## Abstract

We propose and demonstrate a displacement and angular drift simultaneous measurement technique based on a defocus grating. The displacement and angular drift of the incident beam can be detected by monitoring the movements of $\pm 1$ diffraction order spots of the defocus grating. The relationship between drift of the incident beam and movements of $\pm 1$ diffraction order spots is studied in detail. Compared with other methods, this technique eliminates the requirement of two or more detecting systems for measuring displacement and angular drift simultaneously. The proof-of-principle experiment shows that the root-mean-square errors of displacement and angular drift measurements are less than $0.5\text{}\mathrm{\mu m}$ and $0.84\text{}\mathrm{\mu rad}$, respectively.

© 2010 Optical Society of America

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### Equations (21)

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(1)
$$t(x,y)=\sum _{m=-\infty}^{+\infty}{a}_{m}{e}^{i\pi m({x}^{2}+{y}^{2})/\lambda {f}_{g}}{e}^{-i2\pi m{x}_{o}x/\lambda {f}_{g}}{e}^{i\pi m{x}_{o}^{2}/\lambda {f}_{g}},$$
(2)
$$E(x,y,{z}_{1})=\mathrm{exp}[-\frac{(x-{a}_{x}{)}^{2}+(y-{a}_{y}{)}^{2}}{{w}^{2}({z}_{1})}]\phantom{\rule{0ex}{0ex}}\mathrm{exp}\{-ik[\frac{(x-{a}_{x}{)}^{2}+(y-{a}_{y}{)}^{2}}{2R({z}_{1})}+{z}_{1}]-\psi \}\phantom{\rule{0ex}{0ex}}\mathrm{exp}[-ik({c}_{x}x+{c}_{y}y)],$$
(3)
$${E}_{2}({x}_{2},{y}_{2},z)=\frac{-i\mathrm{exp}(ikL)}{\lambda B}\mathrm{exp}\left[\frac{ikD({x}_{2}^{2}+{y}_{2}^{2})}{2B}\right]\phantom{\rule{0ex}{0ex}}\iint E({x}_{1},{y}_{1},{z}_{1})t({x}_{1},{y}_{1})\phantom{\rule{0ex}{0ex}}\mathrm{exp}\{\frac{ik}{2B}[A({x}_{1}^{2}+{y}_{1}^{2})-2({x}_{2}{x}_{1}+{y}_{2}{y}_{1})\left]\right\}\phantom{\rule{0ex}{0ex}}\mathrm{d}{x}_{1}\mathrm{d}{y}_{1},$$
(4)
$${E}_{dm}({x}_{2},{y}_{2},z)={F}_{Edm}\mathrm{exp}\left\{\frac{[\frac{k{x}_{2}i}{B}-\frac{2}{{w}^{2}({z}_{1})}{a}_{x}-\frac{ki}{R({z}_{1})}{a}_{x}+\frac{km{x}_{0}i}{{f}_{g}}{]}^{2}}{\frac{4}{{w}^{2}({z}_{1})}+\frac{2ki}{R({z}_{1})}-\frac{2kAi}{B}-\frac{2kmi}{{f}_{g}}}\right\}\mathrm{exp}\left\{\frac{[\frac{k{y}_{2}i}{B}-\frac{2}{{w}^{2}({z}_{1})}{a}_{y}-\frac{ki}{R({z}_{1})}{a}_{y}{]}^{2}}{\frac{4}{{w}^{2}({z}_{1})}+\frac{2ki}{R({z}_{1})}-\frac{2kAi}{B}-\frac{2kmi}{{f}_{g}}}\right\},$$
