## Numerical reconstruction of digital holograms with variable viewing angles

Optics Express, Vol. 10, Issue 22, pp. 1250-1257 (2002)

http://dx.doi.org/10.1364/OE.10.001250

Acrobat PDF (711 KB)

### Abstract

Here we describe a new method for numerically reconstructing an object with variable viewing angles from its hologram(s) within the Fresnel domain. The proposed algorithm can render the real image of the original object not only with different focal lengths but also with changed viewing angles. Some representative simulation results and demonstrations are presented to verify the effectiveness of the algorithm.

© 2002 Optical Society of America

## 1. Introduction

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

6. L. Yu and L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. A **18**, 1033–1045 (2001). [CrossRef]

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

7. Y. Takaki and H. Ohzu, “Hybrid holographic microscopy: visualization of three-dimensional object information by use of viewing angles,” Appl. Opt. **39**, 5302–5308 (2000). [CrossRef]

8. M. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express **7**, 305–310 (2000). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-305 [CrossRef] [PubMed]

## 2. Principles

*τ*(

*x,y*) is vertically placed in the

*z*=

*0*plane and the observation direction is along the

*z*′ axis. For simplicity we assume the viewing angle

*θ*, the angle between the

*z*′ - axis and the original

*z*axis, to lie in the

*y*-

*z*plane.

*E*(

_{i}*x*,

*y*) with a unitary amplitude of

*E*, shown black in Fig. 1, incident onto the hologram plane; however, the diffracted wave from the hologram is assumed to propagate along the new direction of

_{o}*θ*. The observation plane

*x*-

_{o}*y*is assumed to be perpendicular to the

_{o}*z*′ axis.

*x*-

_{o}*y*, let us first review the Rayleigh-Sommerfeld diffraction integral [9]:

_{o}*λ*is the wavelength,

*k*is termed the wave number and given by

*k*= 2

*π*/

*λ*.

*χ*(

*x,y,x*) is the inclination factor, which will be approximately unitary if the Fresnel approximation is assumed. Thus it is omitted in the following equations. The constant before the integration can also be omitted. The first exponential term is introduced because of the angle between the normal plane of the inclined observation direction and the vertical hologram plane.

_{o},y_{o}10. D. Leseberg and C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. **27**, 3020–3024 (1988). [CrossRef] [PubMed]

*ξ*and

*η*as:

*λ*/2sin(

*α*/ 2) ≅

*λ*/

*α*, where

*α*is the angular size of the object to be clearly reconstructed. In other words, the resolution

*p*of the CCD camera can be given as

*p*=

*λ*/

*α*[4

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

*N*×

*N*is the pixel number of a square CCD,

*δx*and

_{o}*δy*are the pixel sizes in the reconstructed image. From the above Equation we find that when the wavelength selected increases, the spatial frequency bandwidths in both directions decrease, and when the absolute value of the viewing angle

_{o}*θ*increases, the spatial frequency bandwidth in the

*y*direction also decreases. For more detailed digitalization process of CCD-target for digital holography, readers can also refer to some earlier publications [1

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

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

11. L. Xu, J. Miao, and A. Asundi, “Properties of digital holography based on in-line configuration,” Opt. Eng. **39**, 3214–3219 (2000). [CrossRef]

12. T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. **41**, 771–778 (2002). [CrossRef]

*θ*increases, the space-bandwidth product decreases, which also means that the bigger the absolute viewing angle, the more information lost in the reconstructed image. Theoretically speaking, if the absolute viewing angle

*θ*reaches 90 degrees, no information will be reconstructed.

*θ*is limited because of the limited DOE structure size.

*θ*, it would be reasonable to suppose the original incident beam,

*E*(

_{i}*x, y*) with a unitary amplitude (

*E*= 1), to vertically illuminate onto the hologram plane

_{o}*τ*(

*x, y*). Here

*τ*(

*x, y*) is supposed to be a 2D transmittance, as shown in Fig. 1. It could be viewed as paraxial diffraction.

*θ*, because the reconstruction process is virtually and numerically performed, we could slightly change the reconstruction model, where we could assume that the incident beam is illuminated onto the hologram plane with the same inclined angle

*θ*as the observation direction, as shown in Fig. 1 with a dashed blue color. But in this case we should make sure that the diffracted wave field behind the 2D hologram plane is still

*τ*(

*x, y*). This process could also be viewed as paraxial diffraction (along

*z*′ axis). Thus it is feasible to numerically reconstruct from the hologram with a larger viewing angle

*θ*(from

*z*axis). However, as we have discussed, when the absolute value of the viewing angle

*θ*increases, the space-bandwidth product of the system will decreases according to Eq. (9). When the absolute viewing angle

*θ*reaches 90 degrees, no information will be reconstructed at all.

