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In an experiment to measure the photon statistics of thermal light, the radiation from a black- body source is filtered with an interference filter of bandwidth 0.1 nm centered at 500 nm, and allowed to fall on a photon-counting detector. Calculate the number of modes incident on the detector, and hence discuss the type of statistics that would be expected.

The text report the photon number variance for a single mode of the thermal radiation to be $$(\Delta n)^2 = \langle n \rangle + \langle n \rangle^2$$ while for $N_m$ modes it is (under certain approximations) $$(\Delta n)^2 = \langle n \rangle + \frac{\langle n \rangle^2}{N_m}$$

The main difficulty is that I cannot think of a formula to calculate the absolute number of modes, but only a number of modes per unit of volume: $$\frac{N_m}{V} = \frac{8 \pi}{\lambda^4} \Delta \lambda$$ Instead $\langle n \rangle$ should be a pure number: $$\langle n \rangle = \frac{1}{e^{hc/ \lambda K T} - 1}$$

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closed as off-topic by John Rennie, ZeroTheHero, Jon Custer, Kyle Kanos, user191954 Dec 1 '18 at 3:20

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