Coherence is the property of wave-like states that enables them to exhibit interference. It is also the parameter that quantifies the quality of the interference (also known as the degree of coherence). It was originally introduced in connection with Young’s double-slit experiment in optics but is now used in any field that involves waves, such as acoustics, electrical engineering, neuroscience, and quantum physics. In interference, at least two wave-like entities are combined and, depending on the relative phase between them, they can add constructively or subtract destructively. The degree of coherence is equal to the interference visibility, a measure of how perfectly the waves can cancel due to destructive interference. The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and telescope interferometers (astronomical optical interferometers and radio telescopes). Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... In optics, correlation functions are used to characterize the statistical and coherence properties of an electromagnetic field. ... Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ... For the book by Sir Isaac Newton, see Opticks. ... Acoustics is a branch of physics and is the study of sound (mechanical waves in gases, liquids, and solids). ... Electrical Engineers design power systems… … and complex electronic circuits. ... Drawing of the cells in the chicken cerebellum by S. Ramón y Cajal Neuroscience is a field that is devoted to the scientific study of the nervous system. ... Fig. ... This article is about a portion of a periodic process. ... The interferometric visibility (also known as interference visibility or just visibility) quantifies the contrast of interference in any system which has wave-like properties, such as optics, quantum mechanics, water waves, or electrical signals. ... Holography (from the Greek, όλος-hòlòs whole + γραφή-grafè writh) is the science of producing holograms; it is an advanced form of photography that allows an image to be recorded in three dimensions. ... A Sagnac interferometer consists of an optical ring cavity in which two light beams (usually laser beams) are propagating in opposite directions. ... A gyroscope For other uses, see Gyroscope (disambiguation). ... A giant phased-array radar in Alaska In telecommunication, a phased array is a group of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and... Optical coherence tomography tomogram of a fingertip. ... It has been suggested that Optical interferometry be merged into this article or section. ... The 64 meter radio telescope at Parkes Observatory A radio telescope is a form of directional radio antenna most often used in radio astronomy and in tracking and collecting data from satellites and space probes (see Deep Space Network), and are also used in the SETI project. ...

Coherence and correlation

The coherence of two waves follows from how well correlated the waves are as quantified by the cross-correlation function. The cross-correlation quantifies the ability to predict the value of the second wave by knowing the value of the first. As an example, consider two waves perfectly correlated for all times. At any time, if the first wave changes, the second will change in the same way. If combined they can exhibit complete constructive interference at all times. It follows that they are perfectly coherent. As will be discussed below, the second wave need not be a separate entity. It could be the first wave at a different time or position. In this case, sometimes called self-coherence, the measure of correlation is the autocorrelation function. In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar... A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ...

Examples of wave-like states

These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof. The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ...

• Waves in a rope (up and down) or slinky (compression and expansion)
• Surface waves in a liquid
• Electric signals (fields) in transmission cables
• Sound
• Radio and Microwaves
• Light (optics)
• Electrons, atoms, and any other object (as described by quantum physics)

In most of these systems, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one can not measure the electric field directly as it oscillates much faster than any detector’s time resolution. Instead, we measure the intensity of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly. Metal Slinky Rainbow-colored plastic Slinky A Slinky, or Lazy-Spring, is a coil-shaped toy invented by mechanical engineer Richard James in Philadelphia, Pennsylvania. ... In physics, a surface wave is a wave that is guided along the interface between two different media for a mechanical wave, or by a refractive index gradient for an electromagnetic wave. ... Sound is a disturbance of mechanical energy that propagates through matter as a wave. ... Microwave Slang for small waves, like at a beach, often used by surfers. ... This article does not cite any references or sources. ... For the book by Sir Isaac Newton, see Opticks. ... Fig. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... In physics, intensity is a measure of the time-averaged energy flux. ...

Temporal coherence

Figure 1: The amplitude of a single frequency wave as a function of time t (red) and a copy of the same wave delayed by τ(green). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays τ.

Figure 2: The amplitude of a wave whose phase drifts significantly in time τ_c as a function of time t (red) and a copy of the same wave delayed by 2τ_c(green). At any particular time t the wave can interfere perfectly with its delayed copy. But, since half the time the red and green waves are in phase and half the time out of phase, when averaged over t any interference disappears at this delay.

