ABSTRACT 
 
 
 
 
Title of Document: EXPERIMENTAL STUDY OF BEAM HALO 
IN INTENSE CHARGED PARTICLE BEAMS. 
  
 Hao Zhang, Ph.D., 2014 
  
Directed By: Professor Rami Kishek, IREAP 
Professor Patrick O?Shea, Department of ECE 
 
 
 Beam halo is a common phenomenon that occurs in most intense particle 
accelerators, and refers to collections of particles that stray far away from a well-
defined central beam core. Often in high-intensity beams, the space charge force 
induces halo. Even for low intensity accelerators, the beam halo could occur in the 
injection section before the particles are accelerated to relativistic speed. The most 
severe effects from beam halo are emittance growth and beam loss. Emittance growth 
can cause the degradation of beam quality, and beam losses will impose restrictions 
on the beam current. Although one can use a larger aperture to compensate this, the 
overall cost will increase exponentially. In this dissertation, we address the halo 
phenomenon and formation mechanism in intense charged particle beams. Although 
most of the experiment and simulation study of halo is based on the University of 
Maryland Electron Ring, it is applicable to a wide range of accelerators in the same 
intensity regime.  
  
 We first discuss a matching procedure and rotation correction for the beam 
envelope. The gradients of four quadruples in the injection are independently adjust 
to match or mismatch the beam. The gradients of two skew quadruples in the 
injection are independently adjusted to correct the beam rotation. We succeed in 
matching the UMER beams and find out that the envelope mismatch and beam 
skewness are the major sources for halo formation in UMER. Halo could be drive out 
even in very early stage such as in 2 or 3 mismatch oscillations with large mismatch 
or beam rotation. 
 We simulate the halo formation in UMER lattice till about 10 mismatch 
oscillations with higher beam intensity in the frame of two envelope mismatch 
modes. In experiment, we generate envelope mismatch mode with different mismatch 
level (parameter) by adjusting the four quadrupoles in the injection. The agreement of 
the envelope between experiments and simulations is satisfactory for mismatch 
parameter in the range of 0.8-1.2. Emittance and beam width are obtained from 
tomography and adaptive optical masking and imaging method separately for 
comparisons with the simulation as well as the maximum emittance growth predicted 
by a free energy model and maximum particle radius predicted by a particle-core 
model. The experiments confirm the predictions from both the simulation and the 
theory with reasonable agreement.  
 We also further investigate the adaptive masking method for halo imaging, 
and apply it for halo diagnostics at JLAB FEL facility, and for imaging of the injected 
beam at the SLAC SPEAR3 storage ring. 
 
  
 
 
 
 
 
 
 
 
 
 
 
EXPERIMENTAL STUDY OF BEAM HALO IN INTENSE CHARGED 
PARTICLE BEAMS. 
 
 
 
By 
 
 
Hao Zhang 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2014 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Rami Kishek, Chair 
Professor Patrick O?Shea, Co-Chair 
Professor Victor Granatstein 
Professor Thomas Antonsen 
Professor Adil Hassam, Dean?s Representative 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Hao Zhang 
2014 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  ii 
 
Dedication 
???????????????? 
??????????????? 
To my beloved parents, Lang Zhang and Furong Tang.  
In memory of my grandma, Yibi Li. 
  iii 
 
Acknowledgements 
 I am grateful for the years I have spend in University of Maryland. Along the 
process to obtain a degree, many people have inspired me with their wisdom, 
kindness, compassion and creativity. First and foremost is my advisor Professor Rami 
Kishek, who recruits me and lead me into the accelerator field. During these years, he 
has motivated me to challenge myself not only in the safe area which he was familiar 
with but also in the blind spot that we have to explore. I have been influenced deeply 
by his approach of presenting ideas and manner of thinking about problems, which 
will benefit for my whole life. I would like also thank my co-advisor Professor 
Patrick O?Shea, who also executes as Vice President in one of the top public 
university in the nation, but still shares time with me about the blueprint of my 
research during these years. His leadership and insight have encouraged me to 
accomplish this dissertation but also to bigger goal in the future. Dr. Ralph Fiorito 
served as my research advisor with the diagnostics project. His requirement of 
pursuing excellence, attitude of hard working and style of carefulness and 
rigorousness equipped me with full armor for my research. I can?t forget the days we 
work together in an optical table until 3:00 am in the midnight, which would be one 
of my precious memories in my life. 
 I would like to cherish the memories of Professor Martin Reiser, who gave his 
book as gift to me and cared about my research in his last days. I would also thank Dr. 
Brian Beaudoin, Dr. Santiago Bernal, Dr. Dave Sutter, Dr. Timothy Koeth, Dr. Diktys 
Stratakis (BNL) and Dr. Chao Wu for their tremendous help in experimental setup, 
UMER ring maintenance and wonderful suggestions. I would like to thank Dr. Irving 
  iv 
 
Haber, Dr. David Grote (LBL), Dr. Jean-Luc Vay (LBL) and Dr. Steve Lund (LBL) 
for providing the Warp code and their wonderful discussion and suggestion on 
simulation. I would like to thank Dr. Anatoly Shkvarunets, Professor Carsten Welsch 
(University of Liverpool), Dr. Shukui Zhang (JLAB), Dr. Dave Douglas (JLAB), Dr. 
Steven Benson (JLAB), Dr. Jeff Corbett (SLAC), Dr. Kai Tian (SLAC), Dr. Alan 
Fisher (SLAC), Dr. Walter Mok (SLAC), and Dr. Toshiyuki Mitsuhashi (KEK) for 
their help in optics design, opening their facility and also their useful discussion for 
the DMD based experiments. Thanks also to Dr. Donald Feldman, Dr. Massimo 
Cornacchia, Dr. Karen Fiuza, former student Dr. Eric Montgomery, Dr. Jayakar 
Charles Tobin Thangaraj, Dr. Christos Papadopoulos and my colleagues Dr. Yichao 
Mo, Dr. Kamal Poor Rezaei, Dr. Blake Riddick, Dr. Zhigang Pan, Eric Voorhies, 
Will Stem, Kiersten Ruisard who contributed to this dissertation in many ways. 
Finally, I thank my friends in Grace Fellowship and Chinese Christian Church of 
Maryland for their spiritual supports and care during these years.  
  
