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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2022-06-02 Thu 18:06 -->
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<!-- 2022-06-02 Thu 22:25 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>ESRF Double Crystal Monochromator - Metrology</title>
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<meta name="author" content="Dehaeze Thomas" />
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@ -39,49 +39,49 @@
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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="#org6e38443">1. Metrology Concept</a>
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<li><a href="#org5b4a8ea">1. Metrology Concept</a>
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<ul>
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<li><a href="#orga98f1e4">1.1. Sensor Topology</a></li>
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<li><a href="#org1326a32">1.2. Computation of the relative pose between first and second crystals</a></li>
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<li><a href="#orgee0936b">1.1. Sensor Topology</a></li>
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<li><a href="#orgcc7ab6e">1.2. Computation of the relative pose between first and second crystals</a></li>
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</ul>
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</li>
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<li><a href="#orgb627b0b">2. Relation Between Crystal position and X-ray measured displacement</a>
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<li><a href="#orgd1e6cf8">2. Relation Between Crystal position and X-ray measured displacement</a>
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<ul>
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<li><a href="#org50e3742">2.1. Definition of frame</a></li>
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<li><a href="#org469abf4">2.2. Effect of an error in crystal’s distance</a></li>
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<li><a href="#org0ea3b85">2.3. Effect of an error in crystal’s x parallelism</a></li>
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<li><a href="#org0766a20">2.4. Effect of an error in crystal’s y parallelism</a></li>
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<li><a href="#org2d9635c">2.5. Summary</a></li>
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<li><a href="#org65aa9e9">2.6. “Channel cut” Scan</a></li>
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<li><a href="#orgf98dcda">2.1. Definition of frame</a></li>
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<li><a href="#orgcc56e18">2.2. Effect of an error in crystal’s distance</a></li>
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<li><a href="#org4f10916">2.3. Effect of an error in crystal’s x parallelism</a></li>
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<li><a href="#org214d2fb">2.4. Effect of an error in crystal’s y parallelism</a></li>
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<li><a href="#orga01ea08">2.5. Summary</a></li>
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<li><a href="#orgbe05a5f">2.6. “Channel cut” Scan</a></li>
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</ul>
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</li>
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<li><a href="#orgb6410e2">3. Determining relative pose between the crystals using the X-ray</a>
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<li><a href="#orgd1c9f1e">3. Determining relative pose between the crystals using the X-ray</a>
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<ul>
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<li><a href="#orgba5b368">3.1. Determine the \(y\) parallelism - “Rocking Curve”</a></li>
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<li><a href="#orgdad1405">3.2. Determine the \(x\) parallelism - Bragg Scan</a></li>
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<li><a href="#orge213a17">3.3. Determine the \(z\) distance - Bragg Scan</a></li>
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<li><a href="#orgeba2e70">3.4. Use Channel cut scan to determine crystal <code>dry</code> parallelism</a></li>
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<li><a href="#orgddd3c9a">3.5. Effect of an error on Bragg angle</a></li>
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<li><a href="#orgb9a7cbf">3.1. Determine the \(y\) parallelism - “Rocking Curve”</a></li>
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<li><a href="#orgf1f155c">3.2. Determine the \(x\) parallelism - Bragg Scan</a></li>
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<li><a href="#org340a305">3.3. Determine the \(z\) distance - Bragg Scan</a></li>
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<li><a href="#org550643d">3.4. Use Channel cut scan to determine crystal <code>dry</code> parallelism</a></li>
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<li><a href="#org3f02fcd">3.5. Effect of an error on Bragg angle</a></li>
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</ul>
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</li>
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<li><a href="#org1f48d7e">4. Deformations of the Metrology Frame</a>
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<li><a href="#org4eedaad">4. Deformations of the Metrology Frame</a>
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<ul>
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<li><a href="#org5ee730d">4.1. Measurement Setup</a></li>
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<li><a href="#org78a952b">4.2. Simulations</a></li>
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<li><a href="#org053b4f5">4.3. Comparison</a></li>
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<li><a href="#org0aa4bea">4.4. Test</a></li>
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<li><a href="#orgfebab90">4.5. Measured frame deformation</a></li>
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<li><a href="#org12eb5ca">4.6. Test</a></li>
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<li><a href="#org2b6cac9">4.7. Repeatability of frame deformation</a></li>
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<li><a href="#orge197959">4.1. Measurement Setup</a></li>
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<li><a href="#org7cfa17e">4.2. Simulations</a></li>
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<li><a href="#org0c10c30">4.3. Comparison</a></li>
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<li><a href="#org25a32fa">4.4. Test</a></li>
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<li><a href="#orgb7db314">4.5. Measured frame deformation</a></li>
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<li><a href="#org5494ab9">4.6. Test</a></li>
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<li><a href="#org218f95f">4.7. Repeatability of frame deformation</a></li>
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</ul>
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</li>
