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2022-02-15 14:24:11 +01:00
#+TITLE: ESRF Double Crystal Monochromator - Metrology
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
#+BIND: org-latex-image-default-option "scale=1"
#+BIND: org-latex-image-default-width ""
#+LATEX_CLASS: scrreprt
#+LATEX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
#+LATEX_HEADER_EXTRA: \input{preamble.tex}
#+bibliography: dcm_metrology.bib
#+cite_export: csl ~/.local/data/csl-styles/elsevier-harvard.csl
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :tangle no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
#+begin_export html
<hr>
<p>This report is also available as a <a href="./dcm_metrology.pdf">pdf</a>.</p>
<hr>
#+end_export
#+latex: \clearpage
* Introduction :ignore:
In this document, the metrology system is studied.
First, in Section [[sec:metrology_concept]] the goal of the metrology system is stated and the proposed concept is described.
In order to increase the accuracy of the metrology system, two problems are to be dealt with:
- The deformation of the metrology frame under the action of gravity (Section [[sec:frame_deformations]])
- The periodic non-linearity of the interferometers (Section [[sec:dcm_attocube_lut]])
* Metrology Concept
<<sec:metrology_concept>>
** Introduction :ignore:
The goal of the metrology system is to measure the distance and default of parallelism orientation between the first and second crystals
Only 3 degrees of freedom are of interest:
- $d_z$
- $r_y$
- $r_x$
** 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
** Sensor Topology
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.
Three interferometers are used to measured the 3dof of interest for each crystals.
Three additional interferometers are used to measured the relative motion of the metrology frame.
#+name: tab:metrology_notations
#+caption: Notations for the metrology frame
#+attr_latex: :environment tabularx :width 0.4\linewidth :align cX
#+attr_latex: :center t :booktabs t
| Notation | Meaning |
|----------+--------------------------|
| =d= | "Downstream": Positive X |
| =u= | "Upstream": Negative X |
| =h= | "Hall": Positive Y |
| =r= | "Ring": Negative Y |
| =f= | "Frame" |
| =1= | "First Crystals" |
| =2= | "Second Crystals" |
#+name: fig:metrology_schematic
#+caption: Schematic of the Metrology System
[[file:figs/metrology_schematic.png]]
** Crystal's motion computation
From the raw interferometric measurements, the pose between the first and second crystals can be computed.
First, Jacobian matrices can be used to convert raw interferometer measurements to axial displacement and orientation of the crystals and metrology frame.
For the 311 crystals:
#+name: tab:311_crystals_notations_raw_interf
#+caption: Table caption
#+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
#+attr_latex: :center t :booktabs t
| Notation | Description |
|----------+-----------------------------------|
| =um= | Metrology Frame - Upstream |
| =dhm= | Metrology Frame - Downstream Hall |
| =drm= | Metrology Frame - Downstream Ring |
|----------+-----------------------------------|
| =ur1= | First Crystal - Upstream Ring |
| =h1= | First Crystal - Hall |
| =dr1= | First Crystal - Downstream Ring |
|----------+-----------------------------------|
| =ur2= | First Crystal - Upstream Ring |
| =h2= | First Crystal - Hall |
| =dr2= | First Crystal - Downstream Ring |
#+name: tab:311_crystals_notations_converted_interf
#+caption: Table caption
#+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
#+attr_latex: :center t :booktabs t
| Notation | Description |
|----------+--------------------------------|
| =dzm= | Positive: increase of distance |
| =rym= | |
| =rxm= | |
|----------+--------------------------------|
| =dz1= | Positive: decrease of distance |
| =ry1= | |
| =rx1= | |
|----------+--------------------------------|
| =dz2= | Positive: increase of distance |
| =ry2= | |
| =rx2= | |
#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_m.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js_inv) {$\bm{J}_{s,m}^{-1}$};
% Connections and labels
\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{m} \\ dh_{m} \\ dr_{m} \end{bmatrix}$} -- (Js_inv.west);
\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{zm} \\ r_{ym} \\ r_{xm} \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name: fig:schematic_sensor_jacobian_forward_kinematics_m
#+caption: Forward Kinematics for the Metrology frame
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_forward_kinematics_m.png]]
#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_1.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js_inv) {$\bm{J}_{s,1}^{-1}$};
% Connections and labels
\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{r1} \\ h_1 \\ d_{r1} \end{bmatrix}$} -- (Js_inv.west);
\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{z1} \\ r_{y1} \\ r_{x1} \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name: fig:schematic_sensor_jacobian_forward_kinematics_1
#+caption: Forward Kinematics for the 1st crystal
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_forward_kinematics_1.png]]
#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_2.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js_inv) {$\bm{J}_{s,2}^{-1}$};
% Connections and labels
\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{r2} \\ h_2 \\ d_{r2} \end{bmatrix}$} -- (Js_inv.west);
\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{z2} \\ r_{y2} \\ r_{x2} \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name: fig:schematic_sensor_jacobian_forward_kinematics_2
#+caption: Forward Kinematics for the 2nd crystal
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_forward_kinematics_2.png]]
Then, the displacement and orientations can be combined as follows:
\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}
Therefore:
- $d_z$ represents the distance between the two crystals
- $d_{r_y}$ represents the rotation of the second crystal w.r.t. the first crystal around $y$ axis
- $d_{r_x}$ represents the rotation of the second crystal w.r.t. the first crystal around $x$ axis
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.
