# Coherence (physics)

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In physics, two wave sources are coherent if their frequency and waveform are identical. Coherence is an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference. It contains several distinct concepts, which are limiting cases that never quite occur in reality but allow an understanding of the physics of waves, and has become a very important concept in quantum physics. More generally, coherence describes all properties of the correlation between physical quantities of a single wave, or between several waves or wave packets.

## Contents

Interference is the addition, in the mathematical sense, of wave functions. A single wave can interfere with itself, but this is still an addition of two waves (see Young's slits experiment). Constructive or destructive interferences are limit cases, and two waves always interfere, even if the result of the addition is complicated or not remarkable. When interfering, two waves can add together to create a wave of greater amplitude than either one (constructive interference) or subtract from each other to create a wave of lesser amplitude than either one (destructive interference), depending on their relative phase. Two waves are said to be coherent if they have a constant relative phase. The amount of coherence can readily be measured by the interference visibility, which looks at the size of the interference fringes relative to the input waves (as the phase offset is varied); a precise mathematical definition of the degree of coherence is given by means of correlation functions.

Spatial coherence describes the correlation (or predictable relationship) between waves at different points in space, either lateral or longitudinal. [1] Temporal coherence describes the correlation between waves observed at different moments in time. Both are observed in the Michelson–Morley experiment and Young's interference experiment. Once the fringes are obtained in the Michelson interferometer, when one of the mirrors is moved away gradually from the beam-splitter, the time for the beam to travel increases and the fringes become dull and finally disappear, showing temporal coherence. Similarly, in a double-slit experiment, if the space between the two slits is increased, the coherence dies gradually and finally the fringes disappear, showing spatial coherence. In both cases, the fringe amplitude slowly disappears, as the path difference increases past the coherence length.

## Introduction

Coherence was originally conceived in connection with Thomas 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 mechanics. Coherence describes the statistical similarity of a field (electromagnetic field, quantum wave packet etc.) at two points in space or time. [2] 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).

## Mathematical definition

The coherence function between two signals ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$ is defined as [3]

${\displaystyle \gamma _{xy}^{2}(f)={\frac {|S_{xy}(f)|^{2}}{S_{xx}(f)S_{yy}(f)}}}$

where ${\displaystyle S_{xy}(f)}$ is the cross-spectral density of the signal and ${\displaystyle S_{xx}(f)}$ and ${\displaystyle S_{yy}(f)}$ are the power spectral density functions of ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$, respectively. The cross-spectral density and the power spectral density are defined as the Fourier transforms of the cross-correlation and the autocorrelation signals, respectively. For instance, if the signals are functions of time, the cross-correlation is a measure of the similarity of the two signals as a function of the time lag relative to each other and the autocorrelation is a measure of the similarity of each signal with itself in different instants of time. In this case the coherence is a function of frequency. Analogously, if ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$ are functions of space, the cross-correlation measures the similarity of two signals in different points in space and the autocorrelations the similarity of the signal relative to itself for a certain separation distance. In that case, coherence is a function of wavenumber (spatial frequency).

The coherence varies in the interval ${\displaystyle 0\leqslant \gamma _{xy}^{2}(f)\leqslant 1.}$. If ${\displaystyle \gamma _{xy}^{2}(f)=1}$ it means that the signals are perfectly correlated or linearly related and if ${\displaystyle \gamma _{xy}^{2}(f)=0}$ they are totally uncorrelated. If a linear system is being measured, ${\displaystyle x(t)}$ being the input and ${\displaystyle y(t)}$ the output, the coherence function will be unitary all over the spectrum. However, if non-linearities are present in the system the coherence will vary in the limit given above.

A more precise definition is given in the article on degree of coherence.

## Coherence and correlation

The coherence of two waves expresses how well correlated the waves are as quantified by the cross-correlation function. [4] [5] [6] [7] [8] Cross-correlation quantifies the ability to predict the phase of the second wave by knowing the phase of the first. As an example, consider two waves perfectly correlated for all times. At any time, the phase difference will be constant.[ clarification needed ] If, when combined, they exhibit perfect constructive interference, perfect destructive interference, or something in-between but with constant phase difference, then 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, the measure of correlation is the autocorrelation function (sometimes called self-coherence). Degree of correlation involves correlation functions. [9] :545-550

## 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.

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 cannot measure the electric field directly as it oscillates much faster than any detector's time resolution. [10] Instead, one measures 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.

