## Abstract

We have derived the basic spectral relation between a 3-D object and its 2-D diffracted wavefront by interpreting the diffraction calculation in the 3-D Fourier domain. Information on the 3-D object, which is inherent in the diffracted wavefront, becomes clear by using this relation. After the derivation, a method for obtaining the Fourier spectrum that is required to synthesize a hologram with a realistic sampling number for visible light is described. Finally, to verify the validity and the practicality of the above-mentioned spectral relation, fast calculation of a series of wavefronts radially diffracted from a 3-D voxel-based object is demonstrated.

© 2012 Optical Society of America

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### Equations (9)

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(1)
$$f\left(x,y,z\right)=\iiint o\left({x}_{0},{y}_{0},{z}_{0}\right)g\left(x-{x}_{0},y-{y}_{0},z-{z}_{0}\right)\text{d}{x}_{0}\text{d}{y}_{0}\text{d}{z}_{0}$$
(2)
$$=o\left(x,y,z\right)*g\left({x}_{0},{y}_{0},{z}_{0}\right),$$
(3)
$$g\left(x,y,z\right)=\frac{\text{exp}\left(\frac{i2\pi}{\lambda}\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}\right)}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}.$$
(4)
$$G\left(u,v,w\right)=F\left[g\left(x,y,z\right)\right]=\frac{1}{4{\pi}^{2}\left({u}^{2}+{v}^{2}+{w}^{2}-1/{\lambda}^{2}\right)},$$
(5)
$$f\left(x,y,z\right)=\frac{1}{4{\pi}^{2}}\iiint \frac{O\left(u,v,w\right)}{{u}^{2}+{v}^{2}+{w}^{2}-1/{\lambda}^{2}}\text{exp}\{i2\pi \left(ux+uy+wz\right)\}\text{d}u\text{d}v\text{d}w,$$
(6)
$${f\left(x,y\right)|}_{z=R}=\frac{1}{4{\pi}^{2}}\iiint \frac{O\left(u,v,w\right)}{{u}^{2}+{v}^{2}+{w}^{2}-1/{\lambda}^{2}}\text{exp}\left\{i2\pi \left(ux+uy+wR\right)\right\}\text{d}u\text{d}v\text{d}w.$$
(7)
$$\begin{array}{ll}{f\left(x,y\right)|}_{z=R}=\hfill & \frac{i}{4\pi}\iint \frac{O\left(u,v\sqrt{1/{\lambda}^{2}-{u}^{2}-{v}^{2}}\right)}{\sqrt{1/{\lambda}^{2}-{u}^{2}-{v}^{2}}}\text{exp}\left(i2\pi R\sqrt{1/{\lambda}^{2}-{u}^{2}-{v}^{2}}\right)\hfill \\ \hfill & \times \text{exp}\left\{i2\pi \left(ux+uy\right)\right\}\text{d}u\text{d}v.\hfill \end{array}$$
(8)
$${F\left(u,v\right)|}_{z=R}=\frac{i}{4\pi}\frac{O\left(u,v\sqrt{1/{\lambda}^{2}-{u}^{2}-{v}^{2}}\right)}{\sqrt{1/{\lambda}^{2}-{u}^{2}-{v}^{2}}}\text{exp}\left(i2\pi R\sqrt{1/{\lambda}^{2}-{u}^{2}-{v}^{2}}\right)$$
(9)
$${O}^{\prime}\left(u,v,w\right)=F\left[o\left(x,y,z\right)\text{comb}\left(\frac{x}{\delta},\frac{y}{\delta},\frac{z}{\delta}\right)\right]=\frac{1}{W}O\left(u,v,w\right)*\text{comb}\left(\frac{u}{W},\frac{v}{W},\frac{w}{W}\right),$$