Abstract

The set of paraxial optical systems is the manifold of the group of symplectic matrices. The structure of this group is nontrivial: It is not simply connected and is not of an exponential type. Our analysis clarifies the origin of the metaplectic phase and the inherent limitations for optical map fractionalization. We describe, for the first time to our knowledge, an image girator and a cross girator whose geometric and wave implementations are of interest.

© 2000 Optical Society of America

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  1. K. Iwasawa, “On the representations of Lie algebras,” Jpn. J. Math. 19, 513–523 (1948); “On some types of topological groups,” Ann. Math. 50, 507–558 (1948).
  2. V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947);“Group representation in Hilbert spaces of analytic functions,” in Analytical Methods in Mathematical Physics, P. Gilbert, R. G. Newton, eds. (Gordon & Breach, New York, 1970), pp. 27–63.
    [CrossRef]
  3. K. B. Wolf, “The symplectic groups, their parametrization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), App. A, pp. 227–238; “Representations of the algebra sp(2, R),” ibid., pp. 239–247; reprinted in Dynamical Groups and Spectrum Generating Algebras, A. Barut, A. Bohm, Y. Ne’eman, eds. (World Scientific, Singapore, 1989), pp. 1076–1087, 1088–1096.
  4. See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).
  5. R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
    [CrossRef]
  6. R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, V. B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689; G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994);K. Sundar, N. Mukunda, R. Simon, “Coherent mode decomposition of general anisotropic Gaussian Schell model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995);R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
    [CrossRef]
  7. R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
    [CrossRef] [PubMed]
  8. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
  9. D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
    [CrossRef]
  10. G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for squeezed states,” Opt. Commun. 100, 411–414 (1993).
    [CrossRef]
  11. R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
    [CrossRef] [PubMed]
  12. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  13. B. Dutta, N. Mukunda, R. Simon, A. Subramaniam, “Squeezed states, photon number distribution, and U(1) invariance,” J. Opt. Soc. Am. B 10, 253–264 (1993);R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  14. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985); Arvind, B. Dutta, N. Mukunda, R. Simon, “Two-mode quantum systems: invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609–1620 (1995).
    [CrossRef] [PubMed]
  15. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realisation of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  16. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
    [CrossRef]
  17. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their implementations. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993);“Fractional Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993);“Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–750 (1995);D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [CrossRef]
  18. J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
    [CrossRef]
  19. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989);“The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
    [CrossRef] [PubMed]
  20. W. Hamilton, Lectures on Quaternions (Dublin, 1853). An excellent review of Hamilton’s theory of turns for SU(2) can be found in L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Mechanics, Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, Reading, Mass., 1881).
  21. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).
  22. A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980); H. O. Bartelt, K.-H. Brenner, H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980);H. Bartelt, K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260–262 (1980); K.-H. Brenner, H. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  23. R. Simon, N. Mukunda, “Twisted Gaussian Schell model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993);“Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
    [CrossRef]
  24. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  25. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana J. Phys. 32, 769–792 (1989).
    [CrossRef]
  26. R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990);V. Bagini, R. Borghi, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
    [CrossRef]
  27. H. Weyl, The Theory of Groups and Quantum Mechanics, 2nd ed. (Dover, New York, 1930).
  28. M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
    [CrossRef]
  29. O. Castaños, E. López-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182.
  30. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), Chap. 4, pp. 105–158.
    [CrossRef]
  31. K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987);“Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
    [CrossRef]
  32. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]
  33. N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996);A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
    [CrossRef]

1998 (1)

1996 (2)

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996);A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

1995 (1)

1994 (2)

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

1993 (6)

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for squeezed states,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their implementations. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their implementations. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993);“Fractional Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993);“Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–750 (1995);D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
[CrossRef]

B. Dutta, N. Mukunda, R. Simon, A. Subramaniam, “Squeezed states, photon number distribution, and U(1) invariance,” J. Opt. Soc. Am. B 10, 253–264 (1993);R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993);“Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

1990 (1)

R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990);V. Bagini, R. Borghi, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

1989 (2)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana J. Phys. 32, 769–792 (1989).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989);“The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef] [PubMed]

1987 (1)

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987);“Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
[CrossRef]

1985 (2)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985); Arvind, B. Dutta, N. Mukunda, R. Simon, “Two-mode quantum systems: invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609–1620 (1995).
[CrossRef] [PubMed]

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

1984 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1980 (1)

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980); H. O. Bartelt, K.-H. Brenner, H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980);H. Bartelt, K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260–262 (1980); K.-H. Brenner, H. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1948 (1)

K. Iwasawa, “On the representations of Lie algebras,” Jpn. J. Math. 19, 513–523 (1948); “On some types of topological groups,” Ann. Math. 50, 507–558 (1948).