## Temporal coherence

Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by τ, at any pair of times. 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 τc. At a delay of τ=0 the degree of coherence is perfect, whereas it drops significantly as the delay passes τ=τc. The coherence length Lc is defined as the distance the wave travels in time τc. [9] :560, 571–573

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

It can be shown that the larger the range of frequencies Δf a wave contains, the faster the wave decorrelates (and hence the smaller τc is). Thus there is a tradeoff: [9] :358-359, 560

${\displaystyle \tau _{c}\Delta f\gtrsim 1}$.

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. [9] :572

### Examples of temporal coherence

We consider four examples of temporal coherence.

• A wave containing only a single frequency (monochromatic) is perfectly correlated with itself at all time delays, in accordance with 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 high degree of monochromaticity of lasers implies long coherence lengths (up to hundreds of meters). For example, a stabilized and monomode helium–neon laser can easily produce light with coherence lengths of 300 m. [12] Not all lasers have a high degree of monochromaticity, however (e.g. for a mode-locked Ti-sapphire laser, Δλ ≈ 2 nm - 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers.

Holography requires light with a long coherence time. In contrast, optical coherence tomography, in its classical version, uses light with a short coherence time.

### Measurement of temporal coherence

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 visibility of the interference pattern (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 2τc, an infinitely fast detector would measure an intensity that fluctuates significantly over a time t equal to τc. In this case, to find the temporal coherence at 2τc, one would manually time-average the intensity.

## 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 significant interference defines the diameter of the coherence area, Ac [13] (Coherence length, often a feature of a source, is usually an industrial term related to the coherence time of the source, not the coherence area in the medium.) Ac 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.

### Examples

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 ${\displaystyle \tau _{c}}$. Since for a white-light source such as a light-bulb ${\displaystyle \tau _{c}}$ 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.

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.

In February 2011 it was reported that helium atoms, cooled to near absolute zero / Bose–Einstein condensate state, can be made to flow and behave as a coherent beam as occurs in a laser. [14] [15]

## Spectral coherence

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 ${\displaystyle \Delta t}$ is limited by the spectral bandwidth of the light ${\displaystyle \Delta f}$ according to:

${\displaystyle \Delta f\Delta t\geq 1}$,

which follows from the properties of the Fourier transform and results in Küpfmüller's uncertainty principle (for quantum particles it also results in the Heisenberg uncertainty principle).

If the phase depends linearly on the frequency (i.e. ${\displaystyle \theta (f)\propto f}$) then the pulse will have the minimum time duration for its bandwidth (a transform-limited pulse), otherwise it is chirped (see dispersion).

### 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).

## Polarization and coherence

Light also has a polarization, which is the direction in which the electric field oscillates. Unpolarized light is composed of incoherent light waves with random polarization angles. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. An absorbing polarizer rotated to any angle will always transmit half the incident intensity when averaged over time.

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 Poincaré 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

## Applications

### Holography

Coherent superpositions of optical wave fields include holography. Holographic objects are used frequently in daily life in television and credit card security.

### Non-optical wave fields

Further applications concern the coherent superposition of non-optical wave fields. In quantum mechanics for example one considers a probability field, which is related to the wave function ${\displaystyle \psi (\mathbf {r} )}$ (interpretation: density of the probability amplitude). Here the applications concern, among others, the future technologies of quantum computing and the already available technology of quantum cryptography. Additionally the problems of the following subchapter are treated.

Coherence is used to check the quality of the transfer functions (FRFs) being measured. Low coherence can be caused by poor signal to noise ratio, and/or inadequate frequency resolution.

## Quantum coherence

According to quantum mechanics, all objects can 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's wave-function goes through both slits, and hence has two separate split-beams that contribute to the intensity pattern on a screen. According to standard wave theory [16] these two contributions give rise to an intensity pattern of bright bands due to constructive interference, interlaced with dark bands due to destructive interference, on a downstream screen. This ability to interfere and diffract is related to coherence (classical or quantum) of the waves produced at both slits. The association of an electron with a wave is unique to quantum theory.

When the incident beam is represented by a quantum pure state, the split beams downstream of the two slits are represented as a superposition of the pure states representing each split beam. [17] The quantum description of imperfectly coherent paths is called a mixed state. A perfectly coherent state has a density matrix (also called the "statistical operator") that is a projection onto the pure coherent state and is equivalent to a wave function, while a mixed state is described by a classical probability distribution for the pure states that make up the mixture.

