Intuitive idea for chaotic light statistics?

[Affcr]

Does anyone have an intuitive idea for the statistics of chaotic light? I can understand that the power fluctuations in this kind of light can give rise to a second order autocorrelation parameter g(2) higher than 1. However I can not see why the value for this parameter should be g(2)=2. Does anyone have an intuitive explanation for this value and why it is not higher or lower?

 

[Cthugha]

There is very little intuition in there. Just some math. Consider the definition of g(2), which roughly translates into a constant and the photon number variation divided by the squared mean. For thermal light these two are pretty much equal, so you will get a value of two.

To be exact:
$g^{(2)}(0)=\frac{\langle : n^2: \rangle}{\langle n \rangle^2}$

The double stops denote normal ordering which ensures that the detection of the first photon reduces the photon number by 1. That leaves you with:
$g^{(2)}(0)=\frac{\langle n (n-1) \rangle}{\langle n \rangle^2}$

Of course you can represent the instantaneous photon number n as the sum of the mean $\langle n \rangle$ and some fluctuation $\delta$ about the mean. So you get:

$g^{(2)}(0)=\frac{\langle (\langle n \rangle +\delta) (\langle n \rangle+\delta-1) \rangle}{\langle n \rangle^2}$
All terms linear in $\delta$ must of course vanish when taking the expectation value, so you are left with:

$g^{(2)}(0)=\frac{\langle n^2 \rangle - \langle n \rangle + \delta^2}{\langle n \rangle^2}=1-\frac{1}{\langle n \rangle}+\frac{\delta^2}{\langle n \rangle^2}$

The $\delta^2$ term is proportional to the photon number variance. For thermal light, this is $\langle n \rangle^2 +\langle n \rangle$, which just leaves you with two.

 

[Affcr]

Thank you for the explanation, Cthugha. This makes me understant better the concep of chaotic light and why $g^{(2)}(0)=2$, but leads me to further doubts which I would appreciate if you know and can tell me the answer:

Why is the photon number variance of chaotic light $\langle n \rangle^2 +\langle n \rangle$ ? Is this expression just based on experimental evidence?
What is the origin of the intensity fluctuations?
What does the coherence time of cahotic light depend on? Do you know about any experiment in which they can change this time by modifying any property of the source?

 

[Cthugha]

This result is not just based on experiments. You can simulate it yourself, if you have basic knowledge of any programming language or Matlab or something like that. Chaotic light is basically about the physics of the random walk. As a toy model a thermal light source can be described as a large number of harmonic oscillators, each with the same amplitude. The total amplitude is given by vectorial addition of all of these amplitudes. The total intensity is given by the square of the amplitude. Now you set some initial phase for each of these oscillators and simulate the time evolution of the system in discrete steps. You set a small maximal possible phase jump $\phi_m$ per discrete time step and in each time step you create a different random number for each of the many oscillators such that each oscillator undergoes a phase jump somewhere between $- \phi_m$ and $+\phi_m$. Most of the time all of the fields will cancel and you get small intensity. Sometimes the fields will add up to a large amplitude and you will get huge intensity. If you plot the intensity over time, you will get fluctuations that have the variance of thermal light with $\langle n \rangle$ proportional to the number of oscillators you use.

The coherence time depends on the phase jump per step that you use. In experiments you can change this easily. The decay of coherence is proportional to the Fourier transform of the power spectrum of your light field. In easy terms: If you filter your light source spectrally and get a narrower spectrum, you will get longer coherence time.

 

[Affcr]

Thanks again, now I get some intution about the process. Let me just ask a last question. Do you know which can be the physical processes that induce these phase jumps and why should they be small? If they were related to spontaneous emission I would expect the phase jumps to have any value between 0 and $2\pi$, so if it is no like this I guess it has to be related to another phenomenon.