(5)
$${I}_{dm}({x}_{2},{y}_{2},z)={E}_{dm}({x}_{2},{y}_{2},z){{E}_{dm}}^{*}({x}_{2},{y}_{2},z)={F}_{Idm}\mathrm{exp}\left\{\frac{-[{x}_{2}-(A{a}_{x}+\frac{Bm}{{f}_{g}}{a}_{x}-\frac{Bm{x}_{0}}{{f}_{g}}){]}^{2}}{\frac{2{B}^{2}}{{w}^{2}({z}_{1}){k}^{2}}+2{B}^{2}{w}^{2}({z}_{1})[\frac{1}{2R({z}_{1})}-\frac{A}{2B}-\frac{m}{2{f}_{g}}{]}^{2}}\right\}\phantom{\rule{0ex}{0ex}}\mathrm{exp}\left\{\frac{-({y}_{2}-A{a}_{y}-\frac{Bm}{{f}_{g}}{a}_{y}{)}^{2}}{\frac{2{B}^{2}}{w({z}_{1}{)}^{2}{k}^{2}}+2{B}^{2}w({z}_{1}{)}^{2}[\frac{1}{2R({z}_{1})}-\frac{A}{2B}-\frac{m}{2{f}_{g}}{]}^{2}}\right\},$$
(6)
$${F}_{Edm}=\frac{-2{\pi}^{2}\mathrm{sin}\left(\frac{m\pi}{2}\right)\mathrm{sinc}\left(\frac{m}{2}\right)}{\lambda B[\frac{1}{w({z}_{1}{)}^{2}}+i(\frac{k}{2R({z}_{1})}-\frac{kA}{2B}-\frac{km}{2{f}_{g}}\left)\right]}\phantom{\rule{0ex}{0ex}}\mathrm{exp}[-\frac{{a}_{x}^{2}+{a}_{y}^{2}}{{w}^{2}({z}_{1})}-\psi +\frac{ikD({x}_{2}^{2}+{y}_{2}^{2})}{2B}+\frac{i\pi m{x}_{0}^{2}}{\lambda {f}_{g}}\phantom{\rule{0ex}{0ex}}-i\pi m-\frac{ik({a}_{x}^{2}+{a}_{y}^{2})}{2R({z}_{1})}-ik{z}_{1}+ikL],$$
(7)
$${F}_{Idm}=\frac{4{\pi}^{4}{\mathrm{sin}}^{2}\left(\frac{m\pi}{2}\right){\mathrm{sinc}}^{2}\left(\frac{m}{2}\right)}{{\lambda}^{2}{B}^{2}[\frac{1}{{w}^{4}({z}_{1})}+(\frac{k}{2R({z}_{1})}-\frac{kA}{2B}-\frac{km}{2{f}_{g}}{)}^{2}]}\phantom{\rule{0ex}{0ex}}\mathrm{exp}\{\frac{[\frac{{a}_{x}k}{R({z}_{1})}-\frac{{a}_{x}kA}{B}-\frac{{a}_{x}km}{{f}_{g}}{]}^{2}+2[\frac{2{a}_{x}}{{w}^{2}({z}_{1})}{]}^{2}+[\frac{{a}_{y}k}{R({z}_{1})}-\frac{{a}_{y}kA}{B}-\frac{{a}_{y}km}{{f}_{g}}{]}^{2}}{\frac{2}{{w}^{2}({z}_{1})}+2{w}^{2}({z}_{1})[\frac{k}{2R({z}_{1})}-\frac{kA}{2B}-\frac{km}{2{f}_{g}}{]}^{2}}-\frac{2({a}_{x}^{2}+{b}_{x}^{2})}{{w}^{2}({z}_{1})}-2\psi \},$$
(8)
$${E}_{tm}({x}_{2},{y}_{2},z)={F}_{Etm}\mathrm{exp}\left[\frac{-(\frac{k{x}_{2}}{B}+k{c}_{x}+\frac{km{x}_{0}}{{f}_{g}}{)}^{2}}{\frac{4}{{w}^{2}({z}_{1})}+\frac{2ki}{R({z}_{1})}-\frac{2kAi}{B}-\frac{2kmi}{{f}_{g}}}\right]\mathrm{exp}\left[\frac{-(\frac{k{y}_{2}}{B}+k{c}_{y}{)}^{2}}{\frac{4}{{w}^{2}({z}_{1})}+\frac{2ki}{R({z}_{1})}-\frac{2kAi}{B}-\frac{2kmi}{{f}_{g}}}\right],$$
(9)
