## Calculating the Fresnel diffraction of light from a shifted and tilted plane |

Optics Express, Vol. 20, Issue 12, pp. 12949-12958 (2012)

http://dx.doi.org/10.1364/OE.20.012949

Acrobat PDF (1607 KB)

### Abstract

We propose a technique for calculating the diffraction of light in the Fresnel region from a plane that is the light source (source plane) to a plane at which the diffracted light is to be calculated (destination plane). When the wavefield of the source plane is described by a group of points on a grid, this technique can be used to calculate the wavefield of the group of points on a grid on the destination plane. The positions of both planes may be shifted, and the plane normal vectors of both planes may have different directions. Since a scaled Fourier transform is used for the calculation, it can be calculated faster than calculating the diffraction by a Fresnel transform at each point. This technique can be used to calculate and generate planar holograms from computer graphics data.

© 2012 OSA

## 1. Introduction

1. 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**(10), 1567–1574 (2008). [CrossRef] [PubMed]

2. T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A **10**(2), 299–305 (1993). [CrossRef]

5. K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express **18**(17), 18453–18463 (2010). [CrossRef] [PubMed]

6. 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 **15**(4), 857–867 (1998). [CrossRef]

8. R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express **15**(9), 5631–5640 (2007). [CrossRef] [PubMed]

9. L. Yu, U. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express **10**(22), 1250–1257 (2002). [PubMed]

11. L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A **28**(3), 290–295 (2011). [CrossRef]

## 2. Theory

*x*-,

*y*-, and

*z*axes are in the world coordinate system, and coordinates are represented by [

*x,y,z*]

^{T}. Note that [ · ]

^{T}indicates the transpose vector. All coordinates are assumed to be represented in the world coordinate system below. The center

**P****of the source plane is [**

_{0}*x*

_{0},

*y*

_{0},

*z*

_{0}]

^{T}. The vectors

*on this plane.*

**P**_{st}*s*Δ

_{x}*t*+ Δ

_{x}*s*Δ

_{y}*t*+ Δ

_{y}*s*Δ

_{z}*t*= 0. The center

_{z}

**P****of the plane where the diffracted light is to be obtained (destination plane) is [**

_{1}*x*

_{1},

*y*

_{1},

*z*

_{1}]

^{T}. The vectors

*x*- and

*y*-axes respectively, are defined to represent the position

*on this plane.*

**P**_{uv}

**P****and**

_{1}

**P****.**

_{0}*x*,

_{st}*y*,

_{st}*z*]

_{st}^{T}of

*and the coordinates [*

**P**_{st}*x*,

_{uv}*y*,

_{uv}*z*]

_{uv}^{T}of

*are defined as follows. where*

**P**_{uv}*s*and

*t*are real numbers for representing

*and*

**P**_{st}*u*and

*v*are real numbers for representing

*. However, in the actual calculations, these are all handled as integers to make the calculations discrete.*

**P**_{uv}*U*

_{0}diffracts to create the wavefield

*U*

_{1}on the destination plane. The diffracted result

*U*

_{1}can be represented as follows. where

*j*represents the imaginary unit,

*λ*represents the wavelength of the light, and

*k*represents the wave number (

*k*= 2

*π/λ*). Also, the two planes are assumed to be separated by a distance for which the Fresnel diffraction holds. In other words, Eq. (4) is assumed to hold at all

*and*

**P**_{st}*[12]. The diffraction into the destination plane is also assumed to be limited by the angle for which the Fresnel diffraction holds.*

**P**_{uv}13. D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev . **33**(3), 389–404 (1991). [CrossRef]

*a*is the scale parameter.

*s,t,u,v*are all handled as integers to make the calculations discrete in actuality. When you discretize the Eq. (15) and expand it, you can find the terms of discretized scaled Fourier transform.

## 3. Numerical experiment and calculation times

**P****. In other words, the calculation was performed 3 times in the**

_{1}*x*-axis direction and 3 times in the

*y*-axis direction for a total of 9 times. The diffraction from the lens to the image plane was performed 1 time with a size of 768 × 768. To reduce aliasing, the scaled Fourier transform was calculated with a doubled size in both the horizontal and vertical directions. In other words, the calculation was performed, for example, using a scaled Fourier transform with a size of 512 × 512 for the set of 256 × 256 points. Since the surface plane had random phases, speckle noise was conspicuous at the image plane. Therefore, we performed the experiment 30 times while changing the random phase for each condition and let the accumulated value be the experimental result.

