# Diffraction

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Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle. In classical physics, the diffraction phenomenon is described as the interference of waves according to the Huygens–Fresnel principle that treats each point in the wave-front as a collection of individual spherical wavelets. [1] These characteristic behaviors are exhibited when a wave encounters an obstacle or a slit that is comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance. Diffraction has an impact on the acoustic space. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays and radio waves.

In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.

The umbra, penumbra and antumbra are three distinct parts of a shadow, created by any light source after impinging on an opaque object. Assuming no diffraction, for a point source only the umbra is cast.

Classical physics refers to theories of physics 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 previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the realm of "classical physics".

## Contents

Since physical objects have wave-like properties (significantly at the atomic level, invisibly at macro level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1660. [2] [3]

Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

Francesco Maria Grimaldi 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.

While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves whose wavelength is roughly comparable to the dimensions of the diffracting object or slit. If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result. This is due to the addition, or interference, of different parts of a wave that travel to the observer by different paths, where different path lengths result in different phases (see diffraction grating and wave superposition). The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can all be analyzed using diffraction equations.

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

In optics, a diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. The emerging coloration is a form of structural coloration. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as the dispersive element. Because of this, gratings are commonly used in monochromators and spectrometers.

Ultrasonic transducers or ultrasonic sensors are a type of acoustic sensor divided into three broad categories: transmitters, receivers and transceivers. Transmitters convert electrical signals into ultrasound, receivers convert ultrasound into electrical signals, and transceivers can both transmit and receive ultrasound.

## 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 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 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 which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When deli meat appears to be iridescent, that is diffraction off the meat fibers. [4] All these effects are a consequence of the fact that light propagates as a wave.

A hologram is an image that appears to be three dimensional and which can be seen with the naked eye. Holography is the science and practice of making holograms. Typically, a hologram is a photographic recording of a light field, rather than an image formed by a lens. The holograpic medium, i.e., the object produced by a holographic process is usually unintelligible when viewed under diffuse ambient light. It is an encoding of the light field as an interference pattern of variations in the opacity, density, or surface profile of the photographic medium. When suitably lit, the interference pattern diffracts the light into an accurate reproduction of the original light field, and the objects that were in it exhibit visual depth cues such as parallax and perspective that change realistically with the relative position of the observer. That is, the view of the image from different angles represents the subject viewed from similar angles.

Atmospheric diffraction is manifested in the following principal ways:

A speckle pattern is an intensity pattern produced by the mutual interference of a set of wavefronts. This phenomenon has been investigated by scientists since the time of Newton, but speckles have come into prominence since the invention of the laser and have now found a variety of applications. The term speckle pattern is also commonly used in the experimental mechanics community to describe the pattern of physical speckles on a surface which is useful for measuring displacement fields via digital image correlation.

Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree. [5] Diffraction can also be a concern in some technical applications; it sets a fundamental limit to the resolution of a camera, telescope, or microscope.

A jetty is a structure that projects from the land out into water. Often, "jetty" refers to a walkway accessing the centre of an enclosed waterbody. The term is derived from the French word jetée, "thrown", and signifies something thrown out.

The resolution of an optical imaging system – a microscope, telescope, or camera – can be limited by factors such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to the physics of diffraction. An optical system with resolution performance at the instrument's theoretical limit is said to be diffraction-limited.

## History

The effects of diffraction of light were first carefully observed and characterized by Francesco Maria Grimaldi, 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. [6] [7] [8] Isaac Newton 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 to be discovered. [9] Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits. [10] 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 did more definitive studies and calculations of diffraction, made public in 1815 [11] and 1818, [12] and thereby gave great support to the wave theory of light that had been advanced by Christiaan Huygens [13] and reinvigorated by Young, against Newton's particle theory.

Sir Isaac Newton was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Thomas Young FRS was a British polymath and physician. Young made notable scientific contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony, and Egyptology. He "made a number of original and insightful innovations" in the decipherment of Egyptian hieroglyphs before Jean-François Champollion eventually expanded on his work. He was mentioned by, among others, William Herschel, Hermann von Helmholtz, James Clerk Maxwell, and Albert Einstein. Young has been described as "The Last Man Who Knew Everything".

Augustin-Jean Fresnel 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 Newton's corpuscular theory, from the late 1830s  until the end of the 19th century.

## Mechanism

In traditional classical physics diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle 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 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 which describes its path from the emitter through the slit to the screen. The wavefunction (the path the photon will take) 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) the existence of the photon's wavefunction was demonstrated. In the quantum approach the diffraction pattern is created by the distribution of paths, the observation of light and dark bands is the presence or absence of photons in these areas (no interference!). The quantum approach has some striking similarities to the Huygens-Fresnel principle, in that principle the light becomes a series of individually distributed light sources across the slit which is similar to the limited number of paths (or wave functions) available for the photons to travel through the slit.

