From f549fc5313aaede4b340461346618b80883852b6 Mon Sep 17 00:00:00 2001
From: Thomas Dehaeze <dehaeze.thomas@gmail.com>
Date: Thu, 2 Jun 2022 18:06:51 +0200
Subject: [PATCH] Add analysis of errors based on ray-tracing

---
 dcm-metrology.html                            | 1130 ++++++++++-------
 dcm-metrology.org                             |  829 ++++++------
 dcm-metrology.pdf                             |  Bin 703400 -> 724237 bytes
 figs/channel_cut_scan.pdf                     |  Bin 0 -> 21138 bytes
 figs/channel_cut_scan.png                     |  Bin 0 -> 43418 bytes
 figs/motion_beam_drx_error.pdf                |  Bin 16016 -> 16051 bytes
 figs/motion_beam_drx_error.png                |  Bin 25444 -> 26517 bytes
 figs/motion_beam_dry_error.pdf                |  Bin 15348 -> 15334 bytes
 figs/motion_beam_dry_error.png                |  Bin 17330 -> 18139 bytes
 figs/motion_beam_dz_error.pdf                 |  Bin 15093 -> 14798 bytes
 figs/motion_beam_dz_error.png                 |  Bin 19925 -> 19930 bytes
 figs/ray_tracing_channel_cut.pdf              |  Bin 0 -> 6642 bytes
 figs/ray_tracing_channel_cut.png              |  Bin 0 -> 17260 bytes
 figs/ray_tracing_error_drx_overview.pdf       |  Bin 0 -> 5841 bytes
 figs/ray_tracing_error_drx_overview.png       |  Bin 0 -> 5500 bytes
 figs/ray_tracing_error_dry_overview.pdf       |  Bin 0 -> 6680 bytes
 figs/ray_tracing_error_dry_overview.png       |  Bin 0 -> 16390 bytes
 figs/ray_tracing_error_dz_overview.pdf        |  Bin 0 -> 6673 bytes
 figs/ray_tracing_error_dz_overview.png        |  Bin 0 -> 14573 bytes
 matlab/src/getBeamPath.m                      |    8 +-
 ...lane_reflection.m => getPlaneReflection.m} |    6 +-
 matlab/src/plotBeamPath.m                     |    6 +-
 matlab/src/plotCrystals.m                     |   37 +
 23 files changed, 1144 insertions(+), 872 deletions(-)
 create mode 100644 figs/channel_cut_scan.pdf
 create mode 100644 figs/channel_cut_scan.png
 create mode 100644 figs/ray_tracing_channel_cut.pdf
 create mode 100644 figs/ray_tracing_channel_cut.png
 create mode 100644 figs/ray_tracing_error_drx_overview.pdf
 create mode 100644 figs/ray_tracing_error_drx_overview.png
 create mode 100644 figs/ray_tracing_error_dry_overview.pdf
 create mode 100644 figs/ray_tracing_error_dry_overview.png
 create mode 100644 figs/ray_tracing_error_dz_overview.pdf
 create mode 100644 figs/ray_tracing_error_dz_overview.png
 rename matlab/src/{get_plane_reflection.m => getPlaneReflection.m} (57%)
 create mode 100644 matlab/src/plotCrystals.m

diff --git a/dcm-metrology.html b/dcm-metrology.html
index cd27daf..92e55f1 100644
--- a/dcm-metrology.html
+++ b/dcm-metrology.html
@@ -3,7 +3,7 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2022-02-15 mar. 14:25 -->
+<!-- 2022-06-02 Thu 18:06 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <title>ESRF Double Crystal Monochromator - Metrology</title>
 <meta name="author" content="Dehaeze Thomas" />
@@ -39,45 +39,51 @@
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents" role="doc-toc">
 <ul>
-<li><a href="#org958ba4a">1. Metrology Concept</a>
+<li><a href="#org6e38443">1. Metrology Concept</a>
 <ul>
-<li><a href="#org60dec5a">1.1. Sensor Topology</a></li>
-<li><a href="#orgad95b36">1.2. Crystal&rsquo;s motion computation</a></li>
+<li><a href="#orga98f1e4">1.1. Sensor Topology</a></li>
+<li><a href="#org1326a32">1.2. Computation of the relative pose between first and second crystals</a></li>
 </ul>
 </li>
-<li><a href="#org146462e">2. Relation Between Crystal position and X-ray measured displacement</a>
+<li><a href="#orgb627b0b">2. Relation Between Crystal position and X-ray measured displacement</a>
 <ul>
-<li><a href="#org53e570a">2.1. Setup</a></li>
-<li><a href="#org1118f4c">2.2. Relation between second crystal motion and beam motion</a>
-<ul>
-<li><a href="#org48b378e">2.2.1. Axial motion of second crystal</a></li>
-<li><a href="#org9b6ca5e">2.2.2. Ry motion of second crystal</a></li>
-<li><a href="#orge39778e">2.2.3. Rx motion of second crystal</a></li>
+<li><a href="#org50e3742">2.1. Definition of frame</a></li>
+<li><a href="#org469abf4">2.2. Effect of an error in crystal&rsquo;s distance</a></li>
+<li><a href="#org0ea3b85">2.3. Effect of an error in crystal&rsquo;s x parallelism</a></li>
+<li><a href="#org0766a20">2.4. Effect of an error in crystal&rsquo;s y parallelism</a></li>
+<li><a href="#org2d9635c">2.5. Summary</a></li>
+<li><a href="#org65aa9e9">2.6. &ldquo;Channel cut&rdquo; Scan</a></li>
 </ul>
 </li>
-</ul>
-</li>
-<li><a href="#orgc0f14ff">3. Deformations of the Metrology Frame</a>
+<li><a href="#orgb6410e2">3. Determining relative pose between the crystals using the X-ray</a>
 <ul>
-<li><a href="#org54cb1b0">3.1. Measurement Setup</a></li>
-<li><a href="#orga3da277">3.2. Simulations</a></li>
-<li><a href="#orgec833d8">3.3. Comparison</a></li>
-<li><a href="#orga851ac8">3.4. Test</a></li>
-<li><a href="#orgc6c0d26">3.5. Measured frame deformation</a></li>
-<li><a href="#org9ce965b">3.6. Test</a></li>
-<li><a href="#org3613a0c">3.7. Repeatability of frame deformation</a></li>
+<li><a href="#orgba5b368">3.1. Determine the \(y\) parallelism - &ldquo;Rocking Curve&rdquo;</a></li>
+<li><a href="#orgdad1405">3.2. Determine the \(x\) parallelism - Bragg Scan</a></li>
+<li><a href="#orge213a17">3.3. Determine the \(z\) distance - Bragg Scan</a></li>
+<li><a href="#orgeba2e70">3.4. Use Channel cut scan to determine crystal <code>dry</code> parallelism</a></li>
+<li><a href="#orgddd3c9a">3.5. Effect of an error on Bragg angle</a></li>
 </ul>
 </li>
-<li><a href="#orgdedfb8e">4. Attocube - Periodic Non-Linearity</a>
+<li><a href="#org1f48d7e">4. Deformations of the Metrology Frame</a>
 <ul>
-<li><a href="#org324d54e">4.1. Measurement Setup</a></li>
-<li><a href="#org78e73ad">4.2. Choice of the reference signal</a></li>
-<li><a href="#org1965ef7">4.3. Repeatability of the non-linearity</a></li>
-<li><a href="#org3432548">4.4. Simulation</a></li>
-<li><a href="#org018ecfe">4.5. Measurements</a></li>
+<li><a href="#org5ee730d">4.1. Measurement Setup</a></li>
+<li><a href="#org78a952b">4.2. Simulations</a></li>
+<li><a href="#org053b4f5">4.3. Comparison</a></li>
+<li><a href="#org0aa4bea">4.4. Test</a></li>
+<li><a href="#orgfebab90">4.5. Measured frame deformation</a></li>
+<li><a href="#org12eb5ca">4.6. Test</a></li>
+<li><a href="#org2b6cac9">4.7. Repeatability of frame deformation</a></li>
+</ul>
+</li>
+<li><a href="#org27208e8">5. Attocube - Periodic Non-Linearity</a>
+<ul>
+<li><a href="#org62b5863">5.1. Measurement Setup</a></li>
+<li><a href="#orgbaa09c9">5.2. Choice of the reference signal</a></li>
+<li><a href="#org88808b2">5.3. Repeatability of the non-linearity</a></li>
+<li><a href="#org2dfde91">5.4. Simulation</a></li>
+<li><a href="#org1be96dc">5.5. Measurements</a></li>
 </ul>
 </li>
-<li><a href="#org6292dcf">Bibliography</a></li>
 </ul>
 </div>
 </div>
@@ -87,22 +93,22 @@
 
 <p>
 In this document, the metrology system is studied.
-First, in Section <a href="#orgb868c48">1</a> the goal of the metrology system is stated and the proposed concept is described.
+First, in Section <a href="#org883336e">1</a> the goal of the metrology system is stated and the proposed concept is described.
 In order to increase the accuracy of the metrology system, two problems are to be dealt with:
 </p>
 <ul class="org-ul">
-<li>The deformation of the metrology frame under the action of gravity (Section <a href="#orgc876cab">3</a>)</li>
-<li>The periodic non-linearity of the interferometers (Section <a href="#org62f88d9">4</a>)</li>
+<li>The deformation of the metrology frame under the action of gravity (Section <a href="#org8d2959d">4</a>)</li>
+<li>The periodic non-linearity of the interferometers (Section <a href="#org7046a86">5</a>)</li>
 </ul>
 
-<div id="outline-container-org958ba4a" class="outline-2">
-<h2 id="org958ba4a"><span class="section-number-2">1.</span> Metrology Concept</h2>
+<div id="outline-container-org6e38443" class="outline-2">
+<h2 id="org6e38443"><span class="section-number-2">1.</span> Metrology Concept</h2>
 <div class="outline-text-2" id="text-1">
 <p>
-<a id="orgb868c48"></a>
+<a id="org883336e"></a>
 </p>
 <p>
-The goal of the metrology system is to measure the distance and default of parallelism orientation between the first and second crystals
+The goal of the metrology system is to measure the distance and default of parallelism between the first and second crystals.
 </p>
 
 <p>
@@ -114,8 +120,8 @@ Only 3 degrees of freedom are of interest:
 <li>\(r_x\)</li>
 </ul>
 </div>
-<div id="outline-container-org60dec5a" class="outline-3">
-<h3 id="org60dec5a"><span class="section-number-3">1.1.</span> Sensor Topology</h3>
+<div id="outline-container-orga98f1e4" class="outline-3">
+<h3 id="orga98f1e4"><span class="section-number-3">1.1.</span> Sensor Topology</h3>
 <div class="outline-text-3" id="text-1-1">
 <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.
@@ -123,7 +129,13 @@ Three interferometers are used to measured the 3dof of interest for each crystal
 Three additional interferometers are used to measured the relative motion of the metrology frame.
 </p>
 
-<table id="org93de062" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
+
+<p>
+In total, there are 15 interferometers represented in Figure <a href="#org1728a4b">1</a>.
+The measurements are summarized in Table <a href="#org33e1a4e">2</a>.
+</p>
+
+<table id="org755759d" 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>
 
 <colgroup>
@@ -133,8 +145,8 @@ Three additional interferometers are used to measured the relative motion of the
 </colgroup>
 <thead>
 <tr>
-<th scope="col" class="org-left">Notation</th>
-<th scope="col" class="org-left">Meaning</th>
+<th scope="col" class="org-left"><b>Notation</b></th>
+<th scope="col" class="org-left"><b>Meaning</b></th>
 </tr>
 </thead>
 <tbody>
@@ -175,8 +187,118 @@ Three additional interferometers are used to measured the relative motion of the
 </tbody>
 </table>
 