Macroscopic scale quantum coherence leads to novel phenomena, the so-called macroscopic quantum phenomena. For instance, the laser, superconductivity and superfluidity are examples of highly coherent quantum systems whose effects are evident at the macroscopic scale. The macroscopic quantum coherence (off-diagonal long-range order, ODLRO) [18] [19] for superfluidity, and laser light, is related to first-order (1-body) coherence/ODLRO, while superconductivity is related to second-order coherence/ODLRO. (For fermions, such as electrons, only even orders of coherence/ODLRO are possible.) For bosons, a Bose–Einstein condensate is an example of a system exhibiting macroscopic quantum coherence through a multiple occupied single-particle state.

The classical electromagnetic field exhibits macroscopic quantum coherence. The most obvious example is the carrier signal for radio and TV. They satisfy Glauber's quantum description of coherence.

Recently M. B. Plenio and co-workers constructed an operational formulation of quantum coherence as a resource theory. They introduced coherence monotones analogous to the entanglement monotones. [20] Quantum coherence has been shown to be equivalent to quantum entanglement [21] in the sense that coherence can be faithfully described as entanglement, and conversely that each entanglement measure corresponds to a coherence measure.

## Related Research Articles

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. It is defined as the 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 secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.

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.

In physics, coherence length is the propagation distance over which a coherent wave maintains a specified degree of coherence. Wave interference is strong when the paths taken by all of the interfering waves differ by less than the coherence length. A wave with a longer coherence length is closer to a perfect sinusoidal wave. Coherence length is important in holography and telecommunications engineering.

Interferometry is a technique in which waves are superimposed to cause the phenomenon of interference, which is used to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber optics, engineering metrology, optical metrology, oceanography, seismology, spectroscopy, quantum mechanics, nuclear and particle physics, plasma physics, remote sensing, biomolecular interactions, surface profiling, microfluidics, mechanical stress/strain measurement, velocimetry, optometry, and making holograms.

Mode-locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or femtoseconds (10−15 s). A laser operated in this way is sometimes referred to as a femtosecond laser, for example in modern refractive surgery. The basis of the technique is to induce a fixed-phase relationship between the longitudinal modes of the laser's resonant cavity. Constructive interference between these modes can cause the laser light to be produced as a train of pulses. The laser is then said to be 'phase-locked' or 'mode-locked'.

The Michelson interferometer is a common configuration for optical interferometry and was invented by Albert Abraham Michelson. Using a beam splitter, a light source is split into two arms. Each of those light beams is reflected back toward the beamsplitter which then combines their amplitudes using the superposition principle. The resulting interference pattern that is not directed back toward the source is typically directed to some type of photoelectric detector or camera. For different applications of the interferometer, the two light paths can be with different lengths or incorporate optical elements or even materials under test.

Optical coherence tomography (OCT) is an imaging technique that uses low-coherence light to capture micrometer-resolution, two- and three-dimensional images from within optical scattering media. It is used for medical imaging and industrial nondestructive testing (NDT). Optical coherence tomography is based on low-coherence interferometry, typically employing near-infrared light. The use of relatively long wavelength light allows it to penetrate into the scattering medium. Confocal microscopy, another optical technique, typically penetrates less deeply into the sample but with higher resolution.

In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.

In optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are commonly used to estimate the duration of ultrashort pulses produced by modelocked lasers. The laser pulse duration cannot be easily measured by optoelectronic methods, since the response time of photodiodes and oscilloscopes are at best of the order of 200 femtoseconds, yet laser pulses can be made as short as a few femtoseconds.

In physics, the Hanbury Brown and Twiss (HBT) effect is any of a variety of correlation and anti-correlation effects in the intensities received by two detectors from a beam of particles. HBT effects can generally be attributed to the wave–particle duality of the beam, and the results of a given experiment depend on whether the beam is composed of fermions or bosons. Devices which use the effect are commonly called intensity interferometers and were originally used in astronomy, although they are also heavily used in the field of quantum optics.

In quantum optics, correlation functions are used to characterize the statistical and coherence properties of an electromagnetic field. The degree of coherence is the normalized correlation of electric fields; in its simplest form, termed . It is useful for quantifying the coherence between two electric fields, as measured in a Michelson or other linear optical interferometer. The correlation between pairs of fields, , typically is used to find the statistical character of intensity fluctuations. First order correlation is actually the amplitude-amplitude correlation and the second order correlation is the intensity-intensity correlation. It is also used to differentiate between states of light that require a quantum mechanical description and those for which classical fields are sufficient. Analogous considerations apply to any Bose field in subatomic physics, in particular to mesons.

