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# PDF Generation/Building/Compilation # PDF Generation/Building/Compilation
# ====================================================================================== # ======================================================================================
@default_files=('dcm_metrology.tex'); @default_files=('dcm-metrology.tex');
# PDF-generating modes are: # PDF-generating modes are:
# 1: pdflatex, as specified by $pdflatex variable (still largely in use) # 1: pdflatex, as specified by $pdflatex variable (still largely in use)

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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2022-06-02 Thu 22:25 --> <!-- 2022-06-07 Tue 10:58 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>ESRF Double Crystal Monochromator - Metrology</title> <title>ESRF Double Crystal Monochromator - Metrology</title>
<meta name="author" content="Dehaeze Thomas" /> <meta name="author" content="Dehaeze Thomas" />
@ -39,51 +39,52 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents" role="doc-toc"> <div id="text-table-of-contents" role="doc-toc">
<ul> <ul>
<li><a href="#org5b4a8ea">1. Metrology Concept</a> <li><a href="#org2c2c62b">1. Metrology Concept</a>
<ul> <ul>
<li><a href="#orgee0936b">1.1. Sensor Topology</a></li> <li><a href="#org3fa59ba">1.1. Sensor Topology</a></li>
<li><a href="#orgcc7ab6e">1.2. Computation of the relative pose between first and second crystals</a></li> <li><a href="#orgc5b3b8f">1.2. Computation of the relative pose between first and second crystals</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd1e6cf8">2. Relation Between Crystal position and X-ray measured displacement</a> <li><a href="#orgcf5cfb5">2. Relation Between Crystal position and X-ray measured displacement</a>
<ul> <ul>
<li><a href="#orgf98dcda">2.1. Definition of frame</a></li> <li><a href="#org06f9ee5">2.1. Definition of frame</a></li>
<li><a href="#orgcc56e18">2.2. Effect of an error in crystal&rsquo;s distance</a></li> <li><a href="#org4f17ec2">2.2. Effect of an error in crystal&rsquo;s distance</a></li>
<li><a href="#org4f10916">2.3. Effect of an error in crystal&rsquo;s x parallelism</a></li> <li><a href="#orgc5a82df">2.3. Effect of an error in crystal&rsquo;s x parallelism</a></li>
<li><a href="#org214d2fb">2.4. Effect of an error in crystal&rsquo;s y parallelism</a></li> <li><a href="#org7a64dc0">2.4. Effect of an error in crystal&rsquo;s y parallelism</a></li>
<li><a href="#orga01ea08">2.5. Summary</a></li> <li><a href="#org60c73bc">2.5. Summary</a></li>
<li><a href="#orgbe05a5f">2.6. &ldquo;Channel cut&rdquo; Scan</a></li> <li><a href="#orgd3145d9">2.6. &ldquo;Channel cut&rdquo; Scan</a></li>
</ul> </ul>
</li> </li>
<li><a href="#orgd1c9f1e">3. Determining relative pose between the crystals using the X-ray</a> <li><a href="#org11191b2">3. Determining relative pose between the crystals using the X-ray</a>
<ul> <ul>
<li><a href="#orgb9a7cbf">3.1. Determine the \(y\) parallelism - &ldquo;Rocking Curve&rdquo;</a></li> <li><a href="#org0854b62">3.1. Determine the \(y\) parallelism - &ldquo;Rocking Curve&rdquo;</a></li>
<li><a href="#orgf1f155c">3.2. Determine the \(x\) parallelism - Bragg Scan</a></li> <li><a href="#org79c2679">3.2. Determine the \(x\) parallelism - Bragg Scan</a></li>
<li><a href="#org340a305">3.3. Determine the \(z\) distance - Bragg Scan</a></li> <li><a href="#org15cd655">3.3. Determine the \(z\) distance - Bragg Scan</a></li>
<li><a href="#org550643d">3.4. Use Channel cut scan to determine crystal <code>dry</code> parallelism</a></li> <li><a href="#orgb9628d9">3.4. Use Channel cut scan to determine crystal <code>dry</code> parallelism</a></li>
<li><a href="#org3f02fcd">3.5. Effect of an error on Bragg angle</a></li> <li><a href="#orgf6270d8">3.5. Effect of an error on Bragg angle</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4eedaad">4. Deformations of the Metrology Frame</a> <li><a href="#orgb8e523b">4. Deformations of the Metrology Frame</a>
<ul> <ul>
<li><a href="#orge197959">4.1. Measurement Setup</a></li> <li><a href="#org8391240">4.1. Measurement Setup</a></li>
<li><a href="#org7cfa17e">4.2. Simulations</a></li> <li><a href="#org0a584b6">4.2. Simulations</a></li>
<li><a href="#org0c10c30">4.3. Comparison</a></li> <li><a href="#org0820684">4.3. Comparison</a></li>
<li><a href="#org25a32fa">4.4. Test</a></li> <li><a href="#org91e6317">4.4. Test</a></li>
<li><a href="#orgb7db314">4.5. Measured frame deformation</a></li> <li><a href="#org5ca0ac8">4.5. Measured frame deformation</a></li>
<li><a href="#org5494ab9">4.6. Test</a></li> <li><a href="#orgb94c7fb">4.6. Test</a></li>
<li><a href="#org218f95f">4.7. Repeatability of frame deformation</a></li> <li><a href="#org1c4abff">4.7. Repeatability of frame deformation</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org5b8904d">5. Attocube - Periodic Non-Linearity</a> <li><a href="#orgcf35007">5. Attocube - Periodic Non-Linearity</a>
<ul> <ul>
<li><a href="#org9f6a57f">5.1. Measurement Setup</a></li> <li><a href="#org91573e9">5.1. Measurement Setup</a></li>
<li><a href="#orgc06acba">5.2. Choice of the reference signal</a></li> <li><a href="#orgdd79356">5.2. Choice of the reference signal</a></li>
<li><a href="#org26743c0">5.3. Repeatability of the non-linearity</a></li> <li><a href="#org2268c53">5.3. Repeatability of the non-linearity</a></li>
<li><a href="#org15014d3">5.4. Simulation</a></li> <li><a href="#orgadef69b">5.4. Simulation</a></li>
<li><a href="#orga71d790">5.5. Measurements</a></li> <li><a href="#org8ddf260">5.5. Measurements</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org515d8fb">Bibliography</a></li>
</ul> </ul>
</div> </div>
</div> </div>
@ -93,26 +94,26 @@
<p> <p>
In this document, the metrology system is studied. In this document, the metrology system is studied.
