What this calculator does
A polarising filter passes light according to the square of the cosine of the angle between the light's polarisation and the filter's axis. Aligned, everything passes; at 90 degrees, nothing does.
Unpolarised light behaves differently on first contact. It contains every orientation equally, so a first polariser always passes exactly half of it regardless of orientation, and Malus's law applies only from the second filter onward.
The formula
Multiply the incident intensity by the square of the cosine of the angle to the transmission axis. For unpolarised incident light, halve the intensity first to account for the initial polarisation.
| Term | Meaning |
|---|---|
| Polarisation | The orientation of the light wave's oscillation, perpendicular to its direction of travel. |
| Transmission axis | The orientation a polariser passes. Light aligned with it passes fully. |
| Malus's law | The cosine squared relationship between angle and transmitted intensity for already-polarised light. |
The inputs explained
| Field | What to enter |
|---|---|
| Incident intensity (W/m²) | The incident intensity in watts per square metre. |
| Angle to the transmission axis (°) | The angle between the light's polarisation and the filter's transmission axis, in degrees. |
| Incident light | Whether the incoming light is already polarised or unpolarised. Unpolarised light loses half its intensity at the first filter before the angle matters. |
When to use it
Understanding polarised sunglasses
Reflected glare is partially polarised, so a filter oriented to block it cuts glare far more than it dims the rest of the scene.
Working out a two-filter setup
Crossed polarisers block light entirely, and inserting a third at an angle between them surprisingly lets light through again.
Controlling light intensity smoothly
Rotating one polariser against another gives a continuously variable attenuator without changing colour.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How much light passes at each angle?
Polarised light through a filter at a range of angles.
| Angle to axis | Transmitted intensity | Fraction of incident intensity transmitted |
|---|---|---|
| 0° | 100.000 W/m² | 100.0% |
| 30° | 75.000 W/m² | 75.0% |
| 45° | 50.000 W/m² | 50.0% |
| 60° | 25.000 W/m² | 25.0% |
Questions
Why does unpolarised light lose half at the first filter?
Because it contains all orientations equally. Averaging the cosine squared over every possible angle gives exactly one half, whatever direction the filter is turned to.
Why do two crossed polarisers block everything?
Because the first polarises the light and the second is at 90 degrees to that, where the cosine squared is zero. Nothing gets through in the ideal case.
What happens if I add a third filter between two crossed ones?
Light reappears, which seems impossible but is not. The middle filter re-polarises the light to an intermediate angle, so it is no longer at 90 degrees to the final filter. This is a classic demonstration.
Why do polarised sunglasses reduce glare?
Because light reflected off horizontal surfaces is partly horizontally polarised. Lenses with a vertical transmission axis block much of that reflected component while passing the rest of the scene normally.
For light bending at a surface, see the Snell's law calculator. For the refractive index behind it, see the refractive index calculator.