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(N2) in one dimensional or O(N4) 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(N2 log N) in two dimensional cases using non-uniform fast Fourier transform.

© 2012 Optical Society of America

1. Introduction

Diffraction calculations such as Huygens diffraction, Fresnel diffraction and angular spectrum method, are important tools in wide-ranging optics [1, 2], ultrasonic [3], X-ray [4] and so forth. Its applications in optics include computer-generated-hologram (CGH) and digital holography [5], phase retrieval, image encryption and decryption and so forth.

Fast Fourier transform (FFT)-based diffraction calculations according to their convolution or Fourier transform form are used in these applications. The FFT-based diffraction calculations, however, can only be applied to planar surfaces in parallel. In order to apply the methods to a non-parallel planar surface, many methods have been proposed: for example, non-parallel Fresnel diffractions [69] and non-parallel angular spectrum methods [1014]. Unfortunately, these non-parallel diffractions are limited to planar surfaces. If we calculate a source surface with arbitrary shape using non-parallel diffractions, we need to approximate the arbitrary shape surface with many small non-parallel planar surfaces, and then, we need to calculate the non-parallel diffraction per the small non-parallel planar surfaces.

In this paper, we propose Fresnel diffraction calculation on an arbitrary shape surface with complex amplitude. Without approximating an arbitrary shape surface with small non-parallel planar surfaces, this method is capable of calculating Fresnel diffraction from a source arbitrary shape surface 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(N2) in one dimensional or O(N4) 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(N2 log N) in two dimensional cases using non-uniform fast Fourier transform.

In Section 2, we describe the arbitrary shape surface Fresnel diffraction. In Section 3, we present the numerical results. Section 4 concludes this work.

2. Arbitrary shape surface Fresnel diffraction

Let us begin with Huygens diffraction. Huygens diffraction [1] on a planar surface is expressed as:

u2(x2)=z0iλuI(x1)u1(x1)exp(ikr)r2dx1,
where, u1(x1) and u2(x2) are planar source and destination surfaces, x1 and x2 are the position vectors on the source and destination surfaces, λ and k are the wavelength and wave number of light, and r=|x2x1|2+z02, where z0 is the distance between the source and destination surfaces. uI (x1) 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 uI (x1) = 1.

We expand the Huygens diffraction to a source surface with arbitrary shape. As shown in Fig. 1, the source surface with arbitrary shape u1(x1, d1) is defined by the displacement d1 = d1(x1) at the position x1. Note that when the arbitrary shape surface is uniform-sampled, the corresponding coordinate x1 is non-uniform-sampled depending on the slope of u1(x1, d1) at the position x1. Huygens diffraction on an arbitrary shape surface is expressed by:

u(x2)=z0iλuI(x1,d1)u1(x1,d1)exp(ikr)r2dx1,
r=|x2x1|2+(z0d1)2.

 figure: Fig. 1

Fig. 1 Diffraction calculation between source surface with arbitrary and planar destination surface.

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Here, if the incident wave uI (x1, d1) is used as a planar wave that is perpendicular to the optical axis, the incident wave is expressed as uI (x1, d1) = exp(ikd1).

Applying Fresnel approximation to Eq.(3) using r0 = z0d1, we can obtain the following approximation:

rr0+x122r0x1x2r0+x222r0.

And, we also approximate r2z02 in the integral of Eq.(2). Therefore, we obtain Fresnel diffraction on an arbitray surface:

u(x2)=exp(ikz0)iλz0uI(x1,d1)u1(x1,d1)exp(ik(d1+x122r0))exp(ikx1x2r0)exp(ikx222r0)dx1

The above Eqs.(2) and (5) can be treated an arbitrary shape surface. We can readily calculate by the direct integral with regard to these equations; however, the calculation cost is O(N2) in one dimensional or O(N4) in two dimensional cases, where N is the number of sampling points, because we cannot calculate them using Fourier transform.

In order to obtain the Fourier form of Eq.(5), if z0d1, we approximate the third exponential term in the integration as follows:

exp(ikx222r0)exp(ikx222z0).

Eventually, we obtain the following equation:

u(x2)=exp(ik(z0+x222z0))iλz0uI(x1,d1)u1(x1,d1)exp(ik(d1+x122(z0d1)))exp(ikx1x2z0d1)dx1.