The Jacobian matrices are defined as follow:
#+begin_src matlab
%% Sensor Jacobian matrix for the metrology frame
J_m = [1, 0.102, 0
1, -0.088, 0.1275
1, -0.088, -0.1275];
%% Sensor Jacobian matrix for 1st "111" crystal
J_s_111_1 = [-1, -0.036, -0.015
-1, 0, 0.015
-1, 0.036, -0.015];
%% Sensor Jacobian matrix for 2nd "111" crystal
J_s_111_2 = [1, 0.07, 0.015
1, 0, -0.015
1, -0.07, 0.015];
#+end_src
Therefore, the matrix that gives the relative pose of the crystal from the 9 interferometers is:
#+begin_src matlab
%% Compute the transformation matrix
G_111_t = [-inv(J_s_111_1), inv(J_s_111_2), -inv(J_m)];
% Sign convention for the axial motion
G_111_t(1,:) = -G_111_t(1,:);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(G_111_t, ...
{'=dz= [nm]', '=rx= [nrad]', '=ry= [nrad]'}, ...
{'=ur1= [nm]', '=h1= [nm]', '=dr1= [nm]', '=ur2= [nm]', '=h2= [nm]', '=dr1= [nm]', '=um= [nm]', '=dhm= [nm]', '=drm= [nm]'}, ...
' %.3f ');
#+end_src
#+name: tab:transformation_matrix
#+caption: Transformation Matrix
#+attr_latex: :environment tabularx :width \linewidth :align cccccccccc
#+attr_latex: :center t :booktabs t :font \scriptsize
#+RESULTS:
| | =ur1= [nm] | =h1= [nm] | =dr1= [nm] | =ur2= [nm] | =h2= [nm] | =dr1= [nm] | =um= [nm] | =dhm= [nm] | =drm= [nm] |
|-------------+------------+-----------+------------+------------+-----------+------------+-----------+------------+------------|
| =dz= [nm] | -0.25 | -0.5 | -0.25 | -0.25 | -0.5 | -0.25 | 0.463 | 0.268 | 0.268 |
| =rx= [nrad] | 13.889 | 0.0 | -13.889 | 7.143 | 0.0 | -7.143 | -5.263 | 2.632 | 2.632 |
| =ry= [nrad] | 16.667 | -33.333 | 16.667 | 16.667 | -33.333 | 16.667 | 0.0 | -3.922 | 3.922 |
From table [[fig:schematic_sensor_jacobian_forward_kinematics_2]], we can determine the effect of each interferometer on the estimated relative pose between the crystals.
For instance, an error on =dr1= will have much greater impact on =ry= than an error on =drm=.
* Relation Between Crystal position and X-ray measured displacement
** Setup
#+name: fig:calibration_setup
#+caption: Schematic of the setup
[[file:figs/calibration_setup.png]]
Detector:
https://www.baslerweb.com/en/products/cameras/area-scan-cameras/ace/aca1920-40gc/
Pixel size depends on the magnification used (1x, 6x, 12x).
Pixel size of camera is 5.86 um x 5.86 um.
With typical magnification of 6x, pixel size is ~1.44um x 1.44um
Frame rate is: 42 fps
** 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
bragg = pi/180*linspace(5,75,100);
#+end_src
** Relation between second crystal motion and beam motion
*** 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}
* Deformations of the Metrology Frame
<<sec:frame_deformations>>
** Introduction :ignore:
The transformation matrix in Table [[tab:transformation_matrix]] is valid only if the metrology frames are solid bodies.
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.
This can be calibrated.
** Measurement Setup
Two beam viewers:
- one close to the DCM to measure position of the beam
- one far away to the DCM to measure orientation of the beam
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.
