Abstract

The dioptric power of an optical system can be expressed as a four-component dioptric power matrix. We generalize and reformulate the standard matrix approach by utilizing the methods of Lie algebra. This generalization helps one deal with nonlinear problems (such as aberrations) and further extends the standard matrix formulation. Explicit formulas giving the relationship between the incident and the emergent rays are presented. Examples include the general case of thick and thin lenses. The treatment of a graded-index medium is outlined.

© 1998 Optical Society of America

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References

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  1. W. F. Harris, “Dioptric power: its nature and its representation in three and four dimensional space,” Optom. Vision Sci. 74, 349–366 (1997). This issue of the journal (June 1997) is a feature issue on visual optics and contains many related articles.
    [CrossRef]
  2. V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
    [CrossRef]
  3. V. Lakshminarayanan, S. Varadharajan, “Calculation of aberration coefficients: a matrix method,” in Basic and Clinical Applications of Vision Science, V. Lakshminarayanan, ed. (Kluwer, Dordrecht, The Netherlands, 1997), pp. 111–115.
  4. M. Kondo, Y. Takeuchi, “Matrix method for nonlinear transformation and its application to an optical system,” J. Opt. Soc. Am. A 13, 71–89 (1996).
    [CrossRef]
  5. A. J. Dragt, “Lie algebraic theory of geometrical optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
    [CrossRef]
  6. A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
    [CrossRef]
  7. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1986), pp. 105–157. This book contains an extensive overview of Lie group theory and applications in optics. See also the book edited by K. B. Wolf (Ref. 26) for related articles.
  8. See, for example, W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964); A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).
  9. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993). See also A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
  10. A. J. Dragt, “Nonlinear orbit dynamics,” in Physics of High Energy Particle Accelerators, R. A. Carrigan, ed., AIP Conference Proceedings 87 (American Institute of Physics, Woodbury, N.Y., 1982), pp. 147–313.
    [CrossRef]
  11. E. Forest, “Lie algebraic methods for charged particle beams and light optics,” Ph.D. dissertation (University of Maryland, College Park, Md., 1984).
  12. O. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).
  13. For a general treatment of symplectic methods, see V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).
  14. G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Medjias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 325–356.
  15. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  16. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
    [CrossRef]
  17. M. J. Bastianns, “ABCD law for partially coherent Gaussian light, propagating through first order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
    [CrossRef]
  18. W. F. Harris, “Ray vector fields, prismatic effect and thick astigmatic optical systems,” Optom. Vision Sci. 73, 418–423 (1996).
    [CrossRef]
  19. A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math Phys. (N.Y.) 17, 2215–2227 (1976).
    [CrossRef]
  20. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).
  21. A. J. Dragt, E. Forest, “Computation of non-linear behavior of Hamiltonian systems using Lie algebraic methods,” J. Math. Phys. (N.Y.) 24, 2734–2744 (1983).
    [CrossRef]
  22. G. Rangarajan, M. Sachidanand, “Spherical aberration and its correction using Lie algebraic techniques,” Pramana J. Phys. 49, 635–643 (1997).
    [CrossRef]
  23. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  24. D. A. Atchison, G. Smith, “Continuous gradient index and shell models of the human lens,” Vision Res. 35, 2529–2538 (1995).
    [CrossRef] [PubMed]
  25. A computer code, marylie 3.0, a program for charged particle beam transport based on Lie algebraic methods, has been developed by A. J. Dragt and his colleagues. For information contact A. Dragt, Dynamical Systems and Accelerator Theory Group, Department of Physics, University of Maryland, College Park, Maryland 20742-4111.
  26. P. W. Hawkes, “Lie methods in optics: an assessment,” in Lie Methods in Optics II, K. B. Wolf, ed., Vol. 352 of Springer Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1989), pp. 1–17.
    [CrossRef]
  27. C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997).
    [CrossRef]
  28. See, for example, W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966). See also P. Dodwell, Visual Pattern Recognition (Holt, Rinehart & Winston, New York, 1970). The Lie group approach has been used in the invariance coding problem by M. Ferraro, T. Caelli, “Relationship between integral transform invariances and Lie group theory,” J. Opt. Soc. Am. A 5, 738–742 (1988).
    [CrossRef]
  29. V. Lakshminarayanan, T. S. Santhanam, “Representation of rigid stimulus transformations by cortical activity patterns,” in Geometric Representations of Perceptual Phenomena, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Mahwah, N.J., 1995), pp. 61–69.