  v 
 
Table of Contents 
 
 
Dedication ..................................................................................................................... ii 
Acknowledgements ...................................................................................................... iii 
Table of Contents .......................................................................................................... v 
List of Tables .............................................................................................................. vii 
List of Figures .............................................................................................................. ix 
Chapter 1: Introduction ................................................................................................. 1 
1.1 General background of beam halo ...................................................................... 1 
1.2 Previous theoretical and simulation study of beam halo .................................... 2 
1.3 Previous experimental study of beam halo ......................................................... 5 
1.4 Organization of this dissertation ......................................................................... 7 
Chapter 2: Experimental Setup and Simulation Tools .................................................. 8 
2.1 Introduction of UMER ........................................................................................ 8 
2.2 Basic Diagnostics in UMER ............................................................................. 10 
2.3 Imaging System ................................................................................................ 12 
2.4 Phase space tomography ................................................................................... 15 
2.5 Adaptive optical mask method for halo measurement ...................................... 16 
2.5.1 DMD Diffraction Effects ........................................................................... 17 
2.5.2 Spatial Resolution ...................................................................................... 19 
2.5.3 Single Pixel Response ................................................................................ 23 
2.5.4 High Dynamic Range Measurement of the point spread function (PSF) .. 24 
2.6 Simulation tool .................................................................................................. 27 
2.6.1 Trace3D...................................................................................................... 27 
2.5.2 WARP ........................................................................................................ 27 
2.7 Chapter conclusion............................................................................................ 28 
Chapter 3: Transverse Matching of Space Charge Dominated Beam ........................ 29 
3.1 Verify Eddy current of pulsed quadrupole and its compensation ..................... 29 
3.2 General concept of envelope match for UMER beam ...................................... 41 
3.3 Simulation code used for envelope matching ................................................... 44 
3.4 Empirical method using response matrix .......................................................... 46 
3.5 Correction for Skewness ................................................................................... 49 
3.6 Transverse beam matching for 6mA beam ....................................................... 50 
3.7 Transverse beam matching for 21mA beam and 80 mA .................................. 61 
3.8 Chapter conclusion............................................................................................ 68 
Chapter 4: Major sources for halo formation in UMER ............................................. 70 
4. 1 Mismatch and halo formation .......................................................................... 70 
4.2 Injection Quadrupole scan ................................................................................ 75 
4.3 Beam rotation and halo formation .................................................................... 77 
4.4 Chapter conclusion............................................................................................ 81 
Chapter 5:  Halo formation in mismatch modes ......................................................... 82 
5.1 Envelope Mismatch and Mismatch modes ....................................................... 82 
5.1.1 Envelope mismatch mode .......................................................................... 82 
5.1.2 Particle core model .................................................................................... 84 
  vi 
 
5.2 Procedure for generating the pure-mode mismatch[42] ................................... 87 
5.3 Verification of pure mismatch mode using simulation ..................................... 89 
5.4 Maximum extent of halo radius and Emittance growth in simulation .............. 97 
5.5 Experiment study of the mismatch mode ........................................................ 108 
5.6 Chapter conclusion.......................................................................................... 120 
Chapter 6:  DMD based application in accelerators ................................................. 121 
6.1 Halo Experiments at Jefferson National Lab FEL Facility ............................. 121 
6.1.1 Experiment Setup ..................................................................................... 121 
6.1.2 Mask generation ....................................................................................... 122 
6.1.3 Preliminary Results from JLAB............................................................... 124 
6.2 Injected beam measurement in SLAC SPEAR3 ............................................. 125 
6.2.1 Experiment Setup ..................................................................................... 125 
6.2.2 Point spread function Measurement ......................................................... 127 
6.2.3 Injected beam imaging in off-duty test .................................................... 129 
6.2.4 Beam Mismatch Experiment .................................................................... 134 
6.3.5 Injected beam image in on-line test ......................................................... 139 
6.3 Chapter Conclusion ......................................................................................... 142 
Chapter 7:  Conclusion.............................................................................................. 143 
7.1 Summary ......................................................................................................... 143 
7.2 Future work ..................................................................................................... 145 
Appendices A: Sample matching code in Trace3D .................................................. 147 
Appendices B: Sample simulation code in python ................................................... 149 
Bibliography ............................................................................................................. 158 
 
 
 
 
 
 
 
 
 
 
  vii 
 
List of Tables 
 
Table 2.1: UMER design parameters?????????????????..9 
Table 2.2: Parameters of UMER beams????????????????10 
Table 3.1: Initial parameters of 6 mA beam?????????????.....51 
Table 3.2: Envelopes and slopes in 3 location for a matched 6 mA beam from 
Trace3D????????????????????????52 
Table 3.3: Hard edge model of UMER quadrupole??????????..?52 
Table 3.4: Magnets location and matching solution for 6mA from 
Trace3D????????????????????????53 
Table 3.5: Beam sizes of 6mA for 3 cases in Fig. 3.19?????????.?56 
Table 3.6: Rotation angle of the near match case for 6mA????????...57 
Table 3.7: Match beam parameters in each chamber and their statistics for 
6mA?????????????????????????..57 
Table 3.8: Phase space measurement of beam parameters for 6mA beam??...60 
Table 3.9: Initial parameters for 21mA and 80 mA beams????????...61 
Table 3.10: Envelopes and slopes in 3 locations for a matched 21 mA and 80 mA 
beams from Trace3D???????????????????62 
Table 3.11: YQ and QR1 settings for 21 mA and 80 mA beams??????....62 
Table 3.12: Magnet settings after empirical match for 21 mA and 80 mA 
beam????????????????????????.....64 
Table 3.13: Match beam parameters in each chamber and their statistics 
(21mA)????????????????????????.67 
  viii 
 
Table 3.14: Match beam parameters in each chamber and their statistics 
(80mA)????????????????????????.68 
Table 4.1: Beam rotation angles at each screen for three ISkew6 settings 
(6mA)????????????????????????...78 
Table 5.1: Initial conditions for the matched beam (6mA and 21mA) for Warp 
simulation input?????????????????..???.90 
Table 5.2: Initial conditions for Warp input of the breathing mode (6 mA)??.91 
Table 5.3: Initial conditions for Warp input of the quadrupole mode (6 mA)?..91 
Table 5.4: Initial conditions for Warp input of the breathing mode (21 mA)?...92 
Table 5.5: Initial conditions for Warp input of the quadrupole mode (21 
mA)?????????????????????????..92 
Table 5.6: Setting of injection quadrupoles Q2-Q5 for breathing mode mismatch 
(6mA)????????????????????????.109 
Table 5.7: Setting of injection quadrupoles Q2-Q5 for quadrupole mode 
mismatch (6mA)???????????????????....109 
 
