Polarization — Question 3

Question 3

Many small polarization rotations transmit more than one large step

A beam already polarized along x has intensity I0. It passes through N ideal analyzers with successive axes rotated by equal increments pi/(2N), ending along y. Neglect reflection and all loss except ideal projection.

1. Derive the transmitted intensity and evaluate it for N=2 and N=3.

2. Find its limiting behavior for large N using log cos(epsilon) approximately equals minus epsilon squared divided by two.

Question 3 - Solution

Many small polarization rotations transmit more than one large step

(a) Method and calculation

$$\boxed{I_N/I_0=\left[\cos^2\!\left(\frac{\pi}{2N}\right)\right]^N}.$$$$\boxed{I_2/I_0=1/4,\quad I_3/I_0=(3/4)^3=27/64}.$$

(b) Interpretation and further calculation

$$\ln(I_N/I_0)=2N\ln\cos(\pi/2N)\simeq-\pi^2/(4N).$$$$\boxed{I_N/I_0\simeq e^{-\pi^2/(4N)}\longrightarrow1}.$$

Physical check

The initial beam is already polarized, so no initial one-half factor applies; the loss per small step vanishes quadratically.