1997 (4)

G. Rangarajan, M. Sachidanand, “Spherical aberration and its correction using Lie algebraic techniques,” Pramana J. Phys. 49, 635–643 (1997).
[CrossRef]

W. F. Harris, “Dioptric power: its nature and its representation in three and four dimensional space,” Optom. Vision Sci. 74, 349–366 (1997). This issue of the journal (June 1997) is a feature issue on visual optics and contains many related articles.
[CrossRef]

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997).
[CrossRef]

1996 (2)

M. Kondo, Y. Takeuchi, “Matrix method for nonlinear transformation and its application to an optical system,” J. Opt. Soc. Am. A 13, 71–89 (1996).
[CrossRef]

W. F. Harris, “Ray vector fields, prismatic effect and thick astigmatic optical systems,” Optom. Vision Sci. 73, 418–423 (1996).
[CrossRef]

1995 (1)

D. A. Atchison, G. Smith, “Continuous gradient index and shell models of the human lens,” Vision Res. 35, 2529–2538 (1995).
[CrossRef] [PubMed]

1992 (1)

M. J. Bastianns, “ABCD law for partially coherent Gaussian light, propagating through first order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

1987 (1)

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

1985 (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

1983 (1)

A. J. Dragt, E. Forest, “Computation of non-linear behavior of Hamiltonian systems using Lie algebraic methods,” J. Math. Phys. (N.Y.) 24, 2734–2744 (1983).
[CrossRef]

1982 (1)

1976 (1)

A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math Phys. (N.Y.) 17, 2215–2227 (1976).
[CrossRef]

1970 (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

1966 (1)

See, for example, W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966). See also P. Dodwell, Visual Pattern Recognition (Holt, Rinehart & Winston, New York, 1970). The Lie group approach has been used in the invariance coding problem by M. Ferraro, T. Caelli, “Relationship between integral transform invariances and Lie group theory,” J. Opt. Soc. Am. A 5, 738–742 (1988).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

Atchison, D. A.

D. A. Atchison, G. Smith, “Continuous gradient index and shell models of the human lens,” Vision Res. 35, 2529–2538 (1995).
[CrossRef] [PubMed]

Bastianns, M. J.

M. J. Bastianns, “ABCD law for partially coherent Gaussian light, propagating through first order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Brouwer, W.

See, for example, W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964); A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993). See also A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).

Campbell, C.

C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997).
[CrossRef]

Dragt, A. J.

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

A. J. Dragt, E. Forest, “Computation of non-linear behavior of Hamiltonian systems using Lie algebraic methods,” J. Math. Phys. (N.Y.) 24, 2734–2744 (1983).
[CrossRef]

A. J. Dragt, “Lie algebraic theory of geometrical optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
[CrossRef]

A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math Phys. (N.Y.) 17, 2215–2227 (1976).
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1986), pp. 105–157. This book contains an extensive overview of Lie group theory and applications in optics. See also the book edited by K. B. Wolf (Ref. 26) for related articles.

A. J. Dragt, “Nonlinear orbit dynamics,” in Physics of High Energy Particle Accelerators, R. A. Carrigan, ed., AIP Conference Proceedings 87 (American Institute of Physics, Woodbury, N.Y., 1982), pp. 147–313.
[CrossRef]

Finn, J. M.

A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math Phys. (N.Y.) 17, 2215–2227 (1976).
[CrossRef]

Forest, E.

A. J. Dragt, E. Forest, “Computation of non-linear behavior of Hamiltonian systems using Lie algebraic methods,” J. Math. Phys. (N.Y.) 24, 2734–2744 (1983).
[CrossRef]

E. Forest, “Lie algebraic methods for charged particle beams and light optics,” Ph.D. dissertation (University of Maryland, College Park, Md., 1984).

