1. Metrology Concept
+1. Metrology Concept
-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.
@@ -114,8 +120,8 @@ Only 3 degrees of freedom are of interest:
1.1. Sensor Topology
+1.1. 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. @@ -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.
-| Notation | -Meaning | +Notation | +Meaning |
|---|
| Number | +Measurement | +Description | +
|---|---|---|
| 1 | +\(z_{1r,u}\) | +First “Ring” Crystal, “upstream” | +
| 2 | +\(z_{1r,c}\) | +First “Ring” Crystal, “center” | +
| 3 | +\(z_{1r,d}\) | +First “Ring” Crystal, “downstream” | +
| 4 | +\(z_{1h,u}\) | +First “Hall” Crystal, “upstream” | +
| 5 | +\(z_{1h,c}\) | +First “Hall” Crystal, “center” | +
| 6 | +\(z_{1h,d}\) | +First “Hall” Crystal, “downstream” | +
| 7 | +\(z_{2h,u}\) | +Second “Hall” Crystal, “upstream” | +
| 8 | +\(z_{2h,c}\) | +Second “Hall” Crystal, “center” | +
| 9 | +\(z_{2h,d}\) | +Second “Hall” Crystal, “downstream” | +
| 10 | +\(z_{2r,u}\) | +Second “Ring” Crystal, “upstream” | +
| 11 | +\(z_{2r,c}\) | +Second “Ring” Crystal, “center” | +
| 12 | +\(z_{2r,d}\) | +Second “Ring” Crystal, “downstream” | +
| 13 | +\(z_{mf,u}\) | +Metrology Frame, “upstream” | +
| 14 | +\(z_{mf,dr}\) | +Metrology Frame, “downstream-ring” | +
| 15 | +\(z_{mf,dh}\) | +Metrology Frame, “downstream-hall” | +
Figure 1: Schematic of the Metrology System
@@ -184,326 +306,502 @@ Three additional interferometers are used to measured the relative motion of the1.2. Crystal’s motion computation
+1.2. Computation of the relative pose between first and second crystals
-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 this repository (https://gitlab.esrf.fr/dehaeze/dcm-kinematics).
-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 relative pose of the primary and secondary crystals \([d_{h,z},\ r_{h,y},\ r_{h,x}]\) or \([d_{r,z},\ r_{r,y},\ r_{r,x}]\).
+-For the 311 crystals: -
- -| 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 | -
| Notation | -Description | -
|---|---|
dzm |
-Positive: increase of distance | -
rym |
-- |
rxm |
-- |
dz1 |
-Positive: decrease of distance | -
ry1 |
-- |
rx1 |
-- |
dz2 |
-Positive: increase of distance | -
ry2 |
-- |
rx2 |
-- |
-
Figure 2: Forward Kinematics for the Metrology frame
-
-
Figure 3: Forward Kinematics for the 1st crystal
-
-
Figure 4: Forward Kinematics for the 2nd crystal
--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: +The sign conventions for the relative crystal pose are:
-
-
- \(d_z\) 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 +
- An increase of \(d_{h,z}\) means the two crystals are further apart +
- An increase of \(r_{h,}\) means that the second crystals experiences a rotation around \(y\) with respect to the primary crystal +
- An increase of \(r_{h,x}\) means that the second crystals experiences a rotation around \(x\) with respect to the primary crystal +
+The relative pose can be expressed as a function of the interferometers using the Jacobian matrices for the “hall” crystals: +
+\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} + ++As well as for the “ring” crystals: +
+\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} + +
+Values of the matrices can be found in the document describing the kinematics of the DCM (see https://gitlab.esrf.fr/dehaeze/dcm-kinematics).
+
2. Relation Between Crystal position and X-ray measured displacement
++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. +
+ ++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 an external metrology measuring the x-ray beam position/orientation
-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.
-The Jacobian matrices are defined as follow: +In order to easily study that, “ray tracing” techniques are used.