*N*= 32) to 1024 × 1024 (

*N*= 1024). Here,

*N*(

*N*= 32, 64, 128, 256, 512, 1024) is the number of points along an axis of the grid. Note that we doubled the vertical and horizontal sizes for the scaled Fourier transform as we did in experiments 1 and 2. The PC used for the measurements had an Intel Core i7 CPU with Microsoft Windows 7 64-bit edition operating system running single threaded. Table 2 shows the measurement results. From these measurement results, it is apparent that the proposed technique was able shorten the calculation time

*t*to approximately 1/9 (≒ 193/1, 734) that of the Fresnel transform for a 64 × 64 grid and to approximately 1/70 (≒ 99, 918/7, 065, 681) for a 512 × 512 grid.

_{N}*a*and

_{N}*b*as follows. If the index log

_{N}_{10}(

*a*

_{N}/a_{32}) is around 0 for any

*N*, it suggests the computational complexity is

*O*(

*N*

^{4}). In the same way, if the index log

_{10}(

*b*/

_{N}*b*

_{32}) is around 0, it suggests

*O*(

*N*

^{2}log

_{2}

*N*). Figures 8(a) and 8(b) show the indexes at Fresnel transform and at proposed technique respectively. As you can see, the indexes suggest

*O*(

*N*

^{4}) in (a); however, they do not suggest

*O*(

*N*

^{4}) in (b).

*O*(

*N*

^{2}log

_{2}

*N*) so much in (b). It would be the reason that the Eq. (15) is not a two-dimensional scaled Fourier transform, but we are not be confident about it. We have to find out definite reason in the future.

## 4. Conclusions

*s*= 0 and Δ

_{y}*t*= 0), then the calculations could be performed even faster since Eq. (15) would become a two-dimensional scaled Fourier transform. We plan to investigate the advantages and disadvantages of this restriction as well as the computational complexity in the future. In addition, we also plan to confirm the calculation when the rotation is large, for example in the case that the angle between surface plane and destination plane is nearly 90 degree.

_{x}## References and links

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

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

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

4. | S. D. Nicola, A. Finizio, and G. Pierattini, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express |

5. | K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express |

6. | 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 |

7. | R. P. Muffoletto, “Numerical techniques for fresnel diffraction in computational holography,” PhD thesis (Louisiana State University, 2006). |

8. | R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express |

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

10. | J. Miura and T. Shimobaba, “Shifted-fresnel diffraction between two tilted planes |

11. | L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A |

12. | J. W. Goodman, |

13. | D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev . |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(090.1760) Holography : Computer holography

(100.6890) Image processing : Three-dimensional image processing

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 24, 2012

Revised Manuscript: April 17, 2012

Manuscript Accepted: April 27, 2012

Published: May 23, 2012

**Citation**

Kenji Yamamoto, Yasuyuki Ichihashi, Takanori Senoh, Ryutaro Oi, and Taiichiro Kurita, "Calculating the Fresnel diffraction of light from a shifted and tilted plane," Opt. Express **20**, 12949-12958 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12949

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### References

- 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(10), 1567–1574 (2008). [CrossRef] [PubMed]
- T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A10(2), 299–305 (1993). [CrossRef]
- K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on titled planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A20(9), 1755–1762 (2003). [CrossRef]
- S. D. Nicola, A. Finizio, and G. Pierattini, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express13(24), 9935–9940 (2005). [CrossRef] [PubMed]
- K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express18(17), 18453–18463 (2010). [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(4), 857–867 (1998). [CrossRef]
- R. P. Muffoletto, “Numerical techniques for fresnel diffraction in computational holography,” PhD thesis (Louisiana State University, 2006).
- R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express15(9), 5631–5640 (2007). [CrossRef] [PubMed]
- L. Yu, U. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express10(22), 1250–1257 (2002). [PubMed]
- J. Miura and T. Shimobaba, “Shifted-fresnel diffraction between two tilted planes (in Japanese),” Watake Seminar in TohokuYS–6–52 (2008).
- L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A28(3), 290–295 (2011). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, Englewood, CO, 2005), Chap. 4.
- D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev. 33(3), 389–404 (1991). [CrossRef]

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