 
 
  ix 
 
List of Figures 
Figure 1.1: Beam envelope oscillations and the trajectory of a single test particle in 
a mismatched beam???????????..????????..3 
Figure 1.2: Schematic of the beam-halo measurement setup in 
LEDA?????????????????????????.5 
Figure 1.3: Block diagram of the beam-core matching experiment transport line in 
IHEP ???????????????????...????.?..6 
Figure 2.1: Schematic layout of UMER?????????????..?.??8 
Figure 2.2: Screen assembly in UMER????????????????..11 
Figure 2.3: Illustration of air pressed actuator??????????????12 
Figure 2.4: Illustration of Imaging System??????????????.....13 
Figure 2.5: Imaging system GUI??????????????????....14 
Figure 2.6: illustration of the tomography method to reconstruct the phase space: 
top shows the diagram of the configuration; middle shows the 
corresponded phase space change of beam; bottom shows beams in 
different quadrupoles settings and the reconstructed beam picture in  x-
x? space????????????????????????16 
Figure 2.7: Schematic of halo imaging optics using DMD???????...?.17 
Figure 2.8: (a) Single wavelength diffraction pattern; and (b) white light 
diffraction pattern, both formed from a 450 rotated DMD with all 
pixels set at +120????????????????????..18 
Figure 2.9: (a) Resolution target; left: full view, right: (b) Expanded (? 16) 
view?????????????????????????..20 
  x 
 
Figure 2.10: (a) Horizontal and vertical scan of resolution target corner shown 
above in Fig. 6.3 and (b) Related Gaussian functions??????..22 
Figure 2.11: Response of a single DMD pixel to uniform white light 
illumination??????????????????????..23 
Figure 2.12: Sketch of the experimental setup for PSF measurement of the first 
optics channel?????????????????????...25 
Figure 2.13: Measured PSF of the first optical channel......................................?.26 
Figure 3.1: Comparison of beam sizes between experiment and simulation before 
correction???????????????????????..31 
Figure 3.2: Comparison of beam sizes between experiment and simulation with 
peak gradient per ampere for YQ model lowered by 0.67????...32 
Figure 3.3: YQ current profiles at different injection points. The injection pulse 
from the pulse dipole is illustrated as the rectangular box at time 0s. 
The horizontal axis is time and the vertical axis is current????..34 
Figure 3.4: Screen images at RC1, measured at different injection points of the YQ 
pulse for the 6mA beam (as shown in Fig. 3.3)????????...35 
Figure 3.5: X, Y centroids and beam sizes from screen images, measured at 
different injection points of the YQ pulse for the 6mA beam (as shown 
in Fig. 3.3). The horizontal axis is the prepulse time and the vertical 
axis is the x, y centroid or beam size?????????????35 
Figure 3.6: Screen images at RC1, measured at different injection points of the 
QR1 pulse for the 6mA beam (as shown in Fig. 3.3)?????.?..36 
  xi 
 
Figure 3.7: X, Y centroids and beam sizes from screen images, measured at 
different injection points of the QR1 pulse for the 6mA beam (as 
shown in Fig.3.3). The horizontal axis is the prepulse time and the 
vertical axis is the x, y centroid or beam size???????..??.37 
Figure 3.8: Beam current measured at the RC10 wall current monitor, (a) -38 ?s 
pre-pulse with an YQ current of 6.51A; (b) 208 ?s pre-pulse with an 
YQ current of 6.51A. The horizontal axis is time and the vertical axis 
is the uncorrected voltage from the oscilloscope??????...?..38 
Figure 3.9: Beam current measured at the RC10 wall current monitor. (a) -38 ?s 
pre-pulse with an YQ current of 5.34 A, (b) -208 ?s pre-pulse with an 
YQ current of 5.34 A. The horizontal axis is time and the vertical axis 
is the uncorrected voltage from the oscilloscope???????.?39 
Figure 3.10: Beam current measured at the RC10 wall current monitor with a 258 
?s pre-pulse and YQ current of 4.638A and QR1 of 4.754A. The 
horizontal axis is time and the vertical axis is the uncorrected voltage 
from the oscilloscope?????????????????.?..40 
Figure 3.11: Diagram of UMER FODO lattice in one period??????..??41 
Figure 3.12: Magnet components in UMER injection for matching??????44 
Figure 3.13: Simulated beam sizes in screen location before and after empirical 
method: (a) 2*Xrms (b) 2*Yrms????????????????????..???.49 
Figure 3.14: schematic diagram of (a) skew quadrupole and (b) piece of printed 
circuit???????????????????????..?..50 
Figure 3.15: R-R? phase space plot of 6mA beam in aperture (a) and IC1 (b)??.51 
  xii 
 
Figure 3.16: Envelope in injector for a match solution from Trace3D 
simulation??????????????????????.....53 
Figure 3.17: Comparison of beam (a) before and (b) after compensate Eddy current 
in experiment??????????????????..??.?.54 
Figure 3.18: Comparison of beam sizes in each chamber before after beam match in 
experiment:  x axis (upper); y axis (lower)??????????..55 
Figure 3.19: Beam images  for 6 mA beam in each chamber for (a) solution from 
Trace3D, (b) after empirical match, (c)after empirical match and 
rotation correction??????????????????....?56 
Figure 3.20: Comparison of rotation angle (red) before and (blue) after rotation 
correction???????????????????????..57 
Figure 3.21: Plot of 2*Xrms of 6 mA beam in each chamber with 3 cases in Fig. 3.19 
compared with Trace3D prediction????????????..?58 
Figure 3.22: Plot of 2*Xrms of 6 mA beam in each chamber with 3 cases in Fig. 3.19 
compared with Trace3D prediction?????????????..59 
Figure 3.23: Phase space plot (x-x?) of matched beam in screen RC2-RC5???60 
Figure 3.24: x-x? phase space plot of 21 mA beam in aperture (a) and IC1 (b)?...61 
Figure 3.25: x-x? phase space plot of 80 mA beam in aperture (a) and IC1 (b)...?62 
Figure 3.26: Beam images for 21 mA beam in each chamber for (a) solution from 
Trace3D, (b) after empirical match and (c) after empirical match and 
rotation correction????????????????????64 
Figure 3.27: Plot of 2*Xrms of 21 mA beam in each chamber with 3 cases in Fig. 
3.26 compared with Trace3D prediction??????????.?.66 
  xiii 
 