** Simulations
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.
** Comparison
** 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
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
** Test
#+begin_src matlab
aa = importdata("correctInterf-vlm-220201.dat");
#+end_src
#+begin_src matlab
figure;
plot(aa.data(:,1), aa.data(:,24))
#+end_src
** Measured frame deformation
#+begin_src matlab
data = table2array(readtable('itf_polynom.csv','NumHeaderLines',1));
th = pi/180*data(:,1); % [rad]
fj = 0.030427 - 10.51e-3./(2*cos(th)); % [m]
rx2 = 1e-9*data(:,2); % [rad]
ry2 = 1e-9*data(:,3); % [rad]
rx1 = 1e-9*data(:,4); % [rad]
ry1 = 1e-9*data(:,5); % [rad]
#+end_src
#+begin_src matlab
figure;
hold on;
plot(180/pi*th, 1e6*detrend(-rx2, 1), '.')
% plot(180/pi*th, detrend(ry2, 1))
% plot(180/pi*th, detrend(rx1, 1))
% plot(180/pi*th, detrend(ry1, 1))
hold off;
xlabel('Bragg Angle [deg]');
ylabel('Measured $R_x$ [$\mu$rad]')
xlim([10, 75]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/calibration_drx_pres.pdf', 'width', 'full', 'height', 'normal');
#+end_src
#+name: fig:calibration_drx_pres
#+caption: description
#+RESULTS:
[[file:figs/calibration_drx_pres.png]]
Strange that there is correlation between Rx and Ry.
#+begin_src matlab
figure;
hold on;
plot(108/pi*th, 1e9*detrend(rx1, 0), '-', 'DisplayName', '$Rx_1$')
plot(108/pi*th, 1e9*detrend(ry1, 0), '-', 'DisplayName', '$Ry_1$')
hold off;
xlabel('Bragg Angle [deg]'); ylabel('Angle Offset [nrad]');
legend()
#+end_src
#+begin_src matlab
figure;
hold on;
plot(1e3*fj, detrend(rx2, 1), 'DisplayName', '$Rx_1$')
plot(1e3*fj, detrend(ry2, 1), 'DisplayName', '$Ry_1$')
hold off;
xlabel('Fast Jack Displacement [mm]'); ylabel('Angle Offset [nrad]');
legend()
#+end_src
#+begin_src matlab
%% Compute best polynomial fit
f_rx2 = fit(180/pi*th, 1e9*rx2, 'poly4');
f_ry2 = fit(180/pi*th, 1e9*ry2, 'poly4');
f_rx1 = fit(180/pi*th, 1e9*rx1, 'poly4');
f_ry1 = fit(180/pi*th, 1e9*ry1, 'poly4');
#+end_src
#+begin_src matlab
figure;
hold on;
plot(180/pi*th, f_rx2(180/pi*th))
plot(180/pi*th, f_ry2(180/pi*th))
plot(180/pi*th, f_rx1(180/pi*th))
plot(180/pi*th, f_ry1(180/pi*th))
hold off;
#+end_src
#+begin_src matlab
figure;
hold on;
plot(th, f_rx2(th) - rx2)
plot(th, f_ry2(th) - ry2)
plot(th, f_rx1(th) - rx1)
plot(th, f_ry1(th) - ry1)
hold off;
#+end_src
#+begin_src matlab
figure;
hold on;
plot(th, f(th))
plot(th, rx2, '.')