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1986), pp. 105–157. This book contains an extensive overview of Lie group theory and applications in optics. See also the book edited by K. B. Wolf (Ref. 26) for related articles.

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

Guillemin, V.

For a general treatment of symplectic methods, see V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

Harris, W. F.

W. F. Harris, “Dioptric power: its nature and its representation in three and four dimensional space,” Optom. Vision Sci. 74, 349–366 (1997). This issue of the journal (June 1997) is a feature issue on visual optics and contains many related articles.
[CrossRef]

W. F. Harris, “Ray vector fields, prismatic effect and thick astigmatic optical systems,” Optom. Vision Sci. 73, 418–423 (1996).
[CrossRef]

Hawkes, P. W.

P. W. Hawkes, “Lie methods in optics: an assessment,” in Lie Methods in Optics II, K. B. Wolf, ed., Vol. 352 of Springer Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1989), pp. 1–17.
[CrossRef]

Hoffman, W. C.

See, for example, W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966). See also P. Dodwell, Visual Pattern Recognition (Holt, Rinehart & Winston, New York, 1970). The Lie group approach has been used in the invariance coding problem by M. Ferraro, T. Caelli, “Relationship between integral transform invariances and Lie group theory,” J. Opt. Soc. Am. A 5, 738–742 (1988).
[CrossRef]

Kondo, M.

Lakshminarayanan, V.

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

V. Lakshminarayanan, T. S. Santhanam, “Representation of rigid stimulus transformations by cortical activity patterns,” in Geometric Representations of Perceptual Phenomena, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Mahwah, N.J., 1995), pp. 61–69.

V. Lakshminarayanan, S. Varadharajan, “Calculation of aberration coefficients: a matrix method,” in Basic and Clinical Applications of Vision Science, V. Lakshminarayanan, ed. (Kluwer, Dordrecht, The Netherlands, 1997), pp. 111–115.

Mukunda, N.

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Nemes, G.

G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Medjias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 325–356.

Rangarajan, G.

G. Rangarajan, M. Sachidanand, “Spherical aberration and its correction using Lie algebraic techniques,” Pramana J. Phys. 49, 635–643 (1997).
[CrossRef]

Sachidanand, M.

G. Rangarajan, M. Sachidanand, “Spherical aberration and its correction using Lie algebraic techniques,” Pramana J. Phys. 49, 635–643 (1997).
[CrossRef]

Santhanam, T. S.

V. Lakshminarayanan, T. S. Santhanam, “Representation of rigid stimulus transformations by cortical activity patterns,” in Geometric Representations of Perceptual Phenomena, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Mahwah, N.J., 1995), pp. 61–69.

Simon, R.

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Smith, G.

D. A. Atchison, G. Smith, “Continuous gradient index and shell models of the human lens,” Vision Res. 35, 2529–2538 (1995).
[CrossRef] [PubMed]

Stavroudis, O.

O. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

Sternberg, S.

For a general treatment of symplectic methods, see V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

Sudarshan, E. C. G.

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Takeuchi, Y.

Varadharajan, S.

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

V. Lakshminarayanan, S. Varadharajan, “Calculation of aberration coefficients: a matrix method,” in Basic and Clinical Applications of Vision Science, V. Lakshminarayanan, ed. (Kluwer, Dordrecht, The Netherlands, 1997), pp. 111–115.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Wolf, K. B.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1986), pp. 105–157. This book contains an extensive overview of Lie group theory and applications in optics. See also the book edited by K. B. Wolf (Ref. 26) for related articles.

