Heavy Gas Cherenkov Detector Construction
for Hall C at
Thomas Jefferson National Accelerator Facility
A Thesis
Submitted to the Faculty of Graduate Studies and Research
In Partial Fulfilment of the Requirements
for the Degree of
Master of Science
in Physics
University of Regina
By
Wenliang Li
Regina, Saskatchewan

©2012: Wenliang Li

## Abstract

The Thomas Jefferson National Accelerator Facility (JLab) has undertaken the
12 GeV Upgrade to double the accelerating energy of its electron beam. This
attracts many interesting proposals to probe the quark-gluon nature of nuclear
matter at higher energy, therefore a new set of equipment is required.
Experimental Hall C of JLab has planned to construct a new Super High Momentum
Spectrometer (SHMS) to replace the existing Short Orbit Spectrometer (SOS).
The University of Regina is assigned to construct the Heavy Gas Čerenkov (HGC)
Detector as part of the SHMS focal plane detectors. This detector will be used
as critical component to provide reliable $\pi/K$ separation between 3-11
GeV/c central momenta in the SHMS experimental program. In this thesis, we
will report the design, quality control studies and simulated expected
performances of the HGC detector.

## Acknowledgements

I would like to thank my supervisor Garth Huber for everything. I could not
wish for a better supervisor and am extremely excited about the opportunity to
continue my study under his meticulous and patient guidance.

It was such a pleasure to work with Keith Wolbaum, Derek Gervais, Lee
Sichello, Paul Selles and Alex Fischer as a member of the HGC detector
construction team. Collaborators such as Donal Day and Tanja Horn gave much
generous assistance regarding the detector construction and Geant4 simulation.
I would like to acknowledge Chris Ng at the University of Alberta for his
brilliant effort in completing the detector construction drawing. I am very
grateful for the detector construction fund and research assistantships
provided by NSERC of Canada, teaching assistantships provided by the
Department of Physics, and conference travel support provided in part by the
Faculty of Graduate Studies and Research.

My appreciation to the Hall C scientists: Howard Fenker, Dave Gaskell, Mark
Jones, Dave Mack, Steve Wood, and engineers: Michael Fowler, Steve Furches,
Steve Lassiter, Bert Metzger, Eric Sun, for their fantastic support. The
permanent reflectivity setup at Jefferson Lab was a successful collaborative
work between the University of Regina and Jefferson Lab. My gratitude goes to
Brad Sawatzky, Michelle Shinn, Carl Zorn, and other Detector group and FEL
staffs.

Special thanks to all the faculty members and graduate students of the
Department of Physics at the University of Regina. Their suggestions and
criticisms have helped tremendously to achieve the final goal. Many thanks to
the hospitality of the Morrison family and their relatives.

I would like to thank my family and my friends for the support they provided,
in particular Miss Fei Yang, for her understanding and encouragement.

###### Contents

  1. 1 Introduction
    1. 1.1 CEBAF and the 12 GeV Upgrade
    2. 1.2 Physics Program at Hall C After 12 GeV Upgrade
    3. 1.3 Experimental Hall C
      1. 1.3.1 Spectrometer Coordination Definition
      2. 1.3.2 Super High Momentum Spectrometer (SHMS)
      3. 1.3.3 High Momentum Spectrometer (HMS)
      4. 1.3.4 SHMS and HMS Detector Packages
  2. 2 Čerenkov Radiation
  3. 3 HGC Specification and Design
    1. 3.1 Detector Design
    2. 3.2 Čerenkov Radiation Angle
    3. 3.3 Detector Structure
    4. 3.4 Mirrors
      1. 3.4.1 Window Mounting Scheme
    5. 3.5 PMT Hosting Assembly
    6. 3.6 Window Design
  4. 4 Mirror Selection
    1. 4.1 Introduction
    2. 4.2 $X$, $Y$, $Z$ Coordinate Measurement
      1. 4.2.1 Methodology
      2. 4.2.2 Fitting Results
      3. 4.2.3 Fit$-$Measurement Plots
      4. 4.2.4 Uncertainty of Coordinate Measurements
      5. 4.2.5 Coordinate Measurement Result Summary
    3. 4.3 Optical Test
      1. 4.3.1 Methodology
      2. 4.3.2 Optical Test Results
      3. 4.3.3 Optical Test Summary
    4. 4.4 Combined Result
    5. 4.5 Conclusion
  5. 5 Mirror Reflectivity Measurement
    1. 5.1 Introduction
    2. 5.2 Methodology
      1. 5.2.1 Equipment list
      2. 5.2.2 Monochromator
      3. 5.2.3 Setup
      4. 5.2.4 Methodology
      5. 5.2.5 Detector and Signal Schematic Diagram
      6. 5.2.6 Lock-in Method and Monochromator Scanning Setting
    3. 5.3 Results
      1. 5.3.1 Deuterium UV Lamp Thermal Test Results
      2. 5.3.2 UV Detector Response Uniformity
      3. 5.3.3 DUV Flipper Mirror Reflectivity Result
      4. 5.3.4 HGC Mirror #8 Reflectivity Results
    4. 5.4 Conclusion
    5. 5.5 Suggestions for Future Measurement
  6. 6 Geant Simulation
    1. 6.1 Introduction
    2. 6.2 Simulation Setup
    3. 6.3 Particle and Čerenkov Envelopes
    4. 6.4 PMT Optimization
    5. 6.5 Projected Performance
      1. 6.5.1 Čerenkov Photon Detection for $\pi^{+}$ Events
      2. 6.5.2 Signal Threshold
    6. 6.6 Oblate Ellipsoid Versus Spherical Mirrors
    7. 6.7 Remarks and Outlook
  7. 7 Summary and Future Remarks
  8. A Reflectivity Data
    1. A.1 AXUV-100G Detector Response Curve
    2. A.2 Typical Melles Griots DUV Mirror Reflectivity Curve
    3. A.3 ECI Testing Sample Reflectivity Measurements
  9. B Efficiency Documentation
    1. B.1 Corning 7980 Quartz Transmissivity
    2. B.2 PMT Radial Dependent Efficiency
    3. B.3 PMT Quantum Efficiency

###### List of Figures

  1. 1.1 Schematic Diagram of the Jefferson Lab
  2. 1.2 Hall C 12 GeV Overhead View
  3. 1.3 SHMS Side View
  4. 1.4 SHMS Detector Stack
  5. 2.1 Refractive Index and Aborption vs Frequency Sketch
  6. 2.2 Čerenkov Radiation
  7. 3.1 Momentum vs Pressure Plot
  8. 3.2 Čerenkov Angle Plot
  9. 3.3 HGC Vessel Structure
  10. 3.4 Mirror Support Structure
  11. 3.5 Front View of the HGC Detector
  12. 4.1 Slumping Method Demonstration
  13. 4.2 FaroArm Instrument
  14. 4.3 Mirror Placement During Coordinate Measurement
  15. 4.4 Curvatures At Different Conic Constants
  16. 4.5 Mirror Quality Assessment Plot
  17. 4.6 Mirror $\Delta z$ Example Plots
  18. 4.7 Optical Test Setup
  19. 4.8 (Focal length$-$Fitted Radius/2) vs Conic Constant
  20. 4.9 Return Raw Spot Images
  21. 4.10 Processed Return Spot Images
  22. 5.1 McPherson 218 VUV Monochromator
  23. 5.2 Mirror Reflectivity Setup Images
  24. 5.3 Schematic Diagram of the Reflectivity Measurement Modes
  25. 5.4 Reflectivity Measurement Location
  26. 5.5 Signal Output Diagram
  27. 5.6 Thermal Test Result
  28. 5.7 Flipper Mirror Reflectivity
  29. 5.8 Mirror #8 Reflectivity Results
  30. 5.9 Difference Between ECI and Measured Result
  31. 6.1 Geant4 HGC Detector Model
  32. 6.2 Charged Particle and Photon Envelope
  33. 6.3 Focused Čerenkov Radiation Spot onto PMTs
  34. 6.4 Side view of the Reconstructed Photon Path
  35. 6.5 Focused Spot RMS Size vs. Plane Number
  36. 6.6 Photons and Photo-e Distribution
  37. 6.7 Photo-e Distribution for Different Particles
  38. 6.8 Detection Efficiency Plots
  39. A.1 AXUV-100G Photo-diode Detector Response Curve
  40. A.2 Melles Griots DUV Flipper Mirror Theoretical Reflectivity Curve
  41. A.3 ECI Theoretical Reflectivity Curve
  42. A.4 ECI Witness Sample Reflectivity Measurements: 200-600 nm
  43. A.5 ECI Witness Sample Reflectivity Measurements: 200-250 nm
  44. B.1 Quartz Window Transmissivity
  45. B.2 PMT Radial Dependent Efficiency
  46. B.3 PMT Quantum Efficiency

###### List of Tables

  1. 1.1 HMS and SHMS Specification
  2. 3.1 Threshold Refractive Indices Table
  3. 3.2 Material Refractive Indices Table
  4. 4.1 Coordinate Measurement Fitting Results
  5. 4.2 Mirror Assessment Results
  6. 4.3 Optical Test Results
  7. 4.4 Pixel Counting Results
  8. 4.5 Overall Mirror Quality Summary
  9. 5.1 Key Parameters for the Reflectivity Measurement
  10. B.1 Corning 7980 Quartz Transmissivity and Absorption Length

##  Chapter 1 Introduction

The purpose of this thesis is to describe the design, mirror quality control
and projected performance of the Heavy Gas Čerenkov (HGC) detector constructed
at the University of Regina. The HGC detector is an important part of the 12
GeV upgrade installation, used for particle identification in experimental
Hall C at the Thomas Jefferson National Accelerator Facility (Jefferson Lab).

In this chapter, a brief description of Jefferson Lab and the 12 GeV upgrade
project is given. The Hall C experimental apparatus and its upgrade plan is
then described in more detail. Chapter 2 contains a discussion of Čerenkov
radiation in terms of classical electrodynamics. Chapter 3 describes the
specification and design of the HGC detector. The mirror quality control
studies are presented in Chapters 4 and 5. Chapter 4 gives the detailed
description on mirror selection methodology and Chapter 5 presents the
aluminized mirror reflectivity results. Chapter 6 discusses the computer
simulation of the detector performance, and Chapter 7 concludes with a summary
and outlook.

###  1.1 CEBAF and the 12 GeV Upgrade

The Thomas Jefferson National Accelerator Facility (Jefferson Lab) houses the
world’s largest superconducting radio frequency (RF) linear accelerator named
CEBA (Continuous Electron Beam Accelerator). Its reliable continuous electron
beam has become the most effective method to probe the quark-gluon nature of
nuclear matter at high energy.

The CEBA consists of an electron injector, a pair of superconducting linear
accelerators and several bending arcs. The electrons are generated in the
injector, then are grouped into micro-bunches and released into the north
Linac at an energy of 45 MeV with 0.667 ns separation, which corresponds an RF
repetition rate of 1497 MHz. The north and south Linacs have identical design,
each hosting 20 cryomodules with an accelerating gradient of 5 MeV/m. The
Linacs are connected by arcs at both ends so that the entire setup looks like
a racetrack ring (see Fig. 1.1). Each electron may complete as many as five
circles in the racetrack ring before reaching the experimental area; the
maximum energy gain is 1.2 GeV per pass, thus the maximum energy gain for an
electron is 6 GeV. The electron bunches are in turn delivered to each of the
three experimental halls every 2 ns. The current setting of CEBA produces 6
GeV maximum energy at with beam current of 200 mA.

Jefferson Lab currently has three experimental halls. Each of the experimental
halls have different scientific objectives and therefore are equipped with
different experimental apparatus. A short description of each is given below:

HALL A:

A high resolution hall, equipped with a pair of High Resolution Spectrometers
(HRS) which are optimized to study nuclear structure with high precision.

HALL B:

A large solid angle hall, designed to provide full 4$\pi$ solid angle coverage
and broad momentum range for capturing produced charged and neutral particles
produced in nuclear interactions.

HALL C:

A multi-purpose high luminosity hall, equipped with a High Momentum
Spectrometer (HMS) and a Short Orbit Spectrometer (SOS). The specialty of Hall
C is its capability to study rare interactions at high event rate. The
probability of the event detection is directly proportional to the maximum
beam current, where in Hall C it can reach as high as 180 mA which is 90% of
the total beam current.

Jefferson Lab is currently undertaking a 12 GeV upgrade project to double its
accelerating energy and to improve the instrumentation in each of the
experimental halls. The upgrade project will replace three of the existing
cryomodules and install five new cryomodules in each of the accelerators and
improve the energy gain from 1.2 GeV to 2.2 GeV per pass, thus delivering up
to 11 GeV beam to the existing halls. As a result, the maximum electron beam
current after the 12 GeV upgrade will be halved to 100 mA to conserve the
total power output of the accelerator. A new experimental hall, namely Hall D,
will be built to extend the physics program at Jefferson Lab to search
principally for exotic hybrid mesons. A new bending arc is installed to guide
the electron beam to Hall D. Thus, this hall will make use of 5.5 pass
electron beam, with a maximum beam energy of 12 GeV.

Figure 1.1: Schematic diagram of the Jefferson Lab and the 12 GeV upgrade [2].
(Original In Color).

###  1.2 Physics Program at Hall C After 12 GeV Upgrade

Hall C will play a vital role in the overall Jefferson Lab physics program
after 12 GeV upgrade. The planned physics experiments in Hall C require
spectrometers with acceptance for very forward-going particles with momenta
approaching that of the incoming beam (11 GeV/c). Their focal plane detectors
must provide excellent particle identification even at these high energies.
They must be capable of rapid, accurate changes to the kinematic settings with
well understood acceptances allowing experiments to efficiently cover broad
regions of phase space, enabling, for example, precise L/T separations. And
they must possess efficient, highly time-resolved trigger systems and target
and data-acquisition systems suitable for running at high luminosity.

These features are essential for studies such as the pion form factor
experiment [5], which requires precise L/T separations. The long-term interest
in measuring the charged pion form factor ($F_{\pi}$) is due to the rigorous
and unique prediction made by the theory of Quantum Chromodynamics at
asymptotic values of Q2, which indicates the energy limitation where valence
quarks behave as free particles. The pion is ideal choice for this study
because the smaller number of valence quarks in the pion means that the
asymptotic regime will be reached at lower values for $F_{\pi}$ than for the
nucleon form factors. The high quality, continuous electron beam of Jefferson
Lab makes it the only place to seriously pursue these measurements. Other
important examples of the Hall C physics program are color transparency [6],
duality [7] and nucleon form factor measurements [8, 9]. These experiments
will contribute to improve our current understanding of the nucleon structure.

###  1.3 Experimental Hall C

The current permanent setup of Hall C consists of a High Momentum Spectrometer
(HMS) and a Short Orbit Spectrometer (SOS). A Super High Momentum Spectrometer
(SHMS) currently under construction will replace the SOS, becoming the
standard experimental setup coupled with the HMS after the 12 GeV Upgrade.

The Hall C target station is located at the fixed central bearing of the two
spectrometers. The target ensemble consists of a three-loop cryogenic target
stack together with an optics target assembly. The latter is designed for the
calibration of the optics of the spectrometers. The target ensemble is mounted
inside a vacuum scattering chamber in such a way that the stack of cryogenic
cells and optics target can be moved up and down as a whole. The cylindrical
scattering chamber has an inner radius of 61.6 cm and a height of 150 cm. The
spectrometers are not vacuum-coupled to the scattering chamber.

Figure 1.2: Overhead view of standard experiment configuration for Hall C
after 12 GeV upgrade. Key components at labeled on the figure [3]. (Original
In Color).