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<li><a href="#org27208e8">5. Attocube - Periodic Non-Linearity</a>
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<li><a href="#org5b8904d">5. Attocube - Periodic Non-Linearity</a>
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<ul>
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<li><a href="#org62b5863">5.1. Measurement Setup</a></li>
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<li><a href="#orgbaa09c9">5.2. Choice of the reference signal</a></li>
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<li><a href="#org88808b2">5.3. Repeatability of the non-linearity</a></li>
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<li><a href="#org2dfde91">5.4. Simulation</a></li>
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<li><a href="#org1be96dc">5.5. Measurements</a></li>
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<li><a href="#org9f6a57f">5.1. Measurement Setup</a></li>
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<li><a href="#orgc06acba">5.2. Choice of the reference signal</a></li>
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<li><a href="#org26743c0">5.3. Repeatability of the non-linearity</a></li>
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<li><a href="#org15014d3">5.4. Simulation</a></li>
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<li><a href="#orga71d790">5.5. Measurements</a></li>
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</ul>
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</li>
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</ul>
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@ -93,19 +93,26 @@
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<p>
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In this document, the metrology system is studied.
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First, in Section <a href="#org883336e">1</a> the goal of the metrology system is stated and the proposed concept is described.
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First, in Section <a href="#orga8ac352">1</a> the goal of the metrology system is stated and the proposed concept is described.
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</p>
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<p>
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How the relative crystal pose is affecting the pose of the output beam is studied in Section <a href="#org4b9a066">2</a>.
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</p>
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<p>
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In order to increase the accuracy of the metrology system, two problems are to be dealt with:
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</p>
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<ul class="org-ul">
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<li>The deformation of the metrology frame under the action of gravity (Section <a href="#org8d2959d">4</a>)</li>
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<li>The periodic non-linearity of the interferometers (Section <a href="#org7046a86">5</a>)</li>
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<li>The deformation of the metrology frame under the action of gravity (Section <a href="#orgf31717e">4</a>)</li>
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<li>The periodic non-linearity of the interferometers (Section <a href="#org44bdb6d">5</a>)</li>
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</ul>
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<div id="outline-container-org6e38443" class="outline-2">
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<h2 id="org6e38443"><span class="section-number-2">1.</span> Metrology Concept</h2>
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<div id="outline-container-org5b4a8ea" class="outline-2">
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||||
<h2 id="org5b4a8ea"><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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<a id="org883336e"></a>
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<a id="orga8ac352"></a>
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</p>
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<p>
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The goal of the metrology system is to measure the distance and default of parallelism between the first and second crystals.
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@ -120,8 +127,8 @@ Only 3 degrees of freedom are of interest:
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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-orga98f1e4" class="outline-3">
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<h3 id="orga98f1e4"><span class="section-number-3">1.1.</span> Sensor Topology</h3>
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<div id="outline-container-orgee0936b" class="outline-3">
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<h3 id="orgee0936b"><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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@ -131,11 +138,11 @@ Three additional interferometers are used to measured the relative motion of the
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<p>
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In total, there are 15 interferometers represented in Figure <a href="#org1728a4b">1</a>.
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The measurements are summarized in Table <a href="#org33e1a4e">2</a>.
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In total, there are 15 interferometers represented in Figure <a href="#orgce5aaa3">1</a>.
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The measurements are summarized in Table <a href="#org943c065">2</a>.
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</p>
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<table id="org755759d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org5dd8abe" 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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@ -187,7 +194,7 @@ The measurements are summarized in Table <a href="#org33e1a4e">2</a>.
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</tbody>
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</table>
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<table id="org33e1a4e" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="org943c065" 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> List of Interferometer measurements</caption>
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<colgroup>
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@ -298,7 +305,7 @@ The measurements are summarized in Table <a href="#org33e1a4e">2</a>.
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</table>
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<div id="org1728a4b" class="figure">
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<div id="orgce5aaa3" 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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@ -306,8 +313,8 @@ The measurements are summarized in Table <a href="#org33e1a4e">2</a>.
