Use Malus' law I = I0 cos squared theta to compute, in real time, the intensity after each of three stacked polarizers (P1 = 0 deg reference, P2 at theta_a, P3 at theta_b), the total transmittance and the contribution of the middle plate. The "three-polarizer paradox" — light reappearing when a third polarizer is inserted between crossed polarizers — is visualised quantitatively.
Parameters
Incident intensity I0
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Middle polarizer angle theta_a
°
Final polarizer angle theta_b
°
Polarizer leakage ext
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Defaults are I0 = 100, theta_a = 45 deg, theta_b = 90 deg, ext = 0. P1 is the fixed 0 deg reference axis. Natural light loses half its intensity at P1 and emerges vertically polarized. With theta_b = 90 deg, P1 and P3 are crossed and no light passes when P2 is absent, but inserting P2 at theta_a = 45 deg reveals I3 = 12.5 (total transmittance 12.5 percent).
Results
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Final intensity I3
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Total transmittance I3/I0
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Middle intensity I2
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Contribution of middle plate
Three-polarizer stack layout
Left = unpolarized natural light at intensity I0 / after P1 (blue, 0 deg) = vertically polarized I1 = I0/2 / after P2 (green, theta_a) = I2 via Malus' law / after P3 (red, theta_b) = final I3. Arrow length encodes intensity, direction encodes transmission axis.
Final intensity I3 vs middle angle theta_a
Horizontal axis = middle angle theta_a (0 deg to 180 deg) / vertical axis = final I3 with theta_b fixed / blue curve = theoretical cos^2(theta_a) cos^2(theta_b - theta_a) / yellow marker = current theta_a / grey dashed = I3 with the middle plate removed.
Theory & Key Formulas
Malus' law gives the intensity transmitted by a polarizer whose transmission axis makes an angle $\theta$ with the incident linear-polarization direction:
$$I = I_0 \cos^2\theta$$
Unpolarized natural light loses half its intensity at the first polarizer and emerges linearly polarized, so the stage intensities for the three-polarizer stack are:
For real polarizers the transmission function is approximated as $T(\theta) = \cos^2\theta + \mathrm{ext}$ with ext as a leakage parameter. $\theta_b - \theta_a$ is the relative angle of P3 measured from the P2 axis. At $\theta_b = 90^\circ$ the stack is crossed; setting $\theta_a = 0^\circ$ or $90^\circ$ gives $I_3 = 0$, while $\theta_a = 45^\circ$ produces the maximum $I_3 = I_0/8$, reproducing the three-polarizer paradox.
What is the Malus Law Simulator?
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Professor, I heard that two crossed polarizers block all light, but if you slip a third polarizer between them light comes through again. Is that really true? You are adding another absorber.
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It is true — this is the famous three-polarizer paradox. With the defaults of this tool (I0 = 100, theta_a = 45 deg, theta_b = 90 deg, ext = 0), P1 polarizes the natural light to I1 = 50.0, P2 at 45 deg gives I2 = 25.0, and P3 at 90 deg leaves I3 = 12.5 for a total transmittance of 12.5 percent. Meanwhile the "I3 with the middle plate removed" label reads 0.00. So inserting the middle plate adds 12.5 intensity units, which is exactly the "Contribution of middle plate = +12.5" you see in the results card.
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But how can light increase? Does that not violate energy conservation?
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Energy is not created. The crossed pair P1 then P3 (no middle) cuts to half, then to cos squared 90 deg = 0, so everything is absorbed. With P2 at 45 deg, P1 first cuts to 50 percent, P2 projects the vertical polarization onto a 45 deg axis (factor cos squared 45 deg = 0.5), and P3 projects again with a 45 deg relative angle (another factor 0.5). The product 0.5 times 0.5 times 0.5 equals 0.125, so 12.5 percent survives. The middle plate "re-prepares" the polarization. Sweep theta_a in this tool and you will see I3 peak at 45 deg and 135 deg and drop to zero at 0 deg, 90 deg and 180 deg.
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What is the "extinction ratio"? The ext slider goes up to only 0.1 — is it really worth tuning?
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Real polarizers never block perfectly: about 0.1 to 5 percent of the wrong polarization leaks through. ext captures that. Set theta_a = 0 deg and theta_b = 90 deg in this tool, then push ext from 0 to 0.05 — the curve floor rises off zero by about ext times I1/2. In LCD design the contrast ratio is 1 over ext, so ext = 0.001 gives 1000:1 contrast (premium panels) and ext = 0.01 gives 100:1 (low-end). The "black state" of an LCD lifts visibly when ext grows, which is the same effect you observe in the chart.
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One last question: I heard this paradox is connected to quantum spin measurements. Is that right?
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Absolutely. Each polarizer acts as a projection operator in quantum mechanics. Run the experiment with single photons and the statistics are the same: P2 performs a "polarization measurement along the 45 deg axis" that re-prepares the state. The same logic explains why inserting an x-axis Stern-Gerlach measurement between two z-axis measurements changes the output distribution. Malus' cos squared looks classical, but it is the square of the quantum amplitude cos theta — a stepping stone toward Bell's inequalities. Many physicists call the three-polarizer demo the simplest experimental proof that "measurement changes the state".
Frequently Asked Questions
Malus' law I = I0 cos^2 theta gives the intensity transmitted by an ideal linear polarizer when the incident light is linearly polarized at angle theta to the transmission axis. Etienne-Louis Malus discovered it in 1809 while studying light reflected from a water surface. Unpolarized natural light passing through the first polarizer loses half its intensity (I -> I/2) and emerges linearly polarized. With the defaults I0 = 100, theta_a = 45 deg, theta_b = 90 deg, ext = 0 this tool reports I1 = 50.0, I2 = 25.0, I3 = 12.5 and a total transmittance of 12.5 percent.
