Abstract

A phase space description of the fractional Talbot effect, occurring in a one–dimensional Fresnel diffraction from a periodic grating, is presented. Using the phase space formalism a compact summation formula for the Wigner function at rational multiples of the Talbot distance is derived. The summation formula shows that the fractional Talbot image in the phase space is generated by a finite sum of spatially displaced Wigner functions of the source field.

© 1998 Optical Society of America

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  1. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [Crossref]
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [Crossref]
  3. I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: universality in the long–term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A139, 449–453 (1989).
  4. J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
    [Crossref]
  5. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [Crossref]
  6. J. Parker and C. R. Stroud, “Coherence and decay of Rydberg wave–packets,” Phys. Rev. Lett. 56, 716–719 (1986).
    [Crossref] [PubMed]
  7. B. Yurke and D. Stoler, “Generating quantum–mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
    [Crossref] [PubMed]
  8. A. Mecozzi and P. Tombesi, “Distinguishable quantum states generated via nonlinear birefrigerence,” Phys. Rev. Lett. 58, 1055–1058 (1987).
    [Crossref] [PubMed]
  9. K. Tara, G. S. Agarwal, and S. Chaturvedi, “Production of Schrödinger macroscopic quantum-superposition states in a Kerr medium,” Phys. Rev. A 47, 5024–5029 (1993).
    [Crossref] [PubMed]
  10. D. L. Aronstein and C. R. Stroud, “Fractional wave–function revivals in the infinite square well,” Phys. Rev. A 55, 4526–4537 (1997).
    [Crossref]
  11. M. Born and W. Ludwig, “Zur Quantenmechanik des kräftefreien Teilchens,” Z. Phys. 150, 106–117 (1958).
    [Crossref]
  12. P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. 52a, 377–385 (1997).

1997 (2)

D. L. Aronstein and C. R. Stroud, “Fractional wave–function revivals in the infinite square well,” Phys. Rev. A 55, 4526–4537 (1997).
[Crossref]

P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. 52a, 377–385 (1997).

1996 (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

1993 (1)

K. Tara, G. S. Agarwal, and S. Chaturvedi, “Production of Schrödinger macroscopic quantum-superposition states in a Kerr medium,” Phys. Rev. A 47, 5024–5029 (1993).
[Crossref] [PubMed]

1989 (1)

I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: universality in the long–term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A139, 449–453 (1989).

1987 (1)

A. Mecozzi and P. Tombesi, “Distinguishable quantum states generated via nonlinear birefrigerence,” Phys. Rev. Lett. 58, 1055–1058 (1987).
[Crossref] [PubMed]

1986 (2)

J. Parker and C. R. Stroud, “Coherence and decay of Rydberg wave–packets,” Phys. Rev. Lett. 56, 716–719 (1986).
[Crossref] [PubMed]

B. Yurke and D. Stoler, “Generating quantum–mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref] [PubMed]

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

1971 (1)

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[Crossref]

1965 (1)

1958 (1)

M. Born and W. Ludwig, “Zur Quantenmechanik des kräftefreien Teilchens,” Z. Phys. 150, 106–117 (1958).
[Crossref]

Agarwal, G. S.

K. Tara, G. S. Agarwal, and S. Chaturvedi, “Production of Schrödinger macroscopic quantum-superposition states in a Kerr medium,” Phys. Rev. A 47, 5024–5029 (1993).
[Crossref] [PubMed]

Aronstein, D. L.

D. L. Aronstein and C. R. Stroud, “Fractional wave–function revivals in the infinite square well,” Phys. Rev. A 55, 4526–4537 (1997).
[Crossref]

Averbukh, I. Sh.

I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: universality in the long–term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A139, 449–453 (1989).

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

Berry, M. V.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

Born, M.

M. Born and W. Ludwig, “Zur Quantenmechanik des kräftefreien Teilchens,” Z. Phys. 150, 106–117 (1958).
[Crossref]

Chaturvedi, S.

K. Tara, G. S. Agarwal, and S. Chaturvedi, “Production of Schrödinger macroscopic quantum-superposition states in a Kerr medium,” Phys. Rev. A 47, 5024–5029 (1993).
[Crossref] [PubMed]

Guigay, J. P.

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[Crossref]

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

Leichte, C.

P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. 52a, 377–385 (1997).

Ludwig, W.

M. Born and W. Ludwig, “Zur Quantenmechanik des kräftefreien Teilchens,” Z. Phys. 150, 106–117 (1958).
[Crossref]

Marklof, J.

P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. 52a, 377–385 (1997).

Mecozzi, A.

A. Mecozzi and P. Tombesi, “Distinguishable quantum states generated via nonlinear birefrigerence,” Phys. Rev. Lett. 58, 1055–1058 (1987).
[Crossref] [PubMed]

Parker, J.

