Abstract

We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the FT. There is a qualitative increase in the richness of the solution manifold, from U(1) (the circle S1) in the one-dimensional case to U(2) (the four-parameter group of 2 × 2 unitary matrices) in the two-dimensional case [rather than simply U(1)×U(1)]. Our treatment clarifies the situation in the N-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus T2=S1×S1 and two parameters running over the Fourier sphere S2. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the T2 coordinates; the eigenfunctions, only on the S2 coordinates. FrFT’s corresponding to special points on the Fourier sphere have for eigenfunctions the Hermite–Gaussian beams and the Laguerre–Gaussian beams, while those corresponding to generic points are SU(2)-coherent states of these beams. Thus the integral transform produced by every Sp(4, R) first-order system is essentially a FrFT.

© 2000 Optical Society of America

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  1. 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);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993);S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);H. M. Ozaktas, D. Mendlovic, “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]
  2. A. W. Lohmann, D. Mendlovic, G. Shabtay, “Significance of phase and amplitude in the Fourier domain,” J. Opt. Soc. Am. A 14, 2901–2904 (1997);H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
    [CrossRef]
  3. S. C. Pei, M. H. Yeh, “Discrete fractional Fourier transform,” in Proceedings of IEEE International Symposium on Circuits Systems (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 536–539; S. C. Pei, M. H. Yeh, “Improved discrete fractional Fourier transform,” Opt. Lett. 22, 1047–1049 (1997);S. C. Pei, M. H. Yeh, “Two dimensional discrete fractional Fourier transform,” Signal Process. 67, 99–108 (1998);S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1347 (1999).
    [CrossRef] [PubMed]
  4. 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).
    [CrossRef]
  5. H. Weyl, The Theory of Groups and Quantum Mechanics, 2nd ed. (Dover, New York, 1930).
  6. R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]
  7. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  8. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realisation of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  9. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974), Chap. 4.
  10. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chaps. 7 and 9.
  11. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
    [CrossRef]
  12. 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);M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971);M. Moshinsky, “Canonical transformations and quantum mechanics,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 25, 193–203 (1973).
    [CrossRef]
  13. J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
    [CrossRef]
  14. M. Kauderer, Symplectic Matrices, First Order Systems and Special Relativity (World Scientific, Singapore, 1994).
  15. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [CrossRef] [PubMed]
  16. 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]
  17. G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Laser Beam Characterization, P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.
  18. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937). Note that this author uses the kernel exp(ipq) instead of the more common exp(-ipq) that we use here.
    [CrossRef] [PubMed]
  19. 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]
  20. 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]
  21. See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950).
  22. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).
  23. A. Sahin, M. Alper Kutay, H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
    [CrossRef]
  24. D. Han, Y. S. Kim, M. E. Noz, “Jones-matrix formalism as a representation of the Lorentz group,” J. Opt. Soc. Am. A 14, 2290–2298 (1997).
    [CrossRef]
  25. R. Simon, N. Mukunda, “The SO(n, 1) Wigner rotation as an SL(2, R) problem,” Found. Phys. Lett. 3, 425–434 (1990).
    [CrossRef]
  26. 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).
    [CrossRef]
  27. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams: I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
    [CrossRef]
  28. M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
    [CrossRef] [PubMed]
  29. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980);L. F. Ludwig, “General thin-lens action on spatial intensity distribution behaves as non-integer powers of Fourier transform,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 173–176.
    [CrossRef]
  30. See, e.g., J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965), pp. 229–279.
  31. A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).
  32. S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
    [CrossRef] [PubMed]
  33. C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
    [CrossRef]
  34. 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, Heidelberg, 1986), pp. 159–182; K. B. Wolf, “The symplectic groups, their parameterization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 227–238.
  35. R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science V: Algebraic Systems, Their Representation, Realizations, and Physical Applications, B. Gruber, L. C. Biedenharn, H. D. Doebner, eds. (Plenum, New York, 1991), pp. 659–689.
  36. M. J. Padgett, J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
    [CrossRef]
  37. G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,” J. Opt. Soc. Am. A 16, 2914–2916 (1999).
    [CrossRef]

2000 (1)

1999 (2)

1998 (2)

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
[CrossRef]

