Continue the budget error analysis

index.org

@@ -1962,18 +1962,37 @@ We compute the Power Spectral Density of the voltage across the inductance used
 #+end_src
 
 * Budget Error
-** Introduction                                                     :ignore:
-Goals:
-- List all sources of error and their effects on the Attocube measurement
-- Think about how to determine the value of the individual sources of error
+<<sec:budget_error>>
 
-Sources of error for the Attocube measurement:
-- Cercalo rotation error
-- Cercalo unwanted translation perpendicular to its surface
-- Newport rotation error
-- Newport unwanted translation perpendicular to its surface
+** Introduction                                                     :ignore:
+*Goals*:
+- List all sources of error and compute their effects on the Attocube measurement
+- Think about how to determine the value of the individual sources of error
+- Sum all the sources of error and determine the limiting ones
+
+*Sources of error for the Attocube measurement*:
+- Beam non-perpendicularity to the concave mirror is linked to the non-perfect feedback loop:
+  - We have only finite gain / limited bandwidth so the Cercalo mirror angle will not be perfect
+  - The non-perpendicularity is measured by the 4QD and is used as the feedback signal, however this signal is noisy and even with infinite gain, this noise will be transmitted to the angle of the beam
+- Cercalo/Newport unwanted translation perpendicular to its surface.
+  This can be due to:
+  - Non idealities in the mechanics of the Cercalo
+  - Temperature variations
+  - The reproducible part of the perpendicular translation with respect to the angle of the Cercalo can be taken into account and subtracted from the Attocube measurement
 - Temperature variations of the metrology frame
-- Temperature / Pressure / Humidity variations of the air in the beam path
+- Change in the refractive air index in the beam path.
+  This can be due to change of Temperature, Pressure and Humidity of the air in the beam path
+
+*Procedure*:
+- in section [[sec:cercalo_angle_error]]:
+  We estimate the effect of an angle error of the Cercalo mirror on the Attocube measurement
+- in section [[sec:mirror_perpendicular_motion]]:
+  The effect of perpendicular motion of the Newport and Cercalo mirrors on the Attocube measurement is determined.
+- in section [[sec:effect_refractive_index]]:
+  We estimate the expected change of refractive index of the air in the beam path and the resulting Attocube measurement error
+- in section [[sec:feedback_error]]:
+  The feedback system using the 4 quadrant diode and the Cercalo is studied.
+  Sensor noise, actuator noise and their effects on the control error is discussed.
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -1985,10 +2004,16 @@ Sources of error for the Attocube measurement:
 #+end_src
 
 ** Effect of the Cercalo angle error on the measured distance by the Attocube
-To simplify, we suppose that the Newport mirror is a flat mirror.
-The geometry of the setup is shown in Fig. [[fig:angle_error_schematic_cercalo]].
+<<sec:cercalo_angle_error>>
+To simplify, we suppose that the Newport mirror is a flat mirror (instead of a concave one).
 
-We define the angle error range where we want to evaluate the distance error measured by the Attocube.
+The geometry of the setup is shown in Fig. [[fig:angle_error_schematic_cercalo]] 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.
 #+begin_src matlab
   thetas_c = logspace(-7, -4, 100); % [rad]
 #+end_src
@@ -2001,15 +2026,15 @@ The geometrical parameters of the setup are defined below.
 
 #+begin_src latex :file angle_error_schematic_cercalo.pdf :exports results
   \begin{tikzpicture}
-    \draw[->] (0, 0)node[branch](O){}node[below]{$O (0,0)$} -- ++(1, 0) node[above left]{$x$};
+    \draw[->] (0, 0)coordinate(O)node[below]{$O (0,0)$} -- ++(1, 0) node[above left]{$x$};
     \draw[->] (0, 0) -- ++(0, 1) node[below right]{$y$};
-    \draw[] (4, 0)node[branch](S){}node[below right]{$S (L,0)$} -- ++(45:1);
+    \draw[] (4, 0)coordinate(S)node[below right]{$S (L,0)$} -- ++(45:1);
     \draw[] (4, 0) -- ++(225:1);
-    \draw[] (3, 2) --node[midway, branch](X){}node[above]{$X (0,H)$} (5, 2);
-    \draw[<->] ([shift=(30:1.2)]S.center) arc (30:60:1.2) node[midway, above right]{$\theta_c$};
+    \draw[] (3, 2) --coordinate[midway](X)node[above]{$X (0,H)$} (5, 2);
+    \draw[<->] ([shift=(30:1.2)]S.center) arc (30:60:1.2) node[midway, above right]{$\delta \theta_c$};
 
