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Diffraction is defined as the interference or 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 rays that come to a focus in the image plane. An opt ...
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, SJ (2 April 1618 – 28 December 1663) was an Italian Jesuit priest, mathematician and physicist who taught at the Jesuit college in Bologna. He was born in Bologna to Paride Grimaldi and Anna Cattani. Work Between 1 ...
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 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 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 ...
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 deri ...
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, tro ...
, 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 extr ...
, 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 structura ...
), 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 (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 ...
, or when a
sound wave In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
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 of acoustic impedance is the pascal-second per cub ...
– all waves diffract, including
gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s,
water waves In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...
, and other
electromagnetic waves In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
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  picometers to 10  nanometers, corresponding to frequencies in the range 30&nb ...
s and
radio waves Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz (GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (short ...
. Furthermore,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
also demonstrates that matter possesses wave-like properties, and hence, undergoes diffraction (which is measurable at subatomic to molecular levels). The amount of diffraction depends on the size of the gap. Diffraction is greatest when the size of the gap is similar to the wavelength of the wave. In this case, when the waves pass through the gap they become semi-circular.


History

The effects of diffraction of light were first carefully observed and characterized by
Francesco Maria Grimaldi Francesco Maria Grimaldi, SJ (2 April 1618 – 28 December 1663) was an Italian Jesuit priest, mathematician and physicist who taught at the Jesuit college in Bologna. He was born in Bologna to Paride Grimaldi and Anna Cattani. Work Between 1 ...
, 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, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
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 diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
to be discovered. Thomas Young 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 (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Isaac Newton, Newton's co ...
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, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
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 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 ...
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 (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seis ...
. 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-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
. 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, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanics ...
) 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 Kirchhoff-Fresnel diffraction equation which is derived from the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
, the
Fraunhofer diffraction 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 (a distance satisfying Fraunhofer ...
approximation of the Kirchhoff equation which applies to the
far field The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ant ...
, the
Fresnel 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 ...
approximation which applies to the near field and the Feynman path integral formulation. Most configurations cannot be solved analytically, but can yield numerical solutions through
finite element The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...
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 out of proteinaceous spider silk extruded from its spinnerets, generally meant to catch its prey. Spid ...
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 diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
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 a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other Holography#Applications, applic ...
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 Speckle, speckle pattern, or speckle noise is a granular noise texture degrading the quality as a consequence of interference among wavefronts in coherent imaging systems, such as radar, synthetic aperture radar (SAR), medical ultrasound and optic ...
which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When
deli meat Lunch meats—also known as cold cuts, luncheon meats, cooked meats, sliced meats, cold meats, sandwich meats, and deli meats—are precooked or cured meats that are sliced and served cold or hot. They are typically served in sandwiches or on ...
appears to be
iridescent Iridescence (also known as goniochromism) is the phenomenon of certain surfaces that appear to gradually change color as the angle of view or the angle of illumination changes. Examples of iridescence include soap bubbles, feathers, butterfl ...
, that is diffraction off the meat fibers. All these effects are a consequence of the fact that light propagates as a
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 (res ...
. Diffraction can occur with any kind of wave. Ocean waves diffract around
jetties A jetty is a structure that projects from land out into water. A jetty may serve as a breakwater, as a walkway, or both; or, in pairs, as a means of constricting a channel. The term derives from the French word ', "thrown", signifying somet ...
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 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 ...
. An illuminated slit that is wider than a wavelength produces interference effects in the space downstream of the slit. Assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit interference effects can be calculated. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is
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 deri ...
, 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 2\pi 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 \theta_ 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 \theta_ 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 plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
equation as :I(\theta) = I_0 \,\operatorname^2 \left( \frac \sin\theta \right) where * I(\theta) 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 (x) = \begin \frac,&x\neq 0\\ 1,&x=0 \end is the
unnormalized sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the ...
. This analysis applies only to the
far field The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ant ...
(
Fraunhofer diffraction 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 (a distance satisfying Fraunhofer ...
), 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 that describes light propagation in terms of ''rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
. 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: :I(\theta) = I_0 \,\operatorname^2 \left \frac (\sin\theta \pm \sin\theta_i)\right/math> The choice of plus/minus sign depends on the definition of the incident angle \theta_\text.