####  1.3.1 Spectrometer Coordination Definition

The coordinate system used throughout the thesis is based on the axes
convention for charged particle transport in dispersive magnetic systems. The
central ray particles in the SHMS will be bent by 18.4∘ (Table 1.1) vertically
by the superconducting dipole before reaching the detector stack, where the
$z$ axis direction will remain parallel to the central ray after the bending,
so that the $z$ direction is changed by the dipole bending angle. With a
constant dipole magnetic field, the bend angle is smaller for particles with
higher momenta assuming the same particle masses. As a result, when particles
exit the dipole, they will form an envelope with lower momenta particles on
the top quadrants and higher momenta particles on the bottom quadrants. The
direction of the $x$ axis follows the increase in particle momenta (low to
high) according to the transport convention. Thus, $+x$ points down with
respect to the HGC detector frame, which is 18.4∘ from the vertical axis of
the global spectrometer frame. From the direction of the $x$ and $z$ axes. It
can be deduced that the $y$ axis ($\vec{y}=\vec{x}\times\vec{y}$) points away
from the electron beam line (right to left with respect to the beam envelope
after bending).

####  1.3.2 Super High Momentum Spectrometer (SHMS)

The SHMS is an 18.4∘ vertical bend spectrometer and has a maximum central
momentum of 11 GeV/c. The optical design of the SHMS is similar to the HMS and
consists of three super-conducting magnetic quadrupoles ($Q1,Q2,Q3$) and one
vertically bending dipole ($D$). The purpose of the dipole $D$ is to select
the incoming particles depending on their electrical charge and momentum. The
purpose of the quadrupoles is to focus the particles into the dipole,
improving the angular acceptance (solid angle) of the spectrometer. The
combined action of the quadrupoles and dipole is to disperse the charged
particles according to momenta at the focal plane, which is located 18.1 m
from the target station.

The SHMS horizontal scattering angle is as small as 5.5∘ with an acceptance of
$\pm$1.3∘, thus the smallest detectable angle is 4.3∘ from the electron beam
path. In order to achieve such a small scattering angle, the SHMS is in
addition equipped with a horizontal bending super-conducting magnet ($d$) in
front of the first quadrupole $Q1$ and behind of the target station, which can
bend most produced changed particles by 3∘. Thus, the main spectrometer
assembly is centered about 8.5∘. To summarize, the optical configuration of
the SHMS is $dQQQD$. The vertical bending angle is 18.4∘ of the $D$ dipole.
The important performance specifications are listed in Table 1.1. A side view
of the SHMS is shown in Fig. 1.3.

Due to the spatial constraints, the vertical bending dipole $D$ extends into
the shield house of SHMS. The shield house has an asymmetrical trapezoid shape
and is sub-divided into a detector hut and an electronic hut. The detector hut
hosts the focal plane detector stack and the electronic hut protects the
critical electronic modules used for the data acquisition. The two huts are
separated by a 50 cm concrete wall and 2.3 mm of boron/lead; the back and
right sides of the shield house are protected by 64 cm concrete and 2.3 mm
boron/lead; the top and bottom are protected by 50 cm and 70 cm concrete,
respectively, and 2.3mm lead/boron; the left side (close to beam line) has 90
cm concrete and the front side has 100 cm concrete wall. In addition, the left
and front sides have 5 cm lead and boron shield. The top shield bricks are
removable to allow overhead crane access to the detector and electronic huts.

Figure 1.3: Side view of SHMS: $dQQQD$ superconducting magnets configuration
and detector hut geometry [4]. The horizontal bending bending dipole $d$ is
labeled as ‘bender’ in the schematic. (Original In Color).

####  1.3.3 High Momentum Spectrometer (HMS)

The HMS has a $QQQD$ superconducting magnet configuration and has been used
for experiments in Hall C since 1994. The spectrometer has a 25∘ bend in
vertical direction for the central ray particles. The maximum central momentum
is 7.3 GeV/c. The horizontal scattering angle range is between 10.5∘ and 85∘
with $\pm$1.576∘ horizontal acceptance. This corresponds to a smallest
measurable angle of 8.92∘.

All the magnets are supported by a single carriage, which can be moved on
rails around a fixed central bearing near the target station. The quadrupoles
can be moved as a group along the optical axis ($z$ axis). The shield house is
on a separate carriage than the magnets. However, the two carriages are
coupled together. Compared to the SHMS, the HMS shield has a rectangular shape
with much more space for the detectors. The shield roof is not removable. The
left side (away from the beam line) of the shield house is made of concrete
strips, which can be removed to allow access to the detector stack. The
important performance specifications are listed in Table 1.1.

Table 1.1: HMS and SHMS specification [13]. Quantity | Specification
---|---
| HMS | SHMS
Dipole Bend Angle | 25∘ | 18.4∘
Maximum Central Momentum | 7.4 GeV/c | 11 GeV/c
Focal Length | 26.0 m | 18.1 m
Scattering Angular Range | 10.5∘ to 85∘ | 5.5∘ to 40∘
Momentum Acceptance |  $\pm$10% | -10%$<\delta<$+22%
Momentum Resolution |  $<$0.1% | 0.03%-0.08%
Solid Angle Acceptance | 6.7 msr | 4.0 msr
Horizontal Acceptance |  $\pm$27.5 mrad |  $\pm$ 24 mrad
Vertical Acceptance |  $\pm$70 mrad |  $\pm$ 40 mrad
Horizontal Resolution | 0.8 mrad | 0.5-1.2 mrad
Vertical Resolution | 0.9 mrad | 0.3-1.1 mrad
Target Vertex Length |  $\pm$7 cm |  $\pm$15 cm
Target Vertex Reconstruction Accuracy | 1 mm | 0.1-0.3 mm
Figure 1.4: SHMS detector stack layout inside of the detector hut after the
superconducting dipole. Charged particles focused by the SHMS magnetic
elements traverse the stack from right to left in this figure [10]. (Original
In Color).

####  1.3.4 SHMS and HMS Detector Packages

The focal plane instrumentation of the SHMS and HMS consists of a series of
detectors, which are designed to work together to reliably reconstruct the
particle identity and trajectory information of all tracks intercepting them
over a broad momentum range. These individual detectors will be described in
the order that they are traversed by an incident particle.

##### Noble Gas Čerenkov Detector (NGC)

The noble gas Čerenkov (NGC) detector is only used in the SHMS to separate
electrons from heavier charged particles at high central momentum where $p>$ 6
GeV/c. Due to the spatial constraints, it is located in front of the focal
plane, whereas ideally it should be placed behind, so as to not adversely
affect the reconstructed momentum resolution. For experiments with lower
central momenta, the NGC is replaced with a vacuum tank of the same length to
eliminate a source of multiple scattering. The detector is filled with Ar gas
at momentum $p<$ 5.5 GeV/c and He gas at $p>$ 5.5 GeV/c, with the operating
pressure being 1 atm over the full momentum range. The detector housing is
unique due to its rectangular body shape and continuous gas circulation is
required to eliminate O2 contamination. The Čerenkov radiation is reflected by
four curved mirrors and focused onto four PMTs near the top and bottom of the
detector. Due to the good UV transmission characteristics of noble gasses, the
Čerenkov emission band can reach as low as 140 nm wavelength. Such deep UV
light is extremely difficult to detect since the PMT quantum efficiency falls
dramatically below 200 nm wavelength. One possibility being explored is to use
a wavelength shifting technique to increase the Čerenkov radiation wavelength
from the deep UV to a PMT detectable region. The NGC detector is currently
being constructed at the University of Virginia.

##### Drift Chambers

The drift chambers are used to measure the horizontal and vertical angles and
positions of the charged particles before and after the focal plane, in order
to determine their momentum and trajectory. The basic operation principle is
as follows: charged particles induce ionization of the gas atoms inside the
chamber and the free electrons produced due to the ionization process are
captured by sense wires. Good spatial resolution is achieved by measuring the
electron drift time. The electric field inside of the chambers needs to be of
a very specific configuration, which is achieved by surrounding the sense
wires with non-sensed wires at high voltage. The trajectory information of the
two chambers is combined to determine the track of the charged particles
through the focal plane.

Both the SHMS and HMS are equipped with a pair of drift chambers. The focal
plane is sandwiched in between the chambers. Each drift chamber contains six
planes of sense wires. In the SHMS, the wire planes are ordered $u$,
$u^{\prime}$, $x$, $x^{\prime}$, $v$, $v^{\prime}$. There are no $y$ planes,
and $u$ and $v$ plane wires are at $\pm$30∘ with respect to $x$ plane. The
cell spacing is 1 cm and the position resolution is approximately 200 $\mu$m
per plane. As a result, the $y$ resolution of SHMS detector is better than in
the HMS. The two chambers are placed at a distance of 40 cm before and after
the focal plane, respectively. The SHMS drift chambers are currently under
construction at Hampton University.

In the HMS, the wire planes are ordered $x$, $y$, $u$, $v$, $y^{\prime}$,
$x^{\prime}$. The $x$ and $y$ planes measure the vertical and horizontal track
position. $u$ and $v$ plane wires are at $\pm$15∘ with respect to the $x$
plane. The cell spacing is 1 cm and the position resolution is approximately
150 $\mu$m per plane. The HMS has better resolution in the $x$ direction. The
two chambers are placed at a distance of 40 cm before and after the focal
plane.

##### Hodoscopes

In the SHMS, the hodoscopes are only used to generate the trigger for the data
acquisition system. The S1 hodoscopes (see Fig. 1.4) are located before the
heavy gas Čerenkov detector and the S2 are positioned after the Aerogel
Čerenkov detector. Each hodoscope consists of two planes, each plane is read
out by PMTs at both ends. The first plane of each hodoscope is in the
horizontal (S1X, S2X), and second plane is in the vertical direction (S1Y,
S2Y). S1X, S1Y and S2X are made of plastic scintillator and S2Y is made of
quartz. This configuration is used to provide additional confirmation on the
particle identification and reduce the background signal. High momentum
neutron background from the beam dump or beam pipe has high probability to
scatter a proton free, then this secondary proton will generate a strong
signal inside the scintillator detectors through ionization process. However,
in the quartz hodoscope, the secondary proton will have low probability to
generate a Čerenkov signal which exceeds the threshold. The quartz hodoscope
consists of 21 quartz bars, each measuring 125 cm $\times$ 5.5 cm $\times$2.5
cm. There is 0.5 cm overlap between each quartz bar and the total active area
is 205.5 cm$\times$ 115 cm. Each S1 scintillator hodoscope panel consists of
13 paddles, each measuring 100 cm $\times$ 9.8 cm $\times$ 0.5 cm. The S2
scintillator hodoscope panel consists of 14 paddles of the same dimension. The
overlap of the scintillator paddles are 0.5 cm for the S1 and S2 hodoscopes.
Scintillation hodoscopes are being constructed at James Madison University and
the quartz hodoscope is being constructed at the North Carolina A&T
University.

In the HMS, the order of the planes is reversed. Each of the hodoscope panels
in HMS is 1.0 cm thick and 8 cm wide, with 0.5 cm overlap. The HMS hodoscopes
serve two purposes: generating the trigger for the data acquisition system and
measuring the time-of-flight (TOF) to determine the particle velocity. At low
central momenta, the particles’ velocity deviates significantly depending on
the mass. However, at higher momenta the deviation is much smaller as $v$
approaches the light speed, therefore the difference in the TOF would be less
than the hodoscope timing resolution for most particles. For this reason, the
TOF is not expected to be very useful in the SHMS. The data acquisition
systems in the SHMS and HMS are triggered by the coincidence signal from at
least three out of four hodoscope planes. This is to allow the hodoscope
inefficiency to be measured.

##### Heavy Gas Čerenkov Detector (HGC)

The SHMS HGC detector is used to distinguish charged pions from heavier
charged particles. It is filled with C4F8O gas having a refractive index of
1.0014 at a pressure of 1 atm. The detector pressure will vary depending on
the central momentum ($p$) setting of the SHMS for the incoming particles: for
3$<p<$7 GeV/c, detector pressure is set to 0.95 atm; for $p>7$ GeV/c the
detector pressure is reduced. This will be discussed in more detail in Chapter
3. The Čerenkov radiation is reflected by four curved mirrors and focused onto
four PMTs placed on top and bottom of the detector. This detector is being
constructed at the University of Regina.

The role of the HMS HGC detector is different from that in the SHMS, where it
is used to distinguish electrons from other charged particles and is filled
with C4F10 gas at a pressure of 0.78 atm. The resulting refractive index of
1.0011 corresponds to a pion threshold of 3 GeV/c. The Čerenkov light is
reflected by two parabolic mirrors and focused onto two PMTs on the top and
bottom of the detector.

##### Aerogel Čerenkov Detector (ACD)

In the SHMS, the Aerogel Čerenkov Detector (ACD) can be used for pion/kaon
($\pi/K$) separation at low momenta or kaon/proton ($K/p$) separation at high
momenta. The aerogel panel dimension is 90 cm $\times$ 60 cm with a thickness
of 5-10 cm and is constructed of small aerogel bricks. Two sets of aerogel
panels with different refractive indices ($n$=1.030 and $n$=1.015) will be
used for the particle identification at different momenta. The signal readouts
are on both sides. The overall detector dimension is 110 cm $\times$ 100 cm
$\times$ 30 cm. The detector is being constructed jointly by the Catholic
University of America, the University of South Carolina, Florida International
University and the Yerevan Physics Institute.

In the HMS, the aerogel is used for pion-proton separation. The aerogel panel
dimension is 117 cm $\times$ 67 cm with a thickness of 9.5 cm and is being
constructed of 650 small aerogel bricks. The overall detector dimension is 120
cm $\times$ 70 cm $\times$ 34.5 cm.

##### Electromagnetic Calorimeter (ECAL)

The electromagnetic calorimeters are comprised of lead-glass and are used to
provide additional confirmation on the electron-hadron identification. The
SHMS calorimeter consists of two parts: pre-shower and shower. The pre-shower
is constructed using 28 lead glass blocks formerly used in the SOS, each
measuring 10 cm$\times$10 cm$\times$70 cm. The PMT signal readouts are on both
sides. The shower is constructed with 224 lead glass blocks formerly used in
the HERMES experiment at DESY, each measuring 9 cm$\times$ 9 cm$\times$ 50 cm.
The shower blocks are stacked perpendicularly to the pre-shower blocks and the
signal readouts are on the backside. The whole SHMS calorimeter is 140 cm
wide, 144 cm tall and 60 cm thick; total number of readout channels are 252.
The calorimeter will be constructed jointly by the Yerevan Physics Institute
and Jefferson Lab.

The HMS calorimeter is built out of blocks of lead glass, which have
dimensions 10 cm$\times$ 10 cm$\times$ 70 cm. The detection stack is four
layers deep and 13 layers tall. The signal readout is only on one side.

##  Chapter 2 Čerenkov Radiation

All equations in this chapter expressed are in Gaussian Units.

The interactions of charged particles with material take many forms. In this
chapter the properties of Čerenkov radiation are discussed in detail. Several
of these properties are highly relevant for the design and projected
performance of the HGC detector.

Čerenkov radiation is the electromagnetic radiation emitted by charged
particles when their velocity exceeds the light velocity ($c/n$) in a
dielectric medium with refractive index $n$. The moving charged particles
electrically polarize the molecules inside the medium, which then turn back
rapidly to the ground state. The polarization-depolarization process generates
an oscillating electric dipole of angular frequency $\omega$, which emits
electromagnetic radiation as a result.

To determine the energy loss between the charged particle and the dielectric
medium, the fields in the medium are calculated assuming that the mass and
charge of the atoms are uniformly and continuously distributed with a
macroscopic relative permittivity $\epsilon_{r}(\omega)$.
$\epsilon_{r}(\omega)$ can be defined as

$\epsilon_{r}(\omega)=\frac{\epsilon(\omega)}{\epsilon_{0}}$

where $\epsilon(\omega)$ is the absolute permittivity, the measure of the
resistance that is encountered when an electric field forms in a medium, and
$\epsilon_{0}$ is the vacuum permittivity.

