## Digital holography with a point diffraction interferometer

Optics Express, Vol. 13, Issue 6, pp. 1885-1891 (2005)

http://dx.doi.org/10.1364/OPEX.13.001885

Acrobat PDF (163 KB)

### Abstract

We propose a technique to implement digital holography by means of a point diffraction interferometer. The device uses a liquid crystal television display and works under partially coherent illumination which makes it useful for 3D microscopy. We present the theory on which the method is based and the obtained results.

© 2005 Optical Society of America

## 1. Introduction

*et al.*[1

1. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. **33**, 179–181 (1994). [CrossRef] [PubMed]

2. B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. **7**, 1–6 (1996). [CrossRef]

6. B. Javidi and E. Tajahuerce, “Three-dimensional recognition by use of digital holography,” Opt. Lett. **25**, 610–612 (2000). [CrossRef]

*et al.*[7

7. I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. **22**, 1268–1270 (1997). [CrossRef] [PubMed]

5. F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source,” Appl. Opt. **38**, 7085–7094 (1999). [CrossRef]

9. O. Kwon, “Multichannel phase shifted interferometer,” Opt. Lett. **9**, 59–61 (1984). [CrossRef] [PubMed]

10. H. Kadono, N. Takai, and T. Asakura, “A new common path phase shifting interferometer using a polarization technique,” Appl. Opt. **26**, 898–904 (1987). [CrossRef] [PubMed]

11. C. Mercer and K. Creath, “Liquid crystal point diffraction interferometer,” Opt. Lett. **19**, 916–918 (1994). [CrossRef] [PubMed]

12. C. Iemmi, A. Moreno, J. Nicolás, and J. Campos, “Evaluation and correction of aberrations in an optical correlator by phase shifting interferometry,” Opt. Lett. **28**, 1117–1119 (2003). [CrossRef] [PubMed]

*in situ*an arbitrary phase distribution caused by aberrations introduced by the various elements that constitute an optical correlator.

## 2. Description of the method

### 2.1 LCTV point diffraction interferometer

13. A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. , **40**, 2558–2564 (2001). [CrossRef]

*S*illuminates an object

*O*and onto the conjugate plane of the source by means of lens

*L1*the Fourier transform of

*O*is obtained. In that plane is placed a

*LCTV*sandwiched between linear polarizers

*P*

^{’}

*s*and wave plates

*WP*’

*s*that, combined, act as a pure phase filter. This phase modulator can be programmed pixel by pixel offering phase retardations in the range 0–2π. A second convergent lens images the object O onto the final plane Π.

*δ*(

*u*) is the Kronecker function that takes the value 1 when

*u*=0 and the value 0 elsewhere, and 2

*πn*/

*N*is the phase shift introduced in each of the

*N*steps applied to obtain the unknown phase. For simplicity, we carry out 1-D analysis. The intensity transmission of this filter is uniform but the phase of the pixel at the origin is shifted with respect to the other pixels by 2

*πn*/

*N*radians. The amplitude at the final plane is given by the expression

*O*(

*x*)=

*o*(

*x*)

*e*

^{iΨ(x)}is the unknown complex amplitude of the object wave,

*h*(

*x*) is the filter impulse response, the symbol * denote a convolution and the complex constant

*K*=|

*K*|

*e*

^{iµ}is the mean value of

*O*(

*x*).

*A*(

_{n}*x*) is detected by a CCD. To obtain the phase information (

*ψ*(

*x*)) several interference patterns |

*A*|

_{n}^{2}with

*n*=

*0*,…,

*N*-

*1*(

*N*>

*3*) are needed. Following the synchronous detection of interference fringes [15], the measured intensities |

*A*|

_{n}^{2}are multiplied by

*A*given in Eq. (2) we obtain:

_{n}*x*) can be obtained as

*C*=

_{o}*N*|

*K*|

^{2}. The value of

*C*is obtained by evaluating expressions (4) at those points in which

_{o}*o*(

*x*)=

*0*as it is explained in Ref. 12. This implies that the object O should be limited by a field stop, to assure that outside the stop

*o*(

*x*)=

*0*. In the final image plane, the CCD camera should capture part of the field stop image. In this way the constant C

_{0}can be evaluated.

*x*) is calculated as the square root of the object wave intensity.

## 2.2 Refocusing algorithms

*O*(

*x*) represents the object wave in plane Π, i.e., in the input plane of the CCD. We can evaluate the complex amplitude

*O*(

*x*

^{′}) in a plane Π’ parallel to Π and separated by a distance

*d*by means of the Kirchhoff-Fresnel propagation integral. This integral can be expressed in two different forms:

*λ*is the wavelength,

*k*=2

*π*/

*λ*,

*x*is the spatial variable,

*v*is the spatial frequency and

_{x}*𝓕*and

*𝓕*

^{-1}indicate the continuous Fourier transform and the inverse Fourier transform respectively. Although these two expressions are equivalent, Eq. (7) is more convenient for near field refocusing because of its dependence with the distance

*d*in the quadratic phase factors [5

5. F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source,” Appl. Opt. **38**, 7085–7094 (1999). [CrossRef]

*N*is the number of pixels and

*m*,

*m*

^{’}and

*l*are integer numbers that vary from

*0*to

*N*-1.