-    \draw[red, ->-=.7, -<-=0.3] (O.center) -- (S.center);
-    \draw[red, ->-=.7, -<-=0.3] (S.center) -- (X.center);
+    \draw[red, ->-=.7, -<-=0.3] (O) -- (S);
+    \draw[red, ->-=.7, -<-=0.3] (S) -- (X);
   \end{tikzpicture}
 #+end_src
 
@@ -2025,12 +2050,12 @@ The nominal points $O$, $S$ and $X$ are defined.
   X = [0, H];
 #+end_src
 
-Thus, the initial path length is:
+Thus, the initial path length $L$ is:
 #+begin_src matlab
   path_nominal = norm(S-O) + norm(X-S) + norm(S-X) + norm(O-S);
 #+end_src
 
-We now compute the new path length when there is an error angle $\delta \theta_c$ of the Cercalo.
+We now compute the new path length when there is an error angle $\delta \theta_c$ on the Cercalo mirror angle.
 #+begin_src matlab
   path_length = zeros(size(thetas_c));
 
@@ -2044,7 +2069,7 @@ We now compute the new path length when there is an error angle $\delta \theta_c
   end
 #+end_src
 
-We then compute the distance error and we plot it as a function of the Cercalo angle error (Fig. [[]]).
+We then compute the distance error and we plot it as a function of the Cercalo angle error (Fig. [[fig:effect_cercalo_angle_distance_meas]]).
 #+begin_src matlab
   path_error = path_length - path_nominal;
 #+end_src
@@ -2067,7 +2092,7 @@ We then compute the distance error and we plot it as a function of the Cercalo a
 #+CAPTION: Effect of an angle error of the Cercalo on the distance error measured by the Attocube ([[./figs/effect_cercalo_angle_distance_meas.png][png]], [[./figs/effect_cercalo_angle_distance_meas.pdf][pdf]])
 [[file:figs/effect_cercalo_angle_distance_meas.png]]
 
-And we plot the beam path using Matlab for an high angle to verify that the code is working (Fig. [[]]).
+And we plot the beam path using Matlab for an high angle to verify that the code is working (Fig. [[fig:simulation_beam_path_high_angle]]).
 #+begin_src matlab
   theta = 2*2*pi/360; % [rad]
   H = 0.05; % [m]
@@ -2105,14 +2130,336 @@ And we plot the beam path using Matlab for an high angle to verify that the code
 [[file:figs/simulation_beam_path_high_angle.png]]
 
 #+begin_important
-  An angle error $\delta\theta_c$ of the Attocube produces a distance error measured by the attocube of
-  \begin{equation}
-    \delta L = 10^{-6} \cdot \delta\theta_c
-  \end{equation}
-  Thus, $1 \mu \text{rad}$ of angle error corresponds to $1mn$ of distance error.
+Based on Fig. [[fig:effect_cercalo_angle_distance_meas]], we see that an angle error $\delta\theta_c$ of the Cercalo mirror induces a distance error $\delta L$ measured by the Attocube which is dependent of the square of $\delta \theta_c$:
+
+\begin{equation}
+  \delta L = \delta\theta_c^2
+\end{equation}
+with:
+- $\delta L$ expressed in [m]
+- $\delta \theta_c$ in [rad]
+
+Some example are shown in table [[tab:effect_angle_error]].
+
+The tracking error of the feedback system used to position the Cercalo mirror should thus be limited to few micro-meters.
 #+end_important
 
