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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2022-01-05 mer. 15:27 --> <!-- 2022-02-15 mar. 14:25 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>ESRF Double Crystal Monochromator - Metrology</title> <title>ESRF Double Crystal Monochromator - Metrology</title>
<meta name="author" content="Dehaeze Thomas" /> <meta name="author" content="Dehaeze Thomas" />
@ -39,48 +39,67 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc"> <div id="text-table-of-contents" role="doc-toc">
<ul> <ul>
<li><a href="#org138e76d">1. Metrology Concept</a> <li><a href="#org958ba4a">1. Metrology Concept</a>
<ul> <ul>
<li><a href="#orge26c208">1.1. Sensor Topology</a></li> <li><a href="#org60dec5a">1.1. Sensor Topology</a></li>
<li><a href="#orga08f1f5">1.2. Crystal&rsquo;s motion computation</a></li> <li><a href="#orgad95b36">1.2. Crystal&rsquo;s motion computation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orge6e300e">2. Deformations of the Metrology Frame</a> <li><a href="#org146462e">2. Relation Between Crystal position and X-ray measured displacement</a>
<ul> <ul>
<li><a href="#org62287e7">2.1. Measurement Setup</a></li> <li><a href="#org53e570a">2.1. Setup</a></li>
<li><a href="#org9a14028">2.2. Simulations</a></li> <li><a href="#org1118f4c">2.2. Relation between second crystal motion and beam motion</a>
<li><a href="#org2f06b8d">2.3. Comparison</a></li> <ul>
<li><a href="#org48b378e">2.2.1. Axial motion of second crystal</a></li>
<li><a href="#org9b6ca5e">2.2.2. Ry motion of second crystal</a></li>
<li><a href="#orge39778e">2.2.3. Rx motion of second crystal</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org57a7951">3. Attocube - Periodic Non-Linearity</a>
<ul>
<li><a href="#org2e13fe2">3.1. Calibration - Concept</a></li>
<li><a href="#org44715ee">3.2. Measurements</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org3331f0d">Bibliography</a></li> <li><a href="#orgc0f14ff">3. Deformations of the Metrology Frame</a>
<ul>
<li><a href="#org54cb1b0">3.1. Measurement Setup</a></li>
<li><a href="#orga3da277">3.2. Simulations</a></li>
<li><a href="#orgec833d8">3.3. Comparison</a></li>
<li><a href="#orga851ac8">3.4. Test</a></li>
<li><a href="#orgc6c0d26">3.5. Measured frame deformation</a></li>
<li><a href="#org9ce965b">3.6. Test</a></li>
<li><a href="#org3613a0c">3.7. Repeatability of frame deformation</a></li>
</ul>
</li>
<li><a href="#orgdedfb8e">4. Attocube - Periodic Non-Linearity</a>
<ul>
<li><a href="#org324d54e">4.1. Measurement Setup</a></li>
<li><a href="#org78e73ad">4.2. Choice of the reference signal</a></li>
<li><a href="#org1965ef7">4.3. Repeatability of the non-linearity</a></li>
<li><a href="#org3432548">4.4. Simulation</a></li>
<li><a href="#org018ecfe">4.5. Measurements</a></li>
</ul>
</li>
<li><a href="#org6292dcf">Bibliography</a></li>
</ul> </ul>
</div> </div>
</div> </div>
<hr> <hr>
<p>This report is also available as a <a href="./dcm_metrology.pdf">pdf</a>.</p> <p>This report is also available as a <a href="./dcm-metrology.pdf">pdf</a>.</p>
<hr> <hr>
<p> <p>
In this document, the metrology system is studied. In this document, the metrology system is studied.
First, in Section <a href="#orgb946c2b">1</a> the goal of the metrology system is stated and the proposed concept is described. First, in Section <a href="#orgb868c48">1</a> the goal of the metrology system is stated and the proposed concept is described.
In order to increase the accuracy of the metrology system, two problems are to be dealt with: In order to increase the accuracy of the metrology system, two problems are to be dealt with:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>The deformation of the metrology frame under the action of gravity (Section <a href="#orgacc2503">2</a>)</li> <li>The deformation of the metrology frame under the action of gravity (Section <a href="#orgc876cab">3</a>)</li>
<li>The periodic non-linearity of the interferometers (Section <a href="#org3b56a97">3</a>)</li> <li>The periodic non-linearity of the interferometers (Section <a href="#org62f88d9">4</a>)</li>
</ul> </ul>
<div id="outline-container-org138e76d" class="outline-2"> <div id="outline-container-org958ba4a" class="outline-2">
<h2 id="org138e76d"><span class="section-number-2">1.</span> Metrology Concept</h2> <h2 id="org958ba4a"><span class="section-number-2">1.</span> Metrology Concept</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="orgb946c2b"></a> <a id="orgb868c48"></a>
</p> </p>
<p> <p>
The goal of the metrology system is to measure the distance and default of parallelism orientation between the first and second crystals The goal of the metrology system is to measure the distance and default of parallelism orientation between the first and second crystals
@ -95,8 +114,8 @@ Only 3 degrees of freedom are of interest:
<li>\(r_x\)</li> <li>\(r_x\)</li>
</ul> </ul>
</div> </div>
<div id="outline-container-orge26c208" class="outline-3"> <div id="outline-container-org60dec5a" class="outline-3">
<h3 id="orge26c208"><span class="section-number-3">1.1.</span> Sensor Topology</h3> <h3 id="org60dec5a"><span class="section-number-3">1.1.</span> Sensor Topology</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
In order to measure the relative pose of the two crystals, instead of performing a direct measurement which is complicated, the pose of the two crystals are measured from a metrology frame. In order to measure the relative pose of the two crystals, instead of performing a direct measurement which is complicated, the pose of the two crystals are measured from a metrology frame.