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</div>
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</div>
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<div id="outline-container-org1326a32" class="outline-3">
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<h3 id="org1326a32"><span class="section-number-3">1.2.</span> Computation of the relative pose between first and second crystals</h3>
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||||
<div id="outline-container-orgcc7ab6e" class="outline-3">
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||||
<h3 id="orgcc7ab6e"><span class="section-number-3">1.2.</span> Computation of the relative pose between first and second crystals</h3>
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||||
<div class="outline-text-3" id="text-1-2">
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||||
<p>
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||||
To understand how the relative pose between the crystals is computed from the interferometer signals, have a look at <a href="https://gitlab.esrf.fr/dehaeze/dcm-kinematics">this repository</a> (<code>https://gitlab.esrf.fr/dehaeze/dcm-kinematics</code>).
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@ -317,7 +324,7 @@ To understand how the relative pose between the crystals is computed from the in
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Basically, Jacobian matrices are derived from the geometry and are used to convert the 15 interferometer signals to the <b>relative pose</b> of the primary and secondary crystals \([d_{h,z},\ r_{h,y},\ r_{h,x}]\) or \([d_{r,z},\ r_{r,y},\ r_{r,x}]\).
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||||
</p>
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<div class="note" id="org28ac0ce">
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<div class="note" id="org5a06b04">
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<p>
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The sign conventions for the relative crystal pose are:
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</p>
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@ -372,11 +379,11 @@ Values of the matrices can be found in the document describing the kinematics of
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</div>
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</div>
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<div id="outline-container-orgb627b0b" class="outline-2">
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<h2 id="orgb627b0b"><span class="section-number-2">2.</span> Relation Between Crystal position and X-ray measured displacement</h2>
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||||
<div id="outline-container-orgd1e6cf8" class="outline-2">
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||||
<h2 id="orgd1e6cf8"><span class="section-number-2">2.</span> Relation Between Crystal position and X-ray measured displacement</h2>
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||||
<div class="outline-text-2" id="text-2">
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||||
<p>
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<a id="orgbc74a6f"></a>
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||||
<a id="org4b9a066"></a>
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||||
</p>
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||||
<p>
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||||
In this section, the impact of an error in the relative pose between the first and second crystals on the output X-ray beam is studied.
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@ -399,8 +406,8 @@ In order to simplify the problem, the first crystal is supposed to be fixed (i.e
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In order to easily study that, “ray tracing” techniques are used.
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</p>
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||||
</div>
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<div id="outline-container-org50e3742" class="outline-3">
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||||
<h3 id="org50e3742"><span class="section-number-3">2.1.</span> Definition of frame</h3>
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||||
<div id="outline-container-orgf98dcda" class="outline-3">
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||||
<h3 id="orgf98dcda"><span class="section-number-3">2.1.</span> Definition of frame</h3>
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||||
<div class="outline-text-3" id="text-2-1">
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||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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@ -476,15 +483,15 @@ The xy position of the beam is taken in the \(x=0\) plane.
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</div>
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</div>
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<div id="outline-container-org469abf4" class="outline-3">
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<h3 id="org469abf4"><span class="section-number-3">2.2.</span> Effect of an error in crystal’s distance</h3>
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<div id="outline-container-orgcc56e18" class="outline-3">
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<h3 id="orgcc56e18"><span class="section-number-3">2.2.</span> Effect of an error in crystal’s distance</h3>
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<div class="outline-text-3" id="text-2-2">
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||||
<p>
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||||
<a id="org4eba8de"></a>
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<a id="orgfce54e9"></a>
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||||
</p>
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||||
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||||
<p>
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||||
In Figure <a href="#orgf88272f">2</a> is shown the light path for three bragg angles (5, 55 and 85 degrees) when there is an error in the <code>dz</code> position of 1mm.
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||||
In Figure <a href="#orga24d92a">2</a> is shown the light path for three bragg angles (5, 55 and 85 degrees) when there is an error in the <code>dz</code> position of 1mm.