The problem of finding the electric field in the medium due to the incident
fast particle moving with the constant velocity can be solved by Fourier
transforms. If the potential component $A_{\mu}(x)$ and the source density
component $J_{\mu}(x)$ are transformed in space and time according to the
general rule:

$F(\vec{x},t)=\frac{1}{(2\pi)^{2}}\int d^{3}k\int
d\omega\,F(\vec{k},\omega)\,e^{\,i\vec{k}\cdot\vec{x}-i\omega t}\,$ (2.1)

where $k$ is the wavenumber. The transformed wave equations become

$\begin{array}[]{l}\displaystyle\left[k^{2}-\frac{\omega^{2}}{c^{2}}\epsilon_{r}(\omega)\right]\Phi(\vec{k},\omega)=\frac{4\,\pi}{\epsilon_{r}(\omega)}\,\rho(\vec{k},\omega)\\\\[14.22636pt]
\displaystyle\left[k^{2}-\frac{\omega^{2}}{c^{2}}\epsilon_{r}(\omega)\right]\vec{A}(\vec{k},\omega)=\frac{4\,\pi}{c}\,\vec{J}(k,\omega)\,.\end{array}$
(2.2)

where the $\Phi$ is the scalar potential, $\rho$ is the change density,
$\vec{A}$ is the vector potential and $\vec{J}$ is the current density. The
Fourier transforms of

$\begin{array}[]{l}\displaystyle\rho(\vec{x},t)=ze\,\delta(\vec{x}-\vec{v}t)\\\\[14.22636pt]
\displaystyle\vec{J}(\vec{x},t)=\vec{v}\,\rho(\vec{x},t)\,.\end{array}$ (2.3)

are readily found to be

$\begin{array}[]{l}\displaystyle\rho(\vec{k},\omega)=\frac{ze}{2\pi}\,\delta(\omega-\vec{k}\cdot\vec{v})\\\\[14.22636pt]
\displaystyle\vec{J}(\vec{k},\omega)=\vec{v}\,\rho(\vec{k},\omega)\end{array}$
(2.4)

where $\vec{v}$ is the velocity of the charged particle.

From (2.2), the Fourier transforms of the potentials are

$\begin{array}[]{l}\displaystyle\Phi(\vec{k},\omega)=\frac{2ze}{\epsilon_{r}(\omega)}\,\frac{\delta\,(\omega-\vec{k}\cdot\vec{v})}{k^{2}-\frac{\omega^{2}}{c^{2}}\,\epsilon_{r}(\omega)}\\\\[14.22636pt]
\displaystyle\vec{A}(\vec{v},\omega)=\epsilon_{r}(\omega)\,\frac{\vec{v}}{c}\,\Phi(\vec{k},\omega)\end{array}$
(2.5)

The electromagnetic fields in terms of the potentials are

$\begin{array}[]{l}\displaystyle\vec{E}=-\overrightarrow{\nabla}\Phi-\frac{\partial\vec{A}}{\partial
t}\\\ \displaystyle\vec{B}=\overrightarrow{\nabla}\times\vec{A}\,,\end{array}$
(2.6)

substitute (2.5) and obtain:

$\begin{array}[]{l}\displaystyle\vec{E}(\vec{k},\omega)=i\left[\frac{\omega\epsilon_{r}(\omega)}{c}\,\frac{\vec{v}}{c}-\vec{k}\right]\Phi(\vec{k},\omega)\\\\[14.22636pt]
\displaystyle\vec{B}(\vec{k},\omega)=i\,\epsilon_{r}(\omega)\vec{k}\times\frac{\vec{v}}{c}\,\Phi(\vec{k},\omega)\,.\end{array}$
(2.7)

In calculating the energy loss, the Fourier transform in time of the
electromagnetic fields is calculated at a perpendicular distance $b$ from the
path of particle moving along $z$ axis. The required electric field is of the
form:

$\vec{E}=\frac{1}{(2\pi)^{3/2}}\int
d^{3}k\,\vec{E}(\vec{k},\omega)\,e^{ibk_{2}}\,,$ (2.8)

where the observation point has coordinates (0, $b$, 0). Substitute (2.5) and
(2.7) into (2.8), and integrate over first component $dk_{1}$ to obtain:

$\begin{array}[]{lcl}\displaystyle
E_{1}(\omega)&=&\displaystyle-\frac{ize\omega}{\sqrt{2\pi}\,v^{2}}\left[\frac{1}{\epsilon_{r}(\omega)}-\beta^{2}\right]\,\int_{-\infty}^{\infty}\\!\,\,\frac{e^{ibk_{2}}}{(\lambda^{2}+k_{2}^{2})^{1/2}}\,dk_{2}\\\\[14.22636pt]
\displaystyle&=&\displaystyle-\frac{izew}{v^{2}}\left(\frac{2}{\pi}\right)^{1/2}\left[\frac{1}{\epsilon_{r}(\omega)-\beta^{2}}\right]K_{0}(\lambda
b)\\\\[14.22636pt] \displaystyle
E_{2}(\omega)&=&\displaystyle\frac{ze}{v}\,\left(\frac{2}{\pi}\right)^{1/2}\frac{\lambda}{\epsilon_{r}(\omega)}\,K_{1}(\lambda
b)\\\\[14.22636pt] \displaystyle
B_{3}(\omega)&=&\displaystyle\epsilon_{r}(\omega)\,\beta\,E_{2}(\omega)\end{array}$
(2.9)

where $\beta^{2}=v^{2}/c^{2}$, $K_{0}(\lambda b)$ and $K_{1}(\lambda b)$ are
modified Bessel functions. $\lambda^{2}$ can be written as

$\lambda^{2}=\frac{\omega^{2}}{v^{2}}-\frac{\omega^{2}}{c^{2}}\epsilon_{r}(\omega)=\frac{\omega^{2}}{v^{2}}[1-\beta\epsilon_{r}(\omega)]\,.$
(2.10)

If we apply the far field approximation, where $|\lambda b|\gg 1$, the Bessel
functions can be estimated by their asymptotic forms, and the fields (2.9)
become

$\begin{array}[]{l}\displaystyle E_{1}(\omega,b)\rightarrow
i\,\frac{ze\omega}{c^{2}}\left[1-\frac{1}{\beta^{2}\,\epsilon_{r}(\omega)}\right]\,\frac{e^{-\lambda
b}}{\sqrt{\lambda b}}\\\\[14.22636pt] \displaystyle
E_{2}(\omega,b)\rightarrow\frac{ze}{v\,\epsilon_{r}(\omega)}\,\sqrt{\frac{\lambda}{b}}\,e^{-\lambda
b}\\\\[14.22636pt] \displaystyle
B_{3}(\omega,b)\rightarrow\rho\,\epsilon_{r}(\omega)\,E_{2}(\omega,b)\,.\end{array}$
(2.11)

The energy deposited per unit length along of the charged particle path is

$\left(\frac{d\rm{E}}{dx}\right)_{b>a}=-ca\leavevmode\nobreak\
\textrm{Re}\int_{0}^{\infty}B_{3}^{\ast}(\omega)E_{1}(\omega)\,d\omega$ (2.12)

where $a$ is the cylinder radius around the path of the incident particle.
Take the RHS integral in the far field limit and obtain

$\left(\frac{d\rm{E}}{dx}\right)_{b\rightarrow\infty}\rightarrow\frac{z^{2}e^{2}}{c^{2}}\left(-i\sqrt{\frac{\lambda^{\ast}}{\lambda}}\right)\omega\left[1-\frac{1}{\beta\epsilon_{r}(\omega)}\right]e^{-(\lambda+\lambda^{\ast})a}$
(2.13)

where $\lambda^{\ast}$ is the complex conjugate of $\lambda$. The real part of
this expression gives the energy deposited far from the path of the particle.
If $\lambda$ has a positive real part, as is generally true [14], the
exponential factor in (2.13) will cause the expression to vanish rapidly at
large distances, therefore all the energy is deposited near the path. This is
not true when $\lambda$ is purely imaginary. In this case, the exponential
term is 1 and the expression is independent of distance $a$; some energy
escapes to infinity as electromagnetic radiation. From (2.10) it can be seen
that $\lambda$ can be purely imaginary if $\epsilon_{r}(\omega)$ is real (no
absorption) and $\beta^{2}\epsilon_{r}(\omega)>1$, which can be expressed in
the more transparent form

$v>\frac{c}{\sqrt{\epsilon_{r}(\omega)}}=\frac{c}{n(\omega)}\,,$ (2.14)

since the refractive index of a medium can be written as

$n=\sqrt{\epsilon_{r}(\omega)\mu_{r}}$ (2.15)

where $\mu_{r}$ is the relative permeability of the material, and $\mu_{r}=1$
for most natural existing materials at optical frequencies [11]. Equation
(2.14) shows that the velocity of the particle must be larger than the phase
velocity of the electromagnetic fields at frequency $\omega$ in order to emit
Čerenkov radiation at that frequency.

To determine the frequency dependence of the Čerenkov radiation, the relative
permeability can be expressed as [14]

$\epsilon_{r}(\omega)\simeq 1+\frac{4\pi
Ne^{2}}{m}\sum_{j}\,\frac{f_{j}}{\omega_{j}^{2}-w^{2}-i\gamma_{j}\omega}\,,$
(2.16)

where $N$ is the number of molecules per unit volume, $\omega_{j}$ is the
natural resonant frequency of the molecule, $\gamma_{j}$ is the damping ratio
of electromagnetic field in each molecule and $f_{j}$ is the number of
electrons that have the resonant frequency $\omega_{j}$ and damping ratio
$\gamma_{j}$. By substituting Equation (2.16) into Equation (2.15) and
applying the Taylor series expansion one obtains

$n=\sqrt{\epsilon_{r}(\omega)}\simeq 1+\frac{2\pi
Ne^{2}}{m}\sum_{j}\,\frac{f_{j}(\omega_{j}^{2}-\omega^{2})}{(\omega_{j}^{2}-\omega^{2})^{2}+\gamma_{j}^{2}\omega^{2}}\,.$
(2.17)

Fig. 2.1 shows a sketch of the real part of Equation (2.17) versus $\omega$.
The Čerenkov radiation emission band is shaded in red, where $n>\beta^{-1}$.
The absorption coefficient ($\alpha$) of the dielectric medium also depends on
$\omega$ and can be written as [15]

$\alpha\simeq\frac{4\pi
Ne^{2}\omega^{2}}{mc}\sum_{j}\,\frac{f_{j}\gamma_{j}}{(\omega_{j}^{2}-\omega^{2})^{2}+\gamma_{j}^{2}\omega^{2}}\,.$
(2.18)

The absorption curve will become dominant at $\omega>\omega_{1}$ as shown in
Fig .2.1, where the emitted Čerenkov radiation will be absorbed by the medium

Figure 2.1: Sketch of the index of refraction ($n$) and absorption ($\alpha$)
versus angular frequency $\omega$. The index of refraction curve is in blue;
the absorption curve is in black; the Čerenkov radiation band is shaded in
red; the absorption band is shaded in grey; the red solid line indicates
$n=\beta^{-1}$; the dashed blue line indicates $n=1$. $\omega_{1}$ and
$\omega_{2}$ represent the lower and upper boundary for anomalous dispersion
region, where the $n$ drops sharply. Note that the anomalous dispersion region
coincides with the maximum absorption region. (Original In Color).

The direction of propagation of the Čerenkov radiation is given by
$\vec{E}\times\vec{B}$. The emission angle relative to the velocity of the
charged particle is given by:

$\tan\theta_{C}=-\frac{E_{1}}{E_{2}}\,.$ (2.19)

Substitute far fields Equation (2.11) into Equation (2.19) to obtain

$\cos\theta_{C}=\frac{1}{\beta\sqrt{\epsilon_{r}(\omega)}}=\frac{1}{\beta
n}\,.$ (2.20)

Fig. 2.2 is a sketch of one set of spherical wavelets radiated by a charged
particle traveling faster than the speed of light in the medium. For
$v>c/\sqrt{\epsilon_{r}}$, a synchronized electromagnetic “shock” wavefront
occurs, moving in the direction given by the Čerenkov angle. The black circles
indicate the expanding wavefronts of electromagnetic radiation emitted at
previous times. When the particle travels faster than the speed of light in
the medium, the shock fronts coherently add along the Čerenkov cone with angle
$\theta_{C}$.

The energy radiated by the Čerenkov process per unit distance along the path
of the charged particle is [14]:

$\left(\frac{d\rm{E}}{dx}\right)_{rad}=\frac{(ze)^{2}}{c^{2}}\int_{\epsilon_{r}(\omega)>(1/\beta^{2})}\omega\left(1-\frac{1}{\beta^{2}\epsilon_{r}(\omega)}\right)d\omega\,.$
(2.21)

Čerenkov Radiation Charged Particle
---
Trajectory

Figure 2.2: Čerenkov radiation. The black arrows indicate the direction of the
emitted Čerenkov radiation wave front; the red arrow indicates the direction
of the traveling charged particle in the dielectric medium. The black circles
indicate the expanding wavefronts of electromagnetic radiation emitted at
previous times. When the particle travels faster than the speed of light in
the medium, the shock fronts coherently add along the Čerenkov cone with angle
$\theta_{C}$. (Original In Color).

##  Chapter 3 HGC Specification and Design

###  3.1 Detector Design

The interaction between the accelerated electrons and hydrogen nuclei inside
of the target chamber produce a shower of particles such as: electrons (e),
pions ($\pi$), kaons (K), and protons. A small fraction of the produced
particles at specific angles and momenta travels through the apertures inside
the SHMS quadrupoles and dipoles, and eventually reach the detector hut.

Following the 12 GeV upgrade, the charged particles inside of the
spectrometers will have higher momenta and their velocities will approach the
speed of light, so the TOF will be nearly within the resolving time for all
charged particles. The SHMS is designed to detect particles at higher momenta
than the HMS, therefore, more particle identification detectors with different
refractive indices are required to have reliable particle identification over
a wide momentum range.

The purpose of the Heavy Gas Čerenkov (HGC) detector is to provide good
$\pi^{\pm}$ identification over a momentum range of 3-11 GeV/c. Recall from
Chapter 2 the Čerenkov radiation emission threshold condition

$v>c/n\,,$ (3.1)

which can be rewritten in terms of $\beta$ and $n$ as:

$n>\frac{1}{\beta}\,.$ (3.2)

According to special relativity, the momentum ($p$) and energy ($E$) of
relativistic particles are defined as

$\displaystyle\displaystyle p$ $\displaystyle=$ $\displaystyle\gamma mv$
$\displaystyle=$ $\displaystyle\gamma m\beta c$ $\displaystyle\displaystyle E$
$\displaystyle=$ $\displaystyle\gamma mc^{2}$
$\displaystyle\displaystyle\frac{p}{E}$ $\displaystyle=$
$\displaystyle\frac{\gamma m\beta c}{\gamma mc^{2}}$
$\displaystyle\displaystyle\Rightarrow\beta$ $\displaystyle=$
$\displaystyle\frac{pc}{E}\,,$ (3.3)

where $\gamma$ is known as the Lorentz factor, which can be expressed as

$\gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}=\frac{1}{\sqrt{1-\beta^{2}}}\,.$

Since the energy $E$ can also be written as

$E=\sqrt{p^{2}c^{2}+m^{2}c^{4}}\,,$ (3.4)

where $m$ is the mass of the charged particle, expressions (3.2) and (3.3)
become

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{pc}{\sqrt{p^{2}c^{2}+m^{2}c^{4}}}$ (3.5) $\displaystyle n$
$\displaystyle>$ $\displaystyle\frac{\sqrt{p^{2}c^{2}+m^{2}c^{4}}}{pc}\,.$
(3.6)

Table 3.1 shows the Čerenkov threshold refractive indices for different
particles over the SHMS momentum range (3-11 GeV/c). The $n$ values are
directly calculated using Equation (3.6). For 3 GeV/c momentum $\pi^{\pm}$,
the threshold $n$ is 1.00108 and for K is 1.01345. In order to separate them,
the Čerenkov material inside the $\pi$ identification detector has to have a
refractive index value of 1.00108$<n_{detector}<$1.01345. Since the particle
threshold $n$ decreases as the particle central momentum ($p$) increases, the
$n_{detector}$ needs to be adjusted accordingly.