A. Sahin, M. Alper Kutay, H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
[CrossRef]

1997 (2)

1995 (2)

1994 (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]

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).
[CrossRef]

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (1)

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

1990 (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]

R. Simon, N. Mukunda, “The SO(n, 1) Wigner rotation as an SL(2, R) problem,” Found. Phys. Lett. 3, 425–434 (1990).
[CrossRef]

1989 (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]

1988 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

1985 (2)

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, “Anisotropic Gaussian Schell-model beams: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1982 (1)

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980);L. F. Ludwig, “General thin-lens action on spatial intensity distribution behaves as non-integer powers of Fourier transform,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 173–176.
[CrossRef]

1937 (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937). Note that this author uses the kernel exp(ipq) instead of the more common exp(-ipq) that we use here.
[CrossRef] [PubMed]

Agarwal, G. S.

G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,” J. Opt. Soc. Am. A 16, 2914–2916 (1999).
[CrossRef]

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).
[CrossRef]

Aravind, P. K.

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

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, Heidelberg, 1986), pp. 159–182; K. B. Wolf, “The symplectic groups, their parameterization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 227–238.

Cohen, N.

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937). Note that this author uses the kernel exp(ipq) instead of the more common exp(-ipq) that we use here.
[CrossRef] [PubMed]

Courtial, J.

Danakas, S.

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

Dutta, B.

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]

Gilmore, R.

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974), Chap. 4.

Goldstein, H.

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

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

Han, D.

Kauderer, M.

M. Kauderer, Symplectic Matrices, First Order Systems and Special Relativity (World Scientific, Singapore, 1994).

Kim, Y. S.

Kostenbauder, A. G.

G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Laser Beam Characterization, P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.

Kumar, M. Sanjay

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

Kutay, M. Alper

Lohmann, A. W.

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, Heidelberg, 1986), pp. 159–182; K. B. Wolf, “The symplectic groups, their parameterization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 227–238.

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);M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971);M. Moshinsky, “Canonical transformations and quantum mechanics,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 25, 193–203 (1973).
[CrossRef]

Mukunda, N.

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).
[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]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams: I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, “The SO(n, 1) Wigner rotation as an SL(2, R) problem,” Found. Phys. Lett. 3, 425–434 (1990).
[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, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[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, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science V: Algebraic Systems, Their Representation, Realizations, and Physical Applications, B. Gruber, L. C. Biedenharn, H. D. Doebner, eds. (Plenum, New York, 1991), pp. 659–689.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980);L. F. Ludwig, “General thin-lens action on spatial intensity distribution behaves as non-integer powers of Fourier transform,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 173–176.
[CrossRef]

Nazarathy, M.

Nemes, G.

G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Laser Beam Characterization, P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.

Noz, M. E.

Ozaktas, H. M.

Padgett, M. J.

Pei, S. C.

S. C. Pei, M. H. Yeh, “Discrete fractional Fourier transform,” in Proceedings of IEEE International Symposium on Circuits Systems (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 536–539; S. C. Pei, M. H. Yeh, “Improved discrete fractional Fourier transform,” Opt. Lett. 22, 1047–1049 (1997);S. C. Pei, M. H. Yeh, “Two dimensional discrete fractional Fourier transform,” Signal Process. 67, 99–108 (1998);S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1347 (1999).
[CrossRef] [PubMed]

Perelomov, A. M.

A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

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);M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971);M. Moshinsky, “Canonical transformations and quantum mechanics,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 25, 193–203 (1973).
[CrossRef]

Sahin, A.

Schwinger, J.

See, e.g., J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965), pp. 229–279.

Selvadoray, M.

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

Shabtay, G.

Shamir, J.

Sheppard, C. J. R.

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
[CrossRef]

Simon, R.

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

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).
[CrossRef]

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

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).
[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]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams: I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[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, “The SO(n, 1) Wigner rotation as an SL(2, R) problem,” Found. Phys. Lett. 3, 425–434 (1990).
[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, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[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, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science V: Algebraic Systems, Their Representation, Realizations, and Physical Applications, B. Gruber, L. C. Biedenharn, H. D. Doebner, eds. (Plenum, New York, 1991), pp. 659–689.

