Counterintuitive result? Polarizers, Malus's Law

[Guni22]

Malus's Law is an equation that describes how much light intensity that a beam of light will lose if if it goes through a polarizer, dependent on what angle the polarizer is at relative to the light.

It is explained better than I ever could here:
http://en.wikipedia.org/wiki/Polarizer#Malus.27_law_and_other_properties

But what about a scenario like this? If you have unpolarized light, pass it through an ideal polarizer, it loses half of its intensity (explained in the above link) and all the light is now polarized. Now pass the light through another polarizer that's oriented 90 degrees from the original one. By Malus's Law, or just by the nature of EM waves, all the light is gone, and light intensity is effectively 0.

But... if you add a third polarizer that is oriented 45 degrees from the first polarizer, the light seems to lose half its intensity each time (cos(45-0)^2 = 0.5 and cos(90-45)^2 = 0.5), which means after it goes through all 3 polarizers, it has one-fourth the light intensity of the original incident light. Does that mean the addition of another polarizer made it so that more light passes through?

 

[Drakkith]

I believe you are correct. Two polarizers 90 degrees from each other would block all the light. But putting a polarizer in between the first 2 that is 45 degrees would polarize the light and allow some to pass through the last polarizer.

 

[Andy Resnick]

<snip>
But... if you add a third polarizer that is oriented 45 degrees from the first polarizer, the light seems to lose half its intensity each time (cos(45-0)^2 = 0.5 and cos(90-45)^2 = 0.5), which means after it goes through all 3 polarizers, it has one-fourth the light intensity of the original incident light. Does that mean the addition of another polarizer made it so that more light passes through?

The standard explanation is that a state of polarization can always be decomposed to 2 orthogonal basis states. For example, linearly vertically polarized light can be decomposed into 2 circularly polarized states, 2 linearly polarized states oriented at +/- 45 degrees, etc.

So the first polarizer generates linearly polarized light, the middle polarizer transmits one of the 45 degree basis states, which is then decomposed into vertical and horizontal basis states, one of which is transmitted by the third polarizer.

 

[Ken G]

One way you can think of why this is possible is that absorption of a wave is not like absorption of a bullet-- you can absorb a wave by adding another wave to it, 180 degrees out of phase. Indeed, that's just what "sound cancelling headphones" do, they don't take away the wave, they add a second one to the first that has minus its amplitude. If you think of the polarizers as doing the same things, it becomes clearer how adding a third polarizer can result in more getting through-- the third polarizer adds negative waves in such a fashion as to mess up the complete destructive interference of the first two polarizers. It would be vaguely analogous to muffling the output of a sound cancelling headphone-- you could make the noise you hear louder by muffling one of its contributing components.