Photons Have Momentum — Question 10

Question 10

Random reflection gives impulse fluctuations even for a fixed photon count

A pulse contains exactly N normally incident photons, each of momentum p. Each photon independently reflects with probability R or is absorbed with probability $1-R$. Mirror recoil is negligible. The reflected count K is binomial, with $\langle K\rangle=NR$ and $\operatorname{Var}K=NR(1-R)$.

1. Find the mean and variance of the impulse delivered to the surface.

2. For $R=1/2$, find the relative standard deviation.

Question 10 - Solution

Random reflection gives impulse fluctuations even for a fixed photon count

(a) Method and calculation

$$J=2pK+p(N-K)=p(N+K).$$$$\boxed{\langle J\rangle=Np(1+R)},\qquad\boxed{\operatorname{Var}J=Np^2R(1-R)}.$$

(b) Interpretation and further calculation

$$R=1/2:\quad\langle J\rangle=\tfrac32Np,\quad\sigma_J=\tfrac12p\sqrt N.$$$$\boxed{\sigma_J/\langle J\rangle=1/(3\sqrt N)}.$$

Physical check

The incident photon number is fixed here; the fluctuations arise from the random absorption/reflection outcomes, not from Poisson input counts.