Polarized Light Transmission, Malus

Is there an explanation for Malus' Law (which as I understand it is that the intensity of polarized light passed through a subsequent polarizer is diminished by a factor equal to the square of the cosine of the angle between the original polarization direction and the axis of the subsequent polarizer)? Why is this the case instead of, for instance, the intensity diminishing proportionate to the angle (i.e.,�1/3 diminishment at 30 degrees, 1/2 at 45 degrees, 2/3 at 60 degrees)? Is the cosine relationship just a brute fact, discoverable only through observation, or is it explained by another feature of light or of polarization?
- Jeff (age 53)
Washington DC

Great question. Malus' law follows directly from the properties of light polarization and thus is not a separate empirical ingredient of the picture. Here's an argument.

1. Electromagnetic fields follow linear equations (Maxwell's) to extremely high accuracy. Maxwell's equations are thus the basic classical description of propagating electromagnetic waves.

2. Linear equations obey superposition, meaning that the sum of any two solutions is also a solution.

3. The energy content of the wave is proportional to the square of the fields, again a property that could be directly deduced from how they interact with charged particles. (It's related to how kinetic energies go as the square of velocities.)

Now let's put these properties together.

Take a wave propagating in the z direction. Using superposition, express it as the sum of a wave with electric fields in the x direction and one with fields in the y direction. These are the two polarization components. Simple trigonometry gives that one component has magnitude that goes as cos(θ) and the other as sin(θ). Say your polarizer blocks the y component but transmits the x component. The energy transmitted must go as cos²(θ).

Mike W.

(published on 01/06/2016)