Table 3.1: Threshold refractive indices at different particles with momenta as listed. The particle masses are listed in the second row of the table header. Momentum | $n_{e}$ | $n_{\pi}$ | $n_{K}$ | $n_{Proton}$
---|---|---|---|---
(GeV/c) | (0.5 MeV/c2) | (139.57 MeV/$c^{2}$) | (493.67 MeV/$c^{2}$) | (938.27 MeV/c2)
3 | 1.01435 | 1.00108 | 1.01345 | 1.04778
5 | 1.00519 | 1.00039 | 1.00486 | 1.01745
7 | 1.00265 | 1.00020 | 1.00248 | 1.00894
9 | 1.00160 | 1.00012 | 1.00150 | 1.00542
11 | 1.00107 | 1.00008 | 1.00101 | 1.00363
Table 3.2: List of the refractive indices for different materials [24, 25, 26, 28]. STP stands for Standard Temperature Pressure. | Material | $n$ | $\rho$
---|---|---|---
|  |  | (kg/m-3)
| Vacuum | 1.000000 | 0.000
Gases | Air (STP) | 1.000293 | 1.200
Helium (STP) | 1.000036 | 0.179
Hydrogen (STP) | 1.000132 | 0.090
| C4F10 (STP) | 1.001400 | 11.21
| C4F8O (STP) | 1.001389 | 9.190
Liquids | Water (20∘) | 1.333000 | 1,000
Ethanol (20∘) | 1.360000 | 789.0
Solids | Ice | 1.309000 | 916.7
Fused Silica (Quartz) | 1.460000 | 2,203
Crown Glass | 1.520000 | 2,500
Diamond | 2.420000 | 3,500

From Table 3.2, the desired $n_{detector}$ is similar to the refractive
indices for gases. The C4F8O gas at 0∘C and 1 atmospheric (atm) pressure has
$n$ value of 1.001389, this makes it a suitable Čerenkov medium candidate for
$\pi$-K separation. Note that C4F10 has $n$ value of 1.0014, it is the
historically used gas (e.g. HMS Čerernkov detector). Unfortunately, it has
become too expensive, otherwise the HGC detector would use it again, as its
$n$ is even better than C4F8O.

The C4F8O is known as octafluorotetra-hydrofuran, it is a form of freon and
often referred to as heavy gas. The C4F8O gas density at STP is much higher
than normal atmosphere (air), and it has one of the larger $n$ values for a
gas material. It has been used widely as a Čerenkov medium and its stability
was extensively studied at FermiLab. The conclusion of the study states ‘no
measurable amounts of reaction product were observed between the heavy gas and
other materials in the period of 8 years’ [19], which proves the C4F8O gas is
stable.

The connection between $n$ and the gas pressure ($P$) can be written as

$P=\frac{(n-1)}{(n_{\,1\,\bf{atm}}-1)}$ (3.7)

where $n_{\,1\,\bf{atm}}$ is the refractive index at 1 atm pressure, and $P$
is pressure measured in atm. By controlling the operating pressure ($P$), $n$
can be adjusted to any desired value up to the vapor pressure of the gas. Note
that the vapor pressure of the heavy gas is around 2 atm, above this limit the
gas will start to condense into liquid form.

Fig. 3.1 shows the Čerenkov threshold pressure for C4F8O ($P$) vs the particle
momentum ($p$). From Equation (3.7) $n$ and $P$ are directly proportional,
therefore the shape of the $P$ vs $p$ should be same as the $n$ vs $p$ curve.
The black curve is our recommended operating pressure of the HGC detector.
Between 3 and 7 GeV/c central momentum, the detector is operated at 0.95 atm
($n\approx$1.0001389), which is slightly lower than the atmospheric pressure,
to ensure the atmosphere is acting inwards against the vessel. For particles
with $p>$ 7 GeV/c, the detector pressure has to be reduced to separate $\pi$
and K since $n$ is reduced.

From Table 1.1, the SHMS has momentum acceptance of $-13\%<\delta<22\%$, which
means that at a central momentum ($p_{0}$) of 8 GeV/c, the accepted particle
momentum can range from 6.96 to 9.76 GeV/c. Therefore, it is possible for a
6.96 GeV/c pion and 9.76 GeV/c Kaon to both be within the SHMS momentum
acceptance. The $n_{detector}$ at 8 GeV/c momentum must be set lower than the
Kaon threshold $n$ at 9.76 GeV/c to keep this Kaon from generating Čerenkov
radiation. In addition, $(n_{detector}-1)$ is scaled down by an extra 10$\%$
as a safety factor (in case the detector pressure is mis-set). At 11 GeV/c
momentum, the detector pressure is around 0.35 atm, which implies the HGC
detector must be designed as a vacuum vessel to withstand the pressure
difference.

Figure 3.1: Momentum vs pressure plot. The black curve is the C4F8O gas
pressure of the HGC detector. The colored curves represent the Čerenkov
threshold pressure for different particles [17]. (Original In Color). Figure
3.2: Čerenkov angle vs the momentum plot [17].

###  3.2 Čerenkov Radiation Angle

The corresponding pion Čerenkov radiation angle $\theta_{C}$ is calculated
using Equation (2.20) with $n$ values corresponding to the black curve in Fig.
3.1. Fig. 3.2 shows the $\theta_{C}$ vs momentum plot. $n_{detector}$ is kept
constant between 3 and 7 GeV/c, therefore $\theta_{C}$ gradually increases to
a maximum of 2.84∘ at 7 GeV/c; when $p>$ 7 GeV/c, $n_{detector}$ starts to
decrease therefore $\theta_{C}$ also decreases. Since the larger $\theta_{C}$
results in a larger Čerenkov envelope and more divergent light rays, it is
more difficult for the mirrors to focus all of the light onto the PMTs.
Therefore, all optical alignment studies were carried out at 7 GeV/c momentum,
where the $\theta_{C}$ is maximum.

###  3.3 Detector Structure

The HGC detector is the fourth component in the focal plane detector stack of
the SHMS behind the Noble Gas Čerenkov, the Drift Chambers and the Hodoscope
S1. It is followed by the Areogel Čerenkov, Hodoscope S2 and Lead-glass
Calorimeters. The front of the HGC detector is at 18.8 m from the target
chamber in optical ($z$) axis from the target chamber and thus 0.7 m
downstream of the focal plane. Once all beam envelope clearance and vessel
mechanical issues are taken into account, the detector diameter is 1.829 m.
Its length is 1.3 m, to allow a sufficient length of Čerenkov radiator gas,
and room for the light collection optics to operate efficiently [12]. Due to
the dipole configuration in front of the SHMS detector hut, the HGC detector
has 5 cm offset in $+x$ direction, therefore the detector center in 3D is at
(5 cm, 0, 19.476 m). The detector vacuum tight specification is 10-9 atm and
the operating temperature is 20-25∘C.

The HGC vessel structure is shown in Fig. 3.3. The components include: vessel
cylinder, window flanges, lifting lugs, cradle brackets, gas port and PMT
sleeves. The diameter of vessel cylinder is 1.675 m and the wall thickness is
12.7 mm. There are four PMT sleeves on the top and bottom of the vessel: the
top two sleeves are tilted at 42∘ and bottom two are at $-42^{\circ}$ with
respect to the vertical axis.

Figure 3.3: The HGC vessel structure. Components as labelled: 1 and 4. PMT
sleeves; 2. Gas port; 3. Cradle brackets; 5. Lifting lugs; 6. Window flanges;
7 Vessel cylinder [18]. The particles go through the detector vessel from top
to bottom in this picture. (Original In Color).

###  3.4 Mirrors

The dimension of the resultant Čerenkov envelope at 7 GeV/c momentum is 90 cm
$\times$ 80 cm in the $x$-$y$ plane at the mirror location, and a detailed
investigation was carried out to optimize the mirror-PMT arrangement. In the
end, a design of four concave reflecting mirrors with four 5” PMTs was chosen.
Each mirror has dimension: 60 cm$\times$55 cm with radius of curvature of 110
cm and thickness of 3 mm [12]. In order to prevent any possible gaps at the
joint location, the mirrors are interleaved in the order of mirror #: 4, 3, 2,
1, where mirror #4 is in $-x$, $-y$ quadrant, #3 is in $-x$, $+y$ quadrant, #2
is in $+x$, $-y$ quadrant and #1 is in $+x$, $+y$ quadrant. The closest mirror
to mirror approach is between 7-10 mm. There is a 5 cm overlap between mirror
#1 and #2, the same for mirror #3 and #4 in the $y$ direction. The mirrors are
manufactured by Sinclair Glass [16] using the slumping method. A detailed
description on the mirror slumping and quality control method is given in
Chapter 4.

The generated Čerenkov photon wavelength range is between 200 to 600 nm,
therefore the mirrors need be coated with aluminum grain to reflect 70% deep
UV (200nm) photons. The mirror aluminization vendor is ECI [33], and the
reflectivity test results of the sample mirrors are presented in Chapter 5.

Figure 3.4: Diagram of the mirror support structure [29]. (Original In Color).

####  3.4.1 Window Mounting Scheme

Each mirror is supported along its two outer edges, with one free corner near
the center of the detector overlapping the others, in order to minimize the
amount of material inside of the beam path. The supporting edges are
sandwiched between flexible gaskets, and then locked into the metal clamps by
setscrews. The metal clamps are bolted onto the U-shaped supports which are
shown in Fig. 3.4. The clamps will have long bolt holes to allow small mirror
position and angle adjustments.

A half-inch-thick support ring is introduced to hold the upper and lower
U-shaped supports as shown in Fig. 3.4; the ring can be bolted onto the inner
surface of the vessel. On the edge of the support ring, two D-shaped half-
inch-thick plates are attached. Each of the plates has four right angle
brackets bolted onto it (two on each side), the U-shaped supports are then
bolted to the brackets. The bolt holes on the brackets are milled to allow
adjustment on the U-shaped support. A front view of the detector structure,
including the vessel cylinder and mirror support ring assembly are shown in
Fig. 3.5.

Much effort was spent investigating the possibility of using carbon fiber
material to pre-form a backing and then glue it onto the back of the mirror.
Such a scheme would allow to grip the edges of the carbon fiber backings
instead of the mirrors to provide extra strength and stability, since the
strength needed to grip the mirrors directly must be gentle and protective.
However, this scheme raised the following complications:

Out-gassing:

During the heavy gas filling process, the HGC detector is pumped down to a
pressure of 10-6 atm. The out-gassing level of the carbon fiber epoxy needs to
be carefully studied to estimate the possibility for contamination. When the
carbon fiber backing is glued to the back of a mirror using the epoxy, some
air will be trapped inside. It is not obvious how the trapped air bubbles
would affect the mirror-backing bond as they expand under a high vacuum
condition.

Radiation hardness:

When material is exposed to high level of radiation, its properties often
change. Studies are required under intensive radiation environment, to
understand the changes in the epoxy properties and mirror-backing bonding
strength.

Beam contamination:

The mirror thickness used in HGC detector is 3 mm; the thickness of prototype
mirror backing is also 3 mm. This indicates the overall mirror assemble
thickness would be 6mm, which would increase beam contamination probability
for the down stream detectors.

For these reasons, we have since decided to support the mirrors with clamps.

Figure 3.5: Front view of the Heavy Gas Čerenkov Detector [29]. (Original In
Color).

###  3.5 PMT Hosting Assembly

Each mirror focuses the Čerenkov radiation onto the corresponding 5” PMT and
there are four PMTs in total. The PMT front surface is made of UV transparent
glass with a radius of curvature of 13 cm, and are manufactured by Hamamastu
Photonics [20].

To avoid the variable gas pressure causing mechanical strain on the delicate
PMTs, they are designed to be outside of the Čerenkov medium, and are housed
inside the aluminum sleeves wielded on the top and bottom of the vessel
cylinder. Each PMT views the Čerenkov gas enclosure through 1 cm thick quartz
windows, and thus requires a quartz adaptor that has flat surface on one side
to press against the quartz window and a concave surface on the other side to
couple the curved PMT front surface. The center thickness of the adapter is
0.5 cm. In addition, there is a thin (1-2 mm) cookie made of room temperature
vulcanizing (RTV) silicone compound to fill the air gaps in between the quartz
adaptor and PMT surface. Silicon grease is used to couple the smooth surfaces.

The quartz windows and adaptors are made of Corning [21] 7980 quartz, and
machined by Hardin Optical Company [22]. The RTV silicone compound is produced
by Momentive Company [23].

###  3.6 Window Design

The pressure of the HGC detector is purposely set to be 0.95 atm pressure at
low momentum, due to the consideration of the window design. The HGC gas
filling scheme is to pump the vessel to 10-6 atm pressure and then fill the
heavy gas to 0.95 atm pressure. If the detector windows at both ends are flat,
the force induced by the pressure difference will deform and destroy the
windows, however, if the windows are pre-deformed to the desired curvature,
their strength will be increased by an approximate factor of two and they will
not deform further due to the pressure difference. The detector must have
pressure lower than 1 atm during operation to protect the pre-deformed window
curvature. The HGC windows will be made of 1 mm thick 2024-T4 alloy aluminum
and have a diameter of 1.829 m. This soft aluminum alloy is chosen to ensure
the thin windows will rip slowly, rather than shatter, if they are punctured.
The windows will be pre-deformed to a radius of curvature of 4.78 m using the
hydro-forming method at the University of Regina.

##  Chapter 4 Mirror Selection

###  4.1 Introduction

15 HGC mirrors manufactured by Sinclair Glass [16] were received in June 2011.
The specified mirror dimension is: 60 cm $\times$ 55 cm, thickness is 3 mm and
radius of curvature of 110 cm.

The method that Sinclair Glass used to make the mirrors is known as the
slumping technique, and its procedure is shown in Fig. 4.1. A large steel mold
of dimension 1 m $\times$ 1 m $\times$ $40$ cm, was machined to a concaved
surface with radius of curvature of 110 cm. Then, the concave surface was
coated with a layer of release agent to make sure the mirror would not bond to
the mold. A flat soda-lime glass mirror was pre-cut to a rectangular shape but
with slight curvature along the edge, so that the square corners and edges
could form after slumping. After the flat mirror was gently placed onto the
mold, it was moved into an oven where overhead heating was applied to soften
the glass. Then the flat mirror would gradually slump towards the pre-formed
mold curvature.

Air holes were drilled at 10 cm separation on the pre-formed surface. All the
air hole tunnels are inter-jointed into a main gas outlet, which is connected
to a vacuum pump to evacuate the air between the glass and mold. This
eliminates air bubbles trapped by the slumping glass, however, some sequential
imperfections in the form of small dimples were formed because of the air
holes. In order to avoid the imperfections on the mirror back surface, we
specified that the HGC mirrors should not be slumped to completely touch the
metal mold. The ideal curvature would be slightly parabolic with fewer
imperfections.

The level of slumped curvature is dictated by many factors, such as the
heating temperature and time, etc. Therefore, individual mirror qualities
could differ dramatically. We had to develop systematic methods to understand
the quality in term of the radius of curvature and shape for each mirror to
select the final detector mirrors. The methodologies and results for
coordinate measurement and optical test are given in this chapter. The overall
conclusions on mirror quality are drawn using combined results of the two
methods.

$\rightarrow$
$\swarrow$

Figure 4.1: Slumping method procedure demonstration. (Original In Color).

###  4.2 $X$, $Y$, $Z$ Coordinate Measurement

####  4.2.1 Methodology

A local company named Dumur [30] was hired to map out the $x$, $y$, $z$
coordinates for all mirrors. Their setup is shown in Fig. 4.2 and Fig. 4.3.
The equipment in Fig. 4.2 is called a FaroArm [31]; it is a high precision 3D
coordinate measuring system. During the measurement, the mirror was placed
horizontally on top of a grid map and FaroArm sensor was zeroed at a reference
corner of the mirror.

The $x$, $y$ grid map contains 20 $x$ direction and 18 $y$ direction measuring
points, for a total of 360 points with each measuring point being 3 cm apart
in the $x$ and $y$ directions as shown in Fig. 4.3. The measuring points did
not extend to the corners and edges of the mirror and the maximum coverage
area was 90% of the central region. For each measurement, the sensor was
manually placed above a measuring point and descended until it gently touched
the mirror surface, then the FaroArm system was triggered manually to record
the $x$, $y$, $z$ coordinates. Because the sensor was placed manually the grid
point spacing was slightly irregular.

