## 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 Optical Society of America

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### Equations (18)

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(1)
$$\begin{array}{lll}\left[\begin{array}{c}{x}_{st}\\ {y}_{st}\\ {z}_{st}\end{array}\right]\hfill & =\hfill & {\mathit{P}}_{\mathbf{0}}+s\overrightarrow{\Delta s}+t\overrightarrow{\Delta t}\hfill \\ \hfill & =\hfill & \left[\begin{array}{c}{x}_{0}\\ {y}_{0}\\ {z}_{0}\end{array}\right]+s\left[\begin{array}{c}\Delta {s}_{x}\\ \Delta {s}_{y}\\ \Delta {s}_{z}\end{array}\right]+t\left[\begin{array}{c}\Delta {t}_{x}\\ \Delta {t}_{y}\\ \Delta {t}_{z}\end{array}\right]\hfill \end{array}$$
(2)
$$\begin{array}{lll}\left[\begin{array}{c}{x}_{uv}\\ {y}_{uv}\\ {z}_{uv}\end{array}\right]\hfill & =\hfill & {\mathit{P}}_{\mathbf{1}}+u\overrightarrow{\Delta u}+v\overrightarrow{\Delta v}\hfill \\ \hfill & =\hfill & \left[\begin{array}{c}{x}_{1}\\ {y}_{1}\\ {z}_{1}\end{array}\right]+u\left[\begin{array}{c}\Delta u\\ 0\\ 0\end{array}\right]+v\left[\begin{array}{c}0\\ \Delta v\\ 0\end{array}\right]\hfill \end{array}$$
(3)
$${U}_{1}({\mathit{P}}_{uv})=\iint \frac{{U}_{0}({\mathit{P}}_{st})}{j\lambda \left|{z}_{uv}-{z}_{st}\right|}\text{exp}\left\{jk\sqrt{{\left({x}_{uv}-{x}_{st}\right)}^{2}+{\left({y}_{uv}-{y}_{st}\right)}^{2}+{\left({z}_{uv}-{z}_{st}\right)}^{2}}\right\}dsdt$$
(4)
$${\left({z}_{uv}-{z}_{st}\right)}^{3}\gg \frac{\pi}{4\lambda}{\left[{\left({x}_{uv}-{x}_{st}\right)}^{2}+{\left({y}_{uv}-{y}_{st}\right)}^{2}\right]}^{2}$$
(5)
$${z}_{01}={z}_{1}-{z}_{0}$$
(6)
$$\Delta z=-s\Delta {s}_{z}-t\Delta {t}_{z}$$
(7)
$${x}_{01}={x}_{1}-{x}_{0}$$
(8)
$${x}_{st}^{\prime}={\left(s\Delta {s}_{x}\right)}^{2}+{\left(t\Delta {t}_{x}\right)}^{2}-2{x}_{01}s\Delta {s}_{x}-2{x}_{01}t\Delta {t}_{x}+2\Delta {s}_{x}\Delta {t}_{x}st$$
(9)
$${x}_{uv}^{\prime}={x}_{01}{}^{2}+{\left(u\Delta u\right)}^{2}+2{x}_{01}u\Delta u$$
(10)
$${y}_{01}={y}_{1}-{y}_{0}$$
(11)
$${y}_{st}^{\prime}={\left(s\Delta {s}_{y}\right)}^{2}+{\left(t\Delta {t}_{y}\right)}^{2}-2{y}_{01}s\Delta {s}_{y}-2{y}_{01}t\Delta {t}_{y}+2\Delta {s}_{y}\Delta {t}_{y}st$$
(12)
$${y}_{uv}^{\prime}={y}_{01}{}^{2}+{\left(v\Delta v\right)}^{2}+2{y}_{01}v\Delta v$$
(13)
$${\left(\Delta z\right)}^{2}\ll {z}_{01}{}^{2}$$
(14)
$${\left(\Delta z\right)}^{2}\ll {z}_{01}$$
(15)
$$\begin{array}{lll}{U}_{1}({\mathit{P}}_{uv})\hfill & =\hfill & \frac{1}{j\lambda}\text{exp}\left\{jk\frac{{x}_{uv}^{\prime}+{y}_{uv}^{\prime}}{2{z}_{01}}\right\}\hfill \\ \hfill & \hfill & \iint {U}_{0}({\mathit{P}}_{st})\frac{\left|{z}_{01}-\Delta z\right|}{{z}_{01}{}^{2}}\text{exp}\left\{jk({z}_{01}+\Delta z)\right\}\text{exp}\left\{jk\frac{{x}_{st}^{\prime}+{y}_{st}^{\prime}}{2{z}_{01}}\right\}\hfill \\ \hfill & \hfill & \text{exp}\left\{-jk\frac{\left(\Delta u\Delta {s}_{x}u+\Delta v\Delta {s}_{y}v\right)s+\left(\Delta u\Delta {t}_{x}u+\Delta v\Delta {t}_{y}v\right)t}{{z}_{01}}\right\}dsdt\hfill \end{array}$$
(16)
$$F(u)=\int f(s)\text{exp}\left\{-j2\pi aus\right\}ds$$
(17)
$${a}_{N}={t}_{N}/{N}^{4}$$
(18)
$${b}_{N}={t}_{N}/{N}^{2}{\mathit{log}}_{2}N$$