Editing Jones calculus

{{Short description|System for describing optical polarization}}
In [[optics]], [[polarized light]] can be described using the '''Jones calculus''',<ref name=spie/> invented by [[Robert Clark Jones|R. C. Jones]] in 1941. Polarized light is represented by a '''Jones vector''', and linear optical elements are represented by ''Jones [[matrix (mathematics)|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]].

== 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]].

== Jones matrices ==
Jones calculus is a matrix calculus developed in 1941 by [[Henry Hurwitz Jr.]] and [[R. Clark Jones]] and published in the ''[[Journal of the Optical Society of America]]''.<ref>{{Cite journal |last1= Hurwitz |first1= Henry |last2= Jones |first2= R. Clark |authorlink1=Henry Hurwitz Jr.| date= 1941 |title= A new calculus for the treatment of optical systems, II. Proof of three general equivalence theorems |journal= Journal of the Optical Society of America |volume= 31 |issue= 7 |pages= 493&ndash;499 |doi= 10.1364/JOSA.31.000493 |bibcode= 1941JOSA...31..493H }}</ref><ref>{{Cite journal |last= Jones |first= R. Clark |date= 1941 |title= A new calculus for the treatment of optical systems, I. Description and Discussion of the Calculus |journal= Journal of the Optical Society of America |volume= 31 |issue= 7 |pages= 488&ndash;493 |doi= 10.1364/JOSA.31.000488 |bibcode= 1941JOSA...31..488J }}</ref><ref>{{Cite journal |last= Jones |first= R. Clark |date= 1941 |title= A new calculus for the treatment of optical systems, III. The Sohncke Theory of optical activity |journal= Journal of the Optical Society of America |volume= 31 |issue= 7 |pages= 500&ndash;503 |doi= 10.1364/JOSA.31.000500 |bibcode= 1941JOSA...31..500J }}</ref><ref>{{Cite journal |last= Jones |first= R. Clark |date= 1942 |title= A new calculus for the treatment of optical systems, IV. |journal= Journal of the Optical Society of America |volume= 32 |issue= 8 |pages= 486&ndash;493 |doi= 10.1364/JOSA.32.000486 |bibcode= 1942JOSA...32..486C }}</ref>

The Jones matrices are operators that act on the Jones vectors defined above.  These matrices are implemented by various optical elements such as lenses, beam splitters, mirrors, etc.  Each matrix represents projection onto a one-dimensional complex subspace of the Jones vectors. The following table gives examples of Jones matrices for polarizers:
{| class="wikitable" style="text-align:center"
! Optical element !! Jones matrix
|-
| Linear [[polarizer]] with axis of transmission horizontal<ref name="fowles">{{cite book|author=Fowles, G.|title=Introduction to Modern Optics|url=https://archive.org/details/introductiontomo00fowl_441|url-access=limited|edition=2nd|publisher=Dover|date=1989|page=[https://archive.org/details/introductiontomo00fowl_441/page/n44 35]|isbn=9780486659572 }}</ref> ||
<math>\begin{pmatrix}
1 & 0 \\ 0 & 0
\end{pmatrix}</math>
|-
| Linear polarizer with axis of transmission vertical<ref name="fowles" /> ||
<math>\begin{pmatrix}
0 & 0 \\ 0 & 1
\end{pmatrix}</math>
|-
| Linear polarizer with axis of transmission at ±45° with the horizontal<ref name="fowles" /> ||
<math>\frac{1}{2} \begin{pmatrix}
1 & \pm 1 \\ \pm 1 & 1
\end{pmatrix}</math>
|-
| Linear polarizer with axis of transmission angle <math>\theta</math> from the horizontal<ref name="fowles" /> ||
<math> \begin{pmatrix}
\cos^2(\theta) & \cos(\theta)\sin(\theta) \\ \cos(\theta)\sin(\theta) & \sin^2(\theta)
\end{pmatrix}</math>
|-
| Right circular polarizer<ref name="fowles" /> ||
<math>\frac{1}{2} \begin{pmatrix}
1 & i \\ -i & 1
\end{pmatrix}</math>
|-
| Left circular polarizer<ref name="fowles" /> ||
<math>\frac{1}{2} \begin{pmatrix}
1 & -i \\ i & 1
\end{pmatrix}
</math>
|-
|}

