 ABCD optical path generating function

For forward paraxial slopes and a lossless same-index optical system with $AD-BC=1$ and $B\ne 0$, Fermat's principle gives endpoint derivatives ${\partial }_{x}l=-\theta$ and ${\partial }_{x^{\prime }}l={\theta }^{\prime }$. Substituting $\theta =(x^{\prime }-Ax)/B$ and ${\theta }^{\prime }=(D{x}^{\prime }-x)/B$ and integrating proves the displayed optical path. Free space has $B=L>0$ and increases path length by $(x^{\prime }-x{)}^2/(2L)$, fixing the sign. Its quadratic phase generates the associated linear canonical transform; $B=0$ is a separate imaging limit.