Determining the frequency of a monochromatic light source from measurements of irradiance alone?

Question

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).

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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).

Answer 1

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).