- #1
Sigurdsson
- 25
- 1
Homework Statement
Hi guys, appreciate all the help I can get. This has been bugging me for 24 hours now. I'm starting to think I'm missing something in the question.
We are exploring first-order coherence degree. That is, exploring the coherence of two separate signals (wave packets) by using the equation
[tex]g^{(1)}(\tau, t) = \frac{\langle E^\ast (t) E(t + \tau) \rangle}{\langle |E(t)|^2 \rangle }[/tex]
If you are familiar with the Michelson interferometer, then you should be familiar with the equation of fringe visibility
[tex] V = \frac{I_{\text{max}} - I_{\text{min}} }{I_{\text{max}} + I_{\text{min}} } [/tex]
Which equates to
[tex] V = |g^{(1)}(\tau)| [/tex]
So here is the question. We are given the field of two different signals
[tex]\frac{E(x,t)}{E_0} = e^{i(k_1 z - \omega_1 t)} + e^{i(k_2 z - \omega_2 t + \varphi)}[/tex]
with the common amplitude [itex]E_0[/itex] and dephasing difference [itex]\varphi[/itex]. The goal is to show that if [itex]\varphi[/itex] is kept fixed we get
[tex] V = |g^{(1)}(\tau)| = 1 [/tex]
and if it varies randomly between measurements, the averaging should yield
[tex] V = |g^{(1)}(\tau)| = \left|\cos{\left( \frac{1}{2}(\omega_1 - \omega_2) \tau \right) } \right| [/tex]
Homework Equations
This is exercise 2.1 in the book Microcavities by Alexey V. Kavokin.
The Attempt at a Solution
Here is my attempt at the first part if [itex]\varphi[/itex] is kept fixed. Let's put [itex]a = k_1 z - \omega_1 t[/itex] and [itex]b = k_2 z - \omega_2 t + \varphi[/itex] for simplicity's sake. Then we have
[tex] \frac{\langle E^\ast (t) E(t + \tau) \rangle}{\langle |E(t)|^2 \rangle } = \frac{ \langle \left( e^{-ia} + e^{-ib} \right) \left( e^{ia} e^{-i\omega_1 \tau} + e^{ib} e^{-i\omega_2 \tau} \right) \rangle }{ \langle 2 + e^{-i(a -b)} + e^{i(a-b)} \rangle } = \frac{\langle (e^{i(a-b)} + 1) e^{-i\omega_1 \tau} + (e^{-i(a-b)} + 1) e^{-i\omega_2 \tau} \rangle}{\langle 2 + e^{-i(a -b)} + e^{i(a-b)} \rangle} [/tex]
This doesn't look like unity to me. The problem is (I think) is that I'm not sure how to apply the method of "averaging" in this example. Any tips?