1. There are many references describing the customary approach to electron microscopes and ion optics. In the area of electron microscopes they include A. B. El-Kareh and J. C. El-Kareh, “Electron Beams, Lenses, and Optics,” 2 vols. Academic Press, New York, 1970; W. Glaser, “Grundlagen der Elektronenoptik.” Springer-Verlag, Berlin and New York, 1952; P. Grivet and A. Septier, “Electron Optics,” 2nd ed. Pergamon, Oxford, 1972; P. W. Hawkes, “Electron Optics and Electron Microscopy.” Taylor Francis, London, 1972; in “Quadrupole Optics” (G. Hohler, ed.), Springer Tracts Mod. Phys., Vol. 42. Springer-Verlag, Berlin and New York, 1966; “Quadrupoles in Electron Lens Design.” Academic Press, New York, 1969; P. W. Hawkes, ed., “Magnetic Electron Lenses.” Springer-Verlag, Berlin and New York, 1982; “Image Processing and Computer-Aided Design in Electron Optics.” Academic Press, New York, 1973; O. Klemperer and M. E. Barnett, “Electron Optics,” 3rd ed. Cambridge Univ. Press, London and New York, 1971; T. Mulvey and M. J. Wallington, “Rep. Prog. Phys. 36, 347 (1973); H. Rose and U. Petri, Optik 33, 151 (1971); J. C: H. Spence, “Experimental High Resolution Electron Microscopy.” Oxford Univ. Press (Clarendon), London and New York, 1981; P. A. Sturrock, Proc. R. Soc. London, Ser. A 210, 269 (1951); “Static and Dynamic Electron Optics.” Cambridge Univ. Press, London and New York, 1955. In the area of electron microscopes, microprobes, and other optical systems, references include V. E. Cosslett, “Introduction to Electron Optics.” Oxford Univ. Press (Clarendon), London and New York, 1950; G. W. Grime and F. Watt, “Beam Optics of Quadrupole Probe-Forming Systems.” Adam Hilger Ltd., Bristol, 1984; E. Harting and F. H. Read, “Electrostatic Lenses.” Elsevier, Amsterdam, 1976; A. Septier, ed., “Focusing of Charged Particles,” 2 vols. Academic Press, New York, 1967; “Applied Charged Particle Optics,” Parts A and B. Academic Press, New York, 1980; H. Wollnik, ed., “Charged Particle Optics.” North-Holland Publ., Amsterdam, 1981; Ji-ye Ximen, “Principles of Electron and Ion Optics and an Introduction to Aberration Theory” (revised by A. V. Crewe). Enrico Fermi Institute, University of Chicago, Chicago, Illinois, 1984 (book in preparation); V. K. Zworykin, G. A. Morton, E. G. Ramberg, E. G. Hillier, and A. W. Vance, “Electron Optics and the Electron Microscope.” J. Wiley, New York, 1945.
2. Lectures on nonlinear orbit dynamics;Dragt,1976
3. Classical Mechanics,;Goldstein,1950
4. Note that this is indeed a “definition” Had we employed the usual Poisson bracket relations for the quantities on the right-hand side of Eq. (8), then we would have found that the Poisson brackets of the quantities on the left-hand side of Eq. (8) would involve factors of 1/pa
5. Note that the order of an aberration is one unit lower than the degree of its associated polynomial.