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.

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References

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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).<br>
    [CrossRef]
  2. M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).<br>
    [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).<br>
  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).<br>
    [CrossRef]
  5. M. V. Berry and S. Klein, "Integer, fractional and fractal Talbot effects," J. Mod. Opt. 43, 2139-2164 (1996).<br>
    [CrossRef]
  6. J. Parker and C. R. Stroud, Jr., "Coherence and decay of Rydberg wave-packets," Phys. Rev. Lett. 56, 716-719 (1986).<br>
    [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).<br>
    [CrossRef] [PubMed]
  8. A. Mecozzi and P. Tombesi, "Distinguishable quantum states generated via nonlinear birefrigerence," Phys. Rev. Lett. 58, 1055-1058 (1987).<br>
    [CrossRef] [PubMed]
  9. K. Tara, G. S. Agarwal, and S. Chaturvedi, "Production of Schr"odinger macroscopic quantum-superposition states in a Kerr medium," Phys. Rev. A 47, 5024-5029 (1993).<br>
    [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).<br>
    [CrossRef]
  11. M. Born and W. Ludwig, "Zur Quantenmechanik des kr"aftefreien Teilchens," Z. Phys. 150, 106-117 (1958).<br>
    [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).

Other (12)

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).<br>
[CrossRef]

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

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).<br>

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).<br>
[CrossRef]

M. V. Berry and S. Klein, "Integer, fractional and fractal Talbot effects," J. Mod. Opt. 43, 2139-2164 (1996).<br>
[CrossRef]

J. Parker and C. R. Stroud, Jr., "Coherence and decay of Rydberg wave-packets," Phys. Rev. Lett. 56, 716-719 (1986).<br>
[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).<br>
[CrossRef] [PubMed]

A. Mecozzi and P. Tombesi, "Distinguishable quantum states generated via nonlinear birefrigerence," Phys. Rev. Lett. 58, 1055-1058 (1987).<br>
[CrossRef] [PubMed]

K. Tara, G. S. Agarwal, and S. Chaturvedi, "Production of Schr"odinger macroscopic quantum-superposition states in a Kerr medium," Phys. Rev. A 47, 5024-5029 (1993).<br>
[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).<br>
[CrossRef]

M. Born and W. Ludwig, "Zur Quantenmechanik des kr"aftefreien Teilchens," Z. Phys. 150, 106-117 (1958).<br>
[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).

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Equations (16)

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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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