Malus Law Calculator

Turn a second polarizer against the first and the light dies by a cosine squared. Enter the incoming intensity and the angle between the polarizers’ axes, and this calculator returns what survives — Malus’s law, the arithmetic inside every dimmed LCD and crossed-polarizer photograph.

The Cosine, Squared by Geometry

Malus’s law is a projection taken twice: a polarizer passes only the component of the electric field along its axis — a factor of cos θ in amplitude — and intensity goes as amplitude squared, so the transmitted intensity is I = I₀ cos²θ. Étienne-Louis Malus found it in 1809 while looking through calcite at the Louvre’s reflected windows (the discovery story of polarization itself), and the law has survived every revolution since because it is barely physics — it is the geometry of vectors, applied to light. Its consequences run the modern world quietly: two polarizing sheets become a variable dimmer, the LCD pixel is a Malus gate switched by liquid crystals, and the photographer’s polarizer deletes reflections by exactly the angular bookkeeping this page performs.

Worked Example: Thirty Degrees, Three Quarters

The defaults set I₀ = 100 and θ = 30°: cos²30° = 0.750 exactly, so 75.00 units emerge — three quarters of the light for a tilt that looks modest on the protractor, because the square makes small angles cheap and large angles brutal. The table every user should internalise follows from the same line: 45° passes exactly half of the polarized input (cos²45° = 0.5), 60° passes a quarter, and the last degrees are a cliff — 80° passes 3%, 89° passes 0.03%, and 90° is the ideal zero, crossed polarizers, the black of a dead LCD pixel viewed askance. The widget’s note adds the qualifier that textbooks bury: natural light arrives unpolarized, and the first sheet always takes its half off the top, angle-independent — the cosine law governs only the polarized remainder.

Malus in the Wild: Sunglasses, Screens, Stress

Three working habitats: the variable neutral-density filter for cameras is two polarizers and this law with a rotating ring for a user interface (cinematographers ride it mid-shot to hold exposure as clouds move). The photoelastic stress viewer — crossed polarizers around a clear plastic part — turns internal stress into coloured fringes, because stressed plastic rotates polarization by amounts the Malus gate converts to brightness, and engineers have read bridges’ models and gear teeth’s futures in those rainbows for a century. And the LCD: a liquid-crystal cell twists the polarization by a voltage-controlled angle, and the rear polarizer applies cos²θ pixel by pixel — the display in front of you is, literally, a few million Malus calculations refreshed sixty times a second.

Malus at the Movies: 3D Glasses

Modern 3D cinema is Malus’s law in circular dress: the projector throws two images in opposite circular polarizations, and the glasses’ quarter-wave plates convert them back to linear polarizations at right angles, so each eye’s analyzer passes its own image by the cosine-squared rule and extinguishes the other at the crossed zero. Tilt your head in a linear-polarized IMAX and the ghosting creeps in — the axes no longer cross cleanly, cos²θ leaks — while the circular system forgives the tilt, which is why it won. Either way, the extinction that makes the illusion is this page’s 90° case, evaluated per eye, twenty-four times a second, in the dark.

Stress Rainbows in the School Lab

Two sheets of polarizing film and a piece of clear acrylic make the cheapest X-ray in education: hold the sandwich against the light and the acrylic’s frozen-in moulding stresses bloom into coloured bands — each band a contour where the plastic’s birefringence has rotated the polarization by a Malus-measurable amount. The demonstration is a toy only in price; the same bench, scaled with calibrated compensators, is how glassmakers verify annealing and how engineers once certified that a bridge model’s members bore their loads evenly. The calculator’s cosine squared is the decoder key for the whole art: brightness at a point, divided by the aligned brightness, returns cos²θ — and θ is the stress’s signature, written in light the sheet lets through.

