Update incorrect formula
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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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<!-- 2022-06-07 Tue 10:58 -->
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<!-- 2022-06-22 Wed 13:29 -->
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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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@ -39,52 +39,52 @@
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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="#org2c2c62b">1. Metrology Concept</a>
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<li><a href="#org12e0a2f">1. Metrology Concept</a>
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<ul>
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<li><a href="#org3fa59ba">1.1. Sensor Topology</a></li>
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<li><a href="#orgc5b3b8f">1.2. Computation of the relative pose between first and second crystals</a></li>
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<li><a href="#orgd6943c0">1.1. Sensor Topology</a></li>
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<li><a href="#orgbea7abe">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="#orgcf5cfb5">2. Relation Between Crystal position and X-ray measured displacement</a>
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<li><a href="#org4ff5766">2. Relation Between Crystal position and X-ray measured displacement</a>
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<ul>
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<li><a href="#org06f9ee5">2.1. Definition of frame</a></li>
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<li><a href="#org4f17ec2">2.2. Effect of an error in crystal’s distance</a></li>
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<li><a href="#orgc5a82df">2.3. Effect of an error in crystal’s x parallelism</a></li>
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<li><a href="#org7a64dc0">2.4. Effect of an error in crystal’s y parallelism</a></li>
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<li><a href="#org60c73bc">2.5. Summary</a></li>
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<li><a href="#orgd3145d9">2.6. “Channel cut” Scan</a></li>
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<li><a href="#org7c7d2db">2.1. Definition of frame</a></li>
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<li><a href="#org67fb689">2.2. Effect of an error in crystal’s distance</a></li>
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<li><a href="#orgf515721">2.3. Effect of an error in crystal’s x parallelism</a></li>
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<li><a href="#org2fd3524">2.4. Effect of an error in crystal’s y parallelism</a></li>
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<li><a href="#org4ac158b">2.5. Summary</a></li>
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<li><a href="#org86a50e2">2.6. “Channel cut” Scan</a></li>
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</ul>
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</li>
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<li><a href="#org11191b2">3. Determining relative pose between the crystals using the X-ray</a>
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<li><a href="#orgc33f680">3. Determining relative pose between the crystals using the X-ray</a>
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<ul>
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<li><a href="#org0854b62">3.1. Determine the \(y\) parallelism - “Rocking Curve”</a></li>
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<li><a href="#org79c2679">3.2. Determine the \(x\) parallelism - Bragg Scan</a></li>
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<li><a href="#org15cd655">3.3. Determine the \(z\) distance - Bragg Scan</a></li>
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<li><a href="#orgb9628d9">3.4. Use Channel cut scan to determine crystal <code>dry</code> parallelism</a></li>
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<li><a href="#orgf6270d8">3.5. Effect of an error on Bragg angle</a></li>
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<li><a href="#org97fbcd2">3.1. Determine the \(y\) parallelism - “Rocking Curve”</a></li>
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<li><a href="#orga0df616">3.2. Determine the \(x\) parallelism - Bragg Scan</a></li>
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<li><a href="#org01554e4">3.3. Determine the \(z\) distance - Bragg Scan</a></li>
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<li><a href="#org9347512">3.4. Use Channel cut scan to determine crystal <code>dry</code> parallelism</a></li>
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<li><a href="#org7a140bf">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="#orgb8e523b">4. Deformations of the Metrology Frame</a>
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<li><a href="#orgf968de4">4. Deformations of the Metrology Frame</a>
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<ul>
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<li><a href="#org8391240">4.1. Measurement Setup</a></li>
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<li><a href="#org0a584b6">4.2. Simulations</a></li>
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<li><a href="#org0820684">4.3. Comparison</a></li>
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<li><a href="#org91e6317">4.4. Test</a></li>
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<li><a href="#org5ca0ac8">4.5. Measured frame deformation</a></li>
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<li><a href="#orgb94c7fb">4.6. Test</a></li>
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<li><a href="#org1c4abff">4.7. Repeatability of frame deformation</a></li>
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<li><a href="#org573b7e7">4.1. Measurement Setup</a></li>
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<li><a href="#orgbabe313">4.2. Simulations</a></li>
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<li><a href="#orgaf7afd6">4.3. Comparison</a></li>
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<li><a href="#org99a64a4">4.4. Test</a></li>
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<li><a href="#org907c706">4.5. Measured frame deformation</a></li>
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<li><a href="#orgaa71cc1">4.6. Test</a></li>
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<li><a href="#org73400be">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="#orgcf35007">5. Attocube - Periodic Non-Linearity</a>
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<li><a href="#org0b40d1b">5. Attocube - Periodic Non-Linearity</a>
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<ul>
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<li><a href="#org91573e9">5.1. Measurement Setup</a></li>
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<li><a href="#orgdd79356">5.2. Choice of the reference signal</a></li>
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<li><a href="#org2268c53">5.3. Repeatability of the non-linearity</a></li>
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<li><a href="#orgadef69b">5.4. Simulation</a></li>
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<li><a href="#org8ddf260">5.5. Measurements</a></li>
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<li><a href="#orge0a38ec">5.1. Measurement Setup</a></li>
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<li><a href="#org821469c">5.2. Choice of the reference signal</a></li>
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<li><a href="#org3263c1f">5.3. Repeatability of the non-linearity</a></li>
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<li><a href="#org39fabc8">5.4. Simulation</a></li>
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<li><a href="#org373814c">5.5. Measurements</a></li>
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</ul>
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</li>
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<li><a href="#org515d8fb">Bibliography</a></li>
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<li><a href="#orga081f50">Bibliography</a></li>
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</ul>
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</div>
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</div>
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@ -94,26 +94,26 @@
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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="#orga1bc36b">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="#org8b7cd3e">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="#org2865464">2</a>.