#+end_src
** Test
#+begin_src matlab
filename = "/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5";
#+end_src
#+begin_src matlab
data = struct();
data.xtal1_111_u = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM4/data'));
data.xtal1_111_m = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM5/data'));
data.xtal1_111_d = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM6/data'));
data.mframe_u = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM3/data'));
data.mframe_dh = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM4/data'));
data.mframe_dr = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM5/data'));
data.bragg = (pi/180)*double(h5read(filename, '/7.1/instrument/FPGA2_SSIM6/data'));
data.fj_pos = 0.030427 - 10.5e-3./(2*cos(data.bragg));
data.time = double(h5read(filename, '/7.1/instrument/time/data'));
data.rx = 1e-9*double(h5read(filename, '/7.1/instrument/xtal1_111_rx/data'));
data.ry = 1e-9*double(h5read(filename, '/7.1/instrument/xtal1_111_ry/data'));
data.z = 1e-9*double(h5read(filename, '/7.1/instrument/xtal1_111_z/data'));
data.drx = 1e-9*double(h5read(filename, '/7.1/instrument/xtal_111_drx_filter/data'));
data.dry = 1e-9*double(h5read(filename, '/7.1/instrument/xtal_111_dry_filter/data'));
data.dz = 1e-9*double(h5read(filename, '/7.1/instrument/xtal_111_dz_filter/data'));
data.xtal2_111_u = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM10/data'))+10.5e6./(2*cos(data.bragg));
data.xtal2_111_m = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM1/data'))+10.5e6./(2*cos(data.bragg));
data.xtal2_111_d = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM2/data'))+10.5e6./(2*cos(data.bragg));
#+end_src
#+begin_src matlab :exports none
%% Drifts of the metrology as a function of Bragg Angle
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.mframe_dh, 0))
plot(180/pi*data.bragg, detrend(data.mframe_u, 0))
plot(180/pi*data.bragg, detrend(data.mframe_dr, 0))
hold off;
xlabel('Bragg [deg]'); ylabel('Drift [nm]');
ax2 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.mframe_dh, 1))
plot(180/pi*data.bragg, detrend(data.mframe_u, 1))
plot(180/pi*data.bragg, detrend(data.mframe_dr, 1))
hold off;
xlabel('Bragg [deg]'); set(gca, 'YTickLabel',[]);
ax3 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.mframe_dh, 2), 'DisplayName', 'dh')
plot(180/pi*data.bragg, detrend(data.mframe_u, 2), 'DisplayName', 'u')
plot(180/pi*data.bragg, detrend(data.mframe_dr, 2), 'DisplayName', 'dr')
hold off;
xlabel('Bragg [deg]'); set(gca, 'YTickLabel',[]);
legend();
linkaxes([ax1,ax2,ax3], 'xy');
xlim([10, 70]); ylim([-50, 50]);
#+end_src
#+begin_src matlab :exports none
%% Drifts of the first crystal as a function of Bragg Angle
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.xtal1_111_u, 0))
plot(180/pi*data.bragg, detrend(data.xtal1_111_m, 0))
plot(180/pi*data.bragg, detrend(data.xtal1_111_d, 0))
hold off;
xlabel('Bragg [deg]'); ylabel('Drift [nm]');
ax2 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.xtal1_111_u, 1))
plot(180/pi*data.bragg, detrend(data.xtal1_111_m, 1))
plot(180/pi*data.bragg, detrend(data.xtal1_111_d, 1))
hold off;
xlabel('Bragg [deg]'); set(gca, 'YTickLabel',[]);
ax3 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.xtal1_111_u, 2), 'DisplayName', 'u')
plot(180/pi*data.bragg, detrend(data.xtal1_111_m, 2), 'DisplayName', 'm')
plot(180/pi*data.bragg, detrend(data.xtal1_111_d, 2), 'DisplayName', 'd')
hold off;
xlabel('Bragg [deg]'); set(gca, 'YTickLabel',[]);
legend();
linkaxes([ax1,ax2,ax3], 'xy');
xlim([10, 70]); ylim([-1000, 1000]);
#+end_src
#+begin_src matlab :exports none :results none
%% Drifts of the second crystal as a function of Bragg Angle
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.xtal2_111_u, 0))
plot(180/pi*data.bragg, detrend(data.xtal2_111_m, 0))
plot(180/pi*data.bragg, detrend(data.xtal2_111_d, 0))
hold off;
xlabel('Bragg [deg]'); ylabel('Drift [nm]')
ax2 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.xtal2_111_u, 1))
plot(180/pi*data.bragg, detrend(data.xtal2_111_m, 1))
plot(180/pi*data.bragg, detrend(data.xtal2_111_d, 1))
hold off;
xlabel('Bragg [deg]'); set(gca, 'YTickLabel',[]);
ax3 = nexttile();
hold on;
plot(180/pi*data.bragg, detrend(data.xtal2_111_u, 2))
plot(180/pi*data.bragg, detrend(data.xtal2_111_m, 2))
plot(180/pi*data.bragg, detrend(data.xtal2_111_d, 2))
hold off;
xlabel('Bragg [deg]'); set(gca, 'YTickLabel',[]);
linkaxes([ax1,ax2,ax3], 'xy');
xlim([10, 70]); ylim([-1000, 1000]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/drifts_xtal2_detrend.pdf', 'width', 'full', 'height', 'normal');
#+end_src
#+name: fig:drifts_xtal2_detrend
