Electromagnetic waves (light) vibrate in many directions perpendicular to the direction of travel. When all the vibrations are restricted to a single direction, the light is called plane‑polarised.
Think of a polarising filter like a door that only lets people wearing a specific hat (the vibration direction) pass through. Everyone else is turned away.
In a plane‑polarised wave the electric field oscillates in one fixed plane. The intensity of such light is measured by the amplitude of the electric field squared.
When a plane‑polarised wave meets a polarising filter, the filter only lets through the component of the electric field that is aligned with its own axis.
When a plane‑polarised wave passes through a polarising filter whose axis makes an angle θ with the wave’s polarisation direction, the transmitted intensity I is given by Malus’s Law:
\$I = I_0 \cos^2 \theta\$
Here I0 is the intensity before the filter.
Suppose a plane‑polarised beam has intensity I0 = 100 units. It passes through a filter whose axis is at θ = 30°.
So the filter reduces the intensity to 75 % of the original.
Two polarising filters are placed one after the other. The first filter’s axis is aligned with the light (θ1 = 0°). The second filter is rotated by θ2 = 45° relative to the first.
| Step | Calculation | Result |
|---|---|---|
| After first filter | \$I1 = I0 \cos^2 0° = I_0\$ | \$I_1 = 100\$ |
| After second filter | \$I2 = I1 \cos^2 45° = 100 \times (1/\sqrt{2})^2 = 50\$ | \$I_2 = 50\$ |
The final intensity is 50 % of the original.
If a beam passes through n polarising filters with successive angles θ1, θ2, …, θn, the final intensity is:
\$I = I0 \prod{k=1}^{n} \cos^2 \theta_k\$
Each filter multiplies the intensity by the square of the cosine of its relative angle.