## 3. Results

13. A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. , **40**, 2558–2564 (2001). [CrossRef]

_{1}, WP

_{1}) were placed before the object O. After the LCTV an elliptically polarization state must be detected, then a wave plate and a polarizer must be introduced between the LCTV and the detector. In Fig. 1 these elements (P

_{2}, WP

_{2}) were placed just after the LCTV. The used detector was a CCD of 512×512 pixels and with 11 µm pixel size and the object was a graduated scale etched in a piece of glass from an eye piece. The distance between two consecutive numbers of the scale is 1 mm.

## 4. Final discussion

_{0}in Eq (5) part of the field stop should be captured by the CCD. This field stop or an intermediate image should be place on the object.

## Acknowledgments

## References and links

1. | U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

2. | B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. |

3. | E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. |

4. | T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase shifting digital holography,” Opt. Lett. |

5. | F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source,” Appl. Opt. |

6. | B. Javidi and E. Tajahuerce, “Three-dimensional recognition by use of digital holography,” Opt. Lett. |

7. | I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. |

8. | V. Linnik, “Simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. USSR , |

9. | O. Kwon, “Multichannel phase shifted interferometer,” Opt. Lett. |

10. | H. Kadono, N. Takai, and T. Asakura, “A new common path phase shifting interferometer using a polarization technique,” Appl. Opt. |

11. | C. Mercer and K. Creath, “Liquid crystal point diffraction interferometer,” Opt. Lett. |

12. | C. Iemmi, A. Moreno, J. Nicolás, and J. Campos, “Evaluation and correction of aberrations in an optical correlator by phase shifting interferometry,” Opt. Lett. |

13. | A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. , |

14. | J. Nicolás, J. Campos, C. Iemmi, I. Moreno, and M. J. Yzuel, “Convergent optical correlator alignment based on frequency filtering,” Appl. Opt. |

15. | D. Malacara, |

**OCIS Codes**

(090.0090) Holography : Holography

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 3, 2005

Revised Manuscript: March 2, 2005

Published: March 21, 2005

**Citation**

C. Iemmi, A. Moreno, and J. Campos, "Digital holography with a point diffraction interferometer," Opt. Express **13**, 1885-1891 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-1885

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### References

- U. Schnars and W. Jüptner, �??Direct recording of holograms by a CCD target and numerical reconstruction,�?? Appl. Opt. 33, 179-181 (1994). [CrossRef] [PubMed]
- B. Skarman, K. Wozniac, and J. Becker, �??Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,�?? Flow Meas. Instrum. 7, 1-6 (1996). [CrossRef]
- E. Cuche, F. Bevilacqua, and C. Depeursinge, �??Digital holography for quantitative phase contrast imaging�?? Opt. Lett. 24, 291-293 (1999). [CrossRef]
- T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase shifting digital holography,�?? Opt. Lett. 23, 1221-1223 (1998). [CrossRef]
- F. Dubois, L. Joannes, and J. Legros, �??Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source,�?? Appl. Opt. 38, 7085-7094 (1999). [CrossRef]
- B. Javidi, and E. Tajahuerce, �??Three-dimensional recognition by use of digital holography,�?? Opt. Lett. 25, 610-612 (2000). [CrossRef]
- I. Yamaguchi, and T. Zhang, �??Phase shifting digital holography,�?? Opt. Lett. 22, 1268-1270 (1997). [CrossRef] [PubMed]
- V. Linnik, �??Simple interferometer for the investigation of optical systems,�?? C. R. Acad. Sci. USSR, 1, 208- 210 (1933).
- O. Kwon, �??Multichannel phase shifted interferometer,�?? Opt. Lett. 9, 59-61 (1984). [CrossRef] [PubMed]
- H. Kadono, N. Takai, and T. Asakura, �??A new common path phase shifting interferometer using a polarization technique,�?? Appl. Opt. 26, 898-904 (1987). [CrossRef] [PubMed]
- C. Mercer, and K. Creath, �??Liquid crystal point diffraction interferometer,�?? Opt. Lett. 19, 916- 918 (1994). [CrossRef] [PubMed]
- C. Iemmi, A. Moreno, J. Nicolás, and J. Campos, �??Evaluation and correction of aberrations in an optical correlator by phase shifting interferometry,�?? Opt. Lett. 28, 1117-1119 (2003). [CrossRef] [PubMed]
- A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, �??Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model, �?? Opt. Eng., 40, 2558-2564 (2001). [CrossRef]
- J. Nicolás, J. Campos, C. Iemmi, I. Moreno, and M. J. Yzuel, �?? Convergent optical correlator alignment based on frequency filtering,�?? Appl. Opt. 41, 1505-1514 (2002). [CrossRef] [PubMed]
- D. Malacara, Optical Shop Testing (John Wiley & Sons, New York, 1978).

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