The interferometric visibility quantifies the contrast of interference in any system which has wave-like properties, such as optics, quantum mechanics, water waves, sound waves, or electrical signals. Generally, two or more waves are superimposed and as the phase difference between them varies, the power or intensity of the resulting wave oscillates, forming an interference pattern. The pointwise definition may be expanded to a visibility function varying over time or space. For example, the phase difference varies as a function of space in a two-slit experiment. Alternately, the phase difference may be manually controlled by the operator, for example by adjusting a vernier knob in an interferometer. The ratio of the amplitude of the interference pattern to the sum of the powers of the individual waves is the definition of visibility. The interferometric visibility gives a practical way to measure the coherence of two waves. A theoretical definition of the coherence is given by the degree of coherence, using the notion of correlation.

An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave with frequency into two output waves of lower frequency by means of second-order nonlinear optical interaction. The sum of the output waves' frequencies is equal to the input wave frequency: . For historical reasons, the two output waves are called "signal" and "idler", where the output wave with higher frequency is the "signal". A special case is the degenerate OPO, when the output frequency is one-half the pump frequency, , which can result in half-harmonic generation when signal and idler have the same polarization.

Optical heterodyne detection is a method of extracting information encoded as modulation of the phase, frequency or both of electromagnetic radiation in the wavelength band of visible or infrared light. The light signal is compared with standard or reference light from a "local oscillator" (LO) that would have a fixed offset in frequency and phase from the signal if the latter carried null information. "Heterodyne" signifies more than one frequency, in contrast to the single frequency employed in homodyne detection.

A white light scanner (WLS) is a device for performing surface height measurements of an object using coherence scanning interferometry (CSI) with spectrally-broadband, "white light" illumination. Different configurations of scanning interferometer may be used to measure macroscopic objects with surface profiles measuring in the centimeter range, to microscopic objects with surface profiles measuring in the micrometer range. For large-scale non-interferometric measurement systems, see structured-light 3D scanner.

As described here, white light interferometry is a non-contact optical method for surface height measurement on 3-D structures with surface profiles varying between tens of nanometers and a few centimeters. It is often used as an alternative name for coherence scanning interferometry in the context of areal surface topography instrumentation that relies on spectrally-broadband, visible-wavelength light.

Coherence is defined as the ability of waves to interfere. Intuitively, coherent waves have a well-defined constant phase relationship. However, an exclusive and extensive physical definition of coherence is more nuanced. Coherence functions, as introduced by Glauber and others in the 1960s, capture the mathematics behind the intuition by defining correlation between the electric field components as coherence. These correlations between electric field components can be measured to arbitrary orders, hence leading to the concept of different orders of coherence. The coherence encountered in most optical experiments, including the classic Young's double slit experiment and Mach-Zender interferometer, is first order coherence. Robert Hanbury Brown and Richard Q. Twiss performed a correlation experiment in 1956, and brought to light a different kind of correlation between fields, namely the correlation of intensities, which correspond to second order coherence. Higher order coherences become relevant in photon-coincidence counting experiments. Orders of coherence can be measured using classical correlation functions or by using the quantum analogue of those functions, which take quantum mechanical description of electric field (operators) as input. While the quantum coherence functions might yield the same results as the classical functions, the underlying mechanism and description of the physical processes are fundamentally different because quantum interference deals with interference of possible histories while classical interference deals with interference of physical waves.

In quantum physics, light is in a squeezed state if its electric field strength Ԑ for some phases has a quantum uncertainty smaller than that of a coherent state. The term squeezing thus refers to a reduced quantum uncertainty. To obey Heisenberg's uncertainty relation, a squeezed state must also have phases at which the electric field uncertainty is anti-squeezed, i.e. larger than that of a coherent state. Since 2019, the gravitational-wave observatories LIGO and Virgo employ `squeezed' laser light, which has significantly increased the rate of observed gravitational-wave events.

Quantum optical coherence tomography (Q-OCT) is an imaging technique that uses nonclassical(quantum) light sources to generate high-resolution images based on the Hong-Ou-Mandel effect (HOM). Q-OCT is similar to conventional OCT but uses a fourth-order interferometer that incorporates two photodetectors rather than a second-order interferometer with a single photodetector. The primary advantage of Q-OCT over OCT is insensitivity to even-order dispersion in multi-layered and scattering media.

Spectral interferometry (SI) or frequency-domain interferometry is a linear technique used to measure optical pulses, with the condition that a reference pulse that was previously characterized is available. This technique provides information about the intensity and phase of the pulses. SI was first proposed by Claude Froehly and coworkers in the 1970s.

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