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How the relative crystal pose is affecting the pose of the output beam is studied in Section <a href="#orga416efe">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="#org45af7a1">4</a>)</li>
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<li>The periodic non-linearity of the interferometers (Section <a href="#org455a211">5</a>)</li>
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<li>The deformation of the metrology frame under the action of gravity (Section <a href="#org6019b13">4</a>)</li>
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<li>The periodic non-linearity of the interferometers (Section <a href="#org2a9ae01">5</a>)</li>
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</ul>
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<div id="outline-container-org2c2c62b" class="outline-2">
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<h2 id="org2c2c62b"><span class="section-number-2">1.</span> Metrology Concept</h2>
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<div id="outline-container-org12e0a2f" class="outline-2">
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<h2 id="org12e0a2f"><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="orga1bc36b"></a>
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<a id="org8b7cd3e"></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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@ -128,8 +128,8 @@ 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-org3fa59ba" class="outline-3">
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<h3 id="org3fa59ba"><span class="section-number-3">1.1.</span> Sensor Topology</h3>
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<div id="outline-container-orgd6943c0" class="outline-3">
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<h3 id="orgd6943c0"><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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@ -139,11 +139,11 @@ 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="#org0ba932c">1</a>.
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The measurements are summarized in Table <a href="#org9256fe4">2</a>.
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In total, there are 15 interferometers represented in Figure <a href="#org8523680">1</a>.
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The measurements are summarized in Table <a href="#orga3424d2">2</a>.
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</p>
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<table id="orgf3e6978" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orga1dc5ef" 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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@ -195,7 +195,7 @@ The measurements are summarized in Table <a href="#org9256fe4">2</a>.
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</tbody>
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</table>
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<table id="org9256fe4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<table id="orga3424d2" 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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@ -306,7 +306,7 @@ The measurements are summarized in Table <a href="#org9256fe4">2</a>.
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</table>
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<div id="org0ba932c" class="figure">
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<div id="org8523680" 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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@ -314,8 +314,8 @@ The measurements are summarized in Table <a href="#org9256fe4">2</a>.
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</div>
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</div>
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<div id="outline-container-orgc5b3b8f" class="outline-3">
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<h3 id="orgc5b3b8f"><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-orgbea7abe" class="outline-3">
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<h3 id="orgbea7abe"><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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@ -325,7 +325,7 @@ 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="orgc0eb804">
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<div class="note" id="org186878e">
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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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@ -380,11 +380,11 @@ 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-orgcf5cfb5" class="outline-2">
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<h2 id="orgcf5cfb5"><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-org4ff5766" class="outline-2">
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<h2 id="org4ff5766"><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="org2865464"></a>
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<a id="orga416efe"></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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@ -407,8 +407,8 @@ 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-org06f9ee5" class="outline-3">
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<h3 id="org06f9ee5"><span class="section-number-3">2.1.</span> Definition of frame</h3>
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<div id="outline-container-org7c7d2db" class="outline-3">
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<h3 id="org7c7d2db"><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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@ -484,15 +484,15 @@ 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-org4f17ec2" class="outline-3">
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<h3 id="org4f17ec2"><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-org67fb689" class="outline-3">
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<h3 id="org67fb689"><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="orgc160166"></a>
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<a id="org491e37f"></a>
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</p>
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<p>
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In Figure <a href="#org56aadc9">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="#org24baa60">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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@ -500,14 +500,14 @@ 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="org56aadc9" class="figure">
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<div id="org24baa60" class="figure">
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<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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<p>
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The motion of the output beam is displayed as a function of the Bragg angle in Figure <a href="#org72617ff">3</a>.
