Abstract

We examine the evolution in phase space of an N-point signal, produced and sensed at finite arrays transverse to a planar waveguide within the framework of the finite quantization of geometric optics. We use the Kravchuk coherent states provided by the finite oscillator model to evince the nonlinear transformations that elliptic-profile waveguides produce on phase space by means of the SO(3) Wigner function.

© 2011 Optical Society of America

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References

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  1. K. B. Wolf, “Refracting surfaces between fibers,” J. Opt. Soc. Am. A 8, 1389–1398 (1991).
    [CrossRef]
  2. K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
    [CrossRef]
  3. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  4. Ç. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
    [CrossRef]
  5. A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
    [CrossRef]
  6. S.-C. Pei and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
    [CrossRef]
  7. J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).
    [CrossRef]
  8. K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).
  9. N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
    [CrossRef]
  10. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521–555 (2005).
  11. N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
    [CrossRef]
  12. S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “The Wigner 23 function for general Lie groups and the wavelet transform,” Ann. Henri Poincaré 1, 685–714 (2000).
    [CrossRef]
  13. A. J. Dragt, “Lie algebraic theory of geometrical optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
    [CrossRef]
  14. A. J. Dragt, E. Forest, and K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics.,” in Lie Methods in Optics, Lecture Notes in Physics (Springer, 1986), Vol.  250, pp. 105–158.
    [CrossRef]
  15. K. B. Wolf, “Discrete systems and signals on phase space,” Appl. Math. Inf. Sci. 4, 141–181 (2010).
  16. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley-Interscience, 1978).
  17. L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol.  8.
  18. N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
    [CrossRef]
  19. M. Krawtchouk, “Sur une généralization des polinômes d’Hermite,” C. R. Acad. Sci. Paris Ser. IV 189, 620–622 (1929).
  20. A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics (Springer, 1991).
  21. N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1990).
    [CrossRef]
  22. 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]
  23. 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]
  24. S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to a specified accuracy,” SIAM J. Matrix Anal. Appl. 22, 1112–1125 (2001).
    [CrossRef]
  25. L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
    [CrossRef]
  26. C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210(2009).
    [CrossRef]
  27. K. B. Wolf and G. Krötzsch, “Metaxial correction of fractional Fourier transformers,” J. Opt. Soc. Am. A 16, 821–830 (1999).
    [CrossRef]
  28. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  29. H.-W. Lee, “Theory and applications of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]

2010 (2)

J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).
[CrossRef]

K. B. Wolf, “Discrete systems and signals on phase space,” Appl. Math. Inf. Sci. 4, 141–181 (2010).

2009 (1)

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210(2009).
[CrossRef]

2008 (2)

K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

2007 (1)

2005 (1)

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521–555 (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]

2001 (1)

S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to a specified accuracy,” SIAM J. Matrix Anal. Appl. 22, 1112–1125 (2001).
[CrossRef]

2000 (2)

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “The Wigner 23 function for general Lie groups and the wavelet transform,” Ann. Henri Poincaré 1, 685–714 (2000).
[CrossRef]

Ç. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

1999 (3)

S.-C. Pei and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

K. B. Wolf and G. Krötzsch, “Metaxial correction of fractional Fourier transformers,” J. Opt. Soc. Am. A 16, 821–830 (1999).
[CrossRef]

1998 (2)

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[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]

1997 (1)

1995 (1)

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

1991 (1)

1990 (1)

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1990).
[CrossRef]

1982 (1)

1932 (1)

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

1929 (1)

M. Krawtchouk, “Sur une généralization des polinômes d’Hermite,” C. R. Acad. Sci. Paris Ser. IV 189, 620–622 (1929).

Ali, S. T.

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “The Wigner 23 function for general Lie groups and the wavelet transform,” Ann. Henri Poincaré 1, 685–714 (2000).
[CrossRef]

Atakishiyev, N. M.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521–555 (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 23 function for general Lie groups and the wavelet transform,” Ann. Henri Poincaré 1, 685–714 (2000).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[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]

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1990).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol.  8.

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Candan, Ç.

Ç. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Cheng, S. H.

S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to a specified accuracy,” SIAM J. Matrix Anal. Appl. 22, 1112–1125 (2001).
[CrossRef]

Chumakov, S. M.

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “The Wigner 23 function for general Lie groups and the wavelet transform,” Ann. Henri Poincaré 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]

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

Dragt, A. J.

A. J. Dragt, “Lie algebraic theory of geometrical optics and optical aberrations,” J. Opt. Soc. Am. 72, 372–379 (1982).
[CrossRef]

A. J. Dragt, E. Forest, and K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics.,” in Lie Methods in Optics, Lecture Notes in Physics (Springer, 1986), Vol.  250, pp. 105–158.
[CrossRef]

Forest, E.