First, in Section <a href="#orga8ac352">1</a> the goal of the metrology system is stated and the proposed concept is described. First, in Section <a href="#orga1bc36b">1</a> the goal of the metrology system is stated and the proposed concept is described.
</p> </p>
<p> <p>
How the relative crystal pose is affecting the pose of the output beam is studied in Section <a href="#org4b9a066">2</a>. How the relative crystal pose is affecting the pose of the output beam is studied in Section <a href="#org2865464">2</a>.
</p> </p>
<p> <p>
In order to increase the accuracy of the metrology system, two problems are to be dealt with: In order to increase the accuracy of the metrology system, two problems are to be dealt with:
</p> </p>
<ul class="org-ul"> <ul class="org-ul">
<li>The deformation of the metrology frame under the action of gravity (Section <a href="#orgf31717e">4</a>)</li> <li>The deformation of the metrology frame under the action of gravity (Section <a href="#org45af7a1">4</a>)</li>
<li>The periodic non-linearity of the interferometers (Section <a href="#org44bdb6d">5</a>)</li> <li>The periodic non-linearity of the interferometers (Section <a href="#org455a211">5</a>)</li>
</ul> </ul>
<div id="outline-container-org5b4a8ea" class="outline-2"> <div id="outline-container-org2c2c62b" class="outline-2">
<h2 id="org5b4a8ea"><span class="section-number-2">1.</span> Metrology Concept</h2> <h2 id="org2c2c62b"><span class="section-number-2">1.</span> Metrology Concept</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
<a id="orga8ac352"></a> <a id="orga1bc36b"></a>
</p> </p>
<p> <p>
The goal of the metrology system is to measure the distance and default of parallelism between the first and second crystals. The goal of the metrology system is to measure the distance and default of parallelism between the first and second crystals.
@ -127,8 +128,8 @@ Only 3 degrees of freedom are of interest:
<li>\(r_x\)</li> <li>\(r_x\)</li>
</ul> </ul>
</div> </div>
<div id="outline-container-orgee0936b" class="outline-3"> <div id="outline-container-org3fa59ba" class="outline-3">
<h3 id="orgee0936b"><span class="section-number-3">1.1.</span> Sensor Topology</h3> <h3 id="org3fa59ba"><span class="section-number-3">1.1.</span> Sensor Topology</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
In order to measure the relative pose of the two crystals, instead of performing a direct measurement which is complicated, the pose of the two crystals are measured from a metrology frame. In order to measure the relative pose of the two crystals, instead of performing a direct measurement which is complicated, the pose of the two crystals are measured from a metrology frame.
@ -138,11 +139,11 @@ Three additional interferometers are used to measured the relative motion of the
<p> <p>
In total, there are 15 interferometers represented in Figure <a href="#orgce5aaa3">1</a>. In total, there are 15 interferometers represented in Figure <a href="#org0ba932c">1</a>.
The measurements are summarized in Table <a href="#org943c065">2</a>. The measurements are summarized in Table <a href="#org9256fe4">2</a>.
</p> </p>
<table id="org5dd8abe" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orgf3e6978" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Notations for the metrology frame</caption> <caption class="t-above"><span class="table-number">Table 1:</span> Notations for the metrology frame</caption>
<colgroup> <colgroup>
@ -194,7 +195,7 @@ The measurements are summarized in Table <a href="#org943c065">2</a>.
</tbody> </tbody>
</table> </table>
<table id="org943c065" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="org9256fe4" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> List of Interferometer measurements</caption> <caption class="t-above"><span class="table-number">Table 2:</span> List of Interferometer measurements</caption>
<colgroup> <colgroup>
@ -305,7 +306,7 @@ The measurements are summarized in Table <a href="#org943c065">2</a>.
