Coherence (physics)

In physics, coherence expresses the potential for two waves to interfere. Two monochromatic beams from a single source always interfere. Even for wave sources that are not strictly monochromatic, they may still be partly coherent.
When interfering, two waves add together to create a wave of greater amplitude than either one or subtract from each other to create a wave of minima which may be zero, 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. 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 ; a precise mathematical definition of the degree of coherence is given by means of correlation functions. More broadly, coherence describes the statistical similarity of a field, such as an electromagnetic field or quantum wave packet, at different points in space or time.

Qualitative concept

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.
Coherence contains several distinct concepts. Spatial coherence describes the correlation between waves at different points in space, either lateral or longitudinal. 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.

Mathematical definition

The coherence function between two signals and is defined as
where is the cross-spectral density of the signal and and are the power spectral density functions of and, 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 and 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.
The coherence varies in the interval. If it means that the signals are perfectly correlated or linearly related and if they are totally uncorrelated. If a linear system is being measured, being the input and 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.

Coherence and correlation

The coherence of two waves expresses how well correlated the waves are as quantified by the cross-correlation function. 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 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. Degree of correlation involves correlation functions.

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.

Waves in a rope or slinky
Surface waves in a liquid
Electromagnetic signals in transmission lines
Sound
Radio waves and microwaves
Light waves
Matter waves associated with, for examples, electrons and atoms

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. 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, 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 is defined as the coherence time. At a delay of the degree of coherence is perfect, whereas it drops significantly as the delay passes. The coherence length is defined as the distance the wave travels in time.
The coherence time is not the time duration of the signal; the coherence length differs from the coherence area.

The relationship between coherence time and bandwidth

The larger the bandwidth – range of frequencies Δf a wave contains – the faster the wave decorrelates :
Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum to its autocorrelation.
Narrow bandwidth lasers have long coherence lengths. For example, a stabilized and monomode helium–neon laser can easily produce light with coherence lengths of 300 m. Not all lasers have a high monochromaticity, however.
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

Examples of temporal coherence include:

A wave containing only a single frequency is perfectly correlated with itself at all time delays, in accordance with the above relation.
Conversely, a wave whose phase drifts quickly will have a short coherence time.
Similarly, pulses of waves, which naturally have a broad range of frequencies, also have a short coherence time since the amplitude of the wave changes quickly.
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, it is often called incoherent.

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. A detector measures the time-averaged intensity of the light exiting the interferometer. The resulting visibility of the interference pattern gives the temporal coherence at delay. 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, an infinitely fast detector would measure an intensity that fluctuates significantly over a time t equal to. In this case, to find the temporal coherence at, 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 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, .
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.
A distance away from an incoherent source with surface area,
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.