Temporal coherence is the measure of the average correlation between the value of a wave at any pair of times, separated by delay τ. Temporal coherence tells us how monochromatic a source is. In other words, it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount (and hence the correlation decreases by significant amount) is defined as the coherence time τ_c. At τ=0 the degree of coherence is perfect whereas it drops significantly by delay τ_c. The coherence length L_c is defined as the distance the wave travels in time τ_c. Image File history File links Download high resolution version (1459x493, 159 KB)A single frequency wave and a delayed copy of itself. ... Image File history File links Download high resolution version (1459x493, 159 KB)A single frequency wave and a delayed copy of itself. ... Image File history File links Download high resolution version (1465x490, 169 KB)A wave with phase drift and a copy of itself delay by two coherence times. ... Image File history File links Download high resolution version (1465x490, 169 KB)A wave with phase drift and a copy of itself delay by two coherence times. ... For an electromagnetic wave, coherence time is the time over which a propagating wave may be considered coherent. ... In physics, coherence length is the propagation distance from a coherent source to a point where an electromagnetic wave maintains a specified degree of coherence. ...

One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below).

The relationship between coherence time and bandwidth

Since period is the inverse of frequency, it follows that the faster a wave decorrelates (and hence the smaller τ_c is) the larger the range of frequencies Δf the wave contains. Thus there is a tradeoff:

.

In terms of wavelength (fλ = c) this relationship becomes,

Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum (the intensity of each frequency) to its autocorrelation. In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ...

Examples of temporal coherence

We consider four examples of temporal coherence.

• A wave containing only a single frequency (monochromatic) is perfectly correlated at all times according to the above relation. (See Figure 1)
• Conversely, a wave whose phase drifts quickly will have a short coherence time. (See Figure 2)
• Similarly, pulses (wave packets) of waves, which naturally have a broad range of frequencies, also have a short coherence time since the amplitude of the wave changes quickly. (See Figure 3)
• Finally, white light, which has a very broad range of frequencies, is a wave which varies quickly in both amplitude and phase. Since it consequently has a very short coherence time (just 10 periods or so), it is often called incoherent.

The most monochromatic sources are usually lasers, and thus have the longest coherence lengths (up to hundreds of meters). For example, a stabilized helium-neon laser can produce light with coherence lengths in excess of 5 m. Not all lasers are monochromatic, however (e.g. for a Ti-sapphire laser, Δλ ≈ 2 nm - 70 nm). LEDs are less monochromatic (Δλ ≈ 50 nm) than the most monochromatic lasers, and tungsten filament lights are even less monochromatic (Δλ ≈ 300 nm), and so these sources have shorter coherence times than the most monochromatic lasers. The wave packet is one of the most widely misunderstood and misused concepts in physics. ... Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ... A helium-neon laser, usually called a HeNe laser, is a type of small gas laser. ... Part of a Ti:sapphire oscillator. ...

Holography requires light with a long coherence time. In contrast, Optical coherence tomography uses light with a short coherence time. Holography (from the Greek, όλος-hòlòs whole + γραφή-grafè writh) is the science of producing holograms; it is an advanced form of photography that allows an image to be recorded in three dimensions. ... Optical coherence tomography tomogram of a fingertip. ...

Measurement of temporal coherence

Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time τ_c (red) and a copy of the same wave delayed by 2τ_c(green) plotted as a function of time t. At any particular time the red and green waves are uncorrelated; one oscillates while the other is constant and so there will be no interference at this delay. Another way of looking at this is the wavepackets are not overlapped in time and so at any particular time there is only one nonzero field so no interference can occur.

Figure 4: The time-averaged intensity (blue) detected at the output of an interferometer plotted as a function of delay τ for the example waves in Figures 2 and 3. As the delay is changed by half a period, the interference switches between constructive and destructive. The black lines indicate the interference envelope, which gives the degree of coherence. Although the waves in Figures 2 and 3 have different time durations, they have the same coherence time.

In optics, temporal coherence is measured in an interferometer such as the Michelson interferometer or Mach-Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time τ. A detector measures the time-averaged intensity of the light exiting the interferometer. The resulting interference visibility (e.g. see Figure 4) gives the temporal coherence at delay τ. Since for most natural light sources, the coherence time is much shorter than the time resolution of any detector, the detector itself does the time averaging. Consider the example shown in Figure 3. At a fixed delay, here 2τ_c, an infinitely fast detector would measure an intensity that fluctuates significantly over a time t equal to τ_c. In this case, to find the temporal coherence at 2τ_c, one would manually time-average the intensity.
Image File history File links Download high resolution version (1327x485, 91 KB)A wave packet and a copy of itself delay by twice the coherence time. ... Image File history File links Download high resolution version (1327x485, 91 KB)A wave packet and a copy of itself delay by twice the coherence time. ... Image File history File links Download high resolution version (1421x447, 115 KB)The intensity interference pattern observed as the delay between two copies of a wave with a finite coherence time is changed. ... Image File history File links Download high resolution version (1421x447, 115 KB)The intensity interference pattern observed as the delay between two copies of a wave with a finite coherence time is changed. ... In optics, correlation functions are used to characterize the statistical and coherence properties of an electromagnetic field. ... A Michelson interferometer for use on an optical table. ... Mach-Zehnder interferometer. ... In physics, intensity is a measure of the time-averaged energy flux. ...

