## Band-limited double-step Fresnel diffraction and its application to computer-generated holograms |

Optics Express, Vol. 21, Issue 7, pp. 9192-9197 (2013)

http://dx.doi.org/10.1364/OE.21.009192

Acrobat PDF (2345 KB)

### Abstract

Double-step Fresnel diffraction (DSF) is an efficient diffraction calculation in terms of the amount of usage memory and calculation time. This paper describes band-limited DSF, which will be useful for large computer-generated holograms (CGHs) and gigapixel digital holography, mitigating the aliasing noise of the DSF. As the application, we demonstrate a CGH generation with nearly 8K × 4K pixels from texture and depth maps of a three-dimensional scene captured by a depth camera.

© 2013 OSA

## 1. Introduction

1. S. A. Benton and V. M. Bove Jr., *Holographic Imaging* (Wiley-Interscience, 2008) [CrossRef] .

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

3. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer **38**, 46–53 (2005) [CrossRef] .

4. F. Yaras, H. Kang, and L. Onural, “Circular holographic video display system,” Opt. Express **19**, 9147–9156 (2011) [CrossRef] [PubMed] .

3. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer **38**, 46–53 (2005) [CrossRef] .

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

8. Y. Ichihashi, R. Oi, T. Senoh, K. Yamamoto, and T. Kurita, “Real-time capture and reconstruction system with multiple GPUs for a 3D live scene by a generation from 4K IP images to 8K holograms,” Opt. Express **20**, 21645–21655 (2012) [CrossRef] [PubMed] .

7. T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett. **34**, 3133–3135 (2009) [CrossRef] [PubMed] .

8. Y. Ichihashi, R. Oi, T. Senoh, K. Yamamoto, and T. Kurita, “Real-time capture and reconstruction system with multiple GPUs for a 3D live scene by a generation from 4K IP images to 8K holograms,” Opt. Express **20**, 21645–21655 (2012) [CrossRef] [PubMed] .

9. D. J. Brady and S. Lim, “Gigapixel holography,” 2011 ICO International Conference on Information Photonics (IP), 1–2, (2011) [CrossRef] .

11. S. O. Isikman, A. Greenbaum, W. Luo, A.F. Coskun, and A. Ozcan, “Giga-pixel lensfree holographic microscopy and tomography using color image sensors,” PLoS ONE **7**, e45044 (2012) [CrossRef] .

12. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. **29**, 1668–1670 (2004) [CrossRef] [PubMed] .

## 2. Band-limited double-step Fresnel diffraction

*ℱ*[·] and

*ℱ*

^{−1}[·] are the Fourier and inverse Fourier transform, respectively,

*u*

_{1}(

*x*

_{1},

*y*

_{1}) and

*u*

_{2}(

*x*

_{2},

*y*

_{2}) indicate a source and destination planes,

*p*is a point spread function and

_{z}*P*(

_{z}*f*,

_{x}*f*) =

_{y}*ℱ*[

*p*(

_{z}*x*

_{1},

*y*

_{1})] is the transfer function according to propagation distance

*z*. For example, ASM [13] uses

*x*

_{1},

*y*

_{1}) = ((

*m*

_{1}−

*N*/2)

_{x}*p*

_{x1}, (

*n*

_{1}−

*N*/2)

_{x}*p*

_{y1}) and (

*x*

_{2},

*y*

_{2}) = ((

*m*

_{2}−

*N*/2)

_{x}*p*

_{x2}, ((

*n*

_{2}−

*N*/2)

_{y}*p*

_{y2}), where

*m*

_{1},

*m*

_{2}∈ [0,

*N*/2 − 1] and

_{x}*n*

_{1},

*n*

_{2}∈ [0,

*N*/2 − 1], the sampling rates on the source plane are

_{y}*p*

_{x1}and

*p*

_{y1}, and those on the destination plane are

*p*

_{x2}=

*λz*/(

*N*

_{x}p_{x1}) and

*p*

_{y2}=

*λz*/(

*N*

_{y}p_{y1}): where the pixel numbers of the source and destination planes are

*N*×

_{x}*N*.

_{y}*z*by one fast Fourier transform (FFT), so that it does not need zero-padding unlike the convolution-based diffraction. Thus, it is an efficient approach in terms of the memory and the calculation time required; however, the sampling rates on the destination plane are changed by the wavelength and propagation distance.

12. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. **29**, 1668–1670 (2004) [CrossRef] [PubMed] .

