= Solution
Use $\mathbf E(z,t)=\operatorname{Re}[\mathbf J e^{i(kz-\omega t)}]$. A <Jones vector> $\mathbf J=(J_x,J_y)^{\mathsf T}$ records the two transverse <complex amplitudes>. The <amplitude> ratio and relative <phase> determine the <polarization ellipse>; a common nonzero scalar does not change the <polarization>. A fixed <Jones vector> describes fully polarized monochromatic light, not an unpolarized statistical mixture.
<Linear birefringence> gives <orthogonal> principal <polarizations> different <refractive indices>. Put $n_y-n_x=\Delta n>0$, so $x$ is the fast axis. A plate of thickness $d$ gives slow-minus-fast <phase retardation>
$$
\delta(\omega)=\frac{\omega d\Delta n(\omega)}c.
$$
The thinnest <quarter-wave plate> has
$$
\boxed{d=\frac{\pi c}{2\omega_0\Delta n(\omega_0)}=\frac{\lambda_0}{4\Delta n(\omega_0)}.}
$$
Here $\lambda_0=2\pi c/\omega_0$ is the <vacuum> <wavelength>. For linear input at angle $\theta$ to the fast axis, the output at design <frequency> is $(\cos\theta,i\sin\theta)^{\mathsf T}$ up to common <phase>. Axis-aligned input remains linear; equal nonzero components at $45^\circ$ modulo $90^\circ$ give <circular polarization>; other input angles give <elliptical polarization>. There is no need to spatially separate the two components.<The Feynman Lectures on Physics, Volume I by Richard Feynman and others>[§33–3: phase accumulation between orthogonal principal polarizations and quarter-wave conversion]{ref}
For <waveplate retardance detuning>, define
$$
\epsilon=\frac{\omega d\Delta n(\omega)}c-\frac\pi2
\simeq\frac dc[\Delta n_0+\omega_0\Delta n'_0](\omega-\omega_0).
$$
Neglecting material dispersion reduces this to $\epsilon=(\pi/2)(\omega-\omega_0)/\omega_0$. Remove the common fast-axis <phase>. Then $Q=\operatorname{diag}(1,ie^{i\epsilon})$. A similar <half-wave plate> of twice the thickness has twice the retardation, giving $H=\operatorname{diag}(1,-e^{2i\epsilon})$. The shared error parameter presumes the same material and matched thickness ratio.
For the <half-wave and quarter-wave achromatic circularizer>, the first fast-axis angle is $\alpha=15^\circ$ and the second is $\gamma=\alpha+\beta=75^\circ$. With the stated passive rotation convention,
$$
\mathbf J_{\rm out}=R(-\gamma)Q R(\gamma)R(-\alpha)H R(\alpha)
\begin{pmatrix}1\\0\end{pmatrix}.
$$
An explicit expression valid for the given shared error is obtained by putting $z=e^{i\epsilon}$ and defining
$$
v_x=\cos^2\alpha-z^2\sin^2\alpha,\qquad
v_y=\sin\alpha\cos\alpha(1+z^2),
$$
$$
\mathbf J_{\rm out}=\begin{pmatrix}
(\cos^2\gamma+iz\sin^2\gamma)v_x+(1-iz)\sin\gamma\cos\gamma\,v_y\\
(1-iz)\sin\gamma\cos\gamma\,v_x+(\sin^2\gamma+iz\cos^2\gamma)v_y
\end{pmatrix}.
$$
At $\epsilon=0$, the <half-wave plate> rotates the linear input to $30^\circ$, which is $-45^\circ$ relative to the second fast axis. Thus
$$
\boxed{\mathbf J_0=\frac{e^{i75^\circ}}{\sqrt2}\begin{pmatrix}1\\-i\end{pmatrix}.}
$$
This is <circular polarization>. With our time convention its fixed-point electric vector is proportional, after a time-origin shift, to $(\cos\omega t,-\sin\omega t)$; this specifies the rotation without relying on conflicting handedness conventions.
Expanding the exact product to first order gives
$$
\mathbf J_{\rm out}=\begin{pmatrix}0.183+0.683i\\0.683-0.183i\end{pmatrix}
+\epsilon\begin{pmatrix}-0.433+0.116i\\0.116+0.433i\end{pmatrix}+O(\epsilon^2).
$$
These coefficients are rounded to three significant figures. Before rounding, the derivative is exactly $\mathbf J'_0=i\kappa\mathbf J_0$, with $\kappa=(3-\sqrt3)/2$. The <Jones-component ratio sensitivity> therefore vanishes:
$$
\boxed{\frac{J_y}{J_x}=\frac{J_{0y}(1+i\kappa\epsilon)}{J_{0x}(1+i\kappa\epsilon)}+O(\epsilon^2)=-i+O(\epsilon^2).}
$$
Only common <phase> changes at first order. This proves compensation for the prescribed input near the design <frequency>, not exact achromatic performance throughout the visible spectrum for arbitrary input <polarization>.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2012/paper-1-oe2012-retarder-compensation.png]
{title=Original comparison of single-plate and composite polarization errors and the composite electric-vector trajectory}
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