593 lines
19 KiB
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593 lines
19 KiB
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<title>ESRF Double Crystal Monochromator - Metrology</title>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content" class="content">
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<h1 class="title">ESRF Double Crystal Monochromator - Metrology</h1>
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<div id="table-of-contents" role="doc-toc">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents" role="doc-toc">
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<ul>
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<li><a href="#org5fb9a5d">1. Metrology Concept</a>
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<ul>
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<li><a href="#orgdfefe6f">1.1. Sensor Topology</a></li>
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<li><a href="#orgcd986be">1.2. Crystal’s motion computation</a></li>
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</ul>
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</li>
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<li><a href="#org1756962">2. Deformations of the Metrology Frame</a>
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<ul>
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<li><a href="#org7ca31a3">2.1. Measurement Setup</a></li>
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<li><a href="#org7a8ddda">2.2. Simulations</a></li>
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<li><a href="#org29e69e6">2.3. Comparison</a></li>
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</ul>
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</li>
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<li><a href="#orgc6a0286">3. Attocube - Periodic Non-Linearity</a>
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<ul>
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<li><a href="#orgd22a8f7">3.1. Simulations</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<hr>
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<p>This report is also available as a <a href="./dcm_metrology.pdf">pdf</a>.</p>
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<hr>
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<div id="outline-container-org5fb9a5d" class="outline-2">
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<h2 id="org5fb9a5d"><span class="section-number-2">1.</span> Metrology Concept</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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The goal of the metrology system is to measure the distance and default of parallelism orientation between the first and second crystals
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</p>
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<p>
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Only 3 degrees of freedom are of interest:
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</p>
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<ul class="org-ul">
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<li>\(d_z\)</li>
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<li>\(r_y\)</li>
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<li>\(r_x\)</li>
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</ul>
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</div>
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<div id="outline-container-orgdfefe6f" class="outline-3">
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<h3 id="orgdfefe6f"><span class="section-number-3">1.1.</span> Sensor Topology</h3>
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<div class="outline-text-3" id="text-1-1">
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<p>
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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.
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Three interferometers are used to measured the 3dof of interest for each crystals.
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Three additional interferometers are used to measured the relative motion of the metrology frame.
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</p>
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<table id="org63eb22c" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 1:</span> Notations for the metrology frame</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left">Notation</th>
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<th scope="col" class="org-left">Meaning</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left"><code>d</code></td>
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<td class="org-left">“Downstream”: Positive X</td>
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</tr>
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<tr>
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<td class="org-left"><code>u</code></td>
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<td class="org-left">“Upstream”: Negative X</td>
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</tr>
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<tr>
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<td class="org-left"><code>h</code></td>
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<td class="org-left">“Hall”: Positive Y</td>
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</tr>
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<tr>
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<td class="org-left"><code>r</code></td>
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<td class="org-left">“Ring”: Negative Y</td>
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</tr>
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<tr>
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<td class="org-left"><code>f</code></td>
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<td class="org-left">“Frame”</td>
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</tr>
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<tr>
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<td class="org-left"><code>1</code></td>
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<td class="org-left">“First Crystals”</td>
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</tr>
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<tr>
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<td class="org-left"><code>2</code></td>
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<td class="org-left">“Second Crystals”</td>
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</tr>
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</tbody>
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</table>
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<div id="org830789b" class="figure">
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<p><img src="figs/metrology_schematic.png" alt="metrology_schematic.png" />
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</p>
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<p><span class="figure-number">Figure 1: </span>Schematic of the Metrology System</p>
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</div>
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</div>
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</div>
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<div id="outline-container-orgcd986be" class="outline-3">
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<h3 id="orgcd986be"><span class="section-number-3">1.2.</span> Crystal’s motion computation</h3>
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<div class="outline-text-3" id="text-1-2">
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<p>
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From the raw interferometric measurements, the pose between the first and second crystals can be computed.
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</p>
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<p>
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First, Jacobian matrices can be used to convert raw interferometer measurements to axial displacement and orientation of the crystals and metrology frame.