Spatial coherence

In some systems, such as water waves or optics, wave-like states can extend over one or two dimensions. Spatial coherence describes the ability for two points in space, x₁ and x₂, in the extent of a wave to interfere, when averaged over time. More precisely, the spatial coherence is the cross-correlation between two points in a wave for all times. If a wave has only 1 value of amplitude over an infinite length, it is perfectly spatially coherent. The range of separation between the two points over which there is the significant interference is called the coherence area, A_c. This is the relevant type of coherence for the Young’s double-slit interferometer. It is also used in optical imaging systems and particularly in various types of astronomy telescopes. Sometimes people also use “spatial coherence” to refer to the visibility when a wave-like state is combined with a spatially shifted copy of itself. In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar...

Examples of spatial coherence

Figure 5: A plane wave with an infinite coherence length.

• Plane waves with an infinite coherence time have an infinite coherence area. See Figure 5.
• A wave with distorted profile and with an infinite coherence time has an infinite coherence area. See Figure 6.
• A wave with distorted profile and a finite coherence time has a finite coherence area. See Figure 7.
• A wave with finite coherence area is incident on a pinhole (small aperture). The wave will diffract out of the pinhole. Far from the pinhole the emerging spherical wavefronts are approximately flat. The coherence area is now infinite while the coherence length is unchanged. See Figure 8.
• A wave with infinite coherence area is combined with a spatially-shifted copy of itself. Some sections in the wave interfere constructively and some will interfere destructively. Averaging over these sections, a detector with length D will measure reduced interference visibility. See Figure 9.

Consider a tungsten light-bulb filament. Different points in the filament emit light independently and have no fixed phase-relationship. In detail, at any point in time the profile of the emitted light is going to be distorted. The profile will change randomly over the coherence time τ_c. Since for a white-light source such as a light-bulb τ_c is small, the filament is considered a spatially incoherent source. In contrast, a radio antenna array, has large spatial coherence because antennas at opposite ends of the array emit with a fixed phase-relationship. Light waves produced by a laser often have high temporal and spatial coherence (though the degree of coherence depends strongly on the exact properties of the laser). Spatial coherence of laser beams also manifests itself as speckle patterns and diffraction fringes seen at the edges of shadow. Image File history File links Download high resolution version (800x888, 6 KB) Summary I made this Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (800x888, 6 KB) Summary I made this Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... The intensity pattern formed on a screen by diffraction from a square aperture Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. ... The interferometric visibility (also known as interference visibility or just visibility) quantifies the contrast of interference in any system which has wave-like properties, such as optics, quantum mechanics, water waves, or electrical signals. ... A giant phased-array radar in Alaska In telecommunication, a phased array is a group of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and...

Holography requires temporally and spatially coherent light. Its inventor, Dennis Gabor, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the monochromatic light from an emission line of a mercury-vapor lamp through a pinhole spatial filter.

Figure 6: A wave with a varying profile (wavefront) and infinite coherence length.

Figure 7: A wave with a varying profile (wavefront) and finite coherence length.

Figure 8: The wave with finite coherence length from Figure 7 is passed through a pinhole. The emerging wave has infinite coherence area. The coherence length (or coherence time) are unchanged by the pinhole.

Figure 9: The wave with infinite coherence length from Figure 6 is combined with a spatially shifted copy of itself. For example a misaligned Mach-Zehnder interferometer will do this. A detector will measure reduced visibility.

Image File history File links Download high resolution version (800x888, 14 KB) Summary I made this. ... Image File history File links Download high resolution version (800x888, 14 KB) Summary I made this. ... Image File history File links Download high resolution version (850x838, 30 KB) Summary I made this Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (850x838, 30 KB) Summary I made this Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (1500x813, 84 KB) Summary I made this. ... Image File history File links Download high resolution version (1500x813, 84 KB) Summary I made this. ... Image File history File links Download high resolution version (1312x1000, 54 KB) Summary I made this Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (1312x1000, 54 KB) Summary I made this Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Mach-Zehnder interferometer. ...

Spectral coherence

Figure 10: Waves of different frequencies (i.e. colors) interfere to form a pulse if they are coherent.