== Phase retarders ==
A phase retarder is an optical element that produces a phase difference between two orthogonal polarization components of a monochromatic polarized beam of light.<ref name="theocaris">{{cite book|author=P.S. Theocaris|author2=E.E. Gdoutos |title=Matrix Theory of Photoelasticity|series=Springer Series in Optical Sciences |publisher=[[Springer Science+Business Media|Springer-Verlag]]|url=https://link.springer.com/book/10.1007/978-3-540-35789-6|edition=1st|date=1979|volume=11 |doi=10.1007/978-3-540-35789-6 |isbn=978-3-662-15807-4 }}</ref>  Mathematically, using [[Bra–ket notation|kets]] to represent Jones vectors, this means that the action of a phase retarder is to transform light with polarization 
:<math>|P\rangle = c_1 |1\rangle + c_2|2\rangle</math> 
to 
:<math>|P'\rangle = c_1 {\rm e}^{i\eta/2}|1\rangle + c_2 {\rm e}^{-i\eta/2}|2\rangle</math> 
where <math>|1\rangle, |2\rangle</math> are orthogonal polarization components (i.e. <math>\langle 1|2 \rangle =0</math>) that are determined by the physical nature of the phase retarder.  In general, the orthogonal components could be any two basis vectors.  For example, the action of the circular phase retarder is such that
:<math>
|1\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix}
  1 \\
  -i
\end{pmatrix}
\qquad \text{ and } \qquad
|2\rangle =\frac{1}{\sqrt{2}} \begin{pmatrix}
  1 \\
  i
\end{pmatrix}
</math>
However, linear phase retarders, for which <math>|1\rangle, |2\rangle</math> are linear polarizations, are more commonly encountered in discussion and in practice.  In fact, sometimes the term "phase retarder" is used to refer specifically to linear phase retarders.

Linear phase retarders are usually made out of [[birefringent]] [[uniaxial crystal]]s such as [[calcite]], MgF<sub>2</sub> or [[quartz]].  Plates made of these materials for this purpose are referred to as [[waveplate]]s. Uniaxial crystals have one crystal axis that is different from the other two crystal axes (i.e., ''n<sub>i</sub>'' ≠ ''n<sub>j</sub>'' = ''n<sub>k</sub>''). This unique axis is called the extraordinary axis and is also referred to as the [[optic axis of a crystal|optic axis]]. An optic axis can be the fast or the slow axis for the crystal depending on the crystal at hand. Light travels with a higher phase velocity along an axis that has the smallest [[refractive index]] and this axis is called the fast axis. Similarly, an axis which has the largest refractive index is called a slow axis since the [[phase velocity]] of light is the lowest along this axis. "Negative" uniaxial crystals (e.g., [[calcite]] CaCO<sub>3</sub>, [[sapphire]] Al<sub>2</sub>O<sub>3</sub>) have ''n<sub>e</sub>'' < ''n<sub>o</sub>'' so for these crystals, the extraordinary axis (optic axis) is the fast axis, whereas for "positive" uniaxial crystals (e.g., [[quartz]] SiO<sub>2</sub>, [[magnesium fluoride]] MgF<sub>2</sub>, [[rutile]] TiO<sub>2</sub>), ''n<sub>e</sub>'' > ''n<sub>o</sub>'' and thus the extraordinary axis (optic axis) is the slow axis.  Other commercially available linear phase retarders exist and are used in more specialized applications.  The [[Fresnel rhomb]]s is one such alternative.

Any linear phase retarder with its fast axis defined as the x- or y-axis has zero off-diagonal terms and thus can be conveniently expressed as 
:<math>\begin{pmatrix}
  {\rm e}^{i\phi_x} & 0 \\
  0           & {\rm e}^{i\phi_y}
\end{pmatrix} </math>

where <math>\phi_x</math> and <math>\phi_y</math> are the phase offsets of the electric fields in <math>x</math> and <math>y</math> directions respectively. In the phase convention <math>\phi = kz - \omega t</math>, define the relative phase between the two waves as <math>\epsilon = \phi_y - \phi_x</math>. Then a positive <math>\epsilon</math> (i.e. <math>\phi_y</math> > <math>\phi_x</math>) means that <math>E_y</math> doesn't attain the same value as <math>E_x</math> until a later time, i.e. <math>E_x</math> leads <math>E_y</math>. Similarly, if <math>\epsilon < 0</math>, then <math>E_y</math> leads <math>E_x</math>.