</table> </table>
<div id="orgce5aaa3" class="figure"> <div id="org0ba932c" class="figure">
<p><img src="figs/metrology_schematic.png" alt="metrology_schematic.png" /> <p><img src="figs/metrology_schematic.png" alt="metrology_schematic.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Schematic of the Metrology System</p> <p><span class="figure-number">Figure 1: </span>Schematic of the Metrology System</p>
@ -313,8 +314,8 @@ The measurements are summarized in Table <a href="#org943c065">2</a>.
</div> </div>
</div> </div>
<div id="outline-container-orgcc7ab6e" class="outline-3"> <div id="outline-container-orgc5b3b8f" class="outline-3">
<h3 id="orgcc7ab6e"><span class="section-number-3">1.2.</span> Computation of the relative pose between first and second crystals</h3> <h3 id="orgc5b3b8f"><span class="section-number-3">1.2.</span> Computation of the relative pose between first and second crystals</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
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>). 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>).
@ -324,7 +325,7 @@ To understand how the relative pose between the crystals is computed from the in
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 \([d_{r,z},\ r_{r,y},\ r_{r,x}]\). 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 \([d_{r,z},\ r_{r,y},\ r_{r,x}]\).
</p> </p>
<div class="note" id="org5a06b04"> <div class="note" id="orgc0eb804">
<p> <p>
The sign conventions for the relative crystal pose are: The sign conventions for the relative crystal pose are:
</p> </p>
@ -379,11 +380,11 @@ Values of the matrices can be found in the document describing the kinematics of
</div> </div>
</div> </div>
<div id="outline-container-orgd1e6cf8" class="outline-2"> <div id="outline-container-orgcf5cfb5" class="outline-2">
<h2 id="orgd1e6cf8"><span class="section-number-2">2.</span> Relation Between Crystal position and X-ray measured displacement</h2> <h2 id="orgcf5cfb5"><span class="section-number-2">2.</span> Relation Between Crystal position and X-ray measured displacement</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
<a id="org4b9a066"></a> <a id="org2865464"></a>
</p> </p>
<p> <p>
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. 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.
@ -406,8 +407,8 @@ In order to simplify the problem, the first crystal is supposed to be fixed (i.e
In order to easily study that, &ldquo;ray tracing&rdquo; techniques are used. In order to easily study that, &ldquo;ray tracing&rdquo; techniques are used.
</p> </p>
</div> </div>
<div id="outline-container-orgf98dcda" class="outline-3"> <div id="outline-container-org06f9ee5" class="outline-3">
<h3 id="orgf98dcda"><span class="section-number-3">2.1.</span> Definition of frame</h3> <h3 id="org06f9ee5"><span class="section-number-3">2.1.</span> Definition of frame</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
@ -483,15 +484,15 @@ The xy position of the beam is taken in the \(x=0\) plane.
</div> </div>
</div> </div>
<div id="outline-container-orgcc56e18" class="outline-3"> <div id="outline-container-org4f17ec2" class="outline-3">
<h3 id="orgcc56e18"><span class="section-number-3">2.2.</span> Effect of an error in crystal&rsquo;s distance</h3> <h3 id="org4f17ec2"><span class="section-number-3">2.2.</span> Effect of an error in crystal&rsquo;s distance</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<p> <p>
<a id="orgfce54e9"></a> <a id="orgc160166"></a>
</p> </p>
<p> <p>
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. 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.
</p> </p>
<p> <p>
@ -499,14 +500,14 @@ Visually, it is clear that this induce a <code>z</code> offset of the output bea
</p> </p>
<div id="orga24d92a" class="figure"> <div id="org56aadc9" class="figure">
<p><img src="figs/ray_tracing_error_dz_overview.png" alt="ray_tracing_error_dz_overview.png" /> <p><img src="figs/ray_tracing_error_dz_overview.png" alt="ray_tracing_error_dz_overview.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Visual Effect of an error in <code>dz</code> (1mm). Side view.</p> <p><span class="figure-number">Figure 2: </span>Visual Effect of an error in <code>dz</code> (1mm). Side view.</p>
</div> </div>
<p> <p>
The motion of the output beam is displayed as a function of the Bragg angle in Figure <a href="#orgf3acc2d">3</a>. The motion of the output beam is displayed as a function of the Bragg angle in Figure <a href="#org72617ff">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. 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. This offset decreases with the Bragg angle.
</p> </p>
@ -519,7 +520,7 @@ This is indeed equal to:
\end{equation} \end{equation}
<div id="orgf3acc2d" class="figure"> <div id="org72617ff" class="figure">
<p><img src="figs/motion_beam_dz_error.png" alt="motion_beam_dz_error.png" /> <p><img src="figs/motion_beam_dz_error.png" alt="motion_beam_dz_error.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Motion of the output beam with dZ error</p> <p><span class="figure-number">Figure 3: </span>Motion of the output beam with dZ error</p>
@ -527,29 +528,29 @@ This is indeed equal to:
</div> </div>
</div> </div>
<div id="outline-container-org4f10916" class="outline-3"> <div id="outline-container-orgc5a82df" class="outline-3">
<h3 id="org4f10916"><span class="section-number-3">2.3.</span> Effect of an error in crystal&rsquo;s x parallelism</h3> <h3 id="orgc5a82df"><span class="section-number-3">2.3.</span> Effect of an error in crystal&rsquo;s x parallelism</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<p> <p>
<a id="org10b6b9d"></a> <a id="org08f1e7c"></a>
</p> </p>
<p> <p>
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 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).
