Energy Transport and the Poynting Vector — Question 7

Question 7

Amplitude modulation raises mean power through the squared envelope

A vacuum wave at a point is $E(t)=E_0[1+m\cos\Omega t]\cos\omega t$, with $0\le m\le1$ and omega much greater than Omega. Define $I_c=\epsilon_0cE_0^2/2$ as the unmodulated carrier intensity. Use successive fast and slow averages.

1. Find the carrier-averaged intensity envelope and its maximum and minimum.

2. Find the long-time mean intensity and the ratio of maximum envelope intensity to that mean when m=1.

Question 7 - Solution

Amplitude modulation raises mean power through the squared envelope

(a) Method and calculation

$$I_{\mathrm{env}}(t)=\boxed{I_c[1+m\cos\Omega t]^2}.$$$$\boxed{I_{\max}=I_c(1+m)^2,\quad I_{\min}=I_c(1-m)^2}.$$

(b) Interpretation and further calculation

$$\langle I\rangle=I_c[1+2m\langle\cos\Omega t\rangle+m^2\langle\cos^2\Omega t\rangle]=\boxed{I_c(1+m^2/2)}.$$$$m=1:\quad\boxed{I_{\max}/\langle I\rangle=4/(3/2)=8/3}.$$

Physical check

Average field amplitude cannot be squared to obtain mean intensity; averaging must be applied to the squared field.