Degree of coherence

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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 and those for which classical fields are sufficient.

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[edit] Degree of first-order coherence

Figure 1: This is a plot of g(1) as a function of the delay normalized to the coherence length τ/τc. 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).
Figure 1: This is a plot of g(1) as a function of the delay normalized to the coherence length τ/τc. 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).
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 t1, but on the time delay τ = t1 − t2 (or \tau=t_1-t_2-\frac{z_1-z_2}{c} if z_1 \ne 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)(τ) | . 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)(τ) = g(1)( − τ) * .

[edit] 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, ω0 is the central frequency of the light and τc is the coherence time of the light.

[edit] Degree of second-order coherence

Figure 2: This is a plot of g(2) as a function of the delay normalized to the coherence length τ/τc. 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.
Figure 2: This is a plot of g(2) as a function of the delay normalized to the coherence length τ/τc. 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.
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 expressg(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)(τ) = g(2)( − τ) 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 \le g^{(2)}(0) \le \infty and that g^{(2)}(\tau) \le g^{(2)}(0).

[edit] Examples of g(2)

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

Light of a single frequency: g(2)(τ) = 1

[edit] 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 }

[edit] 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

[edit] Generalization to Quantum Fields

Figure 3: This is a plot of g(2) as a function of the delay normalized to the coherence length τ/τc. 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.
Figure 3: This is a plot of g(2) as a function of the delay normalized to the coherence length τ/τc. 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 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^* \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}

[edit] Examples of nonclassical states

[edit] Photon Bunching

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

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

[edit] See also

[edit] Suggested reading

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