A. J. Dragt, E. Forest, and K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics.,” in Lie Methods in Optics, Lecture Notes in Physics (Springer, 1986), Vol.  250, pp. 105–158.
[CrossRef]

Gilmore, R.

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

Healy, J. J.

Higham, N. J.

S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to a specified accuracy,” SIAM J. Matrix Anal. Appl. 22, 1112–1125 (2001).
[CrossRef]

Kenney, C. S.

S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to a specified accuracy,” SIAM J. Matrix Anal. Appl. 22, 1112–1125 (2001).
[CrossRef]

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Krawtchouk, M.

M. Krawtchouk, “Sur une généralization des polinômes d’Hermite,” C. R. Acad. Sci. Paris Ser. IV 189, 620–622 (1929).

Krötzsch, G.

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Ç. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Laub, A. J.

S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to a specified accuracy,” SIAM J. Matrix Anal. Appl. 22, 1112–1125 (2001).
[CrossRef]

Lee, H.-W.

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

Louck, J. D.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol.  8.

Muñoz, C. A.

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210(2009).
[CrossRef]

Nieto, L. M.

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

Nikiforov, A. F.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics (Springer, 1991).

Ozaktas, H. M.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Ç. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Pei, S.-C.

S.-C. Pei and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Pogosyan, G. S.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521–555 (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]

Rueda-Paz, J.

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210(2009).
[CrossRef]

Sheridan, J. T.

Suslov, S. K.

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1990).
[CrossRef]

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics (Springer, 1991).

Tseng, C.-C.

S.-C. Pei and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Uvarov, V. B.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics (Springer, 1991).

Vicent, L. E.

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

Wigner, E. P.

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

Wolf, K. B.

K. B. Wolf, “Discrete systems and signals on phase space,” Appl. Math. Inf. Sci. 4, 141–181 (2010).

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210(2009).
[CrossRef]

K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
[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]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521–555 (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 23 function for general Lie groups and the wavelet transform,” Ann. Henri Poincaré 1, 685–714 (2000).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

K. B. Wolf and G. Krötzsch, “Metaxial correction of fractional Fourier transformers,” J. Opt. Soc. Am. A 16, 821–830 (1999).
[CrossRef]

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[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]

K. B. Wolf, “Refracting surfaces between fibers,” J. Opt. Soc. Am. A 8, 1389–1398 (1991).
[CrossRef]

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

A. J. Dragt, E. Forest, and K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics.,” in Lie Methods in Optics, Lecture Notes in Physics (Springer, 1986), Vol.  250, pp. 105–158.
[CrossRef]

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Ann. Henri Poincaré (1)

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “The Wigner 23 function for general Lie groups and the wavelet transform,” Ann. Henri Poincaré 1, 685–714 (2000).
[CrossRef]

Appl. Math. Inf. Sci. (1)

K. B. Wolf, “Discrete systems and signals on phase space,” Appl. Math. Inf. Sci. 4, 141–181 (2010).

C. R. Acad. Sci. Paris Ser. IV (1)

M. Krawtchouk, “Sur une généralization des polinômes d’Hermite,” C. R. Acad. Sci. Paris Ser. IV 189, 620–622 (1929).

IEEE Trans. Signal Process. (3)

Ç. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

S.-C. Pei and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (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. Comput. Appl. Math. (1)

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[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 (5)

J. Phys. A (3)

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210(2009).
[CrossRef]

K. B. Wolf, “Linear transformations and aberrations in continuous and in finite systems,” J. Phys. A 41, 304026 (2008).
[CrossRef]

Phys. Part. Nucl. (1)

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521–555 (2005).

Phys. Rep. (1)

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

Phys. Rev. (1)

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

SIAM J. Matrix Anal. Appl. (1)

S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to a specified accuracy,” SIAM J. Matrix Anal. Appl. 22, 1112–1125 (2001).
[CrossRef]

Theor. Math. Phys. (1)

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1990).
[CrossRef]

Other (6)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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

A. J. Dragt, E. Forest, and K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics.,” in Lie Methods in Optics, Lecture Notes in Physics (Springer, 1986), Vol.  250, pp. 105–158.
[CrossRef]

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

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol.  8.

A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics (Springer, 1991).

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Figures (8)

Fig. 1
Fig. 1

Planar waveguide where a discrete signal is produced at one end by a transverse linear array of phase-controlled LEDs and read at the other end by a similar array of sensors.

Fig. 2
Fig. 2

Top left, refractive index n ν , μ ( x ) of an elliptic-index profile waveguide [Eq. (6)]; the mathematical boundary is n ν , μ = 0 , while the physical one is n = 1 . Top right, rays starting from the center of the guide at various angles p = ν sin θ < ν ; these trajectories do not reconvene at a single point, so this guide is dispersive. Bottom, evolution of the phase space points ( p ( z ) , x ( z ) ) for 1 4 and 1 2 cycle of the paraxial period [Eq. (10)], z = ν / 2 π μ .

