Lie algebraic treatment of dioptric power and optical aberrations

The dioptric power of an optical system can be expressed as a four-component dioptric power matrix. We generalize and reformulate the standard matrix approach by utilizing the methods of Lie algebra. This generalization helps one deal with nonlinear problems (such as aberrations) and further extends the standard matrix formulation. Explicit formulas giving the relationship between the incident and the emergent rays are presented. Examples include the general case of thick and thin lenses. The treatment of a graded-index medium is outlined.