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</p>
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<p>
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@ -492,14 +499,14 @@ Visually, it is clear that this induce a <code>z</code> offset of the output bea
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</p>
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<div id="orgf88272f" class="figure">
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||||
<div id="orga24d92a" class="figure">
|
||||
<p><img src="figs/ray_tracing_error_dz_overview.png" alt="ray_tracing_error_dz_overview.png" />
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||||
</p>
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||||
<p><span class="figure-number">Figure 2: </span>Visual Effect of an error in <code>dz</code> (1mm). Side view.</p>
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||||
</div>
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||||
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||||
<p>
|
||||
The motion of the output beam is displayed as a function of the Bragg angle in Figure <a href="#org592eac4">3</a>.
|
||||
The motion of the output beam is displayed as a function of the Bragg angle in Figure <a href="#orgf3acc2d">3</a>.
|
||||
It is clear that an error in the distance <code>dz</code> between the crystals only induce a <code>z</code> offset of the output beam.
|
||||
This offset decreases with the Bragg angle.
|
||||
</p>
|
||||
@ -512,7 +519,7 @@ This is indeed equal to:
|
||||
\end{equation}
|
||||
|
||||
|
||||
<div id="org592eac4" class="figure">
|
||||
<div id="orgf3acc2d" class="figure">
|
||||
<p><img src="figs/motion_beam_dz_error.png" alt="motion_beam_dz_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Motion of the output beam with dZ error</p>
|
||||
@ -520,29 +527,29 @@ This is indeed equal to:
|
||||
</div>
|
||||
</div>
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||||
|
||||
<div id="outline-container-org0ea3b85" class="outline-3">
|
||||
<h3 id="org0ea3b85"><span class="section-number-3">2.3.</span> Effect of an error in crystal’s x parallelism</h3>
|
||||
<div id="outline-container-org4f10916" class="outline-3">
|
||||
<h3 id="org4f10916"><span class="section-number-3">2.3.</span> Effect of an error in crystal’s x parallelism</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
<a id="orged7b509"></a>
|
||||
<a id="org10b6b9d"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The effect of an error in <code>rx</code> crystal parallelism on the output beam is visually shown in Figure <a href="#orgad85ff2">4</a> for three bragg angles (5, 55 and 85 degrees).
|
||||
The effect of an error in <code>rx</code> crystal parallelism on the output beam is visually shown in Figure <a href="#orgf3fcd25">4</a> for three bragg angles (5, 55 and 85 degrees).
|
||||
The error is set to one degree, and the top view is shown.
|
||||
It is clear that the output beam experiences some rotation around a vertical axis.
|
||||
The amount of rotation depends on the bragg angle.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgad85ff2" class="figure">
|
||||
<div id="orgf3fcd25" class="figure">
|
||||
<p><img src="figs/ray_tracing_error_drx_overview.png" alt="ray_tracing_error_drx_overview.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Visual Effect of an error in <code>drx</code> (1 degree). Top View.</p>
|
||||
</div>
|
||||
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||||
<p>
|
||||
The effect of <code>drx</code> as a function of the Bragg angle on the output beam pose is computed and shown in Figure <a href="#org785519c">5</a>.
|
||||
The effect of <code>drx</code> as a function of the Bragg angle on the output beam pose is computed and shown in Figure <a href="#org3ce23f8">5</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -561,7 +568,7 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org785519c" class="figure">
|
||||
<div id="org3ce23f8" class="figure">
|
||||
<p><img src="figs/motion_beam_drx_error.png" alt="motion_beam_drx_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Motion of the output beam with drx error</p>
|
||||
@ -569,26 +576,26 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0766a20" class="outline-3">
|
||||
<h3 id="org0766a20"><span class="section-number-3">2.4.</span> Effect of an error in crystal’s y parallelism</h3>
|
||||
<div id="outline-container-org214d2fb" class="outline-3">
|
||||
<h3 id="org214d2fb"><span class="section-number-3">2.4.</span> Effect of an error in crystal’s y parallelism</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
<a id="org5081b73"></a>
|
||||
<a id="orged8dba0"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The effect of an error in <code>ry</code> crystal parallelism on the output beam is visually shown in Figure <a href="#org1ee280e">6</a> for three bragg angles (5, 55 and 85 degrees).
|
||||
The effect of an error in <code>ry</code> crystal parallelism on the output beam is visually shown in Figure <a href="#orgb3ea9d4">6</a> for three bragg angles (5, 55 and 85 degrees).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org1ee280e" class="figure">
|
||||
<div id="orgb3ea9d4" class="figure">
|
||||
<p><img src="figs/ray_tracing_error_dry_overview.png" alt="ray_tracing_error_dry_overview.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Visual Effect of an error in <code>dry</code> (1 degree). Side view.</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The effect of <code>dry</code> as a function of the Bragg angle on the output beam pose is computed and shown in Figure <a href="#org5d3201f">7</a>.
