9.4.3 The Three Polarizer Problem

Consider the set up illustrated below. Do you understand why there is no light after polarizer Y? That’s right. Since polarizers X and Y are aligned perpendicularly to each other, all the light is cut out.

Now. What If we insert a third polarizer Z between X and Y? Oh my god, there is now some light after polarizer Y (as long as Z’s polarization direction is neither vertical nor horizontal). Adding another layer of polarization cuts out LESS light than before! How is that possible?

To understand this surprising outcome, you must realize that a polarizer is more than just a filter. It does not just attenuate the light passing through. It also re-aligns the polarization direction of the light!

By choosing the polarization direction of the 3^rd polarizer (Z) to be neither vertical nor horizontal, we avoid having two consecutive polarizers which are perpendicular to each other. This prevents the light from being totally cut off.

Let’s confirm this with some math. Since X and Z are misaligned by angle θ, the intensity of light after passing through Z is

$\displaystyle {{\mathrm{I}}_{1}}={{\mathrm{I}}_{0}}{{\cos }^{2}}\theta$

Since Z and Y are misaligned by angle $\displaystyle 90^\circ -\theta$ , the intensity of light after passing through Y is

$\displaystyle \begin{aligned}{{\mathrm{I}}_{2}}&={{\mathrm{I}}_{1}}{{\cos }^{2}}(90{}^\circ -\theta )\mathrm{I}\\&={{\mathrm{I}}_{0}}{{\cos }^{2}}\theta {{\cos }^{2}}(90{}^\circ -\theta )\\&={{\mathrm{I}}_{0}}{{\cos }^{2}}\theta {{\sin }^{2}}(\theta )\\&=\frac{{{{\mathrm{I}}_{0}}}}{4}{{\sin }^{2}}2\theta \end{aligned}$

In fact, the maximum intensity of $\displaystyle \displaystyle \frac{{{{\mathrm{I}}_{0}}}}{4}$ is achieved by arranging Z to have polarization direction midway between that of X and Y, i.e. 45°, 135°, 225°, 315°, etc.