The error is set to one degree, and the top view is shown. 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. It is clear that the output beam experiences some rotation around a vertical axis.
The amount of rotation depends on the bragg angle. The amount of rotation depends on the bragg angle.
</p> </p>
<div id="orgf3fcd25" class="figure"> <div id="org64e88a1" class="figure">
<p><img src="figs/ray_tracing_error_drx_overview.png" alt="ray_tracing_error_drx_overview.png" /> <p><img src="figs/ray_tracing_error_drx_overview.png" alt="ray_tracing_error_drx_overview.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Visual Effect of an error in <code>drx</code> (1 degree). Top View.</p> <p><span class="figure-number">Figure 4: </span>Visual Effect of an error in <code>drx</code> (1 degree). Top View.</p>
</div> </div>
<p> <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="#org3ce23f8">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="#orgca99abd">5</a>.
</p> </p>
<p> <p>
@ -568,7 +569,7 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree
</p> </p>
<div id="org3ce23f8" class="figure"> <div id="orgca99abd" class="figure">
<p><img src="figs/motion_beam_drx_error.png" alt="motion_beam_drx_error.png" /> <p><img src="figs/motion_beam_drx_error.png" alt="motion_beam_drx_error.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Motion of the output beam with drx error</p> <p><span class="figure-number">Figure 5: </span>Motion of the output beam with drx error</p>
@ -576,26 +577,26 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree
</div> </div>
</div> </div>
<div id="outline-container-org214d2fb" class="outline-3"> <div id="outline-container-org7a64dc0" class="outline-3">
<h3 id="org214d2fb"><span class="section-number-3">2.4.</span> Effect of an error in crystal&rsquo;s y parallelism</h3> <h3 id="org7a64dc0"><span class="section-number-3">2.4.</span> Effect of an error in crystal&rsquo;s y parallelism</h3>
<div class="outline-text-3" id="text-2-4"> <div class="outline-text-3" id="text-2-4">
<p> <p>
<a id="orged8dba0"></a> <a id="org4e5ab76"></a>
</p> </p>
<p> <p>
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). 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).
</p> </p>
<div id="orgb3ea9d4" class="figure"> <div id="org2cd7448" class="figure">
<p><img src="figs/ray_tracing_error_dry_overview.png" alt="ray_tracing_error_dry_overview.png" /> <p><img src="figs/ray_tracing_error_dry_overview.png" alt="ray_tracing_error_dry_overview.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Visual Effect of an error in <code>dry</code> (1 degree). Side view.</p> <p><span class="figure-number">Figure 6: </span>Visual Effect of an error in <code>dry</code> (1 degree). Side view.</p>
</div> </div>
<p> <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="#org28da5fb">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="#org53f9e3e">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>: 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> </p>
\begin{equation} \begin{equation}
@ -607,7 +608,7 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh
</p> </p>
<div id="org28da5fb" class="figure"> <div id="org53f9e3e" class="figure">
<p><img src="figs/motion_beam_dry_error.png" alt="motion_beam_dry_error.png" /> <p><img src="figs/motion_beam_dry_error.png" alt="motion_beam_dry_error.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>Motion of the output beam with dry error</p> <p><span class="figure-number">Figure 7: </span>Motion of the output beam with dry error</p>
@ -615,16 +616,16 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh
</div> </div>
</div> </div>
<div id="outline-container-orga01ea08" class="outline-3"> <div id="outline-container-org60c73bc" class="outline-3">
<h3 id="orga01ea08"><span class="section-number-3">2.5.</span> Summary</h3> <h3 id="org60c73bc"><span class="section-number-3">2.5.</span> Summary</h3>
<div class="outline-text-3" id="text-2-5"> <div class="outline-text-3" id="text-2-5">
<p> <p>
Effects of crystal&rsquo;s pose errors on the output beam are summarized in Table <a href="#org01cc930">3</a>. Effects of crystal&rsquo;s pose errors on the output beam are summarized in Table <a href="#orga8d650f">3</a>.
Note that the three pose errors are well decoupled regarding their effects on the output beam. 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&rsquo;s distance does not depend on the Bragg angle. Also note that the effect of an error in crystal&rsquo;s distance does not depend on the Bragg angle.
</p> </p>
<table id="org01cc930" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> <table id="orga8d650f" 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&rsquo;s pose on the output beam</caption> <caption class="t-above"><span class="table-number">Table 3:</span> Summary of the effects of the errors in second crystal&rsquo;s pose on the output beam</caption>
<colgroup> <colgroup>
@ -677,30 +678,30 @@ Also note that the effect of an error in crystal&rsquo;s distance does not depen
</div> </div>
</div> </div>
<div id="outline-container-orgbe05a5f" class="outline-3"> <div id="outline-container-orgd3145d9" class="outline-3">
<h3 id="orgbe05a5f"><span class="section-number-3">2.6.</span> &ldquo;Channel cut&rdquo; Scan</h3> <h3 id="orgd3145d9"><span class="section-number-3">2.6.</span> &ldquo;Channel cut&rdquo; Scan</h3>
<div class="outline-text-3" id="text-2-6"> <div class="outline-text-3" id="text-2-6">
<p> <p>
A &ldquo;channel cut&rdquo; scan is a Bragg scan where the distance between the crystals is fixed. A &ldquo;channel cut&rdquo; scan is a Bragg scan where the distance between the crystals is fixed.