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</p>
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<p>
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For the 311 crystals:
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</p>
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<table id="org5eec403" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 2:</span> Table caption</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left">Notation</th>
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<th scope="col" class="org-left">Description</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left"><code>um</code></td>
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<td class="org-left">Metrology Frame - Upstream</td>
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</tr>
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<tr>
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<td class="org-left"><code>dhm</code></td>
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<td class="org-left">Metrology Frame - Downstream Hall</td>
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</tr>
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<tr>
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<td class="org-left"><code>drm</code></td>
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<td class="org-left">Metrology Frame - Downstream Ring</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left"><code>ur1</code></td>
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<td class="org-left">First Crystal - Upstream Ring</td>
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</tr>
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<tr>
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<td class="org-left"><code>h1</code></td>
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<td class="org-left">First Crystal - Hall</td>
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</tr>
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<tr>
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<td class="org-left"><code>dr1</code></td>
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<td class="org-left">First Crystal - Downstream Ring</td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left"><code>ur2</code></td>
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<td class="org-left">First Crystal - Upstream Ring</td>
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</tr>
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<tr>
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<td class="org-left"><code>h2</code></td>
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<td class="org-left">First Crystal - Hall</td>
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</tr>
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<tr>
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<td class="org-left"><code>dr2</code></td>
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<td class="org-left">First Crystal - Downstream Ring</td>
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</tr>
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</tbody>
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</table>
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<table id="org9cbd4a3" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<caption class="t-above"><span class="table-number">Table 3:</span> Table caption</caption>
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<colgroup>
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<col class="org-left" />
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<col class="org-left" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-left">Notation</th>
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<th scope="col" class="org-left">Description</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-left"><code>dzm</code></td>
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<td class="org-left">Positive: increase of distance</td>
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</tr>
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<tr>
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<td class="org-left"><code>rym</code></td>
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<td class="org-left"> </td>
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</tr>
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<tr>
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<td class="org-left"><code>rxm</code></td>
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<td class="org-left"> </td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left"><code>dz1</code></td>
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<td class="org-left">Positive: decrease of distance</td>
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</tr>
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<tr>
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<td class="org-left"><code>ry1</code></td>
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<td class="org-left"> </td>
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</tr>
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<tr>
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<td class="org-left"><code>rx1</code></td>
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<td class="org-left"> </td>
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</tr>
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</tbody>
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<tbody>
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<tr>
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<td class="org-left"><code>dz2</code></td>
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<td class="org-left">Positive: increase of distance</td>
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</tr>
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<tr>
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<td class="org-left"><code>ry2</code></td>
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<td class="org-left"> </td>
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</tr>
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<tr>
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<td class="org-left"><code>rx2</code></td>
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<td class="org-left"> </td>
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</tr>
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</tbody>
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</table>
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<div id="org211c039" class="figure">
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<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_m.png" alt="schematic_sensor_jacobian_forward_kinematics_m.png" />
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</p>
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<p><span class="figure-number">Figure 2: </span>Forward Kinematics for the Metrology frame</p>
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</div>
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<div id="org567b707" class="figure">
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<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_1.png" alt="schematic_sensor_jacobian_forward_kinematics_1.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Forward Kinematics for the 1st crystal</p>
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</div>
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<div id="orgb30fa3c" class="figure">
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<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_2.png" alt="schematic_sensor_jacobian_forward_kinematics_2.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Forward Kinematics for the 2nd crystal</p>
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</div>
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<p>
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Then, the displacement and orientations can be combined as follows:
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</p>
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\begin{align}
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d_{z} &= + d_{z1} - d_{z2} + d_{zm} \\
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d_{r_y} &= - r_{y1} + r_{y2} - r_{ym} \\
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d_{r_x} &= - r_{x1} + r_{x2} - r_{xm}
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\end{align}
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<p>
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Therefore:
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</p>
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<ul class="org-ul">
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<li>\(d_z\) represents the distance between the two crystals</li>
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<li>\(d_{r_y}\) represents the rotation of the second crystal w.r.t. the first crystal around \(y\) axis</li>
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<li>\(d_{r_x}\) represents the rotation of the second crystal w.r.t. the first crystal around \(x\) axis</li>
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</ul>
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<p>
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If \(d_{r_y}\) is positive, the second crystal has a positive rotation around \(y\) w.r.t. the first crystal.
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Therefore, the second crystal should be actuated such that it is making a negative rotation around \(y\) w.r.t. metrology frame.
