Jones vectors and polarization elements

For a paraxial beam propagating along positive $z$, use fixed laboratory axes $x,y$ and the time factor $e^{-i\omega t}$. A normalized Jones vector has unit squared norm, with overall amplitude handled separately. Multiplying both components by one common phase does not change the polarization ellipse. Multiplying just one component by a phase generally changes that ellipse, even when the two component intensities stay exactly the same.

Equal real components describe linear polarization at $45^\circ$. Equal magnitudes in quadrature describe circular polarization. For the explicit vector $(1,i)/\sqrt2$, the real field at a fixed plane rotates from positive $x$ toward positive $y$ as time increases from a suitable zero. Stating that motion avoids incompatible right-versus-left naming conventions based on whether one looks along the beam or toward its source. Reversing the time convention without conjugating the vector would reverse the predicted motion.

An ideal linear polarizer transmits the projection along its real unit transmission axis. The matrix is an outer product, so it is idempotent: applying the same ideal polarizer twice adds no further loss for the already transmitted polarization. For linear input, the squared projection gives Malus's law. Normalizing the output vector before recording its norm would erase the transmitted-power information and produce an incorrect prediction for a sequence of polarizers.

An aligned lossless retarder instead multiplies the components by phases of unit modulus. Define its relative retardance as $\delta=k_0(n_y-n_x)\ell$, dropping the common $x$ phase. The resulting matrix is diagonal with entries one and $e^{i\delta}$. A quarter-wave retardance converts equal in-phase components into quadrature; a half-wave retardance changes the sign of one component. Neither operation by itself removes power in the ideal model.

To orient a retarder by angle $\alpha$, first express the input in its principal axes, apply the diagonal matrix, then return to laboratory axes. With the usual active two-dimensional rotation matrix, this gives $R(\alpha)Q_\delta R(-\alpha)$. The rightmost matrix acts first. Applying a polarizer before a plate therefore usually differs from applying it afterward, because projection and relative-phase transformation need not commute.

A Jones matrix can describe a deterministic element acting on each member of a fluctuating ensemble, but a single Jones vector is insufficient for the ensemble's output in general. Depolarizing samples, unresolved angular mixtures, or broad spectral averaging require a coherency or Mueller description. The transverse approximation also becomes delicate in strong focusing, where longitudinal fields matter. The common misconception is to call any two-component intensity list a Jones vector: phases and mutual coherence are indispensable.