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Polarization describes the orientation of the electric field oscillation. Linear polarization confines the field to a single direction. Circular polarization occurs when two perpendicular components have equal amplitude and 90° phase difference; the field vector rotates as the wave travels. Elliptical polarization is the general case with unequal amplitudes or non-90° phase differences. Malus's law governs transmission through polarizers.
Think of a transverse wave as a rope being shaken — the rope can oscillate up-down, left-right, or anywhere in between. Light is an electromagnetic wave with an oscillating electric field, and polarization describes the direction that field oscillates as the wave travels. When you already understand phase of oscillation, you have the key tool for distinguishing all three polarization states: what makes them different is the phase relationship and amplitude balance between two perpendicular field components.
Linear polarization is the simplest case: the electric field oscillates along a single fixed direction — say, always vertical. Mathematically, this is one non-zero field component and zero in the perpendicular direction. A polarizing filter produces this by blocking all field orientations except one, transmitting only the component aligned with the filter's transmission axis. When unpolarized light (with electric field pointing in all transverse directions randomly) hits a polarizer, roughly half the intensity passes through — the half that happened to be aligned.
Circular polarization arises when you combine two perpendicular components of *equal amplitude* and exactly 90° phase difference. Imagine a vertical component E_y = A sin(kx - ωt) and a horizontal component E_z = A sin(kx - ωt + 90°) = A cos(kx - ωt). At any instant, the combined tip of the electric field vector traces a circle as the wave travels forward — the field doesn't flicker back and forth, it spins. The sign of the phase shift determines whether it's left-handed or right-handed circular polarization, a distinction that matters in optics and chemistry alike.
Elliptical polarization is the general case. When the two perpendicular components have *unequal* amplitudes, or a phase difference that isn't exactly 90°, the field vector traces an ellipse rather than a perfect circle or a line. Linear and circular polarization are limiting cases: a degenerate ellipse (flat line) and a perfectly round ellipse respectively. Most light emerging from crystals or optical waveplates is elliptically polarized.
Malus's law governs what happens when linearly polarized light passes through a second polarizer tilted at angle θ to the polarization direction: I = I₀cos²θ. At θ = 0° all intensity passes; at θ = 90° (crossed polarizers) none does. This squared cosine comes directly from the projection of the electric field vector onto the polarizer axis — intensity is proportional to amplitude squared, and the projected amplitude is the full amplitude times cos θ.