+<table id="org33e1a4e" 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>
 
-<div id="orgaaab71b" class="figure">
+<colgroup>
+<col  class="org-right" />
+
+<col  class="org-left" />
+
+<col  class="org-left" />
+</colgroup>
+<thead>
+<tr>
+<th scope="col" class="org-right"><b>Number</b></th>
+<th scope="col" class="org-left"><b>Measurement</b></th>
+<th scope="col" class="org-left"><b>Description</b></th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="org-right">1</td>
+<td class="org-left">\(z_{1r,u}\)</td>
+<td class="org-left">First &ldquo;Ring&rdquo; Crystal, &ldquo;upstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">2</td>
+<td class="org-left">\(z_{1r,c}\)</td>
+<td class="org-left">First &ldquo;Ring&rdquo; Crystal, &ldquo;center&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">3</td>
+<td class="org-left">\(z_{1r,d}\)</td>
+<td class="org-left">First &ldquo;Ring&rdquo; Crystal, &ldquo;downstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">4</td>
+<td class="org-left">\(z_{1h,u}\)</td>
+<td class="org-left">First &ldquo;Hall&rdquo; Crystal, &ldquo;upstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">5</td>
+<td class="org-left">\(z_{1h,c}\)</td>
+<td class="org-left">First &ldquo;Hall&rdquo; Crystal, &ldquo;center&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">6</td>
+<td class="org-left">\(z_{1h,d}\)</td>
+<td class="org-left">First &ldquo;Hall&rdquo; Crystal, &ldquo;downstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">7</td>
+<td class="org-left">\(z_{2h,u}\)</td>
+<td class="org-left">Second &ldquo;Hall&rdquo; Crystal, &ldquo;upstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">8</td>
+<td class="org-left">\(z_{2h,c}\)</td>
+<td class="org-left">Second &ldquo;Hall&rdquo; Crystal, &ldquo;center&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">9</td>
+<td class="org-left">\(z_{2h,d}\)</td>
+<td class="org-left">Second &ldquo;Hall&rdquo; Crystal, &ldquo;downstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">10</td>
+<td class="org-left">\(z_{2r,u}\)</td>
+<td class="org-left">Second &ldquo;Ring&rdquo; Crystal, &ldquo;upstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">11</td>
+<td class="org-left">\(z_{2r,c}\)</td>
+<td class="org-left">Second &ldquo;Ring&rdquo; Crystal, &ldquo;center&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">12</td>
+<td class="org-left">\(z_{2r,d}\)</td>
+<td class="org-left">Second &ldquo;Ring&rdquo; Crystal, &ldquo;downstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">13</td>
+<td class="org-left">\(z_{mf,u}\)</td>
+<td class="org-left">Metrology Frame, &ldquo;upstream&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">14</td>
+<td class="org-left">\(z_{mf,dr}\)</td>
+<td class="org-left">Metrology Frame, &ldquo;downstream-ring&rdquo;</td>
+</tr>
+
+<tr>
+<td class="org-right">15</td>
+<td class="org-left">\(z_{mf,dh}\)</td>
+<td class="org-left">Metrology Frame, &ldquo;downstream-hall&rdquo;</td>
+</tr>
+</tbody>
+</table>
+
+
+<div id="org1728a4b" class="figure">
 <p><img src="figs/metrology_schematic.png" alt="metrology_schematic.png" />
 </p>
 <p><span class="figure-number">Figure 1: </span>Schematic of the Metrology System</p>
@@ -184,326 +306,502 @@ Three additional interferometers are used to measured the relative motion of the
 </div>
 </div>
 
-<div id="outline-container-orgad95b36" class="outline-3">
-<h3 id="orgad95b36"><span class="section-number-3">1.2.</span> Crystal&rsquo;s motion computation</h3>
+<div id="outline-container-org1326a32" class="outline-3">
+<h3 id="org1326a32"><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">
 <p>
-From the raw interferometric measurements, the pose between the first and second crystals can be computed.
+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>).
 </p>
 
 <p>
-First, Jacobian matrices can be used to convert raw interferometer measurements to axial displacement and orientation of the crystals and metrology frame.
+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>
 
+<div class="note" id="org28ac0ce">
 <p>
-For the 311 crystals:
-</p>
-
-<table id="org3fe0d44" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
-<caption class="t-above"><span class="table-number">Table 2:</span> Table caption</caption>
-
-<colgroup>
-<col  class="org-left" />
-
-<col  class="org-left" />
-</colgroup>
-<thead>
-<tr>
-<th scope="col" class="org-left">Notation</th>
-<th scope="col" class="org-left">Description</th>
-</tr>
-</thead>
-<tbody>
-<tr>
-<td class="org-left"><code>um</code></td>
-<td class="org-left">Metrology Frame - Upstream</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>dhm</code></td>
-<td class="org-left">Metrology Frame - Downstream Hall</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>drm</code></td>
-<td class="org-left">Metrology Frame - Downstream Ring</td>
-</tr>
-</tbody>
-<tbody>
-<tr>
-<td class="org-left"><code>ur1</code></td>
-<td class="org-left">First Crystal - Upstream Ring</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>h1</code></td>
-<td class="org-left">First Crystal - Hall</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>dr1</code></td>
-<td class="org-left">First Crystal - Downstream Ring</td>
-</tr>
-</tbody>
-<tbody>
-<tr>
-<td class="org-left"><code>ur2</code></td>
-<td class="org-left">First Crystal - Upstream Ring</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>h2</code></td>
-<td class="org-left">First Crystal - Hall</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>dr2</code></td>
-<td class="org-left">First Crystal - Downstream Ring</td>
-</tr>
-</tbody>
-</table>
-
-<table id="org1e5a4a0" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
-<caption class="t-above"><span class="table-number">Table 3:</span> Table caption</caption>
-
-<colgroup>
-<col  class="org-left" />
-
-<col  class="org-left" />
-</colgroup>
-<thead>
-<tr>
-<th scope="col" class="org-left">Notation</th>
-<th scope="col" class="org-left">Description</th>
-</tr>
-</thead>
-<tbody>
-<tr>
-<td class="org-left"><code>dzm</code></td>
-<td class="org-left">Positive: increase of distance</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>rym</code></td>
-<td class="org-left">&#xa0;</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>rxm</code></td>
-<td class="org-left">&#xa0;</td>
-</tr>
-</tbody>
-<tbody>
-<tr>
-<td class="org-left"><code>dz1</code></td>
-<td class="org-left">Positive: decrease of distance</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>ry1</code></td>
-<td class="org-left">&#xa0;</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>rx1</code></td>
-<td class="org-left">&#xa0;</td>
-</tr>
-</tbody>
-<tbody>
-<tr>
-<td class="org-left"><code>dz2</code></td>
-<td class="org-left">Positive: increase of distance</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>ry2</code></td>
-<td class="org-left">&#xa0;</td>
-</tr>
-
-<tr>
-<td class="org-left"><code>rx2</code></td>
-<td class="org-left">&#xa0;</td>
-</tr>
-</tbody>
-</table>
-
-
-<div id="org83db6ba" class="figure">
-<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_m.png" alt="schematic_sensor_jacobian_forward_kinematics_m.png" />
-</p>
-<p><span class="figure-number">Figure 2: </span>Forward Kinematics for the Metrology frame</p>
-</div>
-
-
-<div id="orge535a69" class="figure">
-<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_1.png" alt="schematic_sensor_jacobian_forward_kinematics_1.png" />
-</p>
-<p><span class="figure-number">Figure 3: </span>Forward Kinematics for the 1st crystal</p>
-</div>
-
-
-<div id="org03be022" class="figure">
-<p><img src="figs/schematic_sensor_jacobian_forward_kinematics_2.png" alt="schematic_sensor_jacobian_forward_kinematics_2.png" />
-</p>
-<p><span class="figure-number">Figure 4: </span>Forward Kinematics for the 2nd crystal</p>
-</div>
-
-<p>
-Then, the displacement and orientations can be combined as follows:
-</p>
-\begin{align}
-  d_{z}   &= + d_{z1} - d_{z2} + d_{zm} \\
-  d_{r_y} &= - r_{y1} + r_{y2} - r_{ym} \\
-  d_{r_x} &= - r_{x1} + r_{x2} - r_{xm}
-\end{align}
-
-<p>
-Therefore:
+The sign conventions for the relative crystal pose are:
 </p>
 <ul class="org-ul">
-<li>\(d_z\) represents the distance between the two crystals</li>
-<li>\(d_{r_y}\) represents the rotation of the second crystal w.r.t. the first crystal around \(y\) axis</li>
-<li>\(d_{r_x}\) represents the rotation of the second crystal w.r.t. the first crystal around \(x\) axis</li>
+<li>An increase of \(d_{h,z}\) means the two crystals are further apart</li>
+<li>An increase of \(r_{h,}\) means that the second crystals experiences a rotation around \(y\) with respect to the primary crystal</li>
+<li>An increase of \(r_{h,x}\) means that the second crystals experiences a rotation around \(x\) with respect to the primary crystal</li>
+</ul>
+
+</div>
+
+<p>
+The relative pose can be expressed as a function of the interferometers using the Jacobian matrices for the &ldquo;hall&rdquo; crystals:
+</p>
+\begin{equation}
+\boxed{
+\begin{bmatrix} d_{h,z} \\ r_{h,y} \\ r_{h,x} \end{bmatrix}
+=
+\bm{J}_{2h,s}^{-1}
+\begin{bmatrix} z_{2h,u} \\ z_{2h,c} \\ z_{2h,d} \end{bmatrix}
+-
+\bm{J}_{1h,s}^{-1}
+\begin{bmatrix} z_{1h,u} \\ z_{1h,c} \\ z_{1h,d} \end{bmatrix}
+-
+\bm{J}_{mf,s}^{-1}
+\begin{bmatrix} z_{mf,u} \\ z_{mf,dh} \\ z_{mf,dr} \end{bmatrix}
+}
+\end{equation}
+
+<p>
+As well as for the &ldquo;ring&rdquo; crystals:
+</p>
+\begin{equation}
+\boxed{
+\begin{bmatrix} d_{r,z} \\ r_{r,y} \\ r_{r,x} \end{bmatrix}
+=
+\bm{J}_{2r,s}^{-1}
+\begin{bmatrix} z_{2r,u} \\ z_{2r,c} \\ z_{2r,d} \end{bmatrix}
+-
+\bm{J}_{1r,s}^{-1}
+\begin{bmatrix} z_{1r,u} \\ z_{1r,c} \\ z_{1r,d} \end{bmatrix}
+-
+\bm{J}_{mf,s}^{-1}
+\begin{bmatrix} z_{mf,u} \\ z_{mf,dr} \\ z_{mf,dr} \end{bmatrix}
+}
+\end{equation}
+
+<p>
+Values of the matrices can be found in the document describing the kinematics of the DCM (see <code>https://gitlab.esrf.fr/dehaeze/dcm-kinematics</code>).
+</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgb627b0b" class="outline-2">
+<h2 id="orgb627b0b"><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">
+<p>
+<a id="orgbc74a6f"></a>
+</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.
+</p>
+
+<p>
+This is very important in order to:
+</p>
+<ul class="org-ul">
+<li>link a measurement of the x-ray beam position to a default in the crystal position</li>
+<li>understand which pose default will have large impact on the output beam position/orientation</li>
+<li>calibrate the deformations of the metrology frame using an external metrology measuring the x-ray beam position/orientation</li>
 </ul>
 
 <p>
-If \(d_{r_y}\) is positive, the second crystal has a positive rotation around \(y\) w.r.t. the first crystal.
-Therefore, the second crystal should be actuated such that it is making a negative rotation around \(y\) w.r.t. metrology frame.
+In order to simplify the problem, the first crystal is supposed to be fixed (i.e. ideally positioned), and only the motion of the second crystal is studied.
 </p>
 
 <p>
-The Jacobian matrices are defined as follow:
+In order to easily study that, &ldquo;ray tracing&rdquo; techniques are used.
 </p>
-<div class="org-src-container">
-<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for the metrology frame</span>
-J_m = [1,  0.102,  0
-       1, <span class="org-builtin">-</span>0.088,  0.1275
-       1, <span class="org-builtin">-</span>0.088, <span class="org-builtin">-</span>0.1275];
-
-<span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 1st "111" crystal</span>
-J_s_111_1 = [<span class="org-builtin">-</span>1, <span class="org-builtin">-</span>0.036, <span class="org-builtin">-</span>0.015
-             <span class="org-builtin">-</span>1,  0,      0.015
-             <span class="org-builtin">-</span>1,  0.036, <span class="org-builtin">-</span>0.015];
-
-<span class="org-matlab-cellbreak">%% Sensor Jacobian matrix for 2nd "111" crystal</span>
-J_s_111_2 = [1,  0.07,  0.015
-             1,  0,    <span class="org-builtin">-</span>0.015
-             1, <span class="org-builtin">-</span>0.07,  0.015];
-</pre>
 </div>
+<div id="outline-container-org50e3742" class="outline-3">
+<h3 id="org50e3742"><span class="section-number-3">2.1.</span> Definition of frame</h3>
+<div class="outline-text-3" id="text-2-1">
+<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
 
-<p>
-Therefore, the matrix that gives the relative pose of the crystal from the 9 interferometers is:
-</p>
-<div class="org-src-container">
-<pre class="src src-matlab"><span class="org-matlab-cellbreak">%% Compute the transformation matrix</span>
-G_111_t = [<span class="org-builtin">-</span>inv(J_s_111_1), inv(J_s_111_2), <span class="org-builtin">-</span>inv(J_m)];
-
-<span class="org-comment-delimiter">% </span><span class="org-comment">Sign convention for the axial motion</span>
-G_111_t(1,<span class="org-builtin">:</span>) = <span class="org-builtin">-</span>G_111_t(1,<span class="org-builtin">:</span>);
-</pre>
-</div>
-
-<table id="orgdf27d5d" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
-<caption class="t-above"><span class="table-number">Table 4:</span> Transformation Matrix</caption>
 
