## Fast calculation of computer-generated hologram using the circular symmetry of zone plates |

Optics Express, Vol. 20, Issue 25, pp. 27496-27502 (2012)

http://dx.doi.org/10.1364/OE.20.027496

Acrobat PDF (1828 KB)

### Abstract

Computer-Generated Holograms (CGHs) can be generated from three-dimensional objects composed of point light sources by overlapping zone plates. A zone plate is a grating that can focus an incident wave and it has circular symmetry shape. In this study, we propose a fast CGH generating algorithm using the circular symmetry of zone plates and computer graphics techniques. We evaluated the proposed method by numerical simulation.

© 2012 OSA

## 1. Introduction

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

2. K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. **39**, 6587–6594 (2000). [CrossRef]

4. H. Yoshikawa, “Fast Computation of Fresnel Holograms Employing Difference,” Opt. Rev. **8**, 331–335 (2001). [CrossRef]

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

5. Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express **17**, 13895–13902 (2009). [CrossRef] [PubMed]

6. 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]

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

7. S. -C. Kim and E. -S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. **47**, D55–D62 (2008). [CrossRef] [PubMed]

8. S. -C. Kim, J. -M. Kim, and E. -S. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe pattern in computer-generated holograms,” Opt. Express **20**, 12021–12034 (2012). [CrossRef] [PubMed]

8. S. -C. Kim, J. -M. Kim, and E. -S. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe pattern in computer-generated holograms,” Opt. Express **20**, 12021–12034 (2012). [CrossRef] [PubMed]

## 2. Theory

### 2.1. Conventional CGH calculation from PLS-based 3D object

**2**, 28–34 (1993). [CrossRef]

*x*,

_{α}*y*) denotes the coordinates of a CGH plane, (

_{α}*x*,

_{j}*y*,

_{j}*z*) denotes the coordinates of a PLS in three dimensions,

_{j}*x*=

_{α}_{j}*x*−

_{α}*x*,

_{j}*y*=

_{α}_{j}*y*−

_{α}*y*,

_{j}*i*is the imaginary unit,

*ϕ*is the argument of the object wave,

*k*is the wave number of reference light,

*N*is the total number of the PLSs,

*A*is the amplitude of

_{j}*j*-th PLS.

*I*represents the complex amplitude distribution on the CGH for

_{j}*j*-th PLS, and the phase is represented as

### 2.2. Fast calculation of CGH using the circular symmetry of zone plate and LUT

3. T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. **138**, 44–52 (2001). [CrossRef]

*R*= (

*z*/

_{j}*p*) × tan(sin

^{−1}(

*λ*/2

*p*)) where

*p*is the pixel pitch of the device for displaying a CGH and

*λ*is the wave length of a incident wave. These recurrence formulas can calculate an adjacent phase on the CGH only in two additional operations faster than the direct calculation of

*θ*(

_{j}*x*,

_{α}_{j}*y*) where

_{α}_{j}*γ*= 2

*π*

*/*

*λ*

*z*. As the start position of the formula is set as the center of the zone plate, the first terms of the formulas become

_{j}*θ*(0, 0) = 2

_{j}*π*

*z*

_{j}*/*

*λ*,

*δ*(0) =

_{j}*π*

*/*

*λ*

*z*, which is simpler than Ref. [3

_{j}3. T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. **138**, 44–52 (2001). [CrossRef]

10. J. Bresenham, “A linear algorithm for incremental digital display of circular arcs,” Commun. ACM **20**, 100–106 (1977). [CrossRef]

11. E. Andres, “Discrete circles, rings and spheres,” Comput. Graphics **18**, 695–706 (1994). [CrossRef]

11. E. Andres, “Discrete circles, rings and spheres,” Comput. Graphics **18**, 695–706 (1994). [CrossRef]

*x*,

_{α}*y*) and the integer radius

_{α}*r*is defined as

*ε*, the algorithm selects the nearest pixels to draw the circle. Due to

*ε*can be easily updated by simple addition or subtraction as shown in Fig. 2, it can draw a discrete circle rapidly. Therefore, we can obtain a zone plate from the line of the radius quickly, and it is stored in the zone plate table.

