Linear polarization

A wave is linearly polarized when its oscillating electric field always lies along one fixed line. For a transverse plane wave, that line is perpendicular to the wavevector. Unlike circular polarization, the field oscillates along the line rather than rotating through directions.

An ideal linear polarizer transmits the fraction ${\cos }^2\theta$ of incident linearly polarized light when its axis makes angle $\theta$ to the polarization. For a single photon, the corresponding conditional transmission probability is $|⟨\theta |0⟩{|}^2={\cos }^2\theta$, by the Born rule; rejection has probability ${\sin }^2\theta$. The classical field calculation projects its electric-field amplitude onto the transmission direction and squares that amplitude to obtain intensity. This is a conditional probability given the incident polarization and is not automatically the joint probability of passing one analyzer and failing another.