Why Does a Third Polarizer Let Light Through? The Three-Polarizer Paradox and Malus’s Law

Here is a trick you can do with three pairs of polarized sunglass lenses. Hold two of them with their axes at right angles and almost no light gets through. Now slip the third between them, tilted at 45°. The stack brightens. You added a filter and got more light. It looks like a paradox, but it follows from one line of classical optics. Our Diagonix sticker is the character built around this trick. This post builds on our explainer How Are Electricity and Magnetism Connected?, because light is an electromagnetic wave and polarization is the direction its electric field points.

What does a polarizer actually do?

Ordinary light from the Sun or a bulb is unpolarized: it is made of many waves whose electric fields point in all directions. A polarizing filter has an axis that acts like a slit. OpenStax explains that it passes the component of the electric field parallel to its axis, using long molecules aligned perpendicular to that axis, which absorb the other component (OpenStax University Physics, §1.7^[1]). Sheet polarizers of this kind were developed by Edwin Land, founder of the Polaroid Corporation, who described their development in a 1951 paper (Land 1951^[2]; OpenStax College Physics, §27.8^[3]).

When unpolarized light meets the first filter, it loses half its intensity. OpenStax states that the first filter “polarizes the light along its axis, decreasing its intensity by a factor of 2” (OpenStax University Physics, §1.7^[1]).

What is Malus’s law?

Once light is polarized, what happens at the next filter depends only on the angle between the light’s polarization and the filter’s axis. The electric field component that gets through is E cos θ. Intensity is proportional to the square of the field, so

I = I₀ cos²θ

Here I₀ is the intensity of the polarized light arriving at the filter (in watts per square metre, W/m²), I is the intensity after it, and θ is the angle between the polarization direction and the filter axis. This equation is known as Malus’s law (OpenStax University Physics, §1.7^[1]). At θ = 0 everything passes; at θ = 90° nothing does; and, as OpenStax notes, at 45° the intensity “is reduced to 50%” of what arrived.

The law is named after Étienne-Louis Malus (1775–1812), a French physicist and army engineer who in 1809 published a paper on his discovery of the polarization of light by reflection (Britannica^[4]). Our Diagonix sticker carries his cos²θ.

Why do two crossed polarizers block everything?

Call the first filter vertical (0°) and the second horizontal (90°). After the first filter the light is vertically polarized with intensity I₀/2. At the second filter θ = 90°, and cos 90° = 0, so

I = (I₀/2) cos²90° = 0

No light. Real filters leak a little, but ideal ones would give total darkness.

Why does adding a third polarizer let light through?

Now insert a filter at 45° between them. Apply Malus’s law at each step:

1. After filter 1 (0°): I₁ = I₀/2, polarized at 0°.
2. After filter 2 (45°): I₂ = I₁ cos²45° = (I₀/2)(1/2) = I₀/4, now polarized at 45°.
3. After filter 3 (90°): the angle is again 45°, so I₃ = I₂ cos²45° = I₀/8.

I₃ = (I₀/2) cos²45° cos²45° = I₀/8

One eighth of the original light, 12.5%, gets through, where before there was none. The key is in step 2. The middle filter does not just remove light. Whatever passes it is re-polarized along the filter’s own axis, at 45°. The last filter therefore sees light at 45° to its axis, not 90°, and lets half of it through. The middle filter has changed the question the last filter is asked.

What if you use more filters?

Split the 90° turn into N equal steps of 90°/N, and the transmitted fraction becomes

I/I₀ = ½ cos²ᴺ(90°/N)

For N = 2 that is 0.125, for N = 3 it is about 0.211, for N = 10 about 0.390, and for N = 100 about 0.488. With many tiny steps, ideal filters could rotate the polarization by 90° while losing almost nothing beyond the first factor of ½. (Real filters absorb some light at every pass, so in practice the extra filters cost something.)

What does this look like for single photons?

Malus’s law is classical, but the same experiment works one photon at a time. A photon either passes a polarizer or it does not, and cos²θ becomes the probability of passing. After a photon passes the 45° filter, it is in a definite 45° polarization state, and its earlier vertical polarization no longer matters for the next filter. That is a clean example of a measurement re-preparing a quantum state, the theme of our explainer What Is Quantum Measurement?.

Polarizers are also the measuring devices in the most famous tests of quantum entanglement. The Nobel Committee’s popular background to the 2022 Physics Prize describes how pairs of particles with linked polarisation can be investigated “using a filter that allows through polarisation that is oriented in a particular direction” (NobelPrize.org^[5]). The same background explains that John Clauser built an apparatus sending pairs of entangled photons “each towards a filter that tested their polarisation”, and published the result in 1972 with his doctoral student Stuart Freedman (NobelPrize.org^[5]; Freedman & Clauser 1972^[6]). More light-and-field products: our Electromagnetism collection.

FAQ

Is the three-polarizer effect a quantum effect?

No. Malus’s law, a classical wave result, predicts it fully for ordinary light. It also works with single photons, where cos²θ becomes a probability.

What angle gives the most light with three filters?

With the outer filters fixed at 0° and 90°, the middle one at θ transmits (I₀/2) cos²θ sin²θ, which peaks at θ = 45° with I₀/8. At 30°, for example, you get only 3I₀/32.

Why do polarized sunglasses cut glare?

Light reflected from water or roads is partly polarized, the effect Malus discovered in 1809. Sunglasses with vertical axes block much of that polarized glare.

References

1. S. J. Ling, J. Sanny & W. Moebs, University Physics Volume 3 (OpenStax, Rice University), §1.7 “Polarization”. https://openstax.org/books/university-physics-volume-3/pages/1-7-polarization
2. E. H. Land (1951), “Some Aspects of the Development of Sheet Polarizers,” J. Opt. Soc. Am. 41, 957 (DOI 10.1364/JOSA.41.000957; metadata verified via Crossref). https://doi.org/10.1364/JOSA.41.000957
3. P. P. Urone & R. Hinrichs, College Physics 2e (OpenStax, Rice University), §27.8 “Polarization”. https://openstax.org/books/college-physics-2e/pages/27-8-polarization
4. Encyclopaedia Britannica, “Étienne-Louis Malus”. https://www.britannica.com/biography/Etienne-Louis-Malus
5. NobelPrize.org, “How entanglement has become a powerful tool,” popular science background, Nobel Prize in Physics 2022. https://www.nobelprize.org/prizes/physics/2022/popular-information/
6. S. J. Freedman & J. F. Clauser (1972), “Experimental Test of Local Hidden-Variable Theories,” Phys. Rev. Lett. 28, 938–941 (DOI 10.1103/PhysRevLett.28.938; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.28.938

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.