@ -104,7 +123,7 @@ Three interferometers are used to measured the 3dof of interest for each crystal
Three additional interferometers are used to measured the relative motion of the metrology frame. Three additional interferometers are used to measured the relative motion of the metrology frame.
</p> </p>
<table id="orgd5dd455" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org93de062" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Notations for the metrology frame</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Notations for the metrology frame</caption>
<colgroup> <colgroup>
@ -157,7 +176,7 @@ Three additional interferometers are used to measured the relative motion of the
</table> </table>
<div id="org3997ed1" class="figure"> <div id="orgaaab71b" class="figure">
<p><img src="figs/metrology_schematic.png" alt="metrology_schematic.png" /> <p><img src="figs/metrology_schematic.png" alt="metrology_schematic.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Schematic of the Metrology System</p> <p><span class="figure-number">Figure 1: </span>Schematic of the Metrology System</p>
@ -165,8 +184,8 @@ Three additional interferometers are used to measured the relative motion of the
</div> </div>
</div> </div>
<div id="outline-container-orga08f1f5" class="outline-3"> <div id="outline-container-orgad95b36" class="outline-3">
<h3 id="orga08f1f5"><span class="section-number-3">1.2.</span> Crystal&rsquo;s motion computation</h3> <h3 id="orgad95b36"><span class="section-number-3">1.2.</span> Crystal&rsquo;s motion computation</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
From the raw interferometric measurements, the pose between the first and second crystals can be computed. From the raw interferometric measurements, the pose between the first and second crystals can be computed.
@ -180,7 +199,7 @@ First, Jacobian matrices can be used to convert raw interferometer measurements
For the 311 crystals: For the 311 crystals:
</p> </p>
<table id="org75daba0" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org3fe0d44" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Table caption</caption> <caption class="t-above"><span class="table-number">Table 2:</span> Table caption</caption>
<colgroup> <colgroup>
@ -244,7 +263,7 @@ For the 311 crystals:
</tbody> </tbody>
</table> </table>
<table id="orgcc31c85" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org1e5a4a0" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> Table caption</caption> <caption class="t-above"><span class="table-number">Table 3:</span> Table caption</caption>
<colgroup> <colgroup>
@ -309,21 +328,21 @@ For the 311 crystals:
</table> </table>
<div id="orgd029bae" class="figure"> <div id="org83db6ba" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_m.png" alt="schematic_sensor_jacobian_forward_kinematics_m.png" /> <p><img src="figs/schematic_sensor_jacobian_forward_kinematics_m.png" alt="schematic_sensor_jacobian_forward_kinematics_m.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Forward Kinematics for the Metrology frame</p> <p><span class="figure-number">Figure 2: </span>Forward Kinematics for the Metrology frame</p>
</div> </div>
<div id="org8c52c6b" class="figure"> <div id="orge535a69" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_1.png" alt="schematic_sensor_jacobian_forward_kinematics_1.png" /> <p><img src="figs/schematic_sensor_jacobian_forward_kinematics_1.png" alt="schematic_sensor_jacobian_forward_kinematics_1.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Forward Kinematics for the 1st crystal</p> <p><span class="figure-number">Figure 3: </span>Forward Kinematics for the 1st crystal</p>
</div> </div>
<div id="orgabe5ba2" class="figure"> <div id="org03be022" class="figure">
<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_2.png" alt="schematic_sensor_jacobian_forward_kinematics_2.png" /> <p><img src="figs/schematic_sensor_jacobian_forward_kinematics_2.png" alt="schematic_sensor_jacobian_forward_kinematics_2.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Forward Kinematics for the 2nd crystal</p> <p><span class="figure-number">Figure 4: </span>Forward Kinematics for the 2nd crystal</p>
@ -385,7 +404,7 @@ G_111_t(1,<span class="org-builtin">:</span>) = <span class="org-builtin">-</spa
</pre> </pre>
</div> </div>
<table id="orgd893900" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgdf27d5d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Transformation Matrix</caption> <caption class="t-above"><span class="table-number">Table 4:</span> Transformation Matrix</caption>
<colgroup> <colgroup>
@ -466,21 +485,115 @@ G_111_t(1,<span class="org-builtin">:</span>) = <span class="org-builtin">-</spa
</table> </table>
<p> <p>
From table <a href="#orgabe5ba2">4</a>, we can determine the effect of each interferometer on the estimated relative pose between the crystals. From table <a href="#org03be022">4</a>, we can determine the effect of each interferometer on the estimated relative pose between the crystals.
For instance, an error on <code>dr1</code> will have much greater impact on <code>ry</code> than an error on <code>drm</code>. For instance, an error on <code>dr1</code> will have much greater impact on <code>ry</code> than an error on <code>drm</code>.
