9 - Integral Equations

Abstract

This chapter illustrates a general theory of field calculation in systems having a straight optic axis. Two basic ideas, the introduction of azimuthally Fourier series expansions and the formulation of integral equations, have been described in detail, since these are particularly well adapted to the needs of electron optical field calculations. The use of Fourier series expansions results in a sequence of uncoupled mathematical structures of lower dimensions. Since finally interest is in the field of the paraxial domain, the calculation of the Fourier coefficients (axial harmonics) are terminated after the first few orders, which are of most importance. This is thus an economic technique. The use of integral equations rather than partial differential equations further reduces the number of dimensions, since parts of the necessary integrations have already been carried out. In all cases in which the material properties of the pole pieces or electrodes are constant, integral equation methods have proved to be very powerful and efficient.