Abstract from wikipedia
In quantum 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 a 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. First order correlation is actually the amplitude-amplitude correlation and the second order correlation is the intensity-intensity correlation. It is also used to differentiate between states of light that require a quantum mechanical description 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
The normalized first order correlation function is written as:
$$ {g^{(1)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2})={\frac {|\left\langle E^{*}(\mathbf {r} _{1},t_{1})E(\mathbf {r} _{2},t_{2})\right\rangle |}{\left[\left\langle \left|E(\mathbf {r} _{1},t_{1})\right|^{2}\right\rangle \left\langle \left|E(\mathbf {r} _{2},t_{2})\right|^{2}\right\rangle \right]^{1/2}}}}$$Where <> denotes an ensemble (statistical) average.
In optical interferometers such as the Michelson interferometer, Mach-Zehnder interferometer, or Sagnac interferometer, one splits an electric field into two components, introduces a time delay to one of the components, and then recombines them. The intensity of resulting field is measured as a function of the time delay. In this specific case involving two equal input intensities, the visibility of the resulting interference pattern is given by:
$${\displaystyle \nu =|g^{(1)}(\tau )|}\\\\ {\displaystyle \nu =\left|g^{(1)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2})\right|}$$where the second expression involves combining two space-time points from a field. The visibility ranges from zero, for incoherent electric fields, to one, for coherent electric fields. Anything in between is described as partially coherent.
Generally,$${\displaystyle g^{(1)}(0)=1}\quad g^{{(1)}}(0)=1\quad and \quad {\displaystyle g^{(1)}(\tau )=g^{(1)}(-\tau )^{*}}$$ .
Degree of second-order coherence
The normalised second order correlation function is written as:
$$g^{{(2)}}({\mathbf {r}}_{1},t_{1};{\mathbf {r}}_{2},t_{2})={\frac {\left\langle E^{*}({\mathbf {r}}_{1},t_{1})E^{*}({\mathbf {r}}_{2},t_{2})E({\mathbf {r}}_{1},t_{1})E({\mathbf {r}}_{2},t_{2})\right\rangle }{\left\langle \left|E({\mathbf {r}}_{1},t_{1})\right|^{2}\right\rangle \left\langle \left|E({\mathbf {r}}_{2},t_{2})\right|^{2}\right\rangle }}$$Light is said to be bunched if ${\displaystyle g^{(2)}(\tau )< g^{(2)}(0)}$ and antibunched if ${\displaystyle g^{(2)}(\tau )>g^{(2)}(0)}$.