## Abstract

We propose three-dimensional (3D) profilometry based on a Fourier transform in which a two- dimensional (2D) Dammann grating and a cylindrical lens are used to generate structured light. The Dammann grating splits most of the illumination power into a 2D diffractive spot matrix. The cylindrical lens transforms these 2D diffractive spots into one-dimensional fringe lines that are projected on an object. The produced projection fringes have the advantages of high brightness and high contrast and compression ratios. The experiments have verified the proposed 3D profilometry. The 3D profilometry using Dammann grating should be of high interest for practical applications.

© 2009 Optical Society of America

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### Equations (15)

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(1)
$$g(x,y)=r(x,y)\sum _{n=-\infty}^{\infty}{A}_{n}\text{\hspace{0.17em}}\mathrm{exp}\{i[2\pi n{f}_{0}x+n\phi (x,y)]\}\mathrm{.}$$
(2)
$${g}_{0}(x,y)={r}_{0}(x,y)\sum _{n=-\infty}^{\infty}{A}_{n}\text{\hspace{0.17em}}\mathrm{exp}\{i[2\pi n{f}_{0}x+n{\phi}_{0}(x,y)]\},$$
(3)
$$\stackrel{\wedge}{g}(x,y)={A}_{1}r(x,y)\mathrm{exp}[i2\pi {f}_{0}x+\phi (x,y)],$$
(4)
$${\stackrel{\wedge}{g}}_{0}(x,y)={A}_{1}{r}_{0}(x,y)\mathrm{exp}[i2\pi {f}_{0}x+{\phi}_{0}(x,y)]\mathrm{.}$$
(5)
$$\mathrm{\Delta}\phi (x,y)=\mathrm{Im}\{\mathrm{log}[\stackrel{\wedge}{g}(x,y){{\stackrel{\wedge}{g}}_{0}}^{*}(x,y)]\}\mathrm{.}$$
(6)
$$h(x,y)=\frac{{L}_{0}\mathrm{\Delta}\phi (x,y)}{\mathrm{\Delta}\phi (x,y)-2\pi {f}_{0}{d}_{0}}\mathrm{.}$$
(7)
$$h(x,y)\approx \frac{{L}_{0}\mathrm{\Delta}\phi (x,y)}{-2\pi {f}_{0}{d}_{0}}\mathrm{.}$$
(8)
$${p}_{k}(y)=\text{rect}\left(\frac{y-({y}_{k+1}+{y}_{k})/2}{{y}_{k+1}-{y}_{k}}\right),$$
(9)
$$\mathfrak{I}\{{p}_{k}(y)\}=\frac{1}{2n\pi}[(\mathrm{sin}{\alpha}_{k+1}-\mathrm{sin}{\alpha}_{k})+i(\mathrm{cos}{\alpha}_{k+1}-\mathrm{cos}{\alpha}_{k})]\mathrm{.}$$
(10)
$${I}_{n}=\left(\frac{1}{2n\pi}\right)[({P}_{n}{)}_{R}^{2}+({P}_{n}{)}_{I}^{2}],$$
(11)
$$({P}_{n}{)}_{R}=\sum _{k=0}^{K}(-1{)}^{k}(\mathrm{sin}{\alpha}_{k+1}-\mathrm{sin}{\alpha}_{k})=2\sum _{k=1}^{K}(-1{)}^{k+1}\mathrm{sin}{\alpha}_{k}-\mathrm{sin}2n\pi ,$$
(12)
$$({P}_{n}{)}_{\mathrm{I}}=2\sum _{k=1}^{K}(-1{)}^{k+1}(\mathrm{cos}{\alpha}_{k+1}-\mathrm{cos}{\alpha}_{k})=2\sum _{k=1}^{K}(-1{)}^{k+1}\mathrm{cos}{\alpha}_{k}-\mathrm{cos}2n\pi -1.$$
(13)
$${I}_{0}=[1+2\sum _{k=1}^{K}(-1{)}^{k}{y}_{k}{]}^{2},$$
(14)
$${I}_{n}=\left(\frac{1}{n\pi}{)}^{2}\right\{[\sum _{k=1}^{K}(-1{)}^{k}\text{\hspace{0.17em}}\mathrm{sin}{\alpha}_{k}{]}^{2}+[1+\sum _{k=1}^{K}(-1{)}^{k}\text{\hspace{0.17em}}\mathrm{cos}{\alpha}_{k}{]}^{2}\}\mathrm{.}$$
(15)
$${s}_{n}\ge ({z}_{0}+{z}_{1})*\frac{\lambda}{d},$$