</p> </p>
<p> <p>
This is visually shown in Figure <a href="#org278d240">8</a> where it is clear that the output beam experiences some vertical motion. This is visually shown in Figure <a href="#org3e8b9c1">8</a> where it is clear that the output beam experiences some vertical motion.
</p> </p>
<div id="org278d240" class="figure"> <div id="org3e8b9c1" class="figure">
<p><img src="figs/ray_tracing_channel_cut.png" alt="ray_tracing_channel_cut.png" /> <p><img src="figs/ray_tracing_channel_cut.png" alt="ray_tracing_channel_cut.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Visual Effect of a channel cut scan</p> <p><span class="figure-number">Figure 8: </span>Visual Effect of a channel cut scan</p>
</div> </div>
<p> <p>
The \(z\) offset of the beam for several channel cut scans are shown in Figure <a href="#org588c503">9</a>. The \(z\) offset of the beam for several channel cut scans are shown in Figure <a href="#orgfe14da2">9</a>.
</p> </p>
<div id="org588c503" class="figure"> <div id="orgfe14da2" class="figure">
<p><img src="figs/channel_cut_scan.png" alt="channel_cut_scan.png" /> <p><img src="figs/channel_cut_scan.png" alt="channel_cut_scan.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Z motion of the beam during &ldquo;channel cut&rdquo; scans</p> <p><span class="figure-number">Figure 9: </span>Z motion of the beam during &ldquo;channel cut&rdquo; scans</p>
@ -709,8 +710,8 @@ The \(z\) offset of the beam for several channel cut scans are shown in Figure <
</div> </div>
</div> </div>
<div id="outline-container-orgd1c9f1e" class="outline-2"> <div id="outline-container-org11191b2" class="outline-2">
<h2 id="orgd1c9f1e"><span class="section-number-2">3.</span> Determining relative pose between the crystals using the X-ray</h2> <h2 id="org11191b2"><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"> <div class="outline-text-2" id="text-3">
<p> <p>
As Interferometers are only measuring <i>relative</i> displacement, it is mandatory to initialize them correctly. As Interferometers are only measuring <i>relative</i> displacement, it is mandatory to initialize them correctly.
@ -729,21 +730,21 @@ In order to do that, an external metrology using the x-ray is used.
</p> </p>
</div> </div>
<div id="outline-container-orgb9a7cbf" class="outline-3"> <div id="outline-container-org0854b62" class="outline-3">
<h3 id="orgb9a7cbf"><span class="section-number-3">3.1.</span> Determine the \(y\) parallelism - &ldquo;Rocking Curve&rdquo;</h3> <h3 id="org0854b62"><span class="section-number-3">3.1.</span> Determine the \(y\) parallelism - &ldquo;Rocking Curve&rdquo;</h3>
</div> </div>
<div id="outline-container-orgf1f155c" class="outline-3"> <div id="outline-container-org79c2679" class="outline-3">
<h3 id="orgf1f155c"><span class="section-number-3">3.2.</span> Determine the \(x\) parallelism - Bragg Scan</h3> <h3 id="org79c2679"><span class="section-number-3">3.2.</span> Determine the \(x\) parallelism - Bragg Scan</h3>
</div> </div>
<div id="outline-container-org340a305" class="outline-3"> <div id="outline-container-org15cd655" class="outline-3">
<h3 id="org340a305"><span class="section-number-3">3.3.</span> Determine the \(z\) distance - Bragg Scan</h3> <h3 id="org15cd655"><span class="section-number-3">3.3.</span> Determine the \(z\) distance - Bragg Scan</h3>
</div> </div>
<div id="outline-container-org550643d" class="outline-3"> <div id="outline-container-orgb9628d9" 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> <h3 id="orgb9628d9"><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"> <div class="outline-text-3" id="text-3-4">
<p> <p>
Now, let&rsquo;s suppose we want to determine the <code>dry</code> angle between the crystals. Now, let&rsquo;s suppose we want to determine the <code>dry</code> angle between the crystals.
@ -763,16 +764,16 @@ The error is
</div> </div>
</div> </div>
<div id="outline-container-org3f02fcd" class="outline-3"> <div id="outline-container-orgf6270d8" class="outline-3">
<h3 id="org3f02fcd"><span class="section-number-3">3.5.</span> Effect of an error on Bragg angle</h3> <h3 id="orgf6270d8"><span class="section-number-3">3.5.</span> Effect of an error on Bragg angle</h3>
</div> </div>
</div> </div>
<div id="outline-container-org4eedaad" class="outline-2"> <div id="outline-container-orgb8e523b" class="outline-2">
<h2 id="org4eedaad"><span class="section-number-2">4.</span> Deformations of the Metrology Frame</h2> <h2 id="orgb8e523b"><span class="section-number-2">4.</span> Deformations of the Metrology Frame</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
<p> <p>
<a id="orgf31717e"></a> <a id="org45af7a1"></a>
</p> </p>
<p> <p>
The transformation matrices are valid only if the metrology frames are solid bodies. The transformation matrices are valid only if the metrology frames are solid bodies.
