Cercalo Test Bench

The block diagram of the setup to be controlled is shown in Fig. 1.

The transfer functions in the system are:

• Current Amplifier: from the voltage set by the DAC to the current going to the Cercalo's inductors \[ G_i = \begin{bmatrix} G_{i,h} & 0 \\ 0 & G_{i,v} \end{bmatrix} \text{ in } \left[ \frac{A}{V} \right] \] \[ \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} = G_i \begin{bmatrix} U_{c,h} \\ U_{c,v} \end{bmatrix} \]
• Impedance of the Cercalo that converts the current going to the cercalo to the voltage across the cercalo: \[ Z_c = \begin{bmatrix} Z_{c,h} & 0 \\ 0 & Z_{c,v} \end{bmatrix} \text{ in } \left[ \frac{V}{A} \right] \] \[ \begin{bmatrix} \tilde{V}_{c,h} \\ \tilde{V}_{c,v} \end{bmatrix} = Z_c \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} \]
• Voltage Amplifier: from the voltage across the Cercalo inductors to the measured voltage \[ G_a = \begin{bmatrix} G_{a,h} & 0 \\ 0 & G_{a,v} \end{bmatrix} \text{ in } \left[ \frac{V}{V} \right] \] \[ \begin{bmatrix} V_{c,h} \\ V_{c,v} \end{bmatrix} = G_a \begin{bmatrix} \tilde{V}_{c,h} \\ \tilde{V}_{c,v} \end{bmatrix} \]
• Cercalo: Transfer function from the current going through the cercalo inductors to the 4 quadrant measurement \[ G_c = \begin{bmatrix} G_{\frac{V_{p,h}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,h}}{\tilde{U}_{c,v}}} \\ G_{\frac{V_{p,v}}{\tilde{U}_{c,h}}} & G_{\frac{V_{p,v}}{\tilde{U}_{c,v}}} \end{bmatrix} \text{ in } \left[ \frac{V}{A} \right] \] \[ \begin{bmatrix} V_{p,h} \\ V_{p,v} \end{bmatrix} = G_c \begin{bmatrix} I_{c,h} \\ I_{c,v} \end{bmatrix} \]
• Newport Transfer function from the command signal of the Newport to the 4 quadrant measurement \[ G_n = \begin{bmatrix} G_{\frac{V_{p,h}}{U_{n,h}}} & G_{\frac{V_{p,h}}{U_{n,v}}} \\ G_{\frac{V_{p,v}}{U_{n,h}}} & G_{\frac{V_{n,v}}{U_{n,v}}} \end{bmatrix} \text{ in } \left[ \frac{V}{V} \right] \] \[ \begin{bmatrix} V_{p,h} \\ V_{p,v} \end{bmatrix} = G_c \begin{bmatrix} V_{n,h} \\ V_{n,v} \end{bmatrix} \]
• 4 Quadrant Diode: the gain of the 4 quadrant diode in [V/rad] is inverse in order to obtain the physical angle of the beam \[ G_d = \begin{bmatrix} G_{d,h} & 0 \\ 0 & G_{d,v} \end{bmatrix} \text{ in } \left[\frac{V}{rad}\right] \]

The block diagram with each transfer function is shown in Fig. 2.

From the Cercalo documentation, we have the parameters shown on table 1.

The Inductance and DC resistance of the two axis of the Cercalo have been measured:

• \(L_{c,h} = 0.1\ \text{mH}\)
• \(L_{c,v} = 0.1\ \text{mH}\)
• \(R_{c,h} = 9.3\ \Omega\)
• \(R_{c,v} = 8.3\ \Omega\)

Let's first consider the horizontal direction and we try to model the Cercalo by a spring/mass/damper system (Fig. 3).

