= Faraday rotation on a round trip
{c}
{title2=$R(\theta)R(\theta)=R(2\theta)$}
<Faraday rotation> exhibits <optical nonreciprocity>: with the bias <magnetic field> held fixed, a retracing beam gains another rotation in the same laboratory sense. Natural <optical activity> in a reciprocal medium instead unwinds on retracing the path.<RP Photonics Encyclopedia by Rüdiger Paschotta/Faraday effect>[RP Photonics Encyclopedia, Faraday Effect, Rüdiger Paschotta: What is the Faraday Effect?, doubled rotation on a round trip]{ref}
To reconcile this with the signed $B_\parallel$ formula, retain the laboratory $x,y,z$ axes for comparing physical lines. For the backward ray choose a right-handed ray basis $x'=x$, $y'=-y$, $z'=-z$. Its <magnetic flux density> component is $B_\parallel'=-B_z$, so its local rotation increment is $-\theta$. But positive $y'$ is negative laboratory $y$: the backward increment is therefore $+\theta$ in the laboratory orientation plot. Reversing the ray reverses both the longitudinal component and the basis used for its signed <angle>. Reversing only one creates the false prediction of cancellation.
In fixed laboratory transverse components, define the <rotation matrix>
$$
R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}.
$$
For an ideal mirror at <normal incidence> that contributes the same <complex amplitude> factor $r$ to both transverse components, the round-trip map is $rR(\theta)R(\theta)=rR(2\theta)$. The common mirror factor does not change the oscillation line. A reciprocal natural rotator instead gives $rR(-\alpha)R(\alpha)=rI$. This comparison assumes a retraced straight path and no additional polarization-changing elements.
For example, a chosen single-pass $30^\circ$ <Faraday rotation> produces a returned line at $60^\circ$ to the original laboratory $x$ axis. A chosen reciprocal $30^\circ$ natural rotator returns it to $0^\circ$. The same addition makes an ideal <optical isolator> possible: <polarizers> at $0^\circ$ and $45^\circ$ surround a $45^\circ$ Faraday rotator. Forward <light> follows $0^\circ\to45^\circ$ and passes. Returning <light> selected at $45^\circ$ reaches $90^\circ$ at the entrance <polarizer> and is rejected by <Malus's law>. Real transmission loss and incomplete extinction limit the isolation.
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