*z*

_{1}. The sampling rates on the virtual plane are

*p*

_{xv}=

*λz*

_{1}/(

*N*

_{x}p_{x1}) and

*p*

_{yv}=

*λz*

_{1}/(

*N*

_{y}p_{y1}). The second SSF calculates the light propagation between the virtual plane and the destination plane at distance

*z*

_{2}. The sampling rates on the destination plane are

*p*

_{x2}=

*λz*

_{2}/(

*N*

_{x}p_{xv}) = |

*z*

_{2}/

*z*

_{1}|

*p*

_{x1}and

*p*

_{y2}=

*λz*

_{2}/(

*N*

_{y}p_{yv}) = |

*z*

_{2}/

*z*

_{1}|

*p*

_{y1}. The total propagation distance is

*z*=

*z*

_{1}+

*z*

_{2}, where

*z*

_{1}and

*z*

_{2}are acceptable for minus distance. DSF introducing the rectangular function for band limitation, which is referred as to BL-DSF, is expressed as follows:

^{sgn}^{(z)}means forward FFT when the sign of

*z*is plus and inverse FFT when it is minus. The rectangular function is introduced for band-limiting chirp function

### 2.1. Performance

*N*=

_{x}*N*=

_{y}*N*in the following discussion. Convolution-based diffraction needs to extend the size of the source and the destination planes to be at least four times as large as the original ones to avoid circular convolution, so that the calculation time for convolution-based diffraction is proportional to 4

*N*

^{2}log

_{2}2

*N*. On the other hand, the calculation time for BL-DSF diffraction is only proportional to

*N*

^{2}log

_{2}

*N*.

*N*

^{2}bytes and 8

*N*

^{2}bytes, respectively. For instance, when

*N*= 8,192, the angular spectrum method needs 2 GBytes, while BL-DSL needs only 512 MBytes. We could not calculate the case of

*N*= 8,192 using ASM on the GPU because the required memory exceeded the maximum amount of the GPU memory (2 GBytes). While we can calculate the case of

*N*= 8,192 using BL-DSF because BL-DSF requires only the GPU memory of 512 MBytes.

*μ*m, a propagation distance of

*z*=

*z*

_{1}+

*z*

_{2}= 0.02 m and

*N*= 512. Note that the sampling rate on the destination plane is changed by a rate of

*z*

_{2}and

*z*

_{1}. We do not want to change the sampling rates on the source and the destination planes, so we set

*z*

_{1}=

*z*/2 + 500 m and

*z*

_{2}=

*z*/2–500 m. In this case, the sampling rate on the destination plane is almost same as that on the source plane, about 9.996

*μ*m. Figure 1(a) is the case when the rectangular function is absent. Aliasing noise occurs. Meanwhile, Fig. 1(b) shows mitigation of aliasing noise, introducing the rectangular function.

## 3. Application to computer-generated holograms

14. M. Kawakita, K. Iizuka, T. Aida, H. Kikuchi, H. Fujikake, J. Yonai, and K. Takizawa, “Axi-vision camera (real-time distance-mapping camera),” Appl. Opt. **39**, 3931–3939 (2000) [CrossRef] .

8. Y. Ichihashi, R. Oi, T. Senoh, K. Yamamoto, and T. Kurita, “Real-time capture and reconstruction system with multiple GPUs for a 3D live scene by a generation from 4K IP images to 8K holograms,” Opt. Express **20**, 21645–21655 (2012) [CrossRef] [PubMed] .

*μ*m to display amplitude CGHs. To eliminate the 0-th order and conjugate lights inherently arising from the amplitude CGHs, the optical system uses half-zone plate processing and the single-sideband technique [15

15. T. Mishina, F. Okano, and I. Yuyama, “Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones,” Appl. Opt. **38**, 3703–3713 (1999) [CrossRef] .

*tex*(

*m*

_{1},

*n*

_{1}) and depth map

*dep*(

*m*

_{1},

*n*

_{1}) to about 8K × 4K pixels. A pixel value in

*dep*(

*m*

_{1},

*n*

_{1}) indicates a certain depth,

*i*, and the range of the pixel values is 0 to 255. Therefore, the physical distance is expressed as

*z*+

*i*Δ

*(*

_{z}*i*∈ [0, 255]), where Δ

*is the physical spacing between the neighboring pixel values in the depth map.*

_{z}*tex*(

*m*

_{1},

*n*

_{1}) is the texture map (Fig.2(a)) and

*n*(

*m*

_{1},

*n*

_{1}) is the uniform distribution of pseudo-random numbers within 0.0 to 1.0. Function

*mask*(

_{i}*m*

_{1},

*n*

_{1}) is defined by,

*I*(

*m*

_{2},

*n*

_{2}), of the complex amplitude

*u*

_{2}(

*m*

_{2},

*n*

_{2}). In addition, we obtain the final-amplitude CGH by taking ±2

*σ*top increase the brightness of the reconstructed image, where

*σ*is the standard derivation of

*I*(

*m*

_{2},

*n*

_{2}). Figure 3 shows a reconstructed 3D scene from a nearly 8K × 4K CGH using BL-DSF. The left-hand, middle and right-hand figures are photographs by changing focus.