 <colgroup>
 <col  class="org-left" />
 
-<col  class="org-right" />
+<col  class="org-left" />
 
-<col  class="org-right" />
-
-<col  class="org-right" />
-
-<col  class="org-right" />
-
-<col  class="org-right" />
-
-<col  class="org-right" />
-
-<col  class="org-right" />
-
-<col  class="org-right" />
-
-<col  class="org-right" />
+<col  class="org-left" />
 </colgroup>
 <thead>
 <tr>
 <th scope="col" class="org-left">&#xa0;</th>
-<th scope="col" class="org-right"><code>ur1</code> [nm]</th>
-<th scope="col" class="org-right"><code>h1</code> [nm]</th>
-<th scope="col" class="org-right"><code>dr1</code> [nm]</th>
-<th scope="col" class="org-right"><code>ur2</code> [nm]</th>
-<th scope="col" class="org-right"><code>h2</code> [nm]</th>
-<th scope="col" class="org-right"><code>dr1</code> [nm]</th>
-<th scope="col" class="org-right"><code>um</code> [nm]</th>
-<th scope="col" class="org-right"><code>dhm</code> [nm]</th>
-<th scope="col" class="org-right"><code>drm</code> [nm]</th>
+<th scope="col" class="org-left">Position</th>
+<th scope="col" class="org-left">Orientation</th>
 </tr>
 </thead>
 <tbody>
 <tr>
-<td class="org-left"><code>dz</code> [nm]</td>
-<td class="org-right">-0.25</td>
-<td class="org-right">-0.5</td>
-<td class="org-right">-0.25</td>
-<td class="org-right">-0.25</td>
-<td class="org-right">-0.5</td>
-<td class="org-right">-0.25</td>
-<td class="org-right">0.463</td>
-<td class="org-right">0.268</td>
-<td class="org-right">0.268</td>
+<td class="org-left">input beam</td>
+<td class="org-left"><code>x1,y1,z1</code></td>
+<td class="org-left"><code>s1</code></td>
 </tr>
 
 <tr>
-<td class="org-left"><code>rx</code> [nrad]</td>
-<td class="org-right">13.889</td>
-<td class="org-right">0.0</td>
-<td class="org-right">-13.889</td>
-<td class="org-right">7.143</td>
-<td class="org-right">0.0</td>
-<td class="org-right">-7.143</td>
-<td class="org-right">-5.263</td>
-<td class="org-right">2.632</td>
-<td class="org-right">2.632</td>
+<td class="org-left">primary mirror</td>
+<td class="org-left"><code>xp,yp,zp</code></td>
+<td class="org-left"><code>np</code></td>
 </tr>
 
 <tr>
-<td class="org-left"><code>ry</code> [nrad]</td>
-<td class="org-right">16.667</td>
-<td class="org-right">-33.333</td>
-<td class="org-right">16.667</td>
-<td class="org-right">16.667</td>
-<td class="org-right">-33.333</td>
-<td class="org-right">16.667</td>
-<td class="org-right">0.0</td>
-<td class="org-right">-3.922</td>
-<td class="org-right">3.922</td>
+<td class="org-left">reflected beam</td>
+<td class="org-left"><code>x2,y2,z2</code></td>
+<td class="org-left"><code>s2</code></td>
+</tr>
+
+<tr>
+<td class="org-left">secondary mirror</td>
+<td class="org-left"><code>xs,ys,zz</code></td>
+<td class="org-left"><code>ns</code></td>
+</tr>
+
+<tr>
+<td class="org-left">output beam</td>
+<td class="org-left"><code>x3,y3,z3</code></td>
+<td class="org-left"><code>s3</code></td>
+</tr>
+
+<tr>
+<td class="org-left">Dectector</td>
+<td class="org-left"><code>xd,yd,zd</code></td>
+<td class="org-left"><code>nd</code></td>
 </tr>
 </tbody>
 </table>
 
+<ul class="org-ul">
+<li class="off"><code>[&#xa0;]</code> Add schematic</li>
+</ul>
+
 <p>
-From table <a href="#org03be022">4</a>, we can determine the effect of each interferometer on the estimated relative pose between the crystals.
-For instance, an error on <code>dr1</code> will have much greater impact on <code>ry</code> than an error on <code>drm</code>.
+Notations for the pose of the output beam: <code>d_b_y</code>, <code>d_b_z</code>, <code>r_b_y</code>, <code>r_b_z</code>.
+</p>
+
+<p>
+\(d_{b,y}, d_{b,z}, r_{b,y}, r_{b,z}\)
+</p>
+
+<p>
+The xy position of the beam is taken in the \(x=0\) plane.
 </p>
 </div>
 </div>
+
+<div id="outline-container-org469abf4" class="outline-3">
+<h3 id="org469abf4"><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">
+<p>
+<a id="org4eba8de"></a>
+</p>
+
+<p>
+In Figure <a href="#orgf88272f">2</a> is shown the light path for three bragg angles (5, 55 and 85 degrees) when there is an error in the <code>dz</code> position of 1mm.
+</p>
+
+<p>
+Visually, it is clear that this induce a <code>z</code> offset of the output beam.
+</p>
+
+
+<div id="orgf88272f" class="figure">
+<p><img src="figs/ray_tracing_error_dz_overview.png" alt="ray_tracing_error_dz_overview.png" />
+</p>
+<p><span class="figure-number">Figure 2: </span>Visual Effect of an error in <code>dz</code> (1mm). Side view.</p>
 </div>
 
-<div id="outline-container-org146462e" class="outline-2">
-<h2 id="org146462e"><span class="section-number-2">2.</span> Relation Between Crystal position and X-ray measured displacement</h2>
-<div class="outline-text-2" id="text-2">
-</div>
-<div id="outline-container-org53e570a" class="outline-3">
-<h3 id="org53e570a"><span class="section-number-3">2.1.</span> Setup</h3>
-<div class="outline-text-3" id="text-2-1">
+<p>
+The motion of the output beam is displayed as a function of the Bragg angle in Figure <a href="#org592eac4">3</a>.
+It is clear that an error in the distance <code>dz</code> between the crystals only induce a <code>z</code> offset of the output beam.
+This offset decreases with the Bragg angle.
+</p>
 
-<div id="orgbd2501f" class="figure">
+<p>
+This is indeed equal to:
+</p>
+\begin{equation}
+\boxed{d_{b,z} = 2 d_z \cos \theta}
+\end{equation}
+
+
+<div id="org592eac4" class="figure">
+<p><img src="figs/motion_beam_dz_error.png" alt="motion_beam_dz_error.png" />
+</p>
+<p><span class="figure-number">Figure 3: </span>Motion of the output beam with dZ error</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org0ea3b85" class="outline-3">
+<h3 id="org0ea3b85"><span class="section-number-3">2.3.</span> Effect of an error in crystal&rsquo;s x parallelism</h3>
+<div class="outline-text-3" id="text-2-3">
+<p>
+<a id="orged7b509"></a>
+</p>
+
+<p>
+The effect of an error in <code>rx</code> crystal parallelism on the output beam is visually shown in Figure <a href="#orgad85ff2">4</a> for three bragg angles (5, 55 and 85 degrees).
+The error is set to one degree, and the top view is shown.
+It is clear that the output beam experiences some rotation around a vertical axis.
+The amount of rotation depends on the bragg angle.
+</p>
+
+
+<div id="orgad85ff2" class="figure">
+<p><img src="figs/ray_tracing_error_drx_overview.png" alt="ray_tracing_error_drx_overview.png" />
+</p>
+<p><span class="figure-number">Figure 4: </span>Visual Effect of an error in <code>drx</code> (1 degree). Top View.</p>
+</div>
+
+<p>
+The effect of <code>drx</code> as a function of the Bragg angle on the output beam pose is computed and shown in Figure <a href="#org785519c">5</a>.
+</p>
+
+<p>
+It induces a rotation of the output beam in the <code>z</code> direction that depends on the Bragg Angle.
+It is starting at zero for small bragg angles, and it increases with the bragg angle up to 2 times <code>drx</code>.
+This is indeed equal to:
+</p>
+\begin{equation}
+\boxed{r_{b,z} = 2 r_x \cos \theta}
+\end{equation}
+
+<p>
+If also induces a small \(y\) shift of the beam.
+This shift is due to the fact that the rotation point (around which the second crystal is moved) is changing as a function of bragg.
+We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degrees, at which point the center of rotation of the second crystal is at \(x = 0\);
+</p>
+
+
+<div id="org785519c" 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>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org0766a20" class="outline-3">
+<h3 id="org0766a20"><span class="section-number-3">2.4.</span> Effect of an error in crystal&rsquo;s y parallelism</h3>
+<div class="outline-text-3" id="text-2-4">
+<p>
+<a id="org5081b73"></a>
+</p>
+
+<p>
+The effect of an error in <code>ry</code> crystal parallelism on the output beam is visually shown in Figure <a href="#org1ee280e">6</a> for three bragg angles (5, 55 and 85 degrees).
+</p>
+
+
+<div id="org1ee280e" class="figure">
+<p><img src="figs/ray_tracing_error_dry_overview.png" alt="ray_tracing_error_dry_overview.png" />
+</p>
+<p><span class="figure-number">Figure 6: </span>Visual Effect of an error in <code>dry</code> (1 degree). Side view.</p>
+</div>
+
+<p>
+The effect of <code>dry</code> as a function of the Bragg angle on the output beam pose is computed and shown in Figure <a href="#org5d3201f">7</a>.
+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}
+\boxed{r_{b,z} = 2 r_z}
+\end{equation}
+
+<p>
+It also induces a small vertical motion of the beam (at the \(x=0\) location) which is simply due to the fact that the \(x\) coordinate of the impact point on the second crystal changes with the Bragg angle.
+</p>
+
+
+<div id="org5d3201f" 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>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org2d9635c" class="outline-3">
+<h3 id="org2d9635c"><span class="section-number-3">2.5.</span> Summary</h3>
+<div class="outline-text-3" id="text-2-5">
+<p>
+Effects of crystal&rsquo;s pose errors on the output beam are summarized in Table <a href="#org40cfc37">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&rsquo;s distance does not depend on the Bragg angle.
+</p>
+
+<table id="org40cfc37" 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>
+
+<colgroup>
+<col  class="org-left" />
+
+<col  class="org-right" />
+
+<col  class="org-left" />
+
+<col  class="org-left" />
+</colgroup>
+<thead>
+<tr>
+<th scope="col" class="org-left"><b>Beam Motion</b></th>
+<th scope="col" class="org-right"><b>Crystal</b>     \(d_z\)</th>
+<th scope="col" class="org-left"><b>Crystal</b>                      \(r_x\)</th>
+<th scope="col" class="org-left"><b>Crystal</b>    \(r_y\)</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="org-left">\(d_{b,y}\)</td>
+<td class="org-right">0</td>
+<td class="org-left">\(\approx 0\)</td>
+<td class="org-left">0</td>
+</tr>
+
+<tr>
+<td class="org-left">\(d_{b,z}\)</td>
+<td class="org-right">\(\boxed{d_{z} 2 \cos(\theta)}\)</td>
+<td class="org-left">0</td>
+<td class="org-left">\(\approx 0\)</td>
+</tr>
+
+<tr>
+<td class="org-left">\(r_{b,y}\)</td>
+<td class="org-right">0</td>
+<td class="org-left">0</td>
+<td class="org-left">\(\boxed{2 r_{y}}\)</td>
+</tr>
+
+<tr>
+<td class="org-left">\(r_{b,z}\)</td>
+<td class="org-right">0</td>
+<td class="org-left">\(\boxed{- r_x 2 \sin(\theta)}\)</td>
+<td class="org-left">0</td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+
+<div id="outline-container-org65aa9e9" class="outline-3">
+<h3 id="org65aa9e9"><span class="section-number-3">2.6.</span> &ldquo;Channel cut&rdquo; Scan</h3>
+<div class="outline-text-3" id="text-2-6">
+<p>
+A &ldquo;channel cut&rdquo; scan is a Bragg scan where the distance between the crystals is fixed.
+</p>
+
+<p>
+This is visually shown in Figure <a href="#orgbde7e5a">8</a> where it is clear that the output beam experiences some vertical motion.
+</p>
+
+
+<div id="orgbde7e5a" class="figure">
+<p><img src="figs/ray_tracing_channel_cut.png" alt="ray_tracing_channel_cut.png" />
+</p>
+<p><span class="figure-number">Figure 8: </span>Visual Effect of a channel cut scan</p>
+</div>
+
+<p>
+The \(z\) offset of the beam for several channel cut scans are shown in Figure <a href="#org24bc0ad">9</a>.
+</p>
+
+
+<div id="org24bc0ad" 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 &ldquo;channel cut&rdquo; scans</p>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgb6410e2" class="outline-2">
+<h2 id="orgb6410e2"><span class="section-number-2">3.</span> Determining relative pose between the crystals using the X-ray</h2>
+<div class="outline-text-2" id="text-3">
+<p>
+As Interferometers are only measuring <i>relative</i> displacement, it is mandatory to initialize them correctly.
+</p>
+
+<p>
+They should be initialize in such a way that:
+</p>
+<ul class="org-ul">
+<li>\(r_x = 0\) and \(r_y = 0\) when the two crystallographic planes are parallel</li>
+<li>measured \(d_z\) is equal to the distance between the crystallographic planes</li>
+</ul>
+
+<p>
+In order to do that, an external metrology using the x-ray is used.
+</p>
+</div>
+
+<div id="outline-container-orgba5b368" class="outline-3">
+<h3 id="orgba5b368"><span class="section-number-3">3.1.</span> Determine the \(y\) parallelism - &ldquo;Rocking Curve&rdquo;</h3>
+</div>
+
+<div id="outline-container-orgdad1405" class="outline-3">
+<h3 id="orgdad1405"><span class="section-number-3">3.2.</span> Determine the \(x\) parallelism - Bragg Scan</h3>
+</div>
+
+
+<div id="outline-container-orge213a17" class="outline-3">
+<h3 id="orge213a17"><span class="section-number-3">3.3.</span> Determine the \(z\) distance - Bragg Scan</h3>
+</div>
+
+<div id="outline-container-orgeba2e70" class="outline-3">
+<h3 id="orgeba2e70"><span class="section-number-3">3.4.</span> Use Channel cut scan to determine crystal <code>dry</code> parallelism</h3>
+<div class="outline-text-3" id="text-3-4">
+<p>
+Now, let&rsquo;s suppose we want to determine the <code>dry</code> angle between the crystals.
+</p>
+
+<div class="org-src-container">
+<pre class="src src-matlab">dry = 1e<span class="org-builtin">-</span>6; <span class="org-comment-delimiter">% </span><span class="org-comment">[rad]</span>
+</pre>
+</div>
+
+<p>
+The error is
+</p>
+\begin{equation}
+\boxed{d_{b,z} = -2 d_z \cos()}
+\end{equation}
+</div>
+</div>
+
+<div id="outline-container-orgddd3c9a" class="outline-3">
+<h3 id="orgddd3c9a"><span class="section-number-3">3.5.</span> Effect of an error on Bragg angle</h3>
+</div>
+</div>
+
+<div id="outline-container-org1f48d7e" class="outline-2">
+<h2 id="org1f48d7e"><span class="section-number-2">4.</span> Deformations of the Metrology Frame</h2>
+<div class="outline-text-2" id="text-4">
+<p>
+<a id="org8d2959d"></a>
+</p>
+<p>
+The transformation matrices are valid only if the metrology frames are solid bodies.
+</p>
+
+<p>
+The metrology frame itself is experiencing some deformations due to the gravity.
+When the bragg axis is scanned, the effect of gravity on the metrology frame is changing and this introduce some measurement errors.
+</p>
+
+<p>
+This can be calibrated.
+</p>
+</div>
+<div id="outline-container-org5ee730d" class="outline-3">
+<h3 id="org5ee730d"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
+<div class="outline-text-3" id="text-4-1">
+<p>
+Two beam viewers:
+</p>
+<ul class="org-ul">
+<li>one close to the DCM to measure position of the beam</li>
+<li>one far away to the DCM to measure orientation of the beam</li>
+</ul>
+
+<p>
+For each Bragg angle, the Fast Jacks are actuated to that the beam is at the center of the beam viewer.
+Then, then position of the crystals as measured by the interferometers is recorded.
+This position is the wanted position for a given Bragg angle.
+</p>
+
+
+<div id="org3be0478" class="figure">
 <p><img src="figs/calibration_setup.png" alt="calibration_setup.png" />
 </p>
-<p><span class="figure-number">Figure 5: </span>Schematic of the setup</p>
+<p><span class="figure-number">Figure 10: </span>Schematic of the setup</p>
 </div>
 
