## Abstract

A checkerboard phase plate is proposed to be used together with computer generated holograms to eliminate the zero order by working as a convolution function that shifts the zero order away from the center of a reconstructed pattern. By performing a preshift in the desired hologram pattern, it is possible to obtain a reconstructed pattern that is free of zero order. Simulation results have shown that the technique is tolerant of fabrication errors in the hologram. The technique is also shown to effectively reduce the zero order intensity by two orders in the presence of phase depth errors in the checkerboard. Experimental results using a spatial light modulator support the results shown in the simulation.

© 2008 Optical Society of America

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### Equations (9)

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(1)
$${U}_{2}\left(x,y\right)=\frac{{e}^{jkz}{e}^{i\left(k/2z\right)\left({x}^{2}+{y}^{2}\right)}}{j\lambda z}{\displaystyle \int {\int}_{-\infty}^{\infty}{\displaystyle {U}_{1}\left(\xi ,\eta \right)\times \mathrm{exp}\left[-j\text{\hspace{0.17em}}\frac{2\pi}{\lambda z}\left(x\xi +y\eta \right)\right]\mathrm{d}\xi \mathrm{d}\eta}\text{,}}$$
(2)
$$H\left(\xi ,\eta \right)=\left[\mathrm{comb}\left(\frac{\xi}{\Delta},\frac{\eta}{\Delta}\right)p\left(\xi ,\eta \right)\right]\otimes \left[\mathrm{rect}\left(\frac{\xi}{\Delta},\frac{\eta}{\Delta}\right)\right].$$
(3)
$${U}_{2}\left(x,y\right)=\left[\mathrm{comb}\left(x\Delta ,y\Delta \right)\otimes P\left(x,y\right)\right]\left[\mathrm{sinc}\left(x\Delta ,y\Delta \right)\right].$$
(4)
$$k\left(\xi ,\eta \right)=c\left(\xi ,\eta \right)\otimes \mathrm{comb}\left(\frac{\xi}{2\Gamma},\frac{\eta}{2\Gamma}\right).$$
(5)
$$c\left(\xi ,\eta \right)=\mathrm{rect}\left(\frac{\xi}{\Gamma},\frac{\eta}{\Gamma}\right)\otimes \delta \left(\xi -\frac{\Gamma}{2},\eta +\frac{\Gamma}{2}\right)\mathrm{exp}\left(j\pi \right)+\mathrm{rect}\left(\frac{\xi}{\Gamma},\frac{\eta}{\Gamma}\right)\otimes \delta \left(\xi +\frac{\Gamma}{2},\eta +\frac{\Gamma}{2}\right)+\mathrm{rect}\left(\frac{\xi}{\Gamma},\frac{\eta}{\Gamma}\right)\otimes \delta \left(\xi -\frac{\Gamma}{2},\eta -\frac{\eta}{2}\right)+\mathrm{rect}\left(\frac{\xi}{\Gamma},\frac{\eta}{\Gamma}\right)\otimes \delta \left(\xi +\frac{\Gamma}{2},\eta -\frac{\Gamma}{2}\right)\mathrm{exp}\left(j\pi \right).$$
(6)
$$C\left(x,y\right)=\mathrm{sin}\left(\pi \Gamma x\right)\mathrm{sin}\left(\pi \Gamma y\right)\mathrm{sinc}\left(x\Gamma ,y\Gamma \right)\text{,}$$
(7)
$$K\left(x,y\right)=\mathrm{sin}\left(\pi \Gamma x\right)\mathrm{sin}\left(\pi \Gamma y\right)\mathrm{sinc}\left(x\Gamma ,y\Gamma \right)\times \mathrm{comb}\left(x2\Gamma ,y2\Gamma \right)\text{,}$$
(8)
$${U}_{3}\left(\xi ,\eta \right)={U}_{3}\left(\xi ,\eta \right)k\left(\xi ,\eta \right)=\left\{\left[\mathrm{comb}\left(\frac{\xi}{\Delta},\frac{\eta}{\Delta}\right)k\left(\xi ,\eta \right)\right]\otimes \mathrm{rect}\left(\frac{\xi}{\Delta},\frac{\eta}{\Delta}\right)\right\}\times k\left(\xi ,\eta \right).$$
(9)
$${U}_{4}\left(\xi ,\eta \right)=\left\{\left[\mathrm{comb}\left(x\Delta ,y\Delta \right)\otimes P\left(x,y\right)\right]\left[\mathrm{sinc}\left(x\Delta ,y\Delta \right)\right]\right\}\otimes K\left(x,y\right)=\left\{\left[\mathrm{comb}\left(x\Delta ,y\Delta \right)\otimes P\left(x,y\right)\right]\left[\mathrm{sinc}\left(x\Delta ,y\Delta \right)\right]\right\}\otimes \left[\mathrm{sin}\left(\pi \Gamma x\right)\mathrm{sin}\left(\pi \Gamma y\right)\mathrm{sinc}\left(x\Gamma ,y\Gamma \right)\times \mathrm{comb}\left(x2\Gamma ,y2\Gamma \right)\right].$$