|
||||
The effect of <code>dry</code> as a function of the Bragg angle on the output beam pose is computed and shown in Figure <a href="#org28da5fb">7</a>.
|
||||
It is clear that this induces a rotation of the output beam in the <code>y</code> direction equals to 2 times <code>dry</code>:
|
||||
</p>
|
||||
\begin{equation}
|
||||
@ -600,7 +607,7 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org5d3201f" class="figure">
|
||||
<div id="org28da5fb" class="figure">
|
||||
<p><img src="figs/motion_beam_dry_error.png" alt="motion_beam_dry_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Motion of the output beam with dry error</p>
|
||||
@ -608,16 +615,16 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2d9635c" class="outline-3">
|
||||
<h3 id="org2d9635c"><span class="section-number-3">2.5.</span> Summary</h3>
|
||||
<div id="outline-container-orga01ea08" class="outline-3">
|
||||
<h3 id="orga01ea08"><span class="section-number-3">2.5.</span> Summary</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<p>
|
||||
Effects of crystal’s pose errors on the output beam are summarized in Table <a href="#org40cfc37">3</a>.
|
||||
Effects of crystal’s pose errors on the output beam are summarized in Table <a href="#org01cc930">3</a>.
|
||||
Note that the three pose errors are well decoupled regarding their effects on the output beam.
|
||||
Also note that the effect of an error in crystal’s distance does not depend on the Bragg angle.
|
||||
</p>
|
||||
|
||||
<table id="org40cfc37" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<table id="org01cc930" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 3:</span> Summary of the effects of the errors in second crystal’s pose on the output beam</caption>
|
||||
|
||||
<colgroup>
|
||||
@ -670,30 +677,30 @@ Also note that the effect of an error in crystal’s distance does not depen
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org65aa9e9" class="outline-3">
|
||||
<h3 id="org65aa9e9"><span class="section-number-3">2.6.</span> “Channel cut” Scan</h3>
|
||||
<div id="outline-container-orgbe05a5f" class="outline-3">
|
||||
<h3 id="orgbe05a5f"><span class="section-number-3">2.6.</span> “Channel cut” Scan</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<p>
|
||||
A “channel cut” scan is a Bragg scan where the distance between the crystals is fixed.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This is visually shown in Figure <a href="#orgbde7e5a">8</a> where it is clear that the output beam experiences some vertical motion.
|
||||
This is visually shown in Figure <a href="#org278d240">8</a> where it is clear that the output beam experiences some vertical motion.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgbde7e5a" class="figure">
|
||||
<div id="org278d240" class="figure">
|
||||
<p><img src="figs/ray_tracing_channel_cut.png" alt="ray_tracing_channel_cut.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Visual Effect of a channel cut scan</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The \(z\) offset of the beam for several channel cut scans are shown in Figure <a href="#org24bc0ad">9</a>.
|
||||
The \(z\) offset of the beam for several channel cut scans are shown in Figure <a href="#org588c503">9</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org24bc0ad" class="figure">
|
||||
<div id="org588c503" class="figure">
|
||||
<p><img src="figs/channel_cut_scan.png" alt="channel_cut_scan.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>Z motion of the beam during “channel cut” scans</p>
|
||||
@ -702,8 +709,8 @@ The \(z\) offset of the beam for several channel cut scans are shown in Figure <
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb6410e2" class="outline-2">
|
||||
<h2 id="orgb6410e2"><span class="section-number-2">3.</span> Determining relative pose between the crystals using the X-ray</h2>
|
||||
<div id="outline-container-orgd1c9f1e" class="outline-2">
|
||||
<h2 id="orgd1c9f1e"><span class="section-number-2">3.</span> Determining relative pose between the crystals using the X-ray</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
As Interferometers are only measuring <i>relative</i> displacement, it is mandatory to initialize them correctly.
