Energy Transport and the Poynting Vector — Question 5

Question 5

Coherent field addition can change the intensity without changing either input amplitude

Two co-propagating vacuum waves share polarization and electric amplitude E0. Their phase difference is delta. Each alone has intensity $I_1=\epsilon_0cE_0^2/2$.

1. Derive the combined intensity as a function of delta.

2. Find it at delta=0, pi/2, and pi. Then find the mean intensity if delta is uniformly randomized over a full cycle.

Question 5 - Solution

Coherent field addition can change the intensity without changing either input amplitude

(a) Method and calculation

$$E=E_0[\cos\phi+\cos(\phi+\delta)]=2E_0\cos(\delta/2)\cos(\phi+\delta/2).$$$$\boxed{I=4I_1\cos^2(\delta/2)=2I_1(1+\cos\delta)}.$$

(b) Interpretation and further calculation

$$\boxed{I(0)=4I_1,\quad I(\pi/2)=2I_1,\quad I(\pi)=0}.$$$$\langle\cos\delta\rangle_\delta=0\Rightarrow\boxed{\langle I\rangle_\delta=2I_1}.$$

Physical check

The cross term survives only when relative phase is controlled; simple intensity addition assumes it averages away.