Time-dependent perturbation formalism for calculating the aberrations of systems with large ray gradients

A time-dependent perturbation formalism is outlined which enables the calculation of aberrations for systems with large gradients of the particle trajectories in a consistent way. The position of a particle is referred to that of an axial reference particle making the axial coordinate a small quantity. The initial conditions of the trajectory are incorporated by transforming the Lorentz equations in a set of coupled inhomogeneous integral equations for the three positional coordinates of the particle. The integral equations are solved by an iteration procedure which starts from the paraxial approximation. The number of required iteration steps is equal to the rank of the aberration minus one. As an example, the primary and secondary axial aberrations of mirrors are investigated in detail.