Because the coordinate x1 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):

u(x2)=exp(ik(z0+x222z0))iλz0NUF[uI(x1,d1)u1(x1,d1)exp(ik(d1+x122(z0d1)))],
where, NUF[·] denotes NUFT. NUFT of a function f(x1) is defined as:
F(x2)=NUF[f(x1)]=f(x1)exp(iπx1x2)dx1.

Although the form of NUFT is similar to that of uniform Fourier transform, the coordinate x1 is non-uniform-sampled and x2 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 [1518]. NUFFTs are based on the combination of an interpolation and the uniform FFT. In this paper, we used L. Greengard and J. Y. Lee’s NUFFT [18]. For more details, see [18].

3. Result

Let us examine our method using two source surfaces with arbitrary shapes in one dimension: a surface composed of four tilted planar surfaces, and quadratic curve.

We used the distance z0 = 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 μm. We used a planar wave as the incident wave that is perpendicular to the optical axis.

Figure 2 shows a source surface composed of four small planar surfaces with 128 points, which are tilted −30°, −50°, 0° and +50° to x1, respectively. The horizontal axis in (a) indicates the position on x1 in metric units. The horizontal axes in (b)–(e) indicates the position on x2 in metric units. The sampling rates on these small planar surfaces are p, however, the sampling rates on x1 are |p cos(−30°)|, |p cos(−50°)|, |p cos(0°)| and |p cos(+50°)|, respectively. The destination planar surface is not inclined to x2. 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.

 figure: Fig. 2

Fig. 2 Intensity profiles of diffraction results from a source surface composed of four small planar surfaces with 128 points, which are tilted −30°, −50°, 0° and +50° to x1, respectively. (a) source surface (b) diffraction result by Eq.(2) (c) diffraction result by Eq.(5) (d) diffraction result by Eq.(8) (e) Absolute error between (b) and (d). ( Media 1 shows the diffraction result while changing z0=1.0 m to 2.4m)

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Figure 3 shows a source surface with quadratic curve. The sampling rate on x1 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).

 figure: Fig. 3

Fig. 3 Intensity profiles of diffraction results from quadratic curve surface. (a) source surface (b) diffraction result by Eq.(2) (c) diffraction result by Eq.(5) (d) diffraction result by Eq.(8) (e) absolute error between (b) and (d). ( Media 2 shows the diffraction result while changing z0=1.0 m to 2.4m)

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4. Conclusion

We proposed Fresnel diffraction calculation from a source surface with arbitrary shape to a planar destination surface, using one NUFFT. We have had only a direct integral for calculating such diffraction thus far. Unfortunately, it is very time consuming because the calculation cost takes O(N2) in one dimensional or O(N4) in two dimensional cases. In contrast, the calculation cost of our method is O(N log N) in one dimensional or O(N2 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

This work is supported by the Ministry of Internal Affairs and Communications, Strategic Information and Communications R&D Promotion Programme (SCOPE), Japan Society for the Promotion of Science (JSPS) KAKENHI (Young Scientists (B) 23700103) 2011, and the NAKAJIMA FOUNDATION.

References and links

1. J. W. Goodman, Introduction to Fourier Optics (3rd ed.) (Robert & Company, 2005).

2. Okan K. Ersoy, Diffraction, Fourier Optics And Imaging (Wiley-Interscience, 2006).

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]  

7. C. Frere and D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989). [CrossRef]   [PubMed]  

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

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 15, 857–867 (1998). [CrossRef]  

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 20, 1755–1762 (2003). [CrossRef]  

13. G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006). [CrossRef]  

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]  

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

16. 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]  

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. 1, 288–290 (1998).

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

References

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  1. J. W. Goodman, Introduction to Fourier Optics (3rd ed.) (Robert & Company, 2005).
  2. Okan K. Ersoy, Diffraction, Fourier Optics And Imaging (Wiley-Interscience, 2006).
  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]
  7. C. Frere and D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989).
    [Crossref] [PubMed]
  8. L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express 10, 1250–1257 (2002).
    [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]
  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 15, 857–867 (1998).
    [Crossref]
  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 20, 1755–1762 (2003).
    [Crossref]
  13. G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
    [Crossref]
  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]
  15. A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) 14, 1368–1393 (1993).
    [Crossref]
  16. 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]
  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. 1, 288–290 (1998).
  18. L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46, 443–454 (2004).
    [Crossref]

2009 (1)

2008 (1)

2006 (1)

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[Crossref]

2004 (1)

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

2003 (1)

2002 (1)

1998 (3)

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, 857–867 (1998).
[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).