Figure 3.28: Plot of 2*Yrms of 21 mA beam in each chamber with 3 cases in Fig. 
3.26 compared with Trace3D prediction???????????..66 
Figure 3.29: Comparison of rotation angle (red) before and (blue) after rotation 
correction???????????????????????..67 
Figure 3.30: Matched beam images in screens for 80 mA beam???????..68 
Figure 4.1: Images of 6mA beam in RC2, 5, 11 and 12: (a) matched beam, (b) 
mismatched beam, (c) simulation of mismatched beam without lattice 
rotation, and (d) simulation of mismatched beam with lattice rotation. 
Note the mismatch is generated by reduce one of the injection 
quadrupole (Q5)..???????????????????.....71 
Figure 4.2: Normalized beam profile in x axis of matched (blue) and mismatched 
(red) 6 mA beam at RC1?????????????????..71 
Figure 4.3: Normalized beam profile in y axis of matched (blue) and mismatched 
(red) 6 mA beam at RC1?????????????????..72 
Figure 4.4: Normalized beam profile in x axis of matched (blue) and mismatched 
(red) 6 mA beam at RC11?????????????????73 
Figure 4.5: Normalized beam profile in y axis of matched (blue) and mismatched 
(red) 6 mA beam at RC11?????????????????74 
Figure 4.6: Comparison of reconstructed phase space of (a) matched  and (b) 
mismatched 6mA beam??????????????????75 
Figure 4.7: Images of 6 mA beam with injection quadrupole Q4 scan????..76 
Figure 4.8: Images of 6 mA beam with injection quadrupole Q5 scan????..77 
Figure 4.9: Images of 6mA beam with three different initial rotation..????.78 
  xiv 
 
Figure 4.10: Normalized beam profiles in x axis with three setting of Iskew6 at 
RC1?????????????????????????..79 
Figure 4.11: Normalized beam profiles in y axis with three setting of Iskew6 at 
RC1?????????????????????????..79 
Figure 4.12: Normalized beam profiles in x axis with three setting of Iskew6 at 
RC12????????????????????????....80 
Figure 4.13: Normalized beam profiles in y axis with three setting of Iskew6 at 
RC12?????????????????????????80 
Figure 5.1: Poincar? plot of mismatched beam (? = 0.62, ? = 0.5)?????...86 
Figure 5.2: FFT analysis of envelope from the breathing mode for 6 mA (left: x-
axis; right: y-axis)???????????????????....93 
Figure 5.3: FFT analysis of envelope from the quadrupole mode for 6 mA (left: x-
axis; right: y-axis)???????????????????....94 
Figure 5.4: Simulated amplitude of the mismatch oscillation versus mismatch 
parameters for 6mA???????????????????.95 
Figure 5.5: FFT analysis of envelope from the breathing mode for 21 mA (left: x-
axis; right: y-axis)????????????????????96 
Figure 5.6: FFT analysis of envelope from the quadrupole mode for 21 mA (left: 
x-axis; right: y-axis)??????????????????.....97 
Figure 5.7: Simulated amplitude of the mismatch oscillation versus mismatch 
parameters for 21 
mA???????????????????.98 
  xv 
 
Figure 5.8: Comparison of simulations using different number of macro particles 
(6 mA)????????????????????????98 
Figure 5.9: Simulated 6mA beam images at start (upper row), RC1 (middle row) 
and RC12 (lower row) with mismatch parameter from 0.6-1.4 for 
breathing mode?????????????????????.99 
Figure 5.10: Simulated 6mA beam images at start (upper row), RC1 (middle row) 
and RC12 (lower row) with mismatch parameter from 0.6-1.4 for 
quadrupole mode???????????????????.?.99 
Figure 5.11: Simulated number ratio of particles outside 3*RMS region of 6 mA 
beam with mismatch parameter ? =0.6 in breathing mode??...?.101 
Figure 5.12: Simulated number ratio of particles outside the 3*RMS region of 6 
mA beam with different mismatch parameter and different mismatch 
mode at screen in RC12????????????????.?102 
Figure 5.13: Simulated maximum particle radius along the way propagating to 
RC12 versus mismatch parameter compared with prediction from 
particle core model??????????...103 
Figure 5.14: Emittance growth verses mismatch parameter at RC12 for 6mA beam. 
The solid curve shows maximum growth from the free energy 
model????????????????????????.104 
Figure 5.15: Simulated 21mA beam images at start (upper row), RC1 (middle row) 
and RC12 (lower row) with mismatch parameter from 0.6-1.4 for 
breathing mode????????????????????...106 
  xvi 
 
Figure 5.16: Simulated 21mA beam images at start (upper row), RC1 (middle row) 
and RC12 (lower row) with mismatch parameter from 0.6-1.4 for 
quadrupole mode??????????????????..?..106 
Figure 5.17: Simulated 21mA maximum particle radius at screen in RC12 versus 
mismatch parameter??????????????????...107 
Figure 5.18: Emittance growth verses mismatch parameter at RC12 for 21mA beam. 
The solid curve shows maximum growth from the free energy 
model.................................................................................................107 
Figure 5.19: Images of 6 mA beam at RC1, 2, 5, 6, 11, 12 with mismatch parameter 
from 0.6 to 1.4 of breathing mode?????????????.110 
Figure 5.20: Images of 6 mA beam at RC1, 2, 5, 6, 11, 12 with mismatch parameter 
from 0.6 to 1.4 of quadrupole mode???????????..?111 
Figure 5.21: Comparison of Envelope between simulation and experimental data 
for 6mA beam in breathing mode (mismatch parameter ? = 0.8)?.112 
Figure 5.22: Comparison of Envelope between simulation and experimental data 
for 6mA matched beam (mismatch parameter ? = 1.0)???...?.112 
Figure 5.23: Comparison of Envelope between simulation and experimental data 
for 6mA beam in breathing mode (mismatch parameter ? = 
1.2)????????????????????????.....113 
Figure 5.24: Comparison of Envelope between simulation and experimental data 
for 6mA beam in quadrupole mode (mismatch parameter ? = 
0.8)?????????????????????????.113 
  xvii 
 
