## Transmission area and correlated imaging

Optics Express, Vol. 15, Issue 10, pp. 6062-6068 (2007)

http://dx.doi.org/10.1364/OE.15.006062

Acrobat PDF (111 KB)

### Abstract

The relationship between transmission area of an object imaged and the visibility of correlated imaging is investigated in a lensless system. We show that they are not in simple inverse proportion, as usually depicted. The changes of the visibility will be quite different when the transmission area is varied by different manners, which may motivate people to seek a new understanding about the influence factors of the visibility.

© 2007 Optical Society of America

## 1. Introduction

## 2. The model and analytical results

*E*(

*x*). After propagating through two different optical systems, the field has

*G*

^{(2)}(

*u*

_{1},

*u*

_{2})[4

4. 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**, 3600–3603 (1995). [CrossRef] [PubMed]

15. B. E. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A **62**, 043816 (2000). [CrossRef]

*E*(

*u*) (

_{i}*i*= 1,2) is the optical field in the test (reference) detector. Substituting Eq. (1) into Eq. (2), we have

*E*(

*x*

_{1})

*E*(

*x*

_{2})

*E**(

*x*́

_{2})

*E**(

*x*́

_{1})〉is the second-order correlation function at the light source, We represent it by

*G*

^{(2)}(

*x*

_{1},

*x*

_{2},

*x*́

_{2},

*x*́

_{1}) in the following in order to outline the parallelism with the formalism in Eq. (2).

*G*

^{(1)}(

*x*,

_{i}*x*) is the first-order correlation function of the fluctuating source field, and arbitrary order correlation function is thus expressed via the first-order correlation function

_{j}*G*

^{(1)}(

*x*,

_{i}*x*) = [

_{j}*G*

^{(1)}(

*x*,

_{j}*x*)]*. By using Eqs. (4) and (5), we can simplify Eq. (3) as

_{i}17. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) **177**, 27–29 (1956). [CrossRef]

10. Jing Cheng and Shensheng Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. **92**, 093903 (2004). [CrossRef] [PubMed]

*I*

_{1}

*I*

_{2}〉. By subtracting the background term 〈

*I*

_{1}〉〈

*I*

_{2}〉, we can obtain the correlation function of the intensity fluctuations, all information about the object is contained in it

## 3. Numerical results

*G*

_{0}is a normalized constant, a is the transverse size of the source.

*t*(

*x*́)) is located at a distance

*z*

_{1}from the source

*S*and a distance

*z*

_{2}from the detector

*D*. Thus the impulse response function can be expressed in the Fresnel-diffraction approximation as

_{t}*S*to

*D*. Thus the corresponding impulse response function under the paraxial approximation is

_{r}10. Jing Cheng and Shensheng Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. **92**, 093903 (2004). [CrossRef] [PubMed]

*d*= 0.15mm as the object imaged. The transverse size of the source

*a*= 1mm, other parameters are chosen as λ = 532nm,

*z*= 175mm,

*z*

_{1}= 75mm, and

*z*

_{2}=

*z*-

*z*

_{1}.

*u*

_{1}= 0 and substituting Eqs. (9)–(11

11. Da Zhang, Yanhua Zhai, Lingan Wu, and Xihao Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. **30**, 2354–2356 (2005). [CrossRef] [PubMed]

*G*

^{(2)}(

*u*

_{1}= 0,

*u*

_{2}). For simplicity, we assume that the object simply transmits the light over a region of area

*S*and stops it elsewhere. Here, we consider two methods to change the transmission area. Firstly, we increase the number of slits with the same period parameters. The interference-diffraction pattern is given in Fig. 2(a). From our simulations it clearly emerges that, under the given parameters, the visibility decreases slightly with an increase of the number of slits, i.e., the transmission area.

_{obj}*n*. Here, we only depict the dots for finite slits because the interference-diffraction pattern will be deformed when

*n*is much larger. It is clear that the visibility decreases with the increasing

*n*.

*d*= 0.15mm, the results are depicted in Fig. 3(a). An increase of the slit width leads to an increase of the visibility. It should be noticed, by comparing with the curves in Fig. 2(a), the change of the visibility by widening the slits is much bigger than that by increasing the number of slits. That can be simply explained as follows: the high frequency component almost has no changes when the number of slits is increased with the same period parameters, while the increasing slit width of the double-slit will make the high frequency component decrease rapidly, therefore the changes of visibility in Fig. 3 are much greater than those in Fig. 2.

