## Abstract

A computer-generated hologram consisting of *N* × *N*
resolution cells produces a reconstructed image consisting of
*N* × *N* sampling points. Since the width of the
primary peaks in the point-spread function is twice the pitch of the
sampling points, the intensity at intermediate points between the
sampling points depends on the interference between the sampling
points. Carefully controlling the complex amplitudes of the
sampling points makes it possible to control the intensity not only at
the sampling points but also at the intermediate points; the intensity
of the reconstructed image can be controlled at 2*N* ×
2*N* points. Preliminary experiments demonstrating the
generation of high-density intensity patterns were
performed.

© 1999 Optical Society of America

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### Equations (6)

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(1)
$$u\left(x,y\right)=\sum _{k=-N/2}^{N/2-1}\sum _{l=-N/2}^{N/2-1}{g}_{\mathit{kl}}\mathrm{rect}\left(x/a-k,y/a-l\right).$$
(2)
$${g}_{\mathit{kl}}=\sum _{m=-N/2}^{N/2-1}\sum _{n=-N/2}^{N/2-1}{G}_{\mathit{mn}}exp\left[i2\mathrm{\pi}\left(\mathit{km}+\mathrm{ln}\right)/N\right].$$
(3)
$$U\left({\mathit{\nu}}_{x},{\mathit{\nu}}_{y}\right)={a}^{2}\mathrm{sinc}\left(a{\mathit{\nu}}_{x},a{\mathit{\nu}}_{y}\right)\sum _{k=-N/2}^{N/2-1}\sum _{l=-N/2}^{N/2-1}{g}_{\mathit{kl}}\times exp\left[-i2\mathrm{\pi}a\left(k{\mathit{\nu}}_{x}+l{\mathit{\nu}}_{y}\right)\right],$$
(4)
$$U\left({\mathit{\nu}}_{x},{\mathit{\nu}}_{y}\right)={a}^{2}\mathrm{sinc}\left(a{\mathit{\nu}}_{x},a{\mathit{\nu}}_{y}\right)\sum _{m=-N/2}^{N/2-1}\sum _{n=-N/2}^{N/2-1}{G}_{\mathit{mn}}\times \sum _{k=-N/2}^{N/2-1}\sum _{l=-N/2}^{N/2-1}exp\left\{-i2\mathrm{\pi}a\left[\left({\mathit{\nu}}_{x}-m/\mathit{Na}\right)k+\left({\mathit{\nu}}_{y}-n/\mathit{Na}\right)l\right]\right\}={a}^{2}\mathrm{sinc}\left(a{\mathit{\nu}}_{x},a{\mathit{\nu}}_{y}\right)\sum _{m=-N/2}^{N/2-1}\sum _{n=-N/2}^{N/2-1}{G}_{\mathit{mn}}\times F\left({\mathit{\nu}}_{x}-m/\mathit{Na},{\mathit{\nu}}_{y}-n/\mathit{Na}\right),$$
(5)
$$F\left({\mathit{\nu}}_{x},{\mathit{\nu}}_{y}\right)=exp\left[i\mathrm{\pi}a\left({\mathit{\nu}}_{x}+{\mathit{\nu}}_{y}\right)\right]sin\left(\mathrm{\pi}\mathit{Na}{\mathit{\nu}}_{x}\right)sin\left(\mathrm{\pi}\mathit{Na}{\mathit{\nu}}_{y}\right)/sin\left(\mathrm{\pi}a{\mathit{\nu}}_{x}\right)sin\left(\mathrm{\pi}a{\mathit{\nu}}_{y}\right).$$
(6)
$$e\left(t\right)=\sum _{\left(m,n\right)\in S}|{U}_{\mathit{mn}}\left(t\right)-{\mathit{cO}}_{\mathit{mn}}{|}^{2},$$