Figure 5.25: Comparison of Envelope between simulation and experimental data 
for 6mA beam in quadrupole mode (mismatch parameter ? = 
1.3)????????????????????????.?114 
Figure 5.26: Setup of the optical mask method for halo measurement???.?.115 
Figure 5.27: Calibration image of DMD indicating the edges of the DMD in CCD 
coordinate. The DMD is illuminated by a flush light. Camera is in 
shutter mode with 5 ?s exposure time???????????...115 
Figure 5.28: Calibration image of screen. The 80 mA beam is defocused to fill full 
of the screen for this calibration. There is straight light from 
background since we did not shut down all the light for the calibration. 
Camera is in gate mode with gate time 500 ?s for 100 integration 
frames????????????????????????.116 
Figure 5.29: Beam images of full beam (upper row) and after core being masked 
out (lower row) for breathing mode (6mA beam).?????...?.117 
Figure 5.30: Beam images of full beam (upper row) and after core being masked 
out (lower row) for quadrupole mode(6mA beam).??...??..?..117 
Figure 5.31: Beam half widths at 1% level of the maximum intensity versus 
mismatch parameter for both breathing and quadrupole mode 
mismatch. The comparison is between measured beam widths and 
simulated widths at RC12 (6 mA beam).??????????...119 
Figure 5.32: Measured RMS emittance growth for 6mA at RC12 for both breathing 
and quadrupole mode mismatch. The solid curve show maximum 
  xviii 
 
growth from the free energy model and the dashed curve is from 
simulation.??????????????????????..120 
Figure 6.1: Mask generation GUI interface??????????????..122 
Figure 6.2: Image of illuminated DMD with a check board mask for 
calibration??????????????????????..123 
Figure 6.3: Unmasked and masked OSR images of the JLAB FEL beam??...124 
Figure 6.4: Normalized horizontal scans of beam profiles????????...125 
Figure 6.5: Layout of the optical system in the first synchrotron beam line of 
SPEAR3??????????????????????.?126 
Figure 6.6: Log normalized intensity profile of the SPEAR3 stored beam; (a) first 
3 decades; (b) last 4 decades and insert showing aperture function of 
the visible beam line??????????????????..128 
Figure 6.7: Schematic plot of the triggering system???????????.131 
Figure 6.8: Timing table for injected beam imaging??????????....132 
Figure 6.9: Masked with time-gated images of injected beam turn 6 with 
increasing single-bunch current??????????????.133 
Figure 6.10: a) Horizontal intensity profile of the injected beam including PSF 
contribution from stored beam; b) 2?rms injected beam size vs. stored 
beam current????????????????????.?..134 
Figure 6.11: First 20 turns of the injected beam for three cases of matching 
condition??????????????????????....135 
Figure 6.12: Beam centroid motion of the injected beam for the three different BTS 
matching condition. (a) x axis; (b) y axis?????????.?.137 
  xix 
 
Figure 6.13: Beam size evolution of the injected beam with three different BTS 
quadrupole matching conditions. (a) x axis; (b) y axis?????..138 
Figure 6.14: Total intensity of the injected beam versus injected rate for turn 6, 10 
and 18??????????????????????...?..139 
Figure 6.15: Beam images in different turns during normal operation????..140 
Figure 6.16: Turn by turn evolution of beam centroids and beam sizes during 
normal operation???????????????????....141 
 
  1 
 
Chapter 1: Introduction 
1.1 General background of beam halo 
 
 Beam halo is a common phenomenon in particle beams [1, 2], especially for 
modern, advanced, accelerators where high beam intensities lead to strong space charge. 
Although there are many definition of beam halo [3 - 6], there is no well-accepted, 
rigorous one. The halo is generally understood as a population of particles reaching large 
transverse radii relative to a more intense, centralized portion of the beam, which is called 
the ?core?. To reach these larger radii, halo particles must possess higher energies than 
other particles.  Hence, a significant fraction of well-formed halos includes particles that 
are initially in the core, but will be pushed to large radii at a later time because of their 
higher transverse velocities. Beam halo is associated with emittance growth and thus 
decrease the beam quality. Here, the term emittance is a common characteristic of beam 
quality, which is proportional to the phase space volume but dynamically depends on the 
detailed knowledge of phase space [7]. A serious practical implication of halo is that halo 
particles travelling far from the center of the beam can hit the beam pipe, producing a 
number of undesirable effects in addition to the beam loss [8, 9]. For a high energy 
particle beam, the lost particles contribute to the nuclear activation of the wall material, 
increasing the radiation background and possibly causing damage to beam line 
components. Especially for positively charged beams, the lost halo particles can cause 
secondary electron emission, or knock off desorbed neutrals that then get ionized, 
contributing to an electron cloud that complicates the beam dynamics [10, 11].  
  2 
 
Halo mitigation is difficult. Although larger beam pipes can be used to 
accommodate halos, the cost of the required larger magnets, radio frequency cavities, 
etc., can be significant. Moreover, since most halo are actually inside the core with larger 
velocities, simple particle scrapers that remove particles beyond a certain radius are 
ineffective unless used in combinations, with phase-space rotations in between [12]. 
Understanding the causes of halos is crucial, therefore, to guide efforts to eliminate them 
or mitigate their negative effects. 
1.2 Previous theoretical and simulation study of beam halo 
 A number of theories have been developed, backed by simulations, to describe the 
formation mechanism of beam halo [13-15]. These studies have shown that there are 
many factors which can cause halo [16], e.g. intra-beam scattering, collective 
instabilities, misalignments, magnet errors, noise and resonances associated with both 
intrinsic incoherent processes and space charge forces. The most successful model in 
illustrating halo formation is the particle-core model [13, 14], which describes the halo as 
a parametric resonance between single-particle oscillations, and the collective oscillations 
of a mismatched core. Under the assumption of a round continuous beam with a uniform 
spatial density propagating in a uniform beam transport system with azimuthal symmetry 
and a linear radial focusing force, they describe the beam envelope in Eqn. 1.1(also see it 
in  [7]),  
  
    (1.1) 
where R is twice the rms beam radius, ko is the wavenumber corresponding to the 
uniform external focusing force, ? is the RMS emittance, and K the generalized 
2 2
2
02 3- - 0
d R Kk R
ds R R
?+ =
  3 
 
perveance. All these parameters are well defined in Reiser?s book [7]. To be emphasized 
here, k02R represents the external focusing force, ?2 /R3 is a force related to the beam 
emittance, and K/R is the space charge force.  For a perfectly matched beam, these three 
forces should be balanced, meaning 
     
     (1.4) 
When these forces are not balanced, the beam envelope will be mismatched, as the pink 
curve shown in Fig. 1.1. 
 
Figure 1.1:  Beam envelope oscillations and the trajectory of a single test particle in a 
mismatched beam. 
 