|
||||
@ -722,21 +729,21 @@ In order to do that, an external metrology using the x-ray is used.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgba5b368" class="outline-3">
|
||||
<h3 id="orgba5b368"><span class="section-number-3">3.1.</span> Determine the \(y\) parallelism - “Rocking Curve”</h3>
|
||||
<div id="outline-container-orgb9a7cbf" class="outline-3">
|
||||
<h3 id="orgb9a7cbf"><span class="section-number-3">3.1.</span> Determine the \(y\) parallelism - “Rocking Curve”</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdad1405" class="outline-3">
|
||||
<h3 id="orgdad1405"><span class="section-number-3">3.2.</span> Determine the \(x\) parallelism - Bragg Scan</h3>
|
||||
<div id="outline-container-orgf1f155c" class="outline-3">
|
||||
<h3 id="orgf1f155c"><span class="section-number-3">3.2.</span> Determine the \(x\) parallelism - Bragg Scan</h3>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-orge213a17" class="outline-3">
|
||||
<h3 id="orge213a17"><span class="section-number-3">3.3.</span> Determine the \(z\) distance - Bragg Scan</h3>
|
||||
<div id="outline-container-org340a305" class="outline-3">
|
||||
<h3 id="org340a305"><span class="section-number-3">3.3.</span> Determine the \(z\) distance - Bragg Scan</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgeba2e70" class="outline-3">
|
||||
<h3 id="orgeba2e70"><span class="section-number-3">3.4.</span> Use Channel cut scan to determine crystal <code>dry</code> parallelism</h3>
|
||||
<div id="outline-container-org550643d" class="outline-3">
|
||||
<h3 id="org550643d"><span class="section-number-3">3.4.</span> Use Channel cut scan to determine crystal <code>dry</code> parallelism</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
Now, let’s suppose we want to determine the <code>dry</code> angle between the crystals.
|
||||
@ -756,16 +763,16 @@ The error is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgddd3c9a" class="outline-3">
|
||||
<h3 id="orgddd3c9a"><span class="section-number-3">3.5.</span> Effect of an error on Bragg angle</h3>
|
||||
<div id="outline-container-org3f02fcd" class="outline-3">
|
||||
<h3 id="org3f02fcd"><span class="section-number-3">3.5.</span> Effect of an error on Bragg angle</h3>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1f48d7e" class="outline-2">
|
||||
<h2 id="org1f48d7e"><span class="section-number-2">4.</span> Deformations of the Metrology Frame</h2>
|
||||
<div id="outline-container-org4eedaad" class="outline-2">
|
||||
<h2 id="org4eedaad"><span class="section-number-2">4.</span> Deformations of the Metrology Frame</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org8d2959d"></a>
|
||||
<a id="orgf31717e"></a>
|
||||
</p>
|
||||
<p>
|
||||
The transformation matrices are valid only if the metrology frames are solid bodies.
|
||||
@ -780,8 +787,8 @@ When the bragg axis is scanned, the effect of gravity on the metrology frame is
|
||||
This can be calibrated.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org5ee730d" class="outline-3">
|
||||
<h3 id="org5ee730d"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
|
||||
<div id="outline-container-orge197959" class="outline-3">
|
||||
<h3 id="orge197959"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
Two beam viewers:
|
||||
@ -798,7 +805,7 @@ This position is the wanted position for a given Bragg angle.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org3be0478" class="figure">
|
||||
<div id="orgec71cad" class="figure">
|
||||
<p><img src="figs/calibration_setup.png" alt="calibration_setup.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Schematic of the setup</p>
|
||||
@ -828,8 +835,8 @@ Frame rate is: 42 fps
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org78a952b" class="outline-3">
|
||||
<h3 id="org78a952b"><span class="section-number-3">4.2.</span> Simulations</h3>
|
||||
<div id="outline-container-org7cfa17e" class="outline-3">
|
||||
<h3 id="org7cfa17e"><span class="section-number-3">4.2.</span> Simulations</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
The deformations of the metrology frame and therefore the expected interferometric measurements can be computed as a function of the Bragg angle.