+#+name: tab:effect_angle_error
+#+caption: Effect of an angle error $\delta \theta_c$ of the Cercalo's mirror on the measurement error $\delta L$ by the Attocube
+| Angle Error $\delta \theta_c$ | Distance measurement error $\delta L$ |
+|-------------------------------+---------------------------------------|
+| $1\,\mu\text{rad}$            | $1\, nm$                              |
+| $5\,\mu\text{rad}$            | $25\, nm$                             |
+| $10\,\mu\text{rad}$           | $100\, nm$                            |
+
+** Unwanted motion of Cercalo/Newport mirrors perpendicular to its surface
+<<sec:mirror_perpendicular_motion>>
+
+From Figs [[fig:cercalo_perpendicular_motion]] and [[fig:newport_perpendicular_motion]], 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:
+\begin{align}
+  \delta L &= 2 \cdot \delta d_c \\
+  \delta L &= 2 \cdot \delta d_n
+\end{align}
+
+Note here that $\delta L$ denote the change of beam traveled distance.
+The error in measured distance by the Attocube will we $\delta L/2$.
+
+#+begin_src latex :file cercalo_perpendicular_motion.pdf :exports results
+  \begin{tikzpicture}
+    % X-Y axis
+    \draw[->] (0, 0)coordinate(O) -- ++(1, 0) node[above left]{$x$};
+    \draw[->] (0, 0) -- ++(0, 1) node[below right]{$y$};
+
+    % Cercalo Mirror
+    \draw[] ($(4, 0)+(225:1)$)coordinate(a) --node[midway, below, rotate=45]{Cercalo}coordinate[midway](S) ($(4, 0)+(45:1)$);
+    \draw[dashed, name path=Cc--Cd] ($(4, 0)+(135:0.5)+(225:1)$)coordinate(b) -- ($(4, 0)+(135:0.5)+(45:1)$);
+
+    % Mirror displacement
+    \draw[->] (a) --node[midway, below left]{$\delta d_c$} (b);
+
+    % Newport Mirror
+    \draw[] (3, 2) --coordinate[midway](X)node[above]{Newport} (5, 2);
+
+    % Nominal Beam path
+    \draw[red, ->-=.7, -<-=0.3, name path=O--S] (O.center) -- (S.center);
+    \draw[red, ->-=.7, -<-=0.3] (S.center) --coordinate[midway](c1) (X.center);
+
+    % Changed beam path
+    \path [name intersections={of=O--S and Cc--Cd,by=E}];
+    \draw[red, dashed, ->-=.7, -<-=0.3] (E) --coordinate[midway](c2) (E|-X);
+
+    \draw[<->] (c1) --node[midway, above]{$\frac{\delta L}{2}$} (c2);
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:cercalo_perpendicular_motion
+#+caption: Effect of a Perpendicular motion of the Cercalo Mirror
+#+RESULTS:
+[[file:figs/cercalo_perpendicular_motion.png]]
+
+#+begin_src latex :file newport_perpendicular_motion.pdf :exports results
+  \begin{tikzpicture}
+    % X-Y axis
+    \draw[->] (0, 0)coordinate(O) -- ++(1, 0) node[above left]{$x$};
+    \draw[->] (0, 0) -- ++(0, 1) node[below right]{$y$};
+
+    % Cercalo Mirror
+    \draw[] ($(4, 0)+(225:1)$) --node[midway, below, rotate=45]{Cercalo}coordinate[midway](S) ($(4, 0)+(45:1)$);
+
+    % Newport Mirror
+    \draw[] (3, 2)coordinate(a) --coordinate[midway](X) (5, 2);
+    \draw[dashed] (3, 2.5)coordinate(b) --coordinate[midway](X2)node[above]{Newport} (5, 2.5);
+
+    % Mirror displacement
+    \draw[->] (a) --node[midway, left]{$\frac{\delta L}{2} = \delta d_n$} (b);
+
+
+    % Nominal Beam path
+    \draw[red, ->-=.7, -<-=0.3, name path=O--S] (O) -- (S);
+    \draw[red, ->-=.7, -<-=0.3] (S) -- (X);
+    \draw[dashed, red] (X) -- (X2);
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:newport_perpendicular_motion
+#+caption: Effect of a Perpendicular motion of the Newport Mirror
+#+RESULTS:
+[[file:figs/newport_perpendicular_motion.png]]
+
+#+begin_important
+The motion of the both Cercalo's and Newport's mirrors perpendicular to its surface is fully transmitted to the measured distance by the Attocube interferometer.
+
+This motion can be measured and the repeatable part can be compensated.
+However, the non repeatability of this motion should be less than few nano-meters.
+#+end_important
+
+** Change in refractive index of the air in the beam path
