 Jones calculus

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 $2\times 2$ 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 $P=\mathrm{diag}(1,0)$, 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 $Q=\mathrm{diag}(1,i)$. 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.

Jones calculus is a mathematical framework used in optics to describe the polarization state of light and its transformation through optical devices. It was developed by the physicist R.W. Jones in 1941. This calculus uses a two-dimensional complex vector to represent the state of polarization of light, which can include various types of polarization such as linear, circular, and elliptical.