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# Attocube - Test Bench
Author: Dehaeze Thomas

## Table of Contents

- 1. Estimation of the Spectral Density of the Attocube Noise
  - 1.1. Long and Slow measurement
  - 1.2. Short and Fast measurement
  - 1.3. Obtained Amplitude Spectral Density of the measured displacement
- 2. Effect of the “bubble sheet” and Aluminium tube
  - 2.1. Aluminium Tube and Bubble Sheet
  - 2.2. Only Aluminium Tube
  - 2.3. Nothing
  - 2.4. Comparison

## 1 Estimation of the Spectral Density of the Attocube Noise

Figure 1: Test Bench Schematic

Figure 2: Picture of the test bench. The Attocube and mirror are covered by a “bubble sheet”

### 1.1 Long and Slow measurement

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

    load('./mat/long_test2.mat', 'x', 't')
    Ts = 0.1; % [s]

Figure 3: Long measurement time domain data

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*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’

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

    2.0576

Figure 4: Fit of the measurement data with a step response of a first order low pass filter

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 = 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);

Figure 5: Kept data (removed slow drifts during the first hours)

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);

### 1.2 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.

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

    x = detrend(x, 0);

The time domain measurement is shown in Figure 6.

Figure 6: Time domain measurement with the high sampling rate

The Power Spectral Density of the measured displacement is computed

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

### 1.3 Obtained Amplitude Spectral Density of the measured displacement

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

Figure 7: Obtained Amplitude Spectral Density of the measured displacement

## 2 Effect of the “bubble sheet” and Aluminium tube

Figure 8: Aluminium tube used to protect the beam path from disturbances

### 2.1 Aluminium Tube and Bubble Sheet

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

    x = detrend(x, 0);

    win = hann(ceil(length(x)/10));
    [p_1, f_1] = pwelch(x, win, [], [], 1/Ts);

### 2.2 Only Aluminium Tube

    load('./mat/long_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);

### 2.3 Nothing

### 2.4 Comparison

Figure 9: Comparison of the noise ASD with and without bubble sheet

Author: Dehaeze Thomas

Created: 2020-11-02 lun. 16:03