*x*,

_{j}*y*) and then overlapped on the CGH. It is unnecessary for the proposed method to retain the wavefronts for all PLSs in the table. The proposed method retains only the wavefronts for the same depth, so that the required memory for the zone plate table is small. For example, the required memory for the zone plate table is 90 MBytes when the parameters are

_{j}*λ*= 520 nm,

*z*= 0.3 m,

_{j}*p*= 8

*μ*m (in the parameters, the radius of the zone plate is 1,220 pixels). After overlapping zone plates corresponding to the same depth, we recalculate the zone plate table for the other depth. Note that the zone plates are treated as complex amplitude distribution when overlapping them.

12. Z. Yang, Q. Fan, Y. Zhang, J. Liu, and J. Zhou, “A new method for producing computer generated holograms,” J. Opt. **14**, 095702 (2012). [CrossRef]

### 2.3. Discretization error compensation

*θ*on the CGH is calculated by

_{j}*θ*=

_{j}*k*(

*z*+

_{j}*pr*). For example, the phase of a coordinate (0,6) on the line as shown in Fig. 3 is represented as

*θ*=

_{o}*k*(

*z*+

_{j}*pr*)|

_{r}_{=6}. Considering the phase of coordinate (5,4) in Fig. 3, we express an actual phase

*θ*+

_{o}*θ*due to the distance from the center becomes

_{ε}*r*+

*ε*, where

*θ*=

_{ε}*kpε/z*. However, copying the phase

_{j}*θ*to the coordinate (5,4) by the “arithmetical circle algorithm” [11

_{o}11. E. Andres, “Discrete circles, rings and spheres,” Comput. Graphics **18**, 695–706 (1994). [CrossRef]

*θ*.

_{ε}*I*(

*θ*) along the circle by the “arithmetical circle algorithm” is expressed by

_{o}*Re*[

*I*(

*θ*)] = cos(

_{o}*θ*),

_{o}*Im*[

*I*(

*θ*)] = sin(

_{o}*θ*). For the error compensation, applying the additional theorem of the trigonometric function and approximation cosΔ ≈ 1, sinΔ ≈ Δ (when Δ ≪ 1), the compensated complex amplitudes: Note that the additional calculation load is small due to

_{o}*ε*, which is calculated in Arithmetical circle algorithm and cos

*θ*and sin

_{o}*θ*have been already calculated.

_{o}## 3. Results and Discussion

*λ*= 520 nm,

*z*= 0.3 m,

_{j}*p*= 8

*μ*m, the CGH resolution is 1,920 × 1,080 pixels. We used the following environment: the operating system is Microsoft Windows 7 Professional Service pack 1, the CPU is AMD A8-3850 with 2.89 GHz (we used single core) and a memory of 4 GByte, the compilers are Microsoft Visual C++ 2010, the computation precision is set as double. In this figure, the light intensities are normalized at the maximum of the direct calculation of Eq. (1). Compared with the result by the direct calculation of Eq. (1), which is just focused at 0.3 m, the maximum light intensity of our proposed method without compensation decays by about 20% and the focal point is shifted by 400

*μ*m. Applying the error compensation mentioned in subsection 2.3, the maximum light intensity is improved by about 10%; however, the focal point is shifted by 488

*μ*m. Nevertheless, the problem of shifting focal point is not a serious problem because it can be solved easily by changing the depth of PLSs preliminarily.

14. T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. **20**, 3133–3135 (2009). [CrossRef]

18. P. W. M. Tsang, K. W. K Cheung, and T.-C. Poon, “Real-time relighting of digital holograms based on wavefront recording plane method,” Opt. Express **20**, 5962–5967 (2012). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. |

3. | T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun. |

4. | H. Yoshikawa, “Fast Computation of Fresnel Holograms Employing Difference,” Opt. Rev. |

5. | Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express |

6. | 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 |

7. | S. -C. Kim and E. -S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. |

8. | S. -C. Kim, J. -M. Kim, and E. -S. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe pattern in computer-generated holograms,” Opt. Express |

9. | L. Mertz and N. O. Young, “Fresnel Transformation of Images,” in |

10. | J. Bresenham, “A linear algorithm for incremental digital display of circular arcs,” Commun. ACM |

