# Photon statistics of an incandescent light source

We usually calibrate the cameras on our microscopes by capturing 20 images of a blurry (not sharp) fluorescent particle. For each pixel in this stack of 20 images we calculate the intensity variance. By plotting the variance against the intensity we obtain the scale factor that allows us to convert the ADU (arbitrary data unit) into detected photoelectrons.

However, this approach relies on a light source with Poisson statistics.

This is applicable for fluorescence light.

But can we also use light of a halogen lamp instead of fluorescent light?

I heard that photons leaving a heated metal surface have a particular statistic. I think the argument was that they are coming out in bunches and it has something to do with the workfunction. It would be nice if someone could give a good reference on this.

The picture above shows the calibration curves for an EMCCD camera. The left graph corresponds to the readout mode you would use for bright images and has a readnoise of 15 electrons per pixel.

The graph on the right shows the result for the low noise mode. In this mode the charge that has been accumulated under the pixels is transferred through an electron multiplication register. Impact ionization increases the number of electrons. The gain is usually up to 300x.

The inset on the lower right is how a calibration image looks like. Note that this one is fake.

• An incandescent light will have "bunching" of photons (Bose-Einstein statistics) when the intervals under consideration are much shorter than the coherence time (< 1 ps). For larger intervals the distribution is essentially Poisson. See this paper for an experimental overview. – mmc Sep 7 '11 at 1:56

Your methodology assumes that the light source not only has Poissonian statistics but also has the same average intensity for each image. For example if you used a fluorescent tube plugged into a wall socket, its intensity goes up and down at 120Hz (in the US), because that's twice the AC frequency. So if your exposure time (a.k.a. integration time) was less than 10ms, you could big fluctuations not because of photon shot noise, but just because some exposures are taken with a brighter light source and others with a dimmer light source. So you could get the wrong photon conversion.

Poisson statistics is not a concern, every macroscopic light source has Poisson statistics. I am quite confident that a halogen bulb sends out photons with Poisson statistics. I don't know what you heard about photons "coming out in bunches", I haven't heard of that. Anyway even if they do, it would still be Poisson statistics for any given particle. Well, I guess it's important for this measurement that the fluorescent lifetime be much less than the time-separation between photons. So if the light source was really bunchy enough, it could mess up the data--not because the photons are non-Poissonian, but because of a "saturable absorber" effect. I would be surprised if this was an issue.

Frame-to-frame light-intensity fluctuations, on the other hand, may or may not be a problem. I'm not sure what light sources are more or less stable. Of course it certainly depends on the exposure time of each frame and of the whole measurement. You can of course measure this stability directly. But actually in your graph, frame-to-frame-light-intensity-variations would give a variance proportional to the square of intensity, whereas shot noise is linear, and the graph you show is linear. So I guess it's not a problem!

• The stability of the light source is indeed an important problem. Also the baseline of the camera has to be taken into account (usually CCDs have some pixels which are protected from any illumination and their signal, which can vary with sensor temperature, is taken as the baseline). – whoplisp Jul 11 '11 at 10:15
• It may very well be that a halogen lamp produces Poisson statistics. I thought someone who has more experience with Photomultiplier tubes could give a definite answer. – whoplisp Jul 11 '11 at 10:19

I noticed this old answer and I don't think it's right. Surely a laser gives off photons with poisson statistics, but not an incandescent source. The reason is fairly intuitive, to the extent that these types of verbal arguments are ever correct: there are random intensity fluctuations associated with incandescent sources; so if there is a click in the detector, there is an enhanced probability that it occurs during an intensity excursion. Therefore, a second click very close by is more likely than average. This is in contradiction to the definition of Poisson distribution, which is simply that the probability of clicks is independent of the proximity to any other click.

Incandescent (i.e. thermal) light sources exhibit noise which is beyond the standard Poissonian noise limit. The statistics of any particular mode of a thermal light source are given by the Bose-Einstein distribution and therefore fall into the category of super-Poissonnian noise. For light with multiple modes $N_m$, the variance in the number of photons in any given window of time is given by

$$(\Delta n)^2=\bar{n}+\frac{\bar{n}^2}{N_m}$$

where $\bar{n}$ is the average number of photons within the window.

However, macroscopic thermal light sources have a very large number of modes. In this case, the statistics of thermal light sources reduce to Poissonian statistics with

$$(\Delta n)^2=\bar{n}.$$ So, for most practical experiments, an incandescent light source can be treated as a shot noise limited source.

Source: Fox, M. (2006). Quantum Optics: An Introduction (Vol. 6). Oxford University Press. ISBN: 9780198566731 (Specifically Section 5.5.1; Relevant chapters as pdf)