## Interference from a nonlocal double-slit through one-photon process

Optics Express, Vol. 17, Issue 26, pp. 23672-23677 (2009)

http://dx.doi.org/10.1364/OE.17.023672

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### Abstract

In this paper, we report an interference experiment in which a spatially incoherent light source illuminates two spatially separated apertures, whose superposition at the same place forms a double-slit. The experimental result exhibits a well-defined interference fringe solely through intensity measurements, in agreement with the theoretical analysis by means of the first-order spatial interference of the incoherent light. Consequently, the nonlocal double-slit interference with thermal light should be attributed to the first-order spatial correlation of incoherent field.

© 2009 OSA

## 1. Introduction

1. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. **74**(18), 3600–3603 (1995). [CrossRef] [PubMed]

5. E. J. S. Fonseca, P. H. S. Ribeiro, S. Pádua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A **60**(2), 1530–1533 (1999). [CrossRef]

5. E. J. S. Fonseca, P. H. S. Ribeiro, S. Pádua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A **60**(2), 1530–1533 (1999). [CrossRef]

6. R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. **89**(11), 113601 (2002). [CrossRef] [PubMed]

24. L. Gao, J. Xiong, L. F. Lin, W. Wang, S. H. Zhang, and K. G. Wang, “Interference from nonlocal double-slit with pseudo-thermal light,” Opt. Commun. **281**(10), 2838–2841 (2008). [CrossRef]

24. L. Gao, J. Xiong, L. F. Lin, W. Wang, S. H. Zhang, and K. G. Wang, “Interference from nonlocal double-slit with pseudo-thermal light,” Opt. Commun. **281**(10), 2838–2841 (2008). [CrossRef]

25. G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. **96**(6), 063602 (2006). [CrossRef] [PubMed]

26. A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. **98**(3), 039301, discussion 039302 (2007). [CrossRef] [PubMed]

## 2. Experimental results

27. S. H. Zhang, L. Gao, J. Xiong, L. J. Feng, D. Z. Cao, and K. G. Wang, “Spatial interference: from coherent to incoherent,” Phys. Rev. Lett. **102**(7), 073904 (2009). [CrossRef] [PubMed]

27. S. H. Zhang, L. Gao, J. Xiong, L. J. Feng, D. Z. Cao, and K. G. Wang, “Spatial interference: from coherent to incoherent,” Phys. Rev. Lett. **102**(7), 073904 (2009). [CrossRef] [PubMed]

_{1}: one illuminates aperture A

_{1}in path 1 and the other illuminates aperture A

_{2}in path 2. BS

_{2}is a

_{1}, a wire with a diameter of

_{2}, a slit with a width of

_{1}, and their superposition at the same position forms a double-slit of slit width

_{2}and the CCD screen are located in the two focal planes of lens L. The equal-optical-path condition can assure that one photon interferes with itself after passing through the two arms. However, there are different diffraction configurations in the two arms and it is the key in the present scheme since balanced interferometer washes out the information of the object [27

27. S. H. Zhang, L. Gao, J. Xiong, L. J. Feng, D. Z. Cao, and K. G. Wang, “Spatial interference: from coherent to incoherent,” Phys. Rev. Lett. **102**(7), 073904 (2009). [CrossRef] [PubMed]

_{2}, which are recorded by either of two CCD cameras. As the ground glass rotates slowly, the interference patterns in the detection plane fluctuate randomly. However, if we average over a number of frames, a well-defined interference pattern emerges [27

**102**(7), 073904 (2009). [CrossRef] [PubMed]

_{1}and CCD

_{2}, respectively. We will show in the next section that the fringes fit well the Fourier spectrum of a double-slit, modulated by a quadratic phase factor. Since the interference terms at the two outgoing ports have a phase shift

*π*due to the reflection and transmission of the beamsplitter, the two fringes are complementary. For a

^{2}. In this case, the coherence time of the Na lamp is much shorter than the responses time of the CCD camera, so the interference patterns can appear directly on the CCD screen, as shown in Fig. 3 . The patterns are similar to that in Fig. 2, except for the slightly different spacings, owing to the different wavelengths of the two sources.

## 3. Theoretical analysis

_{1}and the field of path

*x*, the transverse position across the beam. The field diffraction in path

*j*is described aswhere the impulse response function

24. L. Gao, J. Xiong, L. F. Lin, W. Wang, S. H. Zhang, and K. G. Wang, “Interference from nonlocal double-slit with pseudo-thermal light,” Opt. Commun. **281**(10), 2838–2841 (2008). [CrossRef]

*k*is the wavenumber of the beam;

*j*and the interference term, respectively.

