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@@ -34,27 +30,27 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgd8ce409">1. Estimation of the Spectral Density of the Attocube Noise</a>
<li><a href="#org91ed2af">1. Estimation of the Spectral Density of the Attocube Noise</a>
<ul>
<li><a href="#org78cd963">1.1. Long and Slow measurement</a></li>
<li><a href="#org46eb0d4">1.2. Short and Fast measurement</a></li>
<li><a href="#org3de4174">1.3. Obtained Amplitude Spectral Density of the measured displacement</a></li>
<li><a href="#org7ed0442">1.1. Long and Slow measurement</a></li>
<li><a href="#org84a3a7f">1.2. Short and Fast measurement</a></li>
<li><a href="#org8263623">1.3. Obtained Amplitude Spectral Density of the measured displacement</a></li>
</ul>
</li>
<li><a href="#orga823efb">2. Effect of the &ldquo;bubble sheet&rdquo; and &ldquo;Aluminium tube&rdquo;</a>
<li><a href="#orgfbcdb24">2. Effect of the &ldquo;bubble sheet&rdquo; and &ldquo;Aluminium tube&rdquo;</a>
<ul>
<li><a href="#orga7204eb">2.1. Aluminium Tube and Bubble Sheet</a></li>
<li><a href="#org15d9822">2.2. Only Aluminium Tube</a></li>
<li><a href="#org2f5206b">2.3. Nothing</a></li>
<li><a href="#orge439118">2.4. Comparison</a></li>
<li><a href="#org2c0e0f6">2.1. Aluminium Tube and Bubble Sheet</a></li>
<li><a href="#orgab2ff04">2.2. Only Aluminium Tube</a></li>
<li><a href="#org5e0cbb8">2.3. Nothing</a></li>
<li><a href="#org0161dc2">2.4. Comparison</a></li>
</ul>
</li>
<li><a href="#org025397b">3. Measurement of the Attocube&rsquo;s non-linearity</a>
<li><a href="#org45e7337">3. Measurement of the Attocube&rsquo;s non-linearity</a>
<ul>
<li><a href="#org01e947d">3.1. Load Data</a></li>
<li><a href="#org32a4e2c">3.2. Time Domain Results</a></li>
<li><a href="#org5be7e09">3.3. Difference between Encoder and Interferometer as a function of time</a></li>
<li><a href="#org75b2a6d">3.4. Difference between Encoder and Interferometer as a function of position</a></li>
<li><a href="#orgf2cd3af">3.1. Load Data</a></li>
<li><a href="#org9caccde">3.2. Time Domain Results</a></li>
<li><a href="#orgb15ec0f">3.3. Difference between Encoder and Interferometer as a function of time</a></li>
<li><a href="#orga0c30bc">3.4. Difference between Encoder and Interferometer as a function of position</a></li>
</ul>
</li>
</ul>
@@ -65,33 +61,33 @@
In this document, few caracteristics of the Attocube Displacement Measuring Interferometer IDS3010 (<a href="https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer">link</a>) are studied:
</p>
<ul class="org-ul">
<li>Section <a href="#org9419f0e">1</a>: the ASD noise of the measured displacement is estimated</li>
<li>Section <a href="#org7cc1f65">2</a>: the effect of two air protections on the stability of the measurement is studied</li>
<li>Section <a href="#org382d51d">3</a>: the cyclic non-linearity of the attocube is estimated using a encoder</li>
<li>Section <a href="#org45bc653">1</a>: the ASD noise of the measured displacement is estimated</li>
<li>Section <a href="#orgf93a547">2</a>: the effect of two air protections on the stability of the measurement is studied</li>
<li>Section <a href="#org421bcba">3</a>: the cyclic non-linearity of the attocube is estimated using a encoder</li>
</ul>
<div id="outline-container-orgd8ce409" class="outline-2">
<h2 id="orgd8ce409"><span class="section-number-2">1</span> Estimation of the Spectral Density of the Attocube Noise</h2>
<div id="outline-container-org91ed2af" class="outline-2">
<h2 id="org91ed2af"><span class="section-number-2">1</span> Estimation of the Spectral Density of the Attocube Noise</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org9419f0e"></a>
<a id="org45bc653"></a>
</p>
<div id="orge10425f" class="figure">
<div id="orgeea45a0" class="figure">
<p><img src="figs/test-bench-schematic.png" alt="test-bench-schematic.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Test Bench Schematic</p>
</div>
<div id="orgac54f33" class="figure">
<div id="org1945c4e" class="figure">
<p><img src="figs/IMG-7865.JPG" alt="IMG-7865.JPG" />
</p>
<p><span class="figure-number">Figure 2: </span>Picture of the test bench. The Attocube and mirror are covered by a &ldquo;bubble sheet&rdquo;</p>
</div>
</div>
<div id="outline-container-org78cd963" class="outline-3">
<h3 id="org78cd963"><span class="section-number-3">1.1</span> Long and Slow measurement</h3>
<div id="outline-container-org7ed0442" class="outline-3">
<h3 id="org7ed0442"><span class="section-number-3">1.1</span> Long and Slow measurement</h3>
<div class="outline-text-3" id="text-1-1">
<p>
The first measurement was made during ~17 hours with a sampling time of \(T_s = 0.1\,s\).
