## Abstract

We present a method for full spatial phase and amplitude control of a laser beam using a twisted nematic LCD combined with a spatial filter. By spatial filtering we combine four neighboring pixels into one superpixel. At each superpixel we are able to independently modulate the phase and the amplitude of light. We experimentally demonstrate the independent phase and amplitude modulation using this novel technique. Our technique does not impose special requirements on the spatial light modulator and allows precise control of fields even with imperfect modulators.

© 2008 Optical Society of America

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### Equations (11)

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(1)
$${E}_{1}={E}_{1r}+i\mathrm{\Delta},$$
(2)
$${E}_{3}={E}_{3r}+i\mathrm{\Delta},$$
(3)
$${E}_{1r}-{E}_{3r}=g,$$
(4)
$${E}_{2}={E}_{2r}+i\u03f5,$$
(5)
$${E}_{4}={E}_{4r}+i\u03f5,$$
(6)
$${E}_{2r}-{E}_{3r}=h.$$
(7)
$${f}_{s}(x,y)={\sum}_{m,n=-\infty}^{\infty}\delta (y-ma/4)[\delta (x-na){E}_{1}(x,y)\phantom{\rule[-0.0ex]{0em}{0.0ex}}+\delta (x-1/4a-na){E}_{2}(x-1/4a,y)\phantom{\rule[-0.0ex]{0em}{0.0ex}}+\delta (x-1/2a-na){E}_{3}(x-1/2a,y)\phantom{\rule[-0.0ex]{0em}{0.0ex}}+\delta (x-3/4a-na){E}_{4}(x-1/4a,y)],$$
(8)
$${F}_{s}({\mathrm{\Omega}}_{x},{\omega}_{y})=\frac{1}{2\pi}{\sum}_{k,l=-\infty}^{\infty}\left\{{\tilde{E}}_{1r}\right({\mathrm{\Omega}}_{xk},{\omega}_{yl})-{\tilde{E}}_{3r}({\mathrm{\Omega}}_{xk},{\omega}_{yl}){e}^{-i\frac{a{\mathrm{\Omega}}_{x}}{2}}\phantom{\rule[-0.0ex]{0em}{0.0ex}}+i[{\tilde{E}}_{2r}({\mathrm{\Omega}}_{xk},{\omega}_{yl})-{\tilde{E}}_{4r}({\mathrm{\Omega}}_{xk},{\omega}_{yl}){e}^{-i\frac{a{\mathrm{\Omega}}_{x}}{2}}]{e}^{-i\frac{a{\mathrm{\Omega}}_{x}}{4}}\phantom{\rule[-0.0ex]{0em}{0.0ex}}+i[\tilde{\mathrm{\Delta}}({\mathrm{\Omega}}_{xk},{\omega}_{yl})+\tilde{\u03f5}({\mathrm{\Omega}}_{xk},{\omega}_{yl}){e}^{-i\frac{a{\mathrm{\Omega}}_{x}}{4}}\left]\right[1-{e}^{-i\frac{a{\mathrm{\Omega}}_{x}}{2}}\left]\right\},$$
(9)
$${f}_{r}(x,y)={\sum}_{m,n=-\infty}^{\infty}\delta (y-ma/4)[\delta (x-na){E}_{1}(x,y)\phantom{\rule[-0.0ex]{0em}{0.0ex}}-\delta (x-1/2a-na){E}_{3}(x-1/2a,y)\phantom{\rule[-0.0ex]{0em}{0.0ex}}+i\delta (x-1/4a-na){E}_{2}(x-1/4a,y)\phantom{\rule[-0.0ex]{0em}{0.0ex}}-i\delta (x-3/4a-na){E}_{4}(x-1/4a,y)]\phantom{\rule[-0.0ex]{0em}{0.0ex}}\otimes \frac{2w}{{a}^{2}}\mathrm{sinc}(\frac{xw}{a},\frac{4y}{a}).$$
(10)
$$I={A}^{2}+{B}^{2}+2AB\mathrm{cos}(\mathrm{\Delta}\theta ).$$
(11)
$$g({V}_{1})={E}_{r}({V}_{1})-{E}_{r}({V}_{L}({E}_{i}({V}_{1}))).$$