For example, if the fast axis of a quarter waveplate is horizontal, then the phase velocity along the horizontal direction is ahead of the vertical direction i.e., <math>E_x</math> leads <math>E_y</math>. Thus, <math>\phi_x < \phi_y</math> which for a quarter waveplate yields <math>\phi_y = \phi_x + \pi/2</math>.

In the opposite convention <math>\phi = \omega t - kz</math>, define the relative phase as <math>\epsilon = \phi_x - \phi_y</math>. Then  <math>\epsilon>0</math> means that <math>E_y</math> doesn't attain the same value as <math>E_x</math> until a later time, i.e. <math> E_x</math> leads <math>E_y</math>.

{| class="wikitable"
! Phase retarders
! Corresponding Jones matrix
|-
| [[Wave plate|Quarter-wave plate]] with fast axis vertical<ref name="hecht">{{cite book|author=Eugene Hecht|title=Optics|url=https://archive.org/details/optics00ehec|url-access=limited|edition=4th|date=2001|page=[https://archive.org/details/optics00ehec/page/n384 378]|publisher=Addison-Wesley |isbn=978-0805385663|author-link=Eugene Hecht}}</ref>{{refn|The prefactor <math>{\rm e}^{i\pi/4}</math> appears only if one defines the phase delays in a symmetric fashion; that is, <math>\phi_x = -\phi_y = \pi/4</math>. This is done in Hecht<ref name="hecht" /> but not in Fowles<ref name="fowles" /> or Peatross and Ware.<ref name="peatrossandware">{{cite book|author=Peatross, Justin|author2=Ware, Michael |title=Physics of Light and Optics|url=https://optics.byu.edu/textbook|edition=2015 edition, Jan. 31, 2025 revision|date=2025|isbn=978-1-312-92927-2 }}</ref> In the latter two references the Jones matrices for a quarter-wave plate have no prefactor.|group=note}}
| <math> {\rm e}^{\frac{i\pi}{4}} \begin{pmatrix}
  1 &  0 \\
  0 & -i
\end{pmatrix} </math>
|-
| [[Wave plate|Quarter-wave plate]] with fast axis horizontal<ref name="hecht" />
| <math> {\rm e}^{-\frac{i\pi}{4}} \begin{pmatrix}
  1 & 0 \\
  0 & i
\end{pmatrix} </math>
|-
| [[wave plate|Quarter-wave plate]] with fast axis at angle <math>\theta</math> w.r.t the horizontal axis 
| <math>{\rm e}^{-\frac{i\pi}{4}}
\begin{pmatrix}
  \cos^2\theta + i\sin^2\theta & (1 - i)\sin\theta \cos\theta \\
  (1 - i)\sin\theta \cos\theta & \sin^2\theta + i\cos^2\theta
\end{pmatrix}</math>
|-
|Half-wave plate rotated by <math>\theta</math> <ref name=spie>{{Cite web |title=Jones Calculus |url=https://spie.org/publications/fg05_p57-61_jones_matrix_calculus?SSO=1 |access-date=2025-05-04 |website=spie.org}}</ref>
|<math>\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}</math>
|-
| [[wave plate|Half-wave plate]] with fast axis at angle <math>\theta</math> w.r.t the horizontal axis<ref>{{cite book|author=Gerald, A.|author2=Burch, J.M.|title=Introduction to Matrix Methods in Optics|edition=1st|publisher=[[John Wiley & Sons]]|date=1975|page=212|isbn=978-0471296850}}</ref>
| <math>{\rm e}^{-\frac{i\pi}{2}} \begin{pmatrix}
  \cos^2\theta - \sin^2\theta &
    2 \cos\theta \sin\theta \\
  2 \cos\theta \sin\theta &
    \sin^2\theta - \cos^2\theta
\end{pmatrix}</math>
|-
| General Waveplate (Linear Phase Retarder)<ref name="theocaris" />
| <math>{\rm e}^{-\frac{i\eta}{2}} \begin{pmatrix}
  \cos^2\theta + {\rm e}^{i\eta} \sin^2\theta &
    \left(1 - {\rm e}^{i\eta}\right) \cos\theta \sin\theta \\
  \left(1 - {\rm e}^{i\eta}\right) \cos\theta \sin\theta &
    \sin^2\theta + {\rm e}^{i\eta} \cos^2\theta
\end{pmatrix}</math>
|-
| Arbitrary birefringent material (Elliptical phase retarder)<ref name="theocaris" /><ref name="jorge">{{cite journal |first1=Jose Jorge |last1=Gill |first2=Eusebio |last2=Bernabeu |year=1987 |title=Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix |journal=[[Optik (journal)|Optik]] |volume=76 |issue=2 |pages=67–71 |issn=0030-4026 }}</ref>
| <math>{\rm e}^{-\frac{i\eta}{2}} \begin{pmatrix}
  \cos^2\theta + {\rm e}^{i\eta} \sin^2\theta &
    \left(1 - {\rm e}^{i\eta}\right) {\rm e}^{-i\phi} \cos\theta \sin\theta \\
  \left(1 - {\rm e}^{i\eta}\right) {\rm e}^{i\phi} \cos\theta \sin\theta &
    \sin^2\theta + {\rm e}^{i\eta} \cos^2\theta
\end{pmatrix}</math>
|}

