## Abstract

A two-wavelength interferometer is described that can be used to measure distances to rough surfaces with micrometer resolution. The measuring range can be made as large as several centimeters. Owing to the small numerical aperture of the object beam, the interferometer is particularly suitable in applications where a long working distance is required. The interferometer uses two single-frequency diode lasers to create a virtual two-wavelength laser. The equivalent wavelength of this laser can be easily tuned by changing the temperature of the laser diodes in opposite directions.

© 1988 Optical Society of America

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### Equations (12)

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(1)
$$\begin{array}{r}{P}_{j}\left(t\right)={P}_{j0}+{\gamma}_{j}^{2}{P}_{j0}+2{\gamma}_{j}{P}_{j0}\phantom{\rule{0.2em}{0ex}}\text{cos}\left[{\omega}_{j}{\tau}_{0}+{\omega}_{j}\widehat{\Delta \tau}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi {f}_{0}t\right)\right],\\ j=1,2.\end{array}$$
(2)
$${P}_{j1}\left(t\right)=-4{\gamma}_{j}{P}_{j0}\phantom{\rule{0.2em}{0ex}}\text{sin}\left({\omega}_{j}{\tau}_{0}\right){J}_{1}\left({\omega}_{j}\widehat{\Delta \tau}\right)\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi {f}_{0}t\right),$$
(3)
$$\begin{array}{r}{P}_{j2}\left(t\right)=4{\gamma}_{j}{P}_{j0}\phantom{\rule{0.2em}{0ex}}\text{cos}\left({\omega}_{j}{\tau}_{0}\right){J}_{2}\left({\omega}_{j}\widehat{\Delta \tau}\right)\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi 2{f}_{0}t\right),\\ j=1,2.\end{array}$$
(4)
$${f}_{j}=\u3008{P}_{j1}\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi {f}_{0}t\right)\u3009={A}_{j1}\phantom{\rule{0.2em}{0ex}}\text{sin}\left({\omega}_{j}{\tau}_{0}\right),$$
(5)
$$\begin{array}{r}{g}_{j}=\u3008{P}_{j2}\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi 2{f}_{0}t\right)\u3009={A}_{j2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left({\omega}_{j}{\tau}_{0}\right),\\ j=1,2.\end{array}$$
(6)
$${g}_{1}{g}_{2}+{f}_{1}{f}_{2}={A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\Delta \omega {\tau}_{0}\right),$$
(7)
$${f}_{1}{g}_{2}-{f}_{2}{g}_{1}={A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\Delta \omega {\tau}_{0}\right),$$
(8)
$$C\left(\Delta \omega {\tau}_{0}\right)=\frac{{A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\Delta \omega {\tau}_{0}\right)}{\left|{A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\Delta \omega {\tau}_{0}\right)\right|+\left|{A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\Delta \omega {\tau}_{0}\right)\right|},$$
(9)
$$S\left(\Delta \omega {\tau}_{0}\right)=\frac{{A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\Delta \omega {\tau}_{0}\right)}{\left|{A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\Delta \omega {\tau}_{0}\right)\right|+\left|{A}_{1}{A}_{2}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\Delta \omega {\tau}_{0}\right)\right|}.$$
(10)
$$p\left({I}_{1},{I}_{2},\varphi \right)=\frac{\text{exp}\left[-\frac{{I}_{1}+{I}_{2}-2\sqrt{{I}_{1}{I}_{2}}\left|\mu \right|\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\varphi \right)}{1-{\left|\mu \right|}^{2}}\right]}{2\pi \left(1-{\left|\mu \right|}^{2}\right)}.$$
(11)
$$\mu =\u3008{A}_{1}^{*}{A}_{2}\u3009=\text{exp}\left(2\pi i\frac{\Delta x}{{\lambda}_{\text{eq}}}\right)\text{exp}\left(-8{\pi}^{2}\frac{{\delta}_{h}^{2}}{{\lambda}_{\text{eq}}^{2}}\right);$$
(12)
$${\delta}_{\varphi}^{2}={\displaystyle {\int}_{0}^{\infty}{\displaystyle {\int}_{0}^{\infty}{\displaystyle {\int}_{-\pi}^{\pi}{\varphi}^{2}\phantom{\rule{0.2em}{0ex}}p\left({I}_{1},{I}_{2},\varphi \right)d\varphi d{I}_{1}d{I}_{2}}}},$$