There are various analytical models which allow the diffracted field to be calculated, including the Kirchhoff-Fresnel diffraction equation which is derived from wave equation, the Fraunhofer diffraction approximation of the Kirchhoff equation which applies to the far field and the Fresnel diffraction approximation which applies to the near field. Most configurations cannot be solved analytically, but can yield numerical solutions through finite element 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.

## Diffraction of light

Some examples of diffraction of light 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.

A slit which 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 ${\displaystyle {\frac {d\sin(\theta )}{2}}}$ so that the minimum intensity occurs at an angle θmin given by

${\displaystyle d\,\sin \theta _{\text{min}}=\lambda }$

where

• d is the width of the slit,
• ${\displaystyle \theta _{\text{min}}}$ is the angle of incidence at which the minimum intensity occurs, and
• ${\displaystyle \lambda }$ 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 θn given by

${\displaystyle d\,\sin \theta _{n}=n\lambda }$

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 equation as

${\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right)}$

where

• ${\displaystyle I(\theta )}$ is the intensity at a given angle,
• ${\displaystyle I_{0}}$ is the original intensity, and
• the unnormalized sinc function above is given by ${\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}}$ if ${\displaystyle x\neq 0}$, and ${\displaystyle \operatorname {sinc} (0)=1}$

This analysis applies only to the far field, that is, at a distance much larger than the width of the slit.

### 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

${\displaystyle d\left(\sin {\theta _{m}}+\sin {\theta _{i}}\right)=m\lambda .}$

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 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. The variation in intensity with angle is given by

${\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2}}$,

where a is the radius of the circular aperture, k is equal to 2π/λ and J1 is a Bessel function. 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 ${\displaystyle \psi }$ at location r that is given by the solution of the frequency domain wave equation for a point source (The Helmholtz Equation),

${\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} )}$

where ${\displaystyle \delta (\mathbf {r} )}$ is the 3-dimensional delta function. The delta function has only radial dependence, so the Laplace operator (a.k.a. scalar Laplacian) in the spherical coordinate system simplifies to (see del in cylindrical and spherical coordinates)

${\displaystyle \nabla ^{2}\psi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(r\psi )}$

By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system (and using the physics time convention ${\displaystyle e^{-i\omega t}}$) is:

${\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}}$

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 ${\displaystyle \mathbf {r} '}$ and the field point is located at the point ${\displaystyle \mathbf {r} }$, then we may represent the scalar Green's function (for arbitrary source location) as:

${\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}}$

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:

${\displaystyle \Psi (r)\propto \iint \limits _{\mathrm {aperture} }E_{\mathrm {inc} }(x',y')~{\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,dx'\,dy',}$

where the source point in the aperture is given by the vector

${\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} }$

In the far field, wherein the parallel rays approximation can be employed, the Green's function,

${\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}}$

simplifies to

${\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}$

as can be seen in the figure to the right (click to enlarge).

The expression for the far-zone (Fraunhofer region) field becomes

${\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }E_{\mathrm {inc} }(x',y')e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}\,dx'\,dy',}$

Now, since

${\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} }$

and

${\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \phi \mathbf {\hat {x}} +\sin \theta ~\sin \phi ~\mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} }$

the expression for the Fraunhofer region field from a planar aperture now becomes,

${\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }E_{\mathrm {inc} }(x',y')e^{-ik\sin \theta (\cos \phi x'+\sin \phi y')}\,dx'\,dy'}$

Letting,

${\displaystyle k_{x}=k\sin \theta \cos \phi \,\!}$

and

${\displaystyle k_{y}=k\sin \theta \sin \phi \,\!}$

the Fraunhofer region field of the planar aperture assumes the form of a Fourier transform

${\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }E_{\mathrm {inc} }(x',y')e^{-i(k_{x}x'+k_{y}y')}\,dx'\,dy',}$

In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture distribution. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern 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).

### Propagation of a laser beam

The way in which the beam profile of a laser beam changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially coherent wave front, it approximates Gaussian beam 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, 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. [14] This may result in a self-focusing 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. 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 having a central spot in the focal plane with radius to first null of

${\displaystyle d=1.22\lambda N,\,}$

where λ is the wavelength of the light and N is the f-number (focal length divided by diameter) of the imaging optics. In object space, the corresponding angular resolution is

${\displaystyle \sin \theta =1.22{\frac {\lambda }{D}},\,}$

where D is the diameter of the entrance pupil 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 specifies that two point sources can be considered to be resolvable 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, and the smaller the wavelength, the finer the resolution of an imaging system. This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see.

