Polarization — Question 2

Question 2

An intermediate polarizer opens a path through crossed end axes

Unpolarized intensity I0 enters three ideal polarizers with axes at 0 degrees, theta, and 90 degrees. Let theta lie between the two end axes.

1. Find the final intensity as a function of theta.

2. Find the maximizing angle and maximum transmitted fraction. Compare with removing the middle polarizer.

Question 2 - Solution

An intermediate polarizer opens a path through crossed end axes

(a) Method and calculation

$$I_1=I_0/2,\quad I_2=I_1\cos^2\theta,\quad I_3=I_2\cos^2(90^\circ-\theta).$$$$\boxed{I_3=\frac{I_0}{8}\sin^2(2\theta)}.$$

(b) Interpretation and further calculation

$$\sin^2(2\theta)\le1\Rightarrow\boxed{\theta=45^\circ,\quad I_{3,\max}=I_0/8}.$$$$I_{\mathrm{without\ middle}}=(I_0/2)\cos^290^\circ=\boxed{0}.$$

Physical check

Each polarizer prepares a new linear polarization; the later projection is taken from that new direction.