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( \sin \pm \sin \right) = m \lambda. where * \theta_ 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 (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
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 In optics, the Airy disk (or Airy disc) 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, ...
. The variation in intensity with angle is given by :I(\theta) = I_0 \left ( \frac \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 and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
. 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 signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signa ...
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
for a point source (the
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...
), :\nabla^2 \psi + k^2 \psi = \delta(\mathbf r) where \delta(\mathbf r) is the 3-dimensional delta function. The delta function has only radial dependence, so the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
(a.k.a. scalar Laplacian) in the
spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
simplifies to (see
del in cylindrical and spherical coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinates, curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11#Coordinate systems, ISO 31- ...
) :\nabla ^2\psi= \frac \frac (r \psi) By direct substitution, the solution to this equation can be readily shown to be the scalar
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
, which in the
spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
(and using the physics time convention e^) is: :\psi(r) = \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' and the field point is located at the point \mathbf r, then we may represent the scalar
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
(for arbitrary source location) as: :\psi(\mathbf r , \mathbf r') = \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, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
: :\Psi(r)\propto \iint\limits_\mathrm E_\mathrm(x',y')~ \frac \,dx'\, dy', where the source point in the aperture is given by the vector :\mathbf' = x' \mathbf + y' \mathbf In the far field, wherein the parallel rays approximation can be employed, the Green's function, :\psi(\mathbf r , \mathbf r') = \frac simplifies to : \psi(\mathbf , \mathbf') = \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(r)\propto \frac \iint\limits_\mathrm E_\mathrm(x',y') e^ \, dx' \,dy', Now, since :\mathbf' = x' \mathbf + y' \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(r)\propto \frac \iint\limits_\mathrm E_\mathrm(x',y') e^ \, dx'\, dy' 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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
:\Psi(r)\propto \frac \iint\limits_\mathrm E_\mathrm(x',y') e^ \,dx'\, dy', In the far-field / Fraunhofer region, this becomes the spatial
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
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 plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
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 optics 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 pr ...
).


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 word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially
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 deri ...
wave front, it approximates
Gaussian beam In optics, a Gaussian beam is a 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 intensity (irradiance) profile. This ...
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 nonlinear optics, non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation. A medium whose refractive index increases with the electric field intensity acts as ...
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 is defined as 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 s ...
. 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 In optics, the Airy disk (or Airy disc) 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, ...
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 ...
(focal length ''f'' divided by aperture diameter D) of the imaging optics; this is strictly accurate for N≫1 (
paraxial In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optic ...
case). In object space, the corresponding
angular resolution 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. ...
is : \theta \approx \sin \theta = 1.22 \frac,\, where ''D'' is the diameter of the
entrance pupil In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front (the object side) of the lens system. The corresponding image of the aperture as seen through the back of the lens system is ...
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 microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. ...
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
microscope objective In optical engineering, the objective is the optical element that gathers light from the object being observed and focuses the light rays to produce a real image. Objectives can be a single lens or mirror, or combinations of several optical elem ...
s require a large
numerical aperture In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the proper ...
(large aperture diameter compared to working distance) in order to obtain the highest possible resolution.


Speckle patterns

The
speckle pattern Speckle, speckle pattern, or speckle noise is a granular noise texture degrading the quality as a consequence of interference among wavefronts in coherent imaging systems, such as radar, synthetic aperture radar (SAR), medical ultrasound and optic ...
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 coherent low-powered laser beam of visible light, intended to be used to highlight something of interest by ...
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 principle In physics, Babinet's principle states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam intensity. It was formulated in the 1800s by French physicist Ja ...
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.


"Knife edge"

The knife-edge effect or knife-edge diffraction is a truncation of a portion of the incident
radiation In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes: * ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
that strikes a sharp well-defined obstacle, such as a mountain range or the wall of a building. The knife-edge effect is explained by
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 ...
, which states that a well-defined obstruction to an electromagnetic wave acts as a secondary source, and creates a new
wavefront 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 ...
. This new wavefront propagates into the geometric shadow area of the obstacle. Knife-edge diffraction is an outgrowth of the " half-plane problem", originally solved by
Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
using a plane wave spectrum formulation. A generalization of the half-plane problem is the "wedge problem", solvable as a boundary value problem in cylindrical coordinates. The solution in cylindrical coordinates was then extended to the optical regime by
Joseph B. Keller Joseph Bishop Keller (July 31, 1923 – September 7, 2016) was an American mathematician who specialized in applied mathematics. He was best known for his work on the "geometrical theory of diffraction" (GTD). Early life and education Born i ...
, who introduced the notion of diffraction coefficients through his geometrical theory of diffraction (GTD). Pathak and Kouyoumjian extended the (singular) Keller coefficients via the
uniform theory of diffraction In numerical analysis, the uniform theory of diffraction (UTD) is a high-frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more than one dimension at the same point. UTD i ...
(UTD). File:Diffraction sharp edge.gif, Diffraction on a sharp metallic edge File:Diffraction softest edge.gif, Diffraction on a soft aperture, with a gradient of conductivity over the image width


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, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
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, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
:\lambda=\frac \, where ''h'' is Planck's constant and ''p'' is the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
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
buckyball Buckminsterfullerene is a type of fullerene with the formula C60. It has a cage-like fused-ring structure (truncated icosahedron) made of twenty hexagons and twelve pentagons, and resembles a soccer ball. Each of its 60 carbon atoms is bonded t ...
s were also shown to diffract.


