General Curvilinear Systems

51 General Curvilinear Systems The feature of particle optics that distinguishes it from particle dynamics in general is the family behaviour of trajectories that remain in the vicinity of some plane or skew curve in space, which we call the optic axis. In the simplest case, this axis is a straight line but there are many devices, spectrometers with magnetic deflecting prisms for example, in which it is a plane curve, often a circular arc. Optic axes in the form of a skew curve are rare but cannot be excluded completely. It is clearly advantageous to match the choice of coordinates to the optic axis. Numerous attempts to establish and study the equations of electron optics in the most general case of an arbitrary skew axis have been made but, as we might expect, the equations rapidly become unmanageable and it is possible to draw only very general conclusions. The analogous situation in which light propagates through a medium with an arbitrarily variable refractive index was studied in some detail by the mathematical ophthalmologist Gullstrand, who established the various categories of firstorder behaviour (e.g. Gullstrand, 1900, 1906, 1915, 1924), and later by Carath6odory (1937), who used mathematical tools more familiar to the reader of today. In electron optics, the paraxial equations for a general system were first explored briefly by Cotte (1938), most of whose work was concerned with systems in which the axis is a plane curve. In 1941 and 1943, the nature of the paraxial imagery to be expected from the various types of systems--general, orthogonal and rotationally symmetric--was discussed by MacColl. Meanwhile, Grinberg (1942, 1943) published his "General theory of the focusing action of electrostatic and magnetostatic fields", using the trajectory method; this work was subsequently re-examined and completed by numerous Russian authors, in particular by Strashkevich and Pilat (1951, 1952) and Strashkevich and Gluzman (1954), who attempted to extend Grinberg's work to include primary aberrations in their calculation of the second-order aberrations (cf. Section 51.6). Some points in Grinberg's work were criticized by Kas'yankov, who devoted an entire book to systems with curved axes (1956), and a lively polemic ensued. A very full study of the aberration question was made at about the same period by Vandakurov. On these matters, see Tsukkerman (1954), Kas'yankov (1956, 1957, 1958a,b), Grinberg (1957a,b) and Vandakurov (1955, 1957), and the textbooks of Strashkevich (1959) and Kel'man and Yavor (1959,

968).