## 3. Numerical Simulations

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

6. L. Yu and L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. A **18**, 1033–1045 (2001). [CrossRef]

*l*=1.2

_{o}*m*, as shown in Fig. 2 (a). In the simulations, the wavelength in recording is supposed to be the same as that in reconstructing, which is set to be 632.8

*nm*, so the real image of the object should lie at the position with

*l*= 1.2

_{i}*m*away from another side of the hologram plane. The object consists of 256×256 pixels with each pixel containing 256 gray levels in the process of numerical reconstruction and the total area of the object is set to be 13.9

*mm*×13.9

*mm*. Since in-line holography is assumed in this paper, the complex amplitude of the object on the hologram plane can be numerically calculated by simulating four phase-shifted holograms [4

**22**, 1268–1270 (1997). [CrossRef] [PubMed]

*l*= 1.2

_{i}*m*and the viewing angle is changed from -75° to 75°. From the above demo we find that all the reconstructed frames have the same ratio in the vertical

*ξ*direction; however, the horizontal

*η*direction becomes more and more shrunk as the absolute viewing angle

*θ*becomes more and more larger. This can be justified from Eqs. (5, 6) and (9) if we substitute different values for

*θ*. Our simulations are carried out with the MATLAB 5.1 software in a P3-866 PC. The computation time of our proposed algorithm is about 16 seconds (to reconstruct a 256×256 image with one selected viewing angle), while the computation time for the Fresnel diffraction formula [1

**33**, 179–181 (1994). [CrossRef] [PubMed]

*l*

_{o1}= 1.0

*m*away from the hologram. Fig. 5(b) shows the texture of layer 2, which is located at the position of

*l*

_{o2}= 1.2

*m*. All the other parameters are the same as those used in the first demo. The complex object information of the new object on the hologram plane could be obtained by the same phase-shifting process, Fig. 6(a) shows the magnitude of the complex object wave and Fig. 6(b) shows the phase image of the complex wave field. Fig. 7(a) shows another demo where the viewing angle is fixed as

*θ*= 45° but the reconstructing distance

*l*is changed between 1.0

_{i}*m*and 1.2

*m*. From this demo we may find that the reconstructed image of layer 1 becomes much more vague since it gets far more out-of-focus as the reconstructing distance

*l*is changed from 1.0

_{i}*m*to 1.2

*m*, and the reconstructed image of layer 2 becomes much clearer. Oppositely, if the reconstructing distance

*l*is changed from 1.2

_{i}*m*to 1.0

*m*, then layer 1 in the reconstructed image will become clearer and clearer, however layer 2 will become more and more vague. Fig. 7(b) shows a third demo when both the viewing angle and the focal length are animatedly adjusted.

## 4. Conclusion

## Acknowledgment

## References and links

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

2. | B. Nilsson and T. E. Carlsson, “Direct three-dimensional shape measurement by digital light-in-flight holography,” Appl. Opt. |

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

4. | Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

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

6. | L. Yu and L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. A |

7. | Y. Takaki and H. Ohzu, “Hybrid holographic microscopy: visualization of three-dimensional object information by use of viewing angles,” Appl. Opt. |

8. | M. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express |

9. | J. W. Goodman, |

10. | D. Leseberg and C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. |

11. | L. Xu, J. Miao, and A. Asundi, “Properties of digital holography based on in-line configuration,” Opt. Eng. |

12. | T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. |

13. | H. Aagedal, F. Wyrowski, and M. Schmid, |

**OCIS Codes**

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(100.3010) Image processing : Image reconstruction techniques

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 17, 2002

Revised Manuscript: October 10, 2002

Published: November 4, 2002

**Citation**

Lingfeng Yu, Yingfei An, and Lilong Cai, "Numerical reconstruction of digital holograms with variable viewing angles," Opt. Express **10**, 1250-1257 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-22-1250

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### References

- O. Schnars andW. Juptner, �??Direct recording of holograms by a CCD target and numerical reconstruction,�?? Appl. Opt. 33, 179-181 (1994). [CrossRef] [PubMed]
- B. Nilsson and T. E. Carlsson, �??Direct three-dimensional shape measurement by digital light-in-flight holography,�?? Appl. Opt. 37, 7954-7959 (1998). [CrossRef]
- E. Cuche, F. Bevilacqua, and C. Depeursinge, �??Digital holography for quantitative phase-contrast imaging,�?? Opt. Lett. 24, 291-293 (1999). [CrossRef]
- Yamaguchi and T. Zhang, �??Phase-shifting digital holography,�?? Opt. Lett. 22, 1268-1270 (1997). [CrossRef] [PubMed]
- F. Dubois, L. Joannes, and J. Legros, �??Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,�?? Appl. Opt. 38, 7085-7094 (1999). [CrossRef]
- L. Yu and L. Cai, �??Iterative algorithm with a constraint condition for numerical reconstruction of a threedimensional object from its hologram,�?? J. Opt. Soc. Am. A 18, 1033-1045 (2001). [CrossRef]
- Y. Takaki and H. Ohzu, �??Hybrid holographic microscopy: visualization of three-dimensional object information by use of viewing angles,�?? Appl. Opt. 39, 5302-5308 (2000). [CrossRef]
- M. Kim, "Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography," Opt. Express 7, 305-310 (2000). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-305">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-305</a> [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
- D. Leseberg and C. Frere, �??Computer-generated holograms of 3-D objects composed of tilted planar segments,�?? Appl. Opt. 27, 3020-3024 (1988). [CrossRef] [PubMed]
- L. Xu, J. Miao, and A. Asundi, �??Properties of digital holography based on in-line configuration,�?? Opt. Eng. 39, 3214-3219 (2000). [CrossRef]
- T. M. Kreis, �??Frequency analysis of digital holography,�?? Opt. Eng. 41, 771-778 (2002). [CrossRef]
- H. Aagedal, F. Wyrowski, and M. Schmid, Diffractive Optics for Industrial and Commercial Applications, J. Turunen and F. Wyrowski, eds. (Akademie Verlag, Berlin, 1997), Chap. 6, pp. 165-188.

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