## Whole optical wavefields reconstruction by Digital Holography

Optics Express, Vol. 9, Issue 6, pp. 294-302 (2001)

http://dx.doi.org/10.1364/OE.9.000294

Acrobat PDF (702 KB)

### Abstract

In this paper, we have investigated on the potentialities of digital holography for whole reconstruction of wavefields. We show that this technique can be efficiently used for obtaining quantitative information from the intensity and the phase distributions of the reconstructed field at different locations along the propagation direction. The basic concept and procedure of wavefield reconstruction for digital in-line holography is discussed. Numerical reconstructions of the wavefield from digitally recorded in-line hologram patterns and from simulated test patterns are presented. The potential of the method for analysing aberrated wave front has been exploited by applying the reconstruction procedure to astigmatic hologram patterns.

© Optical Society of America

## 1. Introduction

1. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. **38**, 6994–7001 (1999). [CrossRef]

2. Y. Takaki and H. Ohzu, “Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,” Appl. Opt. **38**, 2204–2211 (1999). [CrossRef]

3. G. Pedrini, P. Fröning, H. Tiziani, and F. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. **164**, 257–268 (1999). [CrossRef]

4. S. Schedin, G. Pedrini, H. Tiziani, A. K. Aggarwal, and M. E. Gusev, “Highly sensitive pulsed digital holography for built-in defect analysis with a laser excitation,” Appl. Opt. **40**, 100–117 (2001). [CrossRef]

5. T. Kreis, M. Adams, and W. Jüptner, “Digital in-line holography in particle measurement,” SPIE **3744**, 54–64 (1999). [CrossRef]

6. S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Optics & Laser Technology **32**, 567–574 (2000). [CrossRef]

10. A. Stadelmaier and J.H. Massing, “Compensation of lens aberrations in digital holography,” Opt. Lett. **25**, 1630–1632 (2000). [CrossRef]

11. S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. **26**, No.13, 974–976 (2001). [CrossRef]

## 2. Theoretical principle and experimental description

*b′*(

*x′*,

*y′*) in the image plane is obtained by using the well known [7] Fresnel-approximation of the Rayleigh-Sommerfield diffraction formula, namely

*g*(

*ξ,η*)is the impulse response

*N*is the pixel number of the CCD array in each direction. We see that according to the equation (1), the wavefield

*b*(

*x′*,

*y′*;

*d′*)is determined essentially by the two-dimensional Fourier transform of the quantity

*h*(

*ξ*,

*η*)

*r*(

*ξ*,

*η*)

*g*(

*ξ*,

*η*). Equation (1) is employed as the basic governing equation for determining both the light intensity distribution

*I*(

_{d′}*x′*,

*y′*)=

*b*(

*x′*,

*y′*,

*d′*)*

*b*(

*x′*,

*y′*;

*d′*) in the image plane at a distance

*d′*from the hologram plane and the phase distribution

*ψ*(

*x′*,

*y′*;

*d′*)=

*Arg*[

*b*(

*x′*,

*y′*;

*d′*)]. It was pointed out that in the formulation based on equation (1) the reconstructed image is enlarged or contracted according to the reconstruction depth

*d’*. An alternative approach is useful for keeping the size of the reconstructed image constant [7]. In this formulation, the wavefield

*b*(

*x′*,

*y′*;

*d′*)can be computed by

*x′*=Δ

*ξ*,Δ

*y′*=Δ

*η*and one needs one Fourier transform and one inverse Fourier transform each to obtain one two-dimensional reconstructed image at a distance

*d′*. Although the computational procedure is heavier in this case compared to the Fresnel approximation approach of equation (1), this method allows for easy comparison of the reconstructed images at different distances

*d′*since the size does not change with modifying the reconstruction distance. Furthermore, in this case we get an exact solution to the diffraction integral as far as the sampling Nyquist theorem is not violated.

### 2.1 Wavefield intensity reconstruction from digitized experimental holograms

*z*=

*d′*of the image plane along the

*z*-axis propagation direction. A Mach-Zehnder interferometer (see Fig. 1) was used for the observation of in-line hologram patterns.

*λ*=632.8 nm) is divided by the beam splitter BS1 into two beams: one of these, the object beam, is a spherical wave produced by an achromatic doublet of focal length 300 mm (see Fig. 1); the other one is a reference plane wave, interfering with the object beam at the recombining beam splitter BS2. The hologram pattern was digitized by a CCD camera with pixel size Δ

*ξ*=Λ

*η*=11 µm and recorded under two different conditions corresponding to two settings of the frame buffer. The hologram pattern shown in Fig. 2a was recorded with the right setting of the frame buffer corresponding to 736 columns ×572 row. The image shown is a digitized array of

*N*×

*N*=512×512 8-bit encoded numbers. In Fig. 2b the frame buffer setting was intentionally modified to 768 columns ×572 row in order to introduce a slight anamorphism, which changes the aspect ratio of the image [8] from the value 1. The effect of the anamorphism in the recorded hologram of Fig. 2b is to introduce a deformation along the x horizontal direction in the whole fringe pattern, thus obtaining elliptical interference fringes instead of the circular fringes shown in Fig. 2a.

