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# Jones calculus

In optics one can describe polarization 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.

The Jones vector for polarized light is defined by $\begin{pmatrix} E_x(t) \\ E_y(t)\end{pmatrix}$, where Ex(t) and Ey(t) are the x and y components of the electric field of the light wave. It is common to normalize Jones vectors such that the sum of the squares of their components is 1. This discards the amplitude information needed for absorption calculations, but simplifies analysis in other cases. 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.

The following table gives examples of normalized Jones vectors. (i is the imaginary unit, $\sqrt{-1}$.)

 Polarization Corresponding Jones vector Linear polarized in the x-direction $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ Linear polarized in the y-direction $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ Linear polarized at 45° from the x-axis $\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ Right circular polarized $\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ -i \end{pmatrix}$ Left circular polarized $\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ i \end{pmatrix}$

The following table gives examples of Jones matrices.

 Optical element Corresponding Jones matrix Linear polarizer with axis of transmission horizontal $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ Linear polarizer with axis of transmission vertical $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ Linear polarizer with axis of transmission at 45° $\frac12 \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ Linear polarizer with axis of transmission at -45° $\frac12 \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$ Linear polarizer with axis of transmission at angle $\varphi$ $\begin{pmatrix} \cos^2\varphi & \cos\varphi\sin\varphi \\ \sin\varphi\cos\varphi & \sin^2\varphi \end{pmatrix}$ Left circular polarizer $\frac12 \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}$ Right circular polarizer $\frac12 \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}$ Half-wave plate with fast axis in the horizontal direction $\begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}$ Quarter-wave plate with fast axis in the horizontal direction. $e^{i\pi /4} \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$

If an optical element is rotated about the optical axis by angle θ, the Jones matrix for the rotated element, M(θ), is constructed from the matrix for the unrotated element, M, by the transformation

$M(\theta )=R(-\theta )\,M\,R(\theta )$ ,
where $R(\theta ) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$ .

Note that Jones calculus is only applicable to light that is already fully polarized. Light which is unpolarized, partially polarized, or incoherent must be treated using Mueller calculus.

## References

• E. Collett, Field Guide to Polarization, SPIE Field Guides vol. FG05, SPIE (2005). ISBN 0-8194-5868-6.
• E. Hecht, Optics, 2nd ed., Addison-Wesley (1987). ISBN 0-201-11609-X.
• R. C. Jones, "New calculus for the treatment of optical systems," J. Opt. Soc. Am. 31, 488–493, (1941).
• Frank L. Pedrotti, S.J. Leno S. Pedrotti, Introduction to Optics, 2nd ed., Prentice Hall (1993). ISBN 0-13-501545-6