= Jones calculus
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<Jones calculus> represents a fully polarized <monochromatic light> field by a two-component <Jones vector> of transverse <complex amplitudes>. An ideal linear optical element acts by a complex $2\times2$ <matrix>; successive elements act by the ordered product of their <matrices>. The order matters because the transformations need not commute.
For transverse axes $x,y$, an ideal polarizer transmitting $x$ has <matrix> $P=\operatorname{diag}(1,0)$, so $P(E_x,E_y)^{\mathsf T}=(E_x,0)^{\mathsf T}$. A <quarter-wave plate> aligned with those axes can be represented, up to a common <phase factor> and the chosen sign convention, by $Q=\operatorname{diag}(1,i)$. It changes the relative <wave phase>, enabling conversion between <linear polarization> and <elliptical polarization>. Rotating the elements tests these predictions by changes in transmitted <light intensity>. A single <Jones vector> does not describe arbitrary partially polarized <light>; that requires statistics of the field rather than one fixed pair of amplitudes.
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