# Why does stacking polarizers of the same angle still block more and more light?

I have some sheets of polarization film. They came in a big box, all stacked at the same angle. I noticed that the entire stack of them lets almost no light through, even though they're all at the same angle.

I pulled out two, and those two also block more light than just one.

Why?

Is this because I have low-grade polarizers? Or because lining them up at EXACTLY the same angle is impossible? Or because the light that gets through the first one is not really polarized exactly to its angle — it's just that less of it is polarized away from its angle than before?

If it's because these are low-grade polarizers, can anyone recommend a linear polarizer that I can stack several of in a row at the same angle and still have a 100% probability of the light getting through?

I feel like I'm probably just misunderstanding polarization theory so please correct me.

Why? Is this because I have low-grade polarizers?

As any substance, the transmitance (amount of light that passes from one side to the other) of polarizers is not 100%. It depends on the materials and even if you can perfectly align the polarizers, it will not reach 100%.

An important aspect is that transmitance is dependant on the wavelength. So, in your case, your polarizers might not have a very high transmitance in visible light.

As always, if this is a good thing or not, it depends on what you will use it for. The table below indicates a type of polarizer for each priority characteristic. [2]

If it's because these are low-grade polarizers, can anyone recommend a linear polarizer that I can stack several of in a row at the same angle and still have a 100% probability of the light getting through?

100% is not feasible, as it has been said. But if you need higher transmission than you have you should probably choose dichroic glass, as indicated. Downsides are the price (this kinds of polarizer reaches "one thousand dollars for a polarizer with a one inch diameter" according to the document cited), they can not be very large due to manufacturing restrictions and they convert the unwanted light into heat.

• "they convert the unwanted light into heat." - what do other polarizers convert it to? Jan 4, 2017 at 15:05
• @JanDvorak A polarizing beam splitter is a way of polarizing light without producing heat. It simply puts light of the unwanted polarization into a separate beam.
– Yly
Jan 4, 2017 at 20:12
• @Yly: But don't all optical media (not just polarizers) convert some light into heat? Jan 5, 2017 at 15:46

A high-quality supplier of polarizers and other optical equipment would be able to offer you data on the transmission characteristics of even their cheapest polarizers:

I interpret this plot to mean that if you bought two of these devices, aligned their axes parallel to each other, and shone unpolarized $\lambda=550\rm\,nm$ green light on them, you'd only get 40% of your intensity out of the first polarizer, and of that fraction something like $10^{-4}$ still has the "wrong" polarization. From the second polarizer you'd only get $80\% \cdot 40\% = 32\%$ of the original intensity, and another transmission factor 0.8 from any subsequent, also-aligned polarizers. It might be possible to improve the polarization by having multiple parallel filters, at the cost of this lost overall intensity, but you might also run into sneaky laboratory issues.

• As you mention, it the fact that the first polariser 'converts' unpolarised light to polarised light that does the majority of the perceived attenuation. For an interesting test (for the OP who has the sheets..), take the first two sheets and arrange them as a crossed polariser at maximum extinction, then insert a third polariser between them at a 45 degree angle. What happens and Why... ? Jan 4, 2017 at 16:08
• I guess "unpolarized" light has equally distributed random polarization around a circle O. Polarization lets thru the light according to probability (cos 0-θ)^2 where θ was the light's polarization when it hit. Then if the light next goes through a 90-degree polarizer, it will now have a probability (cos 90-θ)^2 of being polarized at an angle θ. Since the waves y = cos 0 and y = cos 90 are opposite phase, they cancel and no light gets through. But with (cos 45-θ)^2 in between, the probability waves don't cancel out. Is that basically correct or no? Jan 4, 2017 at 23:41
• @CommaToast: Unpolarized light has random polarization, but the polarization of light behaves as a vector. Passing light through a filter will pass through the vector component that's parallel to the filter, while removing the component that's perpendicular. Jan 4, 2017 at 23:56
• @supercat I think that's basically what I'm saying, because the vector you speak of represents the probability amplitude (i.e. cosine) of the angle of polarization. In other words the polarization vector is not the actual direction of polarization of the actual photons; it's just a representation of probabilities having to do with their actual angles of polarization having enough of a component in the direction of the polarizer's angle that they can pass through it (after which their actual angles of polarization could have changed to some degree, for all we know). Right? Jan 5, 2017 at 0:25
• @supercat: And it attenuates the component parallel to the filter as well while it's at it. Jan 5, 2017 at 15:49

Even if you had extremely clear glass (like for optical fibers), stacking them would cause them to get more quickly opaque than what would be expected by their transmission coefficient. This is only for completeness because the effect on real polarized filters is dominated by their transmission coefficient as said by Odano.

The reason is that on every boundary surface (seam between two glasses) light is reflected, the amount is approximately 4%. So after 17 glasses only 50% of light is transmitted even with perfectly clear glass.

An ideal (theoretical) polarizer will only let in light along a certain axis.

This is impossible in real life. Any polarizer that you can purchase will let in light in a range of possible polarization angles. Additionally, manufacturing processes cannot guarantee perfect alignment with the apparatus holding the polarizer.

Therefore, when you are aligning them all up, you are not actually accomplishing that. They are still out of alignment, even if in small amounts.

• So once the light comes out of the polarizer, it's fair to say that its polarization angle is not exactly the same as the angle of the polarizer? But that we can be pretty sure it's definitely not the same as an angle perpendicular to the polarizer? It seems to me that some light in a decreasing range of density across other angles must be getting through, since otherwise inserting a 45° polarizer between a 0° and a 90° would reduce the light by less than 1/2 again as much as the first one. Jan 4, 2017 at 6:16
• It seems to me it must be like Trump combing his hair. The more he combs it the straighter it gets but the more falls out. It's never perfectly straight though, even if he thinks "it's the straightest." Jan 4, 2017 at 6:18
• @CommaToast You've got the right idea. It's not that the light doesn't align with the polarizer, it's that the polarizer doesn't have a strict, perfect axis through which it lets light through. In other words, the angle of the polarizer has an uncertainty in it. As the relative angles of the polarizers becomes large, this uncertainty becomes small, so it perfectly reasonable to say that there probably isn't much light going through two polarizers at 90°. Jan 4, 2017 at 6:36
• Careful, you can have reflective perfect polarizers with Brewster's angle. Jan 4, 2017 at 18:23