Symmetry in Lie optics

Optical systems produce nonlinear canonical transformations in optical phase space. Free propagation in a homogeneous medium has Euclidean symmetry and dynamical algebras under the Poisson-Lie bracket. Refracting interfaces between homogeneous media exhibit invariants; in particular, a spherical surface possesses an so(3) symmetry algebra which allows the recursive computation of its aberration coefficients to arbitrarily high order. We present explicit results for aberration order nine.