## Abstract

Computer-generated holograms of plane surfaces tilted and shifted with respect to the hologram plane are considered. The analysis is made in the spatial frequency domain by using the translation and rotation transformations of angular spectra. The frequency approach permits the use of the fast-Fourier-transform algorithm, which decreases the computation time and makes it possible to consider any position of the planes in space. Various configurations of tilted and shifted planes have been investigated, and computer-generated holograms of off-axis planes have been obtained. Computer and optical reconstructions, both of which show the feasibility of the proposed approach, have been carried out.

© 1993 Optical Society of America

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### Equations (12)

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(1)
$$\begin{array}{ccc}\alpha =\mathrm{\lambda}{f}_{x},& \beta =\mathrm{\lambda}{f}_{y},& \gamma =\sqrt{1-{\alpha}^{2}-{\beta}^{2}}\end{array}$$
(2)
$${A}_{1}({f}_{x},{f}_{y},z)={A}_{0}({f}_{x},{f}_{y})exp[2\pi jz{(1/{\mathrm{\lambda}}^{2}-{{f}_{x}}^{2}-{{f}_{y}}^{2})}^{1/2}].$$
(3)
$${(x,y,z)}^{t}=M{(\xi ,\eta ,\zeta )}^{t},$$
(4)
$${({f}_{\xi},{f}_{\eta},{f}_{\zeta})}^{t}={M}^{t}{({f}_{x},{f}_{y},{f}_{z})}^{t},$$
(5)
$${f}_{z}={(1/{\mathrm{\lambda}}^{2}-{{f}_{x}}^{2}-{{f}_{y}}^{2})}^{1/2}.$$
(6)
$${A}_{2}({f}_{\xi},{f}_{\eta},0)=\{\begin{array}{ll}0\hfill & \text{outside}\hspace{0.17em}\text{C}\hfill \\ {A}_{1}({f}_{x},{f}_{y},0){f}_{z}/{f}_{\zeta}\hfill & \text{inside}\hspace{0.17em}\text{C}\hfill \end{array},$$
(7)
$$\begin{array}{cc}{f}_{z}>0,& {f}_{\zeta}>0.\end{array}$$
(8)
$$U(x,y)=T(x,y)exp(2\pi jycos\theta /\mathrm{\lambda}),$$
(9)
$$\begin{array}{ccc}i=n/2+D{f}_{x}+1,& j=n/2+D{f}_{y}+1,& (i,j=1-n).\end{array}$$
(10)
$$\begin{array}{c}j={c}_{1}jj+{c}_{2}{({D}^{2}-i{i}^{2}-j{j}^{2})}^{1/2},\\ i=ii,\hfill \end{array}$$
(11)
$${w}^{\prime}=D\beta /\mathrm{\lambda}-n/2$$
(12)
$${w}^{\prime}=D\beta /\mathrm{\lambda}+n/2$$