Editing Jones calculus (section)

== Jones vector ==
The Jones vector describes the polarization of light in free space or another [[homogeneity (physics)|homogeneous]] [[isotropic]] [[attenuation|non-attenuating]] medium, where the light can be properly described as [[transverse waves]]. Suppose that a monochromatic [[plane wave]] of light is travelling in the positive ''z''-direction, with angular frequency ''ω'' and [[wave vector]] '''k''' = (0,0,''k''), where the [[wavenumber]] ''k'' = ''ω''/''c''. Then the electric and magnetic fields '''E''' and '''H''' are orthogonal to '''k''' at each point; they both lie in the plane "transverse" to the direction of motion. Furthermore, '''H''' is determined from '''E''' by 90-degree rotation and a fixed multiplier depending on the [[wave impedance]] of the medium. So the polarization of the light can be determined by studying '''E'''. The complex amplitude of '''E''' is written: 
:<math>\begin{pmatrix} E_x(t) \\ E_y(t) \\ 0\end{pmatrix}
= \begin{pmatrix} E_{0x} e^{i(kz- \omega t+\phi_x)} \\ E_{0y} e^{i(kz- \omega t+\phi_y)} \\ 0\end{pmatrix}
=\begin{pmatrix} E_{0x} e^{i\phi_x} \\ E_{0y} e^{i\phi_y} \\ 0\end{pmatrix}e^{i(kz- \omega t)}.</math>
Note that the physical '''E''' field is the real part of this vector; the complex multiplier serves up the phase information. Here <math> i </math> is the [[imaginary unit]] with <math>i^2=-1</math>.

The Jones vector is

:<math>\begin{pmatrix}  E_{0x} e^{i\phi_x} \\ E_{0y} e^{i\phi_y} \end{pmatrix}.</math>

Thus, the Jones vector represents the amplitude and phase of the electric field in the ''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 overall phase information that would be needed for calculation of [[Interference (wave propagation)|interference]] with other beams.

Note that all Jones vectors and matrices in this article employ the convention that the phase of the light wave is given by <math>\phi = kz - \omega t</math>, a convention used by [[Eugene Hecht]].<ref name="hecht"/> Under this convention, increase in <math>\phi_x</math> (or <math>\phi_y</math>) indicates retardation (delay) in phase, while decrease indicates advance in phase.  For example, a Jones vectors component of <math>i</math> (<math>=e^{i\pi/2}</math>) indicates retardation by <math> \pi/2</math> (or 90 degrees) compared to 1 (<math>=e^{0}</math>).   Collett<ref>{{cite book |last= Collett |first= E. |date= 2005 |title= Field Guide to Polarization |series= SPIE Field Guides |volume= FG05 |publisher= [[SPIE]] |isbn= 0-8194-5868-6 }}</ref> uses the opposite definition for the phase (<math>\phi = \omega t - kz</math>). Also, Collet and Jones follow different conventions for the definitions of handedness of circular polarization.  Jones' convention is called: "From the point of view of the receiver", while Collett's convention is called: "From the point of view of the source." The reader should be wary of the choice of convention when consulting references on the Jones calculus.

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

{| class="wikitable"
! Polarization !! Jones vector !! Typical [[bra–ket notation|ket]] notation{{Citation needed|date=December 2024}}
|-
| Linear polarized in the ''x'' direction<BR>Typically called "horizontal" || <math>\begin{pmatrix} 1 \\ 0 \end{pmatrix}</math> || <math> |H\rangle </math>
|-
| Linear polarized in the ''y'' direction<BR>Typically called "vertical" || <math>\begin{pmatrix} 0 \\ 1 \end{pmatrix}</math> || <math> |V\rangle </math>
|-
| Linear polarized at 45° from the ''x'' axis<BR>Typically called "diagonal" L+45 || <math>\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ 1 \end{pmatrix}</math> || <math> |D\rangle = \frac{1}{\sqrt2} \big( |H\rangle + |V\rangle \big) </math>
|-
| Linear polarized at −45° from the ''x'' axis<BR>Typically called "anti-diagonal" L−45 || <math>\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ -1 \end{pmatrix}</math> || <math> |A\rangle = \frac{1}{\sqrt2} \big( |H\rangle - |V\rangle \big) </math>
|-
| Right-hand circular polarized<BR>Typically called "RCP" or "RHCP" || <math>\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ -i \end{pmatrix}</math> || <math>| R\rangle = \frac{1}{\sqrt2} \big( |H\rangle - i |V\rangle \big) </math>
|-
| Left-hand circular polarized<BR>Typically called "LCP" or "LHCP" || <math>\frac{1}{\sqrt2} \begin{pmatrix} 1 \\ +i \end{pmatrix}</math> || <math> |L\rangle  = \frac{1}{\sqrt2} \big( |H\rangle + i |V\rangle \big) </math>
|}

A general vector that points to any place on the surface is written as a [[Bra–ket notation|ket]] <math>|\psi\rangle</math>. When employing the [[Poincaré sphere (optics)|Poincaré sphere]] (also known as the [[Bloch sphere]]), the basis kets (<math>|0\rangle</math> and <math>|1\rangle</math>) must be assigned to opposing ([[Antipodal points|antipodal]]) pairs of the kets listed above.  For example, one might assign <math>|0\rangle</math> = <math>|H\rangle</math> and <math>|1\rangle</math> = <math>|V\rangle</math>.  These assignments are arbitrary.  Opposing pairs are

* <math>|H\rangle</math> and <math>|V\rangle</math>
* <math>|D\rangle</math> and <math>|A\rangle</math>
* <math>|R\rangle</math> and <math>|L\rangle</math>

The polarization of any point not equal to <math>|R\rangle</math> or <math>|L\rangle</math> and not on the circle that passes through <math>|H\rangle, |D\rangle, |V\rangle, |A\rangle</math> is known as [[elliptical polarization]].