## A conceptual experiment on single-beam coincidence detection with pseudothermal light

Optics Express, Vol. 15, Issue 19, pp. 12386-12394 (2007)

http://dx.doi.org/10.1364/OE.15.012386

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### Abstract

Ghost imaging produced by pseudothermal light is commonly obtained by correlating the intensities of two separate beams, neither of which conveys information about the shape of the object to be imaged. The single-beam experiment discussed here, while not exploitable for the practical purpose of reconstructing the shape of a real mask, uses the same mathematical machinery as two-beam experiments; it also suggests that image retrieval by classical light ghost imaging is only a product of normal signal processing and does not involve any “ghost”. In addition, the single-beam setup allows simpler calibration procedures in systematic investigations of the efficiency of coincidence imaging.

© 2007 Optical Society of America

## 1. Introduction

## 2. A summary of incoherent light ghost imaging obtained using a bucket detector behind the mask

## 3. Mathematical analysis of the single beam experiment

**(s), whose s**

*W*^{th}component is the sum of the values of

*F*_{S}(i,j) for all pixels (i,j) belonging to

**; using the characteristic funtion of**

*M***(s) can be calculated as follows:**

*M, W***(s) will play the role of statistical weight for the s**

*W*^{th}speckle pattern

**S (i,j).**

*F***and their statistical weights**

*F***. To this end, all values**

*W*

*F*_{S}(i,j) are multiplied by

**(s) to obtain, at each “time” s, the weighted pattern Φ**

*W*_{S}(i,j), i.e.:

_{S}(i,j) is then executed to obtain the final result,

**(i,j):**

*R***(i,j) for the weighted sum:**

*R*_{F}is δ-like:

**R**(i,j) is simply

*χ*M (i,j), namely the shape of the subset

**.**

*M**χ*M of the subset

**that was used in the single-beam experiment;**

*M***; in the two-beam case, a real mask is located in a separate arm but this requires the creation of a twin copy of the speckle pattern passing through the mask. The final results output by the two setups are identical because, in both cases, the same mathematical operations are performed on the same sequences of numbers.**

*M*## 4. Statistical results from the single beam experiment: variance and visibility of the weighted sum

^{th}pixel on the s

^{th}trial W(s) has been defined above

*R*at pixels inside the subset M is higher than outside by the quantity Nµ

^{2}. This difference is the signal to be detected.

^{2}. This result is interesting because it shows that the required ensemble size rapidly increases with the size of the target; the same expression allows to assess the minimum number of independent patterns that must be summed in order to obtain a distinct perception of the “image”. To see whether this request is reasonable, let’s try a subset M consisting of, say, 30 pixels; the order of magnitude of N must be then of the order of 104, an estimate that agrees both with experiments and simulations.

**simply becomes, in two dimensions, the area of the subset M; this prediction, too, has been successfully tested by supplementary experiments described in Sect. 5.2.**

*M*## 5. Experimental results

### 5.1 Reconstruction of a two-hole mask

**can be retrieved by correlating each speckle pattern**

*M***with the total intensity**

*F***of the same speckle falling on**

*W***(lower part of Fig. 2).**

*M*

*F*_{S}(i,j) to which the single-beam processing is applied, was recorded when the ground disk stopped in the position it reached after a pulse of fixed amplitude and random duration (between 0.5s and 10s) was supplied to the driving motor. Moreover, in order to grant the maximum mutual independence of the stored speckle patterns, the whole set consisted of a large number of groups; each of these was obtained by imparting a random lateral shift to the disk’s position.

### 5.2. Experimental dependence of visibility and variance on the area of the subset M

**area)/(coherence area) ranging from about 0.5 up to about 22. The expected behaviour is: Visibility=1/ν**

*M*_{s}where ν

_{s}is the average number of speckles contained within the subset

**.**

*M*_{(in)}(upper graph) and R

_{(out)}(lower graph) depend on the same areal ratio used in Fig. 4. It is natural for R

_{(out)}to give a more satisfactory agreement with theory because it is calculated using an area much more extended than the subset

**. In both graphs, the theoretical fit is made using the variance expressions reported in Table I.**

*M*## 6. Conclusion

## References and links

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

2. | R. S. Bennink, S. J. Bentley, and R. W. Boyd, “two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. |

3. | A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato,” Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation,” Phys. Rev. Lett. |

4. | 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. |

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

6. | A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in Two-Photon Imaging,” Phys. Rev. Lett. |

7. | G. Scarcelli, V. Berardi, Y. Shih, A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, Phys. Rev. Lett.98, 039301 (2007), and Scarcelli, Berardi, and Shih, Reply, Phys. Rev. Lett.98, 039302 (2007). [CrossRef] |

8. | W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. |

9. | L. Basano and P. Ottonello, “Ghost imaging: open secrets and puzzles for undergraduates,” Am. J. Phys. |

10. | In this paper we make reference only to ghost imaging and not to ghost interference or diffraction |

11. | The term “correlation” commonly employed in ghost imaging actually means “zero-delay cross-correlation”; in other words, two sequences are being “correlated” when they are multiplied term by term (without relative shift) and summed. |

**OCIS Codes**

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

(100.0100) Image processing : Image processing

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: May 18, 2007

Revised Manuscript: July 11, 2007

Manuscript Accepted: July 13, 2007

Published: September 14, 2007

**Citation**

Lorenzo Basano and Pasquale Ottonello, "A conceptual experiment on single-beam coincidence detection with pseudothermal light," Opt. Express **15**, 12386-12394 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12386

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### References

- 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, R3429 (1995).
- R. S. Bennink, S. J. Bentley and R. W. Boyd, "two-photon coincidence imaging with a classical source," Phys. Rev. Lett. 89, 113601 (2002). [CrossRef] [PubMed]
- A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato," Ghost Imaging with Thermal Light: Comparing Entanglement and Classical Correlation," Phys. Rev. Lett. 93, 093602 (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, 183602 (2005). [CrossRef] [PubMed]
- A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]
- 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, 123602 (2001). [CrossRef] [PubMed]
- G. Scarcelli, V. Berardi, and Y. Shih, "Can Two-Photon Correlation of Chaotic Light be considered as Correlation of Intensity Fluctuations?," Phys. Rev. Lett. 96, 063602 (2006). Comment by A. Gatti, M. Bondani, L. A.Lugiato, M. G. A. Paris and C. Fabre, Phys. Rev. Lett. 98, 039301 (2007), and Scarcelli, Berardi, and Shih, Reply, Phys. Rev. Lett. 98, 039302 (2007). [CrossRef]
- W. Martienssen and E. Spiller, "Coherence and fluctuations in light beams," Am. J. Phys. 32, 919 (1964). [CrossRef]
- L. Basano, and P. Ottonello, "Ghost imaging: open secrets and puzzles for undergraduates," Am. J. Phys. 75, 343 (2007). [CrossRef]
- In this paper we make reference only to ghost imaging and not to ghost interference or diffraction
- The term "correlation" commonly employed in ghost imaging actually means "zero-delay cross-correlation"; in other words, two sequences are being "correlated" when they are multiplied term by term (without relative shift) and summed.

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