## Abstract

The optical path difference (OPD) and amplitude of a sinusoidal wavelength scanning
(SWS) are controlled with a double feedback control system in an interferometer, so that a ruler marking every wavelength and a ruler with scales smaller than a wavelength are generated. These two rulers enable us to measure an OPD longer than a wavelength. A liquid-crystal Fabry–Perot interferometer (LC-FPI) is adopted as a wavelength-scanning device, and double sinusoidal phase modulation is incorporated in the SWS interferometer. Because of a high resolution of the LC-FPI, the upper limit of the measurement range can be extended to
$280\text{\hspace{0.17em}}\mu \text{m}$ by the use of the phase lock where the amplitude of the SWS is doubled in the feedback control. The ruler marking every wavelength is generated between
$80\text{\hspace{0.17em}}\mu \text{m}$ and
$280\text{\hspace{0.17em}}\mu \text{m}$, and distances are measured with a high accuracy of the order of a nanometer in real time.

© 2007 Optical Society of America

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### Equations (16)

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(1)
$$\lambda \left(t\right)={\lambda}_{0}+b\text{\hspace{0.17em} cos}\left({\omega}_{b}t\right).$$
(2)
$${S}_{D}\left(t\right)=A+B\text{\hspace{0.17em} cos}\left[{Z}_{c}\text{\hspace{0.17em} cos}\left({\omega}_{c}t\right)+{Z}_{b}\text{\hspace{0.17em} cos}\left({\omega}_{b}t\right)+\alpha \right],$$
(3)
$${Z}_{c}=4\pi a/{\lambda}_{0},$$
(4)
$${Z}_{b}=\left(2\pi b/{{\lambda}_{0}}^{2}\right)L,$$
(5)
$$\alpha =-\left(2\pi /{\lambda}_{0}\right)L.$$
(6)
$${S}_{D}\left(t\right)=A+B\text{\hspace{0.17em} cos \hspace{0.17em}}\varphi \left(t\right)\left[{J}_{0}\left({Z}_{c}\right)-2{J}_{2}\left({Z}_{c}\right)\text{cos \hspace{0.17em}}2{\omega}_{c}t+\cdots \right]-B\text{\hspace{0.17em} sin \hspace{0.17em}}\varphi \left(t\right)\left[2{J}_{1}\left({Z}_{c}\right)\text{cos \hspace{0.17em}}{\omega}_{c}t+2{J}_{3}\left({Z}_{c}\right)\text{cos \hspace{0.17em}}3{\omega}_{c}t+\cdots \right],$$
(7)
$$S\left(t\right)=C\text{\hspace{0.17em} sin}\left[{Z}_{b}\text{\hspace{0.17em} cos}\left({\omega}_{b}t\right)+\alpha \right],$$
(8)
$${A}_{1}=C\text{\hspace{0.17em} sin \hspace{0.17em}}\alpha ,$$
(9)
$${L}_{z}=L-{L}_{\alpha}=m{\lambda}_{0}.$$
(10)
$$S\left(t\right)=C\text{\hspace{0.17em} sin}\left[{Z}_{b}\text{\hspace{0.17em} cos}\left({\omega}_{b}t\right)\right],$$
(11)
$${Z}_{b}=\left(2\pi b/{{\lambda}_{0}}^{2}\right){L}_{z}.$$
(12)
$${A}_{2}={S}_{-1}-{S}_{1}=-2B\text{\hspace{0.17em} sin \hspace{0.17em}}{Z}_{b}$$
(13)
$$b=p{{\lambda}_{0}}^{2}/2{L}_{z}=p{\lambda}_{0}/2m\mathrm{.}$$
(14)
$${m}_{c}={L}_{z}/{\lambda}_{0}.$$
(15)
$$L=m{\lambda}_{0}+{L}_{\alpha}.$$
(16)
$$\Delta b=-\left(p{{\lambda}_{0}}^{3}/2{{L}_{z}}^{2}\right).$$