Fig. 3
Fig. 3

Sphere of Kravchuk coherent states ϒ ( θ , ϕ ) j ( x m ) in Eq. (28). This is also the phase-space representation of the finite system, with the axes of position x, momentum p, and pseudoenergy κ (see Appendix A). At the bottom pole, κ = r is the ground state ϒ 0 j ( x m ) = Φ 0 j ( x m ) , and, at the top pole, κ = r is the top state ϒ π j ( x m ) = Φ 2 j j ( x m ) . At the + x pole, the state ϒ π / 2 j ( x m ) is the Kronecker state ( 0 , , 0 , 1 ) ; at the + p pole, it is the z evolution of the latter after a 1 4 cycle, z = 1 2 π .

Fig. 4
Fig. 4

Evolution under U K ( z ) of the coherent state Υ θ j ( m ) for j = 10 and θ = 1 radian. Top to bottom, z = 1 3 π n , n = 0 , 1 , , 6 . Left, each signal on a vertical x m axis; right, the corresponding SO ( 3 ) –Wigner function (see Appendix A). For visibility, the real, imaginary, and absolute values of the discrete signal points are joined by dashed, dotted, and continuous curves, respectively.

Fig. 5
Fig. 5

Eigenvalues of the Hamiltonian matrices and their squares for 0 μ ν . Top row, ν = 1 (the Kravchuk guide); bottom row, ν = 2 for j = 10 ( N = 21 points). Left column, squared eigenvalues { ( η ν , μ ) 2 } of the squared matrices ( H ν , μ ) 2 . Right column, eigenvalues { η ν , μ } of the matrices H ν , μ with the sign assignment scheme given in the text. The ground state eigenvalues ( η 0 ν , μ ) 2 and η 0 ν , μ are marked with a heavy curve. Note that, once separated by the sign, the η n ν , μ curves do not cross. Only the Kravchuk case ν = 1 = μ has equally spaced eigenvalues.

Fig. 6
Fig. 6

Left, pseudoground states Ψ ˜ 0 ( x m ) in Eq. (33) for j = 10 , ν = 2 (for visibility, point values are joined by straight lines), and values of μ = 0.4 , 0.5 , , 1.3 , and μ = 2 marked with a heavy line. Right, pseudotop states Ψ ˜ 2 j ( x m ) in Eq. (34) for the same values of the parameters. Kravchuk ground and top states lie at μ = 1 . Note that widths grow as the waveguide becomes wider (μ decreases).

Fig. 7
Fig. 7

Evolution of the Kravchuk coherent state ϒ θ j ( m ) for θ = 1 radian and j = 10 under the waveguide Hamiltonian H ν , μ in Eq. (30) for ν = 2 and μ = 1 2 2 . Markings for the signals and its Wigner functions are the same as in Fig. 4. In the left column, we show one approximate cycle for z = 1 3 π c n , n = 0 , 1 , , 6 , with the scale factor c = 2.8284 . After several oscillations, in the right column, we show a sequence of two- and three-component catlike states for z = 1 16 315 π + 3 16 π n .

Fig. 8
Fig. 8

Left, polar coordinates ( β , γ ) of the sphere refer to the x axis of positions. Right, projection of the coordinates ( β , γ ) of the sphere onto the rectangle ( 0 β π , π < γ π ) . In this projection, the bottom pole ( 1 2 π , 0 ) is at the center of the rectangle, the x axis of positions is vertical; the boundary ( 0 , γ ) represents the single point x j and ( π , γ ) represents x j . The p axis of momenta is horizontal, with its left and right boundaries identified, so the top pole corresponds to ( 1 2 π , π ) ( 1 2 π , π ) . The heavy lines divide phase space into octants.

Equations (42)