-%% 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]; -
2.1. Definition of frame
+| - | ur1 [nm] |
-h1 [nm] |
-dr1 [nm] |
-ur2 [nm] |
-h2 [nm] |
-dr1 [nm] |
-um [nm] |
-dhm [nm] |
-drm [nm] |
+Position | +Orientation | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
dz [nm] |
--0.25 | --0.5 | --0.25 | --0.25 | --0.5 | --0.25 | -0.463 | -0.268 | -0.268 | +input beam | +x1,y1,z1 |
+s1 |
rx [nrad] |
-13.889 | -0.0 | --13.889 | -7.143 | -0.0 | --7.143 | --5.263 | -2.632 | -2.632 | +primary mirror | +xp,yp,zp |
+np |
ry [nrad] |
-16.667 | --33.333 | -16.667 | -16.667 | --33.333 | -16.667 | -0.0 | --3.922 | -3.922 | +reflected beam | +x2,y2,z2 |
+s2 |
+
| secondary mirror | +xs,ys,zz |
+ns |
+||||||||||
| output beam | +x3,y3,z3 |
+s3 |
+||||||||||
| Dectector | +xd,yd,zd |
+nd |
-
+
[ ]Add schematic
+
-From table 4, 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.
+Notations for the pose of the output beam: d_b_y, d_b_z, r_b_y, r_b_z.
+
+\(d_{b,y}, d_{b,z}, r_{b,y}, r_{b,z}\) +
+ ++The xy position of the beam is taken in the \(x=0\) plane.
+ +2.2. Effect of an error in crystal’s distance
+
+In Figure 2 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.
+
+Visually, it is clear that this induce a z offset of the output beam.
+
+
Figure 2: Visual Effect of an error in dz (1mm). Side view.
2. Relation Between Crystal position and X-ray measured displacement
-2.1. Setup
-
+The motion of the output beam is displayed as a function of the Bragg angle in Figure 3.
+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.
+
+This is indeed equal to: +
+\begin{equation} +\boxed{d_{b,z} = 2 d_z \cos \theta} +\end{equation} + + +
+
Figure 3: Motion of the output beam with dZ error
+2.3. Effect of an error in crystal’s x parallelism
+
+The effect of an error in rx crystal parallelism on the output beam is visually shown in Figure 4 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.
+
+
Figure 4: Visual Effect of an error in drx (1 degree). Top View.
+The effect of drx as a function of the Bragg angle on the output beam pose is computed and shown in Figure 5.
+
+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:
+
+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 \(x = 0\); +
+ + +
+
Figure 5: Motion of the output beam with drx error
+2.4. Effect of an error in crystal’s y parallelism
+
+The effect of an error in ry crystal parallelism on the output beam is visually shown in Figure 6 for three bragg angles (5, 55 and 85 degrees).
+
+
Figure 6: Visual Effect of an error in dry (1 degree). Side view.
+The effect of dry as a function of the Bragg angle on the output beam pose is computed and shown in Figure 7.
+It is clear that this induces a rotation of the output beam in the y direction equals to 2 times dry:
+
+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. +
+ + +
+
Figure 7: Motion of the output beam with dry error
+2.5. Summary
++Effects of crystal’s pose errors on the output beam are summarized in Table 3. +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. +
+ +| 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 | +
2.6. “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 8 where it is clear that the output beam experiences some vertical motion. +
+ + +
+
Figure 8: Visual Effect of a channel cut scan
++The \(z\) offset of the beam for several channel cut scans are shown in Figure 9. +
+ + +
+
Figure 9: Z motion of the beam during “channel cut” scans
+3. Determining relative pose between the crystals using the X-ray
++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. +
+3.1. Determine the \(y\) parallelism - “Rocking Curve”
+3.2. Determine the \(x\) parallelism - Bragg Scan
+3.3. Determine the \(z\) distance - Bragg Scan
+3.4. Use Channel cut scan to determine crystal dry parallelism
+
+Now, let’s suppose we want to determine the dry angle between the crystals.
+
dry = 1e-6; % [rad] ++
+The error is +
+\begin{equation} +\boxed{d_{b,z} = -2 d_z \cos()} +\end{equation} +3.5. Effect of an error on Bragg angle
+4. Deformations of the Metrology Frame
++The transformation matrices are 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. +
+4.1. 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. +
+ + +
Figure 5: Schematic of the setup
+Figure 10: Schematic of the setup
@@ -529,104 +827,10 @@ Frame rate is: 42 fps
2.2. Relation between second crystal motion and beam motion
-2.2.1. 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 \([z^\prime, R_{y^\prime}, R_{x^\prime}]\). -
- -
-
Figure 6: Relation between \(d_{z^\prime}\) motion of the second crystal and vertical motion of the beam
-
-
Figure 7: Relation between vertical motion of the second crystal and vertical motion of the output beam
-2.2.2. Ry motion of second crystal
--with \(D_{\text{vlm}} \approx 10\,m\). -
-2.2.3. Rx motion of second crystal
-3. Deformations of the Metrology Frame
--The transformation matrix in Table 4 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. -
-3.1. 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. -
-3.2. Simulations
-4.2. 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. @@ -634,13 +838,13 @@ This may be done using FE software.