 <p>
@@ -529,104 +827,10 @@ Frame rate is: 42 fps
 </div>
 </div>
 
-<div id="outline-container-org1118f4c" class="outline-3">
-<h3 id="org1118f4c"><span class="section-number-3">2.2.</span> Relation between second crystal motion and beam motion</h3>
-<div class="outline-text-3" id="text-2-2">
-</div>
-<div id="outline-container-org48b378e" class="outline-4">
-<h4 id="org48b378e"><span class="section-number-4">2.2.1.</span> Axial motion of second crystal</h4>
-<div class="outline-text-4" id="text-2-2-1">
-<p>
-Let&rsquo;s consider the relation between the \([y, z]\) motion of the beam and the motion of the second crystal \([z^\prime, R_{y^\prime}, R_{x^\prime}]\).
-</p>
 
-
-<div id="org7214e82" class="figure">
-<p><img src="figs/relation_dz_output_beam.png" alt="relation_dz_output_beam.png" />
-</p>
-<p><span class="figure-number">Figure 6: </span>Relation between \(d_{z^\prime}\) motion of the second crystal and vertical motion of the beam</p>
-</div>
-
-\begin{equation}
-  d_z = d_{z^\prime} 2 \cos \theta
-\end{equation}
-
-
-<div id="org101c11d" class="figure">
-<p><img src="figs/relation_vert_motion_crystal_beam.png" alt="relation_vert_motion_crystal_beam.png" />
-</p>
-<p><span class="figure-number">Figure 7: </span>Relation between vertical motion of the second crystal and vertical motion of the output beam</p>
-</div>
-</div>
-</div>
-
-<div id="outline-container-org9b6ca5e" class="outline-4">
-<h4 id="org9b6ca5e"><span class="section-number-4">2.2.2.</span> Ry motion of second crystal</h4>
-<div class="outline-text-4" id="text-2-2-2">
-\begin{equation}
-d_z = D_{\text{vlm}} d_{R_y^\prime}
-\end{equation}
-
-<p>
-with \(D_{\text{vlm}} \approx 10\,m\).
-</p>
-</div>
-</div>
-
-<div id="outline-container-orge39778e" class="outline-4">
-<h4 id="orge39778e"><span class="section-number-4">2.2.3.</span> Rx motion of second crystal</h4>
-<div class="outline-text-4" id="text-2-2-3">
-\begin{equation}
-d_y = 2 D_{\text{vlm}} \sin \theta \cdot d_{R_x^\prime}
-\end{equation}
-</div>
-</div>
-</div>
-</div>
-
-
-
-<div id="outline-container-orgc0f14ff" class="outline-2">
-<h2 id="orgc0f14ff"><span class="section-number-2">3.</span> Deformations of the Metrology Frame</h2>
-<div class="outline-text-2" id="text-3">
-<p>
-<a id="orgc876cab"></a>
-</p>
-<p>
-The transformation matrix in Table <a href="#orgdf27d5d">4</a> is valid only if the metrology frames are solid bodies.
-</p>
-
-<p>
-The metrology frame itself is experiencing some deformations due to the gravity.
-When the bragg axis is scanned, the effect of gravity on the metrology frame is changing and this introduce some measurement errors.
-</p>
-
-<p>
-This can be calibrated.
-</p>
-</div>
-<div id="outline-container-org54cb1b0" class="outline-3">
-<h3 id="org54cb1b0"><span class="section-number-3">3.1.</span> Measurement Setup</h3>
-<div class="outline-text-3" id="text-3-1">
-<p>
-Two beam viewers:
-</p>
-<ul class="org-ul">
-<li>one close to the DCM to measure position of the beam</li>
-<li>one far away to the DCM to measure orientation of the beam</li>
-</ul>
-
-<p>
-For each Bragg angle, the Fast Jacks are actuated to that the beam is at the center of the beam viewer.
-Then, then position of the crystals as measured by the interferometers is recorded.
-This position is the wanted position for a given Bragg angle.
-</p>
-</div>
-</div>
-
-<div id="outline-container-orga3da277" class="outline-3">
-<h3 id="orga3da277"><span class="section-number-3">3.2.</span> Simulations</h3>
-<div class="outline-text-3" id="text-3-2">
+<div id="outline-container-org78a952b" class="outline-3">
+<h3 id="org78a952b"><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.
 This may be done using FE software.
@@ -634,13 +838,13 @@ This may be done using FE software.
 </div>
 </div>
 
-<div id="outline-container-orgec833d8" class="outline-3">
-<h3 id="orgec833d8"><span class="section-number-3">3.3.</span> Comparison</h3>
+<div id="outline-container-org053b4f5" class="outline-3">
+<h3 id="org053b4f5"><span class="section-number-3">4.3.</span> Comparison</h3>
 </div>
 
-<div id="outline-container-orga851ac8" class="outline-3">
-<h3 id="orga851ac8"><span class="section-number-3">3.4.</span> Test</h3>
-<div class="outline-text-3" id="text-3-4">
+<div id="outline-container-org0aa4bea" class="outline-3">
+<h3 id="org0aa4bea"><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>);
 </pre>
@@ -654,9 +858,9 @@ This may be done using FE software.
 </div>
 </div>
 
-<div id="outline-container-orgc6c0d26" class="outline-3">
-<h3 id="orgc6c0d26"><span class="section-number-3">3.5.</span> Measured frame deformation</h3>
-<div class="outline-text-3" id="text-3-5">
+<div id="outline-container-orgfebab90" class="outline-3">
+<h3 id="orgfebab90"><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));
 th = <span class="org-matlab-math">pi</span><span class="org-builtin">/</span>180<span class="org-builtin">*</span>data(<span class="org-builtin">:</span>,1); <span class="org-comment-delimiter">% </span><span class="org-comment">[rad]</span>
@@ -683,10 +887,10 @@ ry1 = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*</span>dat
 </div>
 
 
-<div id="org2bfeb26" class="figure">
+<div id="org73f1693" class="figure">
 <p><img src="figs/calibration_drx_pres.png" alt="calibration_drx_pres.png" />
 </p>
-<p><span class="figure-number">Figure 8: </span>description</p>
+<p><span class="figure-number">Figure 11: </span>description</p>
 </div>
 
 
@@ -756,9 +960,9 @@ f_ry1 = fit(180<span class="org-builtin">/</span><span class="org-matlab-math">p
 </div>
 </div>
 
-<div id="outline-container-org9ce965b" class="outline-3">
-<h3 id="org9ce965b"><span class="section-number-3">3.6.</span> Test</h3>
-<div class="outline-text-3" id="text-3-6">
+<div id="outline-container-org12eb5ca" class="outline-3">
+<h3 id="org12eb5ca"><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>;
 </pre>
@@ -789,10 +993,10 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
 </div>
 
 
-<div id="org22c7c33" class="figure">
+<div id="orgf09ae2a" class="figure">
 <p><img src="figs/drifts_xtal2_detrend.png" alt="drifts_xtal2_detrend.png" />
 </p>
-<p><span class="figure-number">Figure 9: </span>Drifts of the second crystal as a function of Bragg Angle</p>
+<p><span class="figure-number">Figure 12: </span>Drifts of the second crystal as a function of Bragg Angle</p>
 </div>
 
 <div class="org-src-container">
@@ -806,9 +1010,9 @@ data.xtal2_111_d = double(h5read(filename, <span class="org-string">'/7.1/instru
 </div>
 </div>
 
-<div id="outline-container-org3613a0c" class="outline-3">
-<h3 id="org3613a0c"><span class="section-number-3">3.7.</span> Repeatability of frame deformation</h3>
-<div class="outline-text-3" id="text-3-7">
+<div id="outline-container-org2b6cac9" class="outline-3">
+<h3 id="org2b6cac9"><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>;
 </pre>
@@ -857,15 +1061,15 @@ data_2.dz    = 1e<span class="org-builtin">-</span>9<span class="org-builtin">*<
 </div>
 </div>
 
-<div id="outline-container-orgdedfb8e" class="outline-2">
-<h2 id="orgdedfb8e"><span class="section-number-2">4.</span> Attocube - Periodic Non-Linearity</h2>
-<div class="outline-text-2" id="text-4">
+<div id="outline-container-org27208e8" class="outline-2">
+<h2 id="org27208e8"><span class="section-number-2">5.</span> Attocube - Periodic Non-Linearity</h2>
+<div class="outline-text-2" id="text-5">
 <p>
-<a id="org62f88d9"></a>
+<a id="org7046a86"></a>
 </p>
 <p>
-Interferometers have some periodic nonlinearity (<a href="#citeproc_bib_item_2">Thurner et al., 2015</a>).
-The period is a fraction of the wavelength (usually \(\lambda/2\)) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams (See <a href="#citeproc_bib_item_1">Ducourtieux, 2018, p. 67 to 69</a>; <a href="#citeproc_bib_item_2">Thurner et al., 2015</a>).
+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 amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers.
 </p>
 