#+caption: Drifts of the second crystal as a function of Bragg Angle
#+RESULTS:
[[file:figs/drifts_xtal2_detrend.png]]
#+begin_src matlab :exports none :results none
%% Drifts of the second crystal as a function of Bragg Angle
figure;
tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(1e3*data.dz, detrend(data.xtal2_111_u, 0))
plot(1e3*data.dz, detrend(data.xtal2_111_m, 0))
plot(1e3*data.dz, detrend(data.xtal2_111_d, 0))
hold off;
xlabel('FJ position [mm]'); ylabel('Drift [nrad]')
ax2 = nexttile();
hold on;
plot(1e3*data.dz, detrend(data.xtal2_111_u, 1))
plot(1e3*data.dz, detrend(data.xtal2_111_m, 1))
plot(1e3*data.dz, detrend(data.xtal2_111_d, 1))
hold off;
xlabel('FJ position [mm]'); set(gca, 'YTickLabel',[]);
ax3 = nexttile();
hold on;
plot(1e3*data.dz, detrend(data.xtal2_111_u, 2))
plot(1e3*data.dz, detrend(data.xtal2_111_m, 2))
plot(1e3*data.dz, detrend(data.xtal2_111_d, 2))
hold off;
xlabel('FJ position [mm]'); set(gca, 'YTickLabel',[]);
linkaxes([ax1,ax2,ax3], 'xy');
% xlim([10, 40]); ylim([-1000, 1000]);
#+end_src
#+begin_src matlab
figure;
hold on;
plot(108/pi*data.bragg, data.drx)
plot(108/pi*data.bragg, data.dry)
hold off;
#+end_src
** Repeatability of frame deformation
#+begin_src matlab
filename = "/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5";
#+end_src
#+begin_src matlab
data_1 = struct();
data_1.time = double(h5read(filename, '/7.1/instrument/time/data'));
data_1.bragg = (pi/180)*double(h5read(filename, '/7.1/instrument/FPGA2_SSIM6/data'));
data_1.fj_pos = 0.030427 - 10.5e-3./(2*cos(data_1.bragg));
data_1.xtal1_111_u = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM4/data'));
data_1.xtal1_111_m = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM5/data'));
data_1.xtal1_111_d = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM6/data'));
data_1.xtal2_111_u = double(h5read(filename, '/7.1/instrument/FPGA1_SSIM10/data'))+10.5e6./(2*cos(data_1.bragg));
data_1.xtal2_111_m = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM1/data'))+10.5e6./(2*cos(data_1.bragg));
data_1.xtal2_111_d = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM2/data'))+10.5e6./(2*cos(data_1.bragg));
data_1.mframe_u = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM3/data'));
data_1.mframe_dh = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM4/data'));
data_1.mframe_dr = double(h5read(filename, '/7.1/instrument/FPGA2_SSIM5/data'));
data_1.drx = 1e-9*double(h5read(filename, '/7.1/instrument/xtal_111_drx_filter/data'));
data_1.dry = 1e-9*double(h5read(filename, '/7.1/instrument/xtal_111_dry_filter/data'));
data_1.dz = 1e-9*double(h5read(filename, '/7.1/instrument/xtal_111_dz_filter/data'));
#+end_src
#+begin_src matlab
data_2 = struct();
data_2.time = double(h5read(filename, '/6.1/instrument/time/data'));
data_2.bragg = (pi/180)*double(h5read(filename, '/6.1/instrument/FPGA2_SSIM6/data'));
data_2.fj_pos = 0.030427 - 10.5e-3./(2*cos(data_2.bragg));
data_2.xtal1_111_u = double(h5read(filename, '/6.1/instrument/FPGA1_SSIM4/data'));
data_2.xtal1_111_m = double(h5read(filename, '/6.1/instrument/FPGA1_SSIM5/data'));
data_2.xtal1_111_d = double(h5read(filename, '/6.1/instrument/FPGA1_SSIM6/data'));
data_2.xtal2_111_u = double(h5read(filename, '/6.1/instrument/FPGA1_SSIM10/data'))+10.5e6./(2*cos(data_2.bragg));
data_2.xtal2_111_m = double(h5read(filename, '/6.1/instrument/FPGA2_SSIM1/data'))+10.5e6./(2*cos(data_2.bragg));
data_2.xtal2_111_d = double(h5read(filename, '/6.1/instrument/FPGA2_SSIM2/data'))+10.5e6./(2*cos(data_2.bragg));
data_2.mframe_u = double(h5read(filename, '/6.1/instrument/FPGA2_SSIM3/data'));
data_2.mframe_dh = double(h5read(filename, '/6.1/instrument/FPGA2_SSIM4/data'));
data_2.mframe_dr = double(h5read(filename, '/6.1/instrument/FPGA2_SSIM5/data'));
data_2.drx = 1e-9*double(h5read(filename, '/6.1/instrument/xtal_111_drx_filter/data'));
data_2.dry = 1e-9*double(h5read(filename, '/6.1/instrument/xtal_111_dry_filter/data'));
data_2.dz = 1e-9*double(h5read(filename, '/6.1/instrument/xtal_111_dz_filter/data'));
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(180/pi*data_1.bragg, data_1.mframe_dh)
plot(180/pi*data_2.bragg, data_2.mframe_dh)
hold off;
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(1e3*data_1.fj_pos, data_1.xtal1_111_u)
plot(1e3*data_2.fj_pos, data_2.xtal1_111_u)
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(data_1.bragg, data_1.mframe_u)
plot(data_2.bragg, data_2.mframe_u)
#+end_src
* Attocube - Periodic Non-Linearity
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_attocube_lut.m
:END:
<<sec:dcm_attocube_lut>>
** Introduction :ignore:
Interferometers have some periodic nonlinearity [cite:@thurner15_fiber_based_distan_sensin_inter].
