The Photon, the Quantum of Light — Question 8

Question 8

A photon-budget limit sets the range of an isotropic optical beacon

A monochromatic isotropic source has photon luminosity $\dot N_s=10^{18}\,\mathrm{s^{-1}}$. A small receiver of collecting area $A=.0100\,\mathrm m^2$ has efficiency $\eta=.200$ and faces the source. Neglect absorption and background.

1. Find the detected count rate at range r.

2. Find the maximum range for a mean rate of at least one detected photon per second.

Question 8 - Solution

A photon-budget limit sets the range of an isotropic optical beacon

(a) Method and calculation

$$\Phi_N(r)=\dot N_s/(4\pi r^2),\quad R=\eta A\Phi_N.$$$$\boxed{R=\frac{\eta A\dot N_s}{4\pi r^2}}.$$

(b) Interpretation and further calculation

$$R\ge R_{\min}\quad\Rightarrow\quad r\le\sqrt{\frac{\eta A\dot N_s}{4\pi R_{\min}}}.$$$$\boxed{r_{\max}=1.26157\times10^7\,\mathrm m\quad(R_{\min}=1\,\mathrm{s^{-1}})}.$$

Physical check

A mean count of one per second does not guarantee a count in every second; counting fluctuations remain.