@@ -104,14 +100,14 @@ Ts = 0.1; <span class="org-comment">% [s]</span>
</div>
<div id="org3a063a4" class="figure">
<div id="orgabef257" class="figure">
<p><img src="figs/long_meas_time_domain_full.png" alt="long_meas_time_domain_full.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Long measurement time domain data</p>
</div>
<p>
Let&rsquo;s fit the data with a step response to a first order low pass filter (Figure <a href="#org3619821">4</a>).
Let&rsquo;s fit the data with a step response to a first order low pass filter (Figure <a href="#org3effe27">4</a>).
</p>
<div class="org-src-container">
@@ -135,17 +131,17 @@ The corresponding time constant is (in [h]):
<div id="org3619821" class="figure">
<div id="org3effe27" class="figure">
<p><img src="figs/long_meas_time_domain_fit.png" alt="long_meas_time_domain_fit.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Fit of the measurement data with a step response of a first order low pass filter</p>
</div>
<p>
We can see in Figure <a href="#org3a063a4">3</a> that there is a transient period where the measured displacement experiences some drifts.
We can see in Figure <a href="#orgabef257">3</a> that there is a transient period where the measured displacement experiences some drifts.
This is probably due to thermal effects.
We only select the data between <code>t1</code> and <code>t2</code>.
The obtained displacement is shown in Figure <a href="#org0516899">5</a>.
The obtained displacement is shown in Figure <a href="#orgdb1e675">5</a>.
</p>
<div class="org-src-container">
@@ -159,7 +155,7 @@ t = t <span class="org-type">-</span> t(1);
</div>
<div id="org0516899" class="figure">
<div id="orgdb1e675" class="figure">
<p><img src="figs/long_meas_time_domain_zoom.png" alt="long_meas_time_domain_zoom.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Kept data (removed slow drifts during the first hours)</p>
@@ -194,8 +190,8 @@ f_1 = f_1(f_1 <span class="org-type">&lt;</span> 2);
</div>
</div>
<div id="outline-container-org46eb0d4" class="outline-3">
<h3 id="org46eb0d4"><span class="section-number-3">1.2</span> Short and Fast measurement</h3>
<div id="outline-container-org84a3a7f" class="outline-3">
<h3 id="org84a3a7f"><span class="section-number-3">1.2</span> Short and Fast measurement</h3>
<div class="outline-text-3" id="text-1-2">
<p>
An second measurement is done in order to estimate the high frequency noise of the interferometer.
@@ -214,11 +210,11 @@ Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
</div>
<p>
The time domain measurement is shown in Figure <a href="#orgf3ce8f9">6</a>.
The time domain measurement is shown in Figure <a href="#orgb156e4e">6</a>.
</p>
<div id="orgf3ce8f9" class="figure">
<div id="orgb156e4e" class="figure">
<p><img src="figs/short_meas_time_domain.png" alt="short_meas_time_domain.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Time domain measurement with the high sampling rate</p>
@@ -235,15 +231,15 @@ The Power Spectral Density of the measured displacement is computed
</div>
</div>
<div id="outline-container-org3de4174" class="outline-3">
<h3 id="org3de4174"><span class="section-number-3">1.3</span> Obtained Amplitude Spectral Density of the measured displacement</h3>
<div id="outline-container-org8263623" class="outline-3">
<h3 id="org8263623"><span class="section-number-3">1.3</span> Obtained Amplitude Spectral Density of the measured displacement</h3>
<div class="outline-text-3" id="text-1-3">
<p>
The computed ASD of the two measurements are combined in Figure <a href="#orgcff33dc">7</a>.