@@ -882,9 +1086,9 @@ This process is performed over several periods in order to characterize the erro
 </p>
 </div>
 
-<div id="outline-container-org324d54e" class="outline-3">
-<h3 id="org324d54e"><span class="section-number-3">4.1.</span> Measurement Setup</h3>
-<div class="outline-text-3" id="text-4-1">
+<div id="outline-container-org62b5863" class="outline-3">
+<h3 id="org62b5863"><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.
 </p>
@@ -894,7 +1098,7 @@ 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="orge5737c1">
+<div class="note" id="org95ffe92">
 <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>
@@ -910,7 +1114,7 @@ As any deformations of the metrology frame of deformation of the crystal&rsquo;s
 </div>
 
 <p>
-The setup is schematically with the block diagram in Figure <a href="#orgd56a2ba">10</a>.
+The setup is schematically with the block diagram in Figure <a href="#org6126f2e">13</a>.
 </p>
 
 <p>
@@ -926,10 +1130,10 @@ The PI controller takes care or controlling to position as measured by the strai
 </ul>
 
 
-<div id="orgd56a2ba" class="figure">
+<div id="org6126f2e" class="figure">
 <p><img src="figs/block_diagram_lut_attocube.png" alt="block_diagram_lut_attocube.png" />
 </p>
-<p><span class="figure-number">Figure 10: </span>Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube</p>
+<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>
 </div>
 
 <p>
@@ -938,9 +1142,9 @@ The problem is to estimate the periodic non-linearity of the Attocube from the i
 </div>
 </div>
 
-<div id="outline-container-org78e73ad" class="outline-3">
-<h3 id="org78e73ad"><span class="section-number-3">4.2.</span> Choice of the reference signal</h3>
-<div class="outline-text-3" id="text-4-2">
+<div id="outline-container-orgbaa09c9" class="outline-3">
+<h3 id="orgbaa09c9"><span class="section-number-3">5.2.</span> Choice of the reference signal</h3>
+<div class="outline-text-3" id="text-5-2">
 <p>
 The main specifications for the reference signal are;
 </p>
@@ -963,9 +1167,9 @@ Based on the above discussion, one suitable excitation signal is a sinusoidal sw
 </div>
 </div>
 
-<div id="outline-container-org1965ef7" class="outline-3">
-<h3 id="org1965ef7"><span class="section-number-3">4.3.</span> Repeatability of the non-linearity</h3>
-<div class="outline-text-3" id="text-4-3">
+<div id="outline-container-org88808b2" class="outline-3">
+<h3 id="org88808b2"><span class="section-number-3">5.3.</span> Repeatability of the non-linearity</h3>
+<div 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.
 </p>
@@ -984,9 +1188,9 @@ One way to precisely estimate the laser wavelength is to estimate the non linear
 </div>
 </div>
 
-<div id="outline-container-org3432548" class="outline-3">
-<h3 id="org3432548"><span class="section-number-3">4.4.</span> Simulation</h3>
-<div class="outline-text-3" id="text-4-4">
+<div id="outline-container-org2dfde91" class="outline-3">
+<h3 id="org2dfde91"><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.
 </p>
@@ -1005,18 +1209,18 @@ 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="#org6c0e308">11</a>).
+The non-linear errors are first estimated at the beginning of the stroke (Figure <a href="#org0c7bf9b">14</a>).
 </p>
 
-<div id="org6c0e308" class="figure">
+<div id="org0c7bf9b" class="figure">
 <p><img src="figs/non_linear_errors_start_stroke.png" alt="non_linear_errors_start_stroke.png" />
 </p>
-<p><span class="figure-number">Figure 11: </span>Estimation of the non-linear errors at the beginning of the stroke</p>
+<p><span class="figure-number">Figure 14: </span>Estimation of the non-linear errors at the beginning of the stroke</p>
 </div>
 
 <p>
 From this only measurement, it is not possible to estimate with great accuracy the period of the error.
-To do so, the same measurement is performed with a stroke of several millimeters (Figure <a href="#orgbe11399">12</a>).
+To do so, the same measurement is performed with a stroke of several millimeters (Figure <a href="#org9c29b56">15</a>).
 </p>
 
 <p>
@@ -1025,10 +1229,10 @@ This is due to a mismatch between the estimated period and the true period of th
 </p>
 
 
-<div id="orgbe11399" class="figure">
+<div id="org9c29b56" class="figure">
 <p><img src="figs/non_linear_errors_middle_stroke.png" alt="non_linear_errors_middle_stroke.png" />
 </p>
-<p><span class="figure-number">Figure 12: </span>Estimated non-linear errors at a latter position</p>
+<p><span class="figure-number">Figure 15: </span>Estimated non-linear errors at a latter position</p>
 </div>
 
 <p>
@@ -1058,7 +1262,7 @@ with \(\lambda_{\text{est}}\) the estimated error&rsquo;s period.
 </p>
 
 <p>
-From Figure <a href="#orgbe11399">12</a>, we can see that there is an offset between the two curves.
+From Figure <a href="#org9c29b56">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:
 </p>
 \begin{equation}
@@ -1138,9 +1342,9 @@ The maximum stroke is 2.9 [mm]
 </div>
 </div>
 
-<div id="outline-container-org018ecfe" class="outline-3">
-<h3 id="org018ecfe"><span class="section-number-3">4.5.</span> Measurements</h3>
-<div class="outline-text-3" id="text-4-5">
+<div id="outline-container-org1be96dc" class="outline-3">
+<h3 id="org1be96dc"><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:
 </p>
@@ -1165,20 +1369,10 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).
 </div>
 </div>
 </div>
-
-<div id="outline-container-org6292dcf" class="outline-2">
-<h2 id="org6292dcf">Bibliography</h2>
-<div class="outline-text-2" id="text-org6292dcf">
-<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ducourtieux, S., 2018. Toward high precision position control using laser interferometry: Main sources of error. <a href="https://doi.org/10.13140/rg.2.2.21044.35205">https://doi.org/10.13140/rg.2.2.21044.35205</a></div>
-  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Thurner, K., Quacquarelli, F.P., Braun, P.-F., Dal Savio, C., Karrai, K., 2015. Fiber-based distance sensing interferometry. Applied optics 54, 3051–3063.</div>
-</div>
-</div>
-</div>
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2022-02-15 mar. 14:25</p>
+<p class="date">Created: 2022-06-02 Thu 18:06</p>
 </div>
 </body>
 </html>
diff --git a/dcm-metrology.org b/dcm-metrology.org
index 5569e59..0957ca6 100644
--- a/dcm-metrology.org
+++ b/dcm-metrology.org
@@ -184,10 +184,12 @@ In this section, the impact of an error in the relative pose between the first a
 This is very important in order to:
 - link a measurement of the x-ray beam position to a default in the crystal position
 - understand which pose default will have large impact on the output beam position/orientation
-- calibrate the deformations of the metrology frame using and external metrology measuring the x-ray beam position/orientation
+- calibrate the deformations of the metrology frame using an external metrology measuring the x-ray beam position/orientation
 
 In order to simplify the problem, the first crystal is supposed to be fixed (i.e. ideally positioned), and only the motion of the second crystal is studied.
 
+In order to easily study that, "ray tracing" techniques are used.
+
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
 <<matlab-dir>>
@@ -197,66 +199,6 @@ In order to simplify the problem, the first crystal is supposed to be fixed (i.e
 <<matlab-init>>
 #+end_src
 
-#+begin_src matlab
-bragg = pi/180*linspace(5,75,100);
-#+end_src
-
-** Axial motion of second crystal
-Let's consider the relation between the $[y, z]$ motion of the beam and the motion of the second crystal $[z^\prime, R_{y^\prime}, R_{x^\prime}]$.
-
-#+name: fig:relation_dz_output_beam
-#+caption: Relation between $d_{z^\prime}$ motion of the second crystal and vertical motion of the beam
-[[file:figs/relation_dz_output_beam.png]]
-
-\begin{equation}
-  d_z = d_{z^\prime} 2 \cos \theta
-\end{equation}
-
-#+begin_src matlab :exports none :results none
-%% Relation between vertical motion of the second crystal and vertical motion of the output beam
-figure;
-hold on;
-plot(180/pi*bragg, 2*cos(bragg))
-xlabel('Bragg [deg]'); ylabel('Motion amplification');
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/relation_vert_motion_crystal_beam.pdf', 'width', 'wide', 'height', 'normal');
-#+end_src
-
-#+name: fig:relation_vert_motion_crystal_beam
-#+caption: Relation between vertical motion of the second crystal and vertical motion of the output beam
-#+RESULTS:
-[[file:figs/relation_vert_motion_crystal_beam.png]]
-
-** Ry motion of second crystal
-
-\begin{equation}
-d_z = D_{\text{vlm}} d_{R_y^\prime}
-\end{equation}
-
-with $D_{\text{vlm}} \approx 10\,m$.
-
-** Rx motion of second crystal
-
-\begin{equation}
-d_y = 2 D_{\text{vlm}} \sin \theta \cdot d_{R_x^\prime}
-\end{equation}
-
-
-
-* Ray Tracing
-** Introduction                                                      :ignore:
-
-** Matlab Init                                             :noexport:ignore:
-#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
-<<matlab-dir>>
-#+end_src
-
-#+begin_src matlab :exports none :results silent :noweb yes
-<<matlab-init>>
-#+end_src
-
 #+begin_src matlab :tangle no :noweb yes
 <<m-init-path>>
 #+end_src
@@ -280,271 +222,73 @@ d_y = 2 D_{\text{vlm}} \sin \theta \cdot d_{R_x^\prime}
 | output beam      | =x3,y3,z3= | =s3=        |
 | Dectector        | =xd,yd,zd= | =nd=        |
 
-#+begin_src matlab
-theta = 85*pi/180; % [rad]
-#+end_src
+- [ ] Add schematic
 
-#+begin_src matlab
-thetas = pi/180*[5:1:85];
-#+end_src
+Notations for the pose of the output beam: =d_b_y=, =d_b_z=, =r_b_y=, =r_b_z=.
 
-#+begin_src matlab
-yz = zeros(2, length(thetas));
-#+end_src
+$d_{b,y}, d_{b,z}, r_{b,y}, r_{b,z}$
 
-#+begin_src matlab
-for i = 1:length(thetas)
-    theta = thetas(i);
-#+end_src
+The xy position of the beam is taken in the $x=0$ plane.
 
-#+begin_src matlab
-%% Secondary crystal defaults
-drx = 0; % [rad]
-dry = 0; % [rad]
-dz = 1e-9; % [m]
+** Effect of an error in crystal's distance
+<<sec:ray_tracing_dz>>
 
-% Rotation matrix for drx
-udrx = [cos(theta), 0, -sin(theta)];
-Rdrx = cos(drx)*eye(3)+sin(drx)*[0, -udrx(3), udrx(2); udrx(3), 0, -udrx(1); -udrx(2), udrx(1), 0] + (1-cos(drx))*(udrx'*udrx);
+In Figure [[fig:ray_tracing_error_dz_overview]] is shown the light path for three bragg angles (5, 55 and 85 degrees) when there is an error in the =dz= position of 1mm.
 
