= Faraday rotation from circular phases
{c}
{title2=$\theta=(\phi_+-\phi_-)/2$}
The factor of one half follows from <superposition>, without assigning ambiguous right/left names to the <circular polarizations>. Use the <complex amplitude> convention $\operatorname{Re}[\mathbf E(z)e^{-i\omega t}]$ and fixed transverse <unit vectors>
$$
\mathbf u_+=\frac{\widehat{\mathbf x}-i\widehat{\mathbf y}}{\sqrt2},\qquad
\mathbf u_-=\frac{\widehat{\mathbf x}+i\widehat{\mathbf y}}{\sqrt2},\qquad
\widehat{\mathbf x}=\frac{\mathbf u_++\mathbf u_-}{\sqrt2}.
$$
For equal transmitted <wave amplitudes>, let the accumulated <wave phases> be $\phi_+$ and $\phi_-$. Their sum is
$$
\frac{\mathbf u_+e^{i\phi_+}+\mathbf u_-e^{i\phi_-}}{\sqrt2}
=e^{i\bar\phi}\left(\widehat{\mathbf x}\cos\theta+\widehat{\mathbf y}\sin\theta\right),\qquad
\bar\phi=\frac{\phi_++\phi_-}{2},\quad
\theta=\frac{\phi_+-\phi_-}{2}.
$$
The common <wave phase> $\bar\phi$ sets the timing of the <electric field> oscillation; the real direction in parentheses sets its <linear polarization>. Opposite <circular polarizations> have changed their relative <wave phase>, yet their sum still oscillates along a line. Unequal transmitted <wave amplitudes> would generally give <elliptical polarization> instead.
In a uniform sample, with <refractive indices> $n_+,n_-$ and <vacuum> <wavenumber> $k_0=2\pi/\lambda_0$, the <phase difference> is $(k_+-k_-)L=k_0(n_+-n_-)L$. Thus
$$
\theta=\frac{\pi L}{\lambda_0}(n_+-n_-),\qquad
n_+-n_-=\frac{\lambda_0}{\pi}VB_\parallel.
$$
The second relation identifies the small signed <circular birefringence> responsible for the rotation in this basis. Unlike a <waveplate>, whose separated modes have orthogonal <linear polarizations>, this ideal rotator has modes with opposite <circular polarizations>.
\Image[/faraday-circular-phase-orientation.png]
{title=Equal opposite <circular polarizations> acquire a $60^\circ$ relative <wave phase> and give $30^\circ$ <Faraday rotation>}
{description=Original normalized model: common <wave phase> is removed, and the relative <wave phase> grows linearly with path fraction. The top panels add the real circular-component <electric field> vectors at the same time. Orange lines mark the resulting <linear polarization> direction. The lower panel compares relative <wave phase> with its half-angle rotation. No material measurements are plotted.}
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