Figure 4.2: Sensor equipment used by Dumur to measure a grid of ($x$, $y$,
$z$) coordinates for each mirror. (Original In Color). Figure 4.3: Mirror
placement during the $x$, $y$, $z$ coordinate measurement. (Original In
Color). Figure 4.4: Different types of curvature for various values for conic
constant [32]. (Original In Color).

Once a coordinate map is produced, the conic constant formula is used to fit
the data, where the conic constant formula is given as [32]:

$z=\frac{(x-x_{off})^{2}+(y-y_{off})^{2}}{R+\sqrt{R^{2}-(1-\kappa)[(x-x_{off})^{2}+(y-y_{off})^{2}]}}+z_{off}$
(4.1)

where $x$ $y$ $z$ represent the measured values, $x_{off}$ $y_{off}$ $z_{off}$
are the offset parameters, $\kappa$ and $R$ are conic constant and radius. The
function will fit $x_{off}$, $y_{off}$, $z_{off}$, $R$ and $\kappa$. $R$ and
$\kappa$ will directly indicate the mirror quality.

The conic constant $\kappa$ characterizes the curvature shape, as indicated in
Fig. 4.4:

$0<\kappa<1:$

The curvature is an oblate ellipsoid.

$\kappa=1:$

The curvature is a sphere.

$-1<\kappa<0:$

The curvature is a prolate ellipsoid.

$\kappa=-1:$

The curvature is a paraboloid.

$\kappa<-1:$

The curvature is a hyperboloid.

In order to optimize the focusing and avoid imperfections, we requested that
the detector mirrors be made in a paraboloid shape if possible. Thus, the
expected fitting results are $\kappa\approx-1$ and $R$=110 cm.

For the quality categorization, five sets of fits were applied to the
coordinate data, therefore five sets $R$ and $\kappa$ values are obtained for
each mirror:

90%:

Fits all 360 measuring points, which is close to 90% of the central area
(1$<x<$57 cm and 1$<y<$53 cm).

75% Inner Fit:

Fits 75% of the central area (4$<x<$ 56 cm and 4$<y<$51 cm).

75% Outer Fit:

Fits 25% of the area round the edge ($x<$4 cm or $x>$ 56 cm or $y<$4 cm or
$y>$51 cm).

50% Inner Fit:

Fits 50% of the central area (7.5$<x<$52.5 cm and 8.5$<y<$ 46.5 cm).

50% Outer Fit:

Fits 50% of the area round the edge ($x<$7.5 cm or $x>$52.5 cm and $y<$8.5 cm
or $y>$46.5 cm).

####  4.2.2 Fitting Results

Table 4.1: Radius ($R$) and Conic Constant ($\kappa$) fitting results for 90%, 75% inner, 75% outer, 50% inner and 50% outer fit, for 15 mirrors. Note: gray highlighted mirrors have the worst fitting results, magenta highlighted mirrors have the best fitting results. (Original In Color). Mirror | 90% Fit | 75% Inner Fit | 75% Outer Fit | 50% Inner Fit | 50% Outer Fit
---|---|---|---|---|---
# |  $R$ (cm) | $\kappa$ |  $R$ (cm) | $\kappa$ |  $R$ (cm) | $\kappa$ |  $R$ (cm) | $\kappa$ |  $R$ (cm) | $\kappa$
1 | 118.397 | 1.95 | 118.216 | 1.54 | 115.360 | 1.14 | 120.494 | 2.59 | 120.783 | 2.42
2 | 113.124 | 1.06 | 113.408 | 0.94 | 117.905 | 1.77 | 113.691 | 0.78 | 117.257 | 1.87
3 | 117.110 | 1.70 | 117.286 | 1.57 | 117.289 | 1.50 | 118.475 | 2.34 | 120.665 | 2.39
4 | 122.392 | 2.29 | 122.356 | 1.77 | 119.559 | 1.65 | 123.735 | 2.39 | 121.200 | 2.05
5 | 113.331 | 0.76 | 113.010 | 0.25 | 117.868 | 1.63 | 114.129 | 0.49 | 117.380 | 1.64
6 | 112.906 | 0.94 | 113.239 | 0.86 | 108.735 | -0.22 | 114.824 | 1.61 | 115.172 | 1.30
7 | 113.538 | 1.13 | 114.096 | 1.15 | 110.307 | 0.13 | 116.257 | 2.26 | 116.593 | 1.68
8 | 114.325 | 1.26 | 114.335 | 0.96 | 117.910 | 1.83 | 114.158 | 0.29 | 115.067 | 1.33
9 | 114.372 | 1.30 | 115.151 | 1.39 | 105.942 | -0.67 | 116.887 | 2.44 | 113.365 | 0.94
10 | 112.035 | 0.42 | 111.684 | -0.09 | 109.045 | -0.30 | 112.131 | -0.49 | 109.484 | -0.17
11 | 111.766 | 0.75 | 112.177 | 0.68 | 101.524 | -1.58 | 113.095 | 1.04 | 110.939 | 0.40
12 | 112.117 | 0.84 | 112.264 | 0.62 | 104.243 | -0.98 | 113.636 | 1.26 | 112.183 | 0.72
13 | 122.464 | 2.43 | 123.174 | 2.23 | 113.602 | 0.60 | 126.136 | 2.60 | 116.069 | 1.07
14 | 117.964 | 1.59 | 116.901 | 0.83 | 118.303 | 1.60 | 117.175 | 0.87 | 125.126 | 3.29
15 | 113.674 | 1.17 | 113.717 | 0.92 | 115.336 | 1.31 | 114.701 | 1.25 | 119.361 | 2.39

The results for five different fits are presented in Table 4.1. The ideal
radius ($R$) and conic constant ($\kappa$) should be $R$=110 cm and
$\kappa\approx-$1\. However, the fitted $\kappa$ values range from -0.5 to
2.6, which indicates that most of the mirrors have oblate ellipsoid shape
rather than paraboloid shape. This should increase optical aberrations, but
simulation results described in Chapter 6 indicates that the focusing result
of oblate mirror is comparable to the spherical mirror (Mirror # 6 fitted
parameters were used for the study). The PMT positions need to be adjusted to
optimize the focusing spot for the oblate mirror, as discussed in Chapter 6.

A perfect spherically curved mirror with $R$=110 cm should have $\kappa$=0,
and its focal length ($f$) should be $f$=$R$/2=55 cm. For an oblate ellipsoid
mirror, the focal length is shorter due to the oblateness (see Fig. 4.4). All
the detector mirror candidates have radii larger than 110 cm, and the positive
$\kappa$ value will shorten the focal length to be around 55 cm.

$R$ and $\kappa$ fitting results over different areas vs Mirror # are plotted
in Fig. 4.5. The average value of $R$ or $\kappa$ over all mirrors is plotted
as the solid line and $\pm\sigma$ limitations are plotted as the dashed lines,
where $\sigma$ is the standard deviation. In addition, 6 sets of criteria were
introduced to categorize the mirror quality based on the fitting results:

$R$ Best:

Select mirror when all 5 $R$ values are less than averages (under the solid
line) for 5 area fits.

$\kappa$ Best:

Select mirror when all 5 $\kappa$ values are less than averages for 5 area
fits.

Less Oblate:

Select mirror when all 5 $\kappa$ values are less than 1.

Average Mirror:

Select mirror when all 5 $R$ and $\kappa$ values are less than (averages
$+\frac{\sigma}{2}$) for 5 area fits

Fitting Sum:

Select mirror when the sum of 5 of $R$ and $\kappa$ values are less than
$\sum$averages.

Best Sum:

Select mirror when the sum of 5 of $R$ and $\kappa$ values are less than
$\sum$(averages $-\sigma$).

Table 4.2 shows the criteria and Mirror # that passed the condition. Mirrors
#4, #13 and #14 did not pass any criterion, therefore they have the lowest
quality; Mirror #1 has large $R$ and $\kappa$ value which is also considered
as low quality mirror. Mirrors #2 and #9 are on the boundary line for several
criteria (see Fig. 4.5), and it is necessary to check with the optical test
results to further understand their quality, especially since Mirror #9 passed
the Best Sum criterion which indicates it is a high quality mirror. Mirrors
#6, #7, #10, #11 and #12 passed the most criteria; we conclude these mirrors
have the highest quality in terms of fitted Radius and Conic Constant. Optical
tests were required to confirm the high and low quality mirror results, and
classify the average quality mirrors.

Table 4.2: Criteria and Mirror # passed the condition. Criteria Name | Passed Mirror#
---|---
$R$ Best | 6, 7, 8, 10, 11, 12
$\kappa$ Best | 10, 11, 12
Less Oblate | 10
Average Mirror | 6, 10, 11, 12
Fitting Sum | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 15
Best Sum | 6, 7, 9, 10, 11, 12
Failed All | 4, 13, 14
Figure 4.5: $R$ and $\kappa$ fitting result from coordinate measurements for
all 15 mirrors. From top to bottom, it shows $R$ and $\kappa$ results for 90%,
75% Inner, 75% Outer, 50% Inner and 50% Outer fit. Note that red solid lines
give the average value of all mirrors; red dashed lines give 1 $\sigma$ over
and below the averaged value. The expected radius and conic constant:
$R_{expect}$=110 cm, and $\kappa_{expect}$=-1. (Original In Color).

####  4.2.3 Fit$-$Measurement Plots

Ideally, the detector mirrors should have few imperfections at the reflecting
surface. In order to check imperfections, we computed the difference plot
between the conic constant fit and the experimental results. The fitted radius
and $\kappa$ are used to generate a perfect oblate shape, then its $z$
coordinate is compared with the real measurement at each $x$, $y$ measuring
point. The difference is defined as

$\Delta z=z_{measurement}-z_{generated}\,.$ (4.2)

In general, the central 50% region has small $\Delta z$, whereas large $\Delta
z$ only occur along the edges. The features of all fit$-$measurement plots can
be categorized into four groups.

In the first group, mirrors have uniformly small $\Delta z$ for most areas,
which indicates that the reflecting surface contains few imperfections.
Difference plots for Mirrors #10, #11 and #12 have this feature. Fig. 4.6a
shows the fit$-$measurement plot for Mirror #10.

In the second group, mirrors have large $\Delta z$ ($|\Delta z|>$ 0.6 mm)
along two opposite edges (corners), and small $\Delta z$($<$ 0.6 mm) along
other edges (corners). Mirrors #2, #5, #6, #7, #8 and #9 have this feature.
Fig. 4.6b shows the fit$-$measurement plot for Mirror #6.

In the third group, mirror have large $\Delta z$ ($|\Delta z|>$ 0.6 mm) along
all four edges (corners). Plots for Mirrors #3, #14 and #15 have this feature,
and their imperfections are worse than the second group mirrors. Fig. 4.6d
shows the fit$-$measurement plot for Mirror #14.

In the fourth group, $|\Delta z|$ is not flat near the central region. Plots
for Mirrors #13 and #4 have this feature. Fig. 4.6c shows the
fit$-$measurement plot for Mirror #13.

Based on the Monte Carlo Simulation result, the reflecting area near the edges
are shown to be less important than the central area. Mirrors from the first
and second groups have some imperfections mostly along the edges, while their
$\Delta z$ are relatively small and one of these edges will be outside the
beam envelope. On the other hand, group three mirrors have large $\Delta z$
along all four edges and group four mirrors have imperfections in central
region, and as a result these mirrors have larger scale imperfections.

(a) Mirror #10, an example of a mirror with good uniformity (1st group).

(b) Mirror #6, an example of a mirror with large two edges (2nd group).

(c) Mirror #13, an example of a mirror with large $\Delta z$ along 4 corners
(3rd group).

(d) Mirror #14, an example of a mirror with poor central region (4th group).

Figure 4.6: Mirror $\Delta z$ plots for Mirror #10, #6, #13 and #14. Units of
the plot are mm. (Original In Color).

####  4.2.4 Uncertainty of Coordinate Measurements

We requested that Dumur remeasure Mirrors #10 and #13. This allows the
systematic uncertainty of a single coordinate measurement to be studied. For
the 90% area fit of Mirror #10 the first measurement $a$ has fitted $R_{10a}$
= 112.04 cm and $\kappa_{10a}$ = 0.42, and the second measurement $b$ has
fitted $R_{10b}$ = 110.8 cm and $\kappa_{10b}$ = 0.35. We used Eq. 4.1 to
reconstruct the $z$ value for 810 points across the mirror with both sets of
fitting parameters, then compare the difference $\Delta z=z_{a}-z_{b}$. The
averaged $\Delta z=$ 0.267 mm and $\sigma=$ 0.166 mm. Similarly, for 90% fit
of Mirror #13 the first measurement $a$ has fitted $R_{13a}$ = 122.46 cm and
$\kappa_{13a}$ = 2.43, the second measurement $b$ has fitted $R_{13b}$ =
122.56 cm and $\kappa_{13b}$ = 2.52. The averaged $\Delta z=$ 0.016 mm and
$\sigma=$ 0.033 mm. We used the standard deviation of $\Delta z$ of Mirror #10
(0.166 mm) to estimate the uncertainty of a single coordinate measurement. The
Mirror #10 measurements have larger $\Delta
R=R_{10a}-R_{10b}=(112.04-110.8)\leavevmode\nobreak\ \textrm{cm}=1.8$ cm and
Mirror #13 measurements have larger
$\Delta\kappa=\kappa_{13b}-\kappa_{13a}=0.11$. The uncertainties for $R$ and
$\kappa$ are decided to be $\pm$2 cm and $\pm$0.15, which are very
conservative estimations based on the $\Delta R$ and $\Delta\kappa$. Even
though the mirror uncertainties are slightly overestimated, they are still not
large enough to challenge the mirror quality conclusion.

####  4.2.5 Coordinate Measurement Result Summary

From the coordinate measurement fitting results, we conclude that all
manufactured HGC mirrors have oblate elliptical curvature: $\kappa>$ 0, and
their fitted radius of curvature are slightly larger than desired: $R>$ 110
cm. Mirrors #1, #4, #13 and #14 have the lowest quality in terms of fitted $R$
and $\kappa$. Mirrors #6, #7, #10, #11 and #12 have the highest quality.
Optical tests are needed to categorize the intermediate mirrors. The estimated
systematic uncertainty for a single $z$ coordinate measurement is $\delta
z$=$\pm$0.166 mm; the uncertainties in the fitted $R$ and $\kappa$ are $\pm$2
cm and $\pm$0.15 respectively. From the fit-measurement plots, we learned
Mirrors #10, #11 and #12 have the fewer imperfections; Mirrors #2, #5, #6, #7,
#8 and #9 have some imperfections along two edges (corners).

###  4.3 Optical Test

####  4.3.1 Methodology

In this test a 1 mW laser beam is used as light source, and it is split to
illuminate the whole mirror. A photograph of the reflected pattern is taken on
a small screen at the optimal focusing distance. The equipment used to split
the laser beam is shown in Fig. 4.7: a combination of laser splitter and
concave lens. The distance between the concave lens and the mirrors
$S_{O}=624\leavevmode\nobreak\ cm$ and the mirror to focused image distance
($S_{i}$) is also recorded. Then, the focal length ($f$) can be calculated by
the equation:

$\frac{1}{f}=\frac{1}{S_{O}}+\frac{1}{S_{i}}\,.$ (4.3)

Figure 4.7: Optical test equipment: 1 mW lab laser, a laser splitter and a
concave mirror. (Original In Color).

####  4.3.2 Optical Test Results

In this subsection we present the optical test results. For each mirror,
photographs of the reflected laser beam pattern were taken at the optimal
focus point and the horizontal distance ($S_{i}$) relative to the mirror was
recorded. Due to the various optical aberrations and the fact that the mirror
was not aluminized, reflections were obtained from both the front and back
surfaces of the mirror. The spot shape was complex and not all components
focused at the same distance. Thus, there is some systematic uncertainty in
the image distance measurement, and aluminization will help to reduce this.

Table 4.3 shows $S_{i}$ and calculated focal length (FL) using Eq. 4.3. For a
spherical mirror, the $f$ should be $f=R/2$. For an oblate ellipsoid mirror,
$f<R/2$, and the $\kappa$ value determines the degree of shortness. We divide
the 90% Fitted Radius (FR) from the coordinate measurement by 2, and obtain
the focal length for spherical shaped mirror, then compute the difference
between it and the calculated focal length (FL). The results are listed in
Table 4.3. Fig. 4.8 shows the plot of 90% fit $\kappa$ vs (FL$-$FR/2). From
this plot, we can conclude a strong correlation between $\kappa$ and
(FR/2$-$FL): the more oblate the mirror the shorter the focal length as
determined from the optical test.

