diff --git a/.latexmkrc b/.latexmkrc index 86be11d..4234a51 100644 --- a/.latexmkrc +++ b/.latexmkrc @@ -11,7 +11,7 @@ # PDF Generation/Building/Compilation # ====================================================================================== -@default_files=('dcm_metrology.tex'); +@default_files=('dcm-metrology.tex'); # PDF-generating modes are: # 1: pdflatex, as specified by $pdflatex variable (still largely in use) diff --git a/dcm-metrology.html b/dcm-metrology.html index 9d0f651..7b94d00 100644 --- a/dcm-metrology.html +++ b/dcm-metrology.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + ESRF Double Crystal Monochromator - Metrology @@ -39,51 +39,52 @@

Table of Contents

@@ -93,26 +94,26 @@

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. +First, in Section 1 the goal of the metrology system is stated and the proposed concept is described.

-How the relative crystal pose is affecting the pose of the output beam is studied in Section 2. +How the relative crystal pose is affecting the pose of the output beam is studied in Section 2.

In order to increase the accuracy of the metrology system, two problems are to be dealt with:

-
-

1. Metrology Concept

+
+

1. Metrology Concept

- +

The goal of the metrology system is to measure the distance and default of parallelism between the first and second crystals. @@ -127,8 +128,8 @@ Only 3 degrees of freedom are of interest:

  • \(r_x\)
    • -
    -

    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. @@ -138,11 +139,11 @@ Three additional interferometers are used to measured the relative motion of the

    -In total, there are 15 interferometers represented in Figure 1. -The measurements are summarized in Table 2. +In total, there are 15 interferometers represented in Figure 1. +The measurements are summarized in Table 2.

    - +
    @@ -194,7 +195,7 @@ The measurements are summarized in Table 2.
    Table 1: Notations for the metrology frame
    - +
    @@ -305,7 +306,7 @@ The measurements are summarized in Table 2.
    Table 2: List of Interferometer measurements
    -
    +

    metrology_schematic.png

    Figure 1: Schematic of the Metrology System

    @@ -313,8 +314,8 @@ The measurements are summarized in Table 2.
    -
    -

    1.2. Computation of the relative pose between first and second crystals

    +
    +

    1.2. Computation of the relative pose between first and second crystals

    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). @@ -324,7 +325,7 @@ To understand how the relative pose between the crystals is computed from the in Basically, Jacobian matrices are derived from the geometry and are used to convert the 15 interferometer signals to the relative pose of the primary and secondary crystals \([d_{h,z},\ r_{h,y},\ r_{h,x}]\) or \([d_{r,z},\ r_{r,y},\ r_{r,x}]\).

    -
    +

    The sign conventions for the relative crystal pose are:

    @@ -379,11 +380,11 @@ Values of the matrices can be found in the document describing the kinematics of
    -
    -

    2. Relation Between Crystal position and X-ray measured displacement

    +
    +

    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. @@ -406,8 +407,8 @@ In order to simplify the problem, the first crystal is supposed to be fixed (i.e In order to easily study that, “ray tracing” techniques are used.

    -
    -

    2.1. Definition of frame

    +
    +

    2.1. Definition of frame

    @@ -483,15 +484,15 @@ The xy position of the beam is taken in the \(x=0\) plane. -
    -

    2.2. Effect of an error in crystal’s distance

    +
    +

    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. +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.

    @@ -499,14 +500,14 @@ Visually, it is clear that this induce a z offset of the output bea

    -
    +

    ray_tracing_error_dz_overview.png

    Figure 2: Visual Effect of an error in dz (1mm). Side view.

    -The motion of the output beam is displayed as a function of the Bragg angle in Figure 3. +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.

    @@ -519,7 +520,7 @@ This is indeed equal to: \end{equation} -
    +

    motion_beam_dz_error.png

    Figure 3: Motion of the output beam with dZ error

    @@ -527,29 +528,29 @@ This is indeed equal to:
    -
    -

    2.3. Effect of an error in crystal’s x parallelism

    +
    +

    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 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.

    -
    +

    ray_tracing_error_drx_overview.png

    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. +The effect of drx as a function of the Bragg angle on the output beam pose is computed and shown in Figure 5.

    @@ -568,7 +569,7 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree

    -
    +

    motion_beam_drx_error.png

    Figure 5: Motion of the output beam with drx error

    @@ -576,26 +577,26 @@ We can note that the \(y\) shift is equal to zero for a bragg angle of 45 degree
    -
    -

    2.4. Effect of an error in crystal’s y parallelism

    +
    +

    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). +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).

    -
    +

    ray_tracing_error_dry_overview.png

    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. +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:

    \begin{equation} @@ -607,7 +608,7 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh

    -
    +

    motion_beam_dry_error.png

    Figure 7: Motion of the output beam with dry error

    @@ -615,16 +616,16 @@ It also induces a small vertical motion of the beam (at the \(x=0\) location) wh
    -
    -

    2.5. Summary

    +
    +

    2.5. Summary

    -Effects of crystal’s pose errors on the output beam are summarized in Table 3. +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.