11. | E. Andres, “Discrete circles, rings and spheres,” Comput. Graphics |

12. | Z. Yang, Q. Fan, Y. Zhang, J. Liu, and J. Zhou, “A new method for producing computer generated holograms,” J. Opt. |

13. | J. W. Goodman, |

14. | T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. |

15. | T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express |

16. | P. Tsang, W. -K. Cheung, T. -C. Poon, and C. Zhou, “Holographic video at 40 frames per second for 4-million object points,” Opt. Express |

17. | J. Weng, T. Shimobaba, N Okada, H. Nakayma, M. Oikawa, N. Masuda, and T. Ito, “Generation of real-time large computer generated hologram using wavefront recording method,” Opt. Express |

18. | P. W. M. Tsang, K. W. K Cheung, and T.-C. Poon, “Real-time relighting of digital holograms based on wavefront recording plane method,” Opt. Express |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.2870) Holography : Holographic display

(090.5694) Holography : Real-time holography

**ToC Category:**

Holography

**History**

Original Manuscript: September 18, 2012

Revised Manuscript: November 8, 2012

Manuscript Accepted: November 8, 2012

Published: November 27, 2012

**Citation**

Takashi Nishitsuji, Tomoyoshi Shimobaba, Takashi Kakue, Nobuyuki Masuda, and Tomoyoshi Ito, "Fast calculation of computer-generated hologram using the circular symmetry of zone plates," Opt. Express **20**, 27496-27502 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27496

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### References

- M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging2, 28–34 (1993). [CrossRef]
- K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt.39, 6587–6594 (2000). [CrossRef]
- T. Shimobaba and T. Ito, “An efficient computational method suitable for hardware of computer-generated hologram with phase computation by addition,” Comput. Phys. Commun.138, 44–52 (2001). [CrossRef]
- H. Yoshikawa, “Fast Computation of Fresnel Holograms Employing Difference,” Opt. Rev.8, 331–335 (2001). [CrossRef]
- Y. Ichihashi, H. Nakayama, T. Ito, N. Masuda, T. Shimobaba, A. Shiraki, and T. Sugie, “HORN-6 special-purpose clustered computing system for electroholography,” Opt. Express17, 13895–13902 (2009). [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]
- S. -C. Kim and E. -S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt.47, D55–D62 (2008). [CrossRef] [PubMed]
- S. -C. Kim, J. -M. Kim, and E. -S. Kim, “Effective memory reduction of the novel look-up table with one-dimensional sub-principle fringe pattern in computer-generated holograms,” Opt. Express20, 12021–12034 (2012). [CrossRef] [PubMed]
- L. Mertz and N. O. Young, “Fresnel Transformation of Images,” in Preceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London), 305–310 (1961).
- J. Bresenham, “A linear algorithm for incremental digital display of circular arcs,” Commun. ACM20, 100–106 (1977). [CrossRef]
- E. Andres, “Discrete circles, rings and spheres,” Comput. Graphics18, 695–706 (1994). [CrossRef]
- Z. Yang, Q. Fan, Y. Zhang, J. Liu, and J. Zhou, “A new method for producing computer generated holograms,” J. Opt.14, 095702 (2012). [CrossRef]
- J. W. Goodman, Introduction to Fourier OpticsRoberts & Company (2004).
- T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett.20, 3133–3135 (2009). [CrossRef]
- T. Shimobaba, T. Ito, N. Masuda, Y. Ichihashi, and N. Takada, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express18, 19504–19509 (2010). [CrossRef] [PubMed]
- P. Tsang, W. -K. Cheung, T. -C. Poon, and C. Zhou, “Holographic video at 40 frames per second for 4-million object points,” Opt. Express19, 15205–15211 (2011). [CrossRef] [PubMed]
- J. Weng, T. Shimobaba, N Okada, H. Nakayma, M. Oikawa, N. Masuda, and T. Ito, “Generation of real-time large computer generated hologram using wavefront recording method,” Opt. Express20, 4018–4023 (2012). [CrossRef] [PubMed]
- P. W. M. Tsang, K. W. K Cheung, and T.-C. Poon, “Real-time relighting of digital holograms based on wavefront recording plane method,” Opt. Express20, 5962–5967 (2012). [CrossRef] [PubMed]

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