Polarization — Question 1

Question 1

A single intensity measurement leaves two analyzer orientations

A linearly x-polarized beam of intensity I0 passes through an ideal analyzer whose axis is at angle theta to x. The transmitted intensity is I0/4. Axes are equivalent modulo 180 degrees.

1. Find all analyzer angles in the interval from zero through less than 180 degrees.

2. For each angle, predict the transmitted fraction after a small positive rotation delta, to first order with delta in radians.

Question 1 - Solution

A single intensity measurement leaves two analyzer orientations

(a) Method and calculation

$$I/I_0=\cos^2\theta=1/4\Rightarrow|\cos\theta|=1/2.$$$$\boxed{\theta=60^\circ\ \text{or}\ 120^\circ}.$$

(b) Interpretation and further calculation

$$\frac{I(\theta+\delta)}{I_0}=\cos^2\theta-\sin2\theta\,\delta+O(\delta^2).$$$$\boxed{I(60^\circ+\delta)/I_0\simeq1/4-\sqrt3\delta/2,\quad I(120^\circ+\delta)/I_0\simeq1/4+\sqrt3\delta/2}.$$

Physical check

The intensity alone determines an unsigned projection; the response to a known rotation resolves the two choices.