1947 (1)

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947);“Group representation in Hilbert spaces of analytic functions,” in Analytical Methods in Mathematical Physics, P. Gilbert, R. G. Newton, eds. (Gordon & Breach, New York, 1970), pp. 27–63.
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Agarwal, G. S.

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for squeezed states,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

Atakishiyev, N. M.

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996);A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Bargmann, V.

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947);“Group representation in Hilbert spaces of analytic functions,” in Analytical Methods in Mathematical Physics, P. Gilbert, R. G. Newton, eds. (Gordon & Breach, New York, 1970), pp. 27–63.
[CrossRef]

Castaños, O.

O. Castaños, E. López-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182.

Chumakov, S. M.

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996);A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Cohen, N.

Dragt, A. J.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), Chap. 4, pp. 105–158.
[CrossRef]

Dutta, B.

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), Chap. 4, pp. 105–158.
[CrossRef]

Friberg, A. T.

Gilmore, R.

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).

Goldstein, H.

See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).

Hamilton, W.

W. Hamilton, Lectures on Quaternions (Dublin, 1853). An excellent review of Hamilton’s theory of turns for SU(2) can be found in L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Mechanics, Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, Reading, Mass., 1881).

Iwasawa, K.

K. Iwasawa, “On the representations of Lie algebras,” Jpn. J. Math. 19, 513–523 (1948); “On some types of topological groups,” Ann. Math. 50, 507–558 (1948).

James, D. F. V.

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Lohmann, A.

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980); H. O. Bartelt, K.-H. Brenner, H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980);H. Bartelt, K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260–262 (1980); K.-H. Brenner, H. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

López-Moreno, E.

O. Castaños, E. López-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182.

Mendlovic, D.

Moshinsky, M.

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Twisted Gaussian Schell model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993);“Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

B. Dutta, N. Mukunda, R. Simon, A. Subramaniam, “Squeezed states, photon number distribution, and U(1) invariance,” J. Opt. Soc. Am. B 10, 253–264 (1993);R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990);V. Bagini, R. Borghi, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana J. Phys. 32, 769–792 (1989).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989);“The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985); Arvind, B. Dutta, N. Mukunda, R. Simon, “Two-mode quantum systems: invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609–1620 (1995).
[CrossRef] [PubMed]

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

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, V. B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689; G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994);K. Sundar, N. Mukunda, R. Simon, “Coherent mode decomposition of general anisotropic Gaussian Schell model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995);R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

Ozaktas, H. M.

Quesne, C.

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[CrossRef]

Rivera, A. L.

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996);A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Sahin, A.

Shamir, J.

Simon, R.

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for squeezed states,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

B. Dutta, N. Mukunda, R. Simon, A. Subramaniam, “Squeezed states, photon number distribution, and U(1) invariance,” J. Opt. Soc. Am. B 10, 253–264 (1993);R. Simon, E. C. G. Sudarshan, N. Mukunda, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993);“Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990);V. Bagini, R. Borghi, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989);“The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana J. Phys. 32, 769–792 (1989).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985); Arvind, B. Dutta, N. Mukunda, R. Simon, “Two-mode quantum systems: invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609–1620 (1995).
[CrossRef] [PubMed]

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

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, V. B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689; G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994);K. Sundar, N. Mukunda, R. Simon, “Coherent mode decomposition of general anisotropic Gaussian Schell model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995);R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

Subramaniam, A.

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana J. Phys. 32, 769–792 (1989).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989);“The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985); Arvind, B. Dutta, N. Mukunda, R. Simon, “Two-mode quantum systems: invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609–1620 (1995).
[CrossRef] [PubMed]

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

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Tervonen, E.

Turunen, J.

Weyl, H.

H. Weyl, The Theory of Groups and Quantum Mechanics, 2nd ed. (Dover, New York, 1930).

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Wolf, K. B.