### Speckle patterns

The speckle pattern which is seen when using a laser pointer 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 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 sines 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

Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere 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

${\displaystyle \lambda ={\frac {h}{p}}\,}$

where h is Planck's constant and p is the momentum 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 buckyballs were also shown to diffract. [15]

## Bragg diffraction

Diffraction from a three-dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. 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:

${\displaystyle m\lambda =2d\sin \theta \,}$

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 or matter waves like neutrons (and electrons) whose wavelength is on the order of (or much smaller than) the atomic spacing. [16] The pattern produced gives information of the separations of crystallographic planes d, allowing one to deduce the crystal structure. Diffraction contrast, in electron microscopes 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. [17] :919

The length over which the phase in a beam of light is correlated, is called the coherence length. 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. [18] :71–74 [19] :314–316

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. [18] :74–79

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. [20] :107

## Related Research Articles

In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called interferograms.

Angular resolution or spatial 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. In physics and geosciences, the term spatial resolution refers to the precision of a measurement with respect to space.

Electron 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 at a sample and observing the resulting interference pattern. This phenomenon is commonly known as wave–particle duality, which states that a particle of matter can be described as a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. This technique is similar to X-ray and neutron diffraction.

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

In physics, Bragg's law, or Wulff–Bragg's condition, a special case of Laue diffraction, gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move, as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency, blurred slightly due to a variety of effects; this phenomenon is known as Rayleigh scattering. The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible.

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

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

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

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, , at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

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 approximations may be used, including the Fresnel and Fraunhofer approximations.

A blazed grating – also called echelette grating – is a special type of diffraction grating. It is optimized to achieve maximum grating efficiency in a given diffraction order. For this purpose, maximum optical power is concentrated in the desired diffraction order while the residual power in the other orders is minimized. Since this condition can only exactly be achieved for one wavelength, it is specified for which blaze wavelength the grating is optimized. The direction in which maximum efficiency is achieved is called the blaze angle and is the third crucial characteristic of a blazed grating directly depending on blaze wavelength and diffraction order.

An ultrasonic grating is a type of diffraction grating produced by interfering ultrasonic waves in a medium altering the physical properties of the medium, and hence the refractive index, in a grid-like pattern. The term acoustic grating is a more general term that includes operation at audible frequencies.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

Quantum mechanics was first applied to optics, and interference in particular, by Paul Dirac. Richard Feynman, in his Lectures on Physics, uses Dirac's notation to describe thought experiments on double-slit interference of electrons. Feynman's approach was extended to N-slit interferometers for either single-photon illumination, or narrow-linewidth laser illumination, that is, illumination by indistinguishable photons, by Frank Duarte. The N-slit interferometer was first applied in the generation and measurement of complex interference patterns.

Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light. When the thickness of the film is an odd multiple of one quarter-wavelength of the light on it, the reflected waves from both surfaces interfere to cancel each other. Since the wave cannot be reflected, it is completely transmitted instead. When the thickness is a multiple of a half-wavelength of the light, the two reflected waves reinforce each other, increasing the reflection and reducing the transmission. Thus when white light, which consists of a range of wavelengths, is incident on the film, certain wavelengths (colors) are intensified while others are attenuated. Thin-film interference explains the multiple colors seen in light reflected from soap bubbles and oil films on water. It is also the mechanism behind the action of antireflection coatings used on glasses and camera lenses.

