Gaussian Optics

The electromagnetic field forms an inhomogeneous and anisotropic medium of refraction for charged particles in the most general case. Hence, in the terminology of light optics, electron lenses are gradient-index lenses. The anisotropy results from the magnetic force, which depends on the direction of the particle velocity. Therefore, only electrostatic systems have an isotropic index of refraction. Since all realistic electromagnetic fields are inhomogeneous, the equations for the x- and y-coordinates of a trajectory form a system of two coupled nonlinear differential equations. The solutions of such systems are very involved and exhibit chaotic behavior in many cases. The deleterious effect of the nonlinear terms of the forces remains sufficiently small if the particle beam is confined to the vicinity of the axis, which may be straight or curved. One achieves this in practice by means of apertures, which remove particles with large ray gradients from the beam.

Paraxial conditions prevail approximately if the diameter of the beam within the region of the external fields stays smaller than about one fifth of the diameters of the inner faces of the electrodes and/or magnetic pole pieces. In this case, we can describe the propagation of the particles with a sufficient degree of accuracy by neglecting the nonlinear terms in the trajectory equations. The famous mathematician C.F. Gauss has first introduced this paraxial or Gaussian approximation in light optics. His approximation only considers terms up to the second order in the expansion of the variational function (3.60) with respect to the complex coordinates w, w and their derivatives. The resulting path equations are then two complex second-order linear differential equations whose general solutions are linear combinations of four arbitrary linearly independent particular solutions of the system. This behavior enables one to describe the optical properties of various elements in a simple way by characteristic quantities such as focal length, principal planes, etc. At the beginning of electron optics, one has explored primarily the paraxial properties and the aberrations of axially symmetric systems [60,61]. Later, Melkich [62] investigated the paraxial properties of quadrupole lenses, which became important elements of accelerators due to their strong focusing properties [63]. In electron optics, quadrupoles have first been introduced as stigmators [64] and later as substitutes for axially symmetric lenses. Kawakatsu et al. [65] developed a quadrupole quadruplet, which he substituted for the projector system of an electron microscope. Bauer [66] designed and tested experimentally a quadrupole objective compound lens consisting of a symmetric quadrupole triplet and an antisymmetric doublet. The experiments proved the feasibility of the lens for paraxial imaging. However, the resolution was rather poor due to the extremely large spherical aberration of the lens. In 1947, Scherzer demonstrated that one can eliminate the unavoidable aberrations of round lenses by introducing quadrupoles and octopoles into the system [67]. This finding initiated extensive studies on the properties of quadrupole—octopole systems [68–70]. Hawkes [71] has summarized the results of these investigations in his book on quadrupole optics.