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

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References

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  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).
    [CrossRef]
  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).
    [CrossRef]
  6. V. Arrizón and J. Ojeda-Castañeda, "Fresnel diffraction of substructured gratings: matrix description," Opt. Lett. 20, 118-120 (1995).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]
  9. U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
    [CrossRef] [PubMed]
  10. N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
    [CrossRef]
  11. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, Suppl. 3521-555 (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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  18. M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
    [CrossRef]
  19. W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  26. K. B. Wolf, "Wigner distribution function for paraxial polychromatic optics," Opt. Commun. 132, 343-352 (1996).
    [CrossRef]
  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, 034101(3) (2000).
    [CrossRef]
  28. 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)].
  29. G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]

2007 (1)

2005 (1)

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, Suppl. 3521-555 (2005).

2003 (1)

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]

2000 (2)

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]

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]

1999 (3)

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. 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]

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]

1998 (1)

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

1997 (2)

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]

N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

1996 (3)

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]

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[CrossRef] [PubMed]

K. B. Wolf, "Wigner distribution function for paraxial polychromatic optics," Opt. Commun. 132, 343-352 (1996).
[CrossRef]

1995 (2)

H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
[CrossRef]

V. Arrizón and J. Ojeda-Castañeda, "Fresnel diffraction of substructured gratings: matrix description," Opt. Lett. 20, 118-120 (1995).
[CrossRef] [PubMed]

1987 (1)

W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[CrossRef]

1984 (1)

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]

1981 (1)

G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
[CrossRef]

1980 (1)

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]

1978 (1)

M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

1970 (1)

1956 (1)

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)].

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)].

1932 (1)

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, "Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions," Phys. Rev. A 24, 2889-2896 (1981).
[CrossRef]

Ali, S. T.

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]

Arrizón, V.

Atakishiyev, N. M.

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, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[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]

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[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]

Bastiaans, M. J.

M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

Biedenharn, L. C.

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).

Bradburn, S. C.

Cathey, W. T.

Chumakov, S. M.

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]

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]

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]

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[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]

Collins, S. A.

Frank, A.

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]

García-Calderón, G.

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]

Gilmore, R.

R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications (Wiley Interscience, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hillery, M.

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]

Ibarra, J. G.

Klimov, A. B.

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]

Korkmaz, S.

S. Korkmaz, Harmonic Analysis in Finite Phase Space, M.Sc. thesis (Institute of Engineering and Science of Bilkent University, 2005).

Krötzsch, G.

Lee, H.-W.

H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
[CrossRef]

Leonhardt, U.

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[CrossRef] [PubMed]

Louck, J. D.

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).

Moshinsky, M.

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]

M. Moshinsky and C. Quesne, "Oscillator systems," in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, 1974).

O'Connel, R. F.

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]

Ojeda-Castañeda, J.

O'Neill, J. C.

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]

Pogosyan, G. S.

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, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

Quesne, C.

M. Moshinsky and C. Quesne, "Oscillator systems," in Proceedings of the 15th Solvay Conference in Physics (1970) (Gordon & Breach, 1974).

Rivera, A. L.

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]

Scully, M. O.

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]

Stratonovich, R. L.

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)].

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)].

Wigner, E. P.

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]

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Williams, W. J.

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]

Wolf, K. B.

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]

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, G. S. Pogosyan, and K. B. Wolf, "Contraction of the finite one-dimensional oscillator," Int. J. Mod. Phys. A 18, 317-327 (2003).
[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]

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]

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]

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, "Fractional Fourier-Kravchuk transform," J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[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]

K. B. Wolf, "Wigner distribution function for paraxial polychromatic optics," Opt. Commun. 132, 343-352 (1996).
[CrossRef]

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

Wooters, W. K.

W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[CrossRef]

Ann. Henri Poincare (1)

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]

Ann. Phys. (N.Y.) (1)

W. K. Wooters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[CrossRef]

IEEE Trans. Signal Process. (1)

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]

Int. J. Mod. Phys. A (1)

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]

J. Math. Phys. (1)

N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, "Wigner distribution function for finite systems," J. Math. Phys. 39, 6247-6261 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys. A (1)

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]

Opt. Commun. (2)

K. B. Wolf, "Wigner distribution function for paraxial polychromatic optics," Opt. Commun. 132, 343-352 (1996).
[CrossRef]

M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

Opt. Lett. (1)

Phys. Part. Nucl. (1)

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, "Finite models of the oscillator," Phys. Part. Nucl. 36, Suppl. 3521-555 (2005).

Phys. Rep. (2)

H.-W. Lee, "Theory and applications of the quantum phase-space distribution functions," Phys. Rep. 259, 147-211 (1995).
[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]

Phys. Rev. (1)

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Phys. Rev. A (5)

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]

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]

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

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)].