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The motion of the output beam is displayed as a function of the Bragg angle in Figure <a href="#org692379d">3</a>.
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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.
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This offset decreases with the Bragg angle.
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</p>
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@ -520,7 +520,7 @@ This is indeed equal to:
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\end{equation}
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<div id="org72617ff" class="figure">
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<div id="org692379d" class="figure">
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<p><img src="figs/motion_beam_dz_error.png" alt="motion_beam_dz_error.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Motion of the output beam with dZ error</p>
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@ -528,29 +528,29 @@ This is indeed equal to:
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</div>
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</div>
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<div id="outline-container-orgc5a82df" class="outline-3">
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<h3 id="orgc5a82df"><span class="section-number-3">2.3.</span> Effect of an error in crystal’s x parallelism</h3>
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<div id="outline-container-orgf515721" class="outline-3">
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<h3 id="orgf515721"><span class="section-number-3">2.3.</span> Effect of an error in crystal’s x parallelism</h3>
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<div class="outline-text-3" id="text-2-3">
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<p>
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<a id="org08f1e7c"></a>
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<a id="orgb8a03c5"></a>
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</p>
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<p>
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The effect of an error in <code>rx</code> crystal parallelism on the output beam is visually shown in Figure <a href="#org64e88a1">4</a> for three bragg angles (5, 55 and 85 degrees).
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The effect of an error in <code>rx</code> crystal parallelism on the output beam is visually shown in Figure <a href="#org8f13c50">4</a> for three bragg angles (5, 55 and 85 degrees).
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The error is set to one degree, and the top view is shown.
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It is clear that the output beam experiences some rotation around a vertical axis.
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The amount of rotation depends on the bragg angle.
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</p>
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<div id="org64e88a1" class="figure">
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<div id="org8f13c50" class="figure">
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<p><img src="figs/ray_tracing_error_drx_overview.png" alt="ray_tracing_error_drx_overview.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Visual Effect of an error in <code>drx</code> (1 degree). Top View.</p>
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</div>
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<p>
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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="#orgca99abd">5</a>.
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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="#orgd270718">5</a>.
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</p>
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<p>
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@ -559,7 +559,7 @@ It is starting at zero for small bragg angles, and it increases with the bragg a
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This is indeed equal to:
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</p>
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\begin{equation}
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\boxed{r_{b,z} = 2 r_x \cos \theta}
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\boxed{r_{b,z} = -2 r_x \sin \theta}
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\end{equation}
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<p>
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@ -569,7 +569,7 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree
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</p>
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<div id="orgca99abd" class="figure">
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<div id="orgd270718" 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>
|
||||
@ -577,26 +577,26 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7a64dc0" class="outline-3">
|
||||
<h3 id="org7a64dc0"><span class="section-number-3">2.4.</span> Effect of an error in crystal’s y parallelism</h3>
|
||||
<div id="outline-container-org2fd3524" class="outline-3">
|
||||
<h3 id="org2fd3524"><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="org4e5ab76"></a>
|
||||
<a id="orgc411ea5"></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="#org2cd7448">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="#org5651b67">6</a> for three bragg angles (5, 55 and 85 degrees).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org2cd7448" class="figure">
|
||||
<div id="org5651b67" 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="#org53f9e3e">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="#org94ce447">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}
|
||||
@ -608,7 +608,7 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org53f9e3e" class="figure">
|
||||
<div id="org94ce447" 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>
|
||||
@ -616,16 +616,16 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org60c73bc" class="outline-3">
|
||||
<h3 id="org60c73bc"><span class="section-number-3">2.5.</span> Summary</h3>
|
||||
<div id="outline-container-org4ac158b" class="outline-3">
|
||||
<h3 id="org4ac158b"><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="#orga8d650f">3</a>.