</p> </p>
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-orge6e300e" class="outline-2"> <div id="outline-container-org146462e" class="outline-2">
<h2 id="orge6e300e"><span class="section-number-2">2.</span> Deformations of the Metrology Frame</h2> <h2 id="org146462e"><span class="section-number-2">2.</span> Relation Between Crystal position and X-ray measured displacement</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
</div>
<div id="outline-container-org53e570a" class="outline-3">
<h3 id="org53e570a"><span class="section-number-3">2.1.</span> Setup</h3>
<div class="outline-text-3" id="text-2-1">
<div id="orgbd2501f" class="figure">
<p><img src="figs/calibration_setup.png" alt="calibration_setup.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Schematic of the setup</p>
</div>
<p> <p>
<a id="orgacc2503"></a> Detector:
</p>
<p>
<a href="https://www.baslerweb.com/en/products/cameras/area-scan-cameras/ace/aca1920-40gc/">https://www.baslerweb.com/en/products/cameras/area-scan-cameras/ace/aca1920-40gc/</a>
</p>
<p>
Pixel size depends on the magnification used (1x, 6x, 12x).
</p>
<p>
Pixel size of camera is 5.86 um x 5.86 um.
With typical magnification of 6x, pixel size is ~1.44um x 1.44um
</p>
<p>
Frame rate is: 42 fps
</p>
</div>
</div>
<div id="outline-container-org1118f4c" class="outline-3">
<h3 id="org1118f4c"><span class="section-number-3">2.2.</span> Relation between second crystal motion and beam motion</h3>
<div class="outline-text-3" id="text-2-2">
</div>
<div id="outline-container-org48b378e" class="outline-4">
<h4 id="org48b378e"><span class="section-number-4">2.2.1.</span> Axial motion of second crystal</h4>
<div class="outline-text-4" id="text-2-2-1">
<p>
Let&rsquo;s consider the relation between the \([y, z]\) motion of the beam and the motion of the second crystal \([z^\prime, R_{y^\prime}, R_{x^\prime}]\).
</p>
<div id="org7214e82" class="figure">
<p><img src="figs/relation_dz_output_beam.png" alt="relation_dz_output_beam.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Relation between \(d_{z^\prime}\) motion of the second crystal and vertical motion of the beam</p>
</div>
\begin{equation}
d_z = d_{z^\prime} 2 \cos \theta
\end{equation}
<div id="org101c11d" class="figure">
<p><img src="figs/relation_vert_motion_crystal_beam.png" alt="relation_vert_motion_crystal_beam.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Relation between vertical motion of the second crystal and vertical motion of the output beam</p>
</div>
</div>
</div>
<div id="outline-container-org9b6ca5e" class="outline-4">
<h4 id="org9b6ca5e"><span class="section-number-4">2.2.2.</span> Ry motion of second crystal</h4>
<div class="outline-text-4" id="text-2-2-2">
\begin{equation}
d_z = D_{\text{vlm}} d_{R_y^\prime}
\end{equation}
<p>
with \(D_{\text{vlm}} \approx 10\,m\).
</p>
</div>
</div>
<div id="outline-container-orge39778e" class="outline-4">
<h4 id="orge39778e"><span class="section-number-4">2.2.3.</span> Rx motion of second crystal</h4>
<div class="outline-text-4" id="text-2-2-3">
\begin{equation}
d_y = 2 D_{\text{vlm}} \sin \theta \cdot d_{R_x^\prime}
\end{equation}
</div>
</div>
</div>
</div>
<div id="outline-container-orgc0f14ff" class="outline-2">
<h2 id="orgc0f14ff"><span class="section-number-2">3.</span> Deformations of the Metrology Frame</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="orgc876cab"></a>
</p> </p>
<p> <p>
The transformation matrix in Table <a href="#orgabe5ba2">4</a> is valid only if the metrology frames are solid bodies. The transformation matrix in Table <a href="#orgdf27d5d">4</a> is valid only if the metrology frames are solid bodies.
</p> </p>
<p> <p>
@ -492,9 +605,9 @@ When the bragg axis is scanned, the effect of gravity on the metrology frame is
This can be calibrated. This can be calibrated.
</p> </p>
</div> </div>
<div id="outline-container-org62287e7" class="outline-3"> <div id="outline-container-org54cb1b0" class="outline-3">
<h3 id="org62287e7"><span class="section-number-3">2.1.</span> Measurement Setup</h3> <h3 id="org54cb1b0"><span class="section-number-3">3.1.</span> Measurement Setup</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-3-1">
<p> <p>
Two beam viewers: Two beam viewers:
</p> </p>
@ -511,9 +624,9 @@ This position is the wanted position for a given Bragg angle.
</div> </div>
</div> </div>
<div id="outline-container-org9a14028" class="outline-3"> <div id="outline-container-orga3da277" class="outline-3">
<h3 id="org9a14028"><span class="section-number-3">2.2.</span> Simulations</h3> <h3 id="orga3da277"><span class="section-number-3">3.2.</span> Simulations</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-3-2">
<p> <p>
The deformations of the metrology frame and therefore the expected interferometric measurements can be computed as a function of the Bragg angle. The deformations of the metrology frame and therefore the expected interferometric measurements can be computed as a function of the Bragg angle.
This may be done using FE software. This may be done using FE software.
@ -521,69 +634,513 @@ This may be done using FE software.