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h ( x , p , z ) = p z = n ( x , z ) cos θ = n ( x , z ) 2 p 2 ,
d x d z = h p = { h , x } = p n 2 p 2 = tan θ ,
d p d z = h x = { h , p } = 1 n 2 p 2 n 2 x = 2 n x sec θ ,
d h d z = h z = 1 n 2 p 2 n 2 z = 2 n z sec θ ,
{ f 1 , f 2 } ( x , p ) f 1 x f 2 p f 1 p f 2 x .
n ν , μ ( x ) + ν 2 μ 2 x 2 , ν 1 , μ 0 , | x | ν / μ .
h ν , μ ( x , p ) = ν 2 ( p 2 + μ 2 x 2 ) .
( p ( z ) x ( z ) ) = exp ( z { h , } ) ( p x ) n = 0 ( z ) n n ! { h , } n ( p x ) ,
= ( cos ( μ z / h ( x , p ) ) μ sin ( μ z / h ( x , p ) ) sin ( μ z / h ( x , p ) ) / μ cos ( μ z / h ( x , p ) ) ) ( p x ) ,
exp ( z 2 ν { p 2 + μ 2 x 2 , } ) ( p x ) = ( cos ( μ z / ν ) μ sin ( μ z / ν ) sin ( μ z / ν ) / μ cos ( μ z / ν ) ) ( p x ) .
( p ( z ) x ( z ) ) = ( 1 0 z / ν 2 p 2 1 ) ( p x ) = ( p x + z p / ν 2 p 2 ) .
{ k , x } = p , { k , p } = x .
[ K , X ] = i P , [ K , P ] = i X .
[ X , P ] = i K .
C X 2 + P 2 + K 2 = j ( j + 1 ) 1 ,
X m , m = m δ m , m , m , m { j , j + 1 , , j } ,
P m , m = i 1 2 ( j m ) ( j + m + 1 ) δ m + 1 , m + i 1 2 ( j + m ) ( j m + 1 ) δ m 1 , m ,
K m , m = 1 2 ( j m ) ( j + m + 1 ) δ m + 1 , m + 1 2 ( j + m ) ( j m + 1 ) δ m 1 , m ,
C X 2 + P 2 + K 2 = j ( j + 1 ) 1 .
( f , g ) f g = m = j j f m * g m = ( g , f ) * , | f | = ( f , f ) .
X x m = x m x m , K k κ = κ k κ .
Φ n j ( x m ) ( x m , k n j ) = d n j , m j ( 1 2 π ) ,
( x m , K Φ n j ) = κ Φ n j ( x m ) ,
Φ n j ( x m ) = ( 1 ) n 2 j ( 2 j n ) ( 2 j j + m ) K n ( j + m ; 1 2 , 2 j + 1 ) = Φ m j ( x n ) .
U K ( z ) exp ( i z K ) U ( N ) ,
U m , m K ( z ) = i m m d m , m j ( z ) = e i j z n = 0 2 j Φ n j ( x m ) e i n z Φ n j ( x m )
Φ 0 j ( x m ) = 1 2 j ( 2 j j + m ) = ( 1 ) m Φ 2 j j ( x m ) > 0 .
ϒ θ j ( x m ) m = j j ( exp ( i θ P ) ) m , m Φ 0 j ( x m ) = d j , m j ( 1 2 π + θ ) ,
K = ± j ( j + 1 ) 1 ( P 2 + X 2 ) .
H ν , μ ν 2 C ( P 2 + μ 2 X 2 ) = [ ( ν 2 1 ) C ( μ 2 1 ) X 2 ] + K 2 ,
H 2 Ψ n = ( η n ) 2 Ψ n , Ψ n ( x m ) = ( x m , Ψ n ) ,
1 4 ( j m 1 ) ( j m ) ( j + m + 1 ) ( j + m + 2 ) Ψ η ( m + 2 ) + [ j ( j + 1 ) ( ν 2 1 2 ) m 2 ( μ 2 1 2 ) η 2 ] Ψ η ( m ) + 1 4 ( j + m 1 ) ( j + m ) ( j m + 1 ) ( j m + 2 ) Ψ η ( m 2 ) = 0 .
Ψ ˜ 0 ( x m ) Ψ 0 ( x m ) + Ψ 2 j ( x m ) ,
Ψ ˜ 2 j ( x m ) Ψ 0 ( x m ) Ψ 2 j ( x m ) ,
Ψ H 2 Ψ = D 2 , H ν , μ Ψ D Ψ , H ν , μ Ψ n = η n Ψ n .
U ν , μ ( z ) exp ( i z H ν , μ ) U ( N ) ,
R ( ρ , θ , ϕ ; x , p , κ ) exp [ i ( u x + v p + w κ ) ] , { u = ρ sin θ sin ϕ , v = ρ sin θ cos ϕ , w = ρ cos θ ,
R ( ρ , θ , ϕ ) exp [ i ( u X + v P + w K ) ] ,
W ( x , p , κ ) SO ( 3 ) d ( ρ , θ , ϕ ) exp i [ u ( x X ) + v ( p P ) + w ( κ K ) ] ,
p = r sin β sin γ , κ = r sin β cos γ , x = r cos β .
W SO ( 3 ) ( f | β , γ ) f W ( β , γ ) f = m , m = j j f m * W m , m ( β , γ ) f m .
W m , m ( β , γ ) = e i ( m m ) γ m ¯ = j j d m , m ¯ j ( β ) W ¯ m ¯ j d m ¯ , m j ( β ) ,

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