Except for Mirrors #1 and #13, all other mirrors have focal lengths between 54
and 57 cm, with the measurement uncertainty being 0.5 cm for $S_{i}$, and 2 cm
for $S_{o}$. Although positive $\kappa$ values do not help to focus light,
they bring the focal point closer to the optimised focal point despite the
mirrors having a larger radius.

Figure 4.8: 90% fitted Conic Constant $\kappa$ is plotted against the
difference between Focal Length and half of Fitted Radius (FR/2$-$FL) for 15
mirrors.

The images of the reflected patterns are shown in Fig. 4.9, the processed
version is in Fig. 4.10. GIMP [35] color tools were used to process the
original image: the contrast was increased to +50, brightness was reduced to
-50, red color was filtered out, and the brightest area was turned into blue.
From the original image, it is very difficult to determine the size and shape
of the brightest area, whereas the processed image shows the brightest area
very clearly in blue. Direct observation confirms Mirrors #1, #4 and #13 have
the poorest quality: there are huge tails in Mirror #1 image, and the light
does not focus for Mirrors #4 and #13.

The reflected images include the reflection of the back surface where the
imperfections are much worse due to the slumping process, thus it is
impossible to categorize mirrors based only on their front surface quality.
Some quantitative criteria are introduced to help determine the entire mirror
quality using the processed reflection image. The conditions include the
intensity, size and shape of the blue region, also the size and shape of the
green region.

The size and shape of the brightest (blue) area are the most important factors
to determine the mirror quality. Table 4.4 shows the number of green pixels,
the number of blue pixels and the blue-green ratio for each mirror from Fig.
4.10, where the higher pixel number corresponds to the larger image area.
Mirror #1, #4 and #13 have the largest green pixel number and smallest blue
pixel number, while their blue-green ratio is less than 10%. They are
confirmed to be the low quality mirrors. On the other hand, Mirrors #2 and #3
have the smallest green region and the highest blue-green ratio (larger than
30%). It is difficult to conclude the optical quality of this type of mirrors,
thus other test results are needed. All other mirrors have green and blue
pixel numbers around 40000 and 5000, and the blue-green ratio is around 0.13.
In terms of shape, Mirror #15 has two separate blue regions; Mirror #5 has a
long strip for the blue region. The best blue spots are produced by Mirrors
#6, #7, #9 and #11, they have small circular blue spots with few spikes around
them. Surprisingly, with the best coordinate measurement result, Mirror #10
has a blue triangular spot at low intensity, the spot size is slightly larger
than other high quality mirrors.

Mirrors #1, #4, #13 have the largest green region. Mirrors #5 and #14’s green
regions are much larger than their blue. Mirror #15 has two focused regions on
the image. Other mirrors have reasonable good green regions, especially for
Mirrors #2 and #10, which contain no large tail.

Based on the criteria for the image results, we conclude that Mirrors #6, #7,
#9 and #11 have the best optical quality; Mirrors #2, #8, #10 and #12 have
reasonably good optical quality; Mirrors #3, #5, #14 and #15 have average
quality, Mirrors #1, #4 and #13 have the worst quality.

The shape of the blue spots for high and average quality mirrors can be
classified into two groups: small blue spot and large blue spot. The small
blue spot mirrors include #6, #7, #9, #8, #11 and #12, their characteristics
are a small circular bright blue spot with few spikes and tails in the green
region. The large blue spot mirrors include #2 and #10, their characteristics
are a relatively larger blue spot and no tails in the green region. It is
difficult to determine which group represents the better focusing ability,
therefore, we decided to send one mirror from each group for aluminization.
The optical test will be repeated and the reflection of the back surface
should be completely eliminated. The comparison of the reflected pattern
before and after the aluminization for the testing mirror will give us some
intuitive hints for the front surface quality of other mirrors. Mirrors #2 and
#8 are the perfect candidates for the initial aluminization test, as they are
average quality mirrors which share the same characteristics with the high
quality mirrors, they will contribute to our understanding of the front
surface optical quality for both groups, as well as the ECI aluminization
technique.

####  4.3.3 Optical Test Summary

From the optical test results, we extracted focal lengths corresponding to
full mirror illumination and photographed the reflected pattern of the split
laser beam. All of the high and average quality mirrors from the coordinate
measurements have focal lengths from 54 cm to 57 cm, with uncertainty $\pm$2
cm, which are close to the optimal value. Mirror #6, #7, #9 and #11 have the
highest quality reflected spot, and #2, #8, #10, #12 have reasonably good
reflected spot. Mirror #2 and #8 are selected to be aluminized, and optical
test will be repeated after the aluminization.

Table 4.3:  $S_{i}$ column shows the measurement for mirror to image distance; Focal Length column gives the calculated focal length with $S_{i}$ using Eq. 4.3; Fitted Radius is the 90% fitted radius from coordinate measurement; FR/2 is the Fitted Radius/2; FR/2$-$FL is the difference between half of the Fitted Radius and Focal Length; $\kappa$ is the 90% fitted conic constant from coordinate measurement. Note: gray highlighted mirrors have the worst optical testing results. Mirror | $S_{i}$ | Focal Length | Fitted Radius | FR/2 | FR/2$-$FL | $\kappa$
---|---|---|---|---|---|---
# | (cm) | (cm) | (cm) | (cm) | (cm) |
1 | 64.0 | 58.05 | 118.40 | 59.20 | 1.15 | 1.95
2 | 59.5 | 54.32 | 113.12 | 56.56 | 2.24 | 1.06
3 | 61.0 | 55.57 | 117.11 | 58.56 | 2.99 | 1.70
4 | 62.0 | 56.40 | 122.39 | 61.19 | 4.80 | 2.29
5 | 62.5 | 56.81 | 113.33 | 56.67 | -0.14 | 0.76
6 | 62.5 | 56.81 | 112.91 | 56.45 | -0.36 | 0.94
7 | 61.5 | 55.98 | 113.54 | 56.77 | 0.79 | 1.13
8 | 61.7 | 56.15 | 114.33 | 57.16 | 1.01 | 1.26
9 | 61.0 | 55.57 | 114.37 | 57.19 | 1.62 | 1.30
10 | 62.5 | 56.81 | 112.04 | 56.02 | -0.79 | 0.42
11 | 60.5 | 55.15 | 111.77 | 55.88 | 0.73 | 0.75
12 | 61.3 | 55.82 | 112.12 | 56.06 | 0.24 | 0.84
13 | 64.0 | 58.05 | 122.46 | 61.23 | 3.19 | 2.43
14 | 61.0 | 55.57 | 117.96 | 58.98 | 3.41 | 1.59
15 | 60.0 | 54.74 | 113.67 | 56.83 | 2.10 | 1.17
Table 4.4: Pixel information of Fig. 4.10. The gray highlighted mirrors have the lowest blue-green ratio. Mirror | Green Pixel | Blue Pixel | Blue-Green
---|---|---|---
# | # | # | Ratio
1 | 58072 | 5144 | 0.09
2 | 30201 | 11896 | 0.39
3 | 20590 | 14494 | 0.70
4 | 84718 | 6800 | 0.08
5 | 38860 | 5607 | 0.14
6 | 36418 | 5427 | 0.15
7 | 40475 | 5050 | 0.12
8 | 39362 | 6050 | 0.15
9 | 42580 | 5321 | 0.12
10 | 35126 | 5026 | 0.14
11 | 48764 | 5900 | 0.12
12 | 40120 | 4806 | 0.12
13 | 87806 | 1021 | 0.01
14 | 36801 | 5098 | 0.14
15 | 43228 | 7805 | 0.18
Figure 4.9: Return (raw) spot image for all mirrors. The mirror number is
incremented horizontally, top left corner is Mirror #1 and bottom right corner
is Mirror #15. (Original In Color). Figure 4.10: Processed return spot image
for all mirrors. The mirror number is incremented horizontally, top left
corner is Mirror #1 and bottom right corner is Mirror #15. (Original In
Color).

###  4.4 Combined Result

###### Mirror #1: Bad

Mirror #1 failed nearly all criteria in coordinate measurement test, it has
large fitted $R$ (118.4 cm) and $\kappa$ (1.95). The actual focal length is
too large (58 cm). Spot image shows large tails in the green region, and blue
region is not focused to a circular spot.

###### Mirror #2: Average

According to coordinate measurement result, Mirror #2 failed most of the
criteria, because its fitted parameters are large for 75% outer and 50% outer
fit. On the other hand, it has reasonable fitted $R$ (113.1 cm) and $\kappa$
(1.06) in central region. The optical test for Mirror #2 is better than
expectation, there is no large green tail in the image, and blue spot is
focused to a relatively large region (similar to Mirror #10). Mirror #2 is the
perfect candidate to be sent for the aluminization, to help understand
characteristics of Mirror #10.

###### Mirror #3: Average

Mirror # 3 failed most of the criteria for the coordinate measurement test, it
has large fitted $R$ (117 cm) and $\kappa$ (1.7). From the optical test,
Mirror #3 has long focal length (58.56 cm). From the image, the blue region
failed to focus to a spot.

###### Mirror #4: Bad

Coordinate measurement and optical tests confirmed Mirror# 4 is one of the
worst mirrors: large fitted $R$ and $\kappa$, large imperfection in the
center, the return spot image failed to focus.

###### Mirror #5: Average

Although Mirror #5 failed some strict criteria, its fitted $R$ (113.3 cm) and
$\kappa$ (0.76) look quite reasonable. The optical result is not as good as
the expectation, blue region failed to focus to a spot. All these results may
suggest this mirror has better quality in the center than the edge.

###### Mirror #6: Good

Mirror #6 passed nearly all criteria for the coordinate measurement test, and
its optical test result looks satisfying. It is considered to be one of the
best mirrors.

###### Mirror #7: Reserve

Mirror #7 passed most of the criteria, including the Best Sum criterion, which
indicates it is a high quality mirror from the coordinate measurement test.
The optical focal length is reasonable (56 cm) and the reflected spot image
has high quality.

###### Mirror #8: Average

Mirror #8 has very similar fitted $R$ and $\kappa$ to #7 and #9, however it
did not pass the Best Sum criterion. Mirror #8 has the same spot
characteristics to #6, #7, #8, #9, #11 and #12, and it is the most average
mirror of the group. Mirror #8 will contribute to understand the spot image of
other mirrors.

###### Mirror #9: Reserve

Mirror #9 passed most of the criteria, including the Best Sum criterion, which
indicates it is a high quality mirror from the coordinate measurement test.
The optical focal length is reasonable (55.6 cm) and the reflected spot image
has high quality.

###### Mirror #10: Good

Mirror #10 passed all criteria for the coordinate measurement test, it has the
best fitting result and has fewer imperfections. However, its spot image
looked different from other good mirrors, Mirror#2 is sent for aluminization
to help understand this type of reflected light spot.

###### Mirror #11: Good

Mirror #11 passed nearly all criteria for the coordinate measurement test, has
few imperfections and its optical test result looks satisfying. It is
considered to be one of the best mirrors.

###### Mirror #12: Good

Mirror #12 passed nearly all criteria for the coordinate measurement test, has
few imperfections and its optical test result looks satisfying. It is
considered to be one of the best mirrors.

###### Mirror #13: Bad

Coordinate measurement and optical test both confirmed Mirror #13 is one of
the worst mirrors: large fitted $R$ and $\kappa$, large imperfections in the
center, return spot image does not focus.

###### Mirror #14: Low Average

Despite reasonably good spot image, Mirror #14 has huge fitted $R$ (118 cm)
and $\kappa$ (1.59). Also the focal length from the optical test is very long
(59 cm).

###### Mirror #15: Average

Mirror #15 has average quality according to coordinate measurement test, the
optical image test shows two separate blue regions which indicate certain
parts of the mirror have bad imperfection.

Table 4.5 gives an overall summary for the coordinate measurement and optical
test result, and ranking of all 15 HGC mirrors.

Table 4.5: Overall mirror quality summary from coordinate measurement and optical test results. 90% Fit $R$ and $\kappa$ are form Table 4.1; # of Failed Criterion is deduced from Table 4.2, FR/2$-$FL results are from Table 4.3, Blue-Green ratio results are from Table 4.4. The worst quality mirrors are highlighted in red; the average quality mirrors are highlighted in yellow; the best quality mirrors are highlighted in green. (Original In Color). Mirror | 90% Fit | 90% Fit | # of Failed | FR/2$-$FL | Blue-Green | Overall
---|---|---|---|---|---|---
# |  $R$ (cm) | $\kappa$ | Criterion | (cm) | Ratio | Ranking
1 | 118.397 | 1.95 | 5 | 1.15 | 0.09 | Bad
2 | 113.124 | 1.06 | 5 | 2.24 | 0.39 | Average
3 | 117.110 | 1.70 | 5 | 2.99 | 0.70 | Average
4 | 122.392 | 2.29 | 6 | 4.80 | 0.08 | Bad
5 | 113.331 | 0.76 | 5 | -0.14 | 0.14 | Average
6 | 112.906 | 0.94 | 2 | -0.36 | 0.15 | Good
7 | 113.538 | 1.13 | 3 | 0.79 | 0.12 | Reserve
8 | 114.325 | 1.26 | 4 | 1.01 | 0.15 | Average
9 | 114.372 | 1.30 | 4 | 1.62 | 0.12 | Reserve
10 | 112.035 | 0.42 | 0 | -0.79 | 0.14 | Good
11 | 111.766 | 0.75 | 1 | 0.73 | 0.12 | Good
12 | 112.117 | 0.84 | 1 | 0.24 | 0.12 | Good
13 | 122.464 | 2.43 | 6 | 3.19 | 0.01 | Bad
14 | 117.964 | 1.59 | 6 | 3.41 | 0.14 | Bad
15 | 113.674 | 1.17 | 5 | 2.10 | 0.18 | Average

###  4.5 Conclusion

By combining the results of both tests, we conclude: Mirror #6, #10, #11 and
#12 have excellent quality and are recommended to be the detector mirrors;
Mirror #7 and #9 can be reserved as backups; Mirror #2 and #8 are sent for
test aluminization. The optical test will be repeated after aluminization of
#2 and #8, so that reflected spot can be compared with our first result. The
test aluminization result for Mirror #8 will be discussed in Chapter 5.

##  Chapter 5 Mirror Reflectivity Measurement

###  5.1 Introduction

In November 2011, HGC Mirrors #2 and #8 were shipped to Evaporated Coatings
Inc. (ECI) [33] for an aluminization test based on the conclusion from the
mirror selection study (see Section 4.5). The aluminized mirrors were then
sent to Jefferson Lab. The University of Regina and Jefferson Lab (Hall C,
Detector Group and Free Electron Laser Facility) are collaborating to
construct a permanent setup to measure mirror reflectivity between 165-400nm,
in order to verify the mirror aluminization technique by ECI. In this chapter,
we will describe the permanent reflectivity setup at Jefferson Lab and present
the reflectivity results of the aluminized Mirror #8.

The reflectivity quality control by ECI was not performed on the test mirrors.
A number of 1” diameter test samples were placed at various positions in the
vacuum deposition tank and aluminized the same way as the test mirrors (see
Fig 5.4a). Their reflectivities were later measured by ECI to indicate the
quality of the aluminized test mirrors. At the end of the chapter, the ECI
sample reflectivity results are compared to our measurement.

