New class of characteristic functions in Hamiltonian optics

Hamilton's characteristic functions are frequently reduced to functions of four variables as four coordinates are sufficient to specify a monochromatic ray. For the point characteristic this is usefully achieved by restricting the terminal points of rays to lie on arbitrarily chosen base surfaces, not necessarily planes. Formerly, mixed and angle characteristics with four arguments were defined only for systems with plane base surfaces. It is shown here that the Legendre transforms of the restricted point characteristic of a system with arbitrary base surfaces are further characteristic functions corresponding to mixed and angle characteristics. These new functions (termed restricted mixed characteristics and restricted angle characteristics) complement the existing restricted point characteristic to form a complete set of point, mixed, and angle characteristics all dependent on four variables.