%% Created by Maple V Release 5 (IBM INTEL NT)
%% Source Worksheet: align.mws
%% Generated: Fri Feb 26 15:53:34 1999
\documentclass{article}
\usepackage{maple2e}
\DefineParaStyle{Heading 1}
\DefineParaStyle{Heading 2}
\DefineParaStyle{Heading 3}
\DefineParaStyle{Heading 4}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Maple Plot}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
\begin{document}
\begin{maplegroup}
Conceptual Design of a Laser Alignment system for E614 

                                                                 
{\large  P. Kitching}

{\large We propose to use a semiconductor laser to produce an expanded
parallel beam  illuminating a circular aperture (pinhole) attached to
the detector. The light passing through the pinhole falls on a small
CCD camera with the lens removed. Four sets of such an apparatus are
envisioned, which will overdetermine the position and orientation of
the detector in the x-y plane. The detector position in z can be fixed
by a stop on the track in the solenoid, or by a variation of the
technique proposed here.}

\end{maplegroup}

\section{Apparatus}

The lasers are mounted on the downstream endplug of the magnet yoke.
They will then move with the endplug when the detector package is
withdrawn from the solenoid.

Two pinholes are mounted on the upstream, two on the downstream G10
endplates of detector stack. The CCD cameras are attached to the
upstream endplug of the yoke

\subsection{Layout of detector and alignment system}

{\large Shown below are two figures.  The first Figure shows the
layour of the G10 endplates of the detector, with four possible
locations for the pinholes at the ends of the four protruding
"arms".The second depicts the layout of the E614 experiment with one
laser-pinhole-CCD camera set of alignment tools.}

\subsubsection{Dimensions of E614 system}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Laser_CCD :=3000;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{Laser\_CCD} := 3000
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Det_stack :=1060;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{Det\_stack} := 1060
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Laser_targ_down :=(Laser_CCD-Det_stack)/2;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{Laser\_targ\_down} := 970
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{CCD_targ_down :=
Laser_CCD-Laser_targ_down;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{CCD\_targ\_down} := 2030
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{CCD_targ_up :=(Laser_CCD-Det_stack)/2;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{CCD\_targ\_up} := 970
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Laser_targ_up :=
Laser_CCD-Laser_targ_down;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{Laser\_targ\_up} := 2030
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}














































\mapleinline{active}{1d}{data_list:=[[CCD_targ_up,2.4e+6],[CCD_targ_up,0],[CCD_targ_down,0],[CCD_targ_down,2.4e+6]];
}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{data\_list} := [[970, \,.24\,10^{7}], \,[970, \,0], \,[
2030, \,0], \,[2030, \,.24\,10^{7}]]
\]
\end{maplelatex}

\end{maplegroup}

\subsection{Laser:}

\begin{Heading 2}
{\large With the use of a beam expander a semiconductor laser can
produce beam a few m in extent with a wavelength typically 630 nm and
a divergence of less than 1 mr.}
\end{Heading 2}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{lambda :=630*10^(-6);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\lambda  := {\displaystyle \frac {63}{100000}} 
\]
\end{maplelatex}

\end{maplegroup}

\subsection{CCD Camera:}

{\large We propose to use the "QuickCam" model,which has a 3.2 mm by
2.4 mm CCD with 320 by 240 pixels and a pixel size of 10 microns.}

\section{Theory of light intensity from a circular aperture}

We assume parallel light from the laser falling perpendicular to a
small circular aperture of radius "a".{\large  From Fresnel
diffraction theory, one can show}

\subsection{Intensity distribution along z-axis, perpendicular to
aperture}

\begin{maplegroup}
{\large A circular aperture of radius a gives rise to an intensity
distribution along the z-axis of}

\begin{mapleinput}
\mapleinline{active}{1d}{Intensity:=z ->
(1-cos(Pi*a^2/(z*lambda)))/(2*lambda**2);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{Intensity} := z\rightarrow {\displaystyle \frac {1}{2}} 
\,{\displaystyle \frac {1 - \mathrm{cos}({\displaystyle \frac {
\pi \,a^{2}}{z\,\lambda }} )}{\lambda ^{2}}} 
\]
\end{maplelatex}

\end{maplegroup}

\subsection{Radial intensity distribution}

{\small The theoretical prediction for the intensity in the
diffraction pattern formed by a circular aperture is given by the
function Fresnel(x), where,if r is the radial distance from the z-axis
through the centre of the aperture, then}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{x:=r -> 2*Pi*a*r/(lambda*z);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
x := r\rightarrow 2\,{\displaystyle \frac {\pi \,a\,r}{\lambda \,
z}} 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{Fresnel:=x -> (2*BesselJ(1,x)/x)^2;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{Fresnel} := x\rightarrow 4\,{\displaystyle \frac {
\mathrm{BesselJ}(1, \,x)^{2}}{x^{2}}} 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}