*r*(

*ξ*,

*η*)=1 was carried out for values of the reconstruction distance

*d’*ranging from 170 mm to 200 mm, with spatial discrete step of Δ

*z*=1mm. In the case of the aberrated hologram pattern in Fig. 2b the intensity distribution was determined for

*d’*ranging from 181 mm to 218 mm and with Δ

*z*=1mm. The sequence of intensity distributions were combined to obtain the two clip videos presented in Fig.3.

*d’*=

*D*=180 mm from the hologram plane. According to simple geometrical considerations (see Fig. 4), this distance corresponds to the focusing distance of a converging spherical wavefield produced by the achromatic doublet.

*d’*≅183 mm, corresponding to the tangential focus, and a vertical line image at the sagittal focus reconstructed at a distance

*d’*≅218 mm.

### 2.2 Reconstructing intensity and phase distributions from simulated holograms

*I*(

*ξ*,

*η*)of the recorded hologram in the following form

*z*,

_{x}*z*correspond to the vertical and horizontal focal lines, respectively. Of course, in the case of circular fringes, as those recorded in Fig. 2a, we have simply that

_{y}*z*=

_{x}*z*=

_{y}*z*. The floating-point numbers computed by equation (6), provide a reasonable approximation of the integer-number distribution that occurs from the frame store. Fig. 5a-5b shows respectively the density plot representation of the circular and elliptical fringe patterns computed for

*z*=250 mm in the circular case, z

*=300 mm and z*

_{x}*=250 mm in the elliptical one. The test hologram patterns were digitized as an array*

_{y}*N*×

*N*=512×512 ; we have assumed λ=632.8 nm and step size 11 µm along the

*x*and

*y*directions. Equation (6) can be written in the following form

*h*(

*ξ*,

*η*) being extracted, a reconstruction procedure is employed to determine the complex amplitude of the wavefield. The extraction of the above terms can be carried out by applying for example the four-quadrature-phase shifting reconstruction algorithm as described in the case of the in-line digital holography [13

13. Songcan Lai, Brian King, and Mark A. Neifeld, “Wave front reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. **173**, 155–160 (2000). [CrossRef]

*h*(

*ξ*,

*η*) at the hologram recording plane.

*d’*=180 mm from the hologram plane. In Fig. 6c the reconstruction distance is d’=250 mm. For this distance we have

*z*=

*z*, the spherical wave front focuses at a single point (Fig. 6c) whereas the astigmatic wavefield focuses at a line image corresponding to the tangential focal line (Fig. 6d). These results reproduce quite well those obtained by the reconstruction procedure of the experimental hologram patterns (compare to the movies in Fig. 3a and 3b).

_{y}*π*,

*π*]computed by the numerical reconstruction method at the reconstruction distance

*d’*=180 mm. The density plot representations of the wrapped distributions in Fig. 7a and 7b correspond respectively to the simulated spherical and astigmatic wave fronts shown in Fig. 5a and 5b. Both phase distributions at the reconstructed image plane were computed in the restricted range of 140×140 pixels.

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

*x*-horizontal (straight line) and

*y*-vertical (dashed line) phase distributions for the two considered cases and for different reconstruction distances. In Fig. 9a the two distributions are superposed owing to the spherical symmetry of the wave front, whereas in Fig. 9b they are clearly different due to the astigmatism. The vertical axis in Fig. 9a-9b is the

*z*propagation axis along which the various phase distributions are evaluated for backward reconstruction distances ranging from

*d’*=160 mm to

*d’*=220 mm at step size of 10 mm. The scale of the horizontal axis of Fig. 9 is determined by the pixel size Δ

*x′*=Δ

*ξ*of the reconstructed image, which does not change in the reconstruction method. The plots give a perspective of the wave front phase advance as one proceeds by reconstructing at distances closer to the focus in the case of the spherical wave front or to the tangential focal line in the case of the astigmatic wave front. Determination of the intensity, wrapped phase, unwrapped phase at different planes along the propagation direction of the wave front and wrapped phase show the potential of the DH for whole optical wavefield reconstruction and for qualitative and quantitative analysis of wavefield aberrations. We end this section by pointing out that once we have carried out the numerical procedure for computing sequence of the complex map of the field

*b*(

*x′*,

*y′*;

*d′*)for various reconstruction distances

*d’*, the phase differences Δ

*ψ*(

*x′, y′*, Δ

*z*)at two planes separated by a distance Δ

*z*, can be easily evaluated in terms of the real and imaginary parts of the complex fields

*b*(

*x′, y′;d′*)and

*b*(

*x′, y′; d′*+Δ

*z*) by using the following relationship

*π*. Subsequent application of the unwrapping procedure allows calculation of the unwrapped map of the phase differences Δ

*ψ*(

*x′, y′*, Δ

*z*).