The Jones matrix for an arbitrary birefringent material is the most general form of a polarization transformation in the Jones calculus; it can represent any polarization transformation.  To see this, one can show
:<math> \begin{align}
&{\rm e}^{-\frac{i\eta}{2}} \begin{pmatrix}
  \cos^2\theta + {\rm e}^{i\eta} \sin^2\theta &
    \left(1 - {\rm e}^{i\eta}\right) {\rm e}^{-i\phi} \cos\theta \sin\theta \\
  \left(1 - {\rm e}^{i\eta}\right) {\rm e}^{i\phi} \cos\theta \sin\theta &
    \sin^2\theta + {\rm e}^{i\eta} \cos^2\theta 
\end{pmatrix}
\\
&=
\begin{pmatrix}
  \cos(\eta/2)-i\sin(\eta/2)\cos(2\theta) &
    -\sin(\eta/2)\sin(\phi)\sin(2\theta) - i \sin(\eta/2)\cos(\phi)\sin(2\theta) \\
  \sin(\eta/2)\sin(\phi)\sin(2\theta) - i \sin(\eta/2)\cos(\phi)\sin(2\theta) &
    \cos(\eta/2)+i\sin(\eta/2)\cos(2\theta)
\end{pmatrix}
\end{align}</math>

The above matrix is a general parametrization for the elements of [[Special unitary group|SU(2)]], using the convention 
:<math>\operatorname{SU}(2) = \left\{ \begin{pmatrix} \alpha & -\overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta \in \mathbb{C},\ \ |\alpha|^2 + |\beta|^2 = 1 \right\}~</math>
where the overline denotes [[complex conjugate|complex conjugation]].

Finally, recognizing that the set of [[unitary transformation]]s on <math>\mathbb{C}^2</math> can be expressed as
:<math>\left\{ {\rm e}^{i\gamma}\begin{pmatrix} \alpha & -\overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta \in \mathbb{C},\ \ |\alpha|^2 + |\beta|^2 = 1,\ \ \gamma \in [0,2\pi] \right\}</math>
it becomes clear that the Jones matrix for an arbitrary birefringent material represents any unitary transformation, up to a phase factor <math>{\rm e}^{i\gamma}</math>. Therefore, for appropriate choice of <math>\eta</math>, <math>\theta</math>, and <math>\phi</math>, a transformation between any two Jones vectors can be found, up to a phase factor <math>{\rm e}^{i\gamma}</math>.  However, in the Jones calculus, such phase factors do not change the represented polarization of a Jones vector, so are either considered arbitrary or imposed ad hoc to conform to a set convention.

The special expressions for the phase retarders can be obtained by taking suitable parameter values in the general expression for a birefringent material.<ref name="jorge"/> In the general expression:
*The relative phase retardation induced between the fast axis and the slow axis is given by <math> \eta = \phi_y - \phi_x </math>
*<math>\theta</math> is the orientation of the fast axis with respect to the x-axis.
*<math>\phi</math> is the circularity.

