Arbitrary shape surface Fresnel diffraction.

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.


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 [6,7,8,9] and non-parallel angular spectrum methods [10,11,12,13].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.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(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.
In Section 2, we describe the arbitrary shape surface Fresnel diffraction.In Section 3, we present the numerical results.Section 4 concludes this work.

Arbitrary shape surface Fresnel diffraction
Let us begin with Huygens diffraction.Huygens diffraction [1] on a planar surface is expressed as: where, 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 r = |x 2 − x 1 | 2 + z 2 0 , where 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.
We expand the Huygens diffraction to a source surface with arbitrary shape.As shown in Fig. 1, the source surface with arbitrary shape 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: Here, if the incident wave 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 ).Applying Fresnel approximation to Eq.(3) using r 0 = z 0 − d 1 , we can obtain the following approximation: And, we also approximate r 2 ≈ z 2 0 in the integral of Eq.( 2).Therefore, we obtain Fresnel diffraction on an arbitray surface: 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(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.
In order to obtain the Fourier form of Eq.( 5), we approximate the third exponential term in the integration as follows: Eventually, we obtain the following equation: )) Because the coordinate 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 trans-form (NUFT): where, NUF • denotes NUFT.NUFT of a function f (x 1 ) is defined as: Although the form of NUFT is similar to that of uniform Fourier transform, the coordinate 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 [14,15,16,17].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 [17].For more details, see [17].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.

Result
We used the distance 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 = 10µ m.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 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  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)  Figure 3 shows a source surface with quadratic curve.The sampling rate on 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).

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(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.

Fig. 1 .
Fig. 1.Diffraction calculation between source surface with arbitrary and planar destination surface.
. The absolute error falls into within approximately 0.025.