## Abstract

Interference fringes with different periods are projected on an object surface. There is a constant phase point where the phase of the fringe is kept at a constant value while the period is scanning. Multiple optical fields with different periods on the object surface are made from detected phases of the fringes. The multiple optical fields are backpropagated to the constant phase point of the phase where all of the phases of the multiple backpropagated fields become the same value and the amplitude of the sum of the multiple backpropagated fields becomes maximum. The distance of the backpropagation provides the position of the object surface.
Some experiments show that this method can measure an object surface with discontinuities of several millimeters with high accuracy of several micrometers.

© 2007 Optical Society of America

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### Equations (18)

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(1)
$${\alpha}_{m}\left({z}_{p},{x}_{p}\right)={k}_{m}{x}_{p}+\frac{\pi}{2},$$
(2)
$${\alpha}_{m}\left(x,z\right)={k}_{m}\left(x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}2\beta +z\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}2\beta \right)+\frac{\pi}{2}.$$
(3)
$${I}_{m}\left({x}_{0},t\right)={A}_{m}+{B}_{m}\text{\hspace{0.17em}}\mathrm{cos}\left[a\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\omega}_{c}t+{\alpha}_{m}\left({x}_{0},{z}_{0}\right)\right]\text{,}$$
(4)
$$m=0\text{, \hspace{0.17em} \u2026 \hspace{0.17em} , \hspace{0.17em}}M-1.$$
(5)
$${\alpha}_{mz}\left({z}_{0}\right)={k}_{m}{z}_{0}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}2\beta .$$
(6)
$${D}_{m}\left({x}_{0},{z}_{0}\right)={B}_{m}\text{\hspace{0.17em}}\mathrm{exp}\left[j{\alpha}_{mz}\left({z}_{0}\right)\right].$$
(7)
$${U}_{m}\left({x}_{0},z\right)={D}_{m}\text{\hspace{0.17em}}\mathrm{exp}\left[-j{\alpha}_{mz}\left(z\right)\right],\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}m=0\text{, \hspace{0.17em} \u2026 \hspace{0.17em} , \hspace{0.17em}}M-1.$$
(8)
$${U}_{R}\left(z\right)={\displaystyle \sum _{m=0}^{M-1}{U}_{m}\left(z\right)={\displaystyle \sum _{m=0}^{M-1}{A}_{m}\text{\hspace{0.17em}}\mathrm{exp}[-jS{k}_{m}\left(z-{z}_{0}\right)]}},$$
(9)
$${U}_{R}\left(z\right)=\mathrm{exp}\left\{j[-{k}_{0}+\frac{\left(M-1\right)}{2}\text{\hspace{0.17em}}\Delta k]S{z}_{D}\right\}\frac{\mathrm{sin}\left(\frac{M}{2}\text{\hspace{0.17em}}\Delta k{z}_{D}S\right)}{\mathrm{sin}\left(\frac{1}{2}\text{\hspace{0.17em}}\Delta k{z}_{D}S\right)}={A}_{R}\text{\hspace{0.17em}}\mathrm{exp}\left(j{\varphi}_{R}\right).$$
(10)
$${A}_{R}=\frac{\mathrm{sin}\left(\frac{{B}_{k}}{2}\text{\hspace{0.17em}}{z}_{D}S\right)}{\mathrm{sin}\left(\frac{\Delta k}{2}\text{\hspace{0.17em}}{z}_{D}S\right)},$$
(11)
$${\varphi}_{R}=\left[-{k}_{0}+\frac{\left(M-1\right)}{2}\text{\hspace{0.17em}}\Delta k\right]S{z}_{D}=-{k}_{C}S{z}_{D},$$
(12)
$${x}_{r}={x}_{0}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta +{z}_{0}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta .$$
(13)
$$h\left({x}_{r},{y}_{r}\right)=\left({z}_{0}-{x}_{0}\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\beta \right)\mathrm{cos}\text{\hspace{0.17em}}\beta .$$
(14)
$$\Delta k=\frac{2\pi}{M-1}\left(\frac{1}{{P}_{0}}-\frac{1}{{P}_{M-1}}\right)=\frac{\left({k}_{M-1}-{k}_{0}\right)}{M-1}.$$
(15)
$${z}_{\mathrm{max}}=\frac{2\pi}{\Delta kS}.$$
(16)
$${P}_{C}=\frac{2\pi}{{k}_{C}S}=\frac{2{P}_{0}{P}_{M-1}}{\left({P}_{0}+{P}_{M-1}\right)S}.$$
(17)
$$q=\frac{\partial {\alpha}_{mz}}{\partial {k}_{m}}={z}_{0}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}2\beta .$$
(18)
$${I}_{m}\left(t,{x}_{f}\right)={A}_{m}+{B}_{m}\text{\hspace{0.17em}}\mathrm{cos}\left(a\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\omega}_{c}t+{k}_{m}{x}_{f}\right).$$