|
||||
Effects of crystal’s pose errors on the output beam are summarized in Table <a href="#orgcdf2fe9">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="orga8d650f" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<table id="orgcdf2fe9" 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>
|
||||
@ -678,30 +678,30 @@ Also note that the effect of an error in crystal’s distance does not depen
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd3145d9" class="outline-3">
|
||||
<h3 id="orgd3145d9"><span class="section-number-3">2.6.</span> “Channel cut” Scan</h3>
|
||||
<div id="outline-container-org86a50e2" class="outline-3">
|
||||
<h3 id="org86a50e2"><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="#org3e8b9c1">8</a> where it is clear that the output beam experiences some vertical motion.
|
||||
This is visually shown in Figure <a href="#org02efdb2">8</a> where it is clear that the output beam experiences some vertical motion.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org3e8b9c1" class="figure">
|
||||
<div id="org02efdb2" 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="#orgfe14da2">9</a>.
|
||||
The \(z\) offset of the beam for several channel cut scans are shown in Figure <a href="#org8f490dd">9</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgfe14da2" class="figure">
|
||||
<div id="org8f490dd" 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>
|
||||
@ -710,8 +710,8 @@ The \(z\) offset of the beam for several channel cut scans are shown in Figure <
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org11191b2" class="outline-2">
|
||||
<h2 id="org11191b2"><span class="section-number-2">3.</span> Determining relative pose between the crystals using the X-ray</h2>
|
||||
<div id="outline-container-orgc33f680" class="outline-2">
|
||||
<h2 id="orgc33f680"><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.
|
||||
@ -730,21 +730,21 @@ In order to do that, an external metrology using the x-ray is used.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0854b62" class="outline-3">
|
||||
<h3 id="org0854b62"><span class="section-number-3">3.1.</span> Determine the \(y\) parallelism - “Rocking Curve”</h3>
|
||||
<div id="outline-container-org97fbcd2" class="outline-3">
|
||||
<h3 id="org97fbcd2"><span class="section-number-3">3.1.</span> Determine the \(y\) parallelism - “Rocking Curve”</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org79c2679" class="outline-3">
|
||||
<h3 id="org79c2679"><span class="section-number-3">3.2.</span> Determine the \(x\) parallelism - Bragg Scan</h3>
|
||||
<div id="outline-container-orga0df616" class="outline-3">
|
||||
<h3 id="orga0df616"><span class="section-number-3">3.2.</span> Determine the \(x\) parallelism - Bragg Scan</h3>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org15cd655" class="outline-3">
|
||||
<h3 id="org15cd655"><span class="section-number-3">3.3.</span> Determine the \(z\) distance - Bragg Scan</h3>
|
||||
<div id="outline-container-org01554e4" class="outline-3">
|
||||
<h3 id="org01554e4"><span class="section-number-3">3.3.</span> Determine the \(z\) distance - Bragg Scan</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb9628d9" class="outline-3">
|
||||
<h3 id="orgb9628d9"><span class="section-number-3">3.4.</span> Use Channel cut scan to determine crystal <code>dry</code> parallelism</h3>
|
||||
<div id="outline-container-org9347512" class="outline-3">
|
||||
<h3 id="org9347512"><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.
|
||||
@ -764,16 +764,16 @@ The error is
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf6270d8" class="outline-3">
|
||||
<h3 id="orgf6270d8"><span class="section-number-3">3.5.</span> Effect of an error on Bragg angle</h3>
|
||||
<div id="outline-container-org7a140bf" class="outline-3">
|
||||
<h3 id="org7a140bf"><span class="section-number-3">3.5.</span> Effect of an error on Bragg angle</h3>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb8e523b" class="outline-2">
|
||||
<h2 id="orgb8e523b"><span class="section-number-2">4.</span> Deformations of the Metrology Frame</h2>
|
||||
<div id="outline-container-orgf968de4" class="outline-2">
|
||||
<h2 id="orgf968de4"><span class="section-number-2">4.</span> Deformations of the Metrology Frame</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org45af7a1"></a>
|
||||
<a id="org6019b13"></a>
|
||||
</p>
|
||||
<p>
|
||||
The transformation matrices are valid only if the metrology frames are solid bodies.
|
||||
@ -788,8 +788,8 @@ 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-org8391240" class="outline-3">
|
||||
<h3 id="org8391240"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
|
||||
<div id="outline-container-org573b7e7" class="outline-3">
|
||||
<h3 id="org573b7e7"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
Two beam viewers:
|
||||
@ -806,7 +806,7 @@ This position is the wanted position for a given Bragg angle.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org4769d08" class="figure">
|
||||
<div id="org6080aeb" 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>
|
||||
@ -836,8 +836,8 @@ Frame rate is: 42 fps
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org0a584b6" class="outline-3">
|
||||
<h3 id="org0a584b6"><span class="section-number-3">4.2.</span> Simulations</h3>
|
||||
<div id="outline-container-orgbabe313" class="outline-3">
|
||||
<h3 id="orgbabe313"><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.