J. Parker and C. R. Stroud, “Coherence and decay of Rydberg wave–packets,” Phys. Rev. Lett. 56, 716–719 (1986).
[Crossref] [PubMed]

Perelman, N. F.

I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: universality in the long–term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A139, 449–453 (1989).

Schleich, W. P.

P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. 52a, 377–385 (1997).

Stifter, P.

P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. 52a, 377–385 (1997).

Stoler, D.

B. Yurke and D. Stoler, “Generating quantum–mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref] [PubMed]

Stroud, C. R.

D. L. Aronstein and C. R. Stroud, “Fractional wave–function revivals in the infinite square well,” Phys. Rev. A 55, 4526–4537 (1997).
[Crossref]

J. Parker and C. R. Stroud, “Coherence and decay of Rydberg wave–packets,” Phys. Rev. Lett. 56, 716–719 (1986).
[Crossref] [PubMed]

Tara, K.

K. Tara, G. S. Agarwal, and S. Chaturvedi, “Production of Schrödinger macroscopic quantum-superposition states in a Kerr medium,” Phys. Rev. A 47, 5024–5029 (1993).
[Crossref] [PubMed]

Tombesi, P.

A. Mecozzi and P. Tombesi, “Distinguishable quantum states generated via nonlinear birefrigerence,” Phys. Rev. Lett. 58, 1055–1058 (1987).
[Crossref] [PubMed]

Winthrop, J. T.

Worthington, C. R.

Yurke, B.

B. Yurke and D. Stoler, “Generating quantum–mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref] [PubMed]

J. Mod. Opt. (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[Crossref]

Opt. Commun. (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

Phys. Lett. (1)

I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: universality in the long–term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A139, 449–453 (1989).

Phys. Rev. A (2)

K. Tara, G. S. Agarwal, and S. Chaturvedi, “Production of Schrödinger macroscopic quantum-superposition states in a Kerr medium,” Phys. Rev. A 47, 5024–5029 (1993).
[Crossref] [PubMed]

D. L. Aronstein and C. R. Stroud, “Fractional wave–function revivals in the infinite square well,” Phys. Rev. A 55, 4526–4537 (1997).
[Crossref]

Phys. Rev. Lett. (3)

J. Parker and C. R. Stroud, “Coherence and decay of Rydberg wave–packets,” Phys. Rev. Lett. 56, 716–719 (1986).
[Crossref] [PubMed]

B. Yurke and D. Stoler, “Generating quantum–mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref] [PubMed]

A. Mecozzi and P. Tombesi, “Distinguishable quantum states generated via nonlinear birefrigerence,” Phys. Rev. Lett. 58, 1055–1058 (1987).
[Crossref] [PubMed]

Z. Naturforsch. (1)

P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. 52a, 377–385 (1997).

Z. Phys. (1)

M. Born and W. Ludwig, “Zur Quantenmechanik des kräftefreien Teilchens,” Z. Phys. 150, 106–117 (1958).
[Crossref]

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Equations (16)

Equations on this page are rendered with MathJax. Learn more.

W ( x , u ) = 1 2 π d y E * ( x + y 2 ) e iuy E ( x y 2 ) ,
E ( x ; 0 ) = t ( x ) E 0 = n = t n e 2 πinx / a ,
W ( x , u ; 0 ) = n = t n 2 δ ( u n u 0 )
+ n n t n t n * exp [ 2 π i ( n n ) x / a ] δ ( u n + n 2 u 0 ) .
W ( x , u ; z ) = W ( x λ z 2 π u , u ; 0 ) .
W ( x , u ; z ) = n = t n 2 δ ( u n u 0 )
+ n n t n t n * exp [ 2 π i ( n n ) x / a 2 π i ( θ n θ n ) ] δ ( u n + n 2 u 0 ) .
θ n = z z T n 2 ,
exp ( 2 π i θ n ) = s = 0 l 1 a s exp ( 2 πisn l )
W ( x , u ; p z T q ) = s , s = 0 l 1 a s a s * n , n = t n t n * exp [ 2 πin a ( x s a l ) ]
× exp [ 2 πin a ( x s a l ) ] δ ( u n + n 2 u 0 ) .
W ( x , u ; p z T q ) = s , s = 0 l 1 a s a s * exp [ i u ( s s ) a l ] W ( x ( s + s ) a 2 l , u ; 0 ) .
E ( x ; p z T q ) 2 = d u W ( x , u ; p z T q )
= s = 0 l 1 a s t ( x s a l ; 0 ) 2 ,
W ψ ( x , p ; t ) = n , n ψ n * ψ n exp [ i ( E n E n ) t ħ ] W n n ( x , p ) ,
W nn ( x , p ) = 1 2 π ħ d y φ n * ( x + y 2 ) e ipy ħ φ n ( x y 2 )

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