$${I}_{tm}({x}_{2},{y}_{2},z)={E}_{tm}({x}_{2},{y}_{2},z){{E}_{tm}}^{*}({x}_{2},{y}_{2},z)={F}_{Itm}\mathrm{exp}\left\{\frac{-({x}_{2}+B{c}_{x}+\frac{Bm{x}_{0}}{f}{)}^{2}}{\frac{2{B}^{2}}{{w}^{2}({z}_{1}){k}^{2}}+2{B}^{2}{w}^{2}({z}_{1})[\frac{1}{2R({z}_{1})}-\frac{A}{2B}-\frac{m}{2{f}_{g}}{]}^{2}}\right\}\mathrm{exp}\left\{\frac{-({y}_{2}+B{c}_{y}{)}^{2}}{\frac{2{B}^{2}}{{w}^{2}({z}_{1}){k}^{2}}+2{B}^{2}{w}^{2}({z}_{1})[\frac{1}{2R({z}_{1})}-\frac{A}{2B}-\frac{m}{2{f}_{g}}{]}^{2}}\right\},$$
(10)
$${F}_{Etm}=\frac{-2{\pi}^{2}\mathrm{exp}(-\psi )\mathrm{sin}\left(\frac{m\pi}{2}\right)\mathrm{sinc}\left(\frac{m}{2}\right)}{\lambda B[\frac{1}{{w}^{2}({z}_{1})}+i(\frac{k}{2R({z}_{1})}-\frac{kA}{2B}-\frac{km}{2{f}_{g}}\left)\right]}\phantom{\rule{0ex}{0ex}}\mathrm{exp}[\frac{i\pi m{x}_{0}^{2}}{\lambda {f}_{g}}+\frac{ikD({x}_{2}^{2}+{y}_{2}^{2})}{2B}-ik{z}_{1}-i\pi m+ikL],$$
(11)
$${F}_{Itm}=\frac{4{\pi}^{4}\mathrm{exp}(-2\psi ){\mathrm{sin}}^{2}\left(\frac{m\pi}{2}\right){\mathrm{sinc}}^{2}\left(\frac{m}{2}\right)}{{\lambda}^{2}{B}^{2}[\frac{1}{{w}^{4}({z}_{1})}+(\frac{k}{2R({z}_{1})}-\frac{kA}{2B}-\frac{km}{2{f}_{g}}{)}^{2}]}.$$
(12)
$$\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=\left[\begin{array}{cc}1& f\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -\frac{1}{f}& 1\end{array}\right]=\left[\begin{array}{cc}0& f\\ -\frac{1}{f}& 1\end{array}\right],$$
(13)
$${x}_{dm}=\frac{fm}{{f}_{g}}{a}_{x}-\frac{fm{x}_{0}}{{f}_{g}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}{y}_{dm}=\frac{fm}{{f}_{g}}{a}_{y},$$
(14)
$${x}_{tm}=-f{c}_{x}-\frac{fm{x}_{0}}{{f}_{g}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}{y}_{tm}=-f{c}_{y},$$
(15)
$${a}_{x}=\frac{{f}_{g}}{2f}{d}_{x}+{x}_{o},\phantom{\rule[-0.0ex]{2em}{0.0ex}}{a}_{y}=\frac{{f}_{g}}{2f}{d}_{y},$$
(16)
$${c}_{x}=-\frac{1}{f}{t}_{x}\mp \frac{{x}_{o}}{{f}_{g}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}{c}_{y}=-\frac{1}{f}{t}_{y},$$
(18)
$$d\approx 2L\phi ,$$
(19)
$$\theta \approx d/L.$$
(20)
$${a}_{x}=\frac{{f}_{g}}{2z}{d}_{x}+{x}_{o},\phantom{\rule[-0.0ex]{2em}{0.0ex}}{a}_{y}=\frac{{f}_{g}}{2z}{d}_{y}\mathrm{.}$$
(21)
$${c}_{x}=-\frac{1}{z}{t}_{x}\mp \frac{{x}_{0}}{{f}_{g}},\phantom{\rule[-0.0ex]{2em}{0.0ex}}{c}_{y}=-\frac{1}{z}{t}_{y}\mathrm{.}$$