Electromagnetic waves and polarization

In a source-free vacuum, taking the curl of Faraday's law and using the other Maxwell equations yields a wave equation with speed $c=1/\sqrt{\mu_0\epsilon_0}$. For a single plane wave, the divergence equations require both fields to be perpendicular to its wave vector. The magnetic field is then fixed by the electric field and direction, rather than chosen as an independent oscillation with arbitrary phase.

An infinite plane wave is an idealization with infinite total transverse extent. It is useful locally for a broad beam far from its edges, but focusing and diffraction require more structure. Transversality here refers to each source-free plane-wave component. Fields near charges and antennas need not be purely transverse, so the plane-wave relation between magnitudes should not be applied to every electric or magnetic field in space.

For propagation along z, write $E_x=A\cos\psi$ and $E_y=B\cos(\psi+\delta)$, with $\psi=kz-\omega t$. At one fixed position, the tip of the electric vector generally traces an ellipse. If the relative phase is zero or π, the vector stays on a line. Equal amplitudes and a relative phase of a quarter-cycle produce a circle; unequal amplitudes at that phase produce an ellipse.

Handedness names depend on whether the observer looks along or against propagation and on the time-phase convention. Specify the actual component functions when a sign matters. Polarization does not mean that the beam moves sideways along the ellipse: energy propagates along the Poynting direction. A quarter-wave plate changes relative phase, whereas a linear polarizer projects the field onto an axis and generally discards part of its power.

Two coherent fields along the same direction can add constructively or destructively according to relative phase. Orthogonal components instead determine a vector polarization. After a polarizer projects them onto one common direction, their phases can again affect the measured intensity. This order of operations explains why adding component intensities before specifying the detector can hide observable interference.

For linearly polarized incident light and an ideal linear analyzer, the transmitted field amplitude is the projection onto the analyzer axis. Squaring that amplitude yields Malus's law. This statement concerns a fully polarized input; unpolarized light requires averaging over its polarization statistics. Real polarizers also have finite extinction and absorption, so the ideal cosine-squared dependence is a baseline rather than a complete device model.