## Arbitrary shape surface Fresnel diffraction |

Optics Express, Vol. 20, Issue 8, pp. 9335-9340 (2012)

http://dx.doi.org/10.1364/OE.20.009335

Acrobat PDF (876 KB)

### Abstract

Fresnel diffraction calculation on an arbitrary shape surface is proposed. This method is capable of calculating Fresnel diffraction from a source surface with an arbitrary shape to a planar destination surface. Although such calculation can be readily calculated by the direct integral of a diffraction calculation, the calculation cost is proportional to *O*(*N*^{2}) in one dimensional or *O*(*N*^{4}) in two dimensional cases, where *N* is the number of sampling points. However, the calculation cost of the proposed method is *O*(*N* log *N*) in one dimensional or *O*(*N*^{2} log *N*) in two dimensional cases using non-uniform fast Fourier transform.

© 2012 OSA

## 1. Introduction

3. E. G. Williams, *Fourier Acoustics – Sound Radiation and Nearfield Acoustical Holography* (Academic Press, 1999). [PubMed]

4. D. M. Paganin, *Coherent X-Ray Optics* (Oxford University Press, 2006). [CrossRef]

5. T. C. Poon (ed.), *Digital Holography and Three-Dimensional Display* (Springer, 2006). [CrossRef]

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

9. H. Sakata and Y. Sakamoto, “Fast computation method for a Fresnel hologram using three-dimensional affine transformations in real space,” Appl. Opt. **48**, H212–H221 (2009). [CrossRef] [PubMed]

10. T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. **17**, 556–558 (1992). [CrossRef] [PubMed]

14. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. **47**, 1567–1574 (2008). [CrossRef] [PubMed]

*O*(

*N*

^{2}) in one dimensional or

*O*(

*N*

^{4}) in two dimensional cases, where

*N*is the number of sampling points. However, the calculation cost of the proposed method is

*O*(

*N*log

*N*) in one dimensional or

*O*(

*N*

^{2}log

*N*) in two dimensional cases using non-uniform fast Fourier transform.

## 2. Arbitrary shape surface Fresnel diffraction

*u*

_{1}(

**x**

_{1}

**)**and

*u*

_{2}(

**x**

_{2}) are planar source and destination surfaces,

**x**

_{1}and

**x**

_{2}are the position vectors on the source and destination surfaces,

*λ*and

*k*are the wavelength and wave number of light, and

*z*

_{0}is the distance between the source and destination surfaces.

*u*(

_{I}**x**

_{1}) is the incident wave to the source surface. If the incident wave is used as a planar wave that is perpendicular to the optical axis, we can treat as

*u*(

_{I}**x**

_{1}) = 1.

*u*

_{1}(

**x**,

_{1}*d*

_{1}) is defined by the displacement

*d*

_{1}=

*d*

_{1}(

**x**

_{1}) at the position

**x**

_{1}. Note that when the arbitrary shape surface is uniform-sampled, the corresponding coordinate

**x**

_{1}is non-uniform-sampled depending on the slope of

*u*

_{1}(

**x**,

_{1}*d*

_{1}) at the position

**x**

_{1}. Huygens diffraction on an arbitrary shape surface is expressed by:

*u*(

_{I}**x**

_{1},

*d*

_{1}) is used as a planar wave that is perpendicular to the optical axis, the incident wave is expressed as

*u*(

_{I}**x**

_{1},

*d*

_{1}) = exp(

*ikd*

_{1}).

*r*

_{0}=

*z*

_{0}–

*d*

_{1}, we can obtain the following approximation:

*O*(

*N*

^{2}) in one dimensional or

*O*(

*N*

^{4}) in two dimensional cases, where

*N*is the number of sampling points, because we cannot calculate them using Fourier transform.

*z*

_{0}≫

*d*

_{1}, we approximate the third exponential term in the integration as follows:

**x**

_{1}is sampled by the non-uniform sampling rates, instead of (uniform) Fourier transform, we can calculate the above equation using non-uniform Fourier transform (NUFT): where,

*NUF*[·] denotes NUFT. NUFT of a function

*f*(

**x**

_{1}) is defined as:

**x**

_{1}is non-uniform-sampled and

**x**

_{2}is uniform-sampled, unlike uniform Fourier transform. For the numerical implementation of Eq.(9), it is necessary to use non-uniform fast Fourier transform (NUFFT) which has the complexity of

*O*(

*N*log

*N*). Many methods for NUFFT have been proposed over the course of the past twenty years or so [15

15. A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) **14**, 1368–1393 (1993). [CrossRef]

18. L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. **46**, 443–454 (2004). [CrossRef]

18. L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. **46**, 443–454 (2004). [CrossRef]

18. L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. **46**, 443–454 (2004). [CrossRef]

## 3. Result

*z*

_{0}= 1 m, the wavelength of 633 nm, and the number of sampling points on source and destination

*N*= 1, 024. The sampling rates on the source and destination surface are

*p*= 10

*μ*m. We used a planar wave as the incident wave that is perpendicular to the optical axis.