+<<sec:effect_refractive_index>>
+
+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:
+| Air property Variations | Measurement error |
+|-------------------------+-------------------|
+| $\Delta T = 1^oC$       | 100nm             |
+| $\Delta P = 1hPa$       | 27nm              |
+| $\Delta 10\%RH$         | 10nm              |
+
+An *Environmental Compensation Unit* is used and can compensate for variations or air properties up to:
+| Air property Variations | Measurement error |
+|-------------------------+-------------------|
+| $\Delta T = \pm 0.1^oC$ | 20nm              |
+| $\Delta P = \pm 1hPa$   | 50nm              |
+| $\Delta \pm 2\%RH$      | 4nm               |
+
+#+begin_important
+The total measurement error induced by air properties variations is then:
+\begin{equation}
+  \sqrt{20^2 + 50^2 + 4^2} = 54nm
+\end{equation}
+
+The beam path should be protected using aluminum to minimize the change in refractive index of the air in the beam path.
+#+end_important
+
+** Thermal Expansion of the Metrology Frame
+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 $0.1\,^oC$.
+
 ** Estimation of the Cercalo angle error due to Noise
+<<sec:feedback_error>>
+*** Introduction                                                   :ignore:
+In this section, we seek to estimate the angle error $\delta \theta$
+
+Consider the block diagram in Fig. [[fig:feedback_diagram]] 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
+
+#+begin_src latex :file feedback_diagram.pdf :exports results
+  \begin{tikzpicture}
+    \node[block] (K) at (0,0) {$K$};
+    \node[addb={+}{}{}{}{}, right=1 of K] (addu) {};
+    \node[block, right=1 of addu] (G){$G$};
+    \node[addb={+}{}{}{}{}, right=1 of G] (adddy){};
+    \node[addb={+}{}{}{}{}, below right=1 and 1 of adddy] (addn) {};
+
+    \draw[->] (K.east) -- (addu.west);
+    \draw[->] (addu.east) -- (G.west) node[above left]{$u$};
+    \draw[<-] (addu.north) -- ++(0, 0.7) node[below right]{$d_u$};
+    \draw[->] (G.east) -- (adddy.west);
+    \draw[<-] (addn.east) -- ++(0.8, 0) coordinate[](endpos) node[above left]{$n_\theta$};
+    \draw[->] (adddy.east) -- (G-|endpos) node[above left]{$\delta\theta$};
+    \draw[->] (G-|addn) node[branch]{} -- (addn.north);
+    \draw[->] (addn.west) -| ($(K.west)+(-1, 0)$) -- (K.west) node[above left]{$\delta\theta_m$};
+    \draw[<-] (adddy.north) -- ++(0, 0.7) node[below right]{$d$};
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:feedback_diagram
+#+caption: Block Diagram of the Feedback system
+#+RESULTS:
+[[file:figs/feedback_diagram.png]]
+
+*** Estimation of sources of noise and disturbances
+Let's estimate the values of $d_u$, $d$ and $n_\theta$.
+
+**** ADC Quantization Noise
+The ADC quantization noise is:
+\begin{equation}
+  \Gamma_\text{ADC} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f_s} \text{ in } \left[ \frac{V^2}{Hz} \right]
+\end{equation}
+with:
+- $\Delta V$: is the range of the ADC in $[V]$
+- $n$: is the number of ADC's bits
+- $f_s$: is the sampling frequency in $[Hz]$
+
+For the ADC used:
+- $\Delta V = 20\, V$
+- $n = 16$
+- $f_s = 10\, kHz$
+
+#+begin_important
+\begin{equation}
+  \Gamma_\text{ADC}(f) = 7.76 \cdot 10^{-13}\,\left[ \frac{V^2}{Hz} \right]
+\end{equation}
+#+end_important
+
+**** DAC Quantization Noise
+The DAC quantization noise is:
+\begin{equation}
+  \Gamma_\text{DAC} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f_s} \text{ in } \left[ \frac{V^2}{Hz} \right]
+\end{equation}
+with:
+- $\Delta V$: is the range of the DAC in $[V]$
+- $n$: is the number of DAC's bits
+- $f_s$: is the sampling frequency in $[Hz]$
+
+For the DAC used:
+- $\Delta V = 20\, V$
+- $n = 16$
+- $f_s = 10\, kHz$
+
+#+begin_important
+\begin{equation}
+  \Gamma_\text{DAC}(f) = 7.76 \cdot 10^{-13}\,\left[ \frac{V^2}{Hz} \right]
+\end{equation}