The computed ASD of the two measurements are combined in Figure <a href="#org72c2457">7</a>.
</p>
<div id="orgcff33dc" class="figure">
<div id="org72c2457" class="figure">
<p><img src="figs/psd_combined.png" alt="psd_combined.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Obtained Amplitude Spectral Density of the measured displacement</p>
@@ -252,21 +248,21 @@ The computed ASD of the two measurements are combined in Figure <a href="#orgcff
</div>
</div>
<div id="outline-container-orga823efb" class="outline-2">
<h2 id="orga823efb"><span class="section-number-2">2</span> Effect of the &ldquo;bubble sheet&rdquo; and &ldquo;Aluminium tube&rdquo;</h2>
<div id="outline-container-orgfbcdb24" class="outline-2">
<h2 id="orgfbcdb24"><span class="section-number-2">2</span> Effect of the &ldquo;bubble sheet&rdquo; and &ldquo;Aluminium tube&rdquo;</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org7cc1f65"></a>
<a id="orgf93a547"></a>
</p>
<div id="org54f25e9" class="figure">
<div id="org5a9ad77" class="figure">
<p><img src="figs/IMG-7864.JPG" alt="IMG-7864.JPG" />
</p>
<p><span class="figure-number">Figure 8: </span>Aluminium tube used to protect the beam path from disturbances</p>
</div>
</div>
<div id="outline-container-orga7204eb" class="outline-3">
<h3 id="orga7204eb"><span class="section-number-3">2.1</span> Aluminium Tube and Bubble Sheet</h3>
<div id="outline-container-org2c0e0f6" class="outline-3">
<h3 id="org2c0e0f6"><span class="section-number-3">2.1</span> Aluminium Tube and Bubble Sheet</h3>
<div class="outline-text-3" id="text-2-1">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'short_test_plastic.mat'</span>);
@@ -287,8 +283,8 @@ Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
</div>
</div>
<div id="outline-container-org15d9822" class="outline-3">
<h3 id="org15d9822"><span class="section-number-3">2.2</span> Only Aluminium Tube</h3>
<div id="outline-container-orgab2ff04" class="outline-3">
<h3 id="orgab2ff04"><span class="section-number-3">2.2</span> Only Aluminium Tube</h3>
<div class="outline-text-3" id="text-2-2">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'short_test_alu_tube.mat'</span>);
@@ -302,7 +298,7 @@ Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
</div>
<p>
The time domain measurement is shown in Figure <a href="#orgf3ce8f9">6</a>.
The time domain measurement is shown in Figure <a href="#orgb156e4e">6</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">win = hann(ceil(length(x)<span class="org-type">/</span>10));
@@ -312,8 +308,8 @@ The time domain measurement is shown in Figure <a href="#orgf3ce8f9">6</a>.
</div>
</div>
<div id="outline-container-org2f5206b" class="outline-3">
<h3 id="org2f5206b"><span class="section-number-3">2.3</span> Nothing</h3>
<div id="outline-container-org5e0cbb8" class="outline-3">
<h3 id="org5e0cbb8"><span class="section-number-3">2.3</span> Nothing</h3>
<div class="outline-text-3" id="text-2-3">
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'short_test_without_material.mat'</span>);
@@ -327,7 +323,7 @@ Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
</div>
<p>
The time domain measurement is shown in Figure <a href="#orgf3ce8f9">6</a>.
The time domain measurement is shown in Figure <a href="#orgb156e4e">6</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">win = hann(ceil(length(x)<span class="org-type">/</span>10));
@@ -337,11 +333,11 @@ The time domain measurement is shown in Figure <a href="#orgf3ce8f9">6</a>.