1993 (1)

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

1992 (1)

1989 (1)

1988 (1)

Ahrenberg, L.

An, Y.

Benzie, P.

Bianco, B.

Cai, L.

Delen, N.

Dutt, A.

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

Ersoy, Okan K.

Okan K. Ersoy, Diffraction, Fourier Optics And Imaging (Wiley-Interscience, 2006).

Esmer, G. B.

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[Crossref]

Frere, C.

Frére, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (3rd ed.) (Robert & Company, 2005).

Greengard, L.

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

Hooker, B.

Lee, J. Y.

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

Leseberg, D.

Liu, Q. H.

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).

Magnor, M.

Matsushima, K.

Nguyen, N.

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).

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]

Onural, L.

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[Crossref]

Paganin, D. M.

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

Rokhlin, V.

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

Sakamoto, Y.

Sakata, H.

Schimmel, H.

Tang, X. Y.

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).

Tommasi, T.

Watson, J.

Williams, E. G.

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

Wyrowski, F.

Yu, L.

Appl. Opt. (4)

IEEE Microw. Guid. Wave Lett. (1)

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]

IEEE Trans. Geosci. Remote Sens. (1)

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).

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[Crossref]

SIAM J. Sci. Comput. (USA) (1)

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

SIAM Rev. (1)

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

Other (5)

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]

Supplementary Material (2)

» Media 1: AVI (740 KB)     
» Media 2: AVI (795 KB)     

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Figures (3)

Fig. 1
Fig. 1 Diffraction calculation between source surface with arbitrary and planar destination surface.
Fig. 2
Fig. 2 Intensity profiles of diffraction results from a source surface composed of four small planar surfaces with 128 points, which are tilted −30°, −50°, 0° and +50° to x1, respectively. (a) source surface (b) diffraction result by Eq.(2) (c) diffraction result by Eq.(5) (d) diffraction result by Eq.(8) (e) Absolute error between (b) and (d). ( Media 1 shows the diffraction result while changing z0=1.0 m to 2.4m)
Fig. 3
Fig. 3 Intensity profiles of diffraction results from quadratic curve surface. (a) source surface (b) diffraction result by Eq.(2) (c) diffraction result by Eq.(5) (d) diffraction result by Eq.(8) (e) absolute error between (b) and (d). ( Media 2 shows the diffraction result while changing z0=1.0 m to 2.4m)

Equations (9)

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u 2 ( x 2 ) = z 0 i λ u I ( x 1 ) u 1 ( x 1 ) exp ( i k r ) r 2 d x 1 ,
u ( x 2 ) = z 0 i λ u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k r ) r 2 d x 1 ,
r = | x 2 x 1 | 2 + ( z 0 d 1 ) 2 .
r r 0 + x 1 2 2 r 0 x 1 x 2 r 0 + x 2 2 2 r 0 .
u ( x 2 ) = exp ( i k z 0 ) i λ z 0 u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k ( d 1 + x 1 2 2 r 0 ) ) exp ( i k x 1 x 2 r 0 ) exp ( i k x 2 2 2 r 0 ) d x 1
exp ( i k x 2 2 2 r 0 ) exp ( i k x 2 2 2 z 0 ) .
u ( x 2 ) = exp ( i k ( z 0 + x 2 2 2 z 0 ) ) i λ z 0 u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k ( d 1 + x 1 2 2 ( z 0 d 1 ) ) ) exp ( i k x 1 x 2 z 0 d 1 ) d x 1 .
u ( x 2 ) = exp ( i k ( z 0 + x 2 2 2 z 0 ) ) i λ z 0 NUF [ u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k ( d 1 + x 1 2 2 ( z 0 d 1 ) ) ) ] ,
F ( x 2 ) = NUF [ f ( x 1 ) ] = f ( x 1 ) exp ( i π x 1 x 2 ) d x 1 .

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