## Abstract

In computer-generated Fresnel holography, direct sampling (DS) and simple shading (SS) are two common ways to generate sampled Fresnel zone plates (FZPs) on the hologram plane. Nevertheless, either aliasing or vignetting, or both, will occur in the reconstructed image when the DS method or the SS method is applied. To avoid vignetting together with aliasing in the two sampling methods, either the object size or the object distance must be restricted in generating the holograms. In this paper we propose a mask-shifting (MS) method to generate the sampled FZPs. The main concept of the MS method is that the center of the FZP can be shifted relative to the center of the mask against the FZP when the FZP is at the margin of the hologram. The shifting of the mask will result in only a phase shift and will not change the intensity distribution of the reconstructed point. Thus, by using the MS method, aliasing and vignetting are simultaneously alleviated in any combination of object size and object distance.

© 2014 Optical Society of America

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### Equations (12)

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(1)
$$\mathrm{FZP}={C}_{i}+{A}_{i}\text{\hspace{0.17em}}\mathrm{cos}\left\{\frac{\pi [{(x-{x}_{i})}^{2}+{(y-{y}_{i})}^{2}]}{\lambda {z}_{i}}\right\},$$
(2)
$$2\frac{{|x-{x}_{i}|}_{\mathrm{max}}}{\lambda {z}_{i}}\le \frac{1}{{\mathrm{\Delta}}_{x}},$$
(3)
$${z}_{c}=\frac{({D}_{x}+2{|{x}_{i}|}_{\mathrm{max}}){\mathrm{\Delta}}_{x}}{\lambda},$$
(4)
$${W}_{\mathrm{max}}=\frac{\lambda {z}_{i}}{{\mathrm{\Delta}}_{x}}$$
(5)
$${W}_{ss}={W}_{\mathrm{max}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{i}<\frac{{D}_{x}{\mathrm{\Delta}}_{x}}{\lambda},={D}_{x}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\text{otherwise}.$$
(6)
$$I({x}^{\prime},{y}^{\prime})={({W}_{x}\times {W}_{y})}^{2}\text{\hspace{0.17em}}{\mathrm{sinc}}^{2}\left(\frac{{W}_{x}{x}^{\prime}}{\lambda {z}_{i}}\right){\mathrm{sinc}}^{2}\left(\frac{{W}_{y}{y}^{\prime}}{\lambda {z}_{i}}\right),$$
(7)
$$(\mathrm{a})\text{\hspace{0.17em}}\text{\hspace{0.17em}}{L}_{x}+{W}_{\mathrm{max}}\le {D}_{x},\phantom{\rule{0ex}{0ex}}(\mathrm{b})\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{\mathrm{max}}-{L}_{x}>{D}_{x},\phantom{\rule{0ex}{0ex}}(\mathrm{c})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{otherwise},$$
(8)
$${W}_{x}-(\frac{{D}_{x}}{2}-\frac{{L}_{x}}{2})\le \frac{{W}_{\text{max}}}{2}$$
(9)
$$R=\frac{\lambda {z}_{i}}{{W}_{x}},$$
(10)
$${W}_{\mathrm{ms}}={W}_{\text{max}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{L}_{x}\le {D}_{x}-{W}_{\text{max}},\phantom{\rule{0ex}{0ex}}={D}_{x}\phantom{\rule[-0.0ex]{2.24em}{0.0ex}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{L}_{x}<{W}_{\mathrm{max}}-{D}_{x},\phantom{\rule{0ex}{0ex}}=\frac{{W}_{\text{max}}+{D}_{x}-{L}_{x}}{2}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{otherwise}.$$
(11)
$${H}_{\text{on}}=C+\sum _{i}{W}_{\mathrm{ms}(i)}(x,y)\times {A}_{i}\text{\hspace{0.17em}}\mathrm{cos}\left\{\frac{\pi [{(x-{x}_{i})}^{2}+{(y-{y}_{i})}^{2}]}{\lambda {z}_{i}}\right\},$$
(12)
$${H}_{\text{off}}=\sum _{i}{W}_{i}(x,y)\times {A}_{i}\text{\hspace{0.17em}}\mathrm{cos}\{\frac{\pi [{(x-{x}_{i})}^{2}+{(y-{y}_{i})}^{2}]}{\lambda {z}_{i}}-\frac{2\pi (\mathrm{sin}\text{\hspace{0.17em}}\theta )x}{\lambda}\},$$