The Photon, the Quantum of Light — Question 1

Question 1

Equal optical powers need not carry equal photon counts

Two monochromatic beams have equal power P at wavelengths $\lambda_1=400\,\mathrm{nm}$ and $\lambda_2=800\,\mathrm{nm}$. They are combined without losses. Use $hc=1240\,\mathrm{eV\,nm}$ and $1\,\mathrm{eV}=1.60\times10^{-19}\,\mathrm J$.

1. Find the ratio of photon arrival rates and the fraction of photons from the longer-wavelength beam.

2. Find the mean energy per photon in the combined beam.

Question 1 - Solution

Equal optical powers need not carry equal photon counts

(a) Method and calculation

$$\dot N_i=P/E_i=P\lambda_i/(hc),\quad\boxed{\dot N_2/\dot N_1=2}.$$$$\boxed{\dot N_2/(\dot N_1+\dot N_2)=2/3}.$$

(b) Interpretation and further calculation

$$\overline E=\frac{2P}{P\lambda_1/(hc)+P\lambda_2/(hc)}=\frac{2hc}{\lambda_1+\lambda_2}.$$$$\boxed{\overline E=2.06667\,\mathrm{eV}}.$$

Physical check

Energy fractions are one half each, while photon-number fractions differ. Averaging the two photon energies without count weighting would be incorrect.