Polarized Light and the Three-Polarizer Paradox
Summary
Stack two polarizing1 filters at right angles and they block essentially all light — the view goes black. Now slip a third filter between them, and light reappears, even though you have added yet another barrier. This “three-polarizer paradox” is one of the most surprising results you can produce with pocket-sized equipment. By the end you’ll be able to explain it with Malus’s law2, and see why each filter doesn’t merely block light but re-projects it into a new polarization state — the same idea that underlies quantum measurement.
History
The story of polarization begins with a beautiful accident. In 1669, Erasmus Bartholin3 noticed that crystals of Iceland spar (calcite) split a single ray of light into two. He could not explain why — the wave theory of light did not yet exist. Christiaan Huygens4 returned to the problem in 1678 and found something even stranger: if he stacked two calcite crystals and rotated one, the two beams did not behave the same way. At certain orientations, one beam vanished entirely. Light, he concluded, must have an asymmetry perpendicular to its direction of travel — what we now call polarization. But he had no framework to understand it further.
The decisive step came from an observation by Étienne-Louis Malus5 in 1808. While looking through a calcite crystal at the last light of sunset reflected in the windows of the Luxembourg Palace in Paris, he noticed the reflected light behaved exactly like light that had passed through a crystal. Reflection could polarize light. He worked out the quantitative law relating transmitted intensity to angle — Malus’s Law — and received the Prix de l’Institut for it in 1810. He died two years later, at thirty-six, never knowing the deeper reason his law worked.
That deeper reason came from James Clerk Maxwell6’s equations of electromagnetism in 1865. Light is a transverse wave7 in the electromagnetic field: the electric and magnetic components oscillate perpendicular to the direction of travel. This transverse character is precisely what polarization describes — the orientation of that oscillation.
The practical side of the story belongs to Edwin Land8. In 1928, at nineteen years old and during his first year at Harvard, he grew crystals of quinine iodosulfate and aligned them by dragging them across a surface. When stacked and sealed in plastic, they polarized light efficiently across a large sheet — practical, cheap, and manufacturable. He went on to found the Polaroid Corporation, and polarizing filters became ubiquitous in photography, optics, and eventually LCD screens, which use precisely the paradox you are about to observe.
Hazards & preparation
Safety: This experiment is essentially hazard-free — just filters and everyday light.
- Do not stare at the sun through a polarizer (or ever). Bright indoor lamps and daylight from a window are all you need.
Materials
- 3 linear polarizing filters — photography gray filters, educational polarizer sheets, or filters salvaged from an old LCD screen all work. Three-dimensional cinema glasses use circular polarizers and will not demonstrate Malus’s Law cleanly; avoid them.
- A bright light source — a window in daylight, a bare LED lamp, or a white-LED flashlight. Avoid fluorescent tubes, which emit polarized light and give inconsistent results.
- Tape or a marker (optional) — to mark the transmission axis on each filter.
Procedure
Part 1: Two polarizers
Hold one polarizer in front of the light source. The transmitted light dims noticeably — roughly by half. This is the filter selecting one orientation of the light’s oscillation and blocking everything perpendicular to it.
Hold a second polarizer behind the first, between the first filter and your eye. Both are transmitting — the combined dimming is roughly the same as before.
Slowly rotate the second polarizer while keeping the first fixed. You will find a position where the light brightens slightly relative to the previous step (both transmission axes aligned — parallel polarizers) and a position where the image goes nearly black (crossed polarizers). The transition from bright to dark takes exactly a quarter turn — 90°.
Find the darkest position precisely and hold it. Note this orientation: the two filters are crossed at 90°, and almost no light gets through. This is your starting configuration for Part 2.
Part 2: The three-polarizer paradox
With the two crossed polarizers still held in position — the dark state — take the third polarizer and slide it between the other two. Keep the outer two filters fixed.
With the middle polarizer at a random angle, you will likely see some light return. Rotate it slowly through a full 360° and observe how the transmitted brightness varies.
Find the angle where transmitted light is at its maximum. It occurs at 45° from each of the outer polarizers — halfway between the two crossed axes.
Now remove the middle polarizer. Darkness returns immediately.
Rotate the outer polarizers toward each other, reducing the crossed angle from 90° toward 0°. Observe how the light from the three-polarizer arrangement changes as the geometry changes.