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

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, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[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]

Sundar, K.

Weyl, H.

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

Wolf, K. B.

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chaps. 7 and 9.

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, Heidelberg, 1986), pp. 159–182; K. B. Wolf, “The symplectic groups, their parameterization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 227–238.

Yeh, M. H.

S. C. Pei, M. H. Yeh, “Discrete fractional Fourier transform,” in Proceedings of IEEE International Symposium on Circuits Systems (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 536–539; S. C. Pei, M. H. Yeh, “Improved discrete fractional Fourier transform,” Opt. Lett. 22, 1047–1049 (1997);S. C. Pei, M. H. Yeh, “Two dimensional discrete fractional Fourier transform,” Signal Process. 67, 99–108 (1998);S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1347 (1999).
[CrossRef] [PubMed]

Appl. Opt. (1)

Found. Phys. Lett. (1)

R. Simon, N. Mukunda, “The SO(n, 1) Wigner rotation as an SL(2, R) problem,” Found. Phys. Lett. 3, 425–434 (1990).
[CrossRef]

J. Inst. Math. Its Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980);L. F. Ludwig, “General thin-lens action on spatial intensity distribution behaves as non-integer powers of Fourier transform,” in Spatial Light Modulators and Applications, Vol. 8 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 173–176.
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993);S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994);H. M. Ozaktas, D. Mendlovic, “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]

A. W. Lohmann, D. Mendlovic, G. Shabtay, “Significance of phase and amplitude in the Fourier domain,” J. Opt. Soc. Am. A 14, 2901–2904 (1997);H. M. Ozaktas, M. Alper Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
[CrossRef]

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

G. S. Agarwal, “SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum,” J. Opt. Soc. Am. A 16, 2914–2916 (1999).
[CrossRef]

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).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams: I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

D. Han, Y. S. Kim, M. E. Noz, “Jones-matrix formalism as a representation of the Lorentz group,” J. Opt. Soc. Am. A 14, 2290–2298 (1997).
[CrossRef]

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. (1)

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).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (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]

Phys. Rev. A (5)

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

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, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

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]

Proc. Natl. Acad. Sci. USA (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937). Note that this author uses the kernel exp(ipq) instead of the more common exp(-ipq) that we use here.
[CrossRef] [PubMed]

Other (13)

G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Laser Beam Characterization, P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.

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);M. Moshinsky, C. Quesne, “Canonical transformations and matrix elements,” J. Math. Phys. 12, 1780–1783 (1971);M. Moshinsky, “Canonical transformations and quantum mechanics,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 25, 193–203 (1973).
[CrossRef]

M. Kauderer, Symplectic Matrices, First Order Systems and Special Relativity (World Scientific, Singapore, 1994).

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

S. C. Pei, M. H. Yeh, “Discrete fractional Fourier transform,” in Proceedings of IEEE International Symposium on Circuits Systems (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 536–539; S. C. Pei, M. H. Yeh, “Improved discrete fractional Fourier transform,” Opt. Lett. 22, 1047–1049 (1997);S. C. Pei, M. H. Yeh, “Two dimensional discrete fractional Fourier transform,” Signal Process. 67, 99–108 (1998);S.-C. Pei, M.-H. Yeh, C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1347 (1999).
[CrossRef] [PubMed]

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974), Chap. 4.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chaps. 7 and 9.

See, e.g., J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. van Dam, eds. (Academic, New York, 1965), pp. 229–279.

A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

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, Heidelberg, 1986), pp. 159–182; K. B. Wolf, “The symplectic groups, their parameterization and cover,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 227–238.

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science V: Algebraic Systems, Their Representation, Realizations, and Physical Applications, B. Gruber, L. C. Biedenharn, H. D. Doebner, eds. (Plenum, New York, 1991), pp. 659–689.

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

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

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Figures (3)

Fig. 1
Fig. 1

Circle of FrFT’s Fα, α counted modulo 4. Powers of F3/4 successively fall on (F3/4)4=F-1, (F3/4)8=F2, and (F3/4)12=F. Next would come (F3/4)16=1.