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996);A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987);“Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), Chap. 4, pp. 105–158.
[CrossRef]

O. Castaños, E. López-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182.

K. B. Wolf, “The symplectic groups, their parametrization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), App. A, pp. 227–238; “Representations of the algebra sp(2, R),” ibid., pp. 239–247; reprinted in Dynamical Groups and Spectrum Generating Algebras, A. Barut, A. Bohm, Y. Ne’eman, eds. (World Scientific, Singapore, 1989), pp. 1076–1087, 1088–1096.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

Ann. Math. (1)

V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947);“Group representation in Hilbert spaces of analytic functions,” in Analytical Methods in Mathematical Physics, P. Gilbert, R. G. Newton, eds. (Gordon & Breach, New York, 1970), pp. 27–63.
[CrossRef]

Appl. Opt. (1)

J. Math. Phys. (1)

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987);“Symmetry-adapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226–1232 (1988).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Jpn. J. Math. (1)

K. Iwasawa, “On the representations of Lie algebras,” Jpn. J. Math. 19, 513–523 (1948); “On some types of topological groups,” Ann. Math. 50, 507–558 (1948).

Opt. Acta (1)

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

Opt. Commun. (4)

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980); H. O. Bartelt, K.-H. Brenner, H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980);H. Bartelt, K.-H. Brenner, “The Wigner distribution function: an alternate signal representation in optics,” Isr. J. Technol. 18, 260–262 (1980); K.-H. Brenner, H. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993).
[CrossRef]

D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for squeezed states,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

Phys. Lett. A (2)

R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990);V. Bagini, R. Borghi, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, G. S. Spagnolo, “The Simon–Mukunda polarization gadget,” Eur. J. Phys. 17, 279–284 (1996).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, K. B. Wolf, “On the phase space description of quantum nonlinear dynamics,” Phys. Lett. A 215, 128–134 (1996);A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Phys. Rev. A (3)

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985); Arvind, B. Dutta, N. Mukunda, R. Simon, “Two-mode quantum systems: invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609–1620 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989);“The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

Pramana J. Phys. (1)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana J. Phys. 32, 769–792 (1989).
[CrossRef]

Other (10)

H. Weyl, The Theory of Groups and Quantum Mechanics, 2nd ed. (Dover, New York, 1930).

M. Moshinsky, C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780 (1971);C. Quesne, M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971).
[CrossRef]

O. Castaños, E. López-Moreno, K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159–182.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), Chap. 4, pp. 105–158.
[CrossRef]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, V. B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689; G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994);K. Sundar, N. Mukunda, R. Simon, “Coherent mode decomposition of general anisotropic Gaussian Schell model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995);R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

K. B. Wolf, “The symplectic groups, their parametrization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), App. A, pp. 227–238; “Representations of the algebra sp(2, R),” ibid., pp. 239–247; reprinted in Dynamical Groups and Spectrum Generating Algebras, A. Barut, A. Bohm, Y. Ne’eman, eds. (World Scientific, Singapore, 1989), pp. 1076–1087, 1088–1096.

See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).

W. Hamilton, Lectures on Quaternions (Dublin, 1853). An excellent review of Hamilton’s theory of turns for SU(2) can be found in L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Mechanics, Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, Reading, Mass., 1881).

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).

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Figures (9)

Fig. 1
Fig. 1

Free displacement along the optical axis (left) acts on phase space by slanting the coordinate grid vertically (right); the map is (q, p)=(q+zp, p). The ray angle (p) and the area of phase space are conserved.

Fig. 2
Fig. 2

A lens acts on an incoming bundle of rays (left) through horizontal slanting of the phase-space coordinate grid (right); the map is (q, p)=(q, p-gq). A convex lens of Gaussian power g>0 turns parallel rays (p=0) to cross the z axis at focal distance f=1/g. At the plane of the lens where the transformation takes place, the position q of the rays and the phase-space area are conserved.

Fig. 3
Fig. 3

Left, a pure magnifier built with two convex lenses in the DLDL configuration; right, in phase space, as the position coordinate is squeezed, the momentum coordinate is stretched to conserve areas. The magnifier is called pure because the concomitant slanting of phase space in a DLD configuration is corrected by the rightmost lens, which is coincident with the output screen.

Fig. 4
Fig. 4

Left, a fractional Fourier transformer built with a DLD arrangement; right, it rotates phase space. On the left, solid lines correspond to the object and image screens of the θ40° Fourier transformer whose action is shown on the right; dashed lines serve for the θ=12π Fourier transform.

Fig. 5
Fig. 5

Left, a hyperbolic expander built as DLD with one concave lens stretches phase space along a 45° line (right) and squeezes it along the orthogonal direction. A hyperbolic reducer cannot be built with a single lens.

