Editing Jones calculus (section)

== 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.