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. A. Benton and V. M. Bove Jr., |

2. | U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical
Reconstruction,” Appl.Opt. , |

3. | C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer |

4. | F. Yaras, H. Kang, and L. Onural, “Circular holographic video display system,” Opt. Express |

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

6. | H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-time generation of full color image hologram with compact distance look-up table,” OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2009, DWC4 (2009). |

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

8. | Y. Ichihashi, R. Oi, T. Senoh, K. Yamamoto, and T. Kurita, “Real-time capture and reconstruction system with multiple GPUs for a 3D live scene by a generation from 4K IP images to 8K holograms,” Opt. Express |

9. | D. J. Brady and S. Lim, “Gigapixel holography,” 2011 ICO International Conference on Information Photonics (IP), 1–2, (2011) [CrossRef] . |

10. | J. R. Fienup and A. E. Tippie, “Gigapixel synthetic-aperture digital holography,” Proc. SPIE |

11. | S. O. Isikman, A. Greenbaum, W. Luo, A.F. Coskun, and A. Ozcan, “Giga-pixel lensfree holographic microscopy and tomography using color image sensors,” PLoS ONE |

12. | F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. |

13. | J. W. Goodman, |

14. | M. Kawakita, K. Iizuka, T. Aida, H. Kikuchi, H. Fujikake, J. Yonai, and K. Takizawa, “Axi-vision camera (real-time distance-mapping camera),” Appl. Opt. |

15. | T. Mishina, F. Okano, and I. Yuyama, “Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones,” Appl. Opt. |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.2870) Holography : Holographic display

(090.1995) Holography : Digital holography

(090.5694) Holography : Real-time holography

**ToC Category:**

Holography

**History**

Original Manuscript: February 28, 2013

Revised Manuscript: March 23, 2013

Manuscript Accepted: March 23, 2013

Published: April 5, 2013

**Citation**

Naohisa Okada, Tomoyoshi Shimobaba, Yasuyuki Ichihashi, Ryutaro Oi, Kenji Yamamoto, Minoru Oikawa, Takashi Kakue, Nobuyuki Masuda, and Tomoyoshi Ito, "Band-limited double-step Fresnel diffraction and its application to computer-generated holograms," Opt. Express **21**, 9192-9197 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-9192

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### References

- S. A. Benton and V. M. Bove, Holographic Imaging (Wiley-Interscience, 2008). [CrossRef]
- U. Schnars and W. Juptner, “Direct recording of holograms by a CCD target and numerical Reconstruction,” Appl.Opt., 33, 179–181 (1994). [CrossRef]
- C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer38, 46–53 (2005). [CrossRef]
- F. Yaras, H. Kang, and L. Onural, “Circular holographic video display system,” Opt. Express19, 9147–9156 (2011). [CrossRef] [PubMed]
- M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging, 2, 28–34 (1993). [CrossRef]
- H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-time generation of full color image hologram with compact distance look-up table,” OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2009, DWC4 (2009).
- T. Shimobaba, N. Masuda, and T. Ito, “Simple and fast calculation algorithm for computer-generated hologram with wavefront recording plane,” Opt. Lett.34, 3133–3135 (2009). [CrossRef] [PubMed]
- Y. Ichihashi, R. Oi, T. Senoh, K. Yamamoto, and T. Kurita, “Real-time capture and reconstruction system with multiple GPUs for a 3D live scene by a generation from 4K IP images to 8K holograms,” Opt. Express20, 21645–21655 (2012). [CrossRef] [PubMed]
- D. J. Brady and S. Lim, “Gigapixel holography,” 2011 ICO International Conference on Information Photonics (IP), 1–2, (2011). [CrossRef]
- J. R. Fienup and A. E. Tippie, “Gigapixel synthetic-aperture digital holography,” Proc. SPIE8122, 812203 (2011). [CrossRef]
- S. O. Isikman, A. Greenbaum, W. Luo, A.F. Coskun, and A. Ozcan, “Giga-pixel lensfree holographic microscopy and tomography using color image sensors,” PLoS ONE7, e45044 (2012). [CrossRef]
- F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett.29, 1668–1670 (2004). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (3rd ed.), (Robert & Company, 2005).
- M. Kawakita, K. Iizuka, T. Aida, H. Kikuchi, H. Fujikake, J. Yonai, and K. Takizawa, “Axi-vision camera (real-time distance-mapping camera),” Appl. Opt.39, 3931–3939 (2000). [CrossRef]
- T. Mishina, F. Okano, and I. Yuyama, “Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones,” Appl. Opt.38, 3703–3713 (1999). [CrossRef]

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