## Fractional Talbot effect in phase space: A compact summation formula

Optics Express, Vol. 2, Issue 5, pp. 169-172 (1998)

http://dx.doi.org/10.1364/OE.2.000169

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### 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.

© Optical Society of America

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]

*E*(

*x*) along the axis

*x*perpendicular to the propagation direction:

*u*has the meaning of a spatial frequency. We consider a monochromatic plane wave characterized by its electric field 𝜀(

*x*;

*z*) =

*e*(

^{ikz}E*x*;

*z*) propagated paraxially along the

*z*-axis. At

*z*= 0, where the field starts to propagate, there is an infinite one–dimensional periodic grating with transmittance

*t*(

*x*) =

*t*(

*x*+

*a*). The electric field amplitude after the passage through the grating can be expanded into the Fourier series:

*a*is the period of the grating and

*E*

_{0}is a constant amplitude of the incident wave. The Wigner distribution function of this source field is given by:

*u*

_{0}= 2

*π*/

*a*, given by the reciprocal grid spacing. Each of them is generated by a separate Fourier component of the source field. Coherence between the Fourier components results in nonpositive oscillatory terms of the Wigner function, located precisely in the middle between the contributing frequencies. The nonpositive interference terms are a consequence of the linear superposition principle and the bilinear character of the Wigner function.

*z*is described in the phase space as the following simple transformation of the Wigner distribution function [2

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

*z*=

_{T}*a*

^{2}/2λ is the Talbot distance. It is straightforward to see that for integer multiples of the Talbot distance the original Wigner function of the input field is reproduced.

*θ*play a nontrivial role and the structure of the observed Fresnel images becomes more complex. Nevertheless, they exhibit an interesting regular behavior at rational multiples of the Talbot distance. We will now discuss this effect in terms of the Wigner distribution function. Let us denote

_{n}*z*/

*z*=

_{T}*p*/

*q*, where

*p*and

*q*are coprime integers. The main complication in Eq. (5) are phase factors

*θ*which depend quadratically on

_{n}*n*. We will simplify it with the help of an observation used in the studies of quantum wave packets dynamics [3]: the exponent exp(-2

*πiθ*) is periodic in

_{n}*n*with the period

*l*=

*q*/4 if

*q*is a multiple of 4 and

*l*=

*q*otherwise. The quadratic phase factor can therefore be represented as a finite Fourier sum:

*a*, which have been analyzed in detail in Ref. [3]. Substitution of the above expression yields:

_{s}*sa*/

*l*, where

*s*= 0,1,… ,

*l*–1. A simple rearrangement of the exponent arguments allows one to represent the sum over

*n*and

*n*′ solely in terms of the source field Wigner function:

*W*(

*x*,

*u*;

*pz*/

_{T}*q*) is simply given by a

*finite*sum of spatially displaced Wigner functions of the source field, with some phase factors.

*u*yields the known formula for the field intensity distribution in the observation plane:

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]

*x*,

*p*) phase space:

*ψ*are the amplitudes of the initial wave function

_{n}*φ*(

*x*;0) projected onto the energy eigenstate

*φ*(

_{n}*x*), and where

*E*are given by a quadratic polynomial of

_{n}*n*. This can be either an approximate dependence, as it is for the Rydberg electron wave packets [6

6. J. Parker and C. R. Stroud Jr., “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]

## Acknowledgments

## References

1. | J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. |

2. | M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. |

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. |

4. | J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta |

5. | M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. |

6. | J. Parker and C. R. Stroud Jr., “Coherence and decay of Rydberg wave–packets,” Phys. Rev. Lett. |

7. | B. Yurke and D. Stoler, “Generating quantum–mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. |

8. | A. Mecozzi and P. Tombesi, “Distinguishable quantum states generated via nonlinear birefrigerence,” Phys. Rev. Lett. |

9. | K. Tara, G. S. Agarwal, and S. Chaturvedi, “Production of Schrödinger macroscopic quantum-superposition states in a Kerr medium,” Phys. Rev. A |

10. | D. L. Aronstein and C. R. Stroud, “Fractional wave–function revivals in the infinite square well,” Phys. Rev. A |

11. | M. Born and W. Ludwig, “Zur Quantenmechanik des kräftefreien Teilchens,” Z. Phys. |

12. | P. Stifter, C. Leichte, W. P. Schleich, and J. Marklof, “Das Teilchen im Kasten: Strukturen in der Wahrscheinlichkeitsdichte,” Z. Naturforsch. |

**OCIS Codes**

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

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 14, 1997

Revised Manuscript: November 6, 1997

Published: March 2, 1998

**Citation**

Konrad Banaszek, Krzysztof Wodkiewicz, and Wolfgang Peter Schleich, "Fractional Talbot effect in phase space: A compact
summation formula," Opt. Express **2**, 169-172 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-5-169

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### References

- 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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