</div> </div>
</div> </div>
<div id="outline-container-org2f06b8d" class="outline-3"> <div id="outline-container-orgec833d8" class="outline-3">
<h3 id="org2f06b8d"><span class="section-number-3">2.3.</span> Comparison</h3> <h3 id="orgec833d8"><span class="section-number-3">3.3.</span> Comparison</h3>
</div>
<div id="outline-container-orga851ac8" class="outline-3">
<h3 id="orga851ac8"><span class="section-number-3">3.4.</span> Test</h3>
<div class="outline-text-3" id="text-3-4">
<div class="org-src-container">
<pre class="src src-matlab">aa = importdata(<span class="org-string">"correctInterf-vlm-220201.dat"</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">plot</span>(aa.data(<span class="org-builtin">:</span>,1), aa.data(<span class="org-builtin">:</span>,24))
</pre>
</div>
</div> </div>
</div> </div>
<div id="outline-container-org57a7951" class="outline-2"> <div id="outline-container-orgc6c0d26" class="outline-3">
<h2 id="org57a7951"><span class="section-number-2">3.</span> Attocube - Periodic Non-Linearity</h2> <h3 id="orgc6c0d26"><span class="section-number-3">3.5.</span> Measured frame deformation</h3>
<div class="outline-text-2" id="text-3"> <div class="outline-text-3" id="text-3-5">
<div class="org-src-container">
<pre class="src src-matlab">data = table2array(readtable(<span class="org-string">'itf_polynom.csv'</span>,<span class="org-string">'NumHeaderLines'</span>,1));
th = <span class="org-matlab-math">pi</span><span class="org-builtin">/</span>180<span class="org-builtin">*</span>data(<span class="org-builtin">:</span>,1); <span class="org-comment-delimiter">% </span><span class="org-comment">[rad]</span>
fj = 0.030427 <span class="org-builtin">-</span> 10.51e<span class="org-builtin">-</span>3<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(th)); <span class="org-comment-delimiter">% </span><span class="org-comment">[m]</span>
rx2 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>data(<span class="org-builtin">:</span>,2); <span class="org-comment-delimiter">% </span><span class="org-comment">[rad]</span>
ry2 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>data(<span class="org-builtin">:</span>,3); <span class="org-comment-delimiter">% </span><span class="org-comment">[rad]</span>
rx1 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>data(<span class="org-builtin">:</span>,4); <span class="org-comment-delimiter">% </span><span class="org-comment">[rad]</span>
ry1 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>data(<span class="org-builtin">:</span>,5); <span class="org-comment-delimiter">% </span><span class="org-comment">[rad]</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">on;</span>
<span class="org-builtin">plot</span>(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, 1e6<span class="org-builtin">*</span>detrend(<span class="org-builtin">-</span>rx2, 1), <span class="org-string">'.'</span>)
<span class="org-comment-delimiter">% </span><span class="org-comment">plot(180/pi*th, detrend(ry2, 1))</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">plot(180/pi*th, detrend(rx1, 1))</span>
<span class="org-comment-delimiter">% </span><span class="org-comment">plot(180/pi*th, detrend(ry1, 1))</span>
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">off;</span>
<span class="org-builtin">xlabel</span>(<span class="org-string">'Bragg Angle [deg]'</span>);
<span class="org-builtin">ylabel</span>(<span class="org-string">'Measured $R_x$ [$\mu$rad]'</span>)
<span class="org-builtin">xlim</span>([10, 75]);
</pre>
</div>
<div id="org2bfeb26" class="figure">
<p><img src="figs/calibration_drx_pres.png" alt="calibration_drx_pres.png" />
</p>
<p><span class="figure-number">Figure 8: </span>description</p>
</div>
<p> <p>
<a id="org3b56a97"></a> Strange that there is correlation between Rx and Ry.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">on;</span>
<span class="org-builtin">plot</span>(108<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, 1e9<span class="org-builtin">*</span>detrend(rx1, 0), <span class="org-string">'-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$Rx_1$'</span>)
<span class="org-builtin">plot</span>(108<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, 1e9<span class="org-builtin">*</span>detrend(ry1, 0), <span class="org-string">'-'</span>, <span class="org-string">'DisplayName'</span>, <span class="org-string">'$Ry_1$'</span>)
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">off;</span>
<span class="org-builtin">xlabel</span>(<span class="org-string">'Bragg Angle [deg]'</span>); <span class="org-builtin">ylabel</span>(<span class="org-string">'Angle Offset [nrad]'</span>);
<span class="org-builtin">legend</span>()
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">on;</span>
<span class="org-builtin">plot</span>(1e3<span class="org-builtin">*</span>fj, detrend(rx2, 1), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$Rx_1$'</span>)
<span class="org-builtin">plot</span>(1e3<span class="org-builtin">*</span>fj, detrend(ry2, 1), <span class="org-string">'DisplayName'</span>, <span class="org-string">'$Ry_1$'</span>)
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">off;</span>
<span class="org-builtin">xlabel</span>(<span class="org-string">'Fast Jack Displacement [mm]'</span>); <span class="org-builtin">ylabel</span>(<span class="org-string">'Angle Offset [nrad]'</span>);
<span class="org-builtin">legend</span>()
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Compute best polynomial fit</span>
f_rx2 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, 1e9<span class="org-builtin">*</span>rx2, <span class="org-string">'poly4'</span>);
f_ry2 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, 1e9<span class="org-builtin">*</span>ry2, <span class="org-string">'poly4'</span>);
f_rx1 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, 1e9<span class="org-builtin">*</span>rx1, <span class="org-string">'poly4'</span>);
f_ry1 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, 1e9<span class="org-builtin">*</span>ry1, <span class="org-string">'poly4'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">on;</span>
<span class="org-builtin">plot</span>(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, f_rx2(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th))
<span class="org-builtin">plot</span>(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, f_ry2(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th))
<span class="org-builtin">plot</span>(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, f_rx1(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th))
<span class="org-builtin">plot</span>(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th, f_ry1(180<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>th))
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">off;</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">on;</span>
<span class="org-builtin">plot</span>(th, f_rx2(th) <span class="org-builtin">-</span> rx2)
<span class="org-builtin">plot</span>(th, f_ry2(th) <span class="org-builtin">-</span> ry2)
<span class="org-builtin">plot</span>(th, f_rx1(th) <span class="org-builtin">-</span> rx1)
<span class="org-builtin">plot</span>(th, f_ry1(th) <span class="org-builtin">-</span> ry1)
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">off;</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">on;</span>
<span class="org-builtin">plot</span>(th, f(th))
<span class="org-builtin">plot</span>(th, rx2, <span class="org-string">'.'</span>)
</pre>
</div>
</div>
</div>
<div id="outline-container-org9ce965b" class="outline-3">
<h3 id="org9ce965b"><span class="section-number-3">3.6.</span> Test</h3>
