Jones Calculus - Rotated Elements

Rotated Elements

Assume an optical element has its optic axis perpendicular to the surface vector for the plane of incidence and is rotated about this surface vector by angle θ/2 (.i.e., the principal plane, through which the optic axis passes, makes angle θ/2 with respect to the plane of polarization of the electric field 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

where R(\theta ) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.

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

R(\theta ) = \begin{pmatrix} r & t'\\ t & r' \end{pmatrix}

where the primed and unprimed coefficients represent beams incident from opposite sides of the beam splitter. The reflected and transmitted components acquire a phase θr and θt, respectively. The requirements for a valid representation of the element are

\theta_\text{t} - \theta_\text{r} + \theta_\text{t'} - \theta_\text{r'} = \pm \pi

and

Both of these representations are unitary matrices fitting these requirements; and as such, are both valid.

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