Degree of coherence

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

$g^{(1)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2})={\frac {\left\langle E^{*}(\mathbf {r} _{1},t_{1})E(\mathbf {r} _{2},t_{2})\right\rangle }{\left[\left\langle \left|E(\mathbf {r} _{1},t_{1})\right|^{2}\right\rangle \left\langle \left|E(\mathbf {r} _{2},t_{2})\right|^{2}\right\rangle \right]^{1/2}}}$

Where <> denotes an ensemble (statistical) average. 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 $\mathbf {r} =z$ . In this case, the result for stationary states will not depend on $t_{1}$ , but on the time delay $\tau =t_{1}-t_{2}$ (or $\tau =t_{1}-t_{2}-{\frac {z_{1}-z_{2}}{c}}$ if $z_{1}\neq z_{2}$ ).

This allows us to write a simplified form $g^{(1)}(\tau )={\frac {\left\langle E^{*}(t)E(t+\tau )\right\rangle }{\left\langle \left|E(t)\right|^{2}\right\rangle }}$ 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 $|g^{(1)}(\tau )|$ . More generally, when combining two space-time points from a field

visibility= $\left|g^{(1)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2})\right|.$

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 $g^{(1)}(\tau )=g^{(1)}(-\tau )^{*}$ .

Examples of $g^{(1)}$

For light of a single frequency (e.g. laser light): $g^{(1)}(\tau )=e^{-i\omega _{0}\tau }$

For Lorentzian chaotic light (e.g. collision broadened): $g^{(1)}(\tau )=e^{-i\omega _{0}\tau -(|\tau |/\tau _{c})}$

For Gaussian chaotic light (e.g. Doppler broadened): $g^{(1)}(\tau )=e^{-i\omega _{0}\tau -{\frac {\pi }{2}}(\tau /\tau _{c})^{2}}$

Here, $\omega _{0}$ is the central frequency of the light and $\tau _{c}$ is the coherence time of the light.

Degree of second-order coherence

$g^{(2)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2})={\frac {\left\langle E^{*}(\mathbf {r} _{1},t_{1})E^{*}(\mathbf {r} _{2},t_{2})E(\mathbf {r} _{1},t_{1})E(\mathbf {r} _{2},t_{2})\right\rangle }{\left\langle \left|E(\mathbf {r} _{1},t_{1})\right|^{2}\right\rangle \left\langle \left|E(\mathbf {r} _{2},t_{2})\right|^{2}\right\rangle }}$

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 $g^{(2)}(\tau )={\frac {\left\langle I(t)I(t+\tau )\right\rangle }{\left\langle I(t)\right\rangle ^{2}}}$

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

Examples of $g^{(2)}$

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

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

Degree of nth-order coherence

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

A generalization of the second-order coherence

$g^{(n)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2};...;\mathbf {r} _{n},t_{n})={\frac {\left\langle E^{*}(\mathbf {r} _{1},t_{1})E^{*}(\mathbf {r} _{2},t_{2})...E^{*}(\mathbf {r} _{n},t_{n})E(\mathbf {r} _{1},t_{1})E(\mathbf {r} _{2},t_{2})...E(\mathbf {r} _{n},t_{n})\right\rangle }{\left\langle \left|E(\mathbf {r} _{1},t_{1})\right|^{2}\right\rangle \left\langle \left|E(\mathbf {r} _{2},t_{2})\right|^{2}\right\rangle ...\left\langle \left|E(\mathbf {r} _{n},t_{n})\right|^{2}\right\rangle }}$

Examples of $g^{(n)}$

Using the second definition

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

Light of a single frequency: $g^{(n)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2};...;\mathbf {r} _{n},t_{n})=1$

Generalization to Quantum Fields

The predictions of $g^{(n)}$ for n>1 change when the 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^{*}\rightarrow {\hat {E}}^{-}

E\rightarrow {\hat {E}}^{+}.

With

${\hat {E}}^{-}=-i\left({\frac {\hbar \omega }{2\epsilon _{0}V}}\right)^{1/2}{\hat {a}}^{\dagger }e^{-i(kz-wt)}$

we get

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

Examples of nonclassical states of light

N-photon state (Fock state): $g^{(2)}(0)=1-{\frac {1}{n}}$

Squeezed vacuum (with average photon number <n>): $g^{(2)}(0)=3+{\frac {1}{<n>}}$

Amplitude squeezed light (where s is the squeezing parameter): $g^{(2)}(0)=1+{\frac {e^{-2s}-1}{<n>}}$

Phase squeezed light (where s is the squeezing parameter): $g^{(2)}(0)=1+{\frac {e^{2s}-1}{<n>}}$

Photon Bunching

Light is said to be bunched if $g^{(2)}(\tau )\leq g^{(2)}(0)$ and antibunched if $g^{(2)}(\tau )\geq g^{(2)}(0)$ .

Degree of coherence