Fig. 6
Fig. 6

Astigmatic magnifier built with one DLDL configuration in the x direction and another in the y direction. The two share the same input and output screens. The system is a pure astigmatic magnifier when the two superposed cylindrical lenses at the exit face (an astigmatic lens) are chosen appropriately. With all lens powers and orientations, these arrangements are all negative-definite elements of the solvable Iwasawa subgroup NA of Sp(4, R); the positive-definite elements are obtained by the concatenation of two astigmatic magnifiers.

Fig. 7
Fig. 7

An image gyrator is a paraxial instrument that will rotate the xy planes of phase space. Does it exist? We can build it with two identical pieces (see Fig. 8) rotated at half the gyration angle.

Fig. 8
Fig. 8

The reflector preserves the x axis and inverts the y axis. It is built with inverting DLD subunits (cf. the two-dimensional Fig. 3), two for the x axis and (within the same available length) one on the y axis; at the end there is a common astigmatic lens to correct the slant of four-dimensional phase space.

Fig. 9
Fig. 9

Left, arrangement of a Fourier transformer in the y direction with perfect imaging in the x direction. On the right: Two such arrangements concatenated at an angle will produce cross gyration. Cross gyrators performs fractional Fourier transformations in the qxpy and qypx planes.

Equations (98)

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

{f, g}=fq·gp-fp·gq=fq, fpΩg/qg/p,
Ω=01-10,
dv(z)dz={H, v(z)},H=p22n0-v(q, z),
vv=Mav,
 v=Mbv=MbMav.
T (Ma)T (Mb)=T (MbMa).
MΩM T=Ω.
det M=1.
M=ABCD,
ABT-BATADT-BCTCBT-DATCDT-DCT=01-10.
Sp(2N, R)RN2+N×[SU(N)×U(1)]/ZN.
abcd=e-β0-γe-βeβ cos ωsin ω-sin ωcos ω=MI(γ, β, ω)=MI(γ, β, 0)MI(0, 0, ω)=MI(γ, 0, 0)MI(0, β, 0)MI(0, 0, ω), -<γ,β<,0ω<2π,
e-β=+a2+b2>0,
γ=-ac+bda2+b2,
ω=arg(a+ib).
MI(γ, β, ω)=MI(γ2, β2, ω2)MI(γ1, β1, ω1)=MI(γ2, β2, 0)MI(0, 0, ω2)MI(γ1, β1, 0)×MI(0, 0, ω1)=MI(γ2, β2, 0)MI(γ3, β3, ω3)MI(0, 0, ω1)=MI(γ2+γ3e2β2, β1+β3, ω3+ω1),
e-2β3=(cos ω2e-β1-sin ω2e-β1γ1)2+(sin ω2eβ1)2,
γ3=cos 2ω2e-2β1γ1+12sin 2ω2[e-2β1(γ12-1)+e2β1],
ω3=arg(a3+ib3)=arg(cos ω2e-β1+sin ω2e-β1γ1+i sin ω2eβ1).
D(z)=T[D(z)],D(z)=1z01,
D(z) : qpq+zpp,
L(g)=T[L(g)],L(g)=10-g1,
L(g) : qpqp-gq.
D(z1)L(g)D(z2)=T1-z2gz1+z2-z1gz2-g1-gz1.
M(ζ)=D(z1)L(g)D(z2)L(g)=T[M(ζ)]
M(ζ)=ζ00ζ-1,ζ=1-z2g.
F (θ)=D(z)L(g)D(z)=T[F(θ)],
F(θ)=cos θsin θ-sin θcos θ;
g=sin θ>0,z=tan(θ/2)>0.