Other (7)

R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications (Wiley Interscience, 1978).

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).

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

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).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (9)

Fig. 1
Fig. 1

Fresnel transform F ( z ) shears classical phase space ( x , p ) R 2 by the angle ζ arctan z .

Fig. 2
Fig. 2

Top, rectangle signal f ( x ) = Rect 5 , 5 ( x ) and its Wigner function on phase space W ( f x , p ) for x , p [ 15 , 15 ] and 20 contours. The x axes are drawn vertically, as if the function represented a light signal on an input screen. Bottom, Fresnel transforms of the rectangle function for z = 1 , 2 , 5 , 10 ; real, imaginary, and absolute values are indicated by dashed, dotted, and continuous curves, respectively. These will be compared with their discrete Fresnel versions below.

Fig. 3
Fig. 3

Discrete toroidal phase space of integer positions m and momenta k counted modulo N. (The two circles are supposed to have the same radius; for visibility they are drawn unequal.) The finite Fresnel transformations F [ ] ( z ) classically shear the (continuous) torus by angles ζ arctan z . The torus breaks at the two circles k ± 1 4 N .

Fig. 4
Fig. 4

Left, discrete (and periodic) centered rectangle function Rect 10 , 10 ( m ) for N = 63 . For visibility, the data points are joined by continuous lines. Right, the corresponding Wigner function on the discrete toroidal phase space ( k , m ) , shown as pixellated; the upper and lower boundaries of the square represent the same line, as do the right and left boundaries. The center ( 0 , 0 ) corresponds to the “front end” of the torus in Fig. 3.

Fig. 5
Fig. 5

Left, evolution of the centered discrete rectangle signal Rect 5 , 5 ( m ) for N = 31 points under the first Fresnel transform F [ 1 ] ( z ) for z = 1 , 2 , 5 , 10 . For visibility, the real, imaginary, and absolute values of the discrete points are joined by dashed, dotted, and continuous curves, respectively. Right, their corresponding Wigner functions on the 31 × 31 discrete torus.

Fig. 6
Fig. 6

Left, evolution of the rectangle signal Rect 5 , 5 ( m ) under the second Fresnel transform F m , m [ 2 ] ( z ) , with the same values of z = 1 , 2 , 5 , 10 and indications of real, imaginary, and absolute values. Right, their corresponding Wigner functions on the phase space torus. We note that at z 5 and z 10 , the Wigner function band reconnects with its copies.

Fig. 7
Fig. 7

Left, rectangle function Rect 5 , 5 ( m ) of Figs. 5, 6 under the third Fresnel transform ( F [ 3 ] ) z for integer powers z = 1 , 2 . Right, the corresponding Wigner functions on the phase space torus.

Fig. 8
Fig. 8

Left, geometric picture of the action of the Fresnel–Kravchuk transform F [ K ] ( a ) on the sphere, which is embedded in the meta-phase space ( m , k , μ ) R 3 of position, momentum, and (displaced) energy; the polar angles ( β , γ ) are determined by the position and energy axes. The transform wrings the sphere around the momentum k axis, rotating it differentially by angles 1 2 a k 2 , as shown by the empty arrows. Right: We represent the surface of the sphere by its coordinates ( β , γ ) drawn on a plane to present the su (2) Wigner function in a form directly comparable with the three previous versions of the discrete Fresnel transform on the torus. The center (B) corresponds to the “bottom pole” of ground energy at ( β = 1 2 π , γ = 0 ) . The ± k points ( β = 1 2 π , γ = ± 1 2 π ) are the two extreme “momentum poles.” The left and right lines at γ = ± π are the “top meridian” of the sphere. The top and bottom heavy lines correspond to the singular points β = 0 and π of the polar coordinate system at the upper and lower extremes of the position m axis.

Fig. 9
Fig. 9

Left, evolution of the rectangle function Rect 5 , 5 ( m ) under the Fresnel–Kravchuk transform F [ K ] ( a ) for N = 31 ( = 15 ) , a = z ( π N ) 2 0.010270 z , and z = 0 , 1 , 2 , 5 , 10 . Right, the corresponding Wigner functions on the su ( 2 ) phase space sphere ( β , γ ) ; it is displayed on the ( β , γ ) plane as explained in Fig. 8.

