Light that vibrates in one direction
Ordinary light from the Sun or a bulb vibrates in all directions perpendicular to its path: it is unpolarized. A polarizing filter lets through only the component vibrating along its axis. In 1809, while looking through a calcite crystal at sunlight reflected by the windows of the Luxembourg Palace in Paris, Étienne-Louis Malus noticed that reflected light is partly polarized. He also found how much polarized light passes through a second filter:
I = I₀ cos²θ
- I₀ is the intensity of the polarized light reaching the filter.
- θ is the angle between the light’s polarization direction and the filter’s axis.
- I is the transmitted intensity.
At 0° all the light passes, at 45° half of it, and at 90°, with crossed filters, none.
Unpolarized light
A first polarizer always transmits half of unpolarized light, because the random directions average cos²θ to one half. Choose “Unpolarized” in the calculator to include this first filter: the analyzer then works on the remaining half.
Brewster’s angle
When light reflects off glass or water, the reflection is partly polarized. At one particular angle of incidence, the reflected light is completely polarized parallel to the surface. Scottish physicist David Brewster described it in 1815:
tan θB = n₂ ÷ n₁
At this angle, the reflected and refracted rays are exactly 90° apart.
Worked example
Polarized light of 100 W/m² reaches an analyzer turned 30° from its polarization:
- I = 100 × cos²(30°) = 100 × 0.75 = 75 W/m², or 75% of the light.
For light going from air (n = 1.0) into glass (n = 1.5):
- Brewster angle: arctan(1.5) = 56.31°.
- The refracted ray travels at 33.69° from the normal.
For water (n = 1.33), Brewster’s angle is about 53°.
Everyday uses
- Polarized sunglasses block glare reflected from roads, snow and water, which is largely horizontally polarized.
- LCD screens sit between two polarizers; liquid crystals rotate the light’s polarization to turn each pixel on or off. That is why a screen can look black through polarized sunglasses at certain angles.
- Photography: a circular polarizer darkens blue skies and removes reflections from water and windows.
- Lasers often have windows set at Brewster’s angle, which pass one polarization with no reflection loss.
- 3D cinemas send a different polarization to each eye.