|
||||
@ -838,12 +845,12 @@ This may be done using FE software.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org053b4f5" class="outline-3">
|
||||
<h3 id="org053b4f5"><span class="section-number-3">4.3.</span> Comparison</h3>
|
||||
<div id="outline-container-org0c10c30" class="outline-3">
|
||||
<h3 id="org0c10c30"><span class="section-number-3">4.3.</span> Comparison</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0aa4bea" class="outline-3">
|
||||
<h3 id="org0aa4bea"><span class="section-number-3">4.4.</span> Test</h3>
|
||||
<div id="outline-container-org25a32fa" class="outline-3">
|
||||
<h3 id="org25a32fa"><span class="section-number-3">4.4.</span> Test</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">aa = importdata(<span class="org-string">"correctInterf-vlm-220201.dat"</span>);
|
||||
@ -858,8 +865,8 @@ This may be done using FE software.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfebab90" class="outline-3">
|
||||
<h3 id="orgfebab90"><span class="section-number-3">4.5.</span> Measured frame deformation</h3>
|
||||
<div id="outline-container-orgb7db314" class="outline-3">
|
||||
<h3 id="orgb7db314"><span class="section-number-3">4.5.</span> Measured frame deformation</h3>
|
||||
<div class="outline-text-3" id="text-4-5">
|
||||
<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));
|
||||
@ -887,7 +894,7 @@ ry1 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>dat
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org73f1693" class="figure">
|
||||
<div id="org6aebdab" class="figure">
|
||||
<p><img src="figs/calibration_drx_pres.png" alt="calibration_drx_pres.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>description</p>
|
||||
@ -960,8 +967,8 @@ f_ry1 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">p
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org12eb5ca" class="outline-3">
|
||||
<h3 id="org12eb5ca"><span class="section-number-3">4.6.</span> Test</h3>
|
||||
<div id="outline-container-org5494ab9" class="outline-3">
|
||||
<h3 id="org5494ab9"><span class="section-number-3">4.6.</span> Test</h3>
|
||||
<div class="outline-text-3" id="text-4-6">
|
||||
<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>;
|
||||
@ -993,7 +1000,7 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgf09ae2a" class="figure">
|
||||
<div id="org85bdc09" class="figure">
|
||||
<p><img src="figs/drifts_xtal2_detrend.png" alt="drifts_xtal2_detrend.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Drifts of the second crystal as a function of Bragg Angle</p>
|
||||
@ -1010,8 +1017,8 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2b6cac9" class="outline-3">
|
||||
<h3 id="org2b6cac9"><span class="section-number-3">4.7.</span> Repeatability of frame deformation</h3>
|
||||
<div id="outline-container-org218f95f" class="outline-3">
|
||||
<h3 id="org218f95f"><span class="section-number-3">4.7.</span> Repeatability of frame deformation</h3>
|
||||
<div class="outline-text-3" id="text-4-7">
|
||||
<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>;
|
||||
@ -1061,11 +1068,11 @@ data_2.dz = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*<
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org27208e8" class="outline-2">
|
||||
<h2 id="org27208e8"><span class="section-number-2">5.</span> Attocube - Periodic Non-Linearity</h2>
|
||||
<div id="outline-container-org5b8904d" class="outline-2">
|
||||
<h2 id="org5b8904d"><span class="section-number-2">5.</span> Attocube - Periodic Non-Linearity</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="org7046a86"></a>
|
||||
<a id="org44bdb6d"></a>
|
||||
</p>
|
||||
<p>
|
||||
Interferometers have some periodic nonlinearity (NO_ITEM_DATA:thurner15_fiber_based_distan_sensin_inter).
|
||||
@ -1086,8 +1093,8 @@ This process is performed over several periods in order to characterize the erro
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org62b5863" class="outline-3">
|
||||
<h3 id="org62b5863"><span class="section-number-3">5.1.</span> Measurement Setup</h3>
|
||||
<div id="outline-container-org9f6a57f" class="outline-3">
|
||||
<h3 id="org9f6a57f"><span class="section-number-3">5.1.</span> Measurement Setup</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<p>
|
||||
The metrology that will be compared with the interferometers are the strain gauges incorporated in the PI piezoelectric stacks.
|
||||
@ -1098,7 +1105,7 @@ It is here supposed that the measured displacement by the strain gauges are conv
|
||||
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="org95ffe92">
|
||||
<div class="note" id="org3574ce6">
|
||||
<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>
|
||||
@ -1114,7 +1121,7 @@ As any deformations of the metrology frame of deformation of the crystal’s
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The setup is schematically with the block diagram in Figure <a href="#org6126f2e">13</a>.
