto \frac \iint\limits_\mathrm E_\mathrm\left(x\text{'},y\text{'}\right) e^ \, dx\text{'}\, dy\text{'}$ Letting, :$k_x = k \sin \theta \cos \phi \,\!$ and :$k_y = k \sin \theta \sin \phi \,\!$ the Fraunhofer region field of the planar aperture assumes the form of a
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
:$\Psi\left(r\right)\propto \frac \iint\limits_\mathrm E_\mathrm\left(x\text{'},y\text{'}\right) e^ \,dx\text{'}\, dy\text{'},$ In the far-field / Fraunhofer region, this becomes the spatial
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
of the aperture distribution. Huygens' principle when applied to an aperture simply says that the
far-field diffraction pattern 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 (in the far-field region), and also when it is viewed at the focal plane o ...
is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see
Fourier opticsFourier optics is the study of classical optics Optics is the branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that ...
).

## Propagation of a laser beam

The way in which the beam profile of a
laser beam A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The term "laser" originated as an acronym for "light amplification by stimulated emission of radia ...

changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially coherent wave front, it approximates
Gaussian beam. In optics, a Gaussian beam is a Light beam, beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian irradiance, intensity (irradia ...
profile and has the lowest divergence for a given diameter. The smaller the output beam, the quicker it diverges. It is possible to reduce the divergence of a laser beam by first expanding it with one
convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a #Compound lenses, compound lens consists of several simp ...

, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger diameter, and hence a lower divergence. Divergence of a laser beam may be reduced below the diffraction of a Gaussian beam or even reversed to convergence if the refractive index of the propagation media increases with the light intensity. This may result in a
self-focusing Self-focusing is a non-linear optical process induced by the change in refractive index In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes ...
effect. When the wave front of the emitted beam has perturbations, only the transverse coherence length (where the wave front perturbation is less than 1/4 of the wavelength) should be considered as a Gaussian beam diameter when determining the divergence of the laser beam. If the transverse coherence length in the vertical direction is higher than in horizontal, the laser beam divergence will be lower in the vertical direction than in the horizontal.

## Diffraction-limited imaging

The ability of an imaging system to resolve detail is ultimately limited by
diffraction Diffraction refers to various phenomena that occur when a wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that ...
. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an
Airy disk A computer-generated Airy disk from diffracted white light ( D65 spectrum). Note that the red component is diffracted more than the blue, so that the center appears slightly bluish. In optics Optics is the branch of physics Physics (f ...
having a central spot in the focal plane whose radius (as measured to the first null) is :$\Delta x = 1.22 \lambda N$ where λ is the wavelength of the light and ''N'' is the
f-number In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill Prof ...
(focal length ''f'' divided by aperture diameter D) of the imaging optics; this is strictly accurate for N≫1 ( paraxial case). In object space, the corresponding
angular resolution Angular resolution describes the ability of any image-forming device In optics, an image-forming optical system is a system capable of being used for Image, imaging. The diameter of the aperture of the main objective is a common criterion for c ...

is :$\theta \approx \sin \theta = 1.22 \frac,\,$ where ''D'' is the diameter of the
entrance pupil In an optical Optics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the ...
of the imaging lens (e.g., of a telescope's main mirror). Two point sources will each produce an Airy pattern – see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image. The
Rayleigh criterion Angular resolution describes the ability of any image-forming device such as an optical or radio telescope A radio telescope is a specialized antenna and radio receiver radio in the 1940s. During the golden age of radio, 1925–1955, fa ...
specifies that two point sources are considered "resolved" if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other. Thus, the larger the aperture of the lens compared to the wavelength, the finer the resolution of an imaging system. This is one reason astronomical telescopes require large objectives, and why s require a large
numerical aperture of light goes through a flat plane of glass, its half-angle changes to . Due to Snell's law, the numerical aperture remains the same:\text = n_1 \sin \theta_1 = n_2 \sin\theta_2. In optics Optics is the branch of physics Physics (from ...

(large aperture diameter compared to working distance) in order to obtain the highest possible resolution.

## Speckle patterns

The
speckle pattern A speckle pattern is produced by the mutual interference of a set of coherent wavefronts. Although this phenomenon has been investigated by scientists since the time of Newton, speckles have come into prominence since the invention of the laser. ...

seen when using a
laser pointer A laser pointer or laser pen is a small handheld device with a power source (usually a battery) and a laser diode emitting a very narrow Coherence (physics), coherent low-powered laser beam of visible light, intended to be used to highlight some ...

is another diffraction phenomenon. It is a result of the superposition of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity, varies randomly.

## Babinet's principle

Babinet's principleIn physics, Babinet's principleM. Born and E. Wolf, ''Principles of Optics ''Principles of Optics'', colloquially known as ''Born and Wolf'', is an optics textbook written by Max Born and Emil Wolf that was initially published in 1959 by Pergamon ...
is a useful theorem stating that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape, but with differing intensities. This means that the interference conditions of a single obstruction would be the same as that of a single slit.

# Patterns

Several qualitative observations can be made of diffraction in general: * The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: The smaller the diffracting object, the 'wider' the resulting diffraction pattern, and vice versa. (More precisely, this is true of the
sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of the angles.) * The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object. * When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.

# Particle diffraction

According to quantum theory every particle exhibits wave properties. In particular, massive particles can interfere with themselves and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the
de Broglie wavelength Matter waves are a central part of the theory of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic ...
:$\lambda=\frac \,$ where ''h'' is and ''p'' is the
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

of the particle (mass × velocity for slow-moving particles). For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A sodium atom traveling at about 30,000 m/s would have a De Broglie wavelength of about 50 pico meters. Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins. Relatively larger molecules like s were also shown to diffract.