Two polarizers crossed at 0 deg and 90 deg block all light, but inserting a third polarizer between them at 45 deg lets light through, even though one more absorber has been added. With the defaults this tool reports I3 = 12.5 with the middle plate in place and I3 = 0 with the middle plate removed, so the middle-plate contribution is +12.5 intensity units. The effect is the optical analogue of sequential projective measurements in quantum mechanics: each polarizer re-prepares the polarization state.
ext is a leakage parameter that models real polarizers, which never block light perfectly. An ideal polarizer has T(theta) = cos^2 theta; in practice T(theta) ~ cos^2 theta + ext with ext typically in the range 0.001 to 0.05. High-grade dichroic sheet polarizers reach ext ~ 0.001 (contrast ratio 1000:1), while inexpensive plastic film gives ext ~ 0.05. Sliding ext from 0 to 0.05 in this tool lifts the I3 curve off zero at the crossed-polarizer angles and demonstrates how finite contrast degrades black-state performance in LCD panels.
Liquid-crystal displays modulate brightness with a polarizer pair sandwiching the liquid-crystal cell, and each pixel's transmittance is governed by Malus' law. Polarized sunglasses cut horizontally polarized glare from water and roads; 3D cinema assigns orthogonal polarizations to the two eyes; optical isolators in fiber-optic links suppress back reflections; and photoelastic stress analysis visualizes internal stress through stress-induced birefringence between crossed polarizers. This tool also serves as a quick angular optimiser for LCD designers choosing the middle-layer twist angle.
Real-World Applications
Liquid-crystal display pixel control: Each pixel of an LCD is built from two crossed polarizers with a liquid-crystal layer between them. The liquid crystal twists the polarization direction by up to 90 deg in the unpowered state ("white"); applied voltage straightens the molecules, removes the twist and yields the "black" state. The middle-stage transmittance I2/I1 in this tool plays the role of the LCD pixel, and the cos squared dependence on theta_a is exactly what controls the pixel brightness. The defaults give the maximum I2 = 25 at theta_a = 45 deg and zero at 0 deg or 90 deg, the same on/off behaviour exploited in every phone, monitor and TV using LCD technology.
Polarized sunglasses and 3D cinema: Polarized sunglasses use a vertical transmission axis to suppress the horizontally polarized glare reflected off water, snow and wet roads, dramatically improving visibility for drivers and anglers. Stereoscopic 3D cinema sends orthogonally polarized (linear or circular) images to the two eyes and the glasses separate them via the cos squared 90 deg = 0 absorption you see at theta_a = 0 deg, theta_b = 90 deg in this tool. Modern silver screens preserve the polarization state much better than ordinary cinema screens, which is why active polarization is still preferred for premium 3D experiences.
Photoelasticity (stress-induced birefringence): Many transparent materials become birefringent when stressed. Sandwiching a loaded part between crossed polarizers produces colourful isochromatic fringes that map the principal-stress difference, a technique used to inspect bridges, gears, hip prostheses and dental crowns. The middle "P2" in this tool corresponds to the birefringent specimen, and sweeping theta_a recovers the same fringe-versus-angle modulation. Common engineering examples include checking stress concentrations at gear tooth roots and inspecting contact-lens manufacturing for residual moulding stress.
Optical isolators and quantum cryptography: Fiber-optic communication systems use polarization-based optical isolators (polarizer + Faraday rotator + polarizer) to prevent reflected light from destabilising the laser source — a direct application of Malus' law. The BB84 quantum key distribution protocol encodes single bits onto four polarization states (0 deg, 45 deg, 90 deg, 135 deg). An eavesdropper measuring in the wrong basis introduces statistical errors that the legitimate parties can detect. The three-polarizer stack in this tool is the classical analogue of "sequential measurements in different bases" that underpins this technique.
Common Misconceptions
The most common misconception is that Malus' law cos squared theta applies directly to unpolarized (natural) light. In reality, the first polarizer absorbs half of the unpolarized intensity (I -> I/2) and only the emergent polarized light obeys cos squared theta thereafter. This tool deliberately separates P1 as a fixed 0 deg reference and shows I1 = I0/2 in the results card to keep this distinction visible. If you forget the half-intensity step and apply cos squared theta from the start, you will predict 25 percent transmittance for the default configuration instead of the correct 12.5 percent.
Another popular misconception is that the three-polarizer paradox violates energy conservation. The middle plate does not create light — it just changes how much is absorbed at each subsequent stage. Without the middle plate P1 absorbs 50 percent and P3 absorbs the remaining 50 percent. With the middle plate, P1 absorbs 50 percent, P2 absorbs 25 percent of the original (intensity 25 ends up as heat) and P3 absorbs another 12.5 percent, leaving 12.5 percent in the beam. Look at the I2 = 25 stat card next to "Contribution of middle plate = +12.5" and you can verify every joule is accounted for.
Finally, beginners frequently confuse the angles in Malus' law. The angle theta in cos squared theta is the angle between the incident polarization direction and the polarizer's transmission axis, not an absolute laboratory angle. In this tool P3 uses cos squared (theta_b - theta_a) because the light arriving at P3 is polarized along the P2 axis, not along the absolute 0 deg reference. When designing LCD or photoelastic setups with multiple optical elements, always state explicitly which reference frame each angle is measured from. If your experiment with theta_a = 45 deg and theta_b = 90 deg gives anything other than 12.5 percent, double-check the reference axes first.