Bell Syst. Tech. J. (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

J. Math Phys. (N.Y.) (1)

A. J. Dragt, J. M. Finn, “Lie series and invariant functions for analytic symplectic maps,” J. Math Phys. (N.Y.) 17, 2215–2227 (1976).
[CrossRef]

J. Math. Phys. (N.Y.) (1)

A. J. Dragt, E. Forest, “Computation of non-linear behavior of Hamiltonian systems using Lie algebraic methods,” J. Math. Phys. (N.Y.) 24, 2734–2744 (1983).
[CrossRef]

J. Math. Psychol. (1)

See, for example, W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966). See also P. Dodwell, Visual Pattern Recognition (Holt, Rinehart & Winston, New York, 1970). The Lie group approach has been used in the invariance coding problem by M. Ferraro, T. Caelli, “Relationship between integral transform invariances and Lie group theory,” J. Opt. Soc. Am. A 5, 738–742 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nucl. Instrum. Methods Phys. Res. A (1)

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

Opt. Acta (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Quantum Electron. (1)

M. J. Bastianns, “ABCD law for partially coherent Gaussian light, propagating through first order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

Optom. Vision Sci. (4)

W. F. Harris, “Ray vector fields, prismatic effect and thick astigmatic optical systems,” Optom. Vision Sci. 73, 418–423 (1996).
[CrossRef]

C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997).
[CrossRef]

W. F. Harris, “Dioptric power: its nature and its representation in three and four dimensional space,” Optom. Vision Sci. 74, 349–366 (1997). This issue of the journal (June 1997) is a feature issue on visual optics and contains many related articles.
[CrossRef]

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

Pramana J. Phys. (1)

G. Rangarajan, M. Sachidanand, “Spherical aberration and its correction using Lie algebraic techniques,” Pramana J. Phys. 49, 635–643 (1997).
[CrossRef]

Vision Res. (1)

D. A. Atchison, G. Smith, “Continuous gradient index and shell models of the human lens,” Vision Res. 35, 2529–2538 (1995).
[CrossRef] [PubMed]

Other (14)

A computer code, marylie 3.0, a program for charged particle beam transport based on Lie algebraic methods, has been developed by A. J. Dragt and his colleagues. For information contact A. Dragt, Dynamical Systems and Accelerator Theory Group, Department of Physics, University of Maryland, College Park, Maryland 20742-4111.

P. W. Hawkes, “Lie methods in optics: an assessment,” in Lie Methods in Optics II, K. B. Wolf, ed., Vol. 352 of Springer Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1989), pp. 1–17.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Heidelberg, 1986), pp. 105–157. This book contains an extensive overview of Lie group theory and applications in optics. See also the book edited by K. B. Wolf (Ref. 26) for related articles.

See, for example, W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964); A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993). See also A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).

A. J. Dragt, “Nonlinear orbit dynamics,” in Physics of High Energy Particle Accelerators, R. A. Carrigan, ed., AIP Conference Proceedings 87 (American Institute of Physics, Woodbury, N.Y., 1982), pp. 147–313.
[CrossRef]

E. Forest, “Lie algebraic methods for charged particle beams and light optics,” Ph.D. dissertation (University of Maryland, College Park, Md., 1984).

O. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

For a general treatment of symplectic methods, see V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Medjias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 325–356.

V. Lakshminarayanan, S. Varadharajan, “Calculation of aberration coefficients: a matrix method,” in Basic and Clinical Applications of Vision Science, V. Lakshminarayanan, ed. (Kluwer, Dordrecht, The Netherlands, 1997), pp. 111–115.

V. Lakshminarayanan, T. S. Santhanam, “Representation of rigid stimulus transformations by cortical activity patterns,” in Geometric Representations of Perceptual Phenomena, R. D. Luce, M. D’Zmura, D. Hoffman, G. Iverson, A. K. Romney, eds. (Erlbaum, Mahwah, N.J., 1995), pp. 61–69.

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Figures (3)

Fig. 1
Fig. 1

Coordinate system used in the analysis.

Fig. 2
Fig. 2

Generalized interface between two media. See text for details.

Fig. 3
Fig. 3

Simple lens of thickness t and of uniform refractive index.

Equations (62)

Equations on this page are rendered with MathJax. Learn more.