The equation of motion is:

with:

• \(G_0 = 1/k\) is the gain at DC in rad/N
• \(\xi = \frac{c}{2 \sqrt{km}}\) is the damping ratio of the system
• \(\omega_0 = \sqrt{\frac{k}{m}}\) is the resonance frequency in rad

The force \(F\) applied to the mass is proportional to the current \(I\) flowing through the voice coils: \[ \frac{F}{I} = \alpha \] with \(\alpha\) is in \(N/A\) and is to be determined.

The current \(I\) is also proportional to the voltage at the output of the buffer:

Let's try to determine the equivalent mass and spring values. From table 1, for the horizontal direction: \[ \left| \frac{x}{I} \right|(0) = \left| \alpha \frac{x}{F} \right|(0) = 28.4\ \frac{mA}{deg} = 1.63\ \frac{A}{rad} \]

So: \[ \alpha \frac{1}{k} = 1.63 \Longleftrightarrow k = \frac{\alpha}{1.63} \left[\frac{N}{rad}\right] \]

We also know the resonance frequency: \[ \omega_0 = 411.1\ \text{Hz} = 2583\ \frac{rad}{s} \]

And the gain at resonance:

Thus:

Parameters of the Newport are shown in Fig. 4.

It's dynamics for small angle excitation is shown in Fig. 5.

And we have:

The front view of the 4 quadrant photo-diode is shown in Fig. 6.

Each of the photo-diode is amplified using a 4-channel amplifier as shown in Fig. 7.

Let's compute the theoretical noise of the ADC/DAC.

with \(\Delta V\) the total range of the ADC, \(n\) its number of bits, \(q\) the quantization, \(f_N\) the sampling frequency and \(\Gamma_n\) its theoretical Power Spectral Density.

Prior to any dynamic identification, we would like to be able to determine the meaning of the 4 quadrant diode measurement. For instance, instead of obtaining transfer function in [V/V] from the input of the cercalo to the measurement voltage of the 4QD, we would like to obtain the transfer function in [rad/V]. This will give insight to physical interpretation.

To calibrate the 4 quadrant photo-diode, we can use the metrology included in the Newport. We can choose precisely the angle of the Newport mirror and see what is the value measured by the 4 Quadrant Diode. We then should be able to obtain the "gain" of the 4QD in [V/rad].

We wish here to determine \(G_i\) and \(G_a\) shown in Fig. 1.

We ignore the electro-mechanical coupling.

We now wish to identify the dynamics of the Cercalo identified by \(G_c\) on the block diagram in Fig. 1.

To do so, we inject some noise at the input of the current amplifier \([U_{c,h},\ U_{c,v}]\) (one input after the other) and we measure simultaneously the output of the 4QD \([V_{p,h},\ V_{p,v}]\).

The transfer function obtained will be \(G_c G_i\), and because we have already identified \(G_i\), we can obtain \(G_c\) by multiplying the obtained transfer function matrix by \({G_i}^{-1}\).

We here identify the transfer function from a reference sent to the Newport \([U_{n,h},\ U_{n,v}]\) to the measurement made by the 4QD \([V_{p,h},\ V_{p,v}]\).

To do so, we inject noise to the Newport \([U_{n,h},\ U_{n,v}]\) and we record the 4QD measurement \([V_{p,h},\ V_{p,v}]\).

We now have identified:

• \(G_i\)
• \(G_a\)
• \(G_c\)
• \(G_n\)
• \(G_d\)

We name the input and output of each transfer function:

Gi.InputName = {'Uch', 'Ucv'};
Gi.OutputName = {'Ich', 'Icv'};

Zc.InputName = {'Ich', 'Icv'};
Zc.OutputName = {'Vtch', 'Vtcv'};

Ga.InputName = {'Vtch', 'Vtcv'};
Ga.OutputName = {'Vch', 'Vcv'};

Gc.InputName = {'Ich', 'Icv'};
Gc.OutputName = {'Vpch', 'Vpcv'};

Gn.InputName = {'Unh', 'Unv'};
Gn.OutputName = {'Vpnh', 'Vpnv'};

Gd.InputName = {'Rh', 'Rv'};
Gd.OutputName = {'Vph', 'Vpv'};

Sh = sumblk('Vph = Vpch + Vpnh');
Sv = sumblk('Vpv = Vpcv + Vpnv');

inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};

sys = connect(Gi, Zc, Ga, Gc, Gn, inv(Gd), Sh, Sv, inputs, outputs);

The file mat/plant.mat is accessible here.

save('mat/plant.mat', 'sys', 'Gi', 'Zc', 'Ga', 'Gc', 'Gn', 'Gd');

The Attocube's "Environmental Compensation Unit" does not have a significant effect on the stability of the measurement.

