How this instrument works
Malus's law describes what happens when already-polarized light meets a second polarizer tilted at angle θ from the first. Only the component of the electric field lying along the second polarizer's transmission axis gets through, and that component has amplitude E₀cosθ — a simple projection, the same trigonometry as measuring a shadow's length. Intensity is proportional to amplitude squared, so the surviving intensity is I₀cos²θ, not I₀cosθ. That squaring is why the curve falls faster than it looks: at 45° the amplitude has only dropped to 71% of its start, but the intensity you actually measure has already halved.
The formula assumes Incident intensity is already linearly polarized — light straight from a laser, or light that has already passed through a first polarizing sheet. Feed it truly unpolarized daylight instead and the law does not apply on its own: an unpolarized beam spreads its energy evenly across every possible orientation, so a single polarizer transmits a flat 50% no matter how it is rotated. Only once that first sheet has picked out one polarization state does a second sheet's angle start tracing the cos²θ curve this instrument models.
Rotate through a full turn and the curve repeats every 180°, not 360° — cos²θ cannot tell a polarizer's axis from its exact opposite, since spinning a sheet end-for-end leaves its transmission direction unchanged. At exactly 90° the two axes are crossed and the ideal answer is zero; real sheet polarizers still leak roughly a thousandth of the light through imperfect alignment and manufacturing, while a calcite Glan-Thompson prism suppresses it past one part in 100,000. The arithmetic itself never quite lands on a clean zero either, since a computer's stored approximation of 90° in radians rounds to something vanishingly small rather than nothing at all.
- Enter the light already reaching the second polarizer under Incident intensity — it must already be linearly polarized, not raw daylight.
- Set Angle between polarizer axes in degrees: 0° for axes aligned, 90° for axes fully crossed.
- Read Transmitted intensity in the same unit you used for Incident intensity.
- Raise the angle from 0° and watch the reading fall on a cos²θ curve, not a straight line.
- Values above 90° mirror values below it: 120° and 60° read identically, since the pattern repeats every 180°.
Worked example — a polarizer tilted 30° cuts intensity to 75%
Take a beam already polarized to an incident intensity of 100 — say, milliwatts per square centimetre — and pass it through a second polarizer whose axis sits 30° from the first, the situation a photographer creates by rotating a circular polarizing filter partway across the sky's own polarization band. Malus's law gives cos(30°) = 0.866025, and squaring that yields 0.75. Multiply: I = 100 × 0.75 = 75.0, exactly the reading Transmitted intensity shows for these inputs.
Turn the same filter to 0° and nothing is lost: the reading holds at 100, the two axes fully aligned. Push on past 60° and the drop steepens fast — at 90°, crossed, the transmitted intensity collapses to a number so small it might as well be darkness, which is exactly what happens when two polarizing filters are rotated to cross-purposes in front of a bright window.
Questions
Does Malus's law work directly on unpolarized sunlight?
No — the law assumes Incident intensity is already linearly polarized. Unpolarized light hitting a single polarizer transmits a flat 50% no matter the angle, because its energy is spread evenly across every orientation. Only after that first sheet has picked out one polarization does a second sheet's angle start following the cos²θ curve this calculator models.
Why does intensity depend on cos² of the angle instead of just cos?
Because intensity tracks amplitude squared, not amplitude itself. The transmitted electric field is the projection E₀cosθ onto the second polarizer's axis, but the intensity a sensor or your eye actually registers is proportional to that amplitude squared — hence cos²θ. At 45° the field amplitude is still 71% of its start, yet the intensity has already dropped to 50%.
Does the reading repeat if I rotate the angle past 90°?
Yes, every 180°. Cos²θ cannot distinguish a polarizer's transmission axis from its exact opposite, because turning a sheet through 180° brings the same alignment back around. So 120° reads identically to 60°, and 200° reads identically to 20° — the curve already completes its full pattern within one half-turn.
Why doesn't crossing the polarizers here show exactly zero?
Mathematically it very nearly does — entering 90° returns a transmitted intensity around 3.7 × 10⁻³¹, a rounding artifact of storing 90° in radians as a finite approximation of π/2, not a physically meaningful glow. Real crossed polarizers leak far more than that regardless: ordinary sheet polarizers pass roughly a thousandth of the incident light even fully crossed, while a calcite Glan-Thompson prism can suppress it below one part in 100,000.
What unit should I enter for Incident intensity?
Whatever unit your light source is quoted in — watts per square metre, milliwatts per square centimetre, or an arbitrary reading straight off a photodiode. Malus's law is a ratio, so Transmitted intensity comes out in that same unit; enter irradiance in W/m² and you get W/m² back, enter a raw sensor count and you get a raw sensor count back.
Where does this calculation show up outside a physics classroom?
In every rotating linear polarizer used to control light on purpose: a photographer twisting a circular polarizing filter to darken blue sky, an LCD pixel switching a liquid-crystal layer to steer light between two fixed polarizing sheets, and a laser technician crossing two Glan-Taylor prisms to dial beam power down smoothly instead of stopping it in fixed steps.