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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 5, Iss. 8 — Aug. 1, 1988
  • pp: 1226–1232

Symmetry-adapted classification of aberrations

Kurt Bernardo Wolf  »View Author Affiliations


JOSA A, Vol. 5, Issue 8, pp. 1226-1232 (1988)
http://dx.doi.org/10.1364/JOSAA.5.001226


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Abstract

Optical systems produce canonical transformations on phase space that are nonlinear. When a power expansion of the coordinates is performed around a chosen optical axis, the linear part is the paraxial approximation, and the nonlinear part is the ideal of aberrations. When the optical system has axial symmetry, its linear part is the symplectic group Sp(2, R) represented by 2 × 2 matrices. It is used to provide a classification of aberrations into multiplets of spin that are irreducible under the group, in complete analogy with the quantum harmonic-oscillator states. The “magnetic” axis of the latter may be chosen to adapt to magnifying systems or to optical fiberlike media. There seems to be a significant computational advantage in using the symplectic classification of aberrations.

© 1988 Optical Society of America

Citation
Kurt Bernardo Wolf, "Symmetry-adapted classification of aberrations," J. Opt. Soc. Am. A 5, 1226-1232 (1988)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-5-8-1226


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References

  1. J. Sánchez-Mondragón and K. B. Wolf, eds., Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986). [CrossRef]
  2. A. J. Dragt, "Lie algebraic theory of geometrical optics and optical aberrations," J. Opt. Soc. Am. 72, 372–379 (1982). [CrossRef]
  3. V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).
  4. A. J. Dragt, E. Forest, and K. B. Wolf, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón and K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 105–157. [CrossRef]
  5. T. Sekiguchi and K. B. Wolf, "The Hamiltonian formulation of optics," Am. J. Phys. 55, 830–835 (1987). [CrossRef]
  6. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959).
  7. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).
  8. A. J. Dragt, "Lectures on Nonlinear Orbit Dynamics," AIP Conf. Proc. 87 (1982). [CrossRef]
  9. A. J. Dragt and J. Finn, "Lie series and invariant functions for analytic symplectic maps," J. Math. Phys. 17, 2215–2227 (1976); see also A. J. Dragt and E. Forest, "Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods," J. Math. Phys. 24, 2734–2744 (1983). [CrossRef]
  10. K. B. Wolf, "Symmetry in Lie optics," Ann. Phys. 172, 1–25 (1986). [CrossRef]
  11. K. B. Wolf, "On time-dependent quadratic quantum Hamiltonians," SIAM J. Appl. Math. 40, 419–431 (1981). [CrossRef]
  12. S. Steinberg, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón and K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 45–103. [CrossRef]
  13. M. Navarro-Saad and K. B. Wolf, "Factorization of the phase-space transformation produced by an arbitrary refracting surface," J. Opt. Soc. Am. A 3, 340–346 (1986). [CrossRef]
  14. See Ref. 1, App. A.
  15. H. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).
  16. This aberration coefficient appears as P6 , the missing coefficient, in Ref. 11, Sec. 121, and is treated in Sec. 125 with some incidental remarks on duality (i.e., Fourier conjugation).
  17. M. Navarro-Saad and K. B. Wolf, "The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order," J. Math. Phys. 27, 1449–1457 (1986). [CrossRef]
  18. L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981), Sec. 6.16.
  19. K. B. Wolf, "The group-theoretical treatment of aberrating systems. II. Axis-symmetric inhomogeneous systems and fiber optics in third aberration order," J. Math. Phys. 27, 1458–1465 (1986). [CrossRef]
  20. V. Bargmann, "On a Hilbert space of analytic functions and an associated integral transform," Commun. Pure Appl. Math. 14, 187–214 (1961). [CrossRef]
  21. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
  22. M. Navarro-Saad and K. B. Wolf, "Applications of a factorization theorem for ninth-order aberration optics," J. Symbolic Comp. 1, 235–239 (1985). [CrossRef]
  23. 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). [CrossRef]
  24. K. B. Wolf, "The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations," J. Math. Phys. 28, 2498–2507 (1987). [CrossRef]

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