</div>
</div>
<div id="outline-container-orge439118" class="outline-3">
<h3 id="orge439118"><span class="section-number-3">2.4</span> Comparison</h3>
<div id="outline-container-org0161dc2" class="outline-3">
<h3 id="org0161dc2"><span class="section-number-3">2.4</span> Comparison</h3>
<div class="outline-text-3" id="text-2-4">
<div id="org3d446e2" class="figure">
<div id="orgcaa0ad4" class="figure">
<p><img src="figs/asd_noise_comp_bubble_aluminium.png" alt="asd_noise_comp_bubble_aluminium.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Comparison of the noise ASD with and without bubble sheet</p>
@@ -350,17 +346,17 @@ The time domain measurement is shown in Figure <a href="#orgf3ce8f9">6</a>.
</div>
</div>
<div id="outline-container-org025397b" class="outline-2">
<h2 id="org025397b"><span class="section-number-2">3</span> Measurement of the Attocube&rsquo;s non-linearity</h2>
<div id="outline-container-org45e7337" class="outline-2">
<h2 id="org45e7337"><span class="section-number-2">3</span> Measurement of the Attocube&rsquo;s non-linearity</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org382d51d"></a>
<a id="org421bcba"></a>
</p>
<p>
The measurement setup is shown in Figure <a href="#orgce1d646">10</a>.
The measurement setup is shown in Figure <a href="#org1b67344">10</a>.
</p>
<div class="note" id="orgc9bc822">
<div class="note" id="org1f1e587">
<p>
Here are the equipment used in the test bench:
</p>
@@ -375,7 +371,7 @@ Here are the equipment used in the test bench:
</div>
<div id="orgce1d646" class="figure">
<div id="org1b67344" class="figure">
<p><img src="figs/exp_setup_schematic.png" alt="exp_setup_schematic.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Schematic of the Experiment</p>
@@ -390,8 +386,8 @@ The encoder and the attocube are measure ring the same motion.
As will be shown shortly, this measurement permitted to measure the period non-linearity of the Attocube.
</p>
</div>
<div id="outline-container-org01e947d" class="outline-3">
<h3 id="org01e947d"><span class="section-number-3">3.1</span> Load Data</h3>
<div id="outline-container-orgf2cd3af" class="outline-3">
<h3 id="orgf2cd3af"><span class="section-number-3">3.1</span> Load Data</h3>
<div class="outline-text-3" id="text-3-1">
<p>
The measurement data are loaded and the offset are removed using the <code>detrend</code> command.
@@ -412,11 +408,11 @@ u = detrend(u, 0);
</div>
</div>
<div id="outline-container-org32a4e2c" class="outline-3">
<h3 id="org32a4e2c"><span class="section-number-3">3.2</span> Time Domain Results</h3>
<div id="outline-container-org9caccde" class="outline-3">
<h3 id="org9caccde"><span class="section-number-3">3.2</span> Time Domain Results</h3>
<div class="outline-text-3" id="text-3-2">
<p>
One period of the displacement of the mass as measured by the encoder and interferometer are shown in Figure <a href="#org0f33cd3">11</a>.
One period of the displacement of the mass as measured by the encoder and interferometer are shown in Figure <a href="#orgadbe636">11</a>.
It consist of the sinusoidal motion at 0.5Hz with an amplitude of approximately \(70\mu m\).
</p>
@@ -426,18 +422,18 @@ This should improve the coherence between the measurements made by the encoder a
</p>
<div id="org0f33cd3" class="figure">
<div id="orgadbe636" class="figure">
<p><img src="figs/int_enc_one_cycle.png" alt="int_enc_one_cycle.png" />
</p>
<p><span class="figure-number">Figure 11: </span>One cycle measurement</p>
</div>
<p>
The difference between the two measurements during the same period is shown in Figure <a href="#org5c61cd3">12</a>.
The difference between the two measurements during the same period is shown in Figure <a href="#org448b33f">12</a>.
</p>
<div id="org5c61cd3" class="figure">
<div id="org448b33f" class="figure">
<p><img src="figs/int_enc_one_cycle_error.png" alt="int_enc_one_cycle_error.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Difference between the Encoder and the interferometer during one cycle</p>
@@ -445,8 +441,8 @@ The difference between the two measurements during the same period is shown in F
</div>
</div>
<div id="outline-container-org5be7e09" class="outline-3">
<h3 id="org5be7e09"><span class="section-number-3">3.3</span> Difference between Encoder and Interferometer as a function of time</h3>
<div id="outline-container-orgb15ec0f" class="outline-3">
<h3 id="orgb15ec0f"><span class="section-number-3">3.3</span> Difference between Encoder and Interferometer as a function of time</h3>
<div class="outline-text-3" id="text-3-3">
<p>
The data is filtered using a second order low pass filter with a cut-off frequency \(\omega_0\) as defined below.