To simplify, we suppose that the Newport mirror is a flat mirror (instead of a concave one).

The geometry of the setup is shown in Fig. 33 where:

• \(O\) is the reference surface of the Attocube
• \(S\) is the point where the beam first hits the Cercalo mirror
• \(X\) is the point where the beam first hits the Newport mirror
• \(\delta \theta_c\) is the angle error from its ideal 45 degrees

We define the angle error range \(\delta \theta_c\) where we want to evaluate the distance error \(\delta L\) measured by the Attocube.

thetas_c = logspace(-7, -4, 100); % [rad]

The geometrical parameters of the setup are defined below.

H = 0.05; % [m]
L = 0.05; % [m]

The nominal points \(O\), \(S\) and \(X\) are defined.

O = [-L, 0];
S = [0, 0];
X = [0, H];

Thus, the initial path length \(L\) is:

path_nominal = norm(S-O) + norm(X-S) + norm(S-X) + norm(O-S);

We now compute the new path length when there is an error angle \(\delta \theta_c\) on the Cercalo mirror angle.

path_length = zeros(size(thetas_c));

for i = 1:length(thetas_c)
    theta_c = thetas_c(i);
    Y = [H*tan(2*theta_c), H];
    M = 2*H/(tan(pi/4-theta_c)+1/tan(2*theta_c))*[1, tan(pi/4-theta_c)];
    T = [-L, M(2)+(L+M(1))*tan(4*theta_c)];

    path_length(i) = norm(S-O) + norm(Y-S) + norm(M-Y) + norm(T-M);
end

We then compute the distance error and we plot it as a function of the Cercalo angle error (Fig. 34).

path_error = path_length - path_nominal;

And we plot the beam path using Matlab for an high angle to verify that the code is working (Fig. 35).

theta = 2*2*pi/360; % [rad]
H = 0.05; % [m]
L = 0.05; % [m]

O = [-L, 0];
S = [0, 0];
X = [0, H];
Y = [H*tan(2*theta), H];
M = 2*H/(tan(pi/4-theta)+1/tan(2*theta))*[1, tan(pi/4-theta)];
T = [-L, M(2)+(L+M(1))*tan(4*theta)];

From Figs 36 and 37, it is clear that perpendicular motions of the Cercalo mirror and of the Newport mirror have an impact on the measured distance by the Attocube interferometer.

More precisely, if the note:

• \(\delta d_c\) the perpendicular motion of the Cercalo's mirror
• \(\delta d_n\) the perpendicular motion of the Newport's mirror

We have that:

Note here that \(\delta L\) denote the change of beam traveled distance. The error in measured distance by the Attocube will we \(\delta L/2\).

Three physical properties of the air makes change of the Attocube measurement:

• Temperature: \(K_T \approx 1 ppmK^{-1}\)
• Pressure: \(K_P \approx 0.27 ppm hPa^{-1}\)
• Humidity: \(K_{HR} \approx 0.01 ppm \% RH^{-1}\)

These physical properties should change relatively slowly, however, for a beam path of 100mm:

An Environmental Compensation Unit is used and can compensate for variations or air properties up to:

The material used for the metrology frame is Aluminum. Its linear thermal expansion coefficient is \(\alpha = 23 \cdot 10^{-6} K^{-1}\).