**x**

_{1}, respectively. The horizontal axis in (a) indicates the position on

**x**

_{1}in metric units. The horizontal axes in (b)–(e) indicates the position on

**x**

_{2}in metric units. The sampling rates on these small planar surfaces are

*p*, however, the sampling rates on

**x**

_{1}are |

*p*cos(−30°)|, |

*p*cos(−50°)|, |

*p*cos(0°)| and |

*p*cos(+50°)|, respectively. The destination planar surface is not inclined to

**x**

_{2}. Figures 2(b)–(d) show the intensity profiles of the diffraction results by Eq.(2), Eq.(5) with direct integral and our method (Eq.(8)), respectively. Figure 2(e) depicts the absolute error between (Eq.(2)) and Eq.(8). The absolute error falls into within approximately 0.025.

**x**

_{1}according to the source surface depends on the slope of the quadratic curve. Figure 3(b)–(d) shows the intensity profiles of the diffraction results by Eq.(2), Eq.(5) and our method (Eq.(8)), respectively. Figure 3(e) depicts the absolute error between Eq.(2) and Eq.(8). The absolute error falls into within approximately 0.005. The primary factor of these absolute errors in Figs.2 and 3 is the approximations by Eqs.(4) and (6).

## 4. Conclusion

*O*(

*N*

^{2}) in one dimensional or

*O*(

*N*

^{4}) in two dimensional cases. In contrast, the calculation cost of our method is

*O*(

*N*log

*N*) in one dimensional or

*O*(

*N*

^{2}log

*N*) in two dimensional cases using NUFFT. The method is very useful for calculating a CGH from a three-dimensional object composed of multiple polygons or arbitrary shape surfaces. In our next work, we will show the fast calculation of a CGH from such 3D objects using this method.

## Acknowledgments

## References and links

1. | J. W. Goodman, |

2. | Okan K. Ersoy, |

3. | E. G. Williams, |

4. | D. M. Paganin, |

5. | T. C. Poon (ed.), |

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

7. | C. Frere and D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. |

8. | L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express |

9. | H. Sakata and Y. Sakamoto, “Fast computation method for a Fresnel hologram using three-dimensional affine transformations in real space,” Appl. Opt. |

10. | T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. |

11. | N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A |

12. | 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 |

13. | G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE |

14. | L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. |

15. | A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) |

16. | Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microw. Guid. Wave Lett. |

17. | Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens. |

18. | L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. |

**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 27, 2012

Revised Manuscript: April 1, 2012

Manuscript Accepted: April 2, 2012

Published: April 6, 2012

**Citation**

Tomoyoshi Shimobaba, Nobuyuki Masuda, and Tomoyoshi Ito, "Arbitrary shape surface Fresnel diffraction," Opt. Express **20**, 9335-9340 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-9335

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### References

- J. W. Goodman, Introduction to Fourier Optics (3rd ed.) (Robert & Company, 2005).
- Okan K. Ersoy, Diffraction, Fourier Optics And Imaging (Wiley-Interscience, 2006).
- E. G. Williams, Fourier Acoustics – Sound Radiation and Nearfield Acoustical Holography (Academic Press, 1999). [PubMed]
- D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006). [CrossRef]
- T. C. Poon (ed.), Digital Holography and Three-Dimensional Display (Springer, 2006). [CrossRef]
- D. Leseberg and C. Frére, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt.27, 3020 (1988). [CrossRef] [PubMed]
- C. Frere and D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt.28, 2422–2425 (1989). [CrossRef] [PubMed]
- L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express10, 1250–1257 (2002). [PubMed]
- H. Sakata and Y. Sakamoto, “Fast computation method for a Fresnel hologram using three-dimensional affine transformations in real space,” Appl. Opt.48, H212–H221 (2009). [CrossRef] [PubMed]
- T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett.17, 556–558 (1992). [CrossRef] [PubMed]
- N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A15, 857–867 (1998). [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]
- G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE6252, 62521K (2006). [CrossRef]
- L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt.47, 1567–1574 (2008). [CrossRef] [PubMed]
- A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA)14, 1368–1393 (1993). [CrossRef]
- Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microw. Guid. Wave Lett.8, 18–20 (1998). [CrossRef]
- Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens.1, 288–290 (1998).
- L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev.46, 443–454 (2004). [CrossRef]

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