## Non-linear Young’s double-slit experiment

Optics Express, Vol. 14, Issue 7, pp. 2817-2824 (2006)

http://dx.doi.org/10.1364/OE.14.002817

Acrobat PDF (180 KB)

### Abstract

The Young’s double slit experiment is recreated using intense and short laser pulses. Our experiment evidences the role of the non-linear Kerr effect in the formation of interference patterns. In particular, our results evidence a mixed mechanism in which the zeroth diffraction order of each slit are mainly affected by self-focusing and self-phase modulation, while the higher orders propagate linearly. Despite of the complexity of the general problem of non-linear propagation, we demonstrate that this experiment retains its simplicity and allows for a geometrical interpretation in terms of simple optical paths. In consequence, our results may provide key ideas on experiments on the formation of interference patterns with intense laser fields in Kerr media.

© 2006 Optical Society of America

## 1. Introduction

1. Thomas Young, “Experimental Demonstration of the General Law of the Interference of Light,” Philosophical Transactions of the Royal society of London **94**, 1–16 (1804) [CrossRef]

2. C. JÖnsson, “Elektroneninterferenzen an mehreren künstlich hergestellter Feinspalten,” Zeitschrift fur Physik **161**454’474 (1961) [CrossRef]

3. A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, “Demonstration of single-electron buildup of an interference pattern,” Am. J. Phys. **57**, 117–120 (1989) [CrossRef]

4. A. Zelinger, R. Gähler, C.G. Shull, W. Treimer, and W. Mampe, “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys. **60**1067–1073 (1988) [CrossRef]

5. O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. **66**2689–2692 (1991) [CrossRef] [PubMed]

6. M.W. Noel and C.R. Stroud, “Young’s Double-Slit Interferometry within an Atom,” Phys. Rev. Lett. **75**1252–1255 (1995) [CrossRef] [PubMed]

7. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature **401**680–682, (1999) [CrossRef]

8. A. Einstein,“00FC;ber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt,” Annalen der Physik **17**, 132 (1905) [CrossRef]

9. R.A. Millikan, “A Direct Photoelectric Determination of Planck’s “h”,” Phys. Rev. **7**, 355 (1916) [CrossRef]

10. A.H. Compton, “A Quantum Theory of the Scattering of X-rays by Light Elements,” Phys. Rev.21, 483; 22, 409 (1923) [CrossRef]

*independently*. The relevance of Young’s experiment also relies in its simple geometrical interpretation in terms of optical paths, which gives key ideas to the understanding of more elaborated situations, as diffraction gratings, Fresnel lenses, etcetera. Precisely, the simplicity of this experiment has stimulated us to repeat it in the non-linear regime trying to find more insight on the basic properties of the non-linear propagation of light.

*A priori*we may expect the non-linear propagation to affect most of the fundamental phenomena involved in Young’s experiment. For instance, in Kerr media diffraction is counterbalanced by self-focusing, the lengths of the optical paths depend on self-phase modulation and their trajectories on the field induced refraction index. In addition, the superposition principle does not longer applies, as the field does not result from the sum of its components evolving independently. Fortunately, the double slit scheme turns out to be also useful in this regime. As happens in the linear case, the simplicity of the interference process permits to develop a conceptual analysis which may constitute a basic understanding of the more complex phenomena. In the following section we will describe the experimental setup and results of the double slit experiment in the linear and non-linear regime. Next, we will develop a simple geometrical picture to explain the divergences with the non-linear case.

## 2. Experimental setup and results

12. A detailed description of the system can be obtained at the group web page http:\\optica.usal.es

13. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Optics Letters **20**, 73–75 (1995) [CrossRef] [PubMed]

14. C. Ruiz, J. San Roman, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso,“Observation of Spontaneous Self-Channeling of Light in Air below the Collapse Threshold,” Phys. Rev. Lett. **95**053905 (2005). [CrossRef] [PubMed]

*μ*J after the double-slit) is shown in Fig. 2(a). We have plotted the detected intensity integrated over the vertical direction, at different distances from the slit. As expected from the linear theory, after some distance the propagation approaches the Fraunhofer far field limit and three interference maxima appear. The location of these maxima have been contrasted with the corresponding interference pattern derived theoretically from the Fresnel propagator. This allows us to estimate the initial beam divergence of 5×10

^{-5}

*rad*in our experiment.