@ -787,8 +788,8 @@ When the bragg axis is scanned, the effect of gravity on the metrology frame is
This can be calibrated. This can be calibrated.
</p> </p>
</div> </div>
<div id="outline-container-orge197959" class="outline-3"> <div id="outline-container-org8391240" class="outline-3">
<h3 id="orge197959"><span class="section-number-3">4.1.</span> Measurement Setup</h3> <h3 id="org8391240"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<p> <p>
Two beam viewers: Two beam viewers:
@ -805,7 +806,7 @@ This position is the wanted position for a given Bragg angle.
</p> </p>
<div id="orgec71cad" class="figure"> <div id="org4769d08" class="figure">
<p><img src="figs/calibration_setup.png" alt="calibration_setup.png" /> <p><img src="figs/calibration_setup.png" alt="calibration_setup.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Schematic of the setup</p> <p><span class="figure-number">Figure 10: </span>Schematic of the setup</p>
@ -835,8 +836,8 @@ Frame rate is: 42 fps
</div> </div>
<div id="outline-container-org7cfa17e" class="outline-3"> <div id="outline-container-org0a584b6" class="outline-3">
<h3 id="org7cfa17e"><span class="section-number-3">4.2.</span> Simulations</h3> <h3 id="org0a584b6"><span class="section-number-3">4.2.</span> Simulations</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<p> <p>
The deformations of the metrology frame and therefore the expected interferometric measurements can be computed as a function of the Bragg angle. The deformations of the metrology frame and therefore the expected interferometric measurements can be computed as a function of the Bragg angle.
@ -845,12 +846,12 @@ This may be done using FE software.
</div> </div>
</div> </div>
<div id="outline-container-org0c10c30" class="outline-3"> <div id="outline-container-org0820684" class="outline-3">
<h3 id="org0c10c30"><span class="section-number-3">4.3.</span> Comparison</h3> <h3 id="org0820684"><span class="section-number-3">4.3.</span> Comparison</h3>
</div> </div>
<div id="outline-container-org25a32fa" class="outline-3"> <div id="outline-container-org91e6317" class="outline-3">
<h3 id="org25a32fa"><span class="section-number-3">4.4.</span> Test</h3> <h3 id="org91e6317"><span class="section-number-3">4.4.</span> Test</h3>
<div class="outline-text-3" id="text-4-4"> <div class="outline-text-3" id="text-4-4">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">aa = importdata(<span class="org-string">"correctInterf-vlm-220201.dat"</span>); <pre class="src src-matlab">aa = importdata(<span class="org-string">"correctInterf-vlm-220201.dat"</span>);
@ -865,8 +866,8 @@ This may be done using FE software.
</div> </div>
</div> </div>
<div id="outline-container-orgb7db314" class="outline-3"> <div id="outline-container-org5ca0ac8" class="outline-3">
<h3 id="orgb7db314"><span class="section-number-3">4.5.</span> Measured frame deformation</h3> <h3 id="org5ca0ac8"><span class="section-number-3">4.5.</span> Measured frame deformation</h3>
<div class="outline-text-3" id="text-4-5"> <div class="outline-text-3" id="text-4-5">
<div class="org-src-container"> <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)); <pre class="src src-matlab">data = table2array(readtable(<span class="org-string">'itf_polynom.csv'</span>,<span class="org-string">'NumHeaderLines'</span>,1));
@ -894,7 +895,7 @@ ry1 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>dat
</div> </div>
<div id="org6aebdab" class="figure"> <div id="org96336ed" class="figure">
<p><img src="figs/calibration_drx_pres.png" alt="calibration_drx_pres.png" /> <p><img src="figs/calibration_drx_pres.png" alt="calibration_drx_pres.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>description</p> <p><span class="figure-number">Figure 11: </span>description</p>
@ -967,8 +968,8 @@ f_ry1 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">p
</div> </div>
</div> </div>
<div id="outline-container-org5494ab9" class="outline-3"> <div id="outline-container-orgb94c7fb" class="outline-3">
<h3 id="org5494ab9"><span class="section-number-3">4.6.</span> Test</h3> <h3 id="orgb94c7fb"><span class="section-number-3">4.6.</span> Test</h3>
<div class="outline-text-3" id="text-4-6"> <div class="outline-text-3" id="text-4-6">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">filename = <span class="org-string">"/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5"</span>; <pre 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>;
@ -1000,7 +1001,7 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
</div> </div>
<div id="org85bdc09" class="figure"> <div id="orgb431cdd" class="figure">
<p><img src="figs/drifts_xtal2_detrend.png" alt="drifts_xtal2_detrend.png" /> <p><img src="figs/drifts_xtal2_detrend.png" alt="drifts_xtal2_detrend.png" />
</p> </p>
<p><span class="figure-number">Figure 12: </span>Drifts of the second crystal as a function of Bragg Angle</p> <p><span class="figure-number">Figure 12: </span>Drifts of the second crystal as a function of Bragg Angle</p>
@ -1017,8 +1018,8 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
</div> </div>
</div> </div>
<div id="outline-container-org218f95f" class="outline-3"> <div id="outline-container-org1c4abff" class="outline-3">
<h3 id="org218f95f"><span class="section-number-3">4.7.</span> Repeatability of frame deformation</h3> <h3 id="org1c4abff"><span class="section-number-3">4.7.</span> Repeatability of frame deformation</h3>
<div class="outline-text-3" id="text-4-7"> <div class="outline-text-3" id="text-4-7">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">filename = <span class="org-string">"/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5"</span>; <pre 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>;
@ -1068,15 +1069,15 @@ data_2.dz = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*<
</div> </div>
</div> </div>
<div id="outline-container-org5b8904d" class="outline-2"> <div id="outline-container-orgcf35007" class="outline-2">
<h2 id="org5b8904d"><span class="section-number-2">5.</span> Attocube - Periodic Non-Linearity</h2> <h2 id="orgcf35007"><span class="section-number-2">5.</span> Attocube - Periodic Non-Linearity</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
<p> <p>
<a id="org44bdb6d"></a> <a id="org455a211"></a>
</p> </p>
<p> <p>
Interferometers have some periodic nonlinearity (NO_ITEM_DATA:thurner15_fiber_based_distan_sensin_inter). Interferometers have some periodic nonlinearity (NO_ITEM_DATA:thurner15_fiber_based_distan_sensin_inter).
