Polarization
Polarization describes the orientation of a transverse wave's oscillation. For light, it is conventionally defined by the direction of the electric field. Polarizers make that orientation measurable and controllable.
What it means for light to be polarized
Unpolarized light contains rapidly varying transverse orientations. Linearly polarized light has a definite electric-field direction. Longitudinal sound in air does not have this same polarization degree of freedom.
An ideal linear polarizer
An ideal polarizer transmits the electric-field component along its axis. For unpolarized input, the average transmitted intensity after the first ideal polarizer is I = Iunpol/2.
Malus's law
I = I₀ cos² θ
I₀ is the intensity of light already polarized before the analyzer; θ is the angle between that polarization and the analyzer axis.
Malus-law explorer
Do not automatically apply the one-half factor again if I₀ already describes polarized light incident on the analyzer.
Reflection can polarize light
Reflection from nonmetallic surfaces often favors one polarization, which is why polarized sunglasses can reduce glare from roads and water. At Brewster's angle, the reflected and refracted rays are perpendicular and the reflected light is ideally fully polarized in one orientation.
Real polarizers absorb and scatter some light, so measured transmission is not perfectly ideal.
Three checks before using Malus's law
First identify whether the incident light is already polarized. Second identify the polarization direction immediately before the analyzer. Third measure θ between that direction and the analyzer axis. Only then apply I = I₀ cos²θ.
If the starting light is unpolarized and ideal, the first polarizer gives half the incident intensity on average; Malus's law then applies to subsequent analyzers. Real materials have finite transmission and imperfect extinction, so laboratory values need not reach the ideal 0 or 100% limits.
Worked examples
1. Malus at 60°
Linearly polarized light of intensity 80 W m−2 reaches an analyzer at 60°. Find transmitted intensity.
Solution
I = I₀ cos²60° = 80(0.5)² = 20 W m−2.
2. Unpolarized first polarizer
Ideal unpolarized light of intensity 100 W m−2 passes through one ideal polarizer. What is the average transmitted intensity?
Solution
An ideal first polarizer transmits half the unpolarized intensity on average: 50 W m−2.
3. Two ideal crossed polarizers
Linearly polarized light is aligned with the first polarizer, and a second ideal polarizer is at 90°. What does Malus’s law predict?
Solution
I = I₀ cos²90° = 0 in the ideal model.