In 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 $g^\{(1)\}$, it is useful for quantifying the coherence between two electric fields, as measured in an Michelson or other linear optical interferometer. The correlation between pairs of fields, $g^\{(2)\}$, 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 (QED) and those for which classical fields are sufficient.

Degree of first-order coherence

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$g^\{(1)\}(\; \backslash mathbf\{r\}\_1,t\_1;\backslash mathbf\{r\}\_2,t\_2)=\; \backslash frac\{\backslash left\; \backslash langle\; E^*(\backslash mathbf\{r\}\_1,t\_1)E(\backslash mathbf\{r\}\_2,t\_2)\; \backslash right\; \backslash rangle\}\{\backslash left\backslash left\; \backslash langle\backslash left\; |\; E(\backslash mathbf\{r\}\_1,t\_1)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \backslash left\; \backslash langle\; \backslash left\; |E(\backslash mathbf\{r\}\_2,t\_2)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \backslash right^\{1/2\}\}$

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 $\backslash mathbf\{r\}=z$. In this case, the result for stationary states will not depend on $t\_1$, but on the time delay $\backslash tau=t\_1-t\_2$ (or $\backslash tau=t\_1-t\_2-\backslash frac\{z\_1-z\_2\}\{c\}$ if $z\_1\; \backslash ne\; z\_2$).

This allows us to write a simplified form $g^\{(1)\}(\; \backslash tau)=\; \backslash frac\{\backslash left\; \backslash langle\; E^*(t)E(t+\backslash tau)\; \backslash right\; \backslash rangle\}\{\backslash left\; \backslash langle\backslash left\; |\; E(t)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \}$ where we now average over t.

In optical interferometers such as the Michelson interferometer, Mach-Zehnder interferometer, 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 of the resulting interference pattern is given by $|g^\{(1)\}(\; \backslash tau)|$. More generally, when combining two space-time points from a field

visibility=$\backslash left\; |\; g^\{(1)\}(\; \backslash mathbf\{r\}\_1,t\_1;\backslash mathbf\{r\}\_2,t\_2)\; \backslash right\; |.$

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

Generally, $g^\{(1)\}(0)=1$ and $g^\{(1)\}(\backslash tau)=g^\{(1)\}(-\backslash tau)^*$.

Examples of $g^\{(1)\}$

For light of a single frequency (e.g. laser light): $g^\{(1)\}(\backslash tau)=e^\{-i\backslash omega\_0\backslash tau\}$

For Lorentzian chaotic light (e.g. collision broadened): $g^\{(1)\}(\backslash tau)=e^\{-i\backslash omega\_0\backslash tau-(|\backslash tau|/\backslash tau\_c)\}$

For Gaussian chaotic light (e.g. Doppler broadened): $g^\{(1)\}(\backslash tau)=e^\{-i\backslash omega\_0\backslash tau-\backslash frac\{\backslash pi\}\{2\}(\backslash tau/\backslash tau\_c)^2\}$

Here, $\backslash omega\_0$ is the central frequency of the light and $\backslash tau\_c$ is the coherence time of the light.

Degree of second-order coherence

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$g^\{(2)\}(\; \backslash mathbf\{r\}\_1,t\_1;\backslash mathbf\{r\}\_2,t\_2)=\; \backslash frac\{\backslash left\; \backslash langle\; E^*(\backslash mathbf\{r\}\_1,t\_1)E^*(\backslash mathbf\{r\}\_2,t\_2)E(\backslash mathbf\{r\}\_1,t\_1)E(\backslash mathbf\{r\}\_2,t\_2)\; \backslash right\; \backslash rangle\}\{\backslash left\; \backslash langle\backslash left\; |\; E(\backslash mathbf\{r\}\_1,t\_1)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \backslash left\; \backslash langle\; \backslash left\; |E(\backslash mathbf\{r\}\_2,t\_2)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \}$

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$g^\{(2)\}$ in terms of intensities. A plane parallel wave in a stationary state will have $g^\{(2)\}(\; \backslash tau)=\; \backslash frac\{\backslash left\; \backslash langle\; I(t)I(t+\backslash tau)\; \backslash right\; \backslash rangle\}\{\backslash left\; \backslash langle\; I(t)\; \backslash right\; \backslash rangle^2\; \}$

The above expression is even, $g^\{(2)\}(\backslash tau)=\; g^\{(2)\}(-\backslash tau)$ For classical fields, one can apply Cauchy-Schwarz inequality to the intensities in the above expression (since they are real numbers) to show that $1\; \backslash le\; g^\{(2)\}(0)\; \backslash le\; \backslash infty$ and that $g^\{(2)\}(\backslash tau)\; \backslash le\; g^\{(2)\}(0)$.

Examples of $g^\{(2)\}$

Chaotic light of all kinds: $g^\{(2)\}(\backslash tau)\; =\; 1\; +\; |\; g^\{(1)\}(\backslash tau)|^2$ Note the Hanbury-Brown and Twiss effect uses this fact to find $|\; g^\{(1)\}(\backslash tau)|$ from a measurement of $g^\{(2)\}(\backslash tau)$.

