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< h1 class = "title" > Attocube - Test Bench< / h1 >
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< h2 > Table of Contents< / h2 >
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< li > < a href = "#orgfaed8bc" > 1. Estimation of the Spectral Density of the Attocube Noise< / a >
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< li > < a href = "#org42d8bf8" > 1.1. Long and Slow measurement< / a > < / li >
< li > < a href = "#orgc5a3450" > 1.2. Short and Fast measurement< / a > < / li >
< li > < a href = "#orgc18a5a3" > 1.3. Obtained Amplitude Spectral Density of the measured displacement< / a > < / li >
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< h2 id = "orgfaed8bc" > < span class = "section-number-2" > 1< / span > Estimation of the Spectral Density of the Attocube Noise< / h2 >
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< h3 id = "org42d8bf8" > < span class = "section-number-3" > 1.1< / span > Long and Slow measurement< / h3 >
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< p >
The first measurement was made during ~17 hours with a sampling time of \(T_s = 0.1\,s\).
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< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/long_test2.mat'< / span > , < span class = "org-string" > 'x'< / span > , < span class = "org-string" > 't'< / span > )
Ts = 0.1; < span class = "org-comment" > % [s]< / span >
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< div id = "orgd7563fe" class = "figure" >
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< p > < img src = "figs/long_meas_time_domain_full.png" alt = "long_meas_time_domain_full.png" / >
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< p > < span class = "figure-number" > Figure 1: < / span > Long measurement time domain data< / p >
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< p >
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Let’ s fit the data with a step response to a first order low pass filter (Figure < a href = "#org5d8ea16" > 2< / a > ).
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< pre class = "src src-matlab" > f = @(b,x) b(1)< span class = "org-type" > *< / span > (1 < span class = "org-type" > -< / span > exp(< span class = "org-type" > -< / span > x< span class = "org-type" > /< / span > b(2)));
y_cur = x(t < span class = "org-type" > < < / span > 17< span class = "org-type" > *< / span > 60< span class = "org-type" > *< / span > 60);
t_cur = t(t < span class = "org-type" > < < / span > 17< span class = "org-type" > *< / span > 60< span class = "org-type" > *< / span > 60);
nrmrsd = @(b) norm(y_cur < span class = "org-type" > -< / span > f(b,t_cur)); < span class = "org-comment" > % Residual Norm Cost Function< / span >
B0 = [400e< span class = "org-type" > -< / span > 9, 2< span class = "org-type" > *< / span > 60< span class = "org-type" > *< / span > 60]; < span class = "org-comment" > % Choose Appropriate Initial Estimates< / span >
[B,rnrm] = fminsearch(nrmrsd, B0); < span class = "org-comment" > % Estimate Parameters ‘ B’ < / span >
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The corresponding time constant is (in [h]):
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2.0576
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< p > < img src = "figs/long_meas_time_domain_fit.png" alt = "long_meas_time_domain_fit.png" / >
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< p > < span class = "figure-number" > Figure 2: < / span > Fit of the measurement data with a step response of a first order low pass filter< / p >
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< p >
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We can see in Figure < a href = "#orgd7563fe" > 1< / a > that there is a transient period where the measured displacement experiences some drifts.
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This is probably due to thermal effects.
We only select the data between < code > t1< / code > and < code > t2< / code > .
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The obtained displacement is shown in Figure < a href = "#org9806b5d" > 3< / a > .
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< pre class = "src src-matlab" > t1 = 11; t2 = 17; < span class = "org-comment" > % [h]< / span >
x = x(t < span class = "org-type" > > < / span > t1< span class = "org-type" > *< / span > 60< span class = "org-type" > *< / span > 60 < span class = "org-type" > & < / span > t < span class = "org-type" > < < / span > t2< span class = "org-type" > *< / span > 60< span class = "org-type" > *< / span > 60);
x = x < span class = "org-type" > -< / span > mean(x);
t = t(t < span class = "org-type" > > < / span > t1< span class = "org-type" > *< / span > 60< span class = "org-type" > *< / span > 60 < span class = "org-type" > & < / span > t < span class = "org-type" > < < / span > t2< span class = "org-type" > *< / span > 60< span class = "org-type" > *< / span > 60);
t = t < span class = "org-type" > -< / span > t(1);
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< div id = "org9806b5d" class = "figure" >
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< p > < img src = "figs/long_meas_time_domain_zoom.png" alt = "long_meas_time_domain_zoom.png" / >
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< p > < span class = "figure-number" > Figure 3: < / span > Kept data (removed slow drifts during the first hours)< / p >
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The Power Spectral Density of the measured displacement is computed
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< pre class = "src src-matlab" > win = hann(ceil(length(x)< span class = "org-type" > /< / span > 20));
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[p_1, f_1] = pwelch(x, win, [], [], 1< span class = "org-type" > /< / span > Ts);
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< h3 id = "orgc5a3450" > < span class = "section-number-3" > 1.2< / span > Short and Fast measurement< / h3 >
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< 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.
The measurement is done with a sampling time of \(T_s = 0.1\,ms\) and a duration of ~100s.
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< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/test.mat'< / span > , < span class = "org-string" > 'x'< / span > , < span class = "org-string" > 't'< / span > )
Ts = 1e< span class = "org-type" > -< / span > 4; < span class = "org-comment" > % [s]< / span >
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The time domain measurement is shown in Figure < a href = "#org658d16a" > 4< / a > .
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< div id = "org658d16a" class = "figure" >
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< p > < img src = "figs/short_meas_time_domain.png" alt = "short_meas_time_domain.png" / >
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< p > < span class = "figure-number" > Figure 4: < / span > Time domain measurement with the high sampling rate< / p >
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< p >
The Power Spectral Density of the measured displacement is computed
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< pre class = "src src-matlab" > win = hann(ceil(length(x)< span class = "org-type" > /< / span > 20));
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[p_2, f_2] = pwelch(x, win, [], [], 1< span class = "org-type" > /< / span > Ts);
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< h3 id = "orgc18a5a3" > < span class = "section-number-3" > 1.3< / span > Obtained Amplitude Spectral Density of the measured displacement< / h3 >
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< p >
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The computed ASD of the two measurements are combined in Figure < a href = "#orgfdb6fa9" > 5< / a > .
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< div id = "orgfdb6fa9" class = "figure" >
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< p > < img src = "figs/psd_combined.png" alt = "psd_combined.png" / >
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< p > < span class = "figure-number" > Figure 5: < / span > Obtained Amplitude Spectral Density of the measured displacement< / p >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-10-29 jeu. 11:07< / p >
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