## Vectorial coherence holography |

Optics Express, Vol. 19, Issue 12, pp. 11558-11567 (2011)

http://dx.doi.org/10.1364/OE.19.011558

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### Abstract

Extension of coherence holography to vectorial regime is investigated. A technique for controlling and synthesizing optical fields with desired elements of coherence-polarization matrix is proposed and experimentally demonstrated. The technique uses two separate coherence holograms, each of which is assigned to one of the orthogonal polarization components of the vectorial fields.

© 2011 OSA

## 1. Introduction

1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express **13**(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629. [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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633. [CrossRef] [PubMed]

1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express **13**(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629. [CrossRef] [PubMed]

6. 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]

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

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

5. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109. [CrossRef] [PubMed]

6. 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]

7. E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. **29**(11), 1233–1235 (2004). [CrossRef] [PubMed]

8. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. **208**(1093), 263–277 (1951). [CrossRef]

21. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. **29**(6), 536–538 (2004). [CrossRef] [PubMed]

23. A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express **17**(3), 1746–1752 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-3-1746. [CrossRef] [PubMed]

27. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. **213**(4-6), 201–221 (2002). [CrossRef]

28. D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express **19**(2), 1408–1421 (2011). [CrossRef] [PubMed]

28. D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express **19**(2), 1408–1421 (2011). [CrossRef] [PubMed]

## 2. Principle

1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express **13**(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629. [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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633. [CrossRef] [PubMed]

*x*-polarized light and

*y*-polarized light with their Fourier spectra

*f*so that the fields on the observation becomes

_{T}. We can also think of a similar characteristic of random fields in space domain. Under the assumption of spatial ergodicity, one may replace ensemble averages < > with space averages < >

_{R}. Depending on the averaging process, different experimental schemes can be adopted for the investigation of the stochastic field. As an alternative to generating a temporally fluctuating random field by using a rotating ground glass and taking its time average, one can generate a spatially fluctuating random field by passing light through a static ground glass and take its space average. Assuming that the field is stationary and ergodic in space and noting that

*x*and

*y*components pass through the same ground glass which is free from birefringence, i.e.

## 3. Experiments

_{1}and pinhole S and subsequently collimated by lens L

_{1}. The collimated beam splits into two arms by non-polarizing beam splitter BS

_{1}. The first arm of the interferometer (indicated by a blue line) produces two orthogonally polarized and mutually tilted reference beams with the help of a triangular Sagnac polarization interferometer geometry with a telescopic system formed by lens L2 and L3. The beam linearly polarized at

_{4}.

_{6}. Holograms of the orthogonal polarization components are placed in such a way that both orthogonally polarized beams carry replicas of both holograms as images on the ground glass. Tilt of mirror M

_{6}is adjusted in such a way that both polarization components pick up holograms of different polarization components in overlapping area. Only the light inside this overlapping area is allowed to pass through the ground glass by a small aperture, and consequently holograms of both orthogonal polarization components experience same random structure of the ground glass. Scattering of the beam through the random glass plate produces a speckle pattern with Gaussian statistics. The split ratio of the amplitudes between the orthogonal polarization components on the hologram is adjusted by rotating half wave plate HWP2.

*θ*is the azimuthal angle on the transverse plane. The parameter

*w*represents the size of the dark core, which is selected to be 12 pixels. The helical phase structure of the vortices is represented by integer

*m*called topological charge. The values of

*m*are chosen to be 1 and −1 for

*x*and

*y*polarized objects, respectively. Two different off-axis positions are selected for the case of the numeral objects. For example the objects 0 and 1 are placed at two corners in the first and second quadrants of the rectangular window with size 162X162 pixels. Optical field of the object on the hologram plane is obtained by Fourier transform and this is expressed as:Here

*x*- or

*y*-polarization component of the orthogonally polarized objects, and

*ψ*the phase of the diffracted field on the Fourier plane. In synthesizing the computer generated holograms (CGH), we remove from the interference fringe intensity the term