|
||||
The setup is schematically with the block diagram in Figure <a href="#orgf2d914c">13</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1130,7 +1137,7 @@ The PI controller takes care or controlling to position as measured by the strai
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org6126f2e" class="figure">
|
||||
<div id="orgf2d914c" class="figure">
|
||||
<p><img src="figs/block_diagram_lut_attocube.png" alt="block_diagram_lut_attocube.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube</p>
|
||||
@ -1142,8 +1149,8 @@ The problem is to estimate the periodic non-linearity of the Attocube from the i
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbaa09c9" class="outline-3">
|
||||
<h3 id="orgbaa09c9"><span class="section-number-3">5.2.</span> Choice of the reference signal</h3>
|
||||
<div id="outline-container-orgc06acba" class="outline-3">
|
||||
<h3 id="orgc06acba"><span class="section-number-3">5.2.</span> Choice of the reference signal</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
The main specifications for the reference signal are;
|
||||
@ -1167,8 +1174,8 @@ Based on the above discussion, one suitable excitation signal is a sinusoidal sw
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org88808b2" class="outline-3">
|
||||
<h3 id="org88808b2"><span class="section-number-3">5.3.</span> Repeatability of the non-linearity</h3>
|
||||
<div id="outline-container-org26743c0" class="outline-3">
|
||||
<h3 id="org26743c0"><span class="section-number-3">5.3.</span> Repeatability of the non-linearity</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<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.
|
||||
@ -1188,8 +1195,8 @@ One way to precisely estimate the laser wavelength is to estimate the non linear
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2dfde91" class="outline-3">
|
||||
<h3 id="org2dfde91"><span class="section-number-3">5.4.</span> Simulation</h3>
|
||||
<div id="outline-container-org15014d3" class="outline-3">
|
||||
<h3 id="org15014d3"><span class="section-number-3">5.4.</span> Simulation</h3>
|
||||
<div class="outline-text-3" id="text-5-4">
|
||||
<p>
|
||||
Suppose we have a first approximation of the non-linear period.
|
||||
@ -1209,10 +1216,10 @@ period_nl = period_est <span class="org-builtin">+</span> period_err; <span clas
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The non-linear errors are first estimated at the beginning of the stroke (Figure <a href="#org0c7bf9b">14</a>).
|
||||
The non-linear errors are first estimated at the beginning of the stroke (Figure <a href="#org94b9790">14</a>).
|
||||
</p>
|
||||
|
||||
<div id="org0c7bf9b" class="figure">
|
||||
<div id="org94b9790" 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 14: </span>Estimation of the non-linear errors at the beginning of the stroke</p>
|
||||
@ -1220,7 +1227,7 @@ The non-linear errors are first estimated at the beginning of the stroke (Figure
|
||||
|
||||
<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="#org9c29b56">15</a>).
|
||||
To do so, the same measurement is performed with a stroke of several millimeters (Figure <a href="#org8301473">15</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1229,7 +1236,7 @@ This is due to a mismatch between the estimated period and the true period of th
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org9c29b56" class="figure">
|
||||
<div id="org8301473" 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 15: </span>Estimated non-linear errors at a latter position</p>
|
||||
@ -1262,7 +1269,7 @@ with \(\lambda_{\text{est}}\) the estimated error’s period.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From Figure <a href="#org9c29b56">15</a>, we can see that there is an offset between the two curves.
|
||||
From Figure <a href="#org8301473">15</a>, we can see that there is an offset between the two curves.
|
||||
Let’s call this offset \(\epsilon_x\), we then have:
|
||||
</p>
|
||||
\begin{equation}
|
||||
@ -1342,8 +1349,8 @@ The maximum stroke is 2.9 [mm]
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1be96dc" class="outline-3">
|
||||
<h3 id="org1be96dc"><span class="section-number-3">5.5.</span> Measurements</h3>
|
||||
<div id="outline-container-orga71d790" class="outline-3">
|
||||
<h3 id="orga71d790"><span class="section-number-3">5.5.</span> Measurements</h3>
|
||||
<div class="outline-text-3" id="text-5-5">
|
||||
<p>
|
||||
We have some constrains on the way the motion is imposed and measured:
|
||||
@ -1372,7 +1379,7 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2022-06-02 Thu 18:06</p>
|
||||
<p class="date">Created: 2022-06-02 Thu 22:25</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -54,6 +54,9 @@
|
||||
|
||||
In this document, the metrology system is studied.
|
||||
First, in Section [[sec:metrology_concept]] the goal of the metrology system is stated and the proposed concept is described.
|
||||
|
||||
How the relative crystal pose is affecting the pose of the output beam is studied in Section [[sec:relation_crystal_xray]].
|
||||
|
||||
In order to increase the accuracy of the metrology system, two problems are to be dealt with:
|
||||
- The deformation of the metrology frame under the action of gravity (Section [[sec:frame_deformations]])
|
||||
- The periodic non-linearity of the interferometers (Section [[sec:dcm_attocube_lut]])
|
||||
|
||||
Loading…
Reference in New Issue
Block a user