<div class="outline-text-3" id="text-3-6">
<div class="org-src-container">
<pre class="src src-matlab">filename = <span class="org-string">"/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5"</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">data = struct();
data.xtal1_111_u = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM4/data'</span>));
data.xtal1_111_m = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM5/data'</span>));
data.xtal1_111_d = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM6/data'</span>));
data.mframe_u = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM3/data'</span>));
data.mframe_dh = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM4/data'</span>));
data.mframe_dr = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM5/data'</span>));
data.bragg = (<span class="org-matlab-math">pi</span><span class="org-builtin">/</span>180)<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM6/data'</span>));
data.fj_pos = 0.030427 <span class="org-builtin">-</span> 10.5e<span class="org-builtin">-</span>3<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data.bragg));
data.time = double(h5read(filename, <span class="org-string">'/7.1/instrument/time/data'</span>));
data.rx = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal1_111_rx/data'</span>));
data.ry = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal1_111_ry/data'</span>));
data.z = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal1_111_z/data'</span>));
data.drx = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal_111_drx_filter/data'</span>));
data.dry = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal_111_dry_filter/data'</span>));
data.dz = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal_111_dz_filter/data'</span>));
data.xtal2_111_u = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM10/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data.bragg));
data.xtal2_111_m = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM1/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data.bragg));
data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM2/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data.bragg));
</pre>
</div>
<div id="org22c7c33" class="figure">
<p><img src="figs/drifts_xtal2_detrend.png" alt="drifts_xtal2_detrend.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Drifts of the second crystal as a function of Bragg Angle</p>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-builtin">figure</span>;
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">on;</span>
<span class="org-builtin">plot</span>(108<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>data.bragg, data.drx)
<span class="org-builtin">plot</span>(108<span class="org-builtin">/</span><span class="org-matlab-math">pi</span><span class="org-builtin">*</span>data.bragg, data.dry)
<span class="org-builtin">hold</span> <span class="org-matlab-commanddual-string">off;</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org3613a0c" class="outline-3">
<h3 id="org3613a0c"><span class="section-number-3">3.7.</span> Repeatability of frame deformation</h3>
<div class="outline-text-3" id="text-3-7">
<div class="org-src-container">
<pre class="src src-matlab">filename = <span class="org-string">"/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5"</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">data_1 = struct();
data_1.time = double(h5read(filename, <span class="org-string">'/7.1/instrument/time/data'</span>));
data_1.bragg = (<span class="org-matlab-math">pi</span><span class="org-builtin">/</span>180)<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM6/data'</span>));
data_1.fj_pos = 0.030427 <span class="org-builtin">-</span> 10.5e<span class="org-builtin">-</span>3<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_1.bragg));
data_1.xtal1_111_u = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM4/data'</span>));
data_1.xtal1_111_m = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM5/data'</span>));
data_1.xtal1_111_d = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM6/data'</span>));
data_1.xtal2_111_u = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA1_SSIM10/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_1.bragg));
data_1.xtal2_111_m = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM1/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_1.bragg));
data_1.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM2/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_1.bragg));
data_1.mframe_u = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM3/data'</span>));
data_1.mframe_dh = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM4/data'</span>));
data_1.mframe_dr = double(h5read(filename, <span class="org-string">'/7.1/instrument/FPGA2_SSIM5/data'</span>));
data_1.drx = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal_111_drx_filter/data'</span>));
data_1.dry = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal_111_dry_filter/data'</span>));
data_1.dz = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/7.1/instrument/xtal_111_dz_filter/data'</span>));
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">data_2 = struct();
data_2.time = double(h5read(filename, <span class="org-string">'/6.1/instrument/time/data'</span>));
data_2.bragg = (<span class="org-matlab-math">pi</span><span class="org-builtin">/</span>180)<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA2_SSIM6/data'</span>));
data_2.fj_pos = 0.030427 <span class="org-builtin">-</span> 10.5e<span class="org-builtin">-</span>3<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_2.bragg));
data_2.xtal1_111_u = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA1_SSIM4/data'</span>));
data_2.xtal1_111_m = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA1_SSIM5/data'</span>));
data_2.xtal1_111_d = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA1_SSIM6/data'</span>));
data_2.xtal2_111_u = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA1_SSIM10/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_2.bragg));
data_2.xtal2_111_m = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA2_SSIM1/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_2.bragg));
data_2.xtal2_111_d = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA2_SSIM2/data'</span>))<span class="org-builtin">+</span>10.5e6<span class="org-builtin">./</span>(2<span class="org-builtin">*</span>cos(data_2.bragg));
data_2.mframe_u = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA2_SSIM3/data'</span>));
data_2.mframe_dh = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA2_SSIM4/data'</span>));
data_2.mframe_dr = double(h5read(filename, <span class="org-string">'/6.1/instrument/FPGA2_SSIM5/data'</span>));
data_2.drx = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/6.1/instrument/xtal_111_drx_filter/data'</span>));
data_2.dry = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/6.1/instrument/xtal_111_dry_filter/data'</span>));
data_2.dz = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>double(h5read(filename, <span class="org-string">'/6.1/instrument/xtal_111_dz_filter/data'</span>));
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-orgdedfb8e" class="outline-2">
<h2 id="orgdedfb8e"><span class="section-number-2">4.</span> Attocube - Periodic Non-Linearity</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org62f88d9"></a>
</p> </p>
<p> <p>
(<a href="#citeproc_bib_item_1">Ducourtieux, 2018, p. 11 to 12</a>; See <a href="#citeproc_bib_item_2">Thurner et al., 2015, p. 8</a>) Interferometers have some periodic nonlinearity (<a href="#citeproc_bib_item_2">Thurner et al., 2015</a>).