    -
    +
    @@ -677,30 +678,30 @@ Also note that the effect of an error in crystal’s distance does not depen -
    -

    2.6. “Channel cut” Scan

    +
    +

    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. +This is visually shown in Figure 8 where it is clear that the output beam experiences some vertical motion.

    -
    +

    ray_tracing_channel_cut.png

    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. +The \(z\) offset of the beam for several channel cut scans are shown in Figure 9.

    -
    +

    channel_cut_scan.png

    Figure 9: Z motion of the beam during “channel cut” scans

    @@ -709,8 +710,8 @@ The \(z\) offset of the beam for several channel cut scans are shown in Figure <
    -
    -

    3. Determining relative pose between the crystals using the X-ray

    +
    +

    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. @@ -729,21 +730,21 @@ In order to do that, an external metrology using the x-ray is used.

    -
    -

    3.1. Determine the \(y\) parallelism - “Rocking Curve”

    +
    +

    3.1. Determine the \(y\) parallelism - “Rocking Curve”

    -
    -

    3.2. Determine the \(x\) parallelism - Bragg Scan

    +
    +

    3.2. Determine the \(x\) parallelism - Bragg Scan

    -
    -

    3.3. Determine the \(z\) distance - Bragg Scan

    +
    +

    3.3. Determine the \(z\) distance - Bragg Scan

    -
    -

    3.4. Use Channel cut scan to determine crystal dry parallelism

    +
    +

    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. @@ -763,16 +764,16 @@ The error is

    -
    -

    3.5. Effect of an error on Bragg angle

    +
    +

    3.5. Effect of an error on Bragg angle

    -
    -

    4. Deformations of the Metrology Frame

    +
    +

    4. Deformations of the Metrology Frame

    - +

    The transformation matrices are valid only if the metrology frames are solid bodies. @@ -787,8 +788,8 @@ When the bragg axis is scanned, the effect of gravity on the metrology frame is This can be calibrated.

    -
    -

    4.1. Measurement Setup

    +
    +

    4.1. Measurement Setup

    Two beam viewers: @@ -805,7 +806,7 @@ This position is the wanted position for a given Bragg angle.

    -
    +

    calibration_setup.png

    Figure 10: Schematic of the setup

    @@ -835,8 +836,8 @@ Frame rate is: 42 fps
    -
    -

    4.2. 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. @@ -845,12 +846,12 @@ This may be done using FE software.

    -
    -

    4.3. Comparison

    +
    +

    4.3. Comparison

    -
    -

    4.4. Test

    +
    +

    4.4. Test

    aa = importdata("correctInterf-vlm-220201.dat");
@@ -865,8 +866,8 @@ This may be done using FE software.
    -
    -

    4.5. Measured frame deformation

    +
    +

    4.5. Measured frame deformation

    data = table2array(readtable('itf_polynom.csv','NumHeaderLines',1));
@@ -894,7 +895,7 @@ ry1 = 1e-9*dat
    -
    +

    calibration_drx_pres.png

    Figure 11: description

    @@ -967,8 +968,8 @@ f_ry1 = fit(180/p
    -
    -

    4.6. 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";
@@ -1000,7 +1001,7 @@ data.xtal2_111_d = double(h5read(filename, '/7.1/instru
    -
    +

    drifts_xtal2_detrend.png

    Figure 12: Drifts of the second crystal as a function of Bragg Angle

    @@ -1017,8 +1018,8 @@ data.xtal2_111_d = double(h5read(filename, '/7.1/instru
    -
    -

    4.7. 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";
@@ -1068,15 +1069,15 @@ data_2.dz    = 1e-9*<
    -
    -

    5. Attocube - Periodic Non-Linearity

    +
    +

    5. Attocube - Periodic Non-Linearity

    - +

    Interferometers have some periodic nonlinearity (NO_ITEM_DATA:thurner15_fiber_based_distan_sensin_inter). -The period is a fraction of the wavelength (usually \(\lambda/2\)) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams (See NO_ITEM_DATA:ducourtieux18_towar_high_precis_posit_contr, NO_ITEM_DATA:thurner15_fiber_based_distan_sensin_inter). +The period is a fraction of the wavelength (usually \(\lambda/2\)) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams (Ducourtieux 2018; Thurner et al. 2015). The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers.

    @@ -1093,8 +1094,8 @@ This process is performed over several periods in order to characterize the erro

    -
    -

    5.1. 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. @@ -1105,7 +1106,7 @@ It is here supposed that the measured displacement by the strain gauges are conv It is also supposed that we are at a certain Bragg angle, and that the stepper motors are not moving: only the piezoelectric actuators are used.

    -
    +

    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.

