## Coherence holography by achromatic 3-D field correlation of generic thermal light with an imaging Sagnac shearing interferometer |

Optics Express, Vol. 20, Issue 18, pp. 19658-19669 (2012)

http://dx.doi.org/10.1364/OE.20.019658

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### Abstract

We propose a new technique for achromatic 3-D field correlation that makes use of the characteristics of both axial and lateral magnifications of imaging through a common-path Sagnac shearing interferometer. With this technique, we experimentally demonstrate, for the first time to our knowledge, 3-D image reconstruction of coherence holography with generic thermal light. By virtue of the achromatic axial shearing implemented by the difference in axial magnifications in imaging, the technique enables coherence holography to reconstruct a 3-D object with an axial depth beyond the short coherence length of the thermal light.

© 2012 OSA

## 1. Introduction

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

4. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express **18**(13), 13782–13787 (2010). [CrossRef] [PubMed]

5. W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. **41**(10), 1962–1971 (2002). [CrossRef] [PubMed]

6. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express **14**(25), 12109–12121 (2006). [CrossRef] [PubMed]

7. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. **39**(23), 4107–4111 (2000). [CrossRef] [PubMed]

8. P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. **48**(34), H40–H47 (2009). [CrossRef] [PubMed]

9. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. **96**(7), 073902 (2006). [CrossRef] [PubMed]

2. 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]

4. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express **18**(13), 13782–13787 (2010). [CrossRef] [PubMed]

10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**(1), 156–160 (1982). [CrossRef]

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

11. M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. **3**(7), 853–857 (1964). [CrossRef]

2. 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]

*path difference without time delay*, as will be explained in the following. With the inherent stability of a common path interferometer and controllable magnification introduced by variable shear, it functions as a robust device for correlating optical fields to detect the 3-D coherence function that represents the object recorded in the coherence hologram.

## 2. Principles

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

### 2.1 Generation of phase-shifted coherence holograms

*f.*A conceptual diagram of the generation of the hologram is shown in Fig. 2 .

*f*is the distance between z = 0 and the Fourier transform lens for recording, which is made equal to the focal length of the Fourier transform lens L used in the reconstruction process. The innermost integral inside the curly brace represents the angular spectra of the object field distribution across the plane

*z*with

*m*= 0,1,2,...,N. In synthesizing the computer-generated coherence hologram, we removed from the interference fringe intensity the term

### 2.2 Reconstruction using phase-shift coherence holography

*m*, which allows one to determine the amplitude and the phase of the object

### 2.3 Radial and axial shearing for 3-D field correlation

### 2.4 Measurement of purely spatial coherence of optical field

*c*being the velocity of light. In other words, the lack of temporal coherence manifests itself as the degradation of spatial coherence through the propagation time delay introduced by the longitudinal shear. However, this influence of the short temporal coherence of the light source can be automatically corrected in our imaging Sagnac shearing interferometer. Figure 5 shows two counter propagating beams in the Sagnac interferometer with their directions of propagation made equal and with their common output plane located at

## 3. Experiment

## 4. Results

14. P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas. **49**(6), 1189–1193 (2000). [CrossRef]

*path difference without time delay*has been achieved by virtue of the achromatic common-path imaging Sagnac interferometer.

## 5. Conclusion

## Acknowledgment

## References and links

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

2. | 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 |

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

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

5. | W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. |

6. | Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express |

7. | J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. |

8. | P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. |

9. | W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. |

10. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

11. | M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. |

12. | M. Born and E. Wolf, |

13. | J. W. Goodman, |

14. | P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(090.0090) Holography : Holography

(100.3010) Image processing : Image reconstruction techniques

**ToC Category:**

Holography

**History**

Original Manuscript: June 6, 2012

Manuscript Accepted: July 24, 2012

Published: August 13, 2012

**Citation**

Dinesh N. Naik, Takahiro Ezawa, Rakesh Kumar Singh, Yoko Miyamoto, and Mitsuo Takeda, "Coherence holography by achromatic 3-D field correlation of generic thermal light with an imaging Sagnac shearing interferometer," Opt. Express **20**, 19658-19669 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-19658

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### References

- M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express13(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. Express17(13), 10633–10641 (2009). [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]
- D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express18(13), 13782–13787 (2010). [CrossRef] [PubMed]
- W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt.41(10), 1962–1971 (2002). [CrossRef] [PubMed]
- Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express14(25), 12109–12121 (2006). [CrossRef] [PubMed]
- J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt.39(23), 4107–4111 (2000). [CrossRef] [PubMed]
- P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt.48(34), H40–H47 (2009). [CrossRef] [PubMed]
- W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett.96(7), 073902 (2006). [CrossRef] [PubMed]
- M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982). [CrossRef]
- M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt.3(7), 853–857 (1964). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Chap. 10.
- J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5.
- P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas.49(6), 1189–1193 (2000). [CrossRef]

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