## Abstract

The fabrication of computer-generated holograms (CGH) by e-beam or laser-writing
machine specifically requires using polygon segments to approximate the
continuously smooth fringe pattern of an ideal CGH. Wavefront phase errors
introduced in this process depend on the size of the polygon segments and the
shape of the fringes. In this paper, we propose a method for estimating the
wavefront error and its spatial frequency, allowing optimization of the polygon
sizes for required measurement accuracy. This method is validated with computer
simulation and direct measurements from an interferometer.

© 2013 Optical Society of America

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### Equations (18)

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(1)
$$\mathrm{\Delta}\varphi (x,y)=-m\lambda \frac{\sigma (x,y)}{d(x,y)},$$
(2)
$$\epsilon =\frac{\sigma}{d}.$$
(3)
$$L=4\sqrt{R\xb7\sigma}=4\sqrt{\epsilon d/c},$$
(4)
$$\mathrm{\Upsilon}=\frac{1}{L}=\frac{1}{4}\sqrt{\frac{c}{\epsilon d}}.$$
(5)
$$\nabla f=({f}_{x},{f}_{y}),$$
(6)
$$S(x,y)=|\nabla f|=\sqrt{{f}_{x}^{2}+{f}_{y}^{2}}.$$
(7)
$$\theta (x,y)=\mathrm{arctan}\left(\frac{{f}_{y}}{{f}_{x}}\right).$$
(8)
$$\nabla g=({f}_{y},-{f}_{x}),$$
(9)
$${\kappa}_{c}(x,y)=-\nabla \theta (x,y)\xb7(\nabla g(x,y)/S(x,y)),$$
(10)
$${\kappa}_{c}=-({f}_{y}^{2}{f}_{xx}-2{f}_{x}{f}_{y}{f}_{xy}+{f}_{x}^{2}{f}_{yy})/{S}^{3}.$$
(11)
$$\mathrm{\Upsilon}(x,y)=\frac{1}{4}\sqrt{\frac{c}{\epsilon d}}=\frac{1}{4}\sqrt{\frac{|{\kappa}_{c}|}{\epsilon \lambda /S}}\phantom{\rule{0ex}{0ex}}=\frac{1}{4S}\sqrt{|{f}_{y}^{2}{f}_{xx}-2{f}_{x}{f}_{y}{f}_{xy}+{f}_{x}^{2}{f}_{yy}|/\epsilon \lambda}.$$
(12)
$${f}_{\mathrm{FZP}}(x,y)=\frac{{x}^{2}+{y}^{2}}{2F},$$
(13)
$${\mathrm{\Upsilon}}_{\mathrm{FZP}}=\frac{1}{4\sqrt{\epsilon \lambda F}}.$$
(14)
$$\mathrm{PSD}(u,v)=\frac{1}{A}{|F\{h(x,y)\}|}^{2},$$
(15)
$${\mathrm{PSD}}_{1D}(\rho )={\int}_{0}^{2\pi}\mathrm{PSD}(\rho ,\theta )\mathrm{d}\theta ,$$
(16)
$${\sigma}^{2}=\iint \mathrm{PSD}(u,v)\mathrm{d}u\mathrm{d}v.$$
(17)
$${\sigma}_{\mathrm{RMS}}(x,y)=\frac{\epsilon \lambda}{\sqrt{2}}\sqrt{1-{\left[\frac{S(x,y)/\lambda -{f}_{\text{cutoff}}}{S(x,y)/\lambda -\gamma (x,y)}\right]}^{2}},$$
(18)
$${\sigma}_{\mathrm{RMS}}=\frac{1}{MN}\sum _{x=1}^{M}\sum _{y=1}^{N}{\sigma}_{\mathrm{RMS}}(x,y).$$