Note that for linear retarders, <math>\phi</math> = 0 and for circular retarders, <math>\phi</math> = ± <math>\pi</math>/2, <math>\theta</math> = <math>\pi</math>/4. In general for elliptical retarders, <math>\phi</math> takes on values between - <math>\pi</math>/2 and <math>\pi</math>/2.

== Axially rotated elements ==
Assume an optical element has its optic axis{{clarify|reason=Does this mean (1) the "optic axis" of a (presumably uniaxial) birefringent material, or (2) the "optic axis" (also known as optical axis) of a rotationally symmetric lens system?|date=May 2015}}  perpendicular to the surface vector for the [[plane of incidence]]{{clarify|reason=What is the "surface vector for the plane of incidence"? Is it the normal vector? This would then be tangent to the surface of the refracting material, right?|date=May 2015}} and is rotated about this surface vector by angle ''θ/2'' (i.e., the [[Cardinal point (optics)#Principal planes and points|principal plane]] through which the optic axis passes,{{clarify|reason=What is the geometric relation between a vector and a plane expressed by "passes through"?|date=May 2015}} makes angle ''θ/2'' with respect to the plane of polarization of the electric field{{clarify|reason=What is "the plane of polarization" of the electric field? I thought polarization was expressed by a vector. Does it mean the plane orthogonal to the direction of propagation, in which E can take its values?|date=May 2015}} of the incident TE wave). Recall that a half-wave plate rotates polarization as ''twice'' the angle between incident polarization and optic axis (principal plane). Therefore, the Jones matrix for the rotated polarization state, M(''θ''), is 
:<math>M(\theta )=R(-\theta )\,M\,R(\theta ),</math>
: where <math>R(\theta ) = 
\begin{pmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{pmatrix}.</math>
This agrees with the expression for a half-wave plate in the table above. These rotations are identical to beam unitary splitter transformation in optical physics given by
:<math>R(\theta ) =
\begin{pmatrix}
r & t'\\
t & r'
\end{pmatrix}</math>
where the primed and unprimed coefficients represent beams incident from opposite sides of the beam splitter. The reflected and transmitted components acquire a phase ''θ<sub>r</sub>'' and ''θ<sub>t</sub>'', respectively. The requirements for a valid representation of the element are <ref name=hong-ou-mandel>{{cite journal |first1=Z. Y. |last1=Ou |first2=L. |last2=Mandel |title=Derivation of reciprocity relations for a beam splitter from energy balance |journal=Am. J. Phys. |volume=57 |issue=1 |pages=66 |year=1989 |doi=10.1119/1.15873 |bibcode=1989AmJPh..57...66O }}</ref>
:<math>
\theta_\text{t} - \theta_\text{r} + \theta_\text{t'} - \theta_\text{r'} = \pm \pi
</math>
and
<math>r^*t' + t^*r' = 0.</math>
:Both of these representations are unitary matrices fitting these requirements; and as such, are both valid.

== Arbitrarily rotated elements ==
{{expand section|date=July 2014}}
Finding the Jones matrix, J(''α'', ''β'', ''γ''), for an arbitrary rotation involves a three-dimensional [[rotation matrix]]. In the following notation ''α'', ''β'' and ''γ'' are the [[Yaw pitch roll|yaw, pitch, and roll]] angles (rotation about the z-, y-, and x-axes, with x being the direction of propagation), respectively. The full combination of the 3-dimensional rotation matrices is the following:

:<math>R_{3D}(\theta)=\begin{bmatrix} \cos\alpha\cos\beta &
          \cos\alpha\sin\beta\sin\gamma - \sin\alpha\cos\gamma &
          \cos\alpha\sin\beta\cos\gamma + \sin\alpha\sin\gamma \\
        \sin\alpha\cos\beta &
          \sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\gamma &
          \sin\alpha\sin\beta\cos\gamma - \cos\alpha\sin\gamma \\
       -\sin\beta & \cos\beta\sin\gamma & \cos\beta\cos\gamma \\ \end{bmatrix}</math>

Using the above, for any base Jones matrix J, you can find the rotated state J(''α'', ''β'', ''γ'') using:

:<math>J(\alpha,\beta,\gamma) = R_{3D}(-\alpha,-\beta,-\gamma)\cdot J \cdot R_{3D}(\alpha,\beta,\gamma)</math><ref name=spie/>

The simplest case, where the Jones matrix is for an ideal linear horizontal polarizer, reduces then to:

:<math>J(\alpha, \beta, \gamma) = 
\begin{bmatrix} 
    c^2_{\alpha} c^2_{\beta} & c_{\alpha} c_{\beta} [c_{\alpha} s_{\beta} s_{\gamma} - s_{\alpha} c_{\gamma}] & c_{\alpha} c_{\beta} [c_{\alpha} s_{\beta} c_{\gamma} + s_{\alpha} s_{\gamma}]\\
    s_{\alpha} c_{\alpha} c^2_{\beta} & s_{\alpha} c_{\beta} [c_{\alpha} s_{\beta} s_{\gamma} - s_{\alpha} c_{\gamma}] & s_{\alpha} c_{\beta} [c_{\alpha} s_{\beta} c_{\gamma} + s_{\alpha} s_{\gamma}] \\
    -c_{\alpha} s_{\beta} c_{\beta} & -s_{\beta} [c_{\alpha} s_{\beta} s_{\gamma} - s_{\alpha} c_{\gamma}] & -s_{\beta} [c_{\alpha} s_{\beta} c_{\gamma} + s_{\alpha} s_{\gamma}]\\
\end{bmatrix}
</math>

where c<sub>i</sub> and s<sub>i</sub> represent the cosine or sine of a given angle "i", respectively.


See Russell A. Chipman and Garam Yun for further work done based on this.<ref name="Chipman Lam Young 2018 p.">{{cite book | last1=Chipman | first1=R.A. | last2=Lam | first2=W.S.T. | last3=Young | first3=G. | title=Polarized Light and Optical Systems | publisher=CRC Press | series=Optical Sciences and Applications of Light | year=2018 | isbn=978-1-4987-0057-3 | url=https://books.google.com/books?id=saVuDwAAQBAJ | access-date=2023-01-20 | page=}}</ref><ref>{{cite journal |first=Russell A. |last=Chipman |year=1995 |title=Mechanics of polarization ray tracing |journal=Opt. Eng. |volume=34 |issue=6 |pages=1636–1645 |doi=10.1117/12.202061 |bibcode=1995OptEn..34.1636C }}</ref><ref>{{cite journal |title=Three-dimensional polarization ray-tracing calculus I: definition and diattenuation |journal=[[Applied Optics (journal)|Applied Optics]] |first1=Garam |last1=Yun |first2=Karlton |last2=Crabtree |first3=Russell A. |last3=Chipman |volume=50 |issue= 18|pages=2855–2865 |year=2011 |doi=10.1364/AO.50.002855 |pmid=21691348 |bibcode=2011ApOpt..50.2855Y }}</ref><ref>{{cite journal |title=Three-dimensional polarization ray-tracing calculus II: retardance |journal=Applied Optics |first1=Garam |last1=Yun |first2=Stephen C. |last2=McClain |first3=Russell A. |last3=Chipman |volume=50 |issue= 18|pages=2866–2874 |year=2011 |doi=10.1364/AO.50.002866 |pmid=21691349 |bibcode=2011ApOpt..50.2866Y }}</ref><ref>{{cite thesis |hdl=10150/202979 |first=Garam |last=Yun |title=Polarization Ray Tracing |type=PhD thesis |date=2011 |publisher=University of Arizona }}</ref>

==See also==
* [[Polarization (waves)|Polarization]]
* [[Scattering parameters]]
* [[Stokes parameters]]
* [[Mueller calculus]]
* [[Photon polarization]]