Light of a single frequency: $g^\{(2)\}(\backslash tau)\; =\; 1$

Degree of nth-order coherence

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A generalization of the first-order coherence $g^\{(n)\}(\; \backslash mathbf\{r\}\_1,t\_1;\backslash mathbf\{r\}\_2,t\_2;...;\backslash mathbf\{r\}\_\{2n\},t\_\{2n\})=\; \backslash frac\{\backslash left\; \backslash langle\; E^*(\backslash mathbf\{r\}\_1,t\_1)E^*(\backslash mathbf\{r\}\_2,t\_2)...E^*(\backslash mathbf\{r\}\_n,t\_n)E(\backslash mathbf\{r\}\_\{n+1\},t\_\{n+1\})E(\backslash mathbf\{r\}\_\{n+2\},t\_\{n+2\})...E(\backslash mathbf\{r\}\_\{2n\},t\_\{2n\})\; \backslash right\; \backslash rangle\}\{\backslash left\backslash left\; \backslash langle\backslash left\; |\; E(\backslash mathbf\{r\}\_1,t\_1)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \backslash left\; \backslash langle\; \backslash left\; |E(\backslash mathbf\{r\}\_2,t\_2)\backslash right\; |^2\; \backslash right\; \backslash rangle...\backslash left\; \backslash langle\; \backslash left\; |E(\backslash mathbf\{r\}\_\{2n\},t\_\{2n\})\backslash right\; |^2\; \backslash right\; \backslash rangle\; \backslash right^\{1/2\}\}$

A generalization of the second-order coherence

$g^\{(n)\}(\; \backslash mathbf\{r\}\_1,t\_1;\backslash mathbf\{r\}\_2,t\_2;...;\backslash mathbf\{r\}\_n,t\_n)=\; \backslash frac\{\backslash left\; \backslash langle\; E^*(\backslash mathbf\{r\}\_1,t\_1)E^*(\backslash mathbf\{r\}\_2,t\_2)...E^*(\backslash mathbf\{r\}\_n,t\_n)E(\backslash mathbf\{r\}\_1,t\_1)E(\backslash mathbf\{r\}\_2,t\_2)...E(\backslash mathbf\{r\}\_n,t\_n)\; \backslash right\; \backslash rangle\}\{\backslash left\; \backslash langle\backslash left\; |\; E(\backslash mathbf\{r\}\_1,t\_1)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \backslash left\; \backslash langle\; \backslash left\; |E(\backslash mathbf\{r\}\_2,t\_2)\backslash right\; |^2\; \backslash right\; \backslash rangle...\backslash left\; \backslash langle\; \backslash left\; |E(\backslash mathbf\{r\}\_n,t\_n)\backslash right\; |^2\; \backslash right\; \backslash rangle\; \}$

Examples of $g^\{(n)\}$

Using the second definition

Chaotic light of all kinds: $g^\{(n)\}(0)=\; n!$

Light of a single frequency: $g^\{(n)\}(\; \backslash mathbf\{r\}\_1,t\_1;\backslash mathbf\{r\}\_2,t\_2;...;\backslash mathbf\{r\}\_n,t\_n)=1$

Generalization to Quantum Fields

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The predictions of $g^\{(n)\}$ for n>1 change when the classical fields (complex numbers or c-numbers) 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.

$E^*\; \backslash rightarrow\; \backslash hat\{E\}^-$

$E\; \backslash rightarrow\; \backslash hat\{E\}^+.$

With

$\backslash hat\{E\}^-=\; -i\backslash left\; (\backslash frac\{\backslash hbar\backslash omega\}\{2\backslash epsilon\_0\; V\}\; \backslash right\; )^\{1/2\}\backslash hat\{a\}^\backslash dagger\; e^\{-i(kz-wt)\}$

we get

$g^\{(2)\}\; =\; \backslash frac\{\backslash left\; \backslash langle\; \backslash hat\{a\}^\backslash dagger(t)\; \backslash hat\{a\}^\backslash dagger(t+\backslash tau)\; \backslash hat\{a\}(t+\backslash tau)\; \backslash hat\{a\}(t)\; \backslash right\; \backslash rangle\}\{\backslash left\; \backslash langle\; \backslash hat\{a\}^\backslash dagger(t)\; \backslash hat\{a\}(t)\; \backslash right\; \backslash rangle^2\}$

=== Examples of nonclassical states

Photon Bunching

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Light is said to be bunched if $g^\{(2)\}(\backslash tau)\; \backslash le\; g^\{(2)\}(0)$ and antibunched if $g^\{(2)\}(\backslash tau)\; \backslash ge\; g^\{(2)\}(0)$.

See also

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• Quantum field theory
• Electromagnetic field
• Optical autocorrelation
• Fourier transform spectroscopy

Suggested reading

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• Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), 0198501773

Optics | Quantum optics