The Photographer’s Two-Stop Rule

On location, Malus’s law is carried as folklore: a polarizer costs about two stops of light on unpolarized scenes (the flat half, plus the sheet’s own appetite), and rotating it against a reflection buys extinction at the price of the angle’s cosine squared on everything else in frame. The calculator turns the folklore into exposure math — the landscape photographer who knows the reflection sits at 35° to the filter’s axis can price its suppression before bracketing a single frame. Craft knowledge in every field is a formula that has been carried long enough to feel like instinct; this page is where the instinct can be re-derived when the light refuses to behave as remembered. That re-derivation is the calculator’s real product — not the number 75 it prints for the default, but the recovered ability to price any angle, any sheet, any stack, on any set where the gels are burning and the light is wrong.

Related Tools

More from the Physics Calculators hub: the Brewster’s Angle Calculator — where polarized light comes from — and the Index of Refraction Calculator for the boundary physics both tools share.

Frequently Asked Questions

Why does the first polarizer take exactly half of unpolarized light?

Because unpolarized light is an equal mixture of all field directions, and the average of cos² over all angles is exactly ½ — the sheet keeps the component along its axis from every incoming direction, and the books balance at half the intensity lost (absorbed as heat in the sheet, in the iodine-in-plastic of a Polaroid). This is also why the first sheet’s own orientation cannot matter for unpolarized input: every direction is equally represented, so every axis keeps the same 50%. Direction becomes money only once the light is polarized — which is the state the first sheet leaves it in, and where Malus’s angular rule takes over from the flat tax.

What if I stack three polarizers, the middle one at 45 degrees?

The famous paradox dissolves in multiplication: crossed outer sheets (90°) pass nothing, but insert the middle sheet at 45° and each junction is now a 45° step passing cos²45° = ½ of its polarized input — so unpolarized 100 becomes 50 after the first, 25 after the middle, and 12.5 after the last: light through a “blocked” pair, an eighth of the original, because the middle sheet re-aims the polarization, giving the final sheet a component to pass. No photons sneak and no energy appears; the vector projection is re-cast at each sheet, and the calculator chains exactly this way — run it pairwise, feeding each output forward as the next input.

Do real polarizers reach the ideal zero at 90 degrees?

Not quite — real sheets leak (a good photographic polarizer passes perhaps 0.1–1% at crossed axes, and absorbs some of the wanted component too, which is why two “aligned” sheets transmit nearer 80% than 100%). The law is the ideal the products approach, and the extinction ratio on a polarizer’s datasheet is precisely the measured distance from Malus’s zero. For the calculator’s purposes the ideal is the honest figure; for laboratory polarimetry the leak is calibrated and subtracted — the same relationship to an ideal formula as this hub maintains everywhere: compute exactly, then name the departure.

How does an LCD pixel use this?

Each subpixel is a sandwich: rear polarizer, liquid-crystal layer, front polarizer (crossed, in the common arrangement). With no voltage, the crystal’s helical structure twists the light’s polarization by 90°, so it passes the crossed pair — pixel bright. Apply voltage and the molecules align with the field, the twist unwinds toward 0°, and Malus’s law dims the pixel toward the crossed zero — grey levels are intermediate twist angles priced by cos²θ. The calculator models one such gate with light you supply; the display multiplies it by millions and addresses each by row and column, sixty times a second, in the device you are reading this on.

What angle halves polarized light? And quarters it?

Solve the law backwards: cos²θ = ½ at θ = 45°, cos²θ = ¼ at 60°, cos²θ = ¾ at 30° — the worked example’s figure — and 1% transmission at about 84.3°. The backward direction is the photographer’s real question (how far to rotate the variable ND for two stops? — two stops is a factor of four, 60° from alignment) and it is worth noting the law’s unfairness: the first 30° of rotation buys a 25% cut, the next 30° a further 50 points, and the last few degrees nearly everything. Dimming by rotation is a control with a cliff at the end of its travel, which every cinematographer learns in their wrist.

Who was Malus?

Étienne-Louis Malus (1775–1812) — artillery officer, mathematician, and a survivor of Napoleon’s Egyptian campaign — discovered polarization by reflection in 1808 and quantified its transmission law the following year, in a memoir that won the Paris Academy’s prize; he died of tuberculosis at 36 with the field he founded barely begun. The law’s survival into LCD engineering is the standard immortality physics offers: a cosine squared, found by a soldier looking at sunlight in a window, now evaluated several billion times a day in the screens of the species he could not have imagined.