 
 Even though the space charge force is linear inside a uniform beam, it is nonlinear 
outside. Thus, a single particle that ventures just outside the edge of the beam will 
experience this nonlinear force.  Furthermore, the single-particle oscillation frequency is 
modified by the space charge force, and is different for particles wholly contained within 
the core from that of particles oscillating outside it.  It is thus possible for a single particle 
oscillation to resonate with the core mismatch oscillations and be driven to large 
2
2
0 3
Kk R
R R
?= +
  4 
 
amplitude, as shown by Gluckstern in Ref. [13]. Using particle-core model, Wangler in 
Ref. [14] showed that there is maximum amplitude of particles in a mismatch beam that 
depends on the magnitude of the mismatch parameter, which give us a rough idea of 
beam loss from the halo formation.   
 Another consequence from beam mismatch is emittance growth, which can be 
related to the free energy model developed by Reiser [17]. It can predict the conversion 
of beam free energy from mismatch oscillations into the thermal energy of the beam 
based on the mismatch strength as well as the tune depression ratio of the beam. 
 Other researchers have further developed the particle-core model.  Qiang [18] 
extended it to a 3D mismatched anisotropic beam, and Ikegami [19] studied it in a 
periodic focusing channel.  Kishek [15] found that halo can also arise from skew 
mismatches, i.e. caused by quadrupole rotation errors. The linear coupling between the 
two transverse directions leads to additional mismatch modes that can resonate with 
single-particle trajectories.   
 More recently, Papadopoulos [12] studied the removal methods for beam halo. He 
found that, even for a hypothetical, ideal, collimator that removes all the halo particles in 
phase space, the halo can regenerate if the mismatch oscillations that led to the halo are 
still present.  This implies that, it is necessary to have a detailed theoretical understanding 
of the halo formation mechanism, supported by experimental testing and simulation, 
rather than a reliance on collimators for halo removal. 
 
  5 
 
1.3 Previous experimental study of beam halo 
 There have been few experiments dedicated to the systematic study of halo.  Most 
accelerators lack the flexibility to adjust the beam intensity over a wide enough range the 
effects from space charge, or suitable diagnostics to detect the beam halos. The halo 
measurement of LEDA (Low Energy Demonstration Accelerator) in Los Alamos 
National Laboratory (Fig. 1.2) is an exception [8]. The novel transverse beam-profile 
scanners [20] are used to detect both the beam core and halo profiles over a high dynamic 
range. A proton beam is injected at 6.7 MeV, peak current of 75 mA and a 1 Hz 
repetition rate, with a 30 ?s pulse length.  The beam was matched by adjusting the first 
four quadrupoles such that the RMS beam sizes are unchanged in the last eight scanner 
locations. The mismatch parameter ?, defined as the ratio of the RMS size of the initial 
beam to that of the matched beam is varied by adjusting the first four quadrupoles. The 
profile and maximum extent of the resulting halo, as well as the growth of the RMS 
emittance was measured as a function of the mismatch parameter. The results indicated 
good agreement with predictions from the free energy and the particle-core models.  
 
Figure 1.2: Schematic of the beam-halo measurement setup in LEDA [8] 
 These experiments at LEDA were limited to only about ten mismatch oscillations, 
and thus examined only the early stage of the mismatch oscillations. Moreover, the tune 
  6 
 
depression ratio of the beam is about 0.82-0.95, which is still in the emittance dominated 
region. Additionally, the LEDA experiments suffered from an initial halo, which 
complicated the analysis of the results.  The LEDA machine has since been dismantled, 
so continuing those studies requires the use of a different facility.  
 Similar experiment was carried out recently [21] in IHEP for code validation as 
well as helping design the Accelerator Driven Subcritical System in China. A Proton 
beam was injection from a 3.5 MeV radio frequency quadrupole (RFQ) into 28-
quadrupole FODO transport channel. They succeed in beam match in x axis and the 
simulations are fairly successful in reproducing the rms properties of the measured 
matched beam profile in horizontal projections. Due to lack of beam profile diagnostics 
in y axis, there are some discrepancies between experiment and simulation for 
mismatched cases.  
 
Figure 1.3: Block diagram of the beam-core matching experiment transport line in IHEP  
 In halo formation study, proton beams are used for most of the experiments.  The 
disadvantage using proton beam as tools for halo study is that the diagnostic are very 
limited due to the destructive energy of the proton beam. The wire scanner, one of 
diagnostic used in both cases introduced above, has a reasonable dynamic range but only 
give us a beam profile in one direction. The tune depression ratio of the beams in 
University of Maryland electron ring (UMER)[22] is similar to that of most proton beam 
  7 
 
accelerator nowadays. Since UMER use low energy electrons, which is not so destructive 
for most diagnostics, it is an advantage to study the beam halo. This dissertation will 
focus on the UMER machine and discuss the halo formation phenomenon with 
simulation and experimental results. A novel diagnostic method to measure halo 
developed in my master thesis will be further discussed and used for the halo experiment 
in UMER as well as other experiments in major US national labs. 
1.4 Organization of this dissertation 
 I start in Chapter 2 to introduce the experimental setup in UMER, various 
diagnostic system and methods, and simulation tools. In Chapter 3, I discuss the 
procedure for beam match and then perform beam envelope match for three beams with 
increasing beam currents.  In Chapter 4, I discuss two major sources, transverse envelope 
mismatch and rotational error for halo formation. The latter one is first testified in 
experiment comparing with the discovery in simulation a couple of years ago. In Chapter 
5, the discussion is in frame of mismatch modes. The halo formation is systematically 
studied with mismatch parameters in two envelope mismatch modes. Experiment results 
of maximum particles radius and emittance growth for different mismatch strengths are 
compared with simulation and theory and show a strong support of these theories which 
can be applied for future design of high intensity and high current accelerators.  In 
Chapter 6, I talked about two side projects of halo related measurement by using the 
novel adaptive masking method in JLab FEL facility and in SLAC SPEAR3.  
  8 
 
Chapter 2: Experimental Setup and Simulation Tools 
 
2.1 Introduction of UMER 
 UMER is small compact electron storage ring with a low energy (10 keV) but 
relatively high beam current (~1-100 mA). It is designed to study the physics of electron 
beams from emittance to space charge dominated ones which can be scaled to higher 
beam energy with higher mass. Fig. 2.1 shows a schematic of the UMER layout, and 
Table 2.1 lists its key parameters. 
 
 
Figure 2.1: Schematic layout of UMER. 
 