The period is a fraction of the wavelength (usually \(\lambda/2\)) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams (See NO_ITEM_DATA:ducourtieux18_towar_high_precis_posit_contr, NO_ITEM_DATA:thurner15_fiber_based_distan_sensin_inter). The period is a fraction of the wavelength (usually \(\lambda/2\)) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams (<a href="#citeproc_bib_item_1">Ducourtieux 2018</a>; <a href="#citeproc_bib_item_2">Thurner et al. 2015</a>).
The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers. The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers.
</p> </p>
@ -1093,8 +1094,8 @@ This process is performed over several periods in order to characterize the erro
</p> </p>
</div> </div>
<div id="outline-container-org9f6a57f" class="outline-3"> <div id="outline-container-org91573e9" class="outline-3">
<h3 id="org9f6a57f"><span class="section-number-3">5.1.</span> Measurement Setup</h3> <h3 id="org91573e9"><span class="section-number-3">5.1.</span> Measurement Setup</h3>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-3" id="text-5-1">
<p> <p>
The metrology that will be compared with the interferometers are the strain gauges incorporated in the PI piezoelectric stacks. The metrology that will be compared with the interferometers are the strain gauges incorporated in the PI piezoelectric stacks.
@ -1105,7 +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. 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> </p>
<div class="note" id="org3574ce6"> <div class="note" id="org38e45ca">
<p> <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. Note that the strain gauges are measuring the relative displacement of the piezoelectric stacks while the interferometers are measuring the relative motion between the second crystals and the metrology frame.
</p> </p>
@ -1121,7 +1122,7 @@ As any deformations of the metrology frame of deformation of the crystal&rsquo;s
</div> </div>
<p> <p>
The setup is schematically with the block diagram in Figure <a href="#orgf2d914c">13</a>. The setup is schematically with the block diagram in Figure <a href="#org5df73e7">13</a>.
</p> </p>
<p> <p>
@ -1137,7 +1138,7 @@ The PI controller takes care or controlling to position as measured by the strai
</ul> </ul>
<div id="orgf2d914c" class="figure"> <div id="org5df73e7" class="figure">
<p><img src="figs/block_diagram_lut_attocube.png" alt="block_diagram_lut_attocube.png" /> <p><img src="figs/block_diagram_lut_attocube.png" alt="block_diagram_lut_attocube.png" />
</p> </p>
<p><span class="figure-number">Figure 13: </span>Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube</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>
@ -1149,8 +1150,8 @@ The problem is to estimate the periodic non-linearity of the Attocube from the i
</div> </div>
</div> </div>
<div id="outline-container-orgc06acba" class="outline-3"> <div id="outline-container-orgdd79356" class="outline-3">
<h3 id="orgc06acba"><span class="section-number-3">5.2.</span> Choice of the reference signal</h3> <h3 id="orgdd79356"><span class="section-number-3">5.2.</span> Choice of the reference signal</h3>
<div class="outline-text-3" id="text-5-2"> <div class="outline-text-3" id="text-5-2">
<p> <p>
The main specifications for the reference signal are; The main specifications for the reference signal are;
@ -1174,8 +1175,8 @@ Based on the above discussion, one suitable excitation signal is a sinusoidal sw
</div> </div>
</div> </div>
<div id="outline-container-org26743c0" class="outline-3"> <div id="outline-container-org2268c53" class="outline-3">
<h3 id="org26743c0"><span class="section-number-3">5.3.</span> Repeatability of the non-linearity</h3> <h3 id="org2268c53"><span class="section-number-3">5.3.</span> Repeatability of the non-linearity</h3>
<div class="outline-text-3" id="text-5-3"> <div class="outline-text-3" id="text-5-3">
<p> <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. 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.
@ -1195,8 +1196,8 @@ One way to precisely estimate the laser wavelength is to estimate the non linear
</div> </div>
</div> </div>
<div id="outline-container-org15014d3" class="outline-3"> <div id="outline-container-orgadef69b" class="outline-3">
<h3 id="org15014d3"><span class="section-number-3">5.4.</span> Simulation</h3> <h3 id="orgadef69b"><span class="section-number-3">5.4.</span> Simulation</h3>
<div class="outline-text-3" id="text-5-4"> <div class="outline-text-3" id="text-5-4">
<p> <p>
Suppose we have a first approximation of the non-linear period. Suppose we have a first approximation of the non-linear period.
