Symplectic algebra and Gaussian optics (Q1825978)

Optical systems at small angles are described by linear transformations of optical phase space; the group describing this Gaussian optical regime is the symplectic group Sp(4,R) and, when the system is invariant under rotations around its optical axis, \(Sp(2,R)=SL(2,R)\). Focusable systems relate a pair of conjugate planes as object and image. The author proves (Theorem 2) that generally non-axis-symmetric systems have besides a `touching' pair of planes, either a single other pair, or a one-parameter family of distinct conjugate planes. The latter case corresponds to Gaussian optics with axis-symmetric systems that constitute the generic case of interest. The author concludes stating that, in effect, Gaussian Sp(2,R)-transformation optics is sufficient for the purposes of description of focusable systems, the exceptions being isolated from the main manifold of Gaussian systems.