The period is a fraction of the wavelength (usually $\lambda/2$) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams [cite:See @ducourtieux18_towar_high_precis_posit_contr page 67 to 69;@thurner15_fiber_based_distan_sensin_inter;].
The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers.
In the DCM case, when using Attocube interferometers, the period non-linearity are in the order of several nanometers with a period of $765\,nm$.
This is inducing some positioning errors which are too high.
In order to overcome this issue, the periodic non-linearity of the interferometers have to be calibrated.
To do so, a displacement is imposed and measured both by the interferometers and by another metrology system which does not have this nonlinearity.
By comparing the two measured displacements, the nonlinearity can be calibration.
This process is performed over several periods in order to characterize the error over the full stroke.
** 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
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
** Measurement Setup
The metrology that will be compared with the interferometers are the strain gauges incorporated in the PI piezoelectric stacks.
It is here supposed that the measured displacement by the strain gauges are converted to the displacement at the interferometer locations.
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.
#+begin_note
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.
Only the interferometers measuring the second crystal motion can be calibrated here.
As any deformations of the metrology frame of deformation of the crystal's support can degrade the quality of the calibration, it is better to perform this calibration without any bragg angle motion.
#+end_note
The setup is schematically with the block diagram in Figure [[fig:block_diagram_lut_attocube]].
The signals are:
- $u$: Reference Signal sent to the PI controller (position where we wish to three stacks to be).
The PI controller takes care or controlling to position as measured by the strain gauges such that it is close to the reference position.
- $d$: Disturbances affecting the position of the crystals
- $y$: Displacement of the crystal as measured by one interferometer
- $y_g$: Measurement of the motion in the frame of the interferometer by the strain gauge with some noise $n_g$
- $y_a$: Measurement of the crystal motion by the interferometer with some noise $n_a$
#+begin_src latex :file block_diagram_lut_attocube.pdf
\definecolor{myblue}{rgb}{0, 0.447, 0.741}
\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
\begin{tikzpicture}
\node[block] (G) at (0,0){$G(s)$};
\node[block, align=center, right=1 of G] (non_linearity) {Periodic\\Non-linearity};
\node[addb, right=1 of non_linearity] (addna) {};
\node[addb, below=1.8 of addna] (addnsg) {};
\draw[->] ($(G.west) + (-1.0, 0)$) node[above right]{$u$} -- (G.west);
\draw[->] ($(G.north) + (0, 1.0)$) node[below right]{$d$} -- (G.north);
\draw[->] (G.east) -- (non_linearity.west);
\draw[->] (non_linearity.east) -- (addna.west);
\draw[->] (addna.east) -- ++(1.2, 0) node[above left]{$y_a$};
\draw[->] ($(addna.north) + (0, 1.0)$) node[below right](na){$n_a$} -- (addna.north);
\draw[->] ($(G.east) + (0.4, 0)$)node[branch]{} node[above]{$y$} |- (addnsg.west);
\draw[->] (addnsg.east) -- ++(1.2, 0) node[above left]{$y_g$};
\draw[->] ($(addnsg.north) + (0, 1.0)$) node[below right](nsg){$n_{g}$} -- (addnsg.north);
\begin{scope}[on background layer]
\node[fit={(non_linearity.south west) (na.north east)}, fill=myblue!20!white, draw, inner sep=6pt] (attocube) {};
\node[fit={(non_linearity.west|-addnsg.south) (nsg.north east)}, fill=myred!20!white, draw, inner sep=6pt] (straingauge) {};
\node[below right] at (attocube.north west) {Attocube};
\node[below right] at (straingauge.north west) {Strain Gauge};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:block_diagram_lut_attocube
#+caption: Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube
#+RESULTS:
[[file:figs/block_diagram_lut_attocube.png]]
The problem is to estimate the periodic non-linearity of the Attocube from the imperfect measurements $y_a$ and $y_g$.