 
  9 
 
Table 2.1: UMER design parameters [23] 
Beam Energy  10 keV 
? = v / c 0.2 
Pulse Length  20-120 ns 
Current  0.5-100 mA 
Ring Circumference  11.52 m 
Lap Time  197 ns 
Pulse Repetition Rate  10-60 Hz 
FODO Period  0.32 m 
Zero-current Phase Advance 0.760 
 
A key feature of UMER is the ability to vary the beam intensity. The related 
intensity parameter ?, the ratio between space charge force and external focusing force, 
ranged from 0 (emittance dominant) to 1 (space charge dominant), can be varied to 
generate beams which are dominated by the beam emittance to where intense space 
charge dominates the dynamics. The beam intensity is varied by using different apertures 
to change the beam current. A mechanically rotatable wheel with a number of apertures is 
located right after the electron gun exit to do this. The variable beam intensity allows us 
to study different halo formation mechanisms, including magnet alignment, mismatch, 
resonance, space charge and so on. Table 2.2 shows a list of aperture size, current, 
emittance and intensity parameter.  
 
 
 
 
 
  10 
 
Table 2.2: Parameters of UMER beams [22] 
Aperture# r0?mm? I (mA) ? (?m) ? 
1 0.25 0.6 7.6 0.275 
2 0.875 6 25.5 0.605 
3 1.5 21 30.0 0.901 
4 2.85 78 86.6 0.968 
5 3.2 104 97.3 0.978 
 
 The advantages to conduct a halo experiment in UMER are: 1) the beam is only 
10 keV, which means nothing to worry about the radiation. 2) the schedule is flexible 
since it?s a university based machine; 3) the intensity parameter could be easily varied to 
access the halo dynamics in a wide region; 4) as a ring, the duration time of a single pulse 
is not limited, and a recent success shows UMER can propagate the beam more than 1000 
turns which is 11 km [24]; 5) UMER is a pure research machine, so any new diagnostic 
can be simply implemented.   
 
2.2 Basic Diagnostics in UMER 
 In UMER, I use imaging method to detect the transverse distribution of the beam.  
The image source at UMER is a 31.75 mm diameter glass screen, coated with P-43 
phosphor (Gd2O2S:Tb, with 1.6 ?s response time) or fast phosphor (ZnO:Ga, with <3ns 
response time), which is located in each of the diagnostics chamber as shown in Fig. 2.2.  
  11 
 
 
Figure 2.2: Screen assembly in UMER.  
 
 
 Each chamber contains two diagnostics including the screen in the bottom and the 
button-type BPM in the top. These two diagnostic are interchangeable by a non-magnetic 
actuator shown in Fig. 2.3. Notice that the BPM is non-intercepted while the screen is not. 
The slider, which sets on the rail vertically and holds the flange by its horizontal clamp, is 
powered by high pressure gas instead of the original motor to avoid any magnetic field 
from the motor. There is a switch by the side to lift the screen up for taking images.  The 
phosphor screen is oriented at 90 degrees with respect to the beam direction. The 
fluorescent light is directed out of a vacuum system by a front surfaced mirror at 45? 
angle, and come out from the window.  
  12 
 
 
Figure 2.3: Illustration of air pressed actuator. 
2.3 Imaging System 
 In order to control the image taken process for tomography and empirical 
matching, I designed an imaging system based on Ethernet cameras [25] (GigE Vision 
Flea3 camera manufactured by Point Gray Research). The camera, which features 12 bit, 
variable gain and shutter time, and CCD with 768 ? 468 pixels, is directly used to image 
the beam with a build-in 16mm macro lens.   
 The illustration of the system is shown in Fig. 2.4. The cameras are connected to 
camera control PC through a gigabyte switcher. The camera control PC can control all 
the magnets power supply though the main control PC by sending a command line. 
  13 
 
1 2 3 4 5 6 7 8
UMER 
Magnets
GUI
Ethernet 
Camera
Gigabyte 
Switcher
Camera 
Control PC
Main 
Control PC
 
Figure 2.4: Illustration of Imaging System. 
  
 The whole system is integrated by a self-wrote Matlab GUI code. The interface of 
the GUI is shown in Fig. 2.5. It includes a camera control panel, Magnet control panel, 
image display and calculation panel, and the emittance panel. 
  In the camera control panel, Camera Select is used to specify which camera to 
use. Trigger control specifies whether one runs the camera at trigger mode or shutter 
mode, and which trigger mode one uses (the trigger mode can be referred to the camera 
manual). Gain and Shutter specifies the camera electronical gain and shutter time, which 
is to be set appropriately to prevent the saturation of the CCD and obtain enough beam 
signal from phosphor screen. One can also simply acquire one image by clicking Acquire 
button or runs the camera in continuous focusing mode by enabling the Focus button. The 
image obtained will be shown immediately in the display panel. The SaveAs button is 
used to save an image.  
  14 
 
 
Figure 2.5: Imaging system GUI 
 
 The image display and calculation panel is used to control the image display 
modification such as background subtraction, threshold subtraction, pseudo color and 
screen crop. Each of these corresponds to a checkbox in the panel. The Screen Crop 
checkbox is to remove the background outside the round screen illuminated by scattering. 
In the image, the screen is defined by a circle from calibration, the origin and diameter of 
which are filled in the edit box manually. The Calculate button calculates the beam 
centroid, beam size and rotation angle based on the previous modification while the 
Analysis button is used to do a series of such calculation of similar images. Here, similar 
means the group of images share the same calibration of the screen location.  
 The Magnet control panel is used to communicate with the main control computer. 
One can specify the number and the type of the magnet to be used in one experiment. The 
type of magnets includes solenoid, regular and skew quadrupole, horizontal and vertical 
  15 
 
dipole as well as the pulsed magnets. The current of the power supply for each magnet 
can be set manually by typing a number in the set box and clicking set button. 
Alternatively, one can do a currents auto scan by clicking the AutoPic button. In the auto 
mode, a currents scan file will be required and an image acquiring and saving will be 
initiated after each set of changes of the magnets currents.  
 The emittance panel is used for 2D phase space image generated by tomography. 
It will calculate the beam size, beam divergence, the coupled moments of the two, and the 
emittance.   
 In this dissertation, without special notification, the system describe above is the 
default imaging system. 
 
