Questions: Polarization of Electromagnetic Waves

Question 1

A plane wave has E⃗(z,t) = E₀[cos(kz − ωt) x̂ + cos(kz − ωt + π/2) ŷ]. What is the polarization state, and which way does the electric field vector rotate?

Model answer: This is circular polarization. Since the y-component leads the x-component by π/2, when the x-component is at its maximum the y-component is zero, and when x = 0 the y-component is at maximum. The tip of E⃗ traces a circle. By convention (viewed from the direction of propagation), this is left-circular polarization.

The key is comparing the phases: E_x = E₀ cos(kz−ωt), E_y = E₀ cos(kz−ωt+π/2) = −E₀ sin(kz−ωt). At t=0, z=0: E_x = E₀, E_y = 0. A moment later: E_x decreases, E_y becomes negative — the vector rotates clockwise in the x-y plane when viewed from the +z direction, which is left-circular by the common convention.

Question 2

Unpolarized light of intensity I₀ passes through two polarizers. The first has its transmission axis vertical; the second is rotated 60° from vertical. What fraction of the original intensity exits the second polarizer?

Model answer: After the first polarizer: I₁ = I₀/2 (unpolarized light loses half). After the second polarizer: I₂ = I₁ cos²(60°) = (I₀/2)(1/2)² = I₀/8. One-eighth of the original intensity exits.

Malus's law applies between the two polarizers: the light exiting the first polarizer is fully polarized along vertical, and the second polarizer transmits I₁ cos²θ where θ = 60° is the angle between their axes. cos²(60°) = (0.5)² = 0.25, so I₂ = (I₀/2)(0.25) = I₀/8.