Optical systems produce canonical transformations on phase space that are nonlinear. When a power expansion of the coordinates is performed around a chosen optical axis, the linear part is the paraxial approximation, and the nonlinear part is the ideal of aberrations. When the optical system has axial symmetry, its linear part is the symplectic group Sp(2, R) represented by 2 × 2 matrices. It is used to provide a classification of aberrations into multiplets of spin that are irreducible under the group, in complete analogy with the quantum harmonic-oscillator states. The “magnetic” axis of the latter may be chosen to adapt to magnifying systems or to optical fiberlike media. There seems to be a significant computational advantage in using the symplectic classification of aberrations.
© 1988 Optical Society of America
Kurt Bernardo Wolf, "Symmetry-adapted classification of aberrations," J. Opt. Soc. Am. A 5, 1226-1232 (1988)