###  5.2 Methodology

####  5.2.1 Equipment list

The list of important components used for the setup construction is given
below:

  * •

2 $\times$ AXUV-100G Photo-Diode UV Detectors (#02, #18) with Ceramic
Shoulders manufactured by International Radiation Detectors, Inc. (IRD) [36]

  * •

Thorlab MC100 Optical Chopper System with MC1F2 Chopping Blade

  * •

SR530 Lock-in Amplifier

  * •

The McPherson [38] Model 218 Vacuum Ultraviolet (VUV) Monochromator with
Holographic 200 nm Blaze (1200 Grids/mm) Grating

  * •

1mW Melles Griot [37] Alignment Laser

  * •

3 Watt Hamamastu [20] Deuterium UV Light Source

  * •

Deep Ultraviolet (DUV) Flipper Mirror: Melles Griots DUVA-PM-5010M-UV

  * •

F1.5 Focusing Lens: Edmund Optics UV Plano-Convex 50mm Dia.x75mm FL Uncoated
Lens

  * •

F4 Focusing Lens: Edmund Optics CAF2 PCX50.8 Lens

  * •

1 $\times$ Translation Stage, 1 $\times$ Rotation Stage and 8 $\times$ Stepper
Motors

The mirror reflectivity measurements used HGC Mirror #8 and IRD UV detector
#02. The HGC mirror was clamped along the left and bottom edges with metal
clamps (two along each side). The clamped mirror edges were sandwiched between
rubber gaskets to isolate them from the metal pieces.

####  5.2.2 Monochromator

The monochromator is an optical device that transmits a mechanically
selectable, narrow wavelength band of light. It employs a diffraction grating
to spatially separate the light at different wavelengths, and subsequently a
mechanical device is used to select a specific wavelength and guide the light
out with a reflecting mirror. The basic components of the monochromator
include: two reflecting mirrors, a diffraction grating and a wavelength
adjustment mechanism. However, the inside structure of a monochromator varies
significantly, depending on the purpose and manufacturer.

Figure 5.1: McPherson 218 VUV monochromator. The light path inside the
monochromator is in blue [41]. (Original In Color).

The monochromator used for our reflectivity measurement is the McPherson 218
vacuum ultraviolet (VUV) model as shown in Fig. 5.1. It has adjustable
entrance and exit slits, two reflecting mirrors and a diffraction grating.
During the operation, the incoming light passes though the entrance slit, and
is reflected by the entrance reflecting mirror toward the diffraction grating.
Through the process of light diffraction and interference, a spectrum is
created and projected onto the exit reflecting mirror, then a narrow band of
light is let through by the exit slit. Both the entrance and exit reflecting
mirrors are at fixed positions; the incident angle of the light to the
diffraction grating is adjusted by the wavelength adjusting mechanism, this
will select which part of the light spectrum is reflected by exit mirror and
passes through the exit slit. The complete light path inside the monochromator
is indicated in Fig. 5.1.

The VUV model monochromator is capable of operating under a vacuum of (10-6
torr) or under a purged condition. The wavelength adjustment is achieved using
a remote electronic controller which makes data acquisition much more
efficient. The optimal operating wavelength range is dictated by the
diffraction grating used. To optimize the light signal around 200 nm, a
holographic 200 nm blaze (1200 G/mm) grating is used, so that the coherent
light spectra has a peak intensity around 200 nm. The system must be carefully
calibrated before taking any measurement.

####  5.2.3 Setup

(a)

(b)

Figure 5.2: Images of the mirror reflectivity setup. A: Light source box; B:
Monochromator; C: Focusing lens; D: DUV Flipper Mirror; F: Alignment Laser
Stage; E: Detector; G: Test Mirror; H: Optical Chopper. The light path is
shown on the top image, the monochromator light path is in magenta; the
alignment laser light path is in green; the shared (common) light path is in
yellow. (Original In Color).

The main body of the setup is constructed in three parts, as shown in Figs.
5.2 and 5.3: the light source box, the monochromator and the detector hutch.
In Fig. 5.2 the hutch was not yet installed.

The light box houses the Hamamatsu deuterium lamp and F1.5 focussing lens. The
distance between the lamp and the lens is 4 cm, and the distance between the
lens and the light entering the slit of the monochromator is also 4 cm.

Inside of the detector hutch, the test mirror is clamped onto the hosting
bracket and then mounted to the angle adjusting frame. The stepper motors
attached to the frame are able to change the pitch and yaw angle with an
accuracy of 0.1∘. The whole frame sits upon a three axis translation stage,
which allows the source light from the monochromator to reach any spot on the
sample mirror.

During the reflectivity measurement, the detector position is fixed at a
constant point. This means the sample mirror’s position and angle need to be
adjusted to reflect the source light towards the detector. Since the human eye
does not respond to Medium Ultraviolet (MUV) or Far Ultraviolet (FUV) light,
an alignment laser is introduced to match the monochromator light path, in
order to help guide the mirror adjustment. The laser is installed
perpendicular to the monochromator light path, and a pneumatically controlled
stage device inserts a small reflecting mirror to intercept the light path if
the laser is needed for alignment (see Fig. 5.2).

The air starts to block UV light around 190 nm, so for the reflectivity
measurement at wavelengths below this value, an oxygen free environment is
required. A high vacuum environment would be ideal, but the cost of a large
vacuum chamber is beyond the budget. Instead, a pure N2 environment is used,
because the leak tight requirements for a purged system are much less
demanding. All three parts of the setup can be separately fed with cool N2
gas. The volume of the hutch needs several hours of purging before reaching
the condition necessary for reflectivity measurements. The purging process
continues during the data taking period to maintain a low oxygen level and
also to cool the UV lamp. An oxygen monitor is placed inside of the hutch to
read out the oxygen level. For the HGC mirror reflectivity measurement, the
wavelength range of interest is 190-400 nm, therefore a pure N2 environment is
not required.

####  5.2.4 Methodology

The list below gives the symbols we used to represent the distances between
the important components of the setup throughout the chapter:

$d_{o}$: Distance between beam out slit of monochromator to focusing lens.

$d_{l-f}$: Distance between focusing lens and flipper mirror.

$d_{f-d}$: Distance between flipper mirror and detector.

$d_{l-d}$: Distance between focusing lens and detector.

$d_{f-m}$: Distance between flipper mirror and Mirror #8.

$d_{m-d}$: Distance between Mirror #8 and detector.

$d_{i}$: Total length of light path from the lens and detector (The sum of
$d_{l-f}$, $d_{f-d}$, $d_{l-d}$, $d_{f-m}$ and $d_{m-d}$ for each mode).

$A$: Focused beam spot at the detector at 225 nm wavelength.

$T_{flipper}$ Flipper mirror angle according to the rotation stage angle
indicator.

We performed wavelength scans between 190-400 nm at three different modes to
compute the final reflectivity results and their descriptions are listed as
follows:

No Reflection (NR) Mode:

Light from the monochromator slit was focused by the focusing lens and the
image was directly projected onto the detector. The slit-lens distance
($d_{o}$) was 31 cm, and the lens-detector distance ($d_{l-d}$) was 69 cm. The
wavelength scan at this mode was used as the reference measurement for
determination of the flipper mirror reflectivity. Only one measurement was
taken.

Flipper Mirror Reflection (FMR) Mode:

A schematic diagram is shown in Fig. 5.3a. The $d_{o}$ was the same as for the
NR mode. However, the light was then reflected off the Deep Ultra Violet (DUV)
flipper mirror and projected onto the detector. The total light path of this
mode was equivalent to the NR mode. The flipper mirror was rotated to 10.75∘
according to the rotation stage indicator and the light incident angle to the
flipper mirror was 47.5∘. Other key parameters are listed in Table 5.1. The
wavelength scans at this mode were used to determine the flipper mirror
reflectivity, as well as to provide a reference measurement to compute the
test mirror reflectivity. A total of eight wavelength scans were taken in the
8 hour testing period, which includes: FMR 1 and FMR 2, were measured
immediately after the NR measurement; FMR, Right FMR, Center FMR, Top FMR,
Bottom FMR and Corner FMR, were measured before the each of the Mirror #8
reflection measurements. The measurement names were given based on the Mirror
#8 reflectivity measurement location as shown in Fig. 5.4b.

Mirror #8 Reflection (M8R) Mode:

A schematic digram of this mode is shown in Fig. 5.3b. With $d_{o}$ reduced to
22.2 cm, the flipper mirror was rotated by 95∘, where the rotating stage
indicator should record 105.75∘. After the focusing lens, the light was first
reflected by the flipper mirror towards the test mirror instead of the
detector, then the test mirror reflected the light towards the detector. The
light incident angles to the flipper mirror and test mirror were 47.5∘ and
5.5∘, accordingly. From Table 5.1, the total light path of this mode was two
times longer than the NR and FMR modes. The focused spot size was also larger.
A wavelength scan in this mode was used to compute the test mirror
reflectivity. There were six measurements in total in this mode at the
locations indicated in Fig. 5.4b. Each Mirror #8 reflection measurement was
taken after the corresponding FMR measurement with the same naming convention.

The flipper mirror reflectivity is computed as

$R_{flipper}=\frac{I_{FMR}}{I_{NR}}$ (5.1)

where $I_{FMR}$ is the signal strength at arbitrary wavelength in Flipper
Mirror Reflection Mode, and $I_{NR}$ is the signal strength at the same
arbitrary wavelength in the NR Mode.

The Mirror #8 reflectivity is determined as

$R=\frac{I_{M8R}}{I_{FMR}}$ (5.2)

where $I_{M8R}$ is the signal strength at arbitrary wavelength in M8R Mode,
and $I_{FMR}$ is the signal strength at the same arbitrary wavelength in the
FMR Mode.

Table 5.1 summarizes the distances between the important components in
different measurement modes. The focal length of a lens is dependent on the
wavelength of the incoming light. At all three measurement modes, the focal
length for the optics are optimized for 225 nm where the signal is the
strongest, so a sharp image could be projected on the detector. The focused
image sizes at 225 nm are listed in Table 5.1. A wavelength scan from 190-400
nm would imply for most of the measurements (wavelengths), the projected image
on the detector would be slightly out of focus (blurry). However, the active
area of the detector is 1 cm$\times$1 cm, which is sufficient to detect the
slightly blurry image.

(a) No Reflection Mode

(b) Flipper Mirror Reflection Mode

Figure 5.3: Schematic diagram for different measurement modes. All important
components are labelled with a capital letter. A: Light source box; B:
Monochromator; C: Focusing lens; D: DUV Flipper Mirror; E: Detector; G: Mirror
Holding Frame. Position of the focusing lens (C) and flipper mirror (D) are
different between the two modes [42]. (Original In Color).

(a) Witness sample placement

(b) Mirror Measurement Position

Figure 5.4: (a) Witness sample locations with respect to the main mirror position in the ECI aluminization. (b) Reflectivity measurement locations on Mirror #8. (Original In Color). Table 5.1: The key parameters for all measurement modes. | No Reflection | Flipper Mirror | Real Mirror
---|---|---|---
| Mode | Reflection Mode | Reflection Mode
$d_{o}$ | 31 cm | 31 cm | 22.2 cm
$d_{l-f}$ | - | 42 cm | 43.3 cm
$d_{f-d}$ | - | 27 cm | -
$d_{l-d}$ | 69 cm | - | -
$d_{f-m}$ | - | - | 34 cm
$d_{m-d}$ | - | - | 60 cm
$d_{i}$ | 69 cm | 69 cm | 137.3
$A$ | 6 mm$\times$5 mm | 6 mm$\times$5 mm | 8 mm$\times$7 mm
$T_{flipper}$ | - | 105.75∘ | 10.75∘

####  5.2.5 Detector and Signal Schematic Diagram

The IRD AXUV-100G photo-diode detector has a different signal response
depending on the incoming photon wavelength, with the response curve shown in
Fig. A.1.

For all measurement modes, the continuous light signal was converted to a
pulsed light signal by a Thorlab MC100 optical chopper before reaching the
detector. The chopper had a two-slot chopping blade operated at a frequency of
14 Hz. The chopper generated a reference frequency, which was fed to the
reference channel of the lock-in amplifier. The detector generated a pulsed DC
differential signal through two connecting points, and the two signals were
fed into the input channels A and B of the lock-in amplifier. Then, the lock-
in amplifier would subtract signal B from A, and perform the filtering and
amplifying algorithm. The schematic diagram of the signal treatment is shown
in Fig. 5.5.

Figure 5.5: Systematic diagram for the signal output.

####  5.2.6 Lock-in Method and Monochromator Scanning Setting

The lock-in technique is used to detect and measure very small AC signals. A
lock-in amplifier can make an accurate measurement of small signals, even when
the signals are obscured by background noise thousands of times larger.
Essentially, a lock-in filter is a filter with an arbitrarily narrow bandwidth
which is tuned to the frequency of the signal. Such a filter rejects most
unwanted noise to allow the signal to be measured.

All lock-in measurements share a few basic principles. The technique requires
that the experiment be excited at a fixed frequency in a relatively quiet part
of the noise spectrum. Then, the lock-in can detect the response from the
experiment in a very narrow bandwidth at the excitation frequency.

In this measurement, the SR530 lock-in amplifier was operated in $R,\theta$
display mode, where $R$ stands for the signal magnitude and $\theta$ is the
phase between measured signal and the reference. The reference signal was
generated by the optical chopper, which chopped at the rate of 14 Hz.

There was no special option needed to filter the signal from the detector for
our measurement. During the data taking, the sensitivity and time constant
were the most important parameters that could dictate the data quality. The
sensitivity must be set to not overload the system. The time constant dictated
how fast the RC circuit of the output channel discharges, which would control
signal registration. A shorter time constant would allow a faster response to
the signal change but larger fluctuation in signal strength, and a longer time
constant would register stable signal strength but take longer to react to any
change in signal. The preferred time constant settings during the measurement
were 1 s and 3 s.

For the data taking, the monochromator was configured to perform a wavelength
scan from 190-400 nm at 5 nm steps. The dwell time at each wavelength was 20
s, and the associated time constant was 1 s or 3 s to generate a flat signal
output. If the dwell time was 60 s, the recommended time constant would be 3 s
or 10 s (the time constant does not need to be larger than 10 s).

A baseline (background) measurement should be taken before any measurement.
The monochromator light was diverted away from the detector so that only the
background light falling into the chopping frequency window could be
registered. The baseline (background) signal was measured to be 1.6$\times
10^{-5}$ V consistently, and $R_{offset}$ was set for this value during all
reflectivity measurements.

The signal magnitude $R$ can be can be calculated as [40]

$R=\sqrt{(x+x_{offset})^{2}+(y+y_{offset})^{2}}+R_{offset}$ (5.3)

where $x$ and $y$ are the horizontal and vertical components of the signal
magnitude $R$.

###  5.3 Results

####  5.3.1 Deuterium UV Lamp Thermal Test Results

The 3 Watt Hamamatsu deuterium UV source’s stability has been questioned from
the beginning. It was purchased 20 years ago by the FFL Facility at Jefferson
Lab. A thermal effect test was carried out to investigate the stability of the
UV lamp for long operation at room temperature. The monochromator was set for
225 nm wavelength, corresponding to a maximum signal output of 0.25 mV. The No
Reflection (NR) mode was used for this test, where the light from the
monochromator was directly focused onto the detector by the focusing lens. The
$x$ and $y$ outputs from the lock-in amplifier were monitored separately, as
shown in Fig. 5.6.

Figure 5.6: Lock-in amplifier $x$, $y$ strength monitoring of the Deuterium
lamp for 6 hour testing period. The blue curve is the $x$ output and green
curve is the $y$ output. $x$ and $y$ are the horizontal and vertical component
of the signal magnitude $R$. (Original In Color).

After 6 hours of operation, we observed no fluctuation in either $x$ or $y$
signal output, therefore the signal magnitude $R$ was also constant according
to Eq. 5.3. From this result, we can safely conclude the UV source performance
did not change due to heat and long operation, even without any method of
cooling and ozone removal. For measurement after the facility is fully
commissioned, the light source box will be constantly purged with cold N2
vapour, thus the UV source intensity should be stable during the long data
taking process.

(a) Detector Response

(b) Flipper Mirror Reflectivity

Figure 5.7: Flipper mirror reflectivity results. (a) shows the detector
response; (b) shows the percentage reflectivity results. The standard
deviations (error bars) of flipper mirror reflectivity curve are used as the
systematic errors ($\delta_{measurement}$) for the Mirror #8 reflectivity
curve. (Original In Color).

