Editing Jones calculus (section)

== 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>