The period is a fraction of the wavelength (usually \(\lambda/2\)) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams (See <a href="#citeproc_bib_item_1">Ducourtieux, 2018, p. 67 to 69</a>; <a href="#citeproc_bib_item_2">Thurner et al., 2015</a>).
The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers.
</p> </p>
<p> <p>
The idea is to calibrate the periodic non-linearity of the interferometers, a known displacement must be imposed and the interferometer output compared to this displacement. In the DCM case, when using Attocube interferometers, the period non-linearity are in the order of several nanometers with a period of \(765\,nm\).
This should be performed over several periods in order to characterize the error. This is inducing some positioning errors which are too high.
</p> </p>
<p> <p>
We here suppose that we are already in the frame of the Attocube (the fast-jack displacements are converted to Attocube displacement using the transformation matrices). In order to overcome this issue, the periodic non-linearity of the interferometers have to be calibrated.
We also suppose that we are at a certain Bragg angle, and that the stepper motors are not moving: only the piezoelectric actuators are used. To do so, a displacement is imposed and measured both by the interferometers and by another metrology system which does not have this nonlinearity.
By comparing the two measured displacements, the nonlinearity can be calibration.
This process is performed over several periods in order to characterize the error over the full stroke.
</p>
</div>
<div id="outline-container-org324d54e" class="outline-3">
<h3 id="org324d54e"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
<div class="outline-text-3" id="text-4-1">
<p>
The metrology that will be compared with the interferometers are the strain gauges incorporated in the PI piezoelectric stacks.
</p>
<p>
It is here supposed that the measured displacement by the strain gauges are converted to the displacement at the interferometer locations.
It is also supposed that we are at a certain Bragg angle, and that the stepper motors are not moving: only the piezoelectric actuators are used.
</p>
<div class="note" id="orge5737c1">
<p>
Note that the strain gauges are measuring the relative displacement of the piezoelectric stacks while the interferometers are measuring the relative motion between the second crystals and the metrology frame.
</p>
<p>
Only the interferometers measuring the second crystal motion can be calibrated here.
</p>
<p>
As any deformations of the metrology frame of deformation of the crystal&rsquo;s support can degrade the quality of the calibration, it is better to perform this calibration without any bragg angle motion.
</p>
</div>
<p>
The setup is schematically with the block diagram in Figure <a href="#orgd56a2ba">10</a>.
</p> </p>
<p> <p>
The setup is schematically with the block diagram in Figure <a href="#orga343b78">5</a>.
The signals are: The signals are:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>\(u\): Actuator Signal (position where we wish to go)</li> <li>\(u\): Reference Signal sent to the PI controller (position where we wish to three stacks to be).
<li>\(d\): Disturbances affecting the signal</li> The PI controller takes care or controlling to position as measured by the strain gauges such that it is close to the reference position.</li>
<li>\(y\): Displacement of the crystal</li> <li>\(d\): Disturbances affecting the position of the crystals</li>
<li>\(y_g\): Measurement of the crystal motion by the strain gauge with some noise \(n_g\)</li> <li>\(y\): Displacement of the crystal as measured by one interferometer</li>
<li>\(y_g\): Measurement of the motion in the frame of the interferometer by the strain gauge with some noise \(n_g\)</li>
<li>\(y_a\): Measurement of the crystal motion by the interferometer with some noise \(n_a\)</li> <li>\(y_a\): Measurement of the crystal motion by the interferometer with some noise \(n_a\)</li>
</ul> </ul>
<div id="orga343b78" class="figure"> <div id="orgd56a2ba" class="figure">
<p><img src="figs/block_diagram_lut_attocube.png" alt="block_diagram_lut_attocube.png" /> <p><img src="figs/block_diagram_lut_attocube.png" alt="block_diagram_lut_attocube.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube</p> <p><span class="figure-number">Figure 10: </span>Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube</p>
</div> </div>
<p> <p>
The problem is to estimate the periodic non-linearity of the Attocube from the imperfect measurements \(y_a\) and \(y_g\). The problem is to estimate the periodic non-linearity of the Attocube from the imperfect measurements \(y_a\) and \(y_g\).
