= Half-wave and quarter-wave achromatic circularizer
{title2=$\mathbf J(\epsilon)=[1+i(3-\sqrt3)\epsilon/2]\mathbf J(0)+O(\epsilon^2)$}
Take a <half-wave plate> with fast axis at $15^\circ$ to a linearly polarized input, followed by a <quarter-wave plate> at a further $60^\circ$, so the second axis is at $75^\circ$ to the input. With shared quarter-wave <retardance> error $\epsilon$ and half-wave error $2\epsilon$, use $Q=\operatorname{diag}(1,ie^{i\epsilon})$ and $H=\operatorname{diag}(1,-e^{2i\epsilon})$. Rotating each matrix into the input axes and multiplying gives $\mathbf J(0)=e^{i75^\circ}(1,-i)^{\mathsf T}/\sqrt2$ and the title expansion. Its first-order change is common <phase> only, so the output remains circular to first order. This input-specific local compensation does not prove a frequency-independent quarter-wave operator for arbitrary input across a finite spectral band.
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