Coherence expresses the potential for two waves to interfere. Two monochromatic beams from a single source always interfere. Wave sources are not strictly monochromatic: they may be partly coherent. Beams from different sources are mutually incoherent.
When interfering, two waves add together to create a wave of greater amplitude than either one (constructive interference) or subtract from each other to create a wave of minima which may be zero (destructive interference), depending on their relative phase. Constructive or destructive interference are limit cases, and two waves always interfere, even if the result of the addition is complicated or not remarkable.
Two waves with constant relative phase will be coherent.[1] 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. More generally, coherence describes the statistical similarity of a field (electromagnetic field, quantum wave packet etc.) at two points in space or time.[2]
Coherence controls the visibility or contrast of interference patterns. For example, visibility of the double slit experiment pattern requires that both slits be illuminated by a coherent wave as illustrated in the figure. Large sources without collimation or sources that mix many different frequencies will have lower visibility.[3]
Coherence contains several distinct concepts. Spatial coherence describes the correlation (or predictable relationship) between waves at different points in space, either lateral or longitudinal.[4] 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.
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. 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).
The coherence function between two signals
x(t)
y(t)
| 2 | ||
| \gamma | (f)= | |
| xy |
| |Sxy(f)|2 | |
| Sxx(f)Syy(f) |
where
Sxy(f)
Sxx(f)
Syy(f)
x(t)
y(t)
x(t)
y(t)
The coherence varies in the interval
0\leq
| 2 | |
| \gamma | |
| xy |
(f)\leq1
| 2 | |
| \gamma | |
| xy |
(f)=1
| 2 | |
| \gamma | |
| xy |
(f)=0
x(t)
y(t)
The coherence of two waves expresses how well correlated the waves are as quantified by the cross-correlation function.[6] [7] [8] [9] [10] 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 (by using a monochromatic light source). At any time, the phase difference between the two waves will be constant. If, when they are 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.
These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof.
In system with macroscopic waves, 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.[11] 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 is the measure of the average correlation between the value of a wave and itself delayed by
\tau
\tauc
\tau=0
\tau=\tauc
Lc
\tauc
The coherence time is not the time duration of the signal; the coherence length differs from the coherence area (see below).
The larger the bandwidth – range of frequencies Δf a wave contains – the faster the wave decorrelates (and hence the smaller
\tauc
\tauc\Deltaf\gtrsim1.
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.
Narrow bandwidth lasers have 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.[13] Not all lasers have a high 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.
Examples of temporal coherence include:
Holography requires light with a long coherence time. In contrast, optical coherence tomography, in its classical version, uses light with a short 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
\tau
\tau
2\tau
\tau
2\tauc
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 spatial points 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
lc
Ac
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
\tauc
\tauc
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.[15] [16] Moreover, the coherence properties of the output light from multimode nonlinear optical structures were found to obey the optical thermodynamic theory.[17]
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
\Deltat
\Deltaf
\Deltaf\Deltat\ge1
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.
\theta(f)\proptof
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).
Light also has a polarization, which is the direction in which the electric or magnetic 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.
The signature property of quantum matter waves, wave interference, relies on coherence. While initially patterned after optical coherence, the theory and experimental understanding of quantum coherence greatly expanded the topic.[18]
The simplest extension of optical coherence applies optical concepts to matter waves. For example, when performing the double-slit experiment with atoms in place of light waves, a sufficiently collimated atomic beam creates a coherent atomic wave-function illuminating both slits.[19] Each slit acts as a separate but in-phase beam contributing to the intensity pattern on a screen. 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. Many variations of this experiment have been demonstrated.[20]
As with light, transverse coherence (across the direction of propagation) of matter waves is controlled by collimation. Because light, at all frequencies, travels the same velocity, longitudinal and temporal coherence are linked; in matter waves these are independent. In matter waves, velocity (energy) selection controls longitudinal coherence and pulsing or chopping controls temporal coherence.
The discovery of the Hanbury Brown and Twiss effect – correlation of light upon coincidence – triggered Glauber's creation[21] of uniquely quantum coherence analysis. Classical optical coherence becomes a classical limit for first-order quantum coherence; higher degree of coherence leads to many phenomena in quantum optics.
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)[22] [23] 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.[24] Quantum coherence has been shown to be equivalent to quantum entanglement[25] in the sense that coherence can be faithfully described as entanglement, and conversely that each entanglement measure corresponds to a coherence measure.
Coherent superpositions of optical wave fields include holography. Holographic photographs have been used as art and as difficult to forge security labels.
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
\psi(r)
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.