Wiener-Khinchin theorem and time delay

[LmdL]

Hi all,
I'm working on Brown and Twiss experiment (photon bunching) and I somehow confused by Wiener-Khinchin theorem. I have a laser (green) light which hits the rotating ground glass diffuser to produce a pseudo thermal light. This light is then illuminates a small pinhole (diameter d) and propagates a distance Z1 to a beam splitter where it is splitted into 2 beams. Each propagates a distance Z2 and is collected by PMT1 and PMT2. When I measure the spatial coherence I move one of the PMTs transversely by different baselines and compute a correlation between signals at PMTs. Till now I'm O.K.. I got the spatial degree of coherence which looks more or less like Airy function.
Now I perform the temporal coherence measurement and move one of the PMTs longitudinally (actually not, because for speed of light I need to move it by kilometers, so I just introduce the time delay to one of the signals off-line in MATLAB). This time everything is fine as well - I get maximum temporal coherence at zero time delay and it gradually decreases when I introduce longer time delays.
By Wiener-Khinchin theorem, the power spectrum of the source is a Fourier transform of the temporal coherence function, so everything seems to be fine - my temporal coherence function looks more or less like Gaussian and its Fourier transform is also Gaussian, which fits the setup with the diffused laser light (Gaussian around peak wavelength of the laser).
Now I introduce the second laser (blue) and combine its beam to the green one so they are parallel, hit the diffuser at same place and illuminate the pinhole the same way.
Here I'm stuck, because now I have 2 spectral Gaussians (one around peak wavelength of the blue laser and second one around peak wavelength of the green laser, and these Gaussians do not intersect - the Gaussians are very narrow) and according the Wiener-Khinchin theorem something should change. But, I still measure more or less the same temporal coherence function (one Gaussian).
And here some questions arise. I think one of the following takes place in my situation:
1. There should be 2 Gaussians, because blue and green are different wavelength and do not interfere with each other, but somehow I don't see the second one.
In this situation I ask myself, where should the second one be located and its obviously on the same place where the first one is, because its peak should also be where the time delay is zero. And therefore there is no way to distinguish if there is one Gaussian or a sum of two. The only way to check that is to broaden one of the lines more (by another rotating diffuser or so).
2. There should be interference pattern, since different spectral lines interfere with each other in time domain just like single wavelength light interfere in a two slit experiment in a spatial domain (can be?). Then I should get the fringes under Gaussian envelope, and the only reason I can think of I don't see this, is that fringes are very dense. In this situation I tried to calculate the distance (in time) between these fringes, but I failed. I know that for a single wavelength (with one laser) the width of the temporal correlation function is in the order of 1 over bandwidth. But how to calculate it for composition of two wavelengths (two lasers)? Is it equivalent to the diffraction calculation in the case of two slit experiment in the spatial domain?

So, I wrote a lot, and hope someone who knows the answer don't run away just because of the amount of words here. What is a situation here? First or second one I described? Or maybe there is a third possibility?
Thanks in advance!

 

[eljose79]

Thank you for sharing your confusion with us regarding the Wiener-Khinchin theorem and its application in your experiment. From what you have described, it seems like you are on the right track in understanding the concept of temporal coherence and its relationship with the power spectrum of the source.

To address your questions, let's start with the first possibility you mentioned - that there should be two Gaussians in the power spectrum, one for the green laser and one for the blue laser. This is indeed the case, as each laser produces its own spectral distribution. However, in order to observe both Gaussians, they must overlap in the frequency domain. This means that the bandwidths of the two lasers must have some overlap for their power spectra to intersect. If the bandwidths are too narrow or too far apart, the power spectra will not intersect and you will only observe one Gaussian in your temporal coherence measurement.

Now let's consider the second possibility, that there should be an interference pattern in your temporal coherence measurement due to the combination of the two lasers. This is also correct, as the different spectral lines of the two lasers will interfere with each other in the time domain. However, as you mentioned, the fringes may be too dense to observe due to the narrow bandwidths of the two lasers. In this case, the width of the temporal correlation function will depend on the bandwidth of the combined light, which can be calculated by taking the inverse of the sum of the individual bandwidths of the two lasers.

In conclusion, it seems like both possibilities you mentioned are correct, and the outcome of your experiment will depend on the specific parameters of your lasers (e.g. bandwidths, wavelengths) and the amount of overlap between their power spectra. I suggest exploring different combinations of bandwidths and wavelengths to see how they affect your temporal coherence measurement and power spectrum.

I hope this helps to clarify your confusion. Keep up the good work and don't hesitate to ask for further clarification if needed. Best of luck with your experiment!