Hold two pairs of polarized sunglasses up to a window and look through both at once. Rotate one pair slowly. At some point the world goes nearly black, even though each lens on its own is perfectly transparent. Nothing absorbed the light suddenly — you simply turned one filter until its axis crossed the other's. That dimming is not a gimmick. It is a precise, predictable physical effect, and the rule that describes it is one of the cleanest equations in optics.
This article explains what polarized light is, how a polarizer filters it, how to calculate the transmitted intensity with Malus's law, and why a third polarizer can make blocked light reappear.
Why this calculation matters
Polarizers are everywhere once you start looking. Every liquid-crystal display sandwiches its pixels between two of them; the screen brightness you see is Malus's law playing out millions of times per frame. Photographers rotate a polarizing filter to cut glare off water and glass. Stress engineers use crossed polarizers to make load paths glow inside transparent plastic models. Optical instruments, laser systems, and 3D cinema all lean on controlling the polarization state of light.
In each case the practical question is the same: given the orientation of a polarizer, how much light comes out the other side? Guessing is not good enough. A display designer needs the contrast ratio, an optical engineer needs the power budget, and a measurement setup needs to know how much signal survives each element. Malus's law turns that question into one short calculation, and understanding it also explains a result that surprises almost everyone the first time they see it.
The core formula
Light is a transverse electromagnetic wave: its electric field oscillates perpendicular to the direction of travel. In unpolarized light — sunlight, a bare bulb — that oscillation points in every direction at random. A polarizer is a filter that transmits only the field component aligned with its own axis and removes the rest.
When unpolarized light hits an ideal polarizer, exactly half the intensity gets through, and what emerges is fully polarized along the polarizer's axis. The interesting calculation begins with the second polarizer, because now the incoming light already has a definite polarization direction.
Let theta be the angle between the polarization direction of the incoming light and the axis of the polarizer. Malus's law gives the transmitted intensity:
I = I0 * cos^2(theta)
Here I0 is the intensity of the polarized light arriving at the polarizer and I is the intensity leaving it. The reason for the cosine squared is worth holding onto. The polarizer passes the field component along its axis, and that component scales as cos(theta). Intensity is proportional to the square of the field amplitude, so the transmitted intensity scales as cos squared of the angle.
Two limits anchor the rule. When theta = 0, the axes are aligned, cos(0) = 1, and all the light passes. When theta = 90 degrees, the axes are crossed, cos(90) = 0, and nothing passes — the dark window from the sunglasses experiment.
A worked example
Suppose fully polarized light of intensity I0 strikes a polarizer whose axis is rotated 30 degrees away from the light's polarization direction. How much gets through?
Step 1 — take the cosine of the angle.
cos(30 degrees) = 0.866
Step 2 — square it.
cos^2(30 degrees) = (0.866)^2 = 0.75
Step 3 — apply Malus's law.
I = I0 * cos^2(30 degrees) = I0 * 0.75 = 0.75 * I0
So 75% of the light is transmitted. A 30-degree misalignment costs only a quarter of the intensity — the cosine-squared curve is fairly flat near zero angle, so small misalignments are forgiving.
Now push the angle to 90 degrees. There cos(90) = 0, so I = I0 * 0 = 0: with crossed polarizers the transmitted intensity falls to zero and the field is completely blocked. Between those two points the curve falls steeply — at 45 degrees, cos squared is exactly 0.5, so half the light survives. That midpoint is why a polarizer set at 45 degrees is such a useful "half-power" element.
Common mistakes
Using cosine instead of cosine squared. A frequent slip. The field amplitude scales as cos(theta), but the quantity you measure — intensity or power — scales as cos squared. Forgetting the square overstates the transmitted light, badly at large angles.
Working in degrees and radians inconsistently. Most calculators and programming languages expect radians. Feeding 30 into a function that wants radians returns nonsense. Convert, or set the calculator to degree mode and confirm it.
Applying Malus's law to the first polarizer. The cos-squared rule assumes the incoming light is already polarized. Unpolarized light hitting the first polarizer simply loses half its intensity regardless of orientation; Malus's law governs every polarizer after that.
Expecting an ideal block. Real polarizers leak. A small fraction of light passes even when fully crossed, set by the extinction ratio of the device. For high-contrast work that residual transmission matters.
Misreading the three-polarizer result. Put a polarizer between two crossed polarizers and light reappears, which feels like it should be impossible. It is not a violation of anything. Each polarizer re-projects the polarization onto a new axis, so two 45-degree steps deliver 0.5 times 0.5 = 0.25 of the light, where a single 90-degree step delivers zero.
Try the interactive NovaSolver calculator
The three-polarizer paradox is far easier to believe once you can watch the numbers move. The Malus Law Simulator — Three-Polarizer Paradox on NovaSolver lets you set the input intensity and the angles of three stacked polarizers, then computes the intensity after each plate, the total transmittance, and the contribution of the middle polarizer — so you can dial the centre plate in and out and see crossed-polarizer light switch back on in real time.
Related calculators
- Polarization of Light — for the broader picture of linear, circular and elliptical polarization states.
- Optical Fiber Transmission — to follow polarized light into a guided medium and estimate received power and reach.
- Diffraction Grating Calculator — for another wave-optics workhorse, splitting light by wavelength into its spectral orders.
You can browse the rest in the optics tools hub.
Closing note
Malus's law is small enough to memorize and deep enough to keep teaching you. Intensity through a polarizer equals the incoming intensity times the cosine squared of the angle between the light's polarization and the polarizer's axis — that is the whole rule. It sets the brightness of every LCD pixel, the cut of every polarizing filter, and the surprising glow of the three-polarizer experiment. Remember the square, mind your angle units, and the behaviour of polarized light stops being magic and becomes arithmetic.
Top comments (0)