so I have the following question: I want to calculate the first order coherence function for the light coming from a thermal source. I should note that this is a question I have and not a HW problem.
Relevant Equations
Following "The Quantum Theory of Light" by R. Loudon, this function is given by the Fourier transform of the normalized power spectral density
$$g_1(\tau)=\int_{-\infty}^{+\infty} F(\omega)e^{-i\omega\tau}\,\text{d} \omega$$
where the function $F$ is given by
$$F(\omega)=\frac{f(\omega)}{\int_{-\infty}^{+\infty}f(\omega)\,\text{d}\omega}$$
where $f(\omega)$ is the power spectral density
$$f(\omega)=\frac{|E(\omega)|^2}{T}\simeq\frac{1}{2\pi}\int_{-\infty}^{+\infty}\text{d}\tau\, \left<E^*(t)E(t+\tau)\right>e^{i\omega\tau}$$.
Here $T$ is the time over which we sample the incoming electromagnetic wave, taken to be much larger than the coherence time of the wave, so that the limits of the Fourier transform can be extended to infinity. The brackets denote an ensemble average and $E$ is the electric field. These are Eqs. (3.5.6)-(3.5.10) in the book, if you happen to have it.
Attempt
Now, $f(\omega)$ is really the energy radiated per unit area per unit frequency. For a thermal source I know that this is given by Planck's law ($\hbar=c=k_B=1$ here)
$$\text{energy per unit area per unit frequency}=T\frac{1}{\pi^2}\frac{\omega^3}{\exp(\omega/\Theta)-1}=|E(\omega)|^2$$
where I denote the temperature as $\Theta$ to distinguish it from $T$ which is the total time the detector sees the signal. Here I have multiplied by the time $T$ the usual Planck distribution, so as to convert the "per unit volume" to "per unit area". The time $T$ cancels in the expression $f(\omega)$ as expected. So now
$$f(\omega)=\frac{1}{\pi^2}\frac{\omega^3}{\exp(\omega/\Theta)-1}$$.
I assume that these steps are correct up to here. Please correct me if I'm wrong.
Confusion
Now my problem is that this function $f(\omega)$ is defined only for $\omega>0$, as the Planck spectrum is a frequency spectrum and not the usual symmetric Fourier spectrum that contains negative frequencies. I am not sure how to extend it to negative frequencies to do the $g_1(\tau)$ integral, and I am not even sure if this question makes sense or if I'm grossly misunderstanding something.
The thing is, Loudon says that plugging in the lineshapes $f(\omega)$ for a Lorentzian or a Gaussian distribution (that arises from collision broadening or Doppler broadening of a monochromatic wave, respectively) leads to the correct forms for $g_1(\tau)$, but his lineshapes are not symmetric. In fact, the ones he uses are
$$F(\omega)=\frac{\gamma/\pi}{(\omega-\omega_0)^2+\gamma^2}$$
and
$$F(\omega)=\frac{1}{\sqrt{2\pi\Delta^2}}\exp\left[-\frac{(\omega-\omega_0)^2}{2\Delta^2}\right]$$
respectively, which he integrates from $-\infty$ to $+\infty$ to get the correct result. Thus, there is non-zero support for $\omega<0$.
How would one use the Planck spectrum instead of the Lorentzian/Gaussian ones? Do I put $\omega\to|\omega|$ and integrate from $-\infty$ to $+\infty$, or do I keep the lower limit to $0$, effectively saying that $f(\omega)=0$ for $\omega<0$ for thermal light? Or nothing of the above?
Concrete Questions
The main reason for this question is that I want to understand the dependence of $g_1(\tau)$ on the coherence timescale of a thermal source. This coherence timescale is supposed to be $\Theta^{-1}$, the temperature of the source (e.g. see "The physics of Hanbury Brown-Twiss intensity interferometry", by G. Baym, google-able), which is very different from the usual timescales of collisional or Doppler broadening which depend not only on $\Theta$, but also on the frequency $\omega_0$ and the mass of the atom that emits the spectral line. The numerical values of the timescales are also several order of magnitude different. Therefore, there must be an effect from the fact that we are observing a black body when we look at a star, for example. So here are my questions:
In general, what is the relation of the Planck spectrum to the Fourier transform of the electric field (EM wave) that I measure in the lab? Fourier transforms usually go to negative frequencies and the Planck spectrum does not. Is this even a well-posed question? This sounds very straightforward to me and yet I haven't been able to find such a relation anywhere.
More concretely, what is the $g_1$ function for thermal light? How does it depend on the temperature of the object? The only source I've managed to find on this one is a PhD thesis, which says that the spectrum can be taken approximately as Gaussian centered at the peak frequency. But we do have the Planck spectrum, why don't we use that one (other than the fact that the integration doesn't involve "elementary" functions)? The thesis is titled "THE HIGH-ORDER QUANTUM COHERENCE OF THERMAL LIGHT", by H. Chen, 2014, University of Meryland Baltimore County.
Any reference would be hugely appreciated. I am always confused with spectral densities and Fourier transforms.
Thanks!
