Degree of coherence

Figure 1: This is a plot of the absolute value of g⁽¹⁾ as a function of the delay normalized to the coherence length τ/τ_c. The blue curve is for a coherent state (an ideal laser or a single frequency). The red curve is for Lorentzian chaotic light (e.g. collision broadened). The green curve is for Gaussian chaotic light (e.g. Doppler broadened).

The normalized first order correlation function is written as:[1]

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

where ${\displaystyle \langle \cdots \rangle }$ denotes a (statistical) ensemble 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 to each other waves then ${\displaystyle \mathbf {r} =z}$.

In this case, the result for stationary states will not depend on ${\displaystyle t_{1}}$, but on the time delay ${\displaystyle \tau =t_{1}-t_{2}}$ (or ${\displaystyle \tau =t_{1}-t_{2}={\frac {z_{1}-z_{2}}{c}}}$ if ${\displaystyle z_{1}\neq z_{2}}$).

This allows us to write a simplified form

${\displaystyle g^{(1)}(\tau )={\frac {\left\langle E^{*}(t)E(t+\tau )\right\rangle }{\left\langle \left|E(t)\right|^{2}\right\rangle }},}$

where we have now averaged over t.

The normalised second order correlation function is written as:[1]

Figure 2: This is a plot of g⁽²⁾ as a function of the delay normalized to the coherence length τ/τ_c. The blue curve is for a coherent state (an ideal laser or a single frequency). The red curve is for Lorentzian chaotic light (e.g. collision broadened). The green curve is for Gaussian chaotic light (e.g. Doppler broadened). The chaotic light is super-Poissonian and bunched.

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

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 ${\displaystyle g^{(2)}}$ in terms of intensities. A plane parallel wave in a stationary state will have

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

The above expression is even, ${\displaystyle g^{(2)}(\tau )=g^{(2)}(-\tau )}$. For classical fields, one can apply the Cauchy–Schwarz inequality to the intensities in the above expression (since they are real numbers) to show that ${\displaystyle g^{(2)}(\tau )\leq g^{(2)}(0)}$. The inequality ${\displaystyle \left\langle I(t)I(t)\right\rangle -{\left\langle I(t)\right\rangle }^{2}=\left\langle {\left[I(t)-\left\langle I(t)\right\rangle \right]}^{2}\right\rangle \geq 0}$ shows that ${\displaystyle 1\leq g^{(2)}(0)\leq \infty }$. Assuming independence of intensities when ${\displaystyle \tau \to +\infty }$ leads to ${\displaystyle g^{(2)}(+\infty )=1}$. Nevertheless, the second-order coherence for an average over fringes of complementary interferometer outputs of a coherent state is only 0.5 (even though ${\displaystyle g^{(2)}=1}$ for each output). And ${\displaystyle g^{(2)}}$ (calculated from averages) can be reduced down to zero with a proper discriminating trigger level applied to the signal (within the range of coherence).

A generalization of the first-order coherence

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

A generalization of the second-order coherence

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

or in intensities

${\displaystyle g^{(n)}(\mathbf {r} _{1},t_{1};\mathbf {r} _{2},t_{2};\dots$ ;\mathbf {r} _{n},t_{n})={\frac {\langle I(\mathbf {r} _{1},t_{1})I(\mathbf {r} _{2},t_{2})\cdots I(\mathbf {r} _{n},t_{n})\rangle }{\langle I(\mathbf {r} _{1},t_{1})\rangle \langle I(\mathbf {r} _{2},t_{2})\rangle \cdots \langle I(\mathbf {r} _{n},t_{n})\rangle }}}

Figure 3: This is a plot of g⁽²⁾ as a function of the delay normalized to the coherence length τ/τ_c. A value of g⁽²⁾ below the dashed black line can only occur in a quantum mechanical model of light. The red curve shows the g⁽²⁾ of the antibunched and sub-Poissonian light emitted from a single atom driven by a laser beam.

The predictions of ${\displaystyle g^{(n)}}$ 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.

${\displaystyle {\begin{aligned}E^{*}&\rightarrow {\hat {E}}^{-}\\E&\rightarrow {\hat {E}}^{+}.\end{aligned}}}$

With

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

we get in the case of stationary light:

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

Figure 4: This is a plot of g⁽²⁾ as a function of the delay normalized to the coherence length τ/τ_c. This is an example of a g⁽²⁾ that indicates antibunched light but not sub-Poissonian light.

Figure 5: Photon detections as a function of time for a) antibunching (e.g. light emitted from a single atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). τ_c is the coherence time (the time scale of photon or intensity fluctuations).

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

• Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), ISBN 0-19-850177-3