Principle of Ergodicity in Derivation of Coherence Function

Question

I am a bit confused about the following thing: In the derivation of the coherence function for the electric field at two times $t_1$ and $t_2$, following simplification is made: $$\langle E^*(t_1) E(t_2) \rangle = \langle E^*(t) E(t + \tau) \rangle$$ So the statement is: the correlation function must be homogeneous in time, so it can only be dependent on time differences. As a reason for that, the principle of ergodicity was given without any further explanation.

It's not obvious to me how it follows from that though - maybe someone has a few clarifying words for me.

Answer 1

If a random process – in this case the electric field – is stationary, then average values of functions of the electric field are not dependent on absolute time. By the definition of stationarity the average value does not change over time. Hence, you don't need 2 times $t_1$, $t_2$ to define the average value, but only the time difference $\tau$.

Ergodicity is an even stronger condition on your random process. Besides from being stationary an ergodic process also has the property that ensemble averages equal time averages. This is important for your system, because the $\langle \cdot \rangle$ in the correlation function denotes ensemble average, i.e. you would need many equal realizations of your experiment, while you normally just have one. Under the assumption of ergodicity you can instead average over time.

Further reading about these concepts can be found in Mandel & Wolf – Optical Coherence and Quantum Optics and Goodman – Statistical Optics.