The distance between the Attocube head and the Attocube is approximatively equal to 5cm. \[ \frac{\delta L}{\delta T} \approx 0.05 \cdot 23 \cdot 10^{-6} \approx 1\,\frac{\mu m}{{}^oC} \] If invar is used (\(\alpha = 1.2 \cdot 10^{-6} \, K^{-1}\)): \[ \frac{\delta L}{\delta T} \approx 60 \frac{nm}{{}^oC} \]

Thus, the temperature of the metrology frame should be kept constant to less than \(0.1\,^oC\).

In this section, we seek to estimate the angle error \(\delta \theta\)

Consider the block diagram in Fig. 38 with:

• \(G\): represents the transfer function from a voltage applied by the Speedgoat DAC used for the Cercalo to the Beam angle
• \(K\): is the control law used

The signals are:

• \(\delta \theta\): is the "true" laser beam angle
• \(\delta \theta_m\): is the measured beam angle (\(\delta \theta_m = \delta \theta + n_\theta\))
• \(n_\theta\): is the measurement noise of the laser beam angle using the 4 quadrant diode. It includes:
  • ADC noise
  • \(1/f\) noise, Shot noise, Ambian noise, Intensity noise…
• \(d_u\): is noise at the input of the Cercalo. It includes:
  • DAC noise of the speadgoat
• \(d\): is disturbance on the angle of the beam. It includes:
  • Angle variations of the Newport mirror

The maximum expected stroke is \(y_\text{max} = 3mm \approx 5e^{-2} rad\) at \(1Hz\). The maximum wanted error is \(e_\text{max} = 10 \mu rad\).

Thus, we require the sensitivity function at \(\omega_0 = 1\text{ Hz}\):

In terms of loop gain, this is equivalent to: \[ |L(j\omega_0)| > 5 \cdot 10^{3} \]

The plant is put in a general configuration as shown in Fig. 40.

Active damping does not seems to be applicable here.

Using SISOTOOL, a diagonal controller is designed. The two SISO loop gains are shown in Fig. 41.

Kh = -0.25598*(s+112)*(s^2 + 15.93*s + 6.686e06)/((s^2*(s+352.5)*(1+s/2/pi/2000)));
Kv = 10207*(s+55.15)*(s^2 + 17.45*s + 2.491e06)/(s^2*(s+491.2)*(s+7695));

K = blkdiag(Kh, Kv);
K.InputName  = {'Rh', 'Rv'};
K.OutputName = {'Uch', 'Ucv'};

We then close the loop and we look at the transfer function from the Newport rotation signal to the beam angle (Fig. 42).

inputs  = {'Uch', 'Ucv', 'Unh', 'Unv'};
outputs = {'Vch', 'Vcv', 'Ich', 'Icv', 'Rh', 'Rv', 'Vph', 'Vpv'};

sys_cl = connect(sys, -K, inputs, outputs);

Kd = c2d(K, 1e-4, 'tustin');

The diagonal controller is accessible here.

save('mat/K_diag.mat', 'K', 'Kd');

The plant is basically a constant until frequencies up to the required bandwidth.

We get that constant value.

Gn0 = freqresp(inv(Gd)*Gn, 0);

We design two controller containing 2 integrators and one lead near the crossover frequency set to 10Hz.

h = 2;
w0 = 2*pi*10;

Knh = 1/Gn0(1,1) * (w0/s)^2 * (1 + s/w0*h)/(1 + s/w0/h)/h;
Knv = 1/Gn0(2,2) * (w0/s)^2 * (1 + s/w0*h)/(1 + s/w0/h)/h;

Kn = blkdiag(Knh, Knv);
Knd = c2d(Kn, 1e-4, 'tustin');

The controllers can be downloaded here.

save('mat/K_newport.mat', 'Kn', 'Knd');

Let's verify that the positioning error of the beam is small and what could be the effect on the distance measured by the intereferometer.

