Geometry and dynamics in the Fresnel transforms of discrete systems

Journal of the Optical Society of America A: Optics and Image Science, and Vision

Volumn 24, Issue 9, 2007, Pages 2568-2577

  Wolf, Kurt Bernardo ^a   Krötzsch, Guillermo ^a  

^a INSTITUTO DE CIENCIAS FÍSICAS   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; COMPUTATIONAL GEOMETRY; DISCRETE TIME CONTROL SYSTEMS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; MATHEMATICAL TRANSFORMATIONS; TIME VARYING SYSTEMS;

DISCRETE SYSTEMS; FRESNEL TRANSFORMS; MOMENTUM-SQUARED OPERATOR; WIGNER FUNCTION;

OPTICAL SYSTEMS;

EID: 36448994479     PISSN: 10847529     EISSN: None     Source Type: Journal    
DOI: 10.1364/JOSAA.24.002568     Document Type: Article

References (31)

1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
2. S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
3. M. Moshinsky and C. Quesne, "Oscillator systems," in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, 1974).
4. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).
5. G. García-Calderón and M. Moshinsky, "Wigner distribution function and the representation of canonical transformations in quantum mechanics," J. Phys. A 13, L185-L188 (1980).
6. V. Arrizón and J. Ojeda-Castafieda, "Fresnel diffraction of substructured gratings: matrix description," Opt. Lett. 20, 118-120 (1995).
7. V. Arrizón, J. G. Ibarra, and J. Ojeda-Castañeda, "Matrix formulation of the Fresnel transform of complex transmittance gratings," J. Opt. Soc. Am. A 13, 2414-2422 (1996).
8. S. C. Bradburn, J. Ojeda-Castañeda, and W. T. Cathey, "Matrix description of near-field diffraction and the fractional Fourier transform," J. Opt. Soc. Am. A 16, 316-322 (1999).
9. U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
10. N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
11. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, Suppl. 3 521-555 (2005).
12. N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
13. S. T Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "The Wigner function for general Lie groups and the wavelet transform," Ann. Henri Poincare 1, 685-714 (2000).
14. K. B. Wolf and G. Krötzsch, "Geometry and dynamics in the fractional discrete Fourier transform," J. Opt. Soc. Am. A 24, 651-658 (2007).
15. K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).
16. E. P Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
17. M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984).
18. M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
19. W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
20. J. C. O'Neill and W. J. Williams, "Shift covariant time-frequency distributions of discrete signals," IEEE Trans. Signal Process. 47, 133-146 (1999).
21. S. Korkmaz, Harmonic Analysis in Finite Phase Space, M.Sc. thesis (Institute of Engineering and Science of Bilkent University, 2005).
22. L. C. Biedenharn and J. D. Louck, "Angular Momentum in Quantum Physics," in Encyclopedia of Mathematics and Its Applications, G.-C. Rota, ed. (Addison-Wesley, 1981).
23. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
24. R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications (Wiley Interscience, 1978).
25. H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
26. K. B. Wolf, "Wigner distribution function for paraxial polychromatic optics," Opt. Commun. 132, 343-352 (1996).
27. S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 0341010 (2000).
28. R. L. Stratonovich, "On distributions in representation space," Zh. Eksp. Teor. Fiz. 31 1012-1020 (1956) [Sov. Phys. JETP 4, 891-898 (1957)].
29. G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
30. 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).
31. S. M. Chumakov, A. Frank, and K. B. Wolf, "Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere," Phys. Rev. A 60, 1817-1823 (1999).