@@ -461,7 +457,7 @@ G_lpf = 1<span class="org-type">/</span>(1 <span class="org-type">+</span> 2<spa
</div>
<p>
After filtering, the data is &ldquo;re-shaped&rdquo; such that we can superimpose all the measured periods as shown in Figure <a href="#orgc98f587">13</a>.
After filtering, the data is &ldquo;re-shaped&rdquo; such that we can superimpose all the measured periods as shown in Figure <a href="#org99397ec">13</a>.
This gives an idea of the measurement error as given by the Attocube during a \(70 \mu m\) motion.
</p>
<div class="org-src-container">
@@ -471,7 +467,7 @@ d_err_mean = d_err_mean <span class="org-type">-</span> mean(d_err_mean);
</div>
<div id="orgc98f587" class="figure">
<div id="org99397ec" class="figure">
<p><img src="figs/int_enc_error_mean_time.png" alt="int_enc_error_mean_time.png" />
</p>
<p><span class="figure-number">Figure 13: </span>Difference between the two measurement in the time domain, averaged for all the cycles</p>
@@ -479,17 +475,17 @@ d_err_mean = d_err_mean <span class="org-type">-</span> mean(d_err_mean);
</div>
</div>
<div id="outline-container-org75b2a6d" class="outline-3">
<h3 id="org75b2a6d"><span class="section-number-3">3.4</span> Difference between Encoder and Interferometer as a function of position</h3>
<div id="outline-container-orga0c30bc" class="outline-3">
<h3 id="orga0c30bc"><span class="section-number-3">3.4</span> Difference between Encoder and Interferometer as a function of position</h3>
<div class="outline-text-3" id="text-3-4">
<p>
Figure <a href="#orgc98f587">13</a> gives the measurement error as a function of time.
Figure <a href="#org99397ec">13</a> gives the measurement error as a function of time.
We here wish the compute this measurement error as a function of the position (as measured by the encoer).
</p>
<p>
To do so, all the attocube measurements corresponding to each position measured by the Encoder (resolution of \(1nm\)) are averaged.
Figure <a href="#org1c54e43">14</a> is obtained where we clearly see an error with a period comparable to the motion range and a much smaller period corresponding to the non-linear period errors that we wish the estimate.
Figure <a href="#org0eccff0">14</a> is obtained where we clearly see an error with a period comparable to the motion range and a much smaller period corresponding to the non-linear period errors that we wish the estimate.
</p>
<div class="org-src-container">
<pre class="src src-matlab">[e_sorted, <span class="org-type">~</span>, e_ind] = unique(encoder);
@@ -504,7 +500,7 @@ i_mean_error = (i_mean <span class="org-type">-</span> e_sorted);
</div>
<div id="org1c54e43" class="figure">
<div id="org0eccff0" class="figure">
<p><img src="figs/int_enc_error_mean_position.png" alt="int_enc_error_mean_position.png" />
</p>
<p><span class="figure-number">Figure 14: </span>Difference between the two measurement as a function of the measured position by the encoder, averaged for all the cycles</p>
@@ -526,11 +522,11 @@ e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init<span class="org-typ
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<p>
The obtained periodic non-linearity is shown in Figure <a href="#orgd0a5e3b">15</a>.
The obtained periodic non-linearity is shown in Figure <a href="#orge856c17">15</a>.
</p>
<div id="orgd0a5e3b" class="figure">
<div id="orge856c17" class="figure">
<p><img src="figs/int_non_linearity_period_wavelength.png" alt="int_non_linearity_period_wavelength.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Non-Linearity of the Interferometer over the period of the wavelength</p>
@@ -541,7 +537,7 @@ The obtained periodic non-linearity is shown in Figure <a href="#orgd0a5e3b">15<
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<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-10 mar. 12:50</p>
<p class="date">Created: 2020-11-12 jeu. 10:17</p>
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