load('./mat/plant.mat', 'Gd');

figure;
ax1 = subplot(1, 2, 1);
hold on;
plot(uh.t(1:2*fs), 1e6*uh.Vph(1:2*fs)/freqresp(Gd(1,1), 0), 'DisplayName', '$\theta_{h}$');
plot(uh.t(1:2*fs), 1e6*uh.Vpv(1:2*fs)/freqresp(Gd(2,2), 0), 'DisplayName', '$\theta_{v}$');
hold off;
xlabel('Time [s]'); ylabel('$\theta$ [$\mu$ rad]');
title('Newport Tilt - Horizontal Direction');
legend();

ax2 = subplot(1, 2, 2);
hold on;
plot(uv.t(1:2*fs), 1e6*uv.Vph(1:2*fs)/freqresp(Gd(1,1), 0), 'DisplayName', '$\theta_{h}$');
plot(uv.t(1:2*fs), 1e6*uv.Vpv(1:2*fs)/freqresp(Gd(2,2), 0), 'DisplayName', '$\theta_{v}$');
hold off;
xlabel('Time [s]'); ylabel('$\theta$ [$\mu$ rad]');
title('Newport Tilt - Vertical Direction');
legend();

linkaxes([ax1,ax2],'xy');

Let's compute the PSD of the error to see the frequency content.

[psd_UhRh, f] = pwelch(uh.Vph/freqresp(Gd(1,1), 0), hanning(ceil(1*fs)), [], [], fs);
[psd_UhRv, ~] = pwelch(uh.Vpv/freqresp(Gd(2,2), 0), hanning(ceil(1*fs)), [], [], fs);
[psd_UvRh, ~] = pwelch(uv.Vph/freqresp(Gd(1,1), 0), hanning(ceil(1*fs)), [], [], fs);
[psd_UvRv, ~] = pwelch(uv.Vpv/freqresp(Gd(2,2), 0), hanning(ceil(1*fs)), [], [], fs);

Let's convert that to errors in distance

\[ \Delta L = L^\prime - L = \frac{L}{\cos(\alpha)} - L \approx \frac{L \alpha^2}{2} \] with

• \(L\) is the nominal distance traveled by the beam
• \(L^\prime\) is the distance traveled by the beam with an angle error
• \(\alpha\) is the angle error

L = 0.1; % [m]

[psd_UhLh, f] = pwelch(0.5*L*(uh.Vph/freqresp(Gd(1,1), 0)).^2, hanning(ceil(1*fs)), [], [], fs);
[psd_UhLv, ~] = pwelch(0.5*L*(uh.Vpv/freqresp(Gd(2,2), 0)).^2, hanning(ceil(1*fs)), [], [], fs);
[psd_UvLh, ~] = pwelch(0.5*L*(uv.Vph/freqresp(Gd(1,1), 0)).^2, hanning(ceil(1*fs)), [], [], fs);
[psd_UvLv, ~] = pwelch(0.5*L*(uv.Vpv/freqresp(Gd(2,2), 0)).^2, hanning(ceil(1*fs)), [], [], fs);

Now, compare that with the PSD of the measured distance by the interferometer.

[psd_Lh, f] = pwelch(uh.Va, hanning(ceil(1*fs)), [], [], fs);
[psd_Lv, ~] = pwelch(uv.Va, hanning(ceil(1*fs)), [], [], fs);

First, we get the mean value as measured by the interferometer for each value of the Newport angle.

Vahm = mean(reshape(uh.Va, [fs floor(length(uh.t)/fs)]),2);
Unhm = mean(reshape(uh.Unh, [fs floor(length(uh.t)/fs)]),2);

Vavm = mean(reshape(uv.Va, [fs floor(length(uv.t)/fs)]),2);
Unvm = mean(reshape(uv.Unv, [fs floor(length(uv.t)/fs)]),2);

We then subtract

figure;
hold on;
plot(uh.Unh, 1e9*(uh.Va - repmat(Vam, length(uh.t)/length(Vam),1)))
hold off;
xlabel('$V_{n,h}$ [V]'); ylabel('$V_a$ [nm]');