Attocube - Test Bench

The first measurement was made during ~17 hours with a sampling time of \(T_s = 0.1\,s\).

load('./mat/long_test_plastic.mat', 'x', 't')
Ts = 0.1; % [s]

Let’s fit the data with a step response to a first order low pass filter (Figure 4).

f = @(b,x) b(1)*(1 - exp(-x/b(2)));

y_cur = x(t < 17.5*60*60);
t_cur = t(t < 17.5*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’

The corresponding time constant is (in [h]):

2.0658

We can see in Figure 3 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 5.

t1 = 10.5; t2 = 17.5; % [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);

The Power Spectral Density of the measured displacement is computed

win = hann(ceil(length(x)/20));
[p_1, f_1] = pwelch(x, win, [], [], 1/Ts);

As a low pass filter was used in the measurement process, we multiply the PSD by the square of the inverse of the filter’s norm.

G_lpf = 1/(1 + s/2/pi);
p_1 = p_1./abs(squeeze(freqresp(G_lpf, f_1, 'Hz'))).^2;

Only frequencies below 2Hz are taken into account (high frequency noise will be measured afterwards).

p_1 = p_1(f_1 < 2);
f_1 = f_1(f_1 < 2);

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.

load('./mat/short_test_plastic.mat')
Ts = 1e-4; % [s]

x = detrend(x, 0);

The time domain measurement is shown in Figure 6.

The Power Spectral Density of the measured displacement is computed

win = hann(ceil(length(x)/20));
[p_2, f_2] = pwelch(x, win, [], [], 1/Ts);

The computed ASD of the two measurements are combined in Figure 7.

load('./mat/short_test_alu_tube.mat');
Ts = 1e-4; % [s]

x = detrend(x, 0);

The time domain measurement is shown in Figure 6.

win = hann(ceil(length(x)/10));
[p_2, f_2] = pwelch(x, win, [], [], 1/Ts);

load('./mat/short_test_without_material.mat');
Ts = 1e-4; % [s]

x = detrend(x, 0);

The time domain measurement is shown in Figure 6.

win = hann(ceil(length(x)/10));
[p_3, f_3] = pwelch(x, win, [], [], 1/Ts);

The measurement data are loaded and the offset are removed using the detrend command.

load('mat/int_enc_comp.mat', 'interferometer', 'encoder', 'u', 't');
Ts = 1e-4; % Sampling Time [s]

interferometer = detrend(interferometer, 0);
encoder = detrend(encoder, 0);
u = detrend(u, 0);

One period of the displacement of the mass as measured by the encoder and interferometer are shown in Figure 11. It consist of the sinusoidal motion at 0.5Hz with an amplitude of approximately \(70\mu m\).

The frequency of the motion is chosen such that no resonance in the system is excited. This should improve the coherence between the measurements made by the encoder and interferometer.

The difference between the two measurements during the same period is shown in Figure 12.

The data is filtered using a second order low pass filter with a cut-off frequency \(\omega_0\) as defined below.

w0 = 2*pi*5; % [rad/s]
xi = 0.7;

G_lpf = 1/(1 + 2*xi/w0*s + s^2/w0^2);

After filtering, the data is “re-shaped” such that we can superimpose all the measured periods as shown in Figure 13. This gives an idea of the measurement error as given by the Attocube during a \(70 \mu m\) motion.

d_err_mean = reshape(lsim(G_lpf, encoder - interferometer, t), [2/Ts floor(Ts/2*length(encoder))]);
d_err_mean = d_err_mean - mean(d_err_mean);

Figure 13 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).

To do so, all the attocube measurements corresponding to each position measured by the Encoder (resolution of \(1nm\)) are averaged. Figure 14 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.

[e_sorted, ~, e_ind] = unique(encoder);

i_mean = zeros(length(e_sorted), 1);
for i = 1:length(e_sorted)
  i_mean(i) = mean(interferometer(e_ind == i));
end

i_mean_error = (i_mean - e_sorted);

The period of the non-linearity seems to be equal to \(765 nm\) which corresponds to half the wavelength of the Laser (\(1.53 \mu m\)). For the motion range done here, the non-linearity is measured over ~18 periods which permits to do some averaging.

win_length = 1530/2; % length of the windows (corresponds to 765 nm)
num_avg = floor(length(e_sorted)/win_length); % number of averaging

i_init = ceil((length(e_sorted) - win_length*num_avg)/2); % does not start at the extremity

e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init:i_init+win_length*num_avg-1), [win_length num_avg]), 2);

The obtained periodic non-linearity is shown in Figure 15.