= ABCD optical path generating function
{c}
{title2=$l=L_0+(Ax^2+Dx'^2-2xx')/(2B)$}
For forward paraxial slopes and a lossless same-index optical system with $AD-BC=1$ and $B\ne0$, <Fermat's principle> gives endpoint derivatives $\partial_xl=-\theta$ and $\partial_{x'}l=\theta'$. Substituting $\theta=(x'-Ax)/B$ and $\theta'=(Dx'-x)/B$ and integrating proves the displayed optical path. Free space has $B=L>0$ and increases path length by $(x'-x)^2/(2L)$, fixing the sign. Its quadratic <phase> generates the associated <linear canonical transform>; $B=0$ is a separate imaging limit.
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