Fig. 2
Fig. 2

Torus T2 of xy-separable FT’s F1α,β. The unit transform is at 1=F10,0, and the official FT is at F=F1n1,n2, with n1, n21 mod 4. The α=β circle is composed of central FT’s. For a given (α0, β0), the continuous powers (tα0, tβ0) may or may not pass through the official transform.

Fig. 3
Fig. 3

The Fourier sphere serves to classify U(2) FT’s. The 1 axis (on the equator) corresponds to FT’s separable in the xy coordinates. The 3 axis (north pole) corresponds to central FT’s that also rotate the image (gyrators). The 2 axis corresponds to cross gyrators in the planes (qx, py) and (qy, px). Around the equator the FT is separable in rotated coordinates.

Equations (118)

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(Ff)(qx, qy)=12πR2dqxdqy×exp[-i(qxqx+qyqy)]f(qx, qy).
Fpˆ=qˆF,Fqˆ=-pˆF.
FF=01-10.
U(1)=FαFα=cosπ2αsinπ2α-sinπ2αcosπ2α0α<4,
αm=1n+4mn,m=0, 1,, n-1.
F=00100001-10000-100.
Fα,β=cosπ2α0sinπ2α00cosπ2β0sinπ2β-sinπ2α0cosπ2α00-sinπ2β0cosπ2β.
α=nxn,β=nyn,(nx, n)=1,(ny, n)=1,
MFMT=F,F=01-10,
M=ABCD,
ABT-BAT=0,ADT-BCT=1,
CBT-DAT=-1,CDT-DCT=0.
AAT+BBT=1,ACT+BDT=0,
CAT+DBT=0,CCT+DDT=1.
U(O)=A+iB=D-iC,UU=1=UU,
O(U)=Re UIm U-Im URe U,OOT=1=OTO,
U(O1)U(O2)=U(O1O2),O(U1)O(U2)=O(U1U2).
U(F0α)=exp[i(π/2)α]1,
F0α=cosπ2α1sinπ2α1-sinπ2α1cosπ2α1.
L(g)=10-g1,D(z)=1z101,
(Lgψ)(q)=exp(i12qTgq)ψ(q),
(Dzψ)(q)=exp[-i(z+12π)]2πzR2dq×expi2z(q-q)2ψ(q)
=exp(-iz)exp(iq2/2z)exp(-iπ/2)2πzR2dq×exp-izq·qexp(iq2/2z)ψ(q).
(F0αψ)(q)=exp(iz)exp(i12πα)×[Ltan(π/4)αDsin(π/2)αLtan(π/4)αψ](q)
=R2dqF0α(q, q)ψ(q),
F0α(q, q)=exp[i12π(α-1)]2π sin(π/2)αexp iq22 tan(π/2)α-q·qsin(π/2)α+q22 tan(π/2)α.
U(F1α, β)=expiπ2α00expiπ2β=expiπ4(α+β)×expiπ4(α-β)00exp-iπ4(α-β)=expiπ4(α+β)1001×expiπ4(α-β)100-1,
F1α,β=cosπ2α0sinπ2α00cosπ2β0sinπ2β-sinπ2α0cosπ2α00-sinπ2β0cosπ2β.
(F1α, βψ)(q)=R2dqF1α,β(q, q)ψ(q),
