Editing Jones calculus (section)

== 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.