Degree of coherence

# Degree of coherence

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In optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, correlation functions are used to characterize the statistical and coherence
Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....

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. It is also used to differentiate between states of light that require a quantum mechanical description
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

and those for which classical fields are sufficient. Analogous considerations apply to any Bose field in subatomic physics, in particular to mesons (cf. Bose–Einstein correlations)

## Degree of first-order coherence

Where <> denotes an ensemble (statistical) average. For non-stationary states, such as pulses, the ensemble is made up of many pulses. When one deals with stationary states, where the statistical properties do not change with time, one can replace the ensemble average with a time average. If we restrict ourselves to plane parallel waves then .
In this case, the result for stationary states will not depend on , but on the time delay (or if ).

This allows us to write a simplified form
where we now average over t.

In optical interferometers such as the Michelson interferometer
Michelson interferometer
The Michelson interferometer is the most common configuration for optical interferometry and was invented by Albert Abraham Michelson. An interference pattern is produced by splitting a beam of light into two paths, bouncing the beams back and recombining them...

, Mach-Zehnder interferometer
Mach-Zehnder interferometer
The Mach–Zehnder interferometer is a device used to determine the relative phase shift between two collimated beams from a coherent light source. The interferometer has been used, amongst other things, to measure small phase shifts in one of the two beams caused by a small sample or the change in...

, or Sagnac interferometer, one splits an electric field into two components, time delays one component, and then recombines them. The intensity of resulting field is measured as a function of the time delay. The visibility
Interferometric visibility
The interferometric visibility quantifies the contrast of interference in any system which has wave-like properties, such as optics, quantum mechanics, water waves, or electrical signals. Generally, two or more waves are combined and as the phase between them is changed The interferometric...

of the resulting interference pattern is given by . More generally, when combining two space-time points from a field
visibility=

The visibility ranges from zero, for incoherent electric fields, to one, for coherent electric fields. Anything in between is described as partially coherent.

Generally, and .

### Examples of g(1)

For light of a single frequency (e.g. laser light):

For Lorentzian chaotic light (e.g. collision broadened):

For Gaussian chaotic light (e.g. Doppler broadened):

Here, is the central frequency of the light and is the coherence time
Coherence time
For an electromagnetic wave, the coherence time is the time over which a propagating wave may be considered coherent...

of the light.

## Degree of second-order coherence

Note that this is not a generalization of the first-order coherence

If the electric fields are considered classical, we can reorder them to express in terms of intensities. A plane parallel wave in a stationary state will have

The above expression is even, For classical fields, one can apply Cauchy-Schwarz inequality to the intensities in the above expression (since they are real numbers) to show that and that . Nevertheless the second-order coherence for an average over fringe of complementary interferometer outputs of a coherent state is only 0.5 (even though for each phase). And (calculated from averages) can be reduced down to zero with a proper discriminating trigger
Trigger
-Technology:* Trigger , a mechanism that actuates the firing of firearms* Image trigger, a device used in highspeed cameras* Schmitt trigger, an electronic circuit* Trigger circuit, IBM's name for a flip-flop...

level applied to the signal (within the range of coherence).

### Examples of g(2)

Chaotic light of all kinds: .
Note the Hanbury-Brown and Twiss effect
Hanbury-Brown and Twiss effect
The Hanbury Brown and Twiss 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 dual wave-particle nature of the beam, and the results of a given experiment...

uses this fact to find from a measurement of .

Light of a single frequency:

Photon antibunching
Photon antibunching generally refers to a light field with photons more equally spaced than a coherent laser field and a signal at detectors is anticorrelated. More specifically, it can refer to sub-Poisson photon statistics, that is a photon number distribution for which the variance is less than...

for another use of where for a single photon source because

where is the photon number observable.

## Degree of nth-order coherence

A generalization of the first-order coherence

A generalization of the second-order coherence

or in intensities

### Examples of g(n)

Light of a single frequency:

Using the first definition:
Chaotic light of all kinds:

Using the second definition:
Chaotic light of all kinds:
Chaotic light of all kinds:

## Generalization to quantum fields

The predictions of for n > 1 change when the classical fields (complex numbers or c-number
C-number
The term c-number is an old nomenclature used by Paul Dirac which refers to real and complex numbers. It is used to distinguish from operators in quantum mechanics....

s) are replaced with quantum fields (operators or q-numbers). In general, quantum fields do not necessarily commute, with the consequence that their order in the above expressions can not be simply interchanged.

With

we get

## Photon bunching

Light is said to be bunched if and antibunched if .

• Coherence theory
Coherence theory
In physics, coherence theory is the study of optical effects arising from partially coherent light and radio sources. Partially coherent sources are sources where the coherence time or coherence length are limited by bandwidth, by thermal noise, or by other effect...

• Correlation and dependence
• Optical autocorrelation
Optical autocorrelation
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...

• Fourier transform spectroscopy
Fourier transform spectroscopy
Fourier transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the electromagnetic radiation or other type of radiation....

• Bose–Einstein correlations