** Choice of the reference signal
The main specifications for the reference signal are;
- sweep several periods (i.e. several micrometers)
- stay in the linear region of the strain gauge
- no excitation of mechanical modes (i.e. the frequency content of the signal should be at low frequency)
- no phase shift due to limited bandwidth of both the interferometers and the strain gauge
- the full process should be quite fast
The travel range of the piezoelectric stacks is 15 micrometers, the resolution of the strain gauges is 0.3nm and the maximum non-linearity is 0.15%.
If one non-linear period is swept (765nm), the maximum estimation error of the strain gauge is around 1nm.
Based on the above discussion, one suitable excitation signal is a sinusoidal sweep with a frequency of 10Hz.
** Repeatability of the non-linearity
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.
For that, we need to make sure that the errors are repeatable from one period to the other and also the period should be very precisely estimated (i.e. the wavelength of the laser).
Also, the laser wavelength should be very stable (specified at 50ppb).
One way to precisely estimate the laser wavelength is to estimate the non linear errors of the interferometer at an initial position, and then to estimate the non linear errors at a large offset, say 10mm.
** Simulation
Suppose we have a first approximation of the non-linear period.
#+begin_src matlab
period_est = 765e-9; % Estimated period [m]
#+end_src
And suppose the real period of the non-linear errors is a little bit above (by 0.02nm):
#+begin_src matlab
period_err = 0.02e-9; % Error on the period estimation [m]
period_nl = period_est + period_err; % Period of the non-linear errors [m]
#+end_src
#+begin_src matlab :exports none
displacement_step = 1e-8;
stroke = 0:displacement_step:25e-3; % Measured Stroke [m]
nl_errors = 5e-9*sin(2*pi*stroke/period_nl) + ...
3e-9*sin(2*pi*2*stroke/period_nl) - ...
1e-9*sin(2*pi*3*stroke/period_nl); % Measured non-linear errors [m]
est_errors = 5e-9*sin(2*pi*stroke/period_est) + ...
3e-9*sin(2*pi*2*stroke/period_est) - ...
1e-9*sin(2*pi*3*stroke/period_est); % Estimated non-linear errors [m]
#+end_src
The non-linear errors are first estimated at the beginning of the stroke (Figure [[fig:non_linear_errors_start_stroke]]).
#+begin_src matlab :exports none
%% Measured errors at the begining of the stroke
figure;
plot(1e9*stroke, 1e9*nl_errors);
xlabel('Displacement [nm]'); ylabel('Non linearity [nm]');
xlim([0, 1e9*period_nl]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/non_linear_errors_start_stroke.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:non_linear_errors_start_stroke
#+caption: Estimation of the non-linear errors at the beginning of the stroke
#+RESULTS:
[[file:figs/non_linear_errors_start_stroke.png]]
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 [[fig:non_linear_errors_middle_stroke]]).
It can be seen that there is an offset between the estimated and the measured errors.
This is due to a mismatch between the estimated period and the true period of the error.
#+begin_src matlab :exports none
%% Comparison of the estimated error and measured error
figure;
hold on;
plot(1e3*stroke, 1e9*nl_errors, ...
'DisplayName', 'Measured Errors');
plot(1e3*stroke, 1e9*est_errors, '--', ...
'DisplayName', 'Estimated Errors');
hold off;
xlabel('Displacement [mm]'); ylabel('Non linearity [nm]');
xlim([1e3*(5e-3-period_nl/2), 1e3*(5e-3+period_nl/2)]);
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/non_linear_errors_middle_stroke.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:non_linear_errors_middle_stroke
#+caption: Estimated non-linear errors at a latter position
#+RESULTS:
[[file:figs/non_linear_errors_middle_stroke.png]]
Suppose the non-linear error is characterized by a periodic function $\mathcal{E}$, to simplify let's take a sinusoidal function (this can be generalized by taking the fourier transform of the function):
\begin{equation}
\mathcal{E}(x) = \sin\left(\frac{x}{\lambda}\right)
\end{equation}
with $x$ the displacement and $\lambda$ the period of the error.