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}

==Further reading==
{{more footnotes|date=July 2014}}

* {{Cite journal |last1= Brosseau |first1= Christian |last2= Givens |first2= Clark R. |last3= Kostinski |first3= Alexander B. |date= 1993 |title= Generalized trace condition on the Mueller-Jones polarization matrix |journal= Journal of the Optical Society of America A |volume= 10 |issue= 10 |pages= 2248&ndash;2251 |doi= 10.1364/JOSAA.10.002248 |bibcode= 1993JOSAA..10.2248B }}
* {{Cite journal |last= Fymat |first= A. L. |date= 1971 |title= Jones's Matrix Representation of Optical Instruments. 1: Beam Splitters |journal= Applied Optics |volume= 10 |issue= 11 |pages= 2499&ndash;2505 |doi= 10.1364/AO.10.002499 |pmid= 20111363 |bibcode= 1971ApOpt..10.2499F }}
* {{Cite journal |last= Fymat |first= A. L. |date= 1971 |title= Jones's Matrix Representation of Optical Instruments. 2: Fourier Interferometers (Spectrometers and Spectropolarimeters) |journal= Applied Optics |volume= 10 |issue= 12 |pages= 2711&ndash;2716 |doi= 10.1364/AO.10.002711 |pmid= 20111418 |bibcode= 1971ApOpt..10.2711F }}
* {{Cite journal |last= Fymat |first= A. L. |date= 1972 |title= Polarization Effects in Fourier Spectroscopy. I: Coherency Matrix Representation |journal= Applied Optics |volume= 11 |issue= 1 |pages= 160&ndash;173 |doi= 10.1364/AO.11.000160 |pmid= 20111472 |bibcode= 1972ApOpt..11..160F }}
* {{cite book |last1= Gerald |first1= A. |last2= Burch |first2= J. M. |date= 1975 |title= Introduction to Matrix Methods in Optics |edition= 1st |publisher= John Wiley & Sons |isbn= 0-471-29685-6 }}
* {{Cite journal |last1= Gill |first1= Jose Jorge |last2= Bernabeu |first2= Eusebio |date= 1987 |title= Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix |journal= Optik |volume= 76 |pages= 67&ndash;71 }}
* {{cite book |last1= Goldstein |first1= D. |last2= Collett |first2= E. |date= 2003 |title= Polarized Light |edition= 2nd |publisher= [[CRC Press]] |isbn= 0-8247-4053-X }}
* {{cite book |last= Hecht |first= E. |date= 1987 |title= Optics |edition= 2nd |publisher= Addison-Wesley |isbn= 0-201-11609-X }}
* {{Cite journal |last1= McGuire |first1= James P. |last2= Chipman |first2= Russel A. |date= 1994 |title= Polarization aberrations. 1. Rotationally symmetric optical systems |journal= Applied Optics |volume= 33 |issue= 22 |pages= 5080&ndash;5100 |doi= 10.1364/AO.33.005080 |pmid= 20935891 |bibcode= 1994ApOpt..33.5080M |s2cid= 3805982 }}
* {{Cite journal |last1= Moreno |first1= Ignacio |last2= Yzuel |first2= Maria J. |last3= Campos |first3= Juan |last4= Vargas |first4= Asticio |date= 2004 |title= Jones matrix treatment for polarization Fourier optics |journal= Journal of Modern Optics |volume= 51 |issue= 14 |pages= 2031&ndash;2038 |doi= 10.1080/09500340408232511 |bibcode= 2004JMOp...51.2031M |s2cid= 120169144 |hdl= 10533/175322 |hdl-access= free |author-link2= María Yzuel }}
* {{Cite journal |last= Moreno |first= Ivan |date= 2004 |title= Jones matrix for image-rotation prisms |journal= Applied Optics |volume= 43 |issue= 17 |pages= 3373&ndash;3381 |doi= 10.1364/AO.43.003373 |pmid= 15219016 |bibcode= 2004ApOpt..43.3373M |s2cid= 24268298 }}
* {{cite book |last1= Pedrotti |first1= Frank L. |last2= Leno |first2= S. J. |last3= Pedrotti |first3= S. |date= 1993 |title= Introduction to Optics |edition= 2nd |publisher= Prentice Hall |isbn= 0-13-501545-6 }}
* {{Cite journal |last= Pistoni |first= Natale C. |date= 1995 |title= Simplified approach to the Jones calculus in retracing optical circuits |journal= Applied Optics |volume= 34 |issue= 34 |pages= 7870&ndash;7876 |doi= 10.1364/AO.34.007870 |pmid= 2106888 |bibcode= 1995ApOpt..34.7870P }}
* {{cite book |last= Shurcliff |first= William |date= 1966 |title= Polarized Light: Production and Use |chapter= Chapter 8: Mueller Calculus and Jones Calculus |page= 109 |publisher= [[Harvard University Press]] |author-link= William Shurcliff }}

==External links==
* [http://spie.org/x32380.xml ''Jones Calculus written by E. Collett on Optipedia'']

{{DEFAULTSORT:Jones Calculus}}
[[Category:Optics]]
[[Category:Polarization (waves)]]
[[Category:Matrices (mathematics)]]