Diffraction and the Wave Theory of Light — Question 3

Question 3

Four aperture patches can cancel despite nonzero transmitted power

Four equal aperture patches contribute phasors $E_p e^{ij\pi/2}$ at an observation point, with $j=0,1,2,3$. Let $I_p$ be the intensity of one patch alone. The patches are mutually coherent.

1. Find the resultant field and intensity with all four patches open.

2. Find the intensity after patch $j=3$ is blocked, and explain why this does not imply increased total transmitted power.

Question 3 - Solution

Four aperture patches can cancel despite nonzero transmitted power

(a) Method and calculation

$$E=E_p(1+i-1-i)=\boxed{0}.$$$$\boxed{I=0}\quad\text{although each patch transmits nonzero power}.$$

(b) Interpretation and further calculation

$$E_{\mathrm{three}}=E_p(1+i-1)=iE_p\Longrightarrow\boxed{I_{\mathrm{three}}=I_p}.$$$$P_{\mathrm{transmitted,three}}/P_{\mathrm{transmitted,four}}=\boxed{3/4}\quad\text{for equal nonoverlapping patches}.$$

Physical check

Blocking a patch can brighten one direction by removing cancellation while reducing total transmitted energy.