-% Rotation matrix for dry
-Rdry = [ cos(dry), 0, sin(dry); ...
-         0,        1, 0; ...
-         -sin(dry), 0, cos(dry)];
-#+end_src
-
-#+begin_src matlab
-%% Input Beam
-p1 = [-0.1, 0, 0]; % [m]
-s1 = [ 1, 0, 0];
-
-%% Primary Mirror
-pp = [0, 0, 0]; % [m]
-np = [cos(pi/2-theta), 0, sin(pi/2-theta)];
-
-%% Reflected beam
-[p2, s2] = get_plane_reflection(p1, s1, pp, np);
-
-%% Secondary Mirror
-ps = pp ...
-     + 0.07*[cos(theta), 0, -sin(theta)] ... % x offset (does not matter a lot)
-     - np*10e-3./(2*cos(theta)) ... % Theoretical offset
-     + np*dz; % Add error in distance
-
-ns = [Rdry*Rdrx*[cos(pi/2-theta), 0, sin(pi/2-theta)]']'; % Normal
-
-%% Output Beam
-[p3, s3] = get_plane_reflection(p2, s2, ps, ns);
-
-%% Detector
-pd = [1, 0, 0]; % [m]
-nd = [-1, 0, 0];
-
-%% Get beam position on the decector
-p4 = get_plane_reflection(p3, s3, pd, nd);
-yz(:,i) = p4(2:3);
-end
-#+end_src
-
-#+begin_src matlab
-figure;
-hold on;
-plot(180/pi*thetas, 1e9*(yz(2,:) + 10e-3));
-xlabel('Bragg Angle [deg]'); ylabel('Z position [nm/m/nrad]');
-#+end_src
-
-#+begin_src matlab
-figure;
-hold on;
-plot(180/pi*thetas, 1e9*yz(1,:));
-xlabel('Bragg Angle [deg]'); ylabel('Y position [nm/m/nrad]');
-#+end_src
-
-#+begin_src matlab
-%% Primary crystal plane
-z = np;
-y = [0,1,0];
-x = cross(y,z);
-xtal1_rectangle = [pp + 0.02*y + 0.07*x;
-                   pp - 0.02*y + 0.07*x;
-                   pp - 0.02*y - 0.07*x;
-                   pp + 0.02*y - 0.07*x];
-
-%% Secondary crystal plane
-z = ns;
-y = [0,cos(drx),sin(drx)];
-x = cross(y,z);
-xtal2_rectangle = [ps + 0.02*y + 0.07*x;
-                   ps - 0.02*y + 0.07*x;
-                   ps - 0.02*y - 0.07*x;
-                   ps + 0.02*y - 0.07*x];
-#+end_src
-
-#+begin_src matlab
-figure;
-tiledlayout(2, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
-
-ax1 = nexttile();
-hold on;
-plot3([p1(1), p2(1)],[p1(2), p2(2)], [p1(3), p2(3)])
-plot3([p2(1), p3(1)],[p2(2), p3(2)], [p2(3), p3(3)])
-plot3([p3(1), p3(1)+0.3*s3(1)],[p3(2), p3(2)+0.3*s3(2)], [p3(3), p3(3)+0.3*s3(3)])
-patch(xtal1_rectangle(:,1), xtal1_rectangle(:,2), xtal1_rectangle(:,3), 'k')
-patch(xtal2_rectangle(:,1), xtal2_rectangle(:,2), xtal2_rectangle(:,3), 'k')
-hold off;
-view(0,0)
-axis equal
-xlim([-0.1, 0.15])
-zlim([-0.02, 0.01])
-grid off;
-xlabel('X')
-ylabel('Y')
-zlabel('Z')
-
-ax2 = nexttile();
-hold on;
-plot3([p1(1), p2(1)],[p1(2), p2(2)], [p1(3), p2(3)])
-plot3([p2(1), p3(1)],[p2(2), p3(2)], [p2(3), p3(3)])
-plot3([p3(1), p3(1)+0.3*s3(1)],[p3(2), p3(2)+0.3*s3(2)], [p3(3), p3(3)+0.3*s3(3)])
-patch(xtal1_rectangle(:,1), xtal1_rectangle(:,2), xtal1_rectangle(:,3), 'k')
-patch(xtal2_rectangle(:,1), xtal2_rectangle(:,2), xtal2_rectangle(:,3), 'k')
-hold off;
-view(0,90)
-axis equal
-xlim([-0.1, 0.15])
-zlim([-0.02, 0.01])
-grid off;
-xlabel('X')
-ylabel('Y')
-zlabel('Z')
-#+end_src
-
-#+begin_src matlab
-figure;
-hold on;
-plot3([p1(1), p2(1)],[p1(2), p2(2)], [p1(3), p2(3)])
-plot3([p2(1), p3(1)],[p2(2), p3(2)], [p2(3), p3(3)])
-plot3([p3(1), p3(1)+0.3*s3(1)],[p3(2), p3(2)+0.3*s3(2)], [p3(3), p3(3)+0.3*s3(3)])
-surf(xtal1_x, xtal1_y, xtal1_z)
-patch(xtal2_rectangle(:,1), xtal2_rectangle(:,2), xtal2_rectangle(:,3), 'k')
-hold off;
-view(0,0)
-axis equal
-xlim([-0.1, 0.15])
-zlim([-0.02, 0.01])
-grid off;
-#+end_src
-
-#+begin_src matlab
-figure;
-hold on;
-plot3([p1(1), p2(1)],[p1(2), p2(2)], [p1(3), p2(3)])
-plot3([p2(1), p3(1)],[p2(2), p3(2)], [p2(3), p3(3)])
-plot3([p3(1), p3(1)+0.3*s3(1)],[p3(2), p3(2)+0.3*s3(2)], [p3(3), p3(3)+0.3*s3(3)])
-surf(xtal1_x, xtal1_y, xtal1_z)
-patch(xtal2_rectangle(:,1), xtal2_rectangle(:,2), xtal2_rectangle(:,3), 'k')
-hold off;
-view(90,90)
-axis equal
-xlim([-0.1, 0.15])
-zlim([-0.02, 0.01])
-grid off;
-#+end_src
-
-** Effect of =dry=
-#+begin_src matlab :exports none
-%% Effect of dry
-thetas = 1:1:85;
-Dy = zeros(length(thetas), 1);
-Dz = zeros(length(thetas), 1);
-Ry = zeros(length(thetas), 1);
-Rz = zeros(length(thetas), 1);
-for i = 1:length(thetas)
-    results = getBeamPath(thetas(i)*pi/180, 'dry', 1e-9);
-    Dy(i) = results.p4(2);
-    Dz(i) = results.p4(3);
-    Ry(i) = atan2(results.s3(3), results.s3(1));
-    Rz(i) = atan2(results.s3(2), results.s3(1));
-end
-#+end_src
+Visually, it is clear that this induce a =z= offset of the output beam.
 
 #+begin_src matlab :exports none :results none
-%% Motion of the output beam with dry error
+%% Visual Effect of an error in dz
 figure;
 hold on;
-plot(thetas, 1e9*(Dz+10e-3), '--', 'DisplayName', 'Dz');
-plot(thetas, 1e9*Dy, '--', 'DisplayName', 'Dy');
-plot(thetas, 1e9*Rz, 'DisplayName', 'Rz');
-plot(thetas, 1e9*Ry, 'DisplayName', 'Ry');
+for theta = [5, 55, 85]
+    results = getBeamPath(theta*pi/180, 'dz', 1e-3);
+    set(gca,'ColorOrderIndex',1)
+    plotBeamPath(results);
+    plotCrystals(results);
+end
 hold off;
-xlabel('Bragg [deg]'); ylabel('Offset [nm/m], Tilt [nrad]')
-legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
-xlim([0, 85]);
+view(0,0)
+xlim([-10, 15])
+zlim([-4, 2])
+grid off;
+axis equal
+xlabel('X [cm]')
+ylabel('Y [cm]')
+zlabel('Z [cm]')
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/motion_beam_dry_error.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/ray_tracing_error_dz_overview.pdf', 'width', 'full', 'height', 'normal');
 #+end_src
 
-#+name: fig:motion_beam_dry_error
-#+caption: Motion of the output beam with dry error
+#+name: fig:ray_tracing_error_dz_overview
+#+caption: Visual Effect of an error in =dz= (1mm). Side view.
 #+RESULTS:
-[[file:figs/motion_beam_dry_error.png]]
+[[file:figs/ray_tracing_error_dz_overview.png]]
 
-** Effect of =drx=
-#+begin_src matlab :exports none
-%% Effect of drx
-thetas = 1:1:85;
-Dy = zeros(length(thetas), 1);
-Dz = zeros(length(thetas), 1);
-Ry = zeros(length(thetas), 1);
-Rz = zeros(length(thetas), 1);
-for i = 1:length(thetas)
-    results = getBeamPath(thetas(i)*pi/180, 'drx', 1e-9);
-    Dy(i) = results.p4(2);
-    Dz(i) = results.p4(3);
-    Ry(i) = atan2(results.s3(3), results.s3(1));
-    Rz(i) = atan2(results.s3(2), results.s3(1));
-end
-#+end_src
+The motion of the output beam is displayed as a function of the Bragg angle in Figure [[fig:motion_beam_dz_error]].
+It is clear that an error in the distance =dz= between the crystals only induce a =z= offset of the output beam.
+This offset decreases with the Bragg angle.
 
-#+begin_src matlab :exports none :results none
-%% Motion of the output beam with drx error
-figure;
-hold on;
-plot(thetas, 1e9*(Dz+10e-3), '--', 'DisplayName', 'Dz');
-plot(thetas, 1e9*Dy, '--', 'DisplayName', 'Dy');
-plot(thetas, 1e9*Rz, 'DisplayName', 'Rz');
-plot(thetas, 1e9*Ry, 'DisplayName', 'Ry');
-hold off;
-xlabel('Bragg [deg]'); ylabel('Offset [nm/m], Tilt [nrad]')
-legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
-xlim([0, 85]);
-#+end_src
+This is indeed equal to:
+\begin{equation}
+\boxed{d_{b,z} = 2 d_z \cos \theta}
+\end{equation}
 
-#+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/motion_beam_drx_error.pdf', 'width', 'wide', 'height', 'normal');
-#+end_src
-
-#+name: fig:motion_beam_drx_error
-#+caption: Motion of the output beam with drx error
-#+RESULTS:
-[[file:figs/motion_beam_drx_error.png]]
-
-** Effect of =dz=
 #+begin_src matlab :exports none
 %% Effect of dz
 thetas = 1:1:85;
-Dy = zeros(length(thetas), 1);
-Dz = zeros(length(thetas), 1);
-Ry = zeros(length(thetas), 1);
-Rz = zeros(length(thetas), 1);
+Dy = zeros(length(thetas), 1); % [m]
+Dz = zeros(length(thetas), 1); % [m]
+Ry = zeros(length(thetas), 1); % [rad]
+Rz = zeros(length(thetas), 1); % [rad]
 for i = 1:length(thetas)
     results = getBeamPath(thetas(i)*pi/180, 'dz', 1e-9);
-    Dy(i) = results.p4(2);
-    Dz(i) = results.p4(3);
     Ry(i) = atan2(results.s3(3), results.s3(1));
     Rz(i) = atan2(results.s3(2), results.s3(1));
+    Dy(i) = results.p4(2)-tan(Rz(i));
+    Dz(i) = results.p4(3)+tan(Ry(i));
 end
 #+end_src
 
@@ -552,12 +296,12 @@ end
 %% Motion of the output beam with dZ error
 figure;
 hold on;
-plot(thetas, 1e9*(Dz+10e-3), '--', 'DisplayName', 'Dz');
 plot(thetas, 1e9*Dy, '--', 'DisplayName', 'Dy');
-plot(thetas, 1e9*Rz, 'DisplayName', 'Rz');
+plot(thetas, 1e9*(Dz+10e-3), '--', 'DisplayName', 'Dz');
 plot(thetas, 1e9*Ry, 'DisplayName', 'Ry');
+plot(thetas, 1e9*Rz, 'DisplayName', 'Rz');
 hold off;
-xlabel('Bragg [deg]'); ylabel('Offset [nm/m], Tilt [nrad]')
+xlabel('Bragg [deg]'); ylabel('Offset [m/m] Tilt [rad/m]')
 legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
 xlim([0, 85]);
 #+end_src
@@ -571,98 +315,338 @@ exportFig('figs/motion_beam_dz_error.pdf', 'width', 'wide', 'height', 'normal');
 #+RESULTS:
 [[file:figs/motion_beam_dz_error.png]]
 
-** Test                                                            :noexport:
-#+begin_src matlab
-figure;
-plot(thetas, 1e9*yz(:,1));
-xlabel('Bragg [deg]'); ylabel('Y Displacement [nm/m/nrad]')
-#+end_src
+** Effect of an error in crystal's x parallelism
+<<sec:ray_tracing_rx>>
 
-#+begin_src matlab
-figure;
-plot(thetas, 1e9*yz(:,1));
-xlabel('Bragg [deg]'); ylabel('Y Displacement [nm/m/nrad]')
-#+end_src
+The effect of an error in =rx= crystal parallelism on the output beam is visually shown in Figure [[fig:ray_tracing_error_drx_overview]] for three bragg angles (5, 55 and 85 degrees).
+The error is set to one degree, and the top view is shown.
+It is clear that the output beam experiences some rotation around a vertical axis.
+The amount of rotation depends on the bragg angle.
 