\end{maplegroup}
\begin{maplegroup}
\mapleresult
\begin{maplettyout}
\end{maplettyout}

\end{maplegroup}

\section{Results for E614 Detector Setup}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{focus:=a -> a^2/lambda;}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{focus} := a\rightarrow {\displaystyle \frac {a^{2}}{
\lambda }} 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a_down :=sqrt(lambda*CCD_targ_down);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{a\_down} := {\displaystyle \frac {21}{100}} \,\sqrt{29}
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(a_down);}{%
}
\end{mapleinput}

\begin{mapleinput}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
1.130884609
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a_up :=sqrt(lambda*CCD_targ_up);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{a\_up} := {\displaystyle \frac {3}{100}} \,\sqrt{679}
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{evalf(a_up);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
.7817288532
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a :=0.5*(a_up+a_down):}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{evalf(a);}{%
}
\end{mapleinput}

\begin{mapleinput}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
.9563067313
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
Average hole radius is 0.96 mm

\end{maplegroup}

\subsection{Plots of intensity distributions}

\begin{maplegroup}
\mapleresult
\begin{maplettyout}
\end{maplettyout}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
{\large With optimal hole radius, so that focus is at CCD,}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{r_up:=a_up*x/(2*Pi);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
\mathit{r\_up} := {\displaystyle \frac {3}{200}} \,
{\displaystyle \frac {\sqrt{679}\,x}{\pi }} 
\]
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{evalf(r_up);}{%
}
\end{mapleinput}

\mapleresult
\begin{maplelatex}
\[
.1244160111\,x
\]
\end{maplelatex}

\end{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{r_down:=a_down*x/(2*Pi);}{%
}
\end{mapleinput}

\begin{mapleinput}
\mapleinline{active}{1d}{evalf(r_down);}{%
}
\end{mapleinput}

\begin{maplelatex}
\[
\mathit{r\_down} := {\displaystyle \frac {21}{200}} \,
{\displaystyle \frac {\sqrt{29}\,x}{\pi }} 
\]
\end{maplelatex}

\begin{maplelatex}
\[
.1799858757\,x
\]
\end{maplelatex}

\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{with(plots):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}


\mapleinline{active}{1d}{p1:=plot(Intensity(z),z=700..3000,color=black):
}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{t1:=textplot([1600,2.6e+6,"a=0.96"]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a :=a_up:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{p2:=plot(Intensity(z),z=700..3000,color=red):
}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{t2:=textplot([1000,2.6e+6,"a=.78"]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{a :=a_down:}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}

\mapleinline{active}{1d}{p3:=plot(Intensity(z),z=700..3000,color=blue):
}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{t3:=textplot([2200,2.6e+6,"a=1.13"]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{p5:=pointplot(data_list,style=line):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{t6:=textplot([5,0.8,"r_up=.124x,  r_down=.18x
in mm"]):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{p6:=plot(Fresnel(x),x=0..10,color=black):}{%
}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(p1,t1,p2,t2,p3,t3,p5,title="Intensity
vs z");}{%
}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{align01.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\mapleinline{active}{1d}{display(p6,t6,title="Intensity vs x");}{%
}
\end{mapleinput}

\mapleresult
\begin{center}
\mapleplot{align02.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}

\section{Test of Concept}

\begin{maplegroup}
{\large We have studied a basic system consisting of a semiconductor
laser, a 1mm circular aperture and a "QuickCam" CCD camera in a
student project at the University of Alberta. The overall conclusions
of this study were that }

{\large 1) there is a one-to-one correlation between the position of
the aperture and the diffraction pattern observed in the CCD camera}

{\large 2) the position of the centroid of the diffraction pattern can
be determined to better than 50 microns. Hence the location of the
circular aperture in the X-Y plane can be determined to this kind of
accuracy}

\end{maplegroup}

\section{Conclusions:}

{\large We propose an inexpensive laser-pinhole-CCD system for
aligning the X-Y position  of the E614 detector package. By using four
such systems, two attached to the upstream end of the detector
package, and two to the downstream end, we should be able to
overdetermine the orientation of the detector and its X-Y location.
Its position in z must be found by either using a stop, or by a
variation of the system described here in which the laser beam
impinges at an angle to one end face of the detector and is reflected
by a mirror, after passing through a small circular aperture attached
to the face. The diffraction patter is then detected by another CCD
camera. One the design of detector support structure is finalised, we
can proceed to detailed design of the alignment system}

\end{document}
%% End of Maple V Output