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Let's say we have a monochromatic source of unknown frequency (say, a laser) and some device/sensor that measures the total photon energy absorbed over a fixed time interval (essentially, irradiance).

Since photon counting follows Poisson statistics, we can determine the light frequency via repeat measurements,

$$\langle E\rangle = \langle N \rangle h\nu \\ {\sigma_{E}}^2 = \langle N \rangle h^2 \nu^2$$

where $\langle E\rangle$ is the mean total photon energy across measurements, ${\sigma_{E}}^2$ is the variance, $\langle N \rangle$ is the mean number of photons absorbed across measurements, and $\nu$ is the frequency.

We can find the frequency,

$$ \nu = \frac{{\sigma_{E}}^2}{h \langle E\rangle} $$

This is interesting because you determine the frequency from the variance-to-mean ratio alone without any real calibration required (assuming you can accurately measure the total photon energy).


Has this been meaningfully used in practice or actual research?

My intuition says you could do this at microwave frequencies (with bolometers used to measure total energy). Could you measure a laboratory hydrogen maser this way?

This is probably a terrible estimator compared to just doing spectroscopy - but it's interesting how simple it is (in terms of how few variables it depends on).

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The idea that the coefficient-of-variation in a Poisson process:

$$c_V\equiv \frac{\bar x}{\sigma} = \frac{\alpha N}{\alpha \sqrt N}=\sqrt N $$

can be used to estimate $N$ has been used before. For example, in a gas Cherenkov counter, you can expect 6-ish photons per meter in air. One can estimate the actual number in the apparatus from $c^2_V$ of the PMT output distribution.

In your case, you would measure the mean energy and use:

$$ E = Nh\nu = c^2_Vh\nu $$

Note that $c_V$ is measured entirely from the shape of measured data, without calibration of the axes. Of course, you need to calibrate the device to get $E$.

Some things to watch out for are:

Biased vs. unbiased estimators of $c_V$.

Also compound distributions: you don't aways have direct access to the photons you're measuring, e.g., they may go through a device with a non-zero efficiency. Thus, you're convolving a Poisson creation process with a Bernoulli detection process. The result is a different Poisson process.

If the detection involves amplification (e.g., an electron avalanche), it can convolve the Poisson process with a Gaussian process. This leads to Pólya-distributed data (aka: The Dirichlet-multinomial distribution).

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  • $\begingroup$ Do you know if that has ever been used to obtain the wavelength of e.g. a laser and what the accuracy is? As based entiely on poisson noise, a perfect measurement device (no readout noise etc.) shoud give an exact measurement... however, this works only for monochromatic light... $\endgroup$ Sep 18, 2021 at 23:25

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