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K. B. Wolf, “The symplectic groups, their parametrization and cover,” in Lie Methods in Optics, J. Sa´nchez-Mondrago´n and 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),” pp. 239-247; reprinted in Dynamical Groups and Spectrum Generating Algebras, A. Barut, A. Bohm, and Y. Ne’eman, eds. (World Scientific, Singapore, 1989), pp. 1076-1087, 1088-1096.
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See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).
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Classical Mechanics
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Goldstein, H.1
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Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states
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R. Simon and N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39-45 (1993).
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Opt. Commun
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Simon, R.1
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The two-dimensional symplectic and metaplectic groups and their universal cover
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V. B. Gruber, (Plenum, New York, G. S. Agarwal and 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, and R. Simon, “Coherent mode decomposition of general anisotropic Gaussian Schell model beams,” J. Opt. Soc. Am. A 12, 560-569 (1995); R. Simon and 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)
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R. Simon and 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 and 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, and R. Simon, “Coherent mode decomposition of general anisotropic Gaussian Schell model beams,” J. Opt. Soc. Am. A 12, 560-569 (1995); R. Simon and 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).
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Symmetries in Science
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Simon, R.1
Mukunda, N.2
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7
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Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms
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R. Simon, N. Mukunda, and B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567-1583 (1994).
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Phys. Rev
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Simon, R.1
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9
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The generalized Fresnel transform and its application to optics
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D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207-212 (1996).
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Opt. Commun
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James, D.F.V.1
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10
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An experiment for the study of the Gouy effect for squeezed states
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G. S. Agarwal and R. Simon, “An experiment for the study of the Gouy effect for squeezed states,” Opt. Commun. 100, 411-414 (1993).
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Opt. Commun
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Agarwal, G.S.1
Simon, R.2
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11
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Bargmann invariant and the geometry of the Gouy effect
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R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880-883 (1993).
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Phys. Rev. Lett
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Simon, R.1
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12
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Generalized rays in first order optics: Transformation properties of Gaussian Schell model fields
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R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first order optics: Transformation properties of Gaussian Schell model fields,” Phys. Rev. A 29, 3273-3279 (1984).
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Phys. Rev
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Simon, R.1
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13
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Squeezed states, photon number distribution, and U(1) invariance
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R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322-328 (1988)
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B. Dutta, N. Mukunda, R. Simon, and 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, and N. Mukunda, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322-328 (1988).
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J. Opt. Soc. Am
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Dutta, B.1
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Subramaniam, A.4
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14
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Anisotropic Gaussian Schell model beams: Passage through optical systems and associated invariants
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Arvind, B. Dutta, N. Mukunda, and R. Simon, “Two-mode quantum systems: Invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609-1620 (1995)
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R. Simon, E. C. G. Sudarshan, and 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, and R. Simon, “Two-mode quantum systems: invariant classification of squeezing transformations and squeezed states,” Phys. Rev. A 52, 1609-1620 (1995).
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Phys. Rev
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Simon, R.1
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15
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Realisation of first order optical systems using thin lenses
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E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realisation of first order optical systems using thin lenses,” Opt. Acta 32, 855-872 (1985).
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Opt. Acta
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Sudarshan, E.C.G.1
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Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters
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A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130-2141 (1998).
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H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their implementations. II,” J. Opt. Soc. Am. A 10, 2522-2531 (1993); “FractionalFourier 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, and A. W. Lohmann, “Optical fractional correlation: Experimental results,” J. Opt. Soc. Am. A 12, 1665-1670 (1995)
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D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their implementations. I,” J. Opt. Soc. Am. A 10, 1875-1881 (1993); H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their implementations. II,” J. Opt. Soc. Am. A 10, 2522-2531 (1993); “FractionalFourier 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, and A. W. Lohmann, “Optical fractional correlation: Experimental results,” J. Opt. Soc. Am. A 12, 1665-1670 (1995).
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J. Shamir and N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415-2423 (1995).
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Hamiltons theory of turns generalized to Sp(2, R)
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“The theory of screws—a new geometric representation for the group SU(1, 1),” J. Math. Phys. 30, 1000-1006 (1989)
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R. Simon, N. Mukunda, and 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).
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Simon, R.1
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(Dublin, An excellent review of Hamilton’s theory of turns for SU(2) can be found in L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Mechanics, Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, Reading, Mass., 1881)
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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 and J. D. Louck, Angular Momentum in Quantum Mechanics, Vol. 8 of Encyclopedia of Mathematics and Its Applications (Addison-Wesley, Reading, Mass., 1881).
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(1853)
Lectures on Quaternions
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Hamilton, W.1
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The Wigner function and its optical production
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H. O. Bartelt, K.-H. Brenner, and H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980); H. Bartelt and K.-H. Brenner, “The Wigner distribution function: An alternate signal representation in optics,” Isr. J. Technol. 18, 260-262 (1980); K.-H. Brenner and H. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310-314 (1982)
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A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32-37 (1980); H. O. Bartelt, K.-H. Brenner, and H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32-38 (1980); H. Bartelt and K.-H. Brenner, “The Wigner distribution function: An alternate signal representation in optics,” Isr. J. Technol. 18, 260-262 (1980); K.-H. Brenner and H. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310-314 (1982).
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R. Simon and 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).
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A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818-1826 (1994).
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Hamiltons theory of turns and a new geometrical representation for polarization optics
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R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana J. Phys. 32, 769-792 (1989).
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Pramana J. Phys
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V. Bagini, R. Borghi, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, and G. S. Spagnolo, “The Simon-Mukunda polarization gadget,” Eur. J. Phys. 17, 279-284 (1996)
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R. Simon and 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, and G. S. Spagnolo, “The Simon-Mukunda polarization gadget,” Eur. J. Phys. 17, 279-284 (1996).
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M. Moshinsky and C. Quesne, “Oscillator systems,” in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, New York, 1974); M. Moshinsky and C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772-1780 (1971); C. Quesne and M. Moshinsky, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780-1783 (1971).
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O. Castaños, E. López-Moreno, and K. B. Wolf, “Canonical transforms for paraxial wave optics,” in Lie Methods in Optics, J. Sánchez-Mondragón and K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), pp. 159-182.
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J. Sánchez-Mondragón and K. B. Wolf, Vol, of Lecture Notes in Physics (Springer-Verlag, Berlin, Chap. 4
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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., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1986), Chap. 4, pp. 105-158.
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“Symmetryadapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226-1232 (1988)
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K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498-2507 (1987); “Symmetryadapted classification of aberrations,” J. Opt. Soc. Am. A 5, 1226-1232 (1988).
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A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876-889 (1997)
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N. M. Atakishiyev, S. M. Chumakov, A. L. Rivera, and 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, and K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876-889 (1997).
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