The measured error at $x_0$ is then:
\begin{equation}
\mathcal{E}_m(x_0) = \sin\left( \frac{x_0}{\lambda} \right)
\end{equation}
And the estimated one is:
\begin{equation}
\mathcal{E}_e(x_0) = \sin \left( \frac{x_0}{\lambda_{\text{est}}} \right)
\end{equation}
with $\lambda_{\text{est}}$ the estimated error's period.
From Figure [[fig:non_linear_errors_middle_stroke]], we can see that there is an offset between the two curves.
Let's call this offset $\epsilon_x$, we then have:
\begin{equation}
\mathcal{E}_m(x_0) = \mathcal{E}_e(x_0 + \epsilon_x)
\end{equation}
Which gives us:
\begin{equation}
\sin\left( \frac{x_0}{\lambda} \right) = \sin \left( \frac{x_0 + \epsilon_x}{\lambda_{\text{est}}} \right)
\end{equation}
Finally:
\begin{equation}
\boxed{\lambda = \lambda_{\text{est}} \frac{x_0}{x_0 + \epsilon_x}}
\end{equation}
The estimated delay is computed:
#+begin_src matlab
%% Estimation of the offset between the estimated and measured errors
i_period = stroke > 5e-3-period_nl/2 & stroke < 5e-3+period_nl/2;
epsilon_x = finddelay(nl_errors(i_period), est_errors(i_period)) % [m]
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
ans = sprintf('Estimated delay x0 is %.0f [nm]', 1e9*displacement_step*epsilon_x)
#+end_src
#+RESULTS:
: Estimated delay x0 is -120 [nm]
And the period $\lambda$ can be estimated:
#+begin_src matlab
%% Computation of the period [m]
period_fin = period_est * (5e-3)/(5e-3 + d_offset); % Estimated period after measurement [m]
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
ans = sprintf('The estimated period is %.3f [nm]', 1e9*period_fin)
#+end_src
#+RESULTS:
: The estimated period is 765.020 [nm]
And the results confirms that this method is working on paper.
When doing this computation, we suppose that there are *at most* one half period of offset between the estimated and the measured non-linear (to not have any ambiguity whether the estimated period is too large or too small).
Mathematically this means that the displacement $x_0$ should be smaller than:
\begin{equation}
x_0 < \frac{1}{2} \cdot \lambda \cdot \frac{\lambda}{\epsilon_\lambda}
\end{equation}
With $\epsilon_\lambda$ the absolute estimation error of the period in meters.
For instance, if we estimate the error on the period to be less than 0.1nm, the maximum displacement is:
#+begin_src matlab
%% Estimated maximum stroke [m]
max_x0 = 0.5 * 765e-9 * (765e-9)/(0.1e-9);
#+end_src
#+begin_src matlab :results value replace :exports results :tangle no
ans = sprintf('The maximum stroke is %.1f [mm]', 1e3*max_x0)
#+end_src
#+RESULTS:
: The maximum stroke is 2.9 [mm]
** Measurements
We have some constrains on the way the motion is imposed and measured:
- We want the frequency content of the imposed motion to be at low frequency in order not to induce vibrations of the structure.
We have to make sure the forces applied by the piezoelectric actuator only moves the crystal and not the fast jack below.
Therefore, we have to move much slower than the first resonance frequency in the system.
- As both $y_a$ and $y_g$ should have rather small noise, we have to filter them with low pass filters.
The cut-off frequency of the low pass filter should be high as compared to the motion (to not induce any distortion) but still reducing sufficiently the noise.
Let's say we want the noise to be less than 1nm ($6 \sigma$).
Suppose we have the power spectral density (PSD) of both $n_a$ and $n_g$.
- [ ] Take the PSD of the Attocube
- [ ] Take the PSD of the strain gauge
- [ ] Using 2nd order low pass filter, estimate the required low pass filter cut-off frequency to have sufficiently low noise
* Helping Functions :noexport:
** Initialize Path
#+NAME: m-init-path
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./matlab/mat/'); % Path for data
addpath('./matlab/src/'); % Path for functions
addpath('./matlab/'); % Path for scripts
#+END_SRC
#+NAME: m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./mat/'); % Path for data
addpath('./src/'); % Path for functions
#+END_SRC
** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;
%% Frequency Vector
freqs = logspace(1, 3, 1000);
#+END_SRC
* Bibliography
:PROPERTIES:
:UNNUMBERED: t
:END:
#+print_bibliography:
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