</p> </p>
</div>
</div>
<div id="outline-container-org78e73ad" class="outline-3">
<h3 id="org78e73ad"><span class="section-number-3">4.2.</span> Choice of the reference signal</h3>
<div class="outline-text-3" id="text-4-2">
<p>
The main specifications for the reference signal are;
</p>
<ul class="org-ul">
<li>sweep several periods (i.e. several micrometers)</li>
<li>stay in the linear region of the strain gauge</li>
<li>no excitation of mechanical modes (i.e. the frequency content of the signal should be at low frequency)</li>
<li>no phase shift due to limited bandwidth of both the interferometers and the strain gauge</li>
<li>the full process should be quite fast</li>
</ul>
<p> <p>
The wavelength of the Attocube is 1530nm, therefore the non-linearity has a period of 765nm. The travel range of the piezoelectric stacks is 15 micrometers, the resolution of the strain gauges is 0.3nm and the maximum non-linearity is 0.15%.
The amplitude of the non-linearity can vary from one unit to the other (and maybe from one experimental condition to the other). If one non-linear period is swept (765nm), the maximum estimation error of the strain gauge is around 1nm.
It is typically between 5nm peak to peak and 20nm peak to peak. </p>
<p>
Based on the above discussion, one suitable excitation signal is a sinusoidal sweep with a frequency of 10Hz.
</p> </p>
</div> </div>
<div id="outline-container-org2e13fe2" class="outline-3">
<h3 id="org2e13fe2"><span class="section-number-3">3.1.</span> Calibration - Concept</h3>
</div> </div>
<div id="outline-container-org1965ef7" class="outline-3">
<h3 id="org1965ef7"><span class="section-number-3">4.3.</span> Repeatability of the non-linearity</h3>
<div class="outline-text-3" id="text-4-3">
<p>
Instead of calibrating the non-linear errors of the interferometers over the full fast jack stroke (25mm), one can only calibrate the errors of one period.
</p>
<div id="outline-container-org44715ee" class="outline-3"> <p>
<h3 id="org44715ee"><span class="section-number-3">3.2.</span> Measurements</h3> For that, we need to make sure that the errors are repeatable from one period to the other and also the period should be very precisely estimated (i.e. the wavelength of the laser).
<div class="outline-text-3" id="text-3-2"> </p>
<p>
Also, the laser wavelength should be very stable (specified at 50ppb).
</p>
<p>
One way to precisely estimate the laser wavelength is to estimate the non linear errors of the interferometer at an initial position, and then to estimate the non linear errors at a large offset, say 10mm.
</p>
</div>
</div>
<div id="outline-container-org3432548" class="outline-3">
<h3 id="org3432548"><span class="section-number-3">4.4.</span> Simulation</h3>
<div class="outline-text-3" id="text-4-4">
<p>
Suppose we have a first approximation of the non-linear period.
</p>
<div class="org-src-container">
<pre class="src src-matlab">period_est = 765e<span class="org-builtin">-</span>9; <span class="org-comment-delimiter">% </span><span class="org-comment">Estimated period [m]</span>
</pre>
</div>
<p>
And suppose the real period of the non-linear errors is a little bit above (by 0.02nm):
</p>
<div class="org-src-container">
<pre class="src src-matlab">period_err = 0.02e<span class="org-builtin">-</span>9; <span class="org-comment-delimiter">% </span><span class="org-comment">Error on the period estimation [m]</span>
period_nl = period_est <span class="org-builtin">+</span> period_err; <span class="org-comment-delimiter">% </span><span class="org-comment">Period of the non-linear errors [m]</span>
</pre>
</div>
<p>
The non-linear errors are first estimated at the beginning of the stroke (Figure <a href="#org6c0e308">11</a>).
</p>
<div id="org6c0e308" class="figure">
<p><img src="figs/non_linear_errors_start_stroke.png" alt="non_linear_errors_start_stroke.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Estimation of the non-linear errors at the beginning of the stroke</p>
</div>
<p>
From this only measurement, it is not possible to estimate with great accuracy the period of the error.
To do so, the same measurement is performed with a stroke of several millimeters (Figure <a href="#orgbe11399">12</a>).
</p>
<p>
It can be seen that there is an offset between the estimated and the measured errors.
This is due to a mismatch between the estimated period and the true period of the error.
</p>
<div id="orgbe11399" class="figure">
<p><img src="figs/non_linear_errors_middle_stroke.png" alt="non_linear_errors_middle_stroke.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Estimated non-linear errors at a latter position</p>
</div>
<p>
Suppose the non-linear error is characterized by a periodic function \(\mathcal{E}\), to simplify let&rsquo;s take a sinusoidal function (this can be generalized by taking the fourier transform of the function):
</p>
\begin{equation}
\mathcal{E}(x) = \sin\left(\frac{x}{\lambda}\right)
\end{equation}
<p>
with \(x\) the displacement and \(\lambda\) the period of the error.
</p>
<p>
The measured error at \(x_0\) is then:
</p>
\begin{equation}
\mathcal{E}_m(x_0) = \sin\left( \frac{x_0}{\lambda} \right)
\end{equation}
<p>
And the estimated one is:
</p>
\begin{equation}
\mathcal{E}_e(x_0) = \sin \left( \frac{x_0}{\lambda_{\text{est}}} \right)
\end{equation}
<p>
with \(\lambda_{\text{est}}\) the estimated error&rsquo;s period.