Equations (42)

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( x ( z ) p ( z ) ) = ( 1 z 0 1 ) ( x ( 0 ) p ( 0 ) ) .
F ( z ) exp ( i 1 2 z P 2 ) ,
F ( z ) ( X P ) F ( z ) = ( 1 z 0 1 ) ( X P ) = ( X + z P P ) .
( F ( z ) : f ) ( x ) = 1 2 π i z R d x exp ( i 2 z ( x x ) 2 ) f ( x ) .
W ( f x , p ) 1 2 π R d y f ( x y ) * exp ( 2 i y p ) f ( x + y ) .
W ( F ( z ) : f x , p ) = W ( f x + z p , p ) .
x 2 π m N , for m integer modulo N .
W ( f m , k ) 1 N n f m n * exp ( 4 π i N k n ) f m + n ,
g m = f m + m o W ( g m , k ) = W ( f m + m o , k ) ,
h m = e 2 π i k o m N f m W ( h m , k ) = W ( f m , k + k o ) .
f m = ± f m W ( f m , k ) = W ( f m , k ) ,
f m * = ± f m W ( f m , k ) = W ( f m , k ) .
Hf ( z ) = i d d z f ( z ) f ( z ) = F ( z ) f ( 0 ) , F ( z ) exp ( i z H ) .
Δ circ ( 2 , 1 , 0 , , 0 , 1 ) ,
Δ ̃ Φ Δ Φ 1 = diag ( 4 sin 2 ( π k N ) ) ,
Φ k , m 1 N exp ( 2 π i N k m ) ,
F [ 1 ] ( z ) exp ( i 1 2 z Δ ) = Φ 1 exp ( i 1 2 z Δ ̃ ) Φ = F m , m [ 1 ] ( z ) ,
F m , m [ 1 ] ( z ) = 1 N k exp ( 2 i z sin 2 π N k ) exp ( 2 π i N k ( m m ) ) ,
( 0 1 1 0 ) ( 1 0 z 1 ) ( 0 1 1 0 ) = ( 1 z 0 1 ) .
L ( z ) = diag ( exp [ 2 i z ( π m N ) 2 ] ) ,
F [ 2 ] ( z ) Φ 1 L ( z ) Φ = F m , m [ 2 ] ( z ) ,
F m , m [ 2 ] ( z ) = 1 N k exp [ 2 i z ( π k N ) 2 ] exp ( 2 π i N k ( m m ) ) ,
[ 1 1 0 1 ] = [ 1 0 1 1 ] [ 0 1 1 0 ] [ 1 0 1 1 ] ,
F m , m [ 3 ] 1 N exp ( i π N ( m m ) 2 ) .
position : X ¯ L 1 ,
momentum : P ¯ L 2 .
[ L 3 , X ¯ ] = i P ¯ , [ P ¯ , L 3 ] = i X ¯ , [ X ¯ , P ¯ ] = i L 3 .
H N ¯ + 1 2 1 L 3 + ( + 1 2 ) 1 ,
f m = , m f 1 , m .
K m , m ( α ) , m exp ( i 1 2 π α N ¯ ) , m 1 1 = e i π ( m m ) α 2 d m , m ( 1 2 π α ) ,
d m , m ( θ ) , m e i θ L 2 , m 3 3 = ( j + m ) ! ( j m ) ! ( j + m ) ! ( j m ) ! × n ( 1 ) n ( cos 1 2 θ ) 2 2 n + m m ( sin 1 2 θ ) 2 n m + m n ! ( + m n ) ! ( m n ) ! ( m m + n ) ! ,
F m , m [ K ] ( a ) , m exp ( i a 1 2 P ¯ 2 ) , m 1 1
= k = , m , k 2 1 , k exp ( i a 1 2 P ¯ 2 ) , k 2 2 , k , m 1 2
= ( i ) m + m k = d m , k ( 1 2 π ) ( 1 ) k exp ( i a 1 2 k 2 ) d k , m ( 1 2 π ) ,
i.e. , F [ K ] ( a ) K ( 1 ) diag ( exp ( i a 1 2 k 2 ) ) K ( 1 ) .
W ( v ) su ( 2 ) d g ( w ) exp i [ w 1 ( v 1 L 1 ) + w 2 ( v 2 L 2 ) + w 3 ( v 3 L 3 ) ] ,
w = ψ u ̂ ( θ , ϕ ) = ( w 1 w 2 w 3 ) = [ ψ sin θ sin ϕ ψ sin θ cos ϕ ψ cos θ ] , { 0 ψ < 2 π 0 θ π 0 ϕ < 2 π .
W ( ) ( f v ) f W ( v ) f = m , m = f m * W m , m ( ) ( v ) f m ,
W m , m ( ) ( v , β , γ ) = e i ( m m ) γ m ¯ = d m , m ¯ ( β ) W ¯ m ¯ ( ) ( v ) d m ¯ , m ( β ) .
W ¯ m ¯ ( ) ( v ) = ( 1 ) 2 + 1 π 2 m = 0 π sin β d β × d m ¯ , m ( β ) 2 sin ( 2 π v cos β ) ( v cos β m ) [ ( v cos β m ) 2 1 ] ,
f m = ± f m W ( ) ( f v , β , γ ) = W ( ) ( f v , π β , γ ) ,
f m * = ± f m W ( ) ( f v , β , γ ) = W ( ) ( f v , β , γ ) .

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