# Bragg diffraction

Diffraction from a three-dimensional periodic structure such as atoms in a crystal is called
Bragg diffractionBragg may refer to: Places * Bragg City, Missouri, United States * Bragg, Texas, a ghost town, United States * Bragg, West Virginia, an unincorporated community, United States *Electoral district of Bragg, a state electoral district in South Austral ...

. It is similar to what occurs when waves are scattered from a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...

. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes. The condition of constructive interference is given by ''Bragg's law'': :$m \lambda = 2 d \sin \theta$ where *λ is the wavelength, *''d'' is the distance between crystal planes, *θ is the angle of the diffracted wave. *and ''m'' is an integer known as the ''order'' of the diffracted beam. Bragg diffraction may be carried out using either electromagnetic radiation of very short wavelength like
X-rays An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ...

or matter waves like
neutrons The neutron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass ...
(and
electrons The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has m ...

) whose wavelength is on the order of (or much smaller than) the atomic spacing. The pattern produced gives information of the separations of crystallographic planes ''d'', allowing one to deduce the crystal structure. Diffraction contrast, in
electron microscope An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a high ...

s and x-topography devices in particular, is also a powerful tool for examining individual defects and local strain fields in crystals.

# Coherence

The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent. The length over which the phase in a beam of light is correlated, is called the
coherence length In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...
. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted by an atomic transition, the coherence length is related to the lifetime of the excited state from which the atom made its transition. If waves are emitted from an extended source, this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double slit experiment, this would mean that if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would look like two single slit diffraction patterns. In the case of particles like electrons, neutrons, and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.

# Applications

## Diffraction before destruction

A new way to image single biological particles has emerged over the last few years, utilising the bright X-rays generated by . These femtosecond-duration pulses will allow for the (potential) imaging of single biological macromolecules. Due to these short pulses, radiation damage can be outrun, and diffraction patterns of single biological macromolecules will be able to be obtained.

* Angle-sensitive pixel *
Atmospheric diffraction Atmospheric diffraction is manifested in the following principal ways: * Optical atmospheric diffraction * Radio wave Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared ...
*
Bragg diffractionBragg may refer to: Places * Bragg City, Missouri, United States * Bragg, Texas, a ghost town, United States * Bragg, West Virginia, an unincorporated community, United States *Electoral district of Bragg, a state electoral district in South Austral ...

*
Brocken spectre A Brocken spectre (german: Brockengespenst), also called Brocken bow, mountain spectre, or spectre of the Brocken is the magnified (and apparently enormous) shadow A shadow is a dark area where light Light or visible light is electroma ...
*
Cloud iridescence Cloud iridescence or irisation is a colorful optical phenomenon that occurs in a cloud In meteorology Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major f ...

*
Coherent diffraction imaging Coherent diffractive imaging (CDI) is a "lensless" technique for 2D or 3D reconstruction of the image of nanoscale structures such as nanotubes, nanocrystals, porous nanocrystalline layers, defects, potentially proteins, and more. In CDI, a highly ...
* Diffraction formalism *
Diffraction limit , who approximated the diffraction limit of a microscope as d=\frac, where ''d'' is the resolvable feature size, ''λ'' is the wavelength of light, ''n'' is the index of refraction of the medium being imaged in, and ''θ'' (depicted as ''α'' in the ...
*
Diffraction spike Diffraction spikes from various stars seen on an image taken by the Hubble Space Telescope Diffraction spikes are lines radiating from bright light sources, causing what is known as the starburst effect or sunstars in photographs and in vision. Th ...
* Diffraction vs. interference *
DiffractometerA diffractometer is a measuring instrument for analyzing the structure of a material from the scattering pattern produced when a beam of radiation or particles (such as X-rays or neutrons) interacts with it. Principle Image:2D x-ray diffractometer a ...

*
Dynamical theory of diffractionThe dynamical theory of diffraction describes the interaction of wave In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science th ...
*
Electron diffraction #REDIRECT Electron diffraction#REDIRECT Electron diffractionElectron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons ...

*
Fraunhofer diffraction 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 (in the far-field region), and also when it is viewed at the focal plan ...
*
Fresnel diffraction Augustin-Jean Fresnel ( or ; ; 10 May 1788 – 14 July 1827) was a French civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engineering, professional engi ...

*
Fresnel imager A Fresnel imager is a proposed ultra-lightweight design for a space telescope that uses a Fresnel array as primary optics instead of a typical lens. It focuses light with a thin opaque foil sheet punched with specially shaped holes, thus focusing li ...
*
Fresnel number perfect lens having Fresnel number equal to 100. Adopted wavelength for propagation is 1  µm. Image:half inch perfect lens real amp Fresnel number 001 at focus.png, 210px, Aperture real amplitude as estimated at focus of a half inch perfect ...
*
Fresnel zone A Fresnel zone ( ), named after physicist Augustin-Jean Fresnel, is one of a series of confocal prolate ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings ...

*
Neutron diffraction The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...
*
Prism A prism An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical ...
*
Powder diffraction upright=2, X-ray powder diffraction of Y2Cu2O5 and yttrium_oxide.html"_;"title="Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide">Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide_impurity_(red_tickers). Powder_diff ...
*
Refraction In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

*
ReflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
* Schaefer–Bergmann diffraction * Thinned array curse * X-ray scattering techniques