Malus's Law Calculator (Polarisers)
Light intensity after one, two or three polarisers at any angles, with the classic three-polariser surprise shown on the chart.
Results
The dashed line marks the value you entered.
What this tool does
Malus's law says a polariser passes the fraction cos²θ of light that is already polarised, θ being the angle between the light's polarisation and the polariser's axis; unpolarised light loses half at the first filter whatever its angle. The famous surprise comes with three: two crossed polarisers block everything, yet a third slipped between them at 45° lets an eighth of the original light through. Angles are measured from the first polariser. Real polarisers absorb a little even when aligned, so measured values come out somewhat lower.
Formula
I = I₀ cos² θ · unpolarised light through one polariser: I = I₀ ÷ 2
Variables
| Symbol | Meaning | Unit |
|---|---|---|
i0 | Incoming light intensity | W/m² |
up | Light reaching the first polariser | — |
a1 | Angle of the second polariser | ° |
a3 | Angle of the third polariser | ° |
I1 | After the first polariser | W/m² |
I2 | After the second polariser | W/m² |
I3 | After the third polariser | W/m² |
IX | After the third, without the middle one | W/m² |
TR | Share of the light that gets through | % |
Worked example
- Incoming light intensity100 W/m²
- Light reaching the first polariser1
- Angle of the second polariser45 °
- Angle of the third polariser90 °
- After the first polariser50 W/m²
- After the second polariser25 W/m²
- After the third polariser12.5 W/m²
- After the third, without the middle one1.8747e-31 W/m²
- Share of the light that gets through12.50 %
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.