What you should see
With two filters crossed at 90° the view is nearly black — a deep, even darkness. The moment the third filter slides between them and reaches 45°, a soft glow returns: roughly an eighth of the original brightness, unmistakable against the black. Rotate the middle filter and the glow waxes and wanes twice per full turn, brightest at 45° and 135°, dark again at 0° and 90°. Pull the middle filter out and the darkness snaps back instantly.
| Symptom | Likely cause | Fix |
|---|---|---|
| Crossed pair never goes fully dark | Filters are circular (3-D glasses), not linear | Use linear polarizers — photographic or educational sheets |
| No light returns with the third filter | Middle filter aligned with an outer one (0°/90°) | Rotate the middle filter toward 45° |
| Whole view stays bright | One filter flipped, or a stray light path around the stack | Block other light sources; check all three filters overlap fully |
The Science
Why light polarizes
Light is an electromagnetic wave. Its electric field oscillates perpendicular to the direction the light travels — transversely, not along the direction of propagation as sound does. Unpolarized light from the sun or a lamp contains all orientations of this oscillation simultaneously, averaging out to no preferred direction.
A linear polarizer contains a grid of aligned molecules or structures that absorb the component of the electric field along one direction while transmitting the perpendicular component. Any polarization direction in the incoming light is projected onto the transmission axis. Half the power is absorbed; half passes through, now oscillating in a single plane.
Malus’s Law
When already-polarized light hits a second polarizer, only the component of the electric field aligned with the second filter’s axis is transmitted. If the electric field amplitude is \(E_0\) and the angle between the polarization direction and the filter axis is \(\theta\), the transmitted amplitude is \(E_0 \cos\theta\). Since intensity is proportional to the square of amplitude:
\[I = I_0 \cos^2 \theta\]
This is Malus’s Law. At \(\theta = 0°\) (parallel filters), all light passes: \(\cos^2 0° = 1\). At \(\theta = 90°\) (crossed filters), none passes: \(\cos^2 90° = 0\).
Why the third polarizer works
This is where the result becomes genuinely surprising. Two crossed polarizers transmit zero light. Adding a third filter — blocking more — somehow lets light through. What is happening?
The key is that each polarizer does not merely block — it projects. Whatever polarization state arrives, the filter projects it onto its own axis, producing newly polarized output. The middle polarizer changes the polarization state, not just the intensity.
Following the light through each stage (starting with unpolarized light of intensity \(I_0\)):
| Stage | What happens | Intensity |
|---|---|---|
| First polarizer | Selects one axis from unpolarized light | \(I_0 / 2\) |
| Middle polarizer at 45° | Projects onto 45° axis: \(\cos^2 45° = 1/2\) | \(I_0 / 4\) |
| Last polarizer at 90° from first, so 45° from middle | Projects again: \(\cos^2 45° = 1/2\) | \(I_0 / 8\) |
About 12.5% of the original light emerges. Without the middle polarizer, the last filter sees polarization at exactly 90° to its axis and transmits zero.
The 45° angle turns out to be optimal. The intensity after the middle filter is:
\[I = \frac{I_0}{2} \cos^2\theta \cdot \cos^2(90° - \theta) = \frac{I_0}{2} \cos^2\theta \sin^2\theta = \frac{I_0}{8} \sin^2(2\theta)\]
This is maximized when \(\sin^2(2\theta) = 1\), that is, when \(2\theta = 90°\) and \(\theta = 45°\). The halfway angle is not just intuitive — it is mathematically exact.
A quantum picture
At the scale of individual photons, the story is sharper still. A photon passing the first polarizer is in a definite polarization state — say, horizontal. When it encounters the middle polarizer at 45°, quantum mechanics says it is in a superposition of the two states the 45° filter can transmit or absorb. With probability \(\cos^2 45° = 1/2\), it is transmitted and its polarization is now “45°” — a new, definite state. When this photon then hits the last filter (vertical), it is in a superposition of that filter’s states. Half the time it transmits.
The classical and quantum pictures give the same numerical answer, but the quantum version makes it vivid: each polarizer does not just sort photons by their existing states — it creates a new state. This experiment is one of the simplest physical demonstrations of quantum state projection, and the non-classical multiplication of probabilities is the same arithmetic that underlies Bell’s theorem and quantum cryptography.
Questions to Explore
Finding the optimum. The calculation above shows that 45° is the optimal angle for the middle polarizer. Set up two crossed polarizers and the middle one, rotate the middle filter in small steps, and note the brightness at each angle. Does the maximum occur at exactly 45°?
Hint / answer
Yes. Transmitted intensity follows \(\tfrac{I_0}{8}\sin^2(2\theta)\), which peaks when \(2\theta = 90°\), i.e. \(\theta = 45°\). Because \(\sin^2\) is flat near its maximum, the brightness barely changes between about 40° and 50°, so you’ll find a broad, gentle peak rather than a sharp one.
Two middle polarizers. Replace the single middle polarizer with two, the first at 30° and the second at 60°, between the crossed pair. Calculate the expected fraction with Malus’s law applied at each stage, then verify.
