Abstract
Many important results have now been obtained in particle optics by means of the Lie algebraic methods. We compare these methods with older procedures based on the use of characteristic functions and attempt to bring out the advantages and shortcomings of the various approaches. In particular, we discuss the interrelations between aberration coefficients, concatenation, and the role of computer algebra.
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K.-c. Ai and M. Szilagyi, Fifth-order relativistic aberration theory for electromagnetic focusing systems incluiding spherical cathode lenses, Optik 79, 33–40 (1988).
M. Berz, The method of power series tracking for the mathematical description of beam dynamics, Nucl. Instrum. Meth. Phys. Res. A 258, 431–436 (1987).
M. Berz, Differential algebraic description of beam dynamics to very high orders, Lawrence Berkeley Laboratory Report SSC-152 (1988).
M. Berz and H. Wollnik, The program Hamilton for the analytic solution of the equations of motion through fifth order, Nucl. Instrum. Meth. Phys. Res. A 258, 364–373 (1987).
M Berz, H.C. Hoffman and H. Wollnik, Cosy 5.0 —the fifth order code for corpuscular systems, Nucl. Instrum. Meth. Phys. Res. A 258, 402–406 (1987).
M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1959; 6th ed., 1980).
L. de Broglie, Optique Electronique et Corpuslaire (Hermann, Paris, 1950).
W. Brouwer, The use of matrix algebra in geometrical optics. Dissertation, Delft (1957).
W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York and Amsterdam, 1964).
H.A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, Oxford, 1954 and Dover, New York, 1968).
H.A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, Cambridge, 1970).
O. Castaños, E. López-Moreno, and K.B. Wolf, Canonical transformations for paraxial wave optics, in Lie Methods in Optics, J. Sánchez-Mondragón and K.B. Wolf, Eds. (Springer-Verlag, Berlin, 1986), pp. 159–182.
A.V. Crewe and D. Kopf, Limitations of sextupole correctors, Optik 56, 391–399 (1980).
A.J. Dragt, Lie algebraic theory of geometrical optics and optical aberrations, J. Opt. Soc. Am. 72, 372–379 (1982).
A.J. Dragt, Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics, Nucl. Instrum. Meth. Phys. Res. A 258, 339–354 (1987).
A.J. Dragt and E. Forest, Lie algebraic theory of charged-particle optics and electron microscopes, Advances in Electronics and Electron Physics 67, 65–120 (1986).
A.J. Dragt, E. Forest and K.B. Wolf, Foundations of a Lie algebraic theory of geometrical optics, in Lie Methods in Optics, J. Sánchez-Mondragón and K.B. Wolf, Eds. (Springer-Verlag, Berlin, 1986), pp. 105–157.
A.J. Dragt, F. Neri, G. Rangarajan, D.R. Douglas, L.M. Healy, and R.D. Ryne, Lie algebraic treatment of linear and nonlinear beam dynamics, Ann. Rev. Nucl. Part. Sci. 38, 455–496 (1988).
J. Focke, Die Realisierung der Herzbergerschen Theorie der Fehler fünfter Ordnung in rotationssymmetrischen Systemen mit einer Anwendung auf das Schmidtsche Spiegelteleskop, Jenaer Jahrbuch 89–119 (1951).
J. Focke, Higher order aberration theory, Progress in Optics 4, 1–36 (1965).
A. Foster, Correction of aperture aberrations in magnetic lens spectrometers. Thesis, London (1968).
W. Glaser, Grundlagen der Electronenoptik (Springer-Verlag, Vienna, 1952).
P.W. Hawkes, The geometrical aberrations of general electron optical systems II. The primary (third order) aberrations of orthogonal systems, and the secondary (fifth order) aberrations of round systems, Phil. Trans. Roy. Soc. London A 257, 523–552 (1965).
P.W. Hawkes, Quadrupole Optics (Springer-Verlag, Berlin, 1966).
P.W. Hawkes, Asymptotic aberration coefficients, magnification and object position in electron optics, Optik 27, 287–304 (1968).
P.W. Hawkes, Asymptotic aberration integrals for round lenses, Optik 31, 213–219 (1970).
P.W. Hawkes, Asymptotic aberration integrals for quadrupole systems, Optik 31, 302–314 (1970).
P.W. Hawkes, The addition of round electron lens aberrations, Optik 31, 592–599 (1970).
P.W. Hawkes, The addition of quadrupole aberrations, Optik 32, 5060 (1970).
P.W. Hawkes, Computer calculation of formulae for electron lens aberration coefficients, Optik 48, 29–51 (1977).
