## Fast calculation of computer-generated holograms based on 3-D Fourier spectrum for omnidirectional diffraction from a 3-D voxel-based object |

Optics Express, Vol. 20, Issue 19, pp. 20962-20969 (2012)

http://dx.doi.org/10.1364/OE.20.020962

Acrobat PDF (1286 KB)

### Abstract

We have derived the basic spectral relation between a 3-D object and its 2-D diffracted wavefront by interpreting the diffraction calculation in the 3-D Fourier domain. Information on the 3-D object, which is inherent in the diffracted wavefront, becomes clear by using this relation. After the derivation, a method for obtaining the Fourier spectrum that is required to synthesize a hologram with a realistic sampling number for visible light is described. Finally, to verify the validity and the practicality of the above-mentioned spectral relation, fast calculation of a series of wavefronts radially diffracted from a 3-D voxel-based object is demonstrated.

© 2012 OSA

## 1. Introduction

1. D. Gabor, “A new microscopic principle,” Nature **161**, 777–778 (1948). [CrossRef] [PubMed]

2. L H. Lin and H. L. Beauchamp, “Write-read-erase in situ optical memory using thermoplastic holograms,” Appl. Opt. **9**, 2088–2092 (1970). [CrossRef] [PubMed]

3. X. Zhang, E. Dalsgaard, S. Liu, H. Lai, and J. Chen, “Concealed holographic coding for security applications by using a moiré technique,” Appl. Opt. **36**, 8096–8097 (1997). [CrossRef]

4. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. **6**, 1739–1748 (1967). [CrossRef] [PubMed]

5. T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express **18**, 9955–9960 (2010). [CrossRef] [PubMed]

6. M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging **2**, 28–34 (1993). [CrossRef]

7. T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. **47**, D63–D70 (2008). [CrossRef] [PubMed]

*et al.*[8

8. D. Leseberg and C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. **27**, 3020–3024 (1988). [CrossRef] [PubMed]

*et al.*[9

9. T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A **10**, 299–305 (1993). [CrossRef]

10. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A **20**, 1755–1762 (2003). [CrossRef]

8. D. Leseberg and C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. **27**, 3020–3024 (1988). [CrossRef] [PubMed]

10. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A **20**, 1755–1762 (2003). [CrossRef]

## 2. Basic spectral relation between a 3-D object and its diffracted wavefront in 3-D Fourier domain

### 2.1. Derivation of the basic spectral relation

*o*(

*x*

_{0},

*y*

_{0},

*z*

_{0}) can be assumed as the aggregation of point light sources isotropically emitting spherical wavefronts. Here, a spherical wave

*g*(

*x*

_{0},

*y*

_{0},

*z*

_{0}) is emitted from an individual point light source, and the superposition of spherical waves from all the point light sources is observed at point

*P*(

*x,y,z*) in 3-D space. Therefore, the wavefront

*f*(

*x,y,z*) diffracted from the object

*o*(

*x*

_{0},

*y*

_{0},

*z*

_{0}) can be expressed as follows: where

*λ*represent 3-D convolution and the wavelength of the light sources, respectively.

*g*(

*x,y,z*), expressed by Eq. (3), represents a spherical wave. Because Eq. (3) is a solution of the scalar Helmholtz equation in 3-D free space, Eq. (1) is regarded as a rigorous diffraction integral. The 3-D Fourier spectrum

*G*(

*u,v,w*) of

*g*(

*x,y,z*) is given by [12] where (

*u,v,w*) are the coordinates of (

*x,y,z*) in the Fourier domain and

*F*[·] denotes a 3-D Fourier transform operator. By converting Eq. (2) into the form of a Fourier transform, the following equation is obtained: where

*O*(

*u,v,w*) is the 3-D Fourier spectrum of

*o*(

*x,y,z*). By considering the diffracted wavefront at the plane

*U*(

*x,y,z*=

*R*), Eq. (5) can be written as Next, to analytically calculate the integral in Eq. (6) with respect to the

*w*axis, the integral is performed by considering the complex plane

*w*=

*ξ*+

*iη*. The path of the line integral in the

*w*plane is set to be a closed contour in which only the singularity

*z*

_{0}direction. The radius of the hemicycle

*s*is extended to infinity. By using the residue theorem, this integral is calculated by considering the contribution only from the singularity inside the closed contour:

*u*

^{2}+

*v*

^{2}+

*w*(

*u,v*)

^{2}= 1/

*λ*

^{2}is obtained. Therefore, Eq. (8) implies that the 2-D Fourier spectrum of the diffracted wavefront can be obtained by extracting the spectrum on the surface of the hemisphere whose center is located at the origin and whose radius is 1/

*λ*, from the 3-D spectrum

*O*(

*u,v,w*) and by multiplying the weight factor and the phase components corresponding to the diffraction distance

*R*. In other words, the spectrum on the hemispherical surface in the 3-D spectrum of the 3-D object inheres in the diffracted wavefront. This is a simple but important conclusion, and it clarifies the information contained in the diffracted wavefront, which is equivalent of the holographic information. Therefore, Eq. (8) can be regarded as a basic spectral relation for diffraction.

