## 3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift

Optics Express, Vol. 17, Issue 13, pp. 10633-10641 (2009)

http://dx.doi.org/10.1364/OE.17.010633

Acrobat PDF (1259 KB)

### Abstract

A new image reconstruction scheme for coherence holography using a modified Sagnac-type radial shearing interferometer with geometric phase shift is proposed, and the first experimental demonstration of generic Leith-type coherence holography, which reconstructs off-axis 3-D objects with depth information, is presented. The reconstructed image, represented by a coherence function, can be visualized with a controllable magnification, which opens up a new possibility for a coherence imaging microscope

© 2009 Optical Society of America

## 1. Introduction

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

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

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

4. 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**, 073902 (2006).
[CrossRef] [PubMed]

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

4. 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**, 073902 (2006).
[CrossRef] [PubMed]

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

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**, 12109–12121 (2006).
[CrossRef] [PubMed]

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

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**, 12109–12121 (2006).
[CrossRef] [PubMed]

## 2. Principle

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

*(*

**I**_{S}**r**) rather than the amplitude transmittance of the hologram is here made proportional to the recorded interference fringe intensity. The recording process of a coherence hologram is identical to that of a conventional hologram, in which a coherently illuminated object at point

_{S}**r**is recorded with a reference beam from a point source at

_{P}**r**, as shown in blue in Fig. 1.

_{R}**(**

*J***r**,

_{Q}**r**)=〈

_{R}*E*(

**r**)

_{Q}*E**(

**r**)〉, is described by the van Cittert-Zernike theorem [6, 7]. By virtue of the formal analogy between the van Cittert-Zernike theorem and the diffraction formula, the mutual intensity

_{R}**(**

*J***r**,

_{Q}**r**) has the same distribution as the optical field which would be reconstructed from a conventional hologram illuminated by a coherent beam converging towards the point

_{R}**r**.

_{R}*δ*

**r**=(

**r**-

_{Q}**r**) proportional to the position vector

_{R}**r**such that

*δ*

_{r}=

**r**/

*m*, the reconstructed image represented by a coherence function can be visualized as an interference fringe contrast. It should be noted that, by changing the shearing scale parameter

*m*, we can control the magnification of the reconstructed image. This gives the possibility for an unconventional coherence imaging microscope, which is endowed with an entirely new function of variable coherence zooming enabled by the controllable shearing scale parameter

*m*. In our present experiment we propose the use of a modified Sagnac radial shearing interferometer with a telescopic lens system for this purpose.

## 3. Experiments

8. G. Cochran, “New method of making Fresnel transforms,” J. Opt. Soc. Am. **56**, 1513–1517 (1966).
[CrossRef]

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

*G*(

*x*̂,

*y*̂)=|

*G*(

*x*̂,

*y*̂)exp[

*i*Φ(

*x*̂,

*y*̂)] is given by the Fourier transform of the letter Ψ such that

*G*(

*x*̂,

*y*̂)=

*G*Ψ(

*x*̂,

*y*̂). For the 3D case, the field on the hologram is given by the superposition of the Fourier transforms of the letters U, E, and C, such that

*G*(

*x*̂,

*y*̂)=

*G*(

_{U}*x*̂,

*y*̂)exp[-

*i*Φ

_{Δz}]+

*G*(

_{E}*x*̂,

*y*̂)+

*G*(

_{C}*x*̂,

*y*̂)exp[

*i*Φ

_{Δz}], where Φ

_{Δz}is the quadratic Fresnel phase corresponding to the axial focal shift by Δ

*z*. These optical fields are calculated from the given object letters, and used to generate the coherence hologram

*H*(

*x*̑,

*y*̑). In synthesizing the coherence hologram, we removed from the interference fringe intensity the term

*G*|(

*x*̂,

*y*̂)|

^{2}which becomes the source of unwanted autocorrelation image. Instead, we enhanced the dc term

*C*to make the intensity transmittance nonnegative. We also took the square root of the modulus of the Fourier transform since the reconstruction process involves correlation of optical fields.