F1α,β(q,q)=exp{i(1/2)π[(1/2)(α+β)-1]}2π[sin(π/2)α sin(π/2)β]1/2×exp iqx2+qx22 tan(π/2)α-qxqxsin(π/2)α+qy2+qy22 tan(π/2)β-qyqysin(π/2)β.
U(F3α, β)=expiπ4(α+β)1001×expiπ4(α-β)0i-i0,
U(F3α, β)=expiπ4(α+β)Rπ4(α-β),
R(θ)=cos θsin θ-sin θcos θ,
F3α,β=cos π4(α+β)Rπ4(α-β)sin π4(α+β)Rπ4(α-β)-sin π4(α+β)Rπ4(α-β)cos π4(α+β)Rπ4(α-β).
(Rθψ)(q)=ψ[R(θ)-1q],
F3α,β(q, q)=exp{i(1/2)π[(1/2)(α+β)-1]}2π sin(π/4)(α+β)×exp iq2+q22 tan(π/4)(α+β)
-q·R[(π/4)(α-β)]qsin(π/4)(α+β).
τ0=σ0=1001T0=axax+ayay
=12(px2+py2+qx2+qy2)-1,
τ1=σ3=100-1T1=axax-ayay
=12(px2-py2+qx2-qy2),
τ2=σ1=0110T2=axay+ayax=pxpy+qxqy,
τ3=σ2=0-ii0T3=i(axay-ayax)=pyqx-pxqy.
 Urα,β=exp(iμτ0)exp(iνr·τ)
=exp(iμ)(1 cos ν+ir·τ sin ν)
=exp(iμ)cos ν+ir1 sin ν(r3+ir2)sin ν(-r3+ir2)sin νcos ν-ir1 sin ν.
Frα,β=O(Urα, β)=O[exp(iμ)1]O[exp(iνr·τ)].
Frα,β=F012(α+β)Fr12(α-β),-12(α-β)=cμ1sμ1-sμ1cμ1cνr3sνr1sνr2sν-r3sνcνr2sν-r1sν-r1sν-r2sνcνr3sν-r2sνr1sν-r3sνcν.
qxφqyφpxφpyφ=cφ/2sφ/200-sφ/2cφ/20000cφ/2sφ/200-sφ/2cφ/2qxqypxpy.
Frα,-α:qxφqyφpxφpyφcν0sν00cν0-sν-sν0cν00sν0cνqxφqyφpxφpyφ,
Frα,βFrα,β=Frα+α,β+β,
rsν+ν=rsνcν+rsνcν-(r×r)sν sν,
(Frα,β)-1=Fr-α,-β=F-r-β,-α.
Frγ,δFrα,β(Frγ, δ)-1=FD(r, θ)rα,β,θ=(π/4)(γ-δ),
D(r, θ)r·τ=r·[U(Frγ, δ)τ U(Frγ, δ)-1].
[τ0,τk]=0,[τj,τk]=2ijklτl,j, k, l=1, 2, 3.
[Tˆ0, Tˆk]=0,[Tˆj, Tˆk]=2ijklTˆl, j, k, l=1, 2, 3.
qˆ(α)pˆ(α)=exp-i12παTˆ0qˆpˆexpi12παTˆ0
=cos12παsin12πα-sin12παcos12πα-1qˆpˆ=F0αqˆpˆF0-α.
U(F0α)=exp(i12πατ0)F0α=exp(-i12παTˆ0).
U(F1α, β)=exp[i(π/4)(α+β)τ0+i(π/4)(α-β)τ1]
F1α,β=exp[-i(π/4)(α+β)Tˆ0-i(π/4)(α-β)Tˆ1].
axaxΨnx,ny(q)=nxΨnx,ny(q),
ayayΨnx,ny(q)=nyΨnx,ny(q),
[14(α+β)Tˆ0+14(α-β)Tˆ1]Ψnx,ny(q)
=12(nxα+nyβ)Ψnx,ny(q),
Ψnx,ny(q)=Hnx(qx)Hny(qy)[2nx+nynx!ny!π]1/2exp-12q2.
F1α,β(q, q)=nx,nyΨnx,ny(q)×exp[-i12π(nxα+nyβ)]Ψnx,ny(q).
U(ϑ, φ)=exp(-iφτ1)exp(-iϑτ3)exp(iφτ1)=cos12ϑ-sin12ϑ exp(-iφ)sin12ϑ exp(iφ)cos12ϑ
U(ϑ, φ)=exp(-iφTˆ1)exp(-iϑTˆ3)exp(iφTˆ1).
U(F(ϑ, φ)α, β)=U(ϑ, φ)U(F1α, β)U(ϑ, φ)-1
F(ϑ, φ)α,β=U(ϑ, φ)F1α,βU(ϑ, φ)-1.
