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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2368–2381

Fractional Fourier transforms in two dimensions

R. Simon and Kurt Bernardo Wolf  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2368-2381 (2000)

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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

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(080.2730) Geometric optics : Matrix methods in paraxial optics

Original Manuscript: March 30, 2000
Revised Manuscript: July 13, 2000
Manuscript Accepted: July 13, 2000
Published: December 1, 2000

R. Simon and Kurt Bernardo Wolf, "Fractional Fourier transforms in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000)

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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]

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