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</p>
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<p>
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The Jacobian matrices are defined as follow:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for the metrology frame</span>
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J_m = [1, 0.102, 0
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1, <span class="org-builtin">-</span>0.088, 0.1275
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|
|
1, <span class="org-builtin">-</span>0.088, <span class="org-builtin">-</span>0.1275];
|
||
|
|
|
||
|
|
<span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 1st "111" crystal</span>
|
||
|
|
J_s_111_1 = [<span class="org-builtin">-</span>1, <span class="org-builtin">-</span>0.036, <span class="org-builtin">-</span>0.015
|
||
|
|
<span class="org-builtin">-</span>1, 0, 0.015
|
||
|
|
<span class="org-builtin">-</span>1, 0.036, <span class="org-builtin">-</span>0.015];
|
||
|
|
|
||
|
|
<span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 2nd "111" crystal</span>
|
||
|
|
J_s_111_2 = [1, 0.07, 0.015
|
||
|
|
1, 0, <span class="org-builtin">-</span>0.015
|
||
|
|
1, <span class="org-builtin">-</span>0.07, 0.015];
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Therefore, the matrix that gives the relative pose of the crystal from the 9 interferometers is:
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Compute the transformation matrix</span>
|
||
|
|
G_111_t = [<span class="org-builtin">-</span>inv(J_s_111_1), inv(J_s_111_2), <span class="org-builtin">-</span>inv(J_m)];
|
||
|
|
|
||
|
|
<span class="org-comment-delimiter">% </span><span class="org-comment">Sign convention for the axial motion</span>
|
||
|
|
G_111_t(1,<span class="org-builtin">:</span>) = <span class="org-builtin">-</span>G_111_t(1,<span class="org-builtin">:</span>);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<table id="orgd7b23bf" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||
|
|
<caption class="t-above"><span class="table-number">Table 4:</span> Transformation Matrix</caption>
|
||
|
|
|
||
|
|
<colgroup>
|
||
|
|
<col class="org-left" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
|
||
|
|
<col class="org-right" />
|
||
|
|
</colgroup>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th scope="col" class="org-left"> </th>
|
||
|
|
<th scope="col" class="org-right"><code>ur1</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>h1</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>dr1</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>ur2</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>h2</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>dr1</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>um</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>dhm</code> [nm]</th>
|
||
|
|
<th scope="col" class="org-right"><code>drm</code> [nm]</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><code>dz</code> [nm]</td>
|
||
|
|
<td class="org-right">-0.25</td>
|
||
|
|
<td class="org-right">-0.5</td>
|
||
|
|
<td class="org-right">-0.25</td>
|
||
|
|
<td class="org-right">-0.25</td>
|
||
|
|
<td class="org-right">-0.5</td>
|
||
|
|
<td class="org-right">-0.25</td>
|
||
|
|
<td class="org-right">0.463</td>
|
||
|
|
<td class="org-right">0.268</td>
|
||
|
|
<td class="org-right">0.268</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><code>rx</code> [nrad]</td>
|
||
|
|
<td class="org-right">13.889</td>
|
||
|
|
<td class="org-right">0.0</td>
|
||
|
|
<td class="org-right">-13.889</td>
|
||
|
|
<td class="org-right">7.143</td>
|
||
|
|
<td class="org-right">0.0</td>
|
||
|
|
<td class="org-right">-7.143</td>
|
||
|
|
<td class="org-right">-5.263</td>
|
||
|
|
<td class="org-right">2.632</td>
|
||
|
|
<td class="org-right">2.632</td>
|
||
|
|
</tr>
|
||
|
|
|
||
|
|
<tr>
|
||
|
|
<td class="org-left"><code>ry</code> [nrad]</td>
|
||
|
|
<td class="org-right">16.667</td>
|
||
|
|
<td class="org-right">-33.333</td>
|
||
|
|
<td class="org-right">16.667</td>
|
||
|
|
<td class="org-right">16.667</td>
|
||
|
|
<td class="org-right">-33.333</td>
|
||
|
|
<td class="org-right">16.667</td>
|
||
|
|
<td class="org-right">0.0</td>
|
||
|
|
<td class="org-right">-3.922</td>
|
||
|
|
<td class="org-right">3.922</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
From table <a href="#orgb30fa3c">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>.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org1756962" class="outline-2">
|
||
|
|
<h2 id="org1756962"><span class="section-number-2">2.</span> Deformations of the Metrology Frame</h2>
|
||
|
|
<div class="outline-text-2" id="text-2">
|
||
|
|
<p>
|
||
|
|
The transformation matrix in Table <a href="#orgb30fa3c">4</a> is valid only if the metrology frames are solid bodies.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The metrology frame itself is experiencing some deformations due to the gravity.
