A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
Polarization describes how the electric field vector varies in time as a wave propagates. Linear polarization has E oscillating along a fixed direction. Circular and elliptical polarizations occur when E rotates. Polarization states decompose into orthogonal linear or circular components. Materials interact selectively with different polarizations.
From your study of plane waves in vacuum, you know that a plane wave propagating in the z-direction has E⃗ and B⃗ both transverse — perpendicular to ẑ. This means E⃗ lives in the x-y plane at each point along the wave. Polarization is simply the description of how that transverse E⃗ vector moves as a function of time. The question "what is the polarization state?" is asking: if you stood at a fixed point and watched the tip of the E⃗ arrow, what pattern would it trace?
The simplest case is linear polarization: E⃗ oscillates back and forth along a single fixed direction in the x-y plane. You can write it as E⃗(z,t) = E₀ cos(kz − ωt) x̂, where the tip of the vector traces a straight line along x̂. Think of shaking a jump rope purely up and down — that is linear polarization. If you superpose two linearly polarized waves of equal amplitude but with a 90° phase difference — E_x = E₀ cos(kz − ωt) and E_y = E₀ cos(kz − ωt − π/2) = E₀ sin(kz − ωt) — the resulting vector has constant magnitude E₀ but rotates continuously in the x-y plane. This is circular polarization: the tip of E⃗ traces a circle. Right-circular polarization rotates clockwise when viewed from the direction the wave is traveling; left-circular rotates counterclockwise. The general case of two orthogonal components with arbitrary amplitude ratio and phase difference traces an ellipse — elliptical polarization — of which both linear and circular are special cases.
The reason polarization matters is that materials interact with light in polarization-dependent ways. A polarizer (like a polaroid filter) transmits only the component of E⃗ along a preferred axis, blocking the perpendicular component. When unpolarized light passes through a polarizer, its intensity is cut in half; when polarized light passes through one rotated by angle θ, Malus's law gives transmitted intensity I = I₀ cos²θ. Birefringent crystals have different refractive indices for the two orthogonal polarization components, so they travel at different speeds and accumulate a phase difference — transforming linear polarization into elliptical and vice versa. This effect is used in wave plates (quarter-wave plates convert linear to circular, half-wave plates rotate the polarization direction). At interfaces, reflected and transmitted waves have polarization-dependent reflection coefficients (Fresnel equations), with Brewster's angle giving a condition where reflected light is purely s-polarized.
The decomposition of polarization states into two orthogonal basis states — whether linear or circular — is a linear algebra operation. Any polarization state is a two-component complex vector, and any polarizer or wave plate is a 2×2 complex matrix acting on it. This Jones calculus formalism makes systematic calculations straightforward and foreshadows the way quantum states are written as vectors acted on by operators — the polarization of a photon is, in fact, a direct physical realization of a quantum two-level system.