## 3. Theoretical model and interpretation

*E*(

*x*,

*y*,

*z*,

*t*) =

*U*(

*x*,

*y*,

*z*)

*e*

^{-iωt}at any spatial point is given by the sum of the contributions of each slit,

*U*(

*x*,

*y*,

*z*) =

*U*

_{+}(

*x*,

*y*,

*z*) +

*U*

_{-}(

*x*,

*y*,

*z*), which propagate independently according to (Fresnel approximation)

*k*

_{+}and

*k*

_{-}, although in the linear case

*k*

_{+}=

*k*

_{-}=

*n*

_{0}

*ω*/

*c*with

*n*

_{0}the linear index of refraction of air.

*z*

_{1}from the slit. As the intensity of this component is small, after leaving the central part of the field it travels effectively through a linear medium, until reaching the central field belonging to the neighbor slit, at a distance

*z*>

*z*

_{1}. On the other hand, the central field evolving either to distance

*z*

_{1}or

*z*suffers from the self-phase modulation associated with the changes of its intensity profile. In the non-linear case these two different trajectories have associated a change in the optical paths which can be approximated to

*n*

_{2}= 3.2 × 10

^{-19}

*cm*

^{2}/

*W*defines the non-linear part of the refraction index for air, and

*I*(

*z*) is the effective local intensity at z, averaged in the transversal coordinates. Note that the small value of

*n*

_{2}implies that the effect of self-phase modulation will only be apparent over large distances. Therefore only the z component is relevant. Following these ideas, we may propose a simple extension of the linear propagator to describe the field interference leading to each of the lateral fringes. For instance, the fringe in the positive half-plane (

*x*> 0) results from the overlap between the zeroth diffraction order in

*U*

_{+}and the first diffraction order in

*U*

_{-}. According to Eq. (2), we should replace

*k*

_{+}

*z*in

*U*

_{+}by (

*ω*/

*c*)ΔΓ(

*z*) to account for the field dephase due to the non-linearity.

*n*

_{0}+

*n*

_{2}

*I*(

*z*), corresponding to the zeroth diffraction orders of each slit, and the coordinate

*z*

_{1}in which the first diffraction orders are effectively detached from them. The appropriate values for these factors have to be estimated from the experiment according to a best fit criteria between the model and the experimental findings. To do this, we treat

*z*

_{1}as a fitting parameter, and approximate ΔΓ(

*z*) to first order as ΔΓ(

*z*) =

*α*(

*z*-

*z*

_{1}). We have, therefore, found a best fit of the parameters

*z*

_{1}and

*α*to describe the deviation of the divergence of the lateral fringes in the interference pattern of the non-linear propagation compared with the linear case. The results forz1 = 3.5 m and α = -10

^{-8}are shown in Fig. 3. Note that the value

*z*

_{1}is perfectly consistent with our interpretation as the coordinate where the first order diffraction maxima are detached from the central field (situation close to the represented in Fig. 4(a)). In fact, the actual choice of parameters does not result very critical, except for the sign of

*α*. A positive

*α*leads to an increase of the divergence of the lateral fringes with the intensity, which has not been observed in our experiments. Moreover, a negative

*α*is consistent with the intuition of a decrease of the intensity in the radiation channels with the distance to the slits, which should lead also to a gradual decrease in the non-linear refraction index.

## 4. Discussion

15. D.J. Mitchell, A.W. Snyder, and L. Poladian,“Interacting Self-Guided Beams wiewed as Particles: Lorentz Force Derivation,” Phys. Rev. Lett. **77**271–273 (1996) [CrossRef] [PubMed]

## 5. Conclusion

15. D.J. Mitchell, A.W. Snyder, and L. Poladian,“Interacting Self-Guided Beams wiewed as Particles: Lorentz Force Derivation,” Phys. Rev. Lett. **77**271–273 (1996) [CrossRef] [PubMed]

## References and links

1. | Thomas Young, “Experimental Demonstration of the General Law of the Interference of Light,” Philosophical Transactions of the Royal society of London |

2. | C. JÖnsson, “Elektroneninterferenzen an mehreren künstlich hergestellter Feinspalten,” Zeitschrift fur Physik |

3. | A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, “Demonstration of single-electron buildup of an interference pattern,” Am. J. Phys. |

4. | A. Zelinger, R. Gähler, C.G. Shull, W. Treimer, and W. Mampe, “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys. |