@ -1216,10 +1217,10 @@ period_nl = period_est <span class="org-builtin">+</span> period_err; <span clas
</div> </div>
<p> <p>
The non-linear errors are first estimated at the beginning of the stroke (Figure <a href="#org94b9790">14</a>). The non-linear errors are first estimated at the beginning of the stroke (Figure <a href="#org4cf8064">14</a>).
</p> </p>
<div id="org94b9790" class="figure"> <div id="org4cf8064" class="figure">
<p><img src="figs/non_linear_errors_start_stroke.png" alt="non_linear_errors_start_stroke.png" /> <p><img src="figs/non_linear_errors_start_stroke.png" alt="non_linear_errors_start_stroke.png" />
</p> </p>
<p><span class="figure-number">Figure 14: </span>Estimation of the non-linear errors at the beginning of the stroke</p> <p><span class="figure-number">Figure 14: </span>Estimation of the non-linear errors at the beginning of the stroke</p>
@ -1227,7 +1228,7 @@ The non-linear errors are first estimated at the beginning of the stroke (Figure
<p> <p>
From this only measurement, it is not possible to estimate with great accuracy the period of the error. 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="#org8301473">15</a>). To do so, the same measurement is performed with a stroke of several millimeters (Figure <a href="#orgb57a43c">15</a>).
</p> </p>
<p> <p>
@ -1236,7 +1237,7 @@ This is due to a mismatch between the estimated period and the true period of th
</p> </p>
<div id="org8301473" class="figure"> <div id="orgb57a43c" class="figure">
<p><img src="figs/non_linear_errors_middle_stroke.png" alt="non_linear_errors_middle_stroke.png" /> <p><img src="figs/non_linear_errors_middle_stroke.png" alt="non_linear_errors_middle_stroke.png" />
</p> </p>
<p><span class="figure-number">Figure 15: </span>Estimated non-linear errors at a latter position</p> <p><span class="figure-number">Figure 15: </span>Estimated non-linear errors at a latter position</p>
@ -1269,7 +1270,7 @@ with \(\lambda_{\text{est}}\) the estimated error&rsquo;s period.
</p> </p>
<p> <p>
From Figure <a href="#org8301473">15</a>, we can see that there is an offset between the two curves. From Figure <a href="#orgb57a43c">15</a>, we can see that there is an offset between the two curves.
Let&rsquo;s call this offset \(\epsilon_x\), we then have: Let&rsquo;s call this offset \(\epsilon_x\), we then have:
</p> </p>
\begin{equation} \begin{equation}
@ -1349,8 +1350,8 @@ The maximum stroke is 2.9 [mm]
</div> </div>
</div> </div>
<div id="outline-container-orga71d790" class="outline-3"> <div id="outline-container-org8ddf260" class="outline-3">
<h3 id="orga71d790"><span class="section-number-3">5.5.</span> Measurements</h3> <h3 id="org8ddf260"><span class="section-number-3">5.5.</span> Measurements</h3>
<div class="outline-text-3" id="text-5-5"> <div class="outline-text-3" id="text-5-5">
<p> <p>
We have some constrains on the way the motion is imposed and measured: We have some constrains on the way the motion is imposed and measured:
@ -1376,10 +1377,20 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).
</div> </div>
</div> </div>
</div> </div>
<div id="outline-container-org515d8fb" class="outline-2">
<h2 id="org515d8fb">Bibliography</h2>
<div class="outline-text-2" id="text-org515d8fb">
<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>
</div>
</div>
</div>
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2022-06-02 Thu 22:25</p> <p class="date">Created: 2022-06-07 Tue 10:58</p>
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@ -1069,7 +1069,7 @@ plot(data_2.bragg, data_2.mframe_u)
** Introduction :ignore: ** Introduction :ignore:
Interferometers have some periodic nonlinearity [cite:@thurner15_fiber_based_distan_sensin_inter]. Interferometers have some periodic nonlinearity [cite:@thurner15_fiber_based_distan_sensin_inter].
The period is a fraction of the wavelength (usually $\lambda/2$) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams [cite:See @ducourtieux18_towar_high_precis_posit_contr page 67 to 69;@thurner15_fiber_based_distan_sensin_inter;]. The period is a fraction of the wavelength (usually $\lambda/2$) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams cite:ducourtieux18_towar_high_precis_posit_contr,thurner15_fiber_based_distan_sensin_inter.
The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers. The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers.
In the DCM case, when using Attocube interferometers, the period non-linearity are in the order of several nanometers with a period of $765\,nm$. In the DCM case, when using Attocube interferometers, the period non-linearity are in the order of several nanometers with a period of $765\,nm$.
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patch(100*xtal2_rectangle(:,1), 100*xtal2_rectangle(:,2), 100*xtal2_rectangle(:,3), 'k-') patch(100*xtal2_rectangle(:,1), 100*xtal2_rectangle(:,2), 100*xtal2_rectangle(:,3), 'k-')
#+end_src #+end_src
* Bibliography
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