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

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

## References

1. Wireless Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T. S. Rappaport, Prentice Hall, 2002 pg 126
2. Francesco Maria Grimaldi, Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo (Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665), page 2 Archived 2016-12-01 at the Wayback Machine :
Original : Nobis alius quartus modus illuxit, quem nunc proponimus, vocamusque; diffractionem, quia advertimus lumen aliquando diffringi, hoc est partes eius multiplici dissectione separatas per idem tamen medium in diversa ulterius procedere, eo modo, quem mox declarabimus.
Translation : It has illuminated for us another, fourth way, which we now make known and call "diffraction" [i.e., shattering], because we sometimes observe light break up; that is, that parts of the compound [i.e., the beam of light], separated by division, advance farther through the medium but in different [directions], as we will soon show.
3. Cajori, Florian "A History of Physics in its Elementary Branches, including the evolution of physical laboratories." Archived 2016-12-01 at the Wayback Machine MacMillan Company, New York 1899
4. Arumugam, Nadia. "Food Explainer: Why Is Some Deli Meat Iridescent?". Slate. The Slate Group. Archived from the original on 10 September 2013. Retrieved 9 September 2013.
5. Andrew Norton (2000). Dynamic fields and waves of physics. CRC Press. p. 102. ISBN   978-0-7503-0719-2.
6. Francesco Maria Grimaldi, Physico-mathesis de lumine, coloribus, et iride, aliisque adnexis … [The physical mathematics of light, color, and the rainbow, and other things appended …] (Bologna ("Bonomia"), (Italy): Vittorio Bonati, 1665), pp. 1–11 Archived 2016-12-01 at the Wayback Machine : "Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte." (Proposition 1. Light propagates or spreads not only in a straight line, by refraction, and by reflection, but also by a somewhat different fourth way: by diffraction.) On p. 187, Grimaldi also discusses the interference of light from two sources: "Propositio XXII. Lumen aliquando per sui communicationem reddit obscuriorem superficiem corporis aliunde, ac prius illustratam." (Proposition 22. Sometimes light, as a result of its transmission, renders dark a body's surface, [which had been] previously illuminated by another [source].)
7. Jean Louis Aubert (1760). Memoires pour l'histoire des sciences et des beaux arts. Paris: Impr. de S. A. S.; Chez E. Ganeau. p. 149.
8. Sir David Brewster (1831). A Treatise on Optics. London: Longman, Rees, Orme, Brown & Green and John Taylor. p. 95. Archived from the original on 2016-12-01.
9. Letter from James Gregory to John Collins, dated 13 May 1673. Reprinted in: Correspondence of Scientific Men of the Seventeenth Century …, ed. Stephen Jordan Rigaud (Oxford, England: Oxford University Press, 1841), vol. 2, pp. 251–255, especially p. 254 Archived 2016-12-01 at the Wayback Machine .
10. Thomas Young (1804-01-01). "The Bakerian Lecture: Experiments and calculations relative to physical optics". Philosophical Transactions of the Royal Society of London . 94: 1–16. doi:10.1098/rstl.1804.0001.. (Note: This lecture was presented before the Royal Society on 24 November 1803.)
11. Augustin-Jean Fresnel (1816) "Mémoire sur la Diffraction de la lumière, où l'on examine particulièrement le phénomène des franges colorées que présentent les ombres des corps éclairés par un point lumineux" (Memoir on the diffraction of light, in which is examined particularly the phenomenon of colored fringes that the shadows of bodies illuminated by a point source display), Annales de la Chimie et de Physique, 2nd series, vol. 1, pages 239–281. (Presented before l'Académie des sciences on 15 October 1815.)
12. See:
• Excerpts from Fresnel's paper on diffraction were published in 1819: A. Fresnel (1819) "Mémoire sur la diffraction de la lumière" (Memoir on the diffraction of light), Annales de chimie et de physique, 11 : 246–296 Archived 2016-12-01 at the Wayback Machine and 337–378. Archived 2016-12-01 at the Wayback Machine
• The complete version of Fresnel's paper on diffraction was published in 1821: Augustin-Jean Fresnel (1821) "Mémoire sur la diffraction de la lumière" Archived 2014-09-07 at the Wayback Machine (Memoir on the diffraction of light), Mémoires de l'Académie des sciences de l'Institut de France, 5 : 339–475. (Summitted to l'Académie des sciences of Paris on 20 April 1818.)
13. Christiaan Huygens, Traité de la lumiere Archived 2016-06-16 at the Wayback Machine (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From p. 15 Archived 2016-12-01 at the Wayback Machine : "J'ay donc monstré de quelle façon l'on peut concevoir que la lumiere s'etend successivement par des ondes spheriques, … " (I have thus shown in what manner one can imagine that light propagates successively by spherical waves, … ) (Note: Huygens published his Traité in 1690; however, in the preface to his book, Huygens states that in 1678 he first communicated his book to the French Royal Academy of Sciences.)
14. Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964). "SELF-TRAPPING OF OPTICAL BEAMS". Physical Review Letters. 13 (15): 479–482. Bibcode:1964PhRvL..13..479C. doi:10.1103/PhysRevLett.13.479.
15. Brezger, B.; Hackermüller, L.; Uttenthaler, S.; Petschinka, J.; Arndt, M.; Zeilinger, A. (February 2002). "Matter–Wave Interferometer for Large Molecules" (reprint). Physical Review Letters. 88 (10): 100404. arXiv:. Bibcode:2002PhRvL..88j0404B. doi:10.1103/PhysRevLett.88.100404. PMID   11909334. Archived (PDF) from the original on 2007-08-13. Retrieved 2007-04-30.
16. John M. Cowley (1975) Diffraction physics (North-Holland, Amsterdam) ISBN   0-444-10791-6
17. Halliday, David; Resnick, Robert; Walker, Jerl (2005), Fundamental of Physics (7th ed.), USA: John Wiley and Sons, Inc., ISBN   978-0-471-23231-5
18. Grant R. Fowles (1975). Introduction to Modern Optics. Courier Corporation. ISBN   978-0-486-65957-2.
19. Hecht, Eugene (2002), Optics (4th ed.), United States of America: Addison Wesley, ISBN   978-0-8053-8566-3
20. Ayahiko Ichimiya; Philip I. Cohen (13 December 2004). Reflection High-Energy Electron Diffraction. Cambridge University Press. ISBN   978-0-521-45373-8. Archived from the original on 16 July 2017.