Hint / answer
Starting from polarized light after the first outer filter: \(\cos^2 30° \times \cos^2 30° \times \cos^2 30° = (3/4)^3 \approx 0.42\) of that intensity reaches the far side (each successive filter turns the axis by 30°). That is far more than the single-45° case (\(1/4\) after the first outer filter). More intermediate steps, each a smaller angle, leak more light through — the seed of question 3.
Toward full transmission. With many intermediate polarizers evenly spaced in angle between the crossed pair, the transmission approaches a limit as their number grows. What is that limit, and why?
Hint / answer
With \(N\) filters each rotated by \(90°/N\), the transmission is \(\cos^{2N}(90°/N)\), which tends to 1 (100%) as \(N\to\infty\). Infinitely many infinitesimal rotations turn the polarization smoothly from horizontal to vertical with almost no loss — a striking demonstration that the filters project rather than merely block.
Skylight polarization. On a clear day, light from the blue sky is partially polarized by Rayleigh scattering9. Hold a single polarizer up to different parts of the sky and rotate it. Where is the polarization strongest?
Hint / answer
The polarization is strongest along a band 90° away from the sun across the sky, and weakest looking toward or directly away from the sun. Point your thumb at the sun and your fingers sweep the band of strongest polarization — the sky there noticeably darkens and brightens as you rotate the filter.
Brewster’s angle. Light reflected off a non-metallic surface at Brewster’s angle10 is completely polarized parallel to the surface — about 56° for glass, 53° for water. Hold a single polarizer near a window and look at reflections off a table or floor; rotate it. At what orientation is the glare minimized?
Hint / answer
Glare from a horizontal surface is polarized horizontally, so it is cut most when the polarizer’s transmission axis is vertical. That is exactly how polarized sunglasses work — their axis is vertical to kill horizontally polarized glare off roads and water.
LCD screens. An LCD screen emits polarized light, so viewed through a single polarizer it goes dark at 90° to the screen’s axis. What angle is your phone screen polarized at? What happens through polarized sunglasses when you tilt your head?
Hint / answer
Most phone screens are polarized diagonally (about 45°) so they don’t black out in landscape or portrait through sunglasses — but tilt your head far enough and the screen still dims dramatically, going near-black when your glasses’ axis crosses the screen’s at 90°.
Where the energy goes. Malus’s law gives the transmitted fraction \(\cos^2\theta\); the blocked fraction \(\sin^2\theta\) is absorbed as heat. For a pair at 45°, how much does each filter absorb, and does the order matter?
Hint / answer
Of polarized light hitting the second filter at 45°, half is transmitted and half absorbed — so the second filter absorbs \(\sin^2 45° = 1/2\) of what reaches it. The first filter already absorbed half of the original unpolarized light. Because multiplication commutes, swapping identical absorbers leaves the final transmitted intensity unchanged.
Going further
- Photograph the sky through a polarizer at 90° to the sun and watch it deepen to a dramatic blue-black — the trick landscape photographers use.
- Place clear sticky tape in overlapping layers on a slide between two crossed polarizers: the tape’s birefringence turns white light into vivid colours that shift as you rotate the filters.
- Look at a stressed piece of clear plastic (a ruler, a CD case) between crossed polarizers to see photoelastic stress patterns — the same method engineers use to find stress concentrations in models.
Footnotes
Polarization — The orientation of a transverse wave’s oscillation; for light, the direction in which its electric field vibrates.↩︎
Malus’s law — The transmitted intensity through a polarizer is I = I₀cos²θ, where θ is the angle between the light’s polarization and the filter axis.↩︎
Erasmus Bartholin — Danish scientist (1625–1698) who discovered the double refraction of light by calcite crystals in 1669.↩︎
Christiaan Huygens — Dutch scientist (1629–1695) who proposed that light is a wave and formulated the principle of secondary wavelets.↩︎
Étienne-Louis Malus — French physicist (1775–1812) who discovered polarization by reflection and formulated Malus’s law.↩︎
James Clerk Maxwell — Scottish physicist (1831–1879) whose equations showed light to be an electromagnetic wave.↩︎
Transverse wave — A wave whose oscillation is perpendicular to its direction of travel, as with light on the electromagnetic field.↩︎
Edwin Land — American inventor (1909–1991) who made the first practical sheet polarizer and founded the Polaroid Corporation.↩︎
Rayleigh scattering — The scattering of light by particles much smaller than its wavelength; it polarizes skylight and makes the sky blue.↩︎
Brewster’s angle — The angle of incidence at which light reflected from a surface is completely polarized parallel to that surface.↩︎