## Photon correlation holography |

Optics Express, Vol. 19, Issue 2, pp. 1408-1421 (2011)

http://dx.doi.org/10.1364/OE.19.001408

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### Abstract

Unconventional holography called photon correlation holography is proposed and experimentally demonstrated. Using photon correlation, i.e. intensity correlation or fourth order correlation of optical field, a 3-D image of the object recorded in a hologram is reconstructed stochastically with illumination through a random phase screen. Two different schemes for realizing photon correlation holography are examined by numerical simulations, and the experiment was performed for one of the reconstruction schemes suitable for the experimental proof of the principle. The technique of photon correlation holography provides a new insight into how the information is embedded in the spatial as well as temporal correlation of photons in the stochastic pseudo thermal light.

© 2011 OSA

## 1. Introduction

1. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. **52**(10), 1123–1128 (1962). [CrossRef]

2. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express **13**(23), 9629–9635 (2005). [CrossRef] [PubMed]

3. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express **17**(13), 10633–10641 (2009). [CrossRef] [PubMed]

5. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. **35**(10), 1728–1730 (2010). [CrossRef] [PubMed]

## 2. Principles

### 2.1 Generation of hologram

*λ*is the wavelength of light,

*f*the focal length of the Fourier transform lens L, and the range of integrations extends to

*z*with

### 2.2 Reconstruction of the hologram using fourth order correlation

*t*, respectively, whereas

*t*and

**r**due to the assumption of stationarity of the field

*t*or by space averaging based on integration over

**r**.

### 2.3 Ensemble average replaced by time average

*t*appears only in the instantaneous random phase

*t*. Assuming ideal spatially incoherent illumination with pseudo thermal light that obeys Gaussian statistics, we have

**r**, which indicates stationarity of the statistical process. Hence from Eq. (7), we have for the cross-covariance

### 2.4 Ensemble average replaced by space average

**r**in observation space. In this case, the space average results in a delta function

13. J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. **97**(5), 2963–2978 (1995). [CrossRef]

14. B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory **43**(4), 1273–1276 (1997). [CrossRef]

13. J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. **97**(5), 2963–2978 (1995). [CrossRef]

14. B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory **43**(4), 1273–1276 (1997). [CrossRef]

2. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express **13**(23), 9629–9635 (2005). [CrossRef] [PubMed]

1. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. **52**(10), 1123–1128 (1962). [CrossRef]

## 3. Simulation

### 3.1 Reconstruction using time average

*Γ*and their conjugate images, respectively. Note that 3-D depth information is preserved like conventional holography.

### 3.2 Reconstruction using space average

*Γ*chosen as object are confined to

*Γ*and their conjugate images are reconstructed like conventional holography. The total of about

## 4. Experiment

*Γ*and their conjugate images are reconstructed as predicted by theory and numerical simulations. About

## 5. Conclusions

## Acknowledgement

## References and links

1. | E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. |

2. | M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express |

3. | D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express |

4. | D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express |

5. | D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. |

6. | R. Hanbury Brown and R. Q. Twiss, “Correlations between photons in 2 coherent beams of light,” Nature |

7. | R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” J. Mod. Opt. |

8. | 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 |

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

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

11. | J. W. Goodman, |

12. | J. W. Goodman, |

13. | J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. |

14. | B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(030.6600) Coherence and statistical optics : Statistical optics

(090.0090) Holography : Holography

(100.3010) Image processing : Image reconstruction techniques

**ToC Category:**

Holography

**History**

Original Manuscript: November 17, 2010

Manuscript Accepted: December 27, 2010

Published: January 12, 2011

**Citation**

Dinesh N. Naik, Rakesh Kumar Singh, Takahiro Ezawa, Yoko Miyamoto, and Mitsuo Takeda, "Photon correlation holography," Opt. Express **19**, 1408-1421 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1408

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### References

- E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52(10), 1123–1128 (1962). [CrossRef]
- M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005). [CrossRef] [PubMed]
- D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009). [CrossRef] [PubMed]
- D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010). [CrossRef] [PubMed]
- D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010). [CrossRef] [PubMed]
- R. Hanbury Brown and R. Q. Twiss, “Correlations between photons in 2 coherent beams of light,” Nature 177(4497), 27–29 (1956). [CrossRef]
- R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” J. Mod. Opt. 16(4), 423–451 (1969).
- 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]
- 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]
- 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]
- J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 6.
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006), Chap. 4.
- J. W. Dalle Molle and M. V. Hinich, “Trispectral analysis of stationary random time series,” J. Acoust. Soc. Am. 97(5), 2963–2978 (1995). [CrossRef]
- B. Picinbono, “Ergodicity and fourth-order spectral moments,” IEEE Trans. Inf. Theory 43(4), 1273–1276 (1997). [CrossRef]

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