ESRF Double Crystal Monochromator - Metrology
+Table of Contents
++
This report is also available as a pdf.
++ +
+In this document, the metrology system is studied. +First, in Section 1 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 2) +
- The periodic non-linearity of the interferometers (Section 3) +
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 +
+ ++Only 3 degrees of freedom are of interest: +
+-
+
- \(d_z\) +
- \(r_y\) +
- \(r_x\) +
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. +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. +
+ +| 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” | +
+
Figure 1: Schematic of the Metrology System
+1.2. 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: +
+ +| 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: +
+-
+
- \(d_z\) 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: +
+%% 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]; ++
+Therefore, the matrix that gives the relative pose of the crystal from the 9 interferometers is: +
+%% 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,:); ++
| + | 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 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.
+
2. 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. +
+2.1. 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. +
+2.2. 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. +
+2.3. Comparison
+3. Attocube - Periodic Non-Linearity
++(Ducourtieux, 2018, p. 11 to 12; See Thurner et al., 2015, p. 8) +
+ ++The idea is to calibrate the periodic non-linearity of the interferometers, a known displacement must be imposed and the interferometer output compared to this displacement. +This should be performed over several periods in order to characterize the error. +
+ ++We here suppose that we are already in the frame of the Attocube (the fast-jack displacements are converted to Attocube displacement using the transformation matrices). +We also suppose that we are at a certain Bragg angle, and that the stepper motors are not moving: only the piezoelectric actuators are used. +
+ ++The setup is schematically with the block diagram in Figure 5. +The signals are: +
+-
+
- \(u\): Actuator Signal (position where we wish to go) +
- \(d\): Disturbances affecting the signal +
- \(y\): Displacement of the crystal +
- \(y_g\): Measurement of the crystal motion by the strain gauge with some noise \(n_g\) +
- \(y_a\): Measurement of the crystal motion by the interferometer with some noise \(n_a\) +
+
Figure 5: Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube
++The problem is to estimate the periodic non-linearity of the Attocube from the imperfect measurements \(y_a\) and \(y_g\). +
+ ++The wavelength of the Attocube is 1530nm, therefore the non-linearity has a period of 765nm. +The amplitude of the non-linearity can vary from one unit to the other (and maybe from one experimental condition to the other). +It is typically between 5nm peak to peak and 20nm peak to peak. +
+3.1. Calibration - Concept
+3.2. 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
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