# Jones calculus

 related topics {math, energy, light} {math, number, function}

In optics, polarized light can be described using the Jones calculus, invented by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized, partially polarized, or incoherent must be treated using Mueller calculus.

## Contents

### Jones vectors

The Jones vector describes the polarization of light.

The x and y components of the complex amplitude of the electric field of light travel along z-direction, Ex(t) and Ey(t), are represented as

Here $\begin{pmatrix} E_{0x} e^{i\phi_x} \\ E_{0y} e^{i\phi_y} \end{pmatrix}$ is the Jones vector ($i=\sqrt{-1}$ is the imaginary unit). Thus, the Jones vector represents (relative) amplitude and (relative) phase of electric field in x and y directions.

The sum of the squares of the absolute values of the two components of Jones vectors is proportional to the intensity of light. It is common to normalize it to 1 at the starting point of calculation for simplification. It is also common to constrain the first component of the Jones vectors to be a real number. This discards the phase information needed for calculation of interference with other beams. Note that all Jones vectors and matrices on this page assumes that the phase of the light wave is φ = kz − ωt, which is used by Hecht. In this definition, increase in φx (or φy) indicates retardation (delay) in phase, while decrease indicates advance in phase. For example, a Jones vectors component of i ( = eiπ / 2) indicates retardation by π / 2 (or 90 degree) compared to 1 ( = e0). Collett uses the opposite definition (φ = ωtkz). The reader should be wary when consulting references on Jones calculus.

The following table gives the 6 common examples of normalized Jones vectors.

When applied to the Poincare sphere (also known as the Bloch sphere), the basis kets ($|0\rangle$ and $|1\rangle$) must be assigned to opposing (antipodal) pairs of the kets listed above. For example, one might assign $|0\rangle$ = $|H\rangle$ and $|1\rangle$ = $|V\rangle$. These assignments are arbitrary. Opposing pairs are