Polarization
of Waves

The geometric orientation of a wave's oscillations and its behavior as it passes through filters.

Oscillating in One Plane

An electromagnetic wave consists of electric and magnetic fields oscillating perpendicular to the direction of travel. In a linearly polarized wave, the electric field vector is confined to a single plane along the direction of propagation.

$$ \vec{E}(z,t) = E_0 \cos(kz - \omega t) \hat{n} $$
Real World Analogy

Shaking a Rope

Imagine shaking a rope up and down to create a wave. If you only move your hand vertically, the wave is vertically polarized. If you shake it side-to-side, it is horizontally polarized.

Circular and Elliptical

When two orthogonal linearly polarized waves (e.g., in the x and y directions) superpose, their combined state depends on their relative amplitudes and the phase difference $\delta$ between them. A phase difference of $90^\circ$ ($\pi/2$) with equal amplitudes produces circularly polarized light.

$$ E_x = E_{0x} \cos(\omega t) \quad ; \quad E_y = E_{0y} \cos(\omega t + \delta) $$
Visualizing the Trace

Lissajous Figures

Looking at the wave head-on, the tip of the electric field vector traces out a shape over time. It can be a straight line, a circle, or an ellipse, forming Lissajous figures depending on the phase and amplitude ratios.

Passing Through Polarizers

When linearly polarized light of intensity $I_0$ passes through a polarizing filter (analyzer) whose transmission axis is at an angle $\theta$ relative to the incoming polarization, the transmitted intensity $I$ is reduced according to Malus's Law.

$$ I = I_0 \cos^2(\theta) $$
Real World Analogy

Picket Fence

If you shake a rope to make a wave and pass it through the vertical slats of a picket fence, only vertical waves get through. If you try to pass horizontal waves through the same vertical fence, they are completely blocked.

Academic References