Tˆ0Φj,m(q)=jΦj,m(q),j=0,12, 1,32,,
Tˆ1Φj,m(q)=mΦj,m(q),m=-j,-j+1,,j.
Φj,m(ϑ, φ)(q)=U(ϑ, φ)Φj,m(q).
U(ϑ, φ)Tˆ0U(ϑ, φ)-1=Tˆ0,Tˆ0Φj,m(ϑ, φ)(q)=jΦj,m(ϑ, φ)(q),
U(ϑ, φ)Tˆ1U(ϑ, φ)-1=Tˆ(ϑ, φ),
Tˆ(ϑ, φ)Φj,m(ϑ, φ)(q)=mΦj,m(ϑ, φ)(q).
Φj,m(ϑ, φ)(q)=m=-jjDm,mj(ϑ, φ)Φj,m(q),
D m,mj(ϑ, φ)
=j, m|U(ϑ, φ)|j, m=exp[-iφ(m-m)]dm,mj(ϑ),
dm,mj(ϑ)
=j, m|exp(-iϑTˆ3)|j, m=(-1)m-mν
×(-1)ν[(j+m)!(j-m)!(j+m)!(j-m)!]1/2(j-m-ν)!(j+m-ν)!ν!(m-m+ν)!×(cos12ϑ)2j+m-m-2ν(sin12ϑ)m-m+2ν.
F(ϑ, φ)α,β(q, q)=j=0,1/2,m=-jjΦj,m(ϑ, φ)(q)×exp{-i12π[(j+m)α+(j-m)β]}Φj,m(ϑ, φ)(q)*,
R2d2qΦj,m(ϑ, φ)(q)*Φj,m(ϑ, φ)(q)=δjjδmm,
j=0,1/2,m=-jjΦj,m(ϑ, φ)(q)Φj,m(ϑ, φ)(q)*=δ(q-q).
Tˆ(12π,φ)=cos φTˆ1+sin φTˆ2=12[(pxφˆ)2+(qxφˆ)2]-12[(pyφˆ)2+(qyφˆ)2],
Φj,m((1/2)π, φ)(qx, qy)=Φj,m(qxφ, qyφ)=Hj+m(qx cos12φ+qy sin12φ)Hj-m(qy cos12φ-qx sin12φ)[n2j(j+m)!(j-m)!π]1/2exp-12q2.
Φj,m(0, 0)(q)=(j-|m|)!π(j+|m|)!1/2 exp(2imκ)q2|m|×exp(-12q2)Lj-|m|2|m|(q2).
M=ABCD=10-g1S00S-1XY-YX,
S=ST=(AAT+BBT)1/2 positive definite,
X+iY=(AAT+BBT)-1/2(A+iB)U(2),
g=gT=-(CAT+DBT)(AAT+BBT)-1.
M:ψ(q)  ψM(q)
=1det Sexp-i12qTgqψ(ϑ, φ)α,β(S-1q).
C (M)qˆpˆC (M)-1=M-1qˆpˆ=DˆT-BT-CTATqˆpˆ,
[C(M)f](q)=R2dqC(M)(q, q)f(q),
C(M)(q, q)=-i2πdet B×exp i(12qTDB-1q-qTBT-1q+12qTB-1Aq).
CA0CAT-1f(q)
=1det Aexpi12qT CAT-1qf(A-1q).
F=iC(F),
F(q, q)=iC(F)(q, q)=12πexp(-iqTq),
F0α=exp(i12πα)C(F0α),
F0α(q, q)=exp(i12πα)C(F0α)(q, q).
F1α,β=exp[iπ(α+β)/4]C(F1α, β),
F1α,β(q, q)=exp[iπ(α+β)/4]C(F1α, β)(q, q).
Frα,β=exp(i12πμ)C(F0μFrν,-ν).
 A0=D0=Re Urα,-α=cos νr3 sin ν-r3 sin νcos ν.
B0=-C0=Im Urα,-α=r1r2r2-r1sin ν.
B0-1=1r122 sin νr1r2r2-r1=B0r122 sin2 ν=B0T-1,
A0B0-1=1r122 sin2 ν(B0 cos ν+B0r3 sin ν),
B0-1A0=1r122 sin2 ν(B0 cos ν-B0r3 sin ν),
where B0=r2-r1-r1-r2sin ν
Aμ=A0 cos μ-B0 sin μ,Bμ=A0 sin μ+B0 cos μ.
Bμ-1=Δμ-1(-cos μ B0+sin μA0T),
AμBμ-1=Δμ-1(1 cos μ sin μ+Δ0A0B0-1).
Bμ-1Aμ=Δμ-1(1 cos μ sin μ+Δ0B0-1A0),

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