10. Z. Liu, T. Gemma, J. Rosen, and M. Takeda, “An improved illumination system for spatial coherence control,” Proc. SPIE **5531**, 220–227 (2004).
[CrossRef]

**r**on the image plane, the interference is due to the superposition of fields at the locations

**r**

*α*and

**r**/

*α*of the original beam [9

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

**r**on the image plane, we have a cross-correlation of the fields between two points separated by

*δ*

**r**proportional to

**r**scaled by the factor (

*α*-1/

*α*). The resulting interference gives a 2-D correlation map that reconstructs the image as a coherence function represented by the fringe contrast. The magnification of the reconstructed image can be suitably chosen with the proper choice of the amount of shear. In our present setup with

*α*=1.1, the magnification for reconstruction becomes 5.23 so that the reconstructed image size fits the field aperture of the CCD camera. The half-wave plate 3 (HWP3) is aligned such that the radially sheared interfering beams at the output of the interferometer have equal amplitudes. To detect the reconstructed coherence image represented by the fringe visibility, we need to introduce a phase shift into the common-path Sagnac interferometer. Because of its common path nature, the Sagnac interferometer is insensitive to the conventional PZT-based mechanical mirror movement. We therefore introduced the required phase shift by means of a geometric phase shifter. Inside the interferometer we made use of two quarter wave plates (QWP1 and QWP2) to turn the linearly polarized light into a circularly polarized light and back to linearly polarized light of orthogonal polarization with respect to the initial state of polarization. By rotating a half wave plate 4 (HWP4), placed between QWP1 and QWP2, we introduced a geometric phase into the counter propagating beams [11

11. P. Hariharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. **39**, 1811–1815 (1992).
[CrossRef]

*α*=1.1, formed by lenses L5 (focal length 220mm) and L6 (focal length 200mm), gives a radial shear as the light travels through the interferometer before they are brought back together and relayed by lenses L7 (focal length 500mm) and L8 (focal length 250mm) and then imaged by the CCD. PBS2 allows only those two beams that completed a round trip and got radially sheared to pass out through the interferometer exit. All the other beams generated due to unwanted reflections at BS are blocked at PBS2. At the exit of the interferometer, we made use of a quarter wave plate (QWP1), which transformed the linearly polarized beams with orthogonal polarizations into clockwise and counter-clockwise rotating circularly polarized beams. By rotating the half wave plate (HWP4), we introduced a geometric phase shift, which is used to find the fringe visibility. The other quarter plate QWP2 turned the circularly polarized beams back into linearly polarized beams with orthogonal polarizations. With an analyzer A, with its axis kept at 45° to the orientation of the polarization of the two beams, interference between the two beams was achieved.

## 4. Results

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

## 4. Conclusions

## Acknowledgments

## References and links

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

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

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

4. | 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. |

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. | M. Born and E. Wolf, |

7. | J. W. Goodman, |

8. | G. Cochran, “New method of making Fresnel transforms,” J. Opt. Soc. Am. |

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

10. | Z. Liu, T. Gemma, J. Rosen, and M. Takeda, “An improved illumination system for spatial coherence control,” Proc. SPIE |

11. | P. Hariharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. |

12. | 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: April 1, 2009

Revised Manuscript: May 18, 2009

Manuscript Accepted: May 22, 2009

Published: June 10, 2009

**Citation**

Dinesh N. Naik, Takahiro Ezawa, Yoko Miyamoto, and Mitsuo Takeda, "3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift," Opt. Express **17**, 10633-10641 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10633

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### References

- M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, "Coherence Holography," Opt. Express 23, 9629-9635 (2005). [CrossRef]
- W. Wang, H. Kozaki, J. Rosen, and M. Takeda, "Synthesis of longitudinal coherence function by spatial modulation of an exteded light source: a new interpretation and experimental verifications," Appl. Opt. 41, 1962-1971 (2002). [CrossRef] [PubMed]
- J. Rosen and M. Takeda, "Longitudinal spatial coherence applied for surface profilometry," Appl. Opt. 39, 4107-4111 (2000). [CrossRef]
- 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, 073902 (2006). [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, 12109-12121 (2006). [CrossRef] [PubMed]
- 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.
- G. Cochran, "New method of making Fresnel transforms," J. Opt. Soc. Am. 56, 1513-1517 (1966). [CrossRef]
- M. V. R. K. Murty, "A compact radial shearing interferometer based on the law of refraction," Appl. Opt. 3, 853-857 (1964). [CrossRef]
- Z. Liu, T. Gemma, J. Rosen, and M. Takeda, "An improved illumination system for spatial coherence control," Proc. SPIE 5531, 220-227 (2004). [CrossRef]
- P. Hariharan and M. Roy, "A geometric-phase interferometer," J. Mod. Opt. 39, 1811-1815 (1992). [CrossRef]
- P. Handel, "Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm," IEEE Trans. Instrum. Meas. 49, 1189-1193(2000). [CrossRef]

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