First order coherence classical optics problem

[Azelketh]

Homework Statement

Hi, I am trying to work through exercise 2.1 on page 37 of Microcavities (by alexy kavokin, jeremy baumberg, guillaume malpuech and fabrice laussy)


the problem is to prove
$$| g^{(1)}(\tau) | = | cos( \frac{1}{2}(\omega_1 - \omega_2)\tau) ) |$$


where:

$$g^{(1)}(\tau)=\frac{\langle E^{\ast}(t)E(t+\tau)\rangle}{\langle |E(t)|^2 \rangle}$$
and
$$E(t)=E_0(t)\exp^{i[k_1z-\omega_1t]}+E_0(t)\exp^{i[k_2z-\omega_2t+\varphi]}$$
where
$$\varphi$$ varies randomly between measurements


how do you deal mathmatically with $$\varphi$$ varying??


Also more simply above the exercise the text states a simpler apparently 'trivial' result using the same formula for $$g^{(1)}(\tau)$$ that the sine wave of
$$E(t)=E_0(t)\exp^{i[\omega t - kz + \varphi]}$$
by direct application of the formula for $$g^{(1)}(\tau)$$ yields:

$$g^{(1)}(\tau)= \exp^{i\omega \tau}$$

i cannot show even this 'trivial' application, i find that:
$$\langle E^{\ast}(t)E(t+\tau)\rangle = \langle E_0\exp^{-i\omega \tau}\rangle$$
and
$$\langle |E(t)|^2 \rangle = \langle |E_0^2 \exp^{2i(\omega t -kx + \varphi)} \rangle$$
How does that evaluate to
$$g^{(1)}(\tau)= \exp^{-i\omega \tau}$$ ??
If anyone can give me any pointers( or show me the complete workings of this XD ) then it would much appreciated. Thanks for reading my post.

 

[Azelketh]

problem no longer. Just assumed by $$/varphi$$ varying randomly then all components with $$/varphi$$ cancel in the averageing.