## Geometry and dynamics in the Fresnel transforms of discrete systems

JOSA A, Vol. 24, Issue 9, pp. 2568-2577 (2007)

http://dx.doi.org/10.1364/JOSAA.24.002568

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### Abstract

Free propagation in continuous optical and mechanical systems is generated by the momentum-squared operator and results in a shear of the phase space plane along the position coordinate. We examine three discrete versions of the Fresnel transform in periodic systems through their Wigner function on a toroidal phase space. But since it is topologically impossible to continuously and globally shear a torus, we examine a fourth version of the Fresnel transform on a spherical phase space, in a model based on the Lie algebra of angular momentum, where the corresponding Fresnel transform wrings the sphere.

© 2007 Optical Society of America

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(090.1970) Holography : Diffractive optics

**ToC Category:**

Fourier optics and signal processing

**History**

Original Manuscript: March 6, 2007

Revised Manuscript: April 11, 2007

Manuscript Accepted: April 16, 2007

Published: July 19, 2007

**Citation**

Kurt Bernardo Wolf and Guillermo Krötzsch, "Geometry and dynamics in the Fresnel transforms of discrete systems," J. Opt. Soc. Am. A **24**, 2568-2577 (2007)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-9-2568

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### References

- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
- S. A. Collins, Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970). [CrossRef]
- M. Moshinsky and C. Quesne, "Oscillator systems," in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, 1974).
- K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).
- 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). [CrossRef]
- V. Arrizón and J. Ojeda-Castañeda, "Fresnel diffraction of substructured gratings: matrix description," Opt. Lett. 20, 118-120 (1995). [CrossRef] [PubMed]
- 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). [CrossRef]
- 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). [CrossRef]
- U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996). [CrossRef] [PubMed]
- N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997). [CrossRef]
- N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, Suppl. 3521-555 (2005).
- N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998). [CrossRef]
- 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). [CrossRef]
- K. B. Wolf and G. Krötzsch, "Geometry and dynamics in the fractional discrete Fourier transform," J. Opt. Soc. Am. A 24, 651-658 (2007). [CrossRef]
- K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).
- E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932). [CrossRef]
- M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 259, 121-167 (1984). [CrossRef]
- M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978). [CrossRef]
- W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987). [CrossRef]
- J. C. O'Neill and W. J. Williams, "Shift covariant time-frequency distributions of discrete signals," IEEE Trans. Signal Process. 47, 133-146 (1999). [CrossRef]
- S. Korkmaz, Harmonic Analysis in Finite Phase Space, M.Sc. thesis (Institute of Engineering and Science of Bilkent University, 2005).
- 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).
- 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). [CrossRef]
- R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications (Wiley Interscience, 1978).
- H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995). [CrossRef]
- K. B. Wolf, "Wigner distribution function for paraxial polychromatic optics," Opt. Commun. 132, 343-352 (1996). [CrossRef]
- S. M. Chumakov, A. B. Klimov, and K. B. Wolf, "On the connection of two Wigner functions for spin systems," Phys. Rev. A 61, 034101(3) (2000). [CrossRef]
- R. L. Stratonovich, "On distributions in representation space," Zh. Eksp. Teor. Fiz. 311012-1020 (1956) R. L. Stratonovich, "On distributions in representation space,"[Sov. Phys. JETP 4, 891-898 (1957)].
- G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981). [CrossRef]
- 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). [CrossRef]
- 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). [CrossRef]

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