#+TITLE: Attocube - Test Bench
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:matlab  :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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:END:

* Estimation of the Spectral Density of the Attocube Noise
** Matlab Init                                             :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

** Long and Slow measurement
The first measurement was made during ~17 hours with a sampling time of $T_s = 0.1\,s$.

#+begin_src matlab
  load('./mat/long_test2.mat', 'x', 't')
  Ts = 0.1; % [s]
#+end_src

#+begin_src matlab :exports none
  figure;
  plot(t/60/60, 1e9*x)
  xlabel('Time [h]'); ylabel('Displacement [nm]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/long_meas_time_domain_full.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:long_meas_time_domain_full
#+caption: Long measurement time domain data
#+RESULTS:
[[file:figs/long_meas_time_domain_full.png]]

Let's fit the data with a step response to a first order low pass filter (Figure [[fig:long_meas_time_domain_fit]]).

#+begin_src matlab
  f = @(b,x) b(1)*(1 - exp(-x/b(2)));

  y_cur = x(t < 17*60*60);
  t_cur = t(t < 17*60*60);

  nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
  B0 = [400e-9, 2*60*60];                        % Choose Appropriate Initial Estimates
  [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’
#+end_src

The corresponding time constant is (in [h]):
#+begin_src matlab :results value replace :exports results
  B(2)/60/60
#+end_src

#+RESULTS:
: 2.0576

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(t_cur/60/60, 1e9*y_cur);
  plot(t_cur/60/60, 1e9*f(B, t_cur));
  hold off;
  xlabel('Time [h]'); ylabel('Displacement [nm]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/long_meas_time_domain_fit.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:long_meas_time_domain_fit
#+caption: Fit of the measurement data with a step response of a first order low pass filter
#+RESULTS:
[[file:figs/long_meas_time_domain_fit.png]]

We can see in Figure [[fig:long_meas_time_domain_full]] 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 =t1= and =t2=.
The obtained displacement is shown in Figure [[fig:long_meas_time_domain_zoom]].

#+begin_src matlab
  t1 = 11; t2 = 17; % [h]

  x = x(t > t1*60*60 & t < t2*60*60);
  x = x - mean(x);
  t = t(t > t1*60*60 & t < t2*60*60);
  t = t - t(1);
#+end_src

#+begin_src matlab :exports none
  figure;
  plot(t/60/60, 1e9*x);
  xlabel('Time [h]'); ylabel('Measured Displacement [nm]')
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/long_meas_time_domain_zoom.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:long_meas_time_domain_zoom
#+caption: Kept data (removed slow drifts during the first hours)
#+RESULTS:
[[file:figs/long_meas_time_domain_zoom.png]]

The Power Spectral Density of the measured displacement is computed
#+begin_src matlab
  win = hann(ceil(length(x)/20));
  [p_1, f_1] = pwelch(x, win, [], [], 1/Ts);
#+end_src

** Short and Fast measurement
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.

#+begin_src matlab
  load('./mat/test.mat', 'x', 't')
  Ts = 1e-4; % [s]
#+end_src

The time domain measurement is shown in Figure [[fig:short_meas_time_domain]].

#+begin_src matlab :exports none
  figure;
  plot(t, 1e9*x)
  xlabel('Time [s]'); ylabel('Displacement [nm]');
  xlim([0, 100]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/short_meas_time_domain.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:short_meas_time_domain
#+caption: Time domain measurement with the high sampling rate
#+RESULTS:
[[file:figs/short_meas_time_domain.png]]

The Power Spectral Density of the measured displacement is computed
#+begin_src matlab
  win = hann(ceil(length(x)/20));
  [p_2, f_2] = pwelch(x, win, [], [], 1/Ts);
#+end_src

** Obtained Amplitude Spectral Density of the measured displacement

The computed ASD of the two measurements are combined in Figure [[fig:psd_combined]].

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f_1(8:end), sqrt(p_1(8:end)), 'k-');
  plot(f_2(8:end), sqrt(p_2(8:end)), 'k-');
  hold off;
  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
  ylabel('ASD [$m/\sqrt{Hz}$]'); xlabel('Frequency [Hz]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/psd_combined.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:psd_combined
#+caption: Obtained Amplitude Spectral Density of the measured displacement
#+RESULTS:
[[file:figs/psd_combined.png]]