+#+end_important
+
+**** Noise of the Newport Mirror angle
+Plus, we estimate the effect of DAC quantization noise on the angle error on the Newport mirror.
+
+The gain of the Newport is:
+\begin{align*}
+  \frac{\theta_n}{V_n} &= \frac{26.2}{10}\ \left[ \frac{mrad}{V} \right] \\
+                       &= 2.62 \cdot 10^{-3}\ \left[ \frac{rad}{V} \right]
+\end{align*}
+
+\begin{align*}
+  \Gamma_{\theta_n}(f) &= \left(\frac{\theta_n}{V_n}\right)^2 \cdot \Gamma_\text{DAC}(f) \\
+                       &= (2.62 \cdot 10^{-3})^2 \cdot 7.76 \cdot 10^{-13} \\
+                       &= 3.96 \cdot 10^{-18}\,\left[ \frac{rad^2}{Hz} \right]
+\end{align*}
+
+If we integrate that to obtain an rms value:
+\begin{align*}
+  \theta_{n, rms} &= \sqrt{\int_{-f_s/2}^{f_s/2} \Gamma_{\theta_n}(f) df} \\
+                  &= 0.2\, \mu rad
+\end{align*}
+
+Which is much less than the noise equivalent angle specified by Newport: $3\, \mu rad\,[rms]$.
+Thus, quantization error of the DAC is not a problem.
+
+We expect the angle noise of the Newport mirror to be around $3\, \mu rad\,[rms]$ which is $6\, \mu rad\,[rms]$ for the beam angle.
+
+#+begin_important
+If we suppose a white noise, the power spectral density of the beam angle due to the noise of the Newport mirror corresponds to:
+\begin{align*}
+  \Gamma_{d} &= \frac{(6 \cdot 10^{-6})^2}{f_s}\ \left[ \frac{rad^2}{Hz} \right] \\
+                    &= 3.6 \cdot 10^{-15}\ \left[ \frac{rad^2}{Hz} \right]
+\end{align*}
+#+end_important
+
+**** Disturbances due the Newport Mirror Rotation
+We will rotate the Newport mirror in order to simulate a displacement of the Sample:
+- The angle range for the Newport mirror is $\pm 26.2\ mrad = \pm 1.5^o$
+- The radius of the concave mirror is 200 mm
+
+#+begin_src latex :file newport_angle_concave_mirror.pdf :post pdf2svg(file=*this*, ext="png") :exports results
+  \begin{tikzpicture}
+    % X-Y axis
+    \draw[->] (0, 0)coordinate(O) -- ++(1, 0) node[above left]{$x$};
+    \draw[->] (0, 0) -- ++(0, 1) node[below right]{$y$};
+
+    % Cercalo Mirror
+    \draw[] ($(4, 0)+(225:1)$) --node[midway, below, rotate=45]{Cercalo}coordinate[midway](S) ($(4, 0)+(45:1)$);
+
+    % Concave Newport Mirror
+    \draw[] ([shift=(260:6)]4, 10)coordinate(a) arc (260:280:6)coordinate(b)coordinate[midway](X);
+    \node[branch] (C) at (4, 10){};
+    \draw[dashed, <->] (a) -- node[midway, left]{$R = 200\,mm$} (C);
+
+    \draw[dashed] (X) -- (O|-X);
+    \draw[dashed, <->] (O) -- node[midway, left]{$H = 50\,mm$} (O|-X);
+
+    % Nominal Beam path
+    \draw[red, ->-=.7, -<-=0.3, name path=O--S] (O) -- (S);
+    \draw[red, ->-=.7, -<-=0.3] (S) -- (X);
+    \draw[red, dashed] (X) -- (C);
+
+    \begin{scope}[rotate around={-10:(X)}]
+      \draw[draw=blue!50!white, name path=arcb] ([shift=(260:6)]4, 10) arc (260:280:6) coordinate[midway](Xb);
+      \node[branch, color=blue!50!white] (Cb) at (4, 10){};
+      \draw[dashed, draw=blue!50!white] (Xb) -- (Cb);
+    \end{scope}
+
+    \path[name path=S--Cb] (S) -- (Cb);
+    % Changed beam path
+    \path [name intersections={of=arcb and S--Cb,by=E}];
+    \draw[red, dashed, ->-=.7, -<-=0.3] (S) -- (E);
+    \draw[red, dashed] (E) -- (Cb);
+
+    % Center of rotation
+    \node[branch] at (X){};
+    \node[below left] at (X){$O_M$};
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:newport_angle_concave_mirror
+#+caption: Rotation of the (concave) Newport mirror
+#+RESULTS:
+[[file:figs/newport_angle_concave_mirror.png]]
+
+If we suppose small angles, the corresponding beam deviation is:
+\[ \delta \theta \approx 2*\frac{\alpha R}{H + R} = 1.6 \alpha \]
+where $\alpha$ is the rotation of the Newport mirror.
+
 *** Perfect Control
 If the feedback is perfect, the Cercalo angle error will be equal to the 4 quadrant diode noise.
 Let's estimate the 4QD noise in radians.