Figure 11: Spectrally incoherent light interferes to form continuous light with a randomly varying phase and amplitude

Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see Fourier transform). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise). The temporal duration of the pulse Δt is limited by the spectral bandwidth of the light Δf according to: Image File history File links Download high resolution version (1046x969, 234 KB) Summary I made this myself. ... Image File history File links Download high resolution version (1046x969, 234 KB) Summary I made this myself. ... Image File history File links Download high resolution version (1190x966, 259 KB) Summary I made this myself. ... Image File history File links Download high resolution version (1190x966, 259 KB) Summary I made this myself. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Calculated spectrum of a generated approximation of white noise White noise is a random signal (or process) with a flat power spectral density. ...

,

which follows from the properties of the Fourier transform (for quantum particles it also follows from the Heisenberg uncertainty principle). In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...

If the phase depends linearly on the frequency (i.e. ) then the pulse will have the minimum time duration for its bandwidth (a transform-limited pulse), otherwise it is chirped (see dispersion). Dispersion can mean any of several things: A phenomenon that causes the separation of a wave into components of varying frequency. ...

Measurement of spectral coherence

Measurement of the spectral coherence of light requires a nonlinear optical interferometer, such as an intensity optical correlator, frequency-resolved optical gating (FROG), or Spectral phase interferometry for direct electric-field reconstruction (SPIDER).
Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ... In optics, various autocorrelation functions can be experimentally realized. ... In optics, frequency-resolved optical gating (FROG) is a derivative of autocorrelation, but is far superior in its ability to measure ultrafast optical pulse shapes. ... In ultrafast optics, spectral phase interferometry for direct electric-field reconstruction (SPIDER) is an ultrashort pulse measurement technique. ...

Polarization coherence

Light also has a polarization, which is the direction in which the electric field oscillates. Unpolarized light is composed of two equally intense incoherent light waves with orthogonal polarizations. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. A polarizer rotated to any angle will always transmit half the incident intensity when averaged over time. In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... A polarizer is a device that converts an unpolarized or mixed-polarization beam of electromagnetic waves (e. ...

If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created.

The polarization of a light beam is represented by a vector in the Poincare sphere. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere. In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ...

Quantum coherence

In quantum mechanics, all objects have wave-like properties (see de Broglie waves). For instance, in Young's double-slit experiment electrons can be used in the place of light waves. Each electron can go through either slit and hence has two paths that it can take to a particular final position. In quantum mechanics these two paths interfere. If there is destructive interference, the electron never arrives at that particular position. This ability to interfere is called quantum coherence. Fig. ... In physics, the de Broglie hypothesis is the statement that all matter (any object) has a wave-like nature (wave-particle duality). ...

The quantum description of perfectly coherent paths is called a pure state, in which the two paths are combined in a superposition. The correlation between the two particles exceeds what would be predicted for classical correlation alone (see Bell's inequalities). If this two-particle system is decohered (which would occur in a measurement via Einselection), then there is no longer any phase relationship between the two states. The quantum description of imperfectly coherent paths is called a mixed state, described by a density matrix and is entirely analogous to a classical system of mixed probabilities (the correlations are classical). The term pure state refers to several related concepts in physics, particularly quantum mechanics and in functional analysis. ... The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... This article may be too technical for most readers to understand. ... Einselection is short for environmentally-induced superselection, a nickname coined by Wojciech H. Zurek. ... The term mixed state refers to a concept in physics, particularly quantum mechanics. ... A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ...

Large-scale (macroscopic) quantum coherence leads to very amazing phenomena. For instance, the laser, superconductivity, and superfluidity are examples of highly coherent quantum systems. One example that shows the amazing possibilities of macroscopic quantum coherence is the Schrödinger's cat thought experiment. Another example of quantum coherence is in a Bose-Einstein condensate. Here, all the atoms that make up the condensate are in-phase. They are thus all described by a single quantum wavefunction. Their behavior is communal and inseparable until the coherence is destroyed. Macroscopic is commonly used to describe physical objects that are measurable and observable by the naked eye. ... Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ... A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. ... Superfluidity is a phase of matter characterised by the complete absence of viscosity. ... Schrödingers Cat: If the nucleus in the bottom left decays, the Geiger counter on its right will sense it and trigger the release of the gas. ... A Bose–Einstein condensate is a phase of matter formed by bosons cooled to temperatures very near to absolute zero (0 kelvins or -273. ...

See also

• Atomic coherence

When coherent electro-magnetic fields interact with multi-level atomic systems, coherence between levels may be induced. ...

References

• Rolf G. Winter, Aephraim M. Steinberg, "Coherence", in AccessScience@McGraw-Hill, http://www.accessscience.com, DOI 10.1036/1097-8542.146900, last modified: October 24, 2001.
• M. Born and E. Wolf, Principles of Optics, 7th ed., 1999
• Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), [ISBN 0-19-850177-3]