5. | O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. |

6. | M.W. Noel and C.R. Stroud, “Young’s Double-Slit Interferometry within an Atom,” Phys. Rev. Lett. |

7. | M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature |

8. | A. Einstein,“00FC;ber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt,” Annalen der Physik |

9. | R.A. Millikan, “A Direct Photoelectric Determination of Planck’s “h”,” Phys. Rev. |

10. | A.H. Compton, “A Quantum Theory of the Scattering of X-rays by Light Elements,” Phys. Rev.21, 483; 22, 409 (1923) [CrossRef] |

11. | G. Méchain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz,“Organizing Multiple Femtosecond Filaments in Air,” Phys. Rev. Lett. |

12. | A detailed description of the system can be obtained at the group web page http:\\optica.usal.es |

13. | A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Optics Letters |

14. | C. Ruiz, J. San Roman, C. Mendez, V. Diaz, L. Plaja, I. Arias, and L. Roso,“Observation of Spontaneous Self-Channeling of Light in Air below the Collapse Threshold,” Phys. Rev. Lett. |

15. | D.J. Mitchell, A.W. Snyder, and L. Poladian,“Interacting Self-Guided Beams wiewed as Particles: Lorentz Force Derivation,” Phys. Rev. Lett. |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 1, 2006

Revised Manuscript: March 17, 2006

Manuscript Accepted: March 28, 2006

Published: April 3, 2006

**Citation**

Julio San Roman, Camilo Ruiz, Jose Antonio Perez, Diego Delgado, Cruz Mendez, Luis Plaja, and Luis Roso, "Non-linear Young's double-slit experiment," Opt. Express **14**, 2817-2824 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2817

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### References

- T. Young, "Experimental demonstration of the general law of the interference of light," Philos. Trans. R. Soc. London 94, 1-16 (1804).Q1 [CrossRef]
- C. Jönsson, "Elektroneninterferenzen an mehreren künstlich hergestellter Feinspalten," Zeitschrift fur Physik 161454-474 (1961). [CrossRef]
- A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, "Demonstration of single-electron buildup of an interference pattern," Am. J. Phys. 57, 117-120 (1989). [CrossRef]
- A. Zelinger, R. Gähler, C. G. Shull, W. Treimer and W. Mampe, "Single- and double-slit diffraction of neutrons," Rev. Mod. Phys. 601067-1073 (1988). [CrossRef]
- O. Carnal and J. Mlynek, "Young’s double-slit experiment with atoms: A simple atom interferometer," Phys. Rev. Lett. 662689-2692 (1991). [CrossRef] [PubMed]
- M. W. Noel and C. R. Stroud, "Young’s double-slit interferometry within an Atom," Phys. Rev. Lett. 751252-1255 (1995). [CrossRef] [PubMed]
- M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw and A. Zeilinger, "Wave-particle duality of C60 molecules," Nature 401680-682, (1999). [CrossRef]
- A. Einstein, "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt," Annalen der Physik 17, 132 (1905).Q2 [CrossRef]
- R. A. Millikan, "A direct photoelectric determination of Planck’s "h"," Phys. Rev. 7, 355 (1916). [CrossRef]
- A. H. Compton, "A quantum theory of the scattering of x-rays by light elements," Phys. Rev. 21, 483; 22, 409 (1923). [CrossRef]
- G. Méchain, A. Couairon, M. Franco, B. Prade and A. Mysyrowicz, "Organizing multiple femtosecond filaments in air," Phys. Rev. Lett. 93, 035003 (2004). [CrossRef] [PubMed]
- A detailed description of the system can be obtained at the group web page http:\\optica.usal.es.
- A. Braun, G. Korn, X. Liu, D. Du, J. Squier ang and G. Mourou, "Self-channeling of high-peak-power femtosecond laser pulses in air," Opt. Lett. 20, 73-75 (1995). [CrossRef] [PubMed]
- C. Ruiz, J. San Roman, C. Mendez, V. Diaz, L. Plaja, I. Arias and L. Roso, "Observation of spontaneous self-channeling of light in air below the collapse threshold," Phys. Rev. Lett. 95053905 (2005). [CrossRef] [PubMed]
- D. J. Mitchell, A. W. Snyder and L. Poladian, "Interacting self-guided beams viewed as particles: Lorentz Force Derivation," Phys. Rev. Lett. 77271-273 (1996). [CrossRef] [PubMed]

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