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Say we have a light source emitting $N$ photons per unit area.

In the first case, the photons are reflected at a distance $r$ from the source and measured again at the source. Divergence caused the number of photons per unit area to decline proportional to $r^2$. This means $$\frac{N}{r^2}$$ arrive at the mirror in distance $r$ and $$\frac{\frac{N}{r^2}}{r^2}=\frac{N}{r^4}$$ photons come back in total, after traveling a distance of $2r$.

Now we adjust the setting, instead of reflecting the photons we measure them at twice the distance directly. Therefore, we measure $$\frac{N}{(2r)^2}=\frac{N}{4r^2}.$$

Why is the number of photons per unit area not equal in the two cases?

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  • $\begingroup$ You need to do this calculation as a surface area element on an expanding sphere using the solid angle. $\endgroup$ – boyfarrell Jan 9 at 16:26
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    $\begingroup$ Suppose you have no mirror, but you first measure the photon flux distance $r$, measuring $N/r^2$. Then let the photons propagate further to a distance $2r$. By your initial argument, you would also expect to get a flux of $N/r^4$ in this case. So you can ignore the mirror and just figure out why this calculation is wrong in general. $\endgroup$ – The Photon Jan 9 at 17:11
  • $\begingroup$ Great, @ThePhoton is answering a question about itself. :D But which calculation is wrong then, the one leading to $N/r^4$ or the other one? I suppose $N/4r^2$ is wrong?! $\endgroup$ – kalle Jan 10 at 9:41
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The correct solution is $N/(2r)^2$, which is the same as your $N/4r^2$.

The reason is that after reflecting from the mirror, the photons aren't radiating out from the surface of the mirror, but from the image of the source at a distance $r$ behind the mirror. So the position of the mirror needn't be considered, only the total distance the photons have traveled along whatever optical path.

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