</p>
<p>
From Figure <a href="#orgbe11399">12</a>, we can see that there is an offset between the two curves.
Let&rsquo;s call this offset \(\epsilon_x\), we then have:
</p>
\begin{equation}
\mathcal{E}_m(x_0) = \mathcal{E}_e(x_0 + \epsilon_x)
\end{equation}
<p>
Which gives us:
</p>
\begin{equation}
\sin\left( \frac{x_0}{\lambda} \right) = \sin \left( \frac{x_0 + \epsilon_x}{\lambda_{\text{est}}} \right)
\end{equation}
<p>
Finally:
</p>
\begin{equation}
\boxed{\lambda = \lambda_{\text{est}} \frac{x_0}{x_0 + \epsilon_x}}
\end{equation}
<p>
The estimated delay is computed:
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Estimation of the offset between the estimated and measured errors</span>
i_period = stroke <span class="org-builtin">&gt;</span> 5e<span class="org-builtin">-</span>3<span class="org-builtin">-</span>period_nl<span class="org-builtin">/</span>2 <span class="org-builtin">&amp;</span> stroke <span class="org-builtin">&lt;</span> 5e<span class="org-builtin">-</span>3<span class="org-builtin">+</span>period_nl<span class="org-builtin">/</span>2;
epsilon_x = finddelay(nl_errors(i_period), est_errors(i_period)) <span class="org-comment-delimiter">% </span><span class="org-comment">[m]</span>
</pre>
</div>
<pre class="example">
Estimated delay x0 is -120 [nm]
</pre>
<p>
And the period \(\lambda\) can be estimated:
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Computation of the period [m]</span>
period_fin = period_est <span class="org-builtin">*</span> (5e<span class="org-builtin">-</span>3)<span class="org-builtin">/</span>(5e<span class="org-builtin">-</span>3 <span class="org-builtin">+</span> d_offset); <span class="org-comment-delimiter">% </span><span class="org-comment">Estimated period after measurement [m]</span>
</pre>
</div>
<pre class="example">
The estimated period is 765.020 [nm]
</pre>
<p>
And the results confirms that this method is working on paper.
</p>
<p>
When doing this computation, we suppose that there are <b>at most</b> one half period of offset between the estimated and the measured non-linear (to not have any ambiguity whether the estimated period is too large or too small).
Mathematically this means that the displacement \(x_0\) should be smaller than:
</p>
\begin{equation}
x_0 < \frac{1}{2} \cdot \lambda \cdot \frac{\lambda}{\epsilon_\lambda}
\end{equation}
<p>
With \(\epsilon_\lambda\) the absolute estimation error of the period in meters.
</p>
<p>
For instance, if we estimate the error on the period to be less than 0.1nm, the maximum displacement is:
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Estimated maximum stroke [m]</span>
max_x0 = 0.5 <span class="org-builtin">*</span> 765e<span class="org-builtin">-</span>9 <span class="org-builtin">*</span> (765e<span class="org-builtin">-</span>9)<span class="org-builtin">/</span>(0.1e<span class="org-builtin">-</span>9);
</pre>
</div>
<pre class="example">
The maximum stroke is 2.9 [mm]
</pre>
</div>
</div>
<div id="outline-container-org018ecfe" class="outline-3">
<h3 id="org018ecfe"><span class="section-number-3">4.5.</span> Measurements</h3>
<div class="outline-text-3" id="text-4-5">
<p> <p>
We have some constrains on the way the motion is imposed and measured: We have some constrains on the way the motion is imposed and measured:
</p> </p>
@ -609,9 +1166,9 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).
</div> </div>
</div> </div>
<div id="outline-container-org3331f0d" class="outline-2"> <div id="outline-container-org6292dcf" class="outline-2">
<h2 id="org3331f0d">Bibliography</h2> <h2 id="org6292dcf">Bibliography</h2>
<div class="outline-text-2" id="text-org3331f0d"> <div class="outline-text-2" id="text-org6292dcf">
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body"> <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ducourtieux, S., 2018. Toward high precision position control using laser interferometry: Main sources of error. <a href="https://doi.org/10.13140/rg.2.2.21044.35205">https://doi.org/10.13140/rg.2.2.21044.35205</a></div> <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ducourtieux, S., 2018. Toward high precision position control using laser interferometry: Main sources of error. <a href="https://doi.org/10.13140/rg.2.2.21044.35205">https://doi.org/10.13140/rg.2.2.21044.35205</a></div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Thurner, K., Quacquarelli, F.P., Braun, P.-F., Dal Savio, C., Karrai, K., 2015. Fiber-based distance sensing interferometry. Applied optics 54, 3051–3063.</div> <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Thurner, K., Quacquarelli, F.P., Braun, P.-F., Dal Savio, C., Karrai, K., 2015. Fiber-based distance sensing interferometry. Applied optics 54, 3051–3063.</div>
@ -621,7 +1178,7 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2022-01-05 mer. 15:27</p> <p class="date">Created: 2022-02-15 mar. 14:25</p>
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#+LATEX_HEADER_EXTRA: \input{preamble.tex} #+LATEX_HEADER_EXTRA: \input{preamble.tex}
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<hr> <hr>
<p>This report is also available as a <a href="./dcm_metrology.pdf">pdf</a>.</p> <p>This report is also available as a <a href="./dcm-metrology.pdf">pdf</a>.</p>
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