Coherence (from Latin cohaerere = to be connected) describes in physics a property of waves, that enables stationary (i. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. e. temporally and spatially constant) interference. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern More generally, coherence describes all correlation properties between physical quantities of a wave. In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear A physical Quantity is a physical property that can be quantified

When interfering, waves add constructively or subtract destructively, depending on their relative phase. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 Two waves are said to be coherent if they have a constant relative phase, which also implies that they have the same frequency. The degree of coherence is measured by the interference visibility, a measure of how perfectly the waves can cancel due to destructive interference. In Optics, Correlation functions are used to characterize the statistical and coherence properties of an electromagnetic field The interferometric visibility (also known as "interference visibility" or "fringe visibility" or just "visibility" quantifies the contrast of Interference

## Coherence and correlation

The coherence of two waves follows from how well correlated the waves are as quantified by the cross-correlation function [1][2][3][4][5]. In Signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them 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. Autocorrelation is a mathematical tool for finding repeating patterns such as the presence of a periodic signal which has been buried under noise or identifying the Missing fundamental

## 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 second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves

In most of these systems, one can measure the wave directly. A Slinky is a Coil -shaped Toy invented by mechanical engineer Richard James in Philadelphia Pennsylvania In Physics, surface wave can refer to a Mechanical wave that propagates along the interface between differing media usually two fluids with different densities Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies Radio is the transmission of signals by Modulation of electromagnetic waves with frequencies below those of visible Light. Microwaves are electromagnetic waves with Wavelengths ranging from 1 mm to 1 m or frequencies between 0 Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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. 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 (that can Instead, we measure the intensity of the light. In Physics, intensity is a measure of the time-averaged Energy Flux. 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

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 τc as a function of time t (red) and a copy of the same wave delayed by 2τc(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 τc. For an Electromagnetic wave, coherence time is the Time over which a propagating wave (especially a Laser or Maser beam may be considered At τ=0 the degree of coherence is perfect whereas it drops significantly by delay τc. The coherence length Lc is defined as the distance the wave travels in time τc. In Physics, coherence length is the propagation distance from a coherent source to a point where an Electromagnetic wave maintains a specified

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 τc is) the larger the range of frequencies Δf the wave contains. Thus there is a tradeoff:

$\tau_c \Delta f \approx 1$.

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

$\frac{L_c \Delta \lambda}{\lambda^2} \approx 1$

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. In Mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a Convolution is the Pointwise product This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Autocorrelation is a mathematical tool for finding repeating patterns such as the presence of a periodic signal which has been buried under noise or identifying the Missing fundamental

### 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. In physics a wave packet is an envelope or packet containing an arbitrary number of wave forms (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; such high monochromaticity implies long coherence lengths (up to hundreds of meters). A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. For example, a stabilized helium-neon laser can produce light with coherence lengths in excess of 5 m. A helium-neon laser, usually called a HeNe laser, is a type of small Gas laser. Not all lasers are monochromatic, however (e. g. for a Ti-sapphire laser, Δλ ≈ 2 nm - 70 nm). Tisapphire lasers (also known as TiAl2O3 lasers, titanium-sapphire lasers, or simply Tisapphs) are Tunable lasers which LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 300 nm, so these sources have shorter coherence times than the most monochromatic lasers.

Holography requires light with a long coherence time. Holography (from the Greek, ὅλος - hólos whole + γραφή - grafē writing drawing is a technique that allows the In contrast, Optical coherence tomography uses light with a short coherence time. Optical coherence tomography (OCT is an optical signal acquisition and processing method allowing extremely high-quality micrometre-resolution three-dimensional images from within optical

### Measurement of temporal coherence

Figure 3: The amplitude of a wavepacket whose amplitude changes significantly in time τc (red) and a copy of the same wave delayed by 2τc(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. In Optics, Correlation functions are used to characterize the statistical and coherence properties of an electromagnetic field 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. Michelson interferometer is the most common configuration for optical Interferometry and was invented by Albert Abraham Michelson. The Mach-Zehnder interferometer (named after physicists Ludwig Mach (son of Ernst Mach) and Ludwig Zehnder) is a device used to determine the phase shift 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. In Physics, intensity is a measure of the time-averaged Energy Flux. 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 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. In Signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them 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 is called the coherence area, Ac. 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.

### Examples of spatial coherence

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 τc. Since for a white-light source such as a light-bulb τc 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. This article is about general theory and electromagnetic phased array 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.

## 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). This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and 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). White noise is a random signal (or process with a flat Power spectral density. The temporal duration of the pulse Δt is limited by the spectral bandwidth of the light Δf according to:

$\Delta f\Delta t \ge 1$,

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 states that locating a particle in a small region of space makes the Momentum of the particle uncertain

If the phase depends linearly on the frequency (i. e. $\theta (f) \propto f$) then the pulse will have the minimum time duration for its bandwidth (a transform-limited pulse), otherwise it is chirped (see dispersion). In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its 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 (NLO is the branch of Optics that describes the behaviour of Light in nonlinear media, that is media in which the dielectric polarization 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 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. Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations 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. 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. Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations 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.

## Quantum coherence

In quantum mechanics, all objects have wave-like properties (see de Broglie waves). Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, the de Broglie hypothesis (pronounced /brœj/ as French breuil close to "broy" is the statement that all Matter (any object has a Wave 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.

The quantum description of perfectly coherent paths is called a pure state, in which the two paths are combined in a superposition. In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Quantum superposition is the fundamental law of Quantum mechanics. The correlation between the two particles exceeds what would be predicted for classical correlation alone (see Bell's inequalities). Bell's theorem is a theorem that shows that the predictions of Quantum mechanics (QM are not intuitive and touches upon fundamental philosophical issues that relate to modern 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. Einselection is short for e nvironment - in duced super' selection', a nickname coined by Wojciech H 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).

Large-scale (macroscopic) quantum coherence leads to very amazing phenomena. Macroscopic is commonly used to describe physical objects that are measurable and observable by the Naked eye. For instance, the laser, superconductivity, and superfluidity are examples of highly coherent quantum systems. A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Superfluidity is a phase of matter or description of Heat capacity in which unusual effects are observed when Liquids, typically of Helium-4 One example that shows the amazing possibilities of macroscopic quantum coherence is the Schrödinger's cat thought experiment. Schrödinger's cat is a Thought experiment, often described as a Paradox, devised by Austrian physicist Erwin Schrödinger in 1935 Another example of quantum coherence is in a Bose-Einstein condensate. A Bose–Einstein condensate (BEC is a State of matter of Bosons confined in an external Potential and cooled to Temperatures very near to Here, all the atoms that make up the condensate are in-phase. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 They are thus all described by a single quantum wavefunction. Their behavior is communal and inseparable until the coherence is destroyed.

## References

1. ^ Rolf G. In Physics, atomic coherence is the induced coherence between levels of a multi-level Atomic system sometimes observed when it interacts with In Quantum mechanics a coherent state is a specific kind of quantum state of the Quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour Winter; Aephraim M. Steinberg. Coherence. AccessScience@McGraw-Hill.
2. ^ M. Born; E. Wolf (1999). Principles of Optics, 7.
3. ^ Loudon, Rodney (2000). The Quantum Theory of Light. Oxford University Press.
4. ^ Leonard Mandel (1995). Optical Coherence and Quantum Optics.
5. ^ Arvind Marathay (1982). Elements of Optical Coherence Theory. John Wiley & Sons Inc.