ABCD optical path generating function ($\mathcal{L}={\mathcal{L}}_0+n_0(A{x}^2+D{x}^{\prime 2}-2x{x}^{\prime })/(2B)$)

Consider a lossless optical system in the paraxial approximation with the same refractive index $n_0$ at its input and output. In ray-height/slope coordinates, writeFor $B\ne 0$, its optical path length generating function isThe factor $n_0$ is required for physical optical path length; omitting it defines the index-normalized length $\mathcal{L}/n_0$.

Fermat's principle gives endpoint derivatives ${\partial }_x\mathcal{L}=-n_0\theta$ and ${\partial }_{x^{\prime }}\mathcal{L}=n_0{\theta }^{\prime }$. Substituting $\theta =(x^{\prime }-Ax)/B$ and ${\theta }^{\prime }=(D{x}^{\prime }-x)/B$ and integrating proves the expression. For uniform free space $B=L>0$, the extra optical length is $n_0(x^{\prime }-x{)}^2/(2L)$, fixing both sign and normalization. Its quadratic wave phase generates a linear canonical transform. The case $B=0$ is a separate imaging limit.