### 2.2. Illustration of object information inherent in diffracted wavefronts

8. D. Leseberg and C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. **27**, 3020–3024 (1988). [CrossRef] [PubMed]

10. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A **20**, 1755–1762 (2003). [CrossRef]

## 3. Acquisition of the spectrum on the spherical surface for visible light

*λ*is needed to calculate a wavefront diffracted from the 3-D object according to Eq. (8). It is however impractical to directly calculate a spectrum with a bandwidth of more than 2/

*λ*in each direction by 3-D FFT for visible light. To solve this problem, we consider a 3-D object not as a continuously distributed object

*o*(

*x,y,z*) but as a 3-D object

*o*′(

*x,y,z*) sampled at intervals of

*δ*= 1/

*W*. Its 3-D Fourier spectrum

*O*′(

*u,v,w*) can be written as where comb(

*x,y,z*) is a comb function. When the spectral bandwidth of

*o*(

*x,y,z*) is limited to less than

*W*in each direction, Eq. (9) implies that the original spectrum

*O*(

*u,v,w*) of the 3-D object

*o*(

*x,y,z*) is just repeated with period

*W*in each direction. The full range of the spectrum is obtainable by the simple repetition of the spectrum

*O*(

*u,v,w*) with a bandwidth

*W*. The sampled values of

*O*(

*u,v,w*) are obtained by the 3-D FFT of

*o*(

*δn*,

_{x}*δn*,

_{y}*δn*), where

_{z}*n*,

_{x}*n*and

_{y}*n*are integers. This process facilitates the application of the above-mentioned spectral relation with a feasible sampling number even to visible light.

_{z}## 4. Application of the method of fast calculation of a series of radially diffracted wavefronts

*o*. The diffraction direction is identified by

*θ*and

*ϕ*, which correspond to the latitude and longitude of the globe, respectively. The diffraction distance

*R*is the same as the radius of the spherical object. On the basis of this layout, the diffracted wavefronts are calculated by using FFTs by varying the diffraction direction

*θ*and

*ϕ*. The size and the sampling number of the 3-D computational area as the input are 256 × 256 × 256 and 2mm×2mm×2mm, respectively. The 3-D object

*o*is defined inside this area and its radius

*R*is 0.69mm. The wavelength is set to 632.8nm. Some simulation results obtained under these conditions are shown in Fig. 6. The central parts of Figs. 6(a) and 6(b) are in focus while the parts away from the center are out of focus. This is because the diffraction distance is set to the radius of the spherical object. The focused parts correspond to the diffraction direction; this can be seen by making a comparison with Fig. 5. On the other hand, since the diffraction direction of Fig. 6(c) is opposite to that in Fig. 6(b), Fig. 6(c) is the horizontal reversal of the defocused image of Fig. 6(b). From these results, it is found that our method can calculate diffracted wavefronts properly. Moreover, some results for a series of diffracted wavefronts obtained by continuously varying the diffraction direction are shown as images of movies in Fig. 7. Figures 7(a) ( Media 1) and (b) ( Media 2) show patterns for cases where 360 diffracted images have been calculated by varying

*ϕ*and

*θ*by 1°, respectively. Although the interference patterns between the wavefronts diffracted from the front surface and the rear surface are seen in the movies, only the central parts of the front surface are in focus. The calculation time required for the 360 diffracted images is only about 20s, while it takes about 32h to obtain one diffracted image when the diffraction integral of Eq. (1) is calculated directly without FFTs. This shows that our method involving the use of FFTs practically performs the calculation 2 × 10

^{6}times faster than the method involving the direct calculation of Eq. (1).

## 5. Conclusion

## References and links

1. | D. Gabor, “A new microscopic principle,” Nature |

2. | L H. Lin and H. L. Beauchamp, “Write-read-erase in situ optical memory using thermoplastic holograms,” Appl. Opt. |

3. | X. Zhang, E. Dalsgaard, S. Liu, H. Lai, and J. Chen, “Concealed holographic coding for security applications by using a moiré technique,” Appl. Opt. |

4. | A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. |

5. | T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express |

6. | M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging |

7. | T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. |

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

9. | T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A |

10. | K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A |

11. | J. W. Goodman, |

12. | A. C. Kak and M. Slaney, “Tomographic Imaging with Diffracting Sources,” in |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(090.1760) Holography : Computer holography

(090.2870) Holography : Holographic display

**ToC Category:**

Holography

**History**

Original Manuscript: June 11, 2012

Revised Manuscript: August 9, 2012

Manuscript Accepted: August 9, 2012

Published: August 29, 2012

**Citation**

Yusuke Sando, Daisuke Barada, and Toyohiko Yatagai, "Fast calculation of computer-generated holograms based on 3-D Fourier spectrum for omnidirectional diffraction from a 3-D voxel-based object," Opt. Express **20**, 20962-20969 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-20962

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### References

- D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948). [CrossRef] [PubMed]
- L H. Lin and H. L. Beauchamp, “Write-read-erase in situ optical memory using thermoplastic holograms,” Appl. Opt.9, 2088–2092 (1970). [CrossRef] [PubMed]
- X. Zhang, E. Dalsgaard, S. Liu, H. Lai, and J. Chen, “Concealed holographic coding for security applications by using a moiré technique,” Appl. Opt.36, 8096–8097 (1997). [CrossRef]
- A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt.6, 1739–1748 (1967). [CrossRef] [PubMed]
- T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Fast calculation of computer-generated-hologram on AMD HD5000 series GPU and OpenCL,” Opt. Express18, 9955–9960 (2010). [CrossRef] [PubMed]
- M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging2, 28–34 (1993). [CrossRef]
- T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt.47, D63–D70 (2008). [CrossRef] [PubMed]
- D. Leseberg and C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt.27, 3020–3024 (1988). [CrossRef] [PubMed]
- T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A10, 299–305 (1993). [CrossRef]
- K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A20, 1755–1762 (2003). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
- A. C. Kak and M. Slaney, “Tomographic Imaging with Diffracting Sources,” in Principles of Computerized Tomographic Imaging, R. F. Cotellessa, J. K. Aggarwal, and G. Wade, eds. (Institute of Electrical and Electronics Engineers, 1988), pp. 203–273.

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