-#+begin_src matlab
-%% Effect of dry
-thetas = 1:1:85;
-yz = zeros(length(thetas), 2);
-for i = 1:length(thetas)
-    results = getBeamPath(thetas(i)*pi/180, 'dry', 1e-9);
-    yz(i,:) = results.p4(2:3);
-end
-#+end_src
-
-#+begin_src matlab
-figure;
-plot(thetas, 1e9*(yz(:,2) + 10e-3));
-xlabel('Bragg [deg]'); ylabel('Z Displacement [nm/m/nrad]')
-#+end_src
-
-#+begin_src matlab
-%% Effect of dz
-thetas = 1:1:85;
-yz = zeros(length(thetas), 2);
-for i = 1:length(thetas)
-    results = getBeamPath(thetas(i)*pi/180, 'dz', 1e-9);
-    yz(i,:) = results.p4(2:3);
-end
-#+end_src
-
-#+begin_src matlab
-figure;
-plot(thetas, 1e9*(yz(:,2) + 10e-3));
-xlabel('Bragg [deg]'); ylabel('Z Displacement [nm/m/nm]')
-#+end_src
-
-
-
-#+begin_src matlab
+#+begin_src matlab :exports none :results none
+%% Visual Effect of an error in drx
 figure;
 hold on;
-for theta = 5:5:85
+for theta = [5, 55, 85]
+    results = getBeamPath(theta*pi/180, 'drx', -1*pi/180);
     set(gca,'ColorOrderIndex',1)
-    plotBeamPath(getBeamPath(theta*pi/180, 'dz', 1e-3))
-end
-hold off;
-view(0,0)
-axis equal
-xlim([-0.1, 0.15])
-zlim([-0.02, 0.01])
-grid off;
-#+end_src
-
-#+begin_src matlab
-figure;
-hold on;
-for theta = 5:5:85
-    set(gca,'ColorOrderIndex',1)
-    plotBeamPath(getBeamPath(theta*pi/180, 'dry', 1*pi/180))
-end
-hold off;
-view(0,0)
-axis equal
-xlim([-0.1, 0.15])
-zlim([-0.02, 0.01])
-grid off;
-#+end_src
-
-#+begin_src matlab
-figure;
-hold on;
-for theta = 5:5:85
-    set(gca,'ColorOrderIndex',1)
-    plotBeamPath(getBeamPath(theta*pi/180, 'drx', 1*pi/180))
+    plotBeamPath(results);
+    % plotCrystals(results);
 end
 hold off;
 view(0,90)
 axis equal
-xlim([-0.1, 0.15])
-zlim([-0.02, 0.01])
+xlim([-10, 15])
+ylim([-1, 1])
 grid off;
+xlabel('X [cm]');
+ylabel('Y [cm]');
+zlabel('Z [cm]');
 #+end_src
 
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/ray_tracing_error_drx_overview.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+name: fig:ray_tracing_error_drx_overview
+#+caption: Visual Effect of an error in =drx= (1 degree). Top View.
+#+RESULTS:
+[[file:figs/ray_tracing_error_drx_overview.png]]
+
+The effect of =drx= as a function of the Bragg angle on the output beam pose is computed and shown in Figure [[fig:motion_beam_drx_error]].
+
+It induces a rotation of the output beam in the =z= direction that depends on the Bragg Angle.
+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}
+\end{equation}
+
+If also induces a small $y$ shift of the beam.
+This shift is due to the fact that the rotation point (around which the second crystal is moved) is changing as a function of bragg.
+We can note that the $y$ shift is equal to zero for a bragg angle of 45 degrees, at which point the center of rotation of the second crystal is at $x = 0$;
+
+#+begin_src matlab :exports none
+%% Effect of drx
+thetas = 1:1:85;
+Dy = zeros(length(thetas), 1);
+Dz = zeros(length(thetas), 1);
+Ry = zeros(length(thetas), 1);
+Rz = zeros(length(thetas), 1);
+for i = 1:length(thetas)
+    results = getBeamPath(thetas(i)*pi/180, 'drx', 1e-9);
+    Ry(i) = atan2(results.s3(3), results.s3(1));
+    Rz(i) = atan2(results.s3(2), results.s3(1));
+    Dy(i) = results.p4(2)-tan(Rz(i));
+    Dz(i) = results.p4(3)+tan(Ry(i));
+end
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Motion of the output beam with drx error
+figure;
+hold on;
+plot(thetas, 1e9*Dy, '--', 'DisplayName', 'Dy');
+plot(thetas, 1e9*(Dz+10e-3), '--', 'DisplayName', 'Dz');
+plot(thetas, 1e9*Ry, 'DisplayName', 'Ry');
+plot(thetas, 1e9*Rz, 'DisplayName', 'Rz');
+hold off;
+xlabel('Bragg [deg]'); ylabel('Offset [m/rad] Tilt [rad/rad]')
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+xlim([0, 85]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/motion_beam_drx_error.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:motion_beam_drx_error
+#+caption: Motion of the output beam with drx error
+#+RESULTS:
+[[file:figs/motion_beam_drx_error.png]]
+
+** Effect of an error in crystal's y parallelism
+<<sec:ray_tracing_ry>>
+
+The effect of an error in =ry= crystal parallelism on the output beam is visually shown in Figure [[fig:ray_tracing_error_dry_overview]] for three bragg angles (5, 55 and 85 degrees).
+
+#+begin_src matlab :exports none :results none
+%% Visual Effect of an error in dry
+figure;
+hold on;
+for theta = [5, 55, 85]
+    results = getBeamPath(theta*pi/180, 'dry', -1*pi/180);
+    set(gca,'ColorOrderIndex',1)
+    plotBeamPath(results);
+    plotCrystals(results);
+end
+hold off;
+view(0,0)
+axis equal
+xlim([-10, 15])
+zlim([-4, 2])
+grid off;
+xlabel('X [cm]')
+ylabel('Y [cm]')
+zlabel('Z [cm]')
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/ray_tracing_error_dry_overview.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+name: fig:ray_tracing_error_dry_overview
+#+caption: Visual Effect of an error in =dry= (1 degree). Side view.
+#+RESULTS:
+[[file:figs/ray_tracing_error_dry_overview.png]]
+
+The effect of =dry= as a function of the Bragg angle on the output beam pose is computed and shown in Figure [[fig:motion_beam_dry_error]].
+It is clear that this induces a rotation of the output beam in the =y= direction equals to 2 times =dry=:
+\begin{equation}
+\boxed{r_{b,z} = 2 r_z}
+\end{equation}
+
+It also induces a small vertical motion of the beam (at the $x=0$ location) which is simply due to the fact that the $x$ coordinate of the impact point on the second crystal changes with the Bragg angle.
+
+#+begin_src matlab :exports none
+%% Effect of dry
+thetas = 1:1:85;
+Dy = zeros(length(thetas), 1);
+Dz = zeros(length(thetas), 1);
+Ry = zeros(length(thetas), 1);
+Rz = zeros(length(thetas), 1);
+for i = 1:length(thetas)
+    results = getBeamPath(thetas(i)*pi/180, 'dry', 1e-9);
+    Ry(i) = atan2(results.s3(3), results.s3(1));
+    Rz(i) = atan2(results.s3(2), results.s3(1));
+    Dy(i) = results.p4(2)-tan(Rz(i));
+    Dz(i) = results.p4(3)-tan(Ry(i));
+end
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Motion of the output beam with dry error
+figure;
+hold on;
+plot(thetas, 1e9*Dy, '--', 'DisplayName', 'Dy');
+plot(thetas, 1e9*(Dz+10e-3), '--', 'DisplayName', 'Dz');
+plot(thetas, 1e9*Ry, 'DisplayName', 'Ry');
+plot(thetas, 1e9*Rz, 'DisplayName', 'Rz');
+hold off;
+xlabel('Bragg [deg]'); ylabel('Offset [m/rad] Tilt [rad/rad]')
+legend('location', 'east', 'FontSize', 8, 'NumColumns', 2);
+xlim([0, 85]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/motion_beam_dry_error.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:motion_beam_dry_error
+#+caption: Motion of the output beam with dry error
+#+RESULTS:
+[[file:figs/motion_beam_dry_error.png]]
+
+** Summary
+
+Effects of crystal's pose errors on the output beam are summarized in Table [[tab:crystal_pose_beam_pose]].
+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.
+
+#+name: tab:crystal_pose_beam_pose
+#+caption: Summary of the effects of the errors in second crystal's pose on the output beam
+#+attr_latex: :environment tabularx :width 0.65\linewidth :align X|ccc
+#+attr_latex: :center t :booktabs t
+| *Beam Motion* |          *Crystal*     $d_z$   | *Crystal*                      $r_x$ | *Crystal*    $r_y$ |
+|---------------+--------------------------------+--------------------------------------+--------------------|
+| $d_{b,y}$     |                              0 | $\approx 0$                          | 0                  |
+| $d_{b,z}$     | $\boxed{d_{z} 2 \cos(\theta)}$ | 0                                    | $\approx 0$        |
+| $r_{b,y}$     |                              0 | 0                                    | $\boxed{2 r_{y}}$  |
+| $r_{b,z}$     |                              0 | $\boxed{- r_x 2 \sin(\theta)}$       | 0                  |
+
+** "Channel cut" Scan
+A "channel cut" scan is a Bragg scan where the distance between the crystals is fixed.
+
+This is visually shown in Figure [[fig:ray_tracing_channel_cut]] where it is clear that the output beam experiences some vertical motion.
+
+#+begin_src matlab :exports none :results none
+%% Visual Effect of a channel cut scan
+figure;
+hold on;
+for theta = [30, 40, 50]
+    results = getBeamPath(theta*pi/180, 'dz', 10e-3/(2*cos(pi/180*theta))-10e-3/(2*cos(pi/180*theta_fixed)));
+    set(gca,'ColorOrderIndex',1)
+    plotBeamPath(results);
+    plotCrystals(results);
+end
+hold off;
+view(0,0)
+axis equal
+xlim([-10, 15])
+zlim([-4, 2])
+grid off;
+xlabel('X [cm]')
+ylabel('Y [cm]')
+zlabel('Z [cm]')
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/ray_tracing_channel_cut.pdf', 'width', 'full', 'height', 'normal');
+#+end_src
+
+#+name: fig:ray_tracing_channel_cut
+#+caption: Visual Effect of a channel cut scan
+#+RESULTS:
+[[file:figs/ray_tracing_channel_cut.png]]
+
+The $z$ offset of the beam for several channel cut scans are shown in Figure [[fig:channel_cut_scan]].
+
+#+begin_src matlab :exports none
+%% Computation of Z motion of the beam during "channel cut" scans
+thetas = 5:1:84;
+Dz = zeros(length(thetas), 1);
+
+for i = 1:length(thetas)
+    results = getBeamPath(thetas(i)*pi/180, 'dz', 10e-3/(2*cos(pi/180*thetas(i)))-10e-3/(2*cos(pi/180*(10*fix((5+thetas(i))/10)))));
+    Dz(i) = results.p4(3);
+end
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Dz motion of the beam during "channel cut" scans
+figure;
+hold on;
+for i = 1:8
+    plot(thetas((i-1)*10+1:i*10), 1e3*(Dz((i-1)*10+1:i*10)+10e-3), '-', 'DisplayName', sprintf('$%i^o$: $%.0f\\,\\mu$m/deg', (10*fix((5+thetas((i-1)*10+5))/10)), 1e6*(Dz((i-1)*10+6)-Dz((i-1)*10+5))));
+end
+hold off;
+xlabel('Bragg [deg]'); ylabel('Dz Offset [mm]')
+xlim([0, 90]); ylim([-4, 4]);
+xticks([0:10:90]);
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/channel_cut_scan.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:channel_cut_scan
+#+caption: Z motion of the beam during "channel cut" scans
+#+RESULTS:
+[[file:figs/channel_cut_scan.png]]
+
+* Determining relative pose between the crystals using the X-ray
+** Introduction                                                      :ignore:
+As Interferometers are only measuring /relative/ displacement, it is mandatory to initialize them correctly.
+
+They should be initialize in such a way that:
+- $r_x = 0$ and $r_y = 0$ when the two crystallographic planes are parallel
+- measured $d_z$ is equal to the distance between the crystallographic planes
+
+In order to do that, an external metrology using the x-ray is used.
+
+** Determine the $y$ parallelism - "Rocking Curve"
+
+** Determine the $x$ parallelism - Bragg Scan
+
+
+** Determine the $z$ distance - Bragg Scan
+
+** Use Channel cut scan to determine crystal =dry= parallelism
+
+Now, let's suppose we want to determine the =dry= angle between the crystals.
+
+#+begin_src matlab
+dry = 1e-6; % [rad]
+#+end_src
+
+#+begin_src matlab :exports none
+%% Computation of Z motion of the beam during "channel cut" scans
+thetas = 5:1:84;
+Dz_expected = zeros(length(thetas), 1);
+Dz_obtained = zeros(length(thetas), 1);
+
+for i = 1:length(thetas)
+    results = getBeamPath(thetas(i)*pi/180, 'dz', 10e-3/(2*cos(pi/180*thetas(i)))-10e-3/(2*cos(pi/180*(10*fix((5+thetas(i))/10)))));
+    Dz_expected(i) = results.p4(3);
+
+    results = getBeamPath(thetas(i)*pi/180, 'dz', 10e-3/(2*cos(pi/180*thetas(i)))-10e-3/(2*cos(pi/180*(10*fix((5+thetas(i))/10)))), 'dry', dry);
+    Dz_obtained(i) = results.p4(3);
+end
+#+end_src
+
+The error is
+\begin{equation}
+\boxed{d_{b,z} = -2 d_z \cos()}
+\end{equation}
+
+#+begin_src matlab :exports none :results none
+%% Dz motion of the beam during "channel cut" scans
+figure;
+hold on;
+plot(thetas, 1e6*(Dz_expected-Dz_obtained), '-');
+hold off;
+xlabel('Bragg [deg]'); ylabel('Dz Offset [$\mu$m]')
+xlim([0, 90]);
+xticks([0:10:90]);
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+#+end_src
+
+#+begin_src matlab :exports none :results none
+%% Dz motion of the beam during "channel cut" scans
+figure;
+hold on;
+plot(thetas, 1e3*Dz_expected, '-');
+plot(thetas, 1e3*Dz_obtained, '-');
+hold off;
+xlabel('Bragg [deg]'); ylabel('Dz Offset [mm]')
+xlim([0, 90]);
+xticks([0:10:90]);
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+#+end_src
+
+** Effect of an error on Bragg angle
+
 * Deformations of the Metrology Frame
 <<sec:frame_deformations>>
 ** Introduction                                                      :ignore:
@@ -1396,16 +1380,16 @@ freqs = logspace(1, 3, 1000);
 