_{2}in the interferometer [27

**102**(7), 073904 (2009). [CrossRef] [PubMed]

*σ*characterizes the spot size. Using Eq. (6) and Eq. (4), we calculate 1D interference patterns in Fig. 6 . As we expected, the net interference pattern in Fig. 6 (c) corresponds to the product of the diffraction fields of the two apertures, and the intensity background in Fig. 6 (d) coincides with the sum of the two diffraction patterns of A

_{1}and A

_{2}. No information about the double-slit can be obtained when A

_{1}and A

_{2}are illuminated coherently. The theoretical curves are in a good agreement with the experimental results for the coherent light case of Fig. 4. A slight mismatch of some side-peaks comes from our simple laser model in the theoretical simulation.

## 4. Summary

## Acknowledgment

## References and links

1. | D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. |

2. | T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A |

3. | E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. |

4. | A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. |

5. | E. J. S. Fonseca, P. H. S. Ribeiro, S. Pádua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A |

6. | R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. |

7. | R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. |

8. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A |

9. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. |

10. | J. Cheng and S. S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. |

11. | K. G. Wang and D. Z. Cao, “Subwavelength coincidence interference with classical thermal light,” Phys. Rev. A |

12. | Y. J. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. |

13. | F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. |

14. | D. Z. Cao, J. Xiong, and K. G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A |

15. | A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. |

16. | J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. |

17. | D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. |

18. | Y. H. Zhai, X. H. Chen, D. Zhang, and L. A. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A |

19. | G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. |

20. | L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. |

21. | R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. |

22. | M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of a pure phase object with classical incoherent light,” Phys. Rev. A |

23. | A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. |

24. | L. Gao, J. Xiong, L. F. Lin, W. Wang, S. H. Zhang, and K. G. Wang, “Interference from nonlocal double-slit with pseudo-thermal light,” Opt. Commun. |

25. | G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. |

26. | A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. |

27. | S. H. Zhang, L. Gao, J. Xiong, L. J. Feng, D. Z. Cao, and K. G. Wang, “Spatial interference: from coherent to incoherent,” Phys. Rev. Lett. |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(050.1940) Diffraction and gratings : Diffraction

(090.2880) Holography : Holographic interferometry

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: September 28, 2009

Revised Manuscript: November 7, 2009

Manuscript Accepted: November 30, 2009

Published: December 10, 2009

**Citation**

Shu Gan, Su-Heng Zhang, Jun Xiong, and Kaige Wang, "Interference from a nonlocal double-slit through one-photon process," Opt. Express **17**, 23672-23677 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23672

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### References

- D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995). [CrossRef] [PubMed]
- T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995). [CrossRef] [PubMed]
- E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82(14), 2868–2871 (1999). [CrossRef]
- A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87(12), 123602 (2001). [CrossRef] [PubMed]
- E. J. S. Fonseca, P. H. S. Ribeiro, S. Pádua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A 60(2), 1530–1533 (1999). [CrossRef]
- R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002). [CrossRef] [PubMed]
- R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004). [CrossRef] [PubMed]
- A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70(1), 013802 (2004). [CrossRef]
- A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004). [CrossRef] [PubMed]
- J. Cheng and S. S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004). [CrossRef] [PubMed]
- K. G. Wang and D. Z. Cao, “Subwavelength coincidence interference with classical thermal light,” Phys. Rev. A 70(4), 041801 (2004). [CrossRef]
- Y. J. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29(23), 2716–2718 (2004). [CrossRef] [PubMed]
- F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005). [CrossRef] [PubMed]
- D. Z. Cao, J. Xiong, and K. G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71(1), 013801 (2005). [CrossRef]
- A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005). [CrossRef] [PubMed]
- J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005). [CrossRef] [PubMed]
- D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30(18), 2354–2356 (2005). [CrossRef] [PubMed]
- Y. H. Zhai, X. H. Chen, D. Zhang, and L. A. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72(4), 043805 (2005). [CrossRef]
- G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88(6), 061106 (2006). [CrossRef]
- L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. 89(9), 091109 (2006). [CrossRef]
- R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 96(18), 183901 (2006). [CrossRef] [PubMed]
- M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of a pure phase object with classical incoherent light,” Phys. Rev. A 73(5), 053802 (2006). [CrossRef]
- A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. 53(5-6), 739–760 (2006). [CrossRef]
- L. Gao, J. Xiong, L. F. Lin, W. Wang, S. H. Zhang, and K. G. Wang, “Interference from nonlocal double-slit with pseudo-thermal light,” Opt. Commun. 281(10), 2838–2841 (2008). [CrossRef]
- G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96(6), 063602 (2006). [CrossRef] [PubMed]
- A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. 98(3), 039301, discussion 039302 (2007). [CrossRef] [PubMed]
- S. H. Zhang, L. Gao, J. Xiong, L. J. Feng, D. Z. Cao, and K. G. Wang, “Spatial interference: from coherent to incoherent,” Phys. Rev. Lett. 102(7), 073904 (2009). [CrossRef] [PubMed]

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