## 3. Conclusions

## References and links

1. | E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. |

2. | Y. Takaki and H. Ohzu, “Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,” Appl. Opt. |

3. | G. Pedrini, P. Fröning, H. Tiziani, and F. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. |

4. | S. Schedin, G. Pedrini, H. Tiziani, A. K. Aggarwal, and M. E. Gusev, “Highly sensitive pulsed digital holography for built-in defect analysis with a laser excitation,” Appl. Opt. |

5. | T. Kreis, M. Adams, and W. Jüptner, “Digital in-line holography in particle measurement,” SPIE |

6. | S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Optics & Laser Technology |

7. | J. W. Goodman, |

8. | Shinya Inoué and Kenneth R. Spring, |

9. | S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” to be published, Opt. Las. Eng. (2001). |

10. | A. Stadelmaier and J.H. Massing, “Compensation of lens aberrations in digital holography,” Opt. Lett. |

11. | S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. |

12. | T. M. Kreis and W. P. O. Jüptner, |

13. | Songcan Lai, Brian King, and Mark A. Neifeld, “Wave front reconstruction by means of phase-shifting digital in-line holography,” Opt. Commun. |

14. | Takeda, H Ina, and Kobayashys, “Fourier transform method of fringe pattern analysis for computer based topography and interferometry,” J. Opt. Soc. Am. , |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(090.1760) Holography : Computer holography

(100.2650) Image processing : Fringe analysis

(100.3010) Image processing : Image reconstruction techniques

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 3, 2001

Published: September 10, 2001

**Citation**

Simonetta Grilli, Pietro Ferraro, Sergio De Nicola, A. Finizio, G. Pierattini, and R. Meucci, "Whole optical wavefields reconstruction by digital holography," Opt. Express **9**, 294-302 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-6-294

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### References

- E. Cuche, P. Marquet, and C. Depeursinge, �Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,� Appl. Opt. 38, 6994-7001 (1999). [CrossRef]
- Y. Takaki and H. Ohzu, �Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,� Appl. Opt. 38, 2204-2211 (1999). [CrossRef]
- G. Pedrini, P. Fr�ning, H. Tiziani, F. Santoyo, �Shape measurement of microscopic structures using digital holograms,� Opt. Commun. 164, 257-268 (1999). [CrossRef]
- S. Schedin, G. Pedrini, H. Tiziani, A. K. Aggarwal, and M. E. Gusev, �Highly sensitive pulsed digital holography for built-in defect analysis with a laser excitation,� Appl. Opt. 40, 100-117 (2001). [CrossRef]
- T. Kreis, M. Adams, W. J�ptner, �Digital in-line holography in particle measurement,� SPIE 3744, 54-64 (1999). [CrossRef]
- S. Murata, N. Yasuda, �Potential of digital holography in particle measurement,� Optics & Laser Technology 32, 567-574 (2000). [CrossRef]
- J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Francisco, Calif., 1968, Cap. 5.
- Shinya Inoue and Kenneth R. Spring, Video Microscopy, Second Edition, Cap.7.
- S. De Nicola, P. Ferraro, A. Finizio and G. Pierattini, �Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,� to be published, Opt. Las. Eng. (2001).
- A. Stadelmaier and J. H. Massing, �Compensation of lens aberrations in digital holography,� Opt. Lett. 25, 1630- 1632 (2000). [CrossRef]
- S. De Nicola, P. Ferraro, A. Finizio and G. Pierattini, �Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,� Opt. Lett. 26, No.13, 974-976 (2001). [CrossRef]
- T.M. Kreis, W. P. O. J�ptner, Trends in Optical Non-Destructive testing and Inspection, Editors Pramod Rastogi and Daniele Inaudi, 113-127.
- Songcan Lai, Brian King, MArk A. Neifeld, �Wave front reconstruction by means of phase-shifting digital in-line holography,� Opt. Commun. 173, 155-160 (2000). [CrossRef]
- Takeda, Ina H. and Kobayashys, "Fourier transform method of fringe pattern analysis for computer based topography and interferometry," J. Opt. Soc. Am., 72, 156-160 (1982). [CrossRef]

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