|
||
|
|
When the bragg axis is scanned, the effect of gravity on the metrology frame is changing and this introduce some measurement errors.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
This can be calibrated.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org7ca31a3" class="outline-3">
|
||
|
|
<h3 id="org7ca31a3"><span class="section-number-3">2.1.</span> Measurement Setup</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-1">
|
||
|
|
<p>
|
||
|
|
Two beam viewers:
|
||
|
|
</p>
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li>one close to the DCM to measure position of the beam</li>
|
||
|
|
<li>one far away to the DCM to measure orientation of the beam</li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
For each Bragg angle, the Fast Jacks are actuated to that the beam is at the center of the beam viewer.
|
||
|
|
Then, then position of the crystals as measured by the interferometers is recorded.
|
||
|
|
This position is the wanted position for a given Bragg angle.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org7a8ddda" class="outline-3">
|
||
|
|
<h3 id="org7a8ddda"><span class="section-number-3">2.2.</span> Simulations</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-2">
|
||
|
|
<p>
|
||
|
|
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.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org29e69e6" class="outline-3">
|
||
|
|
<h3 id="org29e69e6"><span class="section-number-3">2.3.</span> Comparison</h3>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgc6a0286" class="outline-2">
|
||
|
|
<h2 id="orgc6a0286"><span class="section-number-2">3.</span> Attocube - Periodic Non-Linearity</h2>
|
||
|
|
<div class="outline-text-2" id="text-3">
|
||
|
|
<p>
|
||
|
|
<a id="org4b25b3a"></a>
|
||
|
|
</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.
|
||
|
|
This should be performed over several periods in order to characterize the error.
|
||
|
|
</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).
|
||
|
|
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.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The setup is schematically with the block diagram in Figure <a href="#org2796bae">5</a>.
|
||
|
|
The signals are:
|
||
|
|
</p>
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li>\(u\): Actuator Signal (position where we wish to go)</li>
|
||
|
|
<li>\(d\): Disturbances affecting the signal</li>
|
||
|
|
<li>\(y\): Displacement of the crystal</li>
|
||
|
|
<li>\(y_g\): Measurement of the crystal motion 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>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org2796bae" class="figure">
|
||
|
|
<p><img src="figs/block_diagram_lut_attocube.png" alt="block_diagram_lut_attocube.png" />
|
||
|
|
</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>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The problem is to estimate the periodic non-linearity of the Attocube from the imperfect measurements \(y_a\) and \(y_g\).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The wavelength of the Attocube is 1530nm, therefore the non-linearity has a period of 765nm.
|
||
|
|
The amplitude of the non-linearity can vary from one unit to the other (and maybe from one experimental condition to the other).
|
||
|
|
It is typically between 5nm peak to peak and 20nm peak to peak.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-orgd22a8f7" class="outline-3">
|
||
|
|
<h3 id="orgd22a8f7"><span class="section-number-3">3.1.</span> Simulations</h3>
|
||
|
|
<div class="outline-text-3" id="text-3-1">
|
||
|
|
<p>
|
||
|
|
We have some constrains on the way the motion is imposed and measured:
|
||
|
|
</p>
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li>We want the frequency content of the imposed motion to be at low frequency in order not to induce vibrations of the structure.
|
||
|
|
We have to make sure the forces applied by the piezoelectric actuator only moves the crystal and not the fast jack below.
|
||
|
|
Therefore, we have to move much slower than the first resonance frequency in the system.</li>
|
||
|
|
<li>As both \(y_a\) and \(y_g\) should have rather small noise, we have to filter them with low pass filters.
|
||
|
|
The cut-off frequency of the low pass filter should be high as compared to the motion (to not induce any distortion) but still reducing sufficiently the noise.
|
||
|
|
Let’s say we want the noise to be less than 1nm (\(6 \sigma\)).</li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<ul class="org-ul">
|
||
|
|
<li class="off"><code>[ ]</code> Take the PSD of the Attocube</li>
|
||
|
|
<li class="off"><code>[ ]</code> Take the PSD of the strain gauge</li>
|
||
|
|
<li class="off"><code>[ ]</code> Using 2nd order low pass filter, estimate the required low pass filter cut-off frequency to have sufficiently low noise</li>
|
||
|
|
</ul>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<div id="postamble" class="status">
|
||
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
||
|
|
<p class="date">Created: 2022-01-05 mer. 10:28</p>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|