ζf=Mζi
ζaf=Ka+bRabζbi+bcTabcζbiζci+
(a, b=1, 2,  2n),
Ma,b=ζafζbi.
J=0I-I0,
MTJM=J,
ρ=Sρ0,
S=ABCD,
F=-C.
αi=ndyidz.
ds=[(dz)2+(dy1)2+(dy2)2]1/2=[1+(y1)2+(y2)2]1/2dz,
yi=dyidz=αin.
A=zizfnds=zizfn[1+(y1)2+(y2)2]1/2dz.
L=n[1+(y1)2+(y2)2]1/2,
ddzLyi-Lyi=0(i=1, 2).
pi=Lyi=nαi(n2+α12+α22)1/2.
H=-[n2(r)-p12-p22]1/2,
{f, g}=ifqigpi-fpigqi.
:f:g={f, g}.
:f:2g={f, {f, g}},
:f:0g=g,
exp(:f:)=n=0:f:nn!,
exp(:f:)g=g+{f, g}+12{f, {f, g}}+13!{f, {f, {f, g}}}+.
M=exp(:f2:)exp(:f3:)exp(:f4:),
f2=-l2n(α12+α22)(n2+α12+α22)=-l2n(p12+p22)
qf=Mqi=exp(-1/2n:p2:)qi=qi+l/npi,
pf=Mpi=pi.
f4=A(p2)2+Bp2(p·q)+C(p·q)2+Dp2q2+E(p·q)q2+F(q2)2.
f=A(p2)2,
qαf=qαi-4Apαipi2,
pαf=pαi.
f=Bp2(p·q),
qαf=qαi-B[2pαi(pi·qi)+pi2qαi]+B22![4pi2(pi·qi)pαi-pi4qαi],
pαf=pαi+Bpi2pαi+B22!3pi4pαi.
f=C(p·q)2,
qαf=qαi-2C(pi·qi)qαi+2C2(pi·qi)2qαi,
pαf=pαi+2C(pi·qi)pαi+2C2(pi·qi)2pαi.
f=Dp2q2,
qαf=qαi-2Dqi2pαi-2D2[pi2qi2qαi-2qi2(pi·qi)pαi],
pαf=pαi+2Dpi2qαi+2D2pi2[2(pi·qi)qαi-qi2pαi].
f=E(p·q)q2,
qαf=qαi-Eqαiqi2+3E22!qi4qαi,
pαf=pαi+E[pαiqi2+2qαi(pi·qi)]+E22![4qi2(qi·pi)qαi-qi4pαi].
f=F(q2)2,
qαf=qαi,
pαf=pαi+4Fqαiqi2.
zs=β2(q12+q22)+β4(q12+q22)2+.
n(z)=n1+(n2-n1)Ө[z-β2(q12+q22)-β4(q12+q22)2-],
dMdz=M:-H(ζi, z):.
M=Pexp-:zizH(ζi, z)dz:,
M=e:f2:e:f4:,
f2=β2(n2-n1)q2,
f4=-β23[(n1-n2)/n1]{n1[2-(β4/β23)]-2n2}(q2)2+2β22[(n1-n2)/n1]q2(p·q)+β2[(n1-n2)/2n1n2]q2p2.
n(z)=n0(z)+n2(z)(q12+q22)+n4(z)(q12+q22)2+
z=z1(q)=γ2(q12+q22)+γ4(q12+q22)2+,
z=z2(q)=β2(q12+q22)+β4(q12+q22)2+.
M=e:g4:eβ2(1-n):q2:e-[t/(2n)]:p2:e-[t/(8n3)]:(p2)2:×eγ2(1-n):q2:e:f4:,
M=eβ2(1-n):q2:e-(t/2n):p2:eγ2(1-n):q2:.
qfqf=Mqiqi=Sqiqi,
S=ABCD=1+2γ2(1-n)t/nt/n2(1-n)[(β2+γ2)+2(1-n)γ2tβ2/n]1+2β2(1-n)t/n.
S=1+(n-1)t/nR1t/n(1-n)1R1-1R2+(n-1)tnR1R21-(n-1)t/nR2.
F=-C=1/f=(n-1)[(1/R1-1/R2)+(n-1)t/nR1R2],

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