## Abstract

A white light extrinsic Fabry–Perot interferometer is implemented as a noncontacting displacement sensor, providing robust, absolute displacement measurements with micrometer accuracy at a sampling rate of 10 Hz. This paper presents a dynamic model of the sensing cavity between the sensor probe and the nearby target surface using a Fabry–Perot etalon approach obtained from straightforward electromagnetic field formulations. Such a model is important for system characterization, as the dynamically changing cavity length imparts a Doppler shift on any signals circulating within the sensing cavity. Contrary to previously published results, Doppler-induced shifting within the low-finesse sensing cavity is shown to significantly distort the measurement signal as recorded by the sensor. Experimental and simulation results are compared, and the direct effects of cavity dynamics on the measurement signal are analyzed along with their indirect impact on sensor performance. This document has been approved by Los Alamos National Laboratory for unlimited public release (LA-UR 12-00301).

© 2012 Optical Society of America

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### Equations (11)

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(1)
$${E}_{\text{circ}}(t)=j{t}_{1}{E}_{\text{inc}}(t)+{g}_{\text{rt}}(t){E}_{\text{circ}}(t-\tau ),$$
(2)
$${g}_{\text{rt}}(t-(n-1)\tau )={r}_{1}{r}_{2}\text{\hspace{0.17em}}\mathrm{exp}[-2{\alpha}_{0}L(t-n\tau /2)\phantom{\rule{0ex}{0ex}}-\frac{j2\omega (t-n\tau )L(t-n\tau /2)}{c}].$$
(3)
$${E}_{\text{circ}}(t)=j{t}_{1}\{{g}_{\text{rt}}(t){E}_{\text{inc},D}(t-\tau )\phantom{\rule{0ex}{0ex}}+{g}_{\text{rt}}(t){g}_{\text{rt}}(t-\tau ){E}_{\text{inc},D}(t-2\tau )+\dots \phantom{\rule{0ex}{0ex}}+{g}_{\text{rt}}(t){g}_{\text{rt}}(t-\tau )\dots {g}_{\text{rt}}(t-(N-1)\tau ){E}_{\text{inc},D}(t-N\tau )\}.\phantom{\rule{0ex}{0ex}}$$
(4)
$${E}_{\text{inc},D}(t-n\tau )={E}_{0}\text{\hspace{0.17em}}\mathrm{exp}[jD(t-n\tau /2)(k(t-n\tau )x\phantom{\rule{0ex}{0ex}}-\omega (t-n\tau )(t-n\tau ))],$$
(5)
$$D(t-n\tau /2)=1+2\frac{L(t-n\tau /2)}{c},$$
(6)
$${E}_{\text{refl}}(t)={r}_{1}{E}_{\text{inc}}(t)+j\frac{{t}_{1}}{{r}_{1}}{E}_{\text{circ}}(t),$$
(7)
$${E}_{\text{refl}}(t)={r}_{1}{E}_{0}\text{\hspace{0.17em}}\mathrm{exp}[-j\omega (t)t]\phantom{\rule{0ex}{0ex}}-\frac{{t}_{1}^{2}}{{r}_{1}}\sum _{i=1}^{10{\tau}_{S}/\tau}[{g}_{\text{rt}}{(t-(i-1)\tau )}^{i}{E}_{0}\text{\hspace{0.17em}}\phantom{\rule{0ex}{0ex}}\mathrm{exp}[-j{D}^{i}(t-i\tau /2)\omega (t-i\tau )(t-i\tau )]].$$
(8)
$${I}_{\text{refl}}(t)=\frac{1}{2}c{\epsilon}_{0}{E}_{\text{refl}}{E}_{\text{refl}}^{*},$$
(9)
$${\tau}_{S}=-\frac{\tau}{\mathrm{ln}({r}_{1}{r}_{2}\text{\hspace{0.17em}}\mathrm{exp}(-2{\alpha}_{0}L))}.$$
(10)
$$I(t)=\frac{{t}_{f}^{4}}{1+{r}_{f}^{4}-2{r}_{f}\text{\hspace{0.17em}}\mathrm{cos}(2\omega (t){L}_{f}(t)/c)}.$$
(11)
$$\text{Finesse}=\frac{\pi \sqrt{{r}_{1}{r}_{2}}}{1-{r}_{1}{r}_{2}},$$