|
||||
@ -846,12 +846,12 @@ This may be done using FE software.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0820684" class="outline-3">
|
||||
<h3 id="org0820684"><span class="section-number-3">4.3.</span> Comparison</h3>
|
||||
<div id="outline-container-orgaf7afd6" class="outline-3">
|
||||
<h3 id="orgaf7afd6"><span class="section-number-3">4.3.</span> Comparison</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org91e6317" class="outline-3">
|
||||
<h3 id="org91e6317"><span class="section-number-3">4.4.</span> Test</h3>
|
||||
<div id="outline-container-org99a64a4" class="outline-3">
|
||||
<h3 id="org99a64a4"><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>);
|
||||
@ -866,8 +866,8 @@ This may be done using FE software.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5ca0ac8" class="outline-3">
|
||||
<h3 id="org5ca0ac8"><span class="section-number-3">4.5.</span> Measured frame deformation</h3>
|
||||
<div id="outline-container-org907c706" class="outline-3">
|
||||
<h3 id="org907c706"><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));
|
||||
@ -895,7 +895,7 @@ ry1 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>dat
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org96336ed" class="figure">
|
||||
<div id="org85f32b7" 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>
|
||||
@ -968,8 +968,8 @@ f_ry1 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">p
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb94c7fb" class="outline-3">
|
||||
<h3 id="orgb94c7fb"><span class="section-number-3">4.6.</span> Test</h3>
|
||||
<div id="outline-container-orgaa71cc1" class="outline-3">
|
||||
<h3 id="orgaa71cc1"><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>;
|
||||
@ -1001,7 +1001,7 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orgb431cdd" class="figure">
|
||||
<div id="orgef4ecac" 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>
|
||||
@ -1018,8 +1018,8 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1c4abff" class="outline-3">
|
||||
<h3 id="org1c4abff"><span class="section-number-3">4.7.</span> Repeatability of frame deformation</h3>
|
||||
<div id="outline-container-org73400be" class="outline-3">
|
||||
<h3 id="org73400be"><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>;
|
||||
@ -1069,11 +1069,11 @@ data_2.dz = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*<
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgcf35007" class="outline-2">
|
||||
<h2 id="orgcf35007"><span class="section-number-2">5.</span> Attocube - Periodic Non-Linearity</h2>
|
||||
<div id="outline-container-org0b40d1b" class="outline-2">
|
||||
<h2 id="org0b40d1b"><span class="section-number-2">5.</span> Attocube - Periodic Non-Linearity</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="org455a211"></a>
|
||||
<a id="org2a9ae01"></a>
|
||||
</p>
|
||||
<p>
|
||||
Interferometers have some periodic nonlinearity (NO_ITEM_DATA:thurner15_fiber_based_distan_sensin_inter).
|
||||
@ -1094,8 +1094,8 @@ This process is performed over several periods in order to characterize the erro
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org91573e9" class="outline-3">
|
||||
<h3 id="org91573e9"><span class="section-number-3">5.1.</span> Measurement Setup</h3>
|
||||
<div id="outline-container-orge0a38ec" class="outline-3">
|
||||
<h3 id="orge0a38ec"><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.
|
||||
@ -1106,7 +1106,7 @@ 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="org38e45ca">
|
||||
<div class="note" id="orgfc7f1df">
|
||||
<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>
|
||||
@ -1122,7 +1122,7 @@ 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="#org5df73e7">13</a>.
|
||||
The setup is schematically with the block diagram in Figure <a href="#org5485724">13</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1138,7 +1138,7 @@ The PI controller takes care or controlling to position as measured by the strai
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org5df73e7" class="figure">
|
||||
<div id="org5485724" 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>
|
||||
@ -1150,8 +1150,8 @@ The problem is to estimate the periodic non-linearity of the Attocube from the i
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdd79356" class="outline-3">
|
||||
<h3 id="orgdd79356"><span class="section-number-3">5.2.</span> Choice of the reference signal</h3>
|
||||
<div id="outline-container-org821469c" class="outline-3">
|
||||
<h3 id="org821469c"><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;
|
||||
@ -1175,8 +1175,8 @@ Based on the above discussion, one suitable excitation signal is a sinusoidal sw
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2268c53" class="outline-3">
|
||||
<h3 id="org2268c53"><span class="section-number-3">5.3.</span> Repeatability of the non-linearity</h3>
|
||||
<div id="outline-container-org3263c1f" class="outline-3">
|
||||
<h3 id="org3263c1f"><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.