3.3. Comparison
+4.3. Comparison
3.4. Test
-4.4. Test
+aa = importdata("correctInterf-vlm-220201.dat");
@@ -654,9 +858,9 @@ This may be done using FE software.
3.5. Measured frame deformation
-4.5. Measured frame deformation
+data = table2array(readtable('itf_polynom.csv','NumHeaderLines',1)); th = pi/180*data(:,1); % [rad] @@ -683,10 +887,10 @@ ry1 = 1e-9*dat
Figure 8: description
+Figure 11: description
3.6. Test
-4.6. Test
+filename = "/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5";
@@ -789,10 +993,10 @@ data.xtal2_111_d = double(h5read(filename, '/7.1/instru
Figure 9: Drifts of the second crystal as a function of Bragg Angle
+Figure 12: Drifts of the second crystal as a function of Bragg Angle
3.7. Repeatability of frame deformation
-4.7. Repeatability of frame deformation
+filename = "/home/thomas/mnt/data_id21/22Jan/blc13550/id21/test_xtal1_interf/test_xtal1_interf_0001/test_xtal1_interf_0001.h5";
@@ -857,15 +1061,15 @@ data_2.dz = 1e-9*<
4. Attocube - Periodic Non-Linearity
-5. Attocube - Periodic Non-Linearity
+-Interferometers have some periodic nonlinearity (Thurner et al., 2015). -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 Ducourtieux, 2018, p. 67 to 69; Thurner et al., 2015). +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.
@@ -882,9 +1086,9 @@ This process is performed over several periods in order to characterize the erro4.1. Measurement Setup
-5.1. Measurement Setup
+The metrology that will be compared with the interferometers are the strain gauges incorporated in the PI piezoelectric stacks.
@@ -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. -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.
@@ -910,7 +1114,7 @@ As any deformations of the metrology frame of deformation of the crystal’s-The setup is schematically with the block diagram in Figure 10. +The setup is schematically with the block diagram in Figure 13.
@@ -926,10 +1130,10 @@ The PI controller takes care or controlling to position as measured by the strai -
Figure 10: Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube
+Figure 13: Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube
@@ -938,9 +1142,9 @@ The problem is to estimate the periodic non-linearity of the Attocube from the i
4.2. Choice of the reference signal
-5.2. Choice of the reference signal
+The main specifications for the reference signal are;
@@ -963,9 +1167,9 @@ Based on the above discussion, one suitable excitation signal is a sinusoidal sw4.3. Repeatability of the non-linearity
-5.3. 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.
@@ -984,9 +1188,9 @@ One way to precisely estimate the laser wavelength is to estimate the non linear4.4. Simulation
-5.4. Simulation
+Suppose we have a first approximation of the non-linear period.
@@ -1005,18 +1209,18 @@ period_nl = period_est + period_err;-The non-linear errors are first estimated at the beginning of the stroke (Figure 11). +The non-linear errors are first estimated at the beginning of the stroke (Figure 14).
-
Figure 11: Estimation of the non-linear errors at the beginning of the stroke
+Figure 14: Estimation of the non-linear errors at the beginning of the stroke
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 12). +To do so, the same measurement is performed with a stroke of several millimeters (Figure 15).
@@ -1025,10 +1229,10 @@ This is due to a mismatch between the estimated period and the true period of th
-
Figure 12: Estimated non-linear errors at a latter position
+Figure 15: Estimated non-linear errors at a latter position
@@ -1058,7 +1262,7 @@ with \(\lambda_{\text{est}}\) the estimated error’s period.
-From Figure 12, we can see that there is an offset between the two curves. +From Figure 15, we can see that there is an offset between the two curves. Let’s call this offset \(\epsilon_x\), we then have:
\begin{equation} @@ -1138,9 +1342,9 @@ The maximum stroke is 2.9 [mm]4.5. Measurements
-5.5. Measurements
+We have some constrains on the way the motion is imposed and measured:
@@ -1165,20 +1369,10 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).Bibliography
-Created: 2022-02-15 mar. 14:25
+Created: 2022-06-02 Thu 18:06