*z*

_{0}from the thermal source in Ref. [14

14. 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**, 739–760 (2006). [CrossRef]

14. 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**, 739–760 (2006). [CrossRef]

19. 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**, 053802 (2006). [CrossRef]

15. B. E. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A **62**, 043816 (2000). [CrossRef]

*I*

_{1})〉〈

*I*

_{2}〉 in our experiment, we can draw the same conclusion that different variety manners of the transmission area will produce different effects on the visibility.

## 4. Conclusion

## Acknowledgements

## References and links

1. | P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringers with photon coincidence measurements,” Phys. Rev. A |

2. | A. V. Belinsky and D. N. Klyshko, “2-photon optics-diffraction, holography and transformation of 2-dimensional signals,” Sov. Phys. JETP |

3. | 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 |

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

5. | A. F. Abouraddy, B. E. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B |

6. | A. Gatti, E. Brambilla, and L. A. Lugiatio, “Correlated imaging, quantum and classical,” Phys. Rev. A |

7. | G. A. Barbosa, “Quantum images in double-slit experiments with spontaneous down-conversion light,” Phys. Rev. A |

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

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

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

11. | Da Zhang, Yanhua Zhai, Lingan Wu, and Xihao Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. |

12. | Yangjian Cai and Shiyao Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E |

13. | Yangjian Cai, Qiang Lin, and Shiyao Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. |

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

15. | B. E. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A |

16. | J. W. Goodman, |

17. | R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) |

18. | M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, Y. Bai, and S. Han, “Sub-wavelength Fourier-transform imaging of a pure-phase object with thermal light,” Phys. Lett. A will be published; arXiv.org e-Print archive, arXiv. quant-ph/0612060, (2006). |

19. | 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 |

**OCIS Codes**

(100.0100) Image processing : Image processing

(110.0110) Imaging systems : Imaging systems

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 31, 2007

Revised Manuscript: April 18, 2007

Manuscript Accepted: April 25, 2007

Published: May 2, 2007

**Citation**

Yanfeng Bai, Honglin Liu, and Shensheng Han, "Transmission area and correlated imaging," Opt. Express **15**, 6062-6068 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6062

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### References

- P. H. S. Ribeiro, S. Padua, J. C. Machado da Silva, and G. A. Barbosa, "Controlling the degree of visibility of Young’s fringers with photon coincidence measurements," Phys. Rev. A 49, 4176-4179 (1994). [CrossRef] [PubMed]
- A. V. Belinsky and D. N. Klyshko, "2-photon optics-diffraction, holography and transformation of 2-dimensional signals," Sov. Phys. JETP 78, 259-262 (1994).
- 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 53, R3429-R3432 (1995). [CrossRef]
- 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, 3600-3603 (1995). [CrossRef] [PubMed]
- A. F. Abouraddy, B. E. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002). [CrossRef]
- A. Gatti, E. Brambilla, and L. A. Lugiatio, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004). [CrossRef]
- G. A. Barbosa, "Quantum images in double-slit experiments with spontaneous down-conversion light," Phys. Rev. A 54, 4473-4478 (1996). [CrossRef] [PubMed]
- A. F. Abouraddy, B. E. Saleh, A. V. Sergienko, and M. C. Teich, "Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001). [CrossRef] [PubMed]
- R. S. Bennink, S. J. Bentley, R.W. Boyd, and J. C. Howell, "Quantum and classical coincidence imaging," Phys. Rev. Lett. 92, 033601 (2004). [CrossRef] [PubMed]
- Jing Cheng and Shensheng Han, "Incoherent coincidence imaging and its applicability in X-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004). [CrossRef] [PubMed]
- Da Zhang, Yanhua Zhai, Lingan Wu, and Xihao Chen, "Correlated two-photon imaging with true thermal light," Opt. Lett. 30, 2354-2356 (2005). [CrossRef] [PubMed]
- Yangjian Cai and Shiyao Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005). [CrossRef]
- Yangjian Cai, Qiang Lin, and Shiyao Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005). [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, 739-760 (2006). [CrossRef]
- B. E. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, "Duality between partial coherence and partial entanglement," Phys. Rev. A 62, 043816 (2000). [CrossRef]
- J. W. Goodman, Statictical Optics (Wiley, New York, 1985).
- R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in two coherent beams of light," Nature (London) 177, 27-29 (1956). [CrossRef]
- M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, Y. Bai, and S. Han, "Sub-wavelength Fourier-transform imaging of a pure-phase object with thermal light," Phys. Lett. A will be published; arXiv.org e-Print archive, arXiv. quant-ph/0612060, (2006).
- 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, 053802 (2006). [CrossRef]

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