P.W. Hawkes, Lie algebraic theory of geometrical optics and optical aberrations: a comment, J. Opt. Soc. Am. 73, 122 (1983).
P.W. Hawkes, Aberration structure, Nucl. Instrum. Meth. Phys. Res. A 258, 462–465 (1987).
P.W. Hawkes and E. Kasper, Principles of Electron Optics, 2 vols. (Academic Press, London and San Diego, 1989).
A.C.S. van Heel, Calcul des aberrations du cinquième ordre et projects tenant compte de ces aberrations, in La Théorie des Images Optiques, P. Fleury, Ed. (Editions Revue d'Optique, Paris, 1949), pp. 32–67.
A.C.S. van Heel, Inleiding in de Optica (Martinus Nijhoff, The Hague, 1964).
Y. Li and W.-x. Ni, Relativistic fifth order geometrical aberrations of a focusing system, Optik 78, 45–47 (1988).
H. Marx, Theorie der geometrisch-optischen Bildfehler, in Handbuch der Physik, S. Flügge, Ed. (Springer, Berlin & New York, 1967), Vol. 29, Optische Instrumente, pp. 68–191.
P. Meads, The theory of aberrations of quadrupole focusing arrays. Thesis, University of California (1963); UCRL-10807.
R. Meinel and S. Thiem, A complete set of independent asymptotic aberration coefficients for deflection magnets, Optik 74, 1–2 (1986).
R. J. Regis, The modern development of Hamiltonian optics, Progress in Optics 1, 1–29 (1961).
H. Rose, Über die Berechnung der Bildfehler elektronenoptischer Systeme mit gerader Achse, Optik 27, 466–474 and 497–514 (1968).
H. Rose, Der Zusammenhang der Bildfehler-Koeffizienten mit den Entwicklungs-Koeffizienten des Eikonals, Optik 28, 462–274 (1968/9).
H. Rose, Correction of aperture aberrations in magnetic systems with threefold symmetry, Nucl. Instr. Meth. 187, 187–199 (1981).
H. Rose, Hamiltonian magnetic optics, Nucl. Instrum. Meth. Phys. Res. A 258, 374–406 (1987).
H. Rose and U. Petri, Zur systematischen Berechnung elektronenoptischer Bildfehler, Optik 33, 151–165 (1971).
J. Sänchez-Mondragón and K.B. Wolf, Eds., Lie Methods in Optics, (Springer-Verlag, Berlin, 1986); Lecture Notes in Physics Vol. 250.
W. Schempp, Analog radar signal design and digital signal processing — a Heisenberg nilpotent Lie group approach, in Lie Methods in Optics, J. Sánchez-Mondragón and K.B. Wolf, Eds. (Springer-Verlag, Berlin, 1986), pp. 1–27.
T. Smith, The changes in aberrations when the object and stop are moved, Trans. Opt. Soc. London 23, 311–322 (1921/2).
T. Smith, The addition of aberrations, Trans. Opt. Soc. London 25, 177–199 (1923/4).
T. Soma, Relativistic aberration formulas for combined electricmagnetic focusing-deflection system, Optik 49, 255–262 (1977).
S. Steinberg, Lie series, Lie transformations, and their applications, in Lie Methods in Optics, J. Sánchez-Mondragón and K.B. Wolf, Eds. (Springer-Verlag, Berlin, 1986), pp. 45–103.
W.G. Stephan, Practische Toepassingen op het Gebied der algebraische Optica. Dissertation, Delft (1947).
P.A. Sturrock, Perturbation characteristic functions and their application to electron optics, Proc. Roy. Soc. London A 210, 269–289 (1951).
P.A. Sturrock, Static and Dynamic Electron Optics (Cambridge University Press, Cambridge, 1955).
C.H.F. Velzel and J.L.F. de Meijere, Characteristic functions and the aberrations of symmetric optical systems. 1. Transverse aberrations when the eikonal is given. 11. Addition of aberrations, J. Opt. Soc. Am. A 5, 246–250 and 251–256 (1988).
J.L. Verster, On the use of gauzes in electron optics, Philips Res. Repts. 18, 465–605 (1963).
H. Wollnik, Optics of Charged Particles (Academic Press, Orlando & London, 1987).
H. Wollnik and M. Berz, Relations between elements of transfer matrices due to the condition of symplecticity, Nucl. Instrum. Meth. Phys. Res. A 238, 127–140 (1985).
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Hawkes, P.W. (1989). Lie methods in optics: An assessment. In: Wolf, K.B. (eds) Lie Methods in Optics II. Lecture Notes in Physics, vol 352. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0012742
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DOI: https://doi.org/10.1007/BFb0012742
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