####  5.3.2 UV Detector Response Uniformity

The UV detector response uniformity was tested with a focused light beam at
225 nm wavelength over an area of 6 mm $\times$4 mm. Five measurements were
taken with the light beam shining at different locations: one at the center
and four at the corners with 1 mm clearance to the edges. The response signal
differences were around 1-2%. During the reflectivity measurement, after
changing the measurement location on the mirror, the focused light beam was
not guaranteed to be returned to the same location on the detector. The
detector uniformity would contribute an uncertainty in the mirror reflectivity
result. However, we think it was already included in the total error estimated
by data reproducibility from the eight repetition measurements at FMR Mode.

####  5.3.3 DUV Flipper Mirror Reflectivity Result

In Fig. 5.7a, one measurement of NR mode and eight measurements of FMR mode
wavelength scan curves are plotted. Measurements FMR1 and FMR2 were taken
immediately after the NR measurement. The other six FMR measurements were
taken separately within a 6 hour period. The ratios between each of the FMR
curves and the corresponding NR curves are plotted in Fig. 5.7b. The averaged
reflectivity curve and the standard deviations (error bars) are also shown.
The standard deviation of the eight reflectivity measurements is between 1-2%
at most of measured wavelength, excepting at 190 nm, where the signal is too
close to the baseline to be trusted. The standard deviations (error bars) of
flipper mirror reflectivity curve in Fig. 5.7b represent the reproducibility
error of a measurement, they are used as the systematic uncertainties
($\delta_{measurement}$) for Mirror #8 reflectivity curve in Fig. 5.8b.

Between 220-400 nm, the averaged reflectivity gradually increases as the
wavelength goes deeper into the UV region, peaks around 235 nm at 89.5%, and
then starts to decrease. At 200 nm the reflectivity is 86.5%. The measured
curve is different from the typical reflectivity curve for the Melles Griots
DUV flipper mirror, see App. A.2. The typical curve peaks around 197 nm at
94.5%, and gradually decreases as the wavelength increases. At 250 nm the
reflectivity is 91%. The difference between the measured and typical
reflectivity curves is around 3% between 220-250 nm; this difference keeps
increasing as the wavelength gets shorter; at 200 nm, the difference is 8%.

The possible explanations for such discrepancy are stated as following: the
typical DUV mirror reflectivity curve is a theoretical prediction under the
ideal condition, whereas the quality of individual DUV mirror could deviate;
also the mirror surface was scratched during the alignment process and the
scratches would result in some scattered light.

####  5.3.4 HGC Mirror #8 Reflectivity Results

Fig. 5.8a shows six sets of FMR and M8R detector response curves. Each set of
FMR and M8R measurements were taken after the mirror was adjusted to a new
position for testing. For the FMR measurements, the signal output was
unrelated to the mirror position because the light was only reflected from the
flipper mirror. For the M8R measurements, the lens and flipper mirror
positions needed to be adjusted, so the light could reflect of the flipper
mirror and Mirror #8. The FMR and M8R measurements were taken back-to-back,
and the lens and flipper mirror positions were adjusted remotely, so that the
testing environment was not disturbed. The orientation of the measured
position (Left FMR, Right FMR and so on) on the mirror are demonstrated in
Fig. 5.4b.

Fig. 5.8b shows the reflectivity curves of Mirror #8, which are the ratios
between the M8R and FMR measurements of the same label, for example (Left M8R
curve)/(Left FMR curve). From the results it is seen that between 200-300 nm
the reflectivity at the center is worse than at the corners; between 300-400
nm the reflectivity at the center is marginally better than at the corners.
The systematic uncertainties for the Mirror #8 reflectivity curve in Fig 5.8b
are taken as the same as that for the flipper mirror reflectivity curve, which
is shown in Fig. 5.7b.

(a) Detector Response

(b) Mirror #8 Reflectivity Results

Figure 5.8: Mirror #8 reflectivity results. In (a), the top set of
measurements are in FMR mode and bottom set of measurements are in M8R mode.
(Original In Color).

During the aluminization process, ECI placed nine 1′′ witness samples at
different locations across the test mirror as shown in Fig. 5.4a. The samples’
reflectivities were tested for product quality control, and these measurements
are attached in App. A.3. We did not measure the reflectivity of the witness
samples because the dispersed UV light beam from the monochromator is too
large to be completely reflected by them.

The ECI sample results have significant jitter, as shown in Fig. A.5. The
systematic uncertainty can only be estimated depending on the variation level
of the jitter. The estimated error below 230 nm is $\delta_{ECI}$ = $\pm$1%
and above 230 nm is $\delta_{ECI}$ = $\pm$0.5%.

Witness sample reflectivity curves #4, #5 and #7 were chosen to compare with
the measured reflectivity curves at positions Left, Center and Corner between
200-250 nm wavelength. The mirror testing positions of #4 and Left, #5 and
Center, #7 and Corner are similar. The witness sample curves were subtracted
from the measured curves, and the difference curve is plotted in Fig. 5.9. For
most of the points, the difference between the ECI witness sample and Mirror
#8 reflectivity is between 0-3%, which means the measured Mirror #8
reflectivity is higher than the witness sample reflectivity provided by ECI.
At the corner, the differences are slightly below 0, but 0 is still within the
error bar. In the ECI witness sample results there is a sharp peak at 205 nm
which does not exist in our measurement.

The error bar in Fig. 5.9 is calculated by

$\delta_{Difference}=\sqrt{\delta_{ECI}^{2}+\delta_{Measurement}^{2}}$ (5.4)

where $\delta_{ECI}$ is the estimated ECI witness sample curve error and
$\delta_{Measurement}$ is our measurement error.

Figure 5.9: The difference between the witness samples and the Mirror #8
measurement reflectivity results. (Original In Color).

###  5.4 Conclusion

Based upon our HGC Mirror #8 reflectivity measurements, we confirm that the
ECI witness sample measurement represents the reflectivity performance of the
aluminized mirror surface between 200-400 nm. At 200 nm wavelength, the
reflectivity is 10% lower than the original ECI claimed value of 76% with the
worst witness sample results. However, this difference decreases rapidly as
the wavelength increases. At 250 nm, the difference between the sample curves
and theoretical curves ranges between 0-5%. Based on the Geant4 simulation
results described in Chapter 6, the difference between the theoretical curve
and worst sample curve corresponds to a decrease of 2.5% in terms of the
detected photo-electrons and 0.13% in terms of detected pion events. This
effect is insignificant compared to other significant contributions to the UV
absorption in the deep UV region, i.e. the C4F8O gas absorption etc. We
believe the ECI aluminization quality meets our performance specifications and
is sufficient for the aluminization of mirrors to be actually used in the HGC.

We are also delighted to report that the reflectivity setup at Jefferson Lab
is capable of measuring the reflectivity of large size optics between 200-400
nm wavelength at normal atmospheric condition and is potentially able to
measure the reflectivity down to 165 nm wavelength under the N2 purged
environment.

###  5.5 Suggestions for Future Measurement

It is not obvious if the current UV source is suitable for UV measurements
below 190 nm. Further testing is required to confirm that the signal strength
is sufficiently strong and stable at such low wavelength. One of the possible
options is to replace the 3 Watt Hamamastu deuterium lamp with a Newport Oriel
30 Watt deuterium lamp, which has much higher intensity at lower wavelength.

At the moment, the setup is in a semi-automatic state, where some manual fine
adjustments are required. Also, the cameras which can help with the alignment
and status check are yet to be installed. For the Noble Gas Čerenkov mirror
reflectivity measurements, the detector hutch enclosure with pure N2
environment will forbid any manual adjustment and alignment. Some level of
automation is needed. However, it is not obvious at the moment how accurate
automatic alignment can be achieved. Manpower needs to be committed to resolve
this issue.

##  Chapter 6 Geant Simulation

###  6.1 Introduction

In order to better understand the optical alignment and expected performance
of the HGC, a computer simulation model was constructed using the standard
software toolkit for high energy physics known as Geant4 [43, 44]. The
original detector simulation platform was developed by the University of
Virginia and many customizations have been implemented to study the HGC
detector.

In this chapter, we will describe the basic setup of the HGC simulation model
and important results for the simulated performances.

There are three main detector performance criteria that we are studying: the
photon focusing efficiency, the photon detection efficiency and the charged
particle identification efficiency.

The photon focusing efficiency detection is defined as

$\eta_{\gamma}=\frac{\textrm{Number of photons within 12\leavevmode\nobreak\
cm diameter of 4 PMTs}}{\textrm{Total Number of Photons}}\,,$ (6.1)

this is the ratio between the number of photons focused onto the 12 cm
diameter active area for all four PMTs and the total number of Čerenkov
photons. $\eta_{\gamma}$ is used to study the alignment of the optical
components.

The photon detection efficiency is defined as

$\eta_{pe}=\frac{\textrm{Number of Photo Electrons in all 4
PMTs}}{\textrm{Total Number of Photons}}\,,$ (6.2)

this is the ratio between the number of photo-electrons output by all four
PMTs and the total number of Čerenkov photons. $\eta_{pe}$ is used to study
the effect of threshold cuts to optimize the $\pi/K$ separation in the
analysis of the simulation data. The quantities $\eta_{\gamma}$ and
$\eta_{pe}$ are closely related, and they will be explained in detail in
Section 6.4 where the PMT optimization results are presented.

The charged particle identification efficiency is defined as

$\eta_{particle}=\frac{\textrm{Number Identified Particles}}{\textrm{Total
Number of Particles}}\,,$ (6.3)

this is the ratio between the number of identified charged particles passing
the simulated HGC analysis threshold and the total number of charged particles
passing through the SHMS. $\eta_{particle}$ is used to study the particle
identification efficiency, which is of primary concern to the experiments that
will make use of the HGC. It is explained in more detail in Section 6.5, where
the overall projected performance results are presented.

Figure 6.1: HGC Geant4 model. The planes for Mirrors #1, #2 and #4 are
switched off to give better view to the photon trajectories in green. See the
text for further details. (Original In Color).

###  6.2 Simulation Setup

In the simulation, the HGC detector dimension and optics arrangement
corresponds to the real specification described in Chapter 3, where the inner
diameter of HGC detector is 170 cm, and length 117.1 cm. The center of the
detector is offset 5 cm in the $+x$ direction. The detector is configured to
contain C4F8O gas at 0.95 atm pressure, which has refractive index of 1.00137.

The center corner of the four mirrors are interleaved in the order of #: 4, 3,
2, 1, where mirror #4 is in $-x$, $-y$ quadrant (magenta), #3 is in $-x$, $+y$
quadrant (yellow), #2 is in $+x$, $-y$ quadrant (red) and #1 is in $+x$, $+y$
quadrant (blue). The closest mirror to mirror approach is between 7-10 mm.
There is a 5 cm overlap between Mirrors #1 and #2 and the same for mirrors #3
and #4 in the $y$ direction. The coordinate convention used in this chapter is
the same one as described in Section 1.3.1. In the HGC detector frame, the $z$
axis is parallel to trajectory of the charged particle, the x axis points down
to represent increasing momentum in particle distribution, and the $y$ axis
points to the left of the detector. In this chapter, the bottom two quadrant
mirrors are referred as $+x$ mirrors and top two as $-x$ mirrors. The $+x$
mirrors are tilted at 20∘ and the $-x$ mirrors are at -16.2∘ with respective
to the $x$ axis.

The simulation can run with three different types of mirror curvature:
parabolic ($\kappa<0$), spherical ($\kappa=0$) or oblate elliptical
($\kappa>0$), where $\kappa$ is the conic constant defined in Section 4.2.1.
From Chapter 4, we concluded that all of the manufactured HGC mirrors are
oblate ellipsoid. Thus, the fitted radius R$=$112.96 cm and conic constant
$\kappa=$0.94 for Mirror #6 were chosen to create the oblate elliptical
mirrors to provide more realistic simulation to study the detector
performance. The Mirror #6 quality is regarded as the average among the high
quality mirrors based on the mirror selection study results (see Section 4.5).
All optical alignment and projected performance results presented in this
chapter are with the oblate mirrors and the difference between the oblate and
spherical mirror performance is also presented herein.

Each of the HGC mirrors reflect the Čerenkov radiation towards the
corresponding 5” PMT. In the simulation, the PMTs are tilted at $\pm$42∘ with
respect to the x-axis. In the Geant4 model, the PMT assemblies consist of four
parts: cathode, quartz window and flange. Quartz window properties such as the
chemical composition and absorption length were configured according to the
properties of Corning 7980 quartz [21], where the material absorption length
are converted from the transmissivity curve from the manufacturer, see App. B.
The 10% photon loss was included in addition to the reflection due to the
Fresnel reflection at the quartz surface as a safety factor in the simulation.
The position of the cathode in the model corresponds to the real PMT cathode
position, and the wavelength dependent quantum efficiency and radial dependent
efficiency provided by the manufacturer [20] were implemented to predict the
PMT performance and particle identification efficiency.

In Fig. 6.1, a single Čerenkov event for a $\pi^{+}$ with 7 GeV/c momentum is
shown. The pion trajectory is in blue and enters the detector at the tip of
the green Čerenkov cone. The pion trajectory is mostly overlapped by the
Čerenkov photon trajectories shown in green. The generated Čerenkov photons
are reflected by the curved mirror towards the PMT assembly, and a small
portion of the photons are reflected back towards the mirror by the quartz
window; eventually these photons will be absorbed by the detector wall. The
position and energy of all photons are recorded and later plotted.

By default, there are 7 tracking planes to record the photon trajectory near
the PMT location: 3 planes before and 3 planes after cathode plane, as shown
in Fig. 6.1. The planes for Mirrors #1, #2 and #4 are switched off in this
figure to give a better view of the photon trajectories in green. The planes
are separated by 2 cm in the $z$ direction and are parallel to each other. The
PMTs can be turned on or off in the simulation; the PMT-on mode is used to
study the particle identification efficiency, and the photons will stop at the
cathode, therefore three tracking planes after the cathode plane are not used;
the PMT-off mode is used to study the optical alignment, therefore all 7
planes are used.

The HGC simulation is capable of studying the particle performances of
$\pi^{\pm}$, $K^{\pm}$ and protons, and the maximum sample size for a single
simulation is 0.5 M events. In order to make sure the particle distribution at
SHMS focal plane is accurate, 0.5 M charged particle events were generated for
our study by a collaborator [45] using the single arm SHMS Monte-Carlo (MC) of
Hall C. This is a detailed MC of the SHMS magnetic optics, and the
trajectories of 0.5 M charged particles are accurately tracked through the
magnetic field elements. The generated particle distribution file records for
every event: position and momentum in the $x$, $y$ directions in front of the
HGC detector, and the momentum ratio $\delta$ with range of $-13<\delta<22$%,
where $\delta=\frac{\Delta P}{P}$ is the percentage value used to reconstruct
the momentum acceptance based upon the SHMS central momentum. The incident
particle momentum for each event is computed as

$P_{Real}=P_{Central}(1+\delta/100)$

where $P_{Central}$ is the specified central momentum of the SHMS and it is
set for 3 and 7 GeV/c in our study.

(a) Particle Distribution

(b) Total Čerenkov Photon Envelope

(c) Captured Čerenkov Photon Envelope

(d) Missed Čerenkov Photon Envelope

Figure 6.2: All plots are generated with 7 GeV/c central momentum $\pi^{+}$.
(a) the $\pi$ envelope in front of the HGC detector; (b) the Čerenkov photon
envelope at the mirror plane; (c) the focused Čerenkov photon envelope at the
mirror plane; (d) missed focused photon envelope at the mirror plane. Note
that all plots show the events distribution in x-y plane. The photons in (c)
are reflected by the mirrors and optically focused onto the PMT active area
when all PMT efficiencies are switched off in the simulation. Thus, photons
which are missed due to the PMT inefficiency are not included in this plot,
only those due to the imperfect optics. The photons in (d) are also reflected
the mirror but not focused onto the PMT active area. Approximate mirror
boundaries are shown in (b), (c) and (d). (Original In Color).

###  6.3 Particle and Čerenkov Envelopes

The generated particle distribution by the single arm SHMS MC at the HGC
detector front surface is plotted in Fig. 6.2a, where the detector front
surface is 18.80 m and the SHMS focal plane is $z$=18.10 m from the scattering