Bragg diffraction

Diffraction from a three-dimensional periodic structure such as atoms in a crystal is called
Bragg diffraction In physics and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave ...
. 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 diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
. 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. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
or matter waves like
neutrons 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 behave ...
(and
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
) 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 hi ...
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, coherence length is the propagation distance over which a coherent wave (e.g. an electromagnetic wave) maintains a specified degree of coherence. Wave interference is strong when the paths taken by all of the interfering waves differ ...
. 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 Atomic electron transition is a change (or jump) of an electron from one energy level to another within an atom or artificial atom. It appears discontinuous as the electron "jumps" from one quantized energy level to another, typically in a few na ...
, 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 X-ray free electron lasers. 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.


See also

*
Angle-sensitive pixel An angle-sensitive pixel (ASP) is a CMOS sensor with a sensitivity to incoming light that is sinusoidal in incident angle. Principles of operation ASPs are typically composed of two gratings (a diffraction grating and an analyzer grating) abo ...
*
Atmospheric diffraction Atmospheric diffraction is manifested in the following principal ways: * Optical atmospheric diffraction * Radio wave diffraction is the scattering of radio frequency or lower frequencies from the Earth's ionosphere, resulting in the ability to ...
*
Brocken spectre A Brocken spectre (British English; American spelling Brocken specter; german: Brockengespenst), also called Brocken bow, mountain spectre, or spectre of the Brocken is the magnified (and apparently enormous) shadow of an observer cast in mid ai ...
*
Cloud iridescence Cloud iridescence or irisation is a colorful optical phenomenon that occurs in a cloud and appears in the general proximity of the Sun or Moon. The colors resemble those seen in soap bubbles and oil on a water surface. It is a type of photom ...
*
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 highl ...
*
Diffraction from slits Diffraction processes affecting waves are amenable to quantitative description and analysis. Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength. Numerical approximatio ...
*
Diffraction spike Diffraction spikes are lines radiating from bright light sources, causing what is known as the starburst effect or sunstars in photographs and in vision. They are artifacts caused by light diffracting around the support vanes of the secondary m ...
*
Diffraction vs. interference The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
*
Diffractometer A 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 Because it is relatively eas ...
*
Dynamical theory of diffraction The dynamical theory of diffraction describes the interaction of waves with a regular lattice. The wave fields traditionally described are X-rays, neutrons or electrons and the regular lattice are atomic crystal structures or nanometer-scale multi ...
*
Fraunhofer diffraction 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 (a distance satisfying Fraunhofer ...
*
Fresnel imager A Fresnel imager is a proposed ultra-lightweight design for a space telescope that uses a Fresnel lens, Fresnel array as primary optics instead of a typical lens. It focuses light with a thin opaque foil sheet punched with specially shaped holes, t ...
*
Fresnel number The Fresnel number (''F''), named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in diffraction, scalar diffraction theory. Definition For an electromagnetic wave passing through an apertu ...
*
Fresnel zone A Fresnel zone ( ), named after physicist Augustin-Jean Fresnel, is one of a series of confocal prolate ellipsoidal regions of space between and around a transmitter and a receiver. The primary wave will travel in a relative straight line from ...
* Point spread function *
Powder diffraction Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is call ...
*
Quasioptics Quasioptics concerns the propagation of electromagnetic radiation where the wavelength is comparable to the size of the optical components (e.g. lenses, mirrors, and apertures) and hence diffraction effects may become significant. It commonly descr ...
*
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 phenomeno ...
*
Reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
* Schaefer–Bergmann diffraction *
Thinned-array curse The thinned-array curse (sometimes, sparse-array curse) is a theorem in electromagnetic theory of antennas. It states that a transmitting antenna which is synthesized from a coherent phased array of smaller antenna apertures that are spaced apa ...
*
X-ray scattering techniques X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystal structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observ ...


References


External links


The Feynman Lectures on Physics Vol. I Ch. 30: Diffraction
* {{Authority control Physical phenomena