|
||||
@ -1196,8 +1196,8 @@ One way to precisely estimate the laser wavelength is to estimate the non linear
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgadef69b" class="outline-3">
|
||||
<h3 id="orgadef69b"><span class="section-number-3">5.4.</span> Simulation</h3>
|
||||
<div id="outline-container-org39fabc8" class="outline-3">
|
||||
<h3 id="org39fabc8"><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.
|
||||
@ -1217,10 +1217,10 @@ 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="#org4cf8064">14</a>).
|
||||
The non-linear errors are first estimated at the beginning of the stroke (Figure <a href="#org49c2d00">14</a>).
|
||||
</p>
|
||||
|
||||
<div id="org4cf8064" class="figure">
|
||||
<div id="org49c2d00" 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>
|
||||
@ -1228,7 +1228,7 @@ 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="#orgb57a43c">15</a>).
|
||||
To do so, the same measurement is performed with a stroke of several millimeters (Figure <a href="#org8c5db79">15</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1237,7 +1237,7 @@ This is due to a mismatch between the estimated period and the true period of th
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgb57a43c" class="figure">
|
||||
<div id="org8c5db79" 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>
|
||||
@ -1270,7 +1270,7 @@ with \(\lambda_{\text{est}}\) the estimated error’s period.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From Figure <a href="#orgb57a43c">15</a>, we can see that there is an offset between the two curves.
|
||||
From Figure <a href="#org8c5db79">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}
|
||||
@ -1350,8 +1350,8 @@ The maximum stroke is 2.9 [mm]
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8ddf260" class="outline-3">
|
||||
<h3 id="org8ddf260"><span class="section-number-3">5.5.</span> Measurements</h3>
|
||||
<div id="outline-container-org373814c" class="outline-3">
|
||||
<h3 id="org373814c"><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:
|
||||
@ -1378,9 +1378,9 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org515d8fb" class="outline-2">
|
||||
<h2 id="org515d8fb">Bibliography</h2>
|
||||
<div class="outline-text-2" id="text-org515d8fb">
|
||||
<div id="outline-container-orga081f50" class="outline-2">
|
||||
<h2 id="orga081f50">Bibliography</h2>
|
||||
<div class="outline-text-2" id="text-orga081f50">
|
||||
<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, Sebastien. 2018. “Toward High Precision Position Control Using Laser Interferometry: Main Sources of Error.” doi:<a href="https://doi.org/10.13140/rg.2.2.21044.35205">10.13140/rg.2.2.21044.35205</a>.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” <i>Applied Optics</i> 54 (10). Optical Society of America: 3051–63.</div>
|
||||
@ -1390,7 +1390,7 @@ 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: 2022-06-07 Tue 10:58</p>
|
||||
<p class="date">Created: 2022-06-22 Wed 13:29</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -362,7 +362,7 @@ It induces a rotation of the output beam in the =z= direction that depends on th
|
||||
It is starting at zero for small bragg angles, and it increases with the bragg angle up to 2 times =drx=.
|
||||
This is indeed equal to:
|
||||
\begin{equation}
|
||||
\boxed{r_{b,z} = 2 r_x \cos \theta}
|
||||
\boxed{r_{b,z} = -2 r_x \sin \theta}
|
||||
\end{equation}
|
||||
|
||||
If also induces a small $y$ shift of the beam.
|
||||
@ -661,8 +661,14 @@ When the bragg axis is scanned, the effect of gravity on the metrology frame is
|
||||
|
||||
This can be calibrated.
|
||||
|
||||
** Measurement Setup
|
||||
** New idea to calibrate deformations :noexport:
|
||||
For each Bragg angle $\theta$:
|
||||
- Correct parallelism (Dry) between crystals by maximizing the intensity.
|
||||
Save =fjsry=.
|
||||
- Correct for (Y,Z) position on the detector by modifying Drx and Dz.
|
||||
Save =fjsrx= and =fjsz=.
|
||||
|
||||
** Measurement Setup
|
||||
Two beam viewers:
|
||||
- one close to the DCM to measure position of the beam
|
||||
- one far away to the DCM to measure orientation of the beam
|
||||
|
||||
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Reference in New Issue
Block a user