 ** Intersection between Ray and Plane
 :PROPERTIES:
-:header-args:matlab+: :tangle matlab/src/get_plane_reflection.m
+:header-args:matlab+: :tangle matlab/src/getPlaneReflection.m
 :header-args:matlab+: :comments none :mkdirp yes :eval no
 :END:
-<<sec:get_plane_reflection>>
+<<sec:getPlaneReflection>>
 
 #+begin_src matlab
-function [p_reflect, s_reflect] = get_plane_reflection(p_in, s_in, p_plane, n_plane)
-% get_plane_reflection -
+function [p_reflect, s_reflect] = getPlaneReflection(p_in, s_in, p_plane, n_plane)
+% getPlaneReflection -
 %
-% Syntax: [p_reflect, s_reflect] = get_plane_reflection(p_in, s_in, p_plane, n_plane)
+% Syntax: [p_reflect, s_reflect] = getPlaneReflection(p_in, s_in, p_plane, n_plane)
 %
 % Inputs:
 %    - p_in, s_in, p_plane, n_plane -
@@ -1474,25 +1458,25 @@ pp = [0, 0, 0]; % [m]
 np = [cos(pi/2-theta), 0, sin(pi/2-theta)];
 
 %% Reflected beam
-[p2, s2] = get_plane_reflection(p1, s1, pp, np);
+[p2, s2] = getPlaneReflection(p1, s1, pp, np);
 
 %% Secondary Mirror
 ps = pp ...
-     + 0.07*[cos(theta), 0, -sin(theta)] ... % x offset (does not matter a lot)
+     + 0.032*[cos(theta), 0, -sin(theta)] ... % x offset (does not matter a lot)
      - np*10e-3./(2*cos(theta)) ... % Theoretical offset
      + np*args.dz; % Add error in distance
 
 ns = [Rdry*Rdrx*[cos(pi/2-theta), 0, sin(pi/2-theta)]']'; % Normal
 
 %% Output Beam
-[p3, s3] = get_plane_reflection(p2, s2, ps, ns);
+[p3, s3] = getPlaneReflection(p2, s2, ps, ns);
 
 %% Detector
 pd = [1, 0, 0]; % [m]
 nd = [-1, 0, 0];
 
 %% Get beam position on the decector
-p4 = get_plane_reflection(p3, s3, pd, nd);
+p4 = getPlaneReflection(p3, s3, pd, nd);
 #+end_src
 
 #+begin_src matlab
@@ -1535,8 +1519,65 @@ function [] = plotBeamPath(results)
 #+end_src
 
 #+begin_src matlab
-plot3([results.p1(1), results.p2(1)],[results.p1(2), results.p2(2)], [results.p1(3), results.p2(3)])
-plot3([results.p2(1), results.p3(1)],[results.p2(2), results.p3(2)], [results.p2(3), results.p3(3)])
-plot3([results.p3(1), results.p4(1)],[results.p3(2), results.p4(2)], [results.p3(3), results.p4(3)])
+plot3(100*[results.p1(1), results.p2(1)],100*[results.p1(2), results.p2(2)], 100*[results.p1(3), results.p2(3)], 'linewidth', 0.5)
+plot3(100*[results.p2(1), results.p3(1)],100*[results.p2(2), results.p3(2)], 100*[results.p2(3), results.p3(3)], 'linewidth', 0.5)
+plot3(100*[results.p3(1), results.p4(1)],100*[results.p3(2), results.p4(2)], 100*[results.p3(3), results.p4(3)], 'linewidth', 0.5)
 #+end_src
 
+
+** Plot Crystal
+:PROPERTIES:
+:header-args:matlab+: :tangle matlab/src/plotCrystals.m
+:header-args:matlab+: :comments none :mkdirp yes :eval no
+:END:
+<<sec:plotCrystals>>
+
+#+begin_src matlab
+function [] = plotCrystals(results, args)
+% plotCrystals -
+%
+% Syntax: [in_data] = plotCrystals(drx, dry, dz, theta, )
+%
+% Inputs:
+%    - drx, dry, dz, theta,  -
+%
+% Outputs:
+%    - in_data -
+#+end_src
+
+#+begin_src matlab
+arguments
+    results
+    args.color (3,1) double {mustBeNumeric} = [1;1;1]
+    args.alpha (1,1) double {mustBeNumeric} = 1
+end
+#+end_src
+
+#+begin_src matlab
+z = results.np;
+y = [0,1,0];
+x = cross(y,z);
+xtal1_rectangle = [results.pp + 0.02/2*y + 0.035/2*x;
+                   results.pp - 0.02/2*y + 0.035/2*x;
+                   results.pp - 0.02/2*y - 0.035/2*x;
+                   results.pp + 0.02/2*y - 0.035/2*x];
+#+end_src
+
+#+begin_src matlab
+patch(100*xtal1_rectangle(:,1), 100*xtal1_rectangle(:,2), 100*xtal1_rectangle(:,3), 'k-')
+#+end_src
+
+#+begin_src matlab
+z = results.ns;
+% y = [0,cos(drx),sin(drx)];
+y = [0,1,0];
+x = cross(y,z);
+xtal2_rectangle = [results.ps + 0.02/2*y + 0.07/2*x;
+                   results.ps - 0.02/2*y + 0.07/2*x;
+                   results.ps - 0.02/2*y - 0.07/2*x;
+                   results.ps + 0.02/2*y - 0.07/2*x];
+#+end_src
+
+#+begin_src matlab
+patch(100*xtal2_rectangle(:,1), 100*xtal2_rectangle(:,2), 100*xtal2_rectangle(:,3), 'k-')
+#+end_src
diff --git a/dcm-metrology.pdf b/dcm-metrology.pdf
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diff --git a/figs/motion_beam_dry_error.pdf b/figs/motion_beam_dry_error.pdf
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diff --git a/figs/motion_beam_dry_error.png b/figs/motion_beam_dry_error.png
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diff --git a/figs/motion_beam_dz_error.pdf b/figs/motion_beam_dz_error.pdf
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literal 0
HcmV?d00001

diff --git a/matlab/src/getBeamPath.m b/matlab/src/getBeamPath.m
index bffa885..10d834f 100644
--- a/matlab/src/getBeamPath.m
+++ b/matlab/src/getBeamPath.m
@@ -34,25 +34,25 @@ pp = [0, 0, 0]; % [m]
 np = [cos(pi/2-theta), 0, sin(pi/2-theta)];
 
 %% Reflected beam
-[p2, s2] = get_plane_reflection(p1, s1, pp, np);
+[p2, s2] = getPlaneReflection(p1, s1, pp, np);
 
 %% Secondary Mirror
 ps = pp ...
-     + 0.07*[cos(theta), 0, -sin(theta)] ... % x offset (does not matter a lot)
+     + 0.032*[cos(theta), 0, -sin(theta)] ... % x offset (does not matter a lot)
      - np*10e-3./(2*cos(theta)) ... % Theoretical offset
      + np*args.dz; % Add error in distance
 
 ns = [Rdry*Rdrx*[cos(pi/2-theta), 0, sin(pi/2-theta)]']'; % Normal
 
 %% Output Beam
-[p3, s3] = get_plane_reflection(p2, s2, ps, ns);
+[p3, s3] = getPlaneReflection(p2, s2, ps, ns);
 
 %% Detector
 pd = [1, 0, 0]; % [m]
 nd = [-1, 0, 0];
 
 %% Get beam position on the decector
-p4 = get_plane_reflection(p3, s3, pd, nd);
+p4 = getPlaneReflection(p3, s3, pd, nd);
 
 results = struct();
 % Beam position and orientations
diff --git a/matlab/src/get_plane_reflection.m b/matlab/src/getPlaneReflection.m
similarity index 57%
rename from matlab/src/get_plane_reflection.m
rename to matlab/src/getPlaneReflection.m
index b65c203..56cc19d 100644
--- a/matlab/src/get_plane_reflection.m
+++ b/matlab/src/getPlaneReflection.m
@@ -1,7 +1,7 @@
-function [p_reflect, s_reflect] = get_plane_reflection(p_in, s_in, p_plane, n_plane)
-% get_plane_reflection -
+function [p_reflect, s_reflect] = getPlaneReflection(p_in, s_in, p_plane, n_plane)
+% getPlaneReflection -
 %
-% Syntax: [p_reflect, s_reflect] = get_plane_reflection(p_in, s_in, p_plane, n_plane)
+% Syntax: [p_reflect, s_reflect] = getPlaneReflection(p_in, s_in, p_plane, n_plane)
 %
 % Inputs:
 %    - p_in, s_in, p_plane, n_plane -
diff --git a/matlab/src/plotBeamPath.m b/matlab/src/plotBeamPath.m
index 066468b..7120a74 100644
--- a/matlab/src/plotBeamPath.m
+++ b/matlab/src/plotBeamPath.m
@@ -9,6 +9,6 @@ function [] = plotBeamPath(results)
 % Outputs:
 %    - in_data -
 
-plot3([results.p1(1), results.p2(1)],[results.p1(2), results.p2(2)], [results.p1(3), results.p2(3)])
-plot3([results.p2(1), results.p3(1)],[results.p2(2), results.p3(2)], [results.p2(3), results.p3(3)])
-plot3([results.p3(1), results.p4(1)],[results.p3(2), results.p4(2)], [results.p3(3), results.p4(3)])
+plot3(100*[results.p1(1), results.p2(1)],100*[results.p1(2), results.p2(2)], 100*[results.p1(3), results.p2(3)], 'linewidth', 0.5)
+plot3(100*[results.p2(1), results.p3(1)],100*[results.p2(2), results.p3(2)], 100*[results.p2(3), results.p3(3)], 'linewidth', 0.5)
+plot3(100*[results.p3(1), results.p4(1)],100*[results.p3(2), results.p4(2)], 100*[results.p3(3), results.p4(3)], 'linewidth', 0.5)
diff --git a/matlab/src/plotCrystals.m b/matlab/src/plotCrystals.m
new file mode 100644
index 0000000..f3e7770
--- /dev/null
+++ b/matlab/src/plotCrystals.m
@@ -0,0 +1,37 @@
+function [] = plotCrystals(results, args)
+% plotCrystals -
+%
+% Syntax: [in_data] = plotCrystals(drx, dry, dz, theta, )
+%
+% Inputs:
+%    - drx, dry, dz, theta,  -
+%
+% Outputs:
+%    - in_data -
+
+arguments
+    results
+    args.color (3,1) double {mustBeNumeric} = [1;1;1]
+    args.alpha (1,1) double {mustBeNumeric} = 1
+end
+
+z = results.np;
+y = [0,1,0];
+x = cross(y,z);
+xtal1_rectangle = [results.pp + 0.02/2*y + 0.035/2*x;
+                   results.pp - 0.02/2*y + 0.035/2*x;
+                   results.pp - 0.02/2*y - 0.035/2*x;
+                   results.pp + 0.02/2*y - 0.035/2*x];
+
+patch(100*xtal1_rectangle(:,1), 100*xtal1_rectangle(:,2), 100*xtal1_rectangle(:,3), 'k-')
+
+z = results.ns;
+% y = [0,cos(drx),sin(drx)];
+y = [0,1,0];
+x = cross(y,z);
+xtal2_rectangle = [results.ps + 0.02/2*y + 0.07/2*x;
+                   results.ps - 0.02/2*y + 0.07/2*x;
+                   results.ps - 0.02/2*y - 0.07/2*x;
+                   results.ps + 0.02/2*y - 0.07/2*x];
+
+patch(100*xtal2_rectangle(:,1), 100*xtal2_rectangle(:,2), 100*xtal2_rectangle(:,3), 'k-')