H(ζ)=D(z)L(g)D(z)=T[H(ζ)],
H(ζ)=cosh ζsinh ζsinh ζcosh ζ;
g=-sinh ζ<0,z=tanh12ζ>0.
D(g)=F (±12π)L(g)F (±12π),
L(z)=F (±12π)D(z)F (±12π),
H(-θ)=F (±12π)H(θ)F (±12π).
M(ζ)=F14πH(ζ)F-14π.
a>0b<0c0d>0,a>00c0a-1.
τ1=σ3=100-1,
τ2=σ1=0110,
τ0=iσ2=01-10;
τ+=12(τ2+τ0)=0100,
τ-=12(τ2-τ0)=0010.
D(z)=exp(zτ+),z0,
L(g)=exp(-gτ-),
F(θ)=exp(θτ0),θ0,
H(ζ)=exp(ζτ2),ζ0.
M[x1, x2, x0]=exp(x1τ1+x2τ2+x0τ0)=expx1x2+x0x2-x0-x1,
abcdM[x]abcd-1=M[x],
x1x2x0=ad+bccd-ab-cd-abbd-ac12(a2-b2-c2+d2)12(a2-b2+c2-d2)-bd-ac12(a2+b2-c2-d2)12(a2+b2+c2+d2) x1x2x0.
-x12-x22+x02=σχ2,
σ=+1(timelike)or
σ=0(lightlike)or
σ=-1(spacelike).
σ=+1elliptic:rotatorM[0, 0, θ],σ=0parabolic:displacementM[0, -12z, 12z],lensM[0, 12g, 12g],σ=-1hyperbolic:expanderM[0, -ζ, 0],magnifierM[-ζ, 0, 0].
σ=+1elliptic:tr M=2 cos θ,(-2, 2),σ=0parabolic:tr M=+2,σ=-1hyperbolic:tr M=2 cosh ζ,(2, ).
tr M[x]=2 cosdet T[x],det T[x]=σχ2,
-1-z0-1,-cosh ζsinh ζsinh ζ-cosh ζ
tr M<-2,ortr M=-2exceptM=-1.
M(ξ)
=exp(ξu·τ)=1 cos ξ+u·τ sin ξtr M=2 cos ξ(-2, 2)1+ξu·τtr M=21 cosh ξ+u·τ sinh ξtr M=2 cosh ξ>2,
|m|<2  u·τ=M-12m1[1-(12m)2]1/2,
ξ=arccos12m+2πnw,
m=2  u·τ=M-1M2,1-M1,2,
ξ=M2,1-M1,2,
m>2  u·τ=M-12m1[(12m)2-1]1/2,
ξ=arccosh12m.
D(z)=1z101,z0.
L(g, κ)=10-g cos2 κg cos κ sin κ1g cos κ sin κ-g cos2 κ.
L(g)=10-g1,g=gxxgxygxygyy.
M(z1, z2, z1)
=diag(1-z2gx, 1-z2gy, 1-z1gx, 1-z1gy).
cos θ00cos θ+i-sin θ00-sin θ=e-iθ1U(1).
F(θ)=cos θ1sin θ1-sin θ1cos θ1
G(ϕ)= cos ϕsin ϕ0-sin ϕcos ϕ cos ϕsin ϕ0-sin ϕcos ϕ=cos ϕ1+sin ϕτ000cos ϕ1+sin ϕτ0.
J (f)=D(2 f)L(1/f)D(2 f)=T-10-1/f-1,
[J (f)]2=T102/f1.
J (f)L(1/f)=T-100-1,
[J (f)]2L(2/f)=T1001.
I0=Jx(f)Jy(2 f)Jx(f)Lx(2/f)Ly(12f)=T (I0)=D(2 f)Lx(1/f)D(4f)Lx(1/f)D(2 f)Lx(2/f)D(4f)Ly(1/2 f)D(4f)Ly(1/2 f),
I0=diag(1,-1, 1,-1).
Iϕ=G(ϕ)I0G(-ϕ).
G(ϕ)=Iϕ0+(1/2)ϕIϕ0.
X(ϕ)=cos ϕ00sin ϕ0cos ϕsin ϕ00-sin ϕcos ϕ0-sin ϕ00cos ϕ=cos ϕ1sin ϕτ2sin ϕτ2cos ϕ1,
Fy(12π)=D(12)Lx(4)D(1)Lx(4)D(12)Lx(8)D(1)Ly(1)D(1)
Fy(12π)=1000000100100-100,
X(ϕ)=Fy(12π)G(ϕ)Fy(-12π)
Xγ(α)=G(12γ)X(α)G(-12γ)
Y(β)=Xπ/2(β)=G(14π)X(β)G(-14π)=T[Y(β)],
Y(β)=cos β0sin β00cos β0-sin β-sin β0cos β00sin β0cos β=cos β1sin βτ1-sin βτ1cos β1.
U0(ω)=F(ω)=exp(-iωJ0)exp(-iω1),
U1(α)=X(α)=exp(-iαJ1)exp(-iασ1),
U2(β)=Y(β)=exp(-iβJ2)exp(-iβσ2),
U3(γ)=G(γ)=exp(-iγJ3)exp(-iγσ3).
J0=-σ21,J1=-σ2σ1,
J2=-σ2σ3,J3=-1σ2.
[Jj, Jk]=2ijklJl,[J0, Jk]=0,j, k=1, 2, 3.
tr M<-4ortr M=-4exceptM=-1,
U(θ1, θ2, θ3)=Xθ1(12π)Xθ2(14π)Xθ3(14π),

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