    @@ -1121,7 +1122,7 @@ As any deformations of the metrology frame of deformation of the crystal’s

    -The setup is schematically with the block diagram in Figure 13. +The setup is schematically with the block diagram in Figure 13.

    @@ -1137,7 +1138,7 @@ The PI controller takes care or controlling to position as measured by the strai -

    +

    block_diagram_lut_attocube.png

    Figure 13: Block Diagram schematic of the setup used to measure the periodic non-linearity of the Attocube

    @@ -1149,8 +1150,8 @@ The problem is to estimate the periodic non-linearity of the Attocube from the i
    -
    -

    5.2. Choice of the reference signal

    +
    +

    5.2. Choice of the reference signal

    The main specifications for the reference signal are; @@ -1174,8 +1175,8 @@ Based on the above discussion, one suitable excitation signal is a sinusoidal sw

    -
    -

    5.3. 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. @@ -1195,8 +1196,8 @@ One way to precisely estimate the laser wavelength is to estimate the non linear

    -
    -

    5.4. Simulation

    +
    +

    5.4. Simulation

    Suppose we have a first approximation of the non-linear period. @@ -1216,10 +1217,10 @@ period_nl = period_est + period_err;

    -The non-linear errors are first estimated at the beginning of the stroke (Figure 14). +The non-linear errors are first estimated at the beginning of the stroke (Figure 14).

    -
    +

    non_linear_errors_start_stroke.png

    Figure 14: Estimation of the non-linear errors at the beginning of the stroke

    @@ -1227,7 +1228,7 @@ The non-linear errors are first estimated at the beginning of the stroke (Figure

    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 15). +To do so, the same measurement is performed with a stroke of several millimeters (Figure 15).

    @@ -1236,7 +1237,7 @@ This is due to a mismatch between the estimated period and the true period of th

    -
    +

    non_linear_errors_middle_stroke.png

    Figure 15: Estimated non-linear errors at a latter position

    @@ -1269,7 +1270,7 @@ with \(\lambda_{\text{est}}\) the estimated error’s period.

    -From Figure 15, 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} @@ -1349,8 +1350,8 @@ The maximum stroke is 2.9 [mm]
    -
    -

    5.5. Measurements

    +
    +

    5.5. Measurements

    We have some constrains on the way the motion is imposed and measured: @@ -1376,10 +1377,20 @@ Suppose we have the power spectral density (PSD) of both \(n_a\) and \(n_g\).

    + +
    +

    Bibliography

    +
    +
    +
    Ducourtieux, Sebastien. 2018. “Toward High Precision Position Control Using Laser Interferometry: Main Sources of Error.” doi:10.13140/rg.2.2.21044.35205.
    +
    Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” Applied Optics 54 (10). Optical Society of America: 3051–63.
    +
    +
    +

    Author: Dehaeze Thomas

    -

    Created: 2022-06-02 Thu 22:25

    +

    Created: 2022-06-07 Tue 10:58

    diff --git a/dcm-metrology.org b/dcm-metrology.org index c9c74e7..49b719a 100644 --- a/dcm-metrology.org +++ b/dcm-metrology.org @@ -16,7 +16,7 @@ #+LaTeX_CLASS: scrreprt #+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full] #+LaTeX_HEADER_EXTRA: \input{preamble.tex} -#+LATEX_HEADER_EXTRA: \bibliography{ref} +#+LATEX_HEADER_EXTRA: \addbibresource{dcm-metrology.bib} #+PROPERTY: header-args:matlab :session *MATLAB* #+PROPERTY: header-args:matlab+ :comments org @@ -1069,7 +1069,7 @@ plot(data_2.bragg, data_2.mframe_u) ** 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 period is a fraction of the wavelength (usually $\lambda/2$) and can be due to polarization mixing, non perfect alignment of the optical components and unwanted reflected beams cite:ducourtieux18_towar_high_precis_posit_contr,thurner15_fiber_based_distan_sensin_inter. The amplitude of the nonlinearity can vary from a fraction of a nanometer to tens of nanometers. In the DCM case, when using Attocube interferometers, the period non-linearity are in the order of several nanometers with a period of $765\,nm$. @@ -1584,3 +1584,10 @@ xtal2_rectangle = [results.ps + 0.02/2*y + 0.07/2*x; #+begin_src matlab patch(100*xtal2_rectangle(:,1), 100*xtal2_rectangle(:,2), 100*xtal2_rectangle(:,3), 'k-') #+end_src + +* Bibliography +:PROPERTIES: +:UNNUMBERED: t +:END: +#+latex: \printbibliography[heading=none] +[[bibliography:dcm-metrology.bib]] diff --git a/dcm-metrology.pdf b/dcm-metrology.pdf index b8e415b..18d647b 100644 Binary files a/dcm-metrology.pdf and b/dcm-metrology.pdf differ
    Table 3: Summary of the effects of the errors in second crystal’s pose on the output beam