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

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

## 4. Results

*x*and

*y*polarized objects for orthogonally polarized CGHs and the objects were reconstructed in the form of the spatial distributions of the elements of the coherence-polarization matrix as shown in Fig. 4. Due to Eq. (5) for the reconstruction based on spatial averaging and Eq. (8) for the hologram transmittancethe x and y polarized objects are reconstructed as the 2-D amplitude distributions of

*x*and

*y*components

*f*are mutually translated by

*x*and

*y*polarized objects with unit and opposite topological charge. These results due to the cross-correlation of the

*x*and

*y*polarized field, which leads to two side lobes for this particular condition. Position and number of lobes in non-diagonal terms of the polarization- coherence matrix is controlled by topological charge of the vortex encoded into holograms.

## 5. Conclusions

## Acknowledgments

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

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

5. | 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 |

6. | 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. |

7. | E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. |

8. | H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

9. | H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

10. | H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. |

11. | A. W. McCollough and G. M. Gallatin, “Illumination system with spatially controllable partial coherence compensation for line width variance in a photolithographic system,” US Patent 6628370 B1 (2003). |

12. | D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express |

13. | E. Wolf, |

14. | J. W. Goodman, |

15. | J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett. |

16. | J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express |

17. | F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. |

18. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

19. | J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express |

20. | O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. |

21. | J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. |

22. | F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. |

23. | A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express |

24. | T. Shirai, “Some consequences of the van Cittert-Zernike theorem for partially polarized stochastic electromagnetic fields,” Opt. Lett. |

25. | G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. |

26. | M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A |

27. | M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. |

28. | D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express |

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

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

31. | D. N. Naik, R. K. Singh, H. Itou, Y. Miyamoto, and M. Takeda, “A highly stable interferometric technique for polarization measurement,” 156, Photonics (December 2010), IIT Guwahati, India. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(090.2880) Holography : Holographic interferometry

(260.5430) Physical optics : Polarization

**ToC Category:**

Holography

**History**

Original Manuscript: March 31, 2011

Revised Manuscript: May 16, 2011

Manuscript Accepted: May 16, 2011

Published: May 31, 2011

**Citation**

Rakesh Kumar Singh, Dinesh N. Naik, Hitoshi Itou, Yoko Miyamoto, and Mitsuo Takeda, "Vectorial coherence holography," Opt. Express **19**, 11558-11567 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11558

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### References

- M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629 . [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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633 . [CrossRef] [PubMed]
- J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000). [CrossRef]
- P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. 48(34), H40–H47 (2009). [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. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109 . [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]
- E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29(11), 1233–1235 (2004). [CrossRef] [PubMed]
- H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951). [CrossRef]
- H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953). [CrossRef]
- H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).
- A. W. McCollough and G. M. Gallatin, “Illumination system with spatially controllable partial coherence compensation for line width variance in a photolithographic system,” US Patent 6628370 B1 (2003).
- D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-25-20418 . [CrossRef] [PubMed]
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts-Company, 2006).
- J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett. 34(16), 2429–2431 (2009). [CrossRef] [PubMed]
- J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-19-20105 . [CrossRef] [PubMed]
- F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
- J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-4-1063 . [CrossRef] [PubMed]
- O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). [CrossRef] [PubMed]
- J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004). [CrossRef] [PubMed]
- F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000). [CrossRef]
- A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express 17(3), 1746–1752 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-3-1746 . [CrossRef] [PubMed]
- T. Shirai, “Some consequences of the van Cittert-Zernike theorem for partially polarized stochastic electromagnetic fields,” Opt. Lett. 34(23), 3761–3763 (2009). [CrossRef] [PubMed]
- G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]
- M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009). [CrossRef]
- M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002). [CrossRef]
- D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011). [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]
- M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
- D. N. Naik, R. K. Singh, H. Itou, Y. Miyamoto, and M. Takeda, “A highly stable interferometric technique for polarization measurement,” 156, Photonics (December 2010), IIT Guwahati, India.

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