2.4 Phase space tomography 
 Similar to the computerized tomography used in medical and industrial 
applications, a method developed previous in UMER [26, 27] to reconstruct the phase 
space of the beam though the projections of the 4D beam into the configuration space 
with different angles as shown in Fig. 2.6. Three or four quadrupoles are used to produce 
these angles to cover a whole 180 degree range, so that any noise artifact is minimized. 
The reconstruction process is written in code using MATLAB, detail of which can be 
referred to [26, 27]. One can calculate the RMS emittance though the reconstructed phase 
space image and the error is usually below 7% [27]. 
  16 
 
17
Electron 
Beam
QR 1 ScreenQR 2 QR 3
X
Z
X?
X
?=46.3o ?=-13.3o?=163.2o
x
y
Phase space 
change  
x
x?
 
Figure 2.6: illustration of the tomography method to reconstruct the phase space: top 
shows the diagram of the configuration; middle shows the corresponded phase space 
change of beam; bottom shows beams in different quadrupoles settings and the 
reconstructed beam picture in  x-x? space  
 
2.5 Adaptive optical mask method for halo measurement 
 In accelerator related diagnostic, it requires a high dynamic range measurement 
for beam profile.  In my master?s thesis [28], I have developed a novel imaging 
diagnostic using adaptive masking method to measure beam halo with very high dynamic 
range based on the digital micro-array device (DMD). Based on the properties of DMD, 
my colleagues and I have designed a flexible imaging system which is easy to employ 
and apply to image the beam halo in most of the accelerators [29-33].  The essential 
features of the design are schematically shown in Fig. 2.7. 
  17 
 
Scheimpflug
compensation 
angle, ?
Laser
Beam 
splitter
24o
DMD
M1
L1
L2
Light
Source
(target)
CCD sensor 
M: mirror
L:  lens
1st image plane 
2nd image plane 
 
Figure 2.7: Schematic of halo imaging optics using DMD. 
 
The setup shown in Fig. 2.7 can be considered as two optical channels: 1) the 
source, lens L1 and the DMD surface, which is oriented perpendicular to the optical axis 
and is the first image plane; and 2) the tilted DMD image plane, considered as a new 
source, lens L2 and the CCD sensor, which is the second image plane. Note that the 
DMD in the first channel is perpendicular to the optical axis. This allows us to easily 
align the entire system will the help of a laser. The DMD is mounted on a combination 
rotation and 2 axis mirrors mount to facilitate the alignment. 
Mirror M1 further facilitates the alignment of the second channel.  Two rotational 
compensations [33] are required to use this system to image the source: 1) the DMD must 
be rotated about the optical axis; 2) the camera sensor must be rotated in the horizontal 
plane by 24 degree.  
2.5.1 DMD Diffraction Effects 
 The DMD behaves like a 2D optical grating. If illuminated by a single wavelength 
laser source, a cross like diffraction pattern similar to that of a rectangular mesh will be 
  18 
 
observed. In addition when all the micro-mirrors are rotated by +12? the DMD becomes a 
blazed grating with the central order reflected in the direction +24? in the horizontal 
plane with respect to the incident laser beam.  When the DMD is rotated by 45? the 
diffraction pattern also correspondingly rotates.  The resulting pattern is shown in Fig. 2.8 
(a).  Note that the central order has been suppressed so as not to saturate the imager. 
When a uniform source of white light illuminates this ?blazed? grating, the light is further 
dispersed in the horizontal plane producing the Fraunhofer pattern shown in Fig. 2.8 (b).  
These pictures were obtained by imaging the light diffracted by from the DMD (all pixels 
?on?) in the focal plane of a 200 mm focal length lens. 
(a) (b)
 
Figure 2.8: (a) Single wavelength diffraction pattern; and (b) white light diffraction 
pattern, both formed from a 450 rotated DMD with all pixels set at +120. 
  
 Note that for the white light diffraction pattern all the orders are smeared in the 
horizontal plane. This is due to the effect of both wavelength dispersion and overlapping 
of the light from the central spot and nearby first order diffraction spots. The latter effect 
is particularly evident in the central and first orders.  The picture further makes it clear 
that most of the light reflected by the DMD is contained in these two orders.  
 If the incident light on any grating is non-uniform but has a known distribution, 
i.e. is an ?object?, any order of the light diffracted by the grating can be used to image the 
  19 
 
?object?. Any aperture in the optics will filter out higher order diffraction spots which 
ultimately will reduce the imaging resolution so this must be checked for a given 
application. However, if the numerical aperture of the second channel optics is sufficient 
to accept the central two orders, most of the light diffracted by the DMD will be relayed 
into the second channel. This is indeed the case for our optics.  We have traced the rays 
corresponding to the angles of the central and first order diffraction spots with an optics 
code to insure that there is no vignetting of these rays through the optics of channel two.  
 
2.5.2 Spatial Resolution 
 There are several standard methods to measure the spatial resolution of an optical 
system.  We measured the spatial resolution of our optical system by imaging a ?knife 
edge? resolution target. This was constructed from a rectangular piece of, black anodized 
aluminum foil (Cinefoil) mounted on a white card, onto the DMD. The card is backlit 
with an adjustable intensity ?white light? source (i.e. an incandescent lamp). In these 
bench tests we make use of a PIMAX2 camera with a 1024 ? 1024 pixels CCD array.  
Each pixel of the CCD sensor is 13 ?m ? 13 ?m.  
 To insure that the DMD plane is in good focus on the camera to begin with, we 
program the DMD to accept a well known test image, i.e. a black and white checkerboard, 
which is included in the software supplied by the manufacturer to control the DMD. This 
pattern is ideal for adjusting the focus of the second channel since it originates with a few 
microns of the surface plane of the DMD chip and has multiple sharp black-white 
boundaries, i.e. the checkers. Once this source is focused onto the CCD, we turn all pixels 
to the ?on? state to reflect the resolution target image from the first channel into the 
  20 
 
camera light path. We then adjust the focus of the first lens system L1 and L2 to produce 
the best focus of the resolution target on the camera, without moving lenses L3 or L4. 
Figure 2.9 shows two views of the resolution target; the left hand side is a full view of the 
entire target, the right hand side a magnified view of the corner of the black rectangle 
portion of the target.   
(a) (b)
 
Figure 2.9: (a) Resolution target; left: full view, right: (b) Expanded (? 16) view. 
 
 Vertical and horizontal line scans of the corner of  the black corner of the target at 
the pixel level show an ?S? shaped dispersion curve. Figure 2.10 shows a horizontal scan 
which is averaged over 4 vertical pixels (see the  white box shown in Fig. 2.9(b)).  To 
analyze the resolution of the image we assume that the point spread function can be 
represented by a 2D Gaussian and convolve this function with the source intensity 
distribution A(x,y). An line scan across the image, e.g. I(x), the brightness along the linear 
  21 
 
scan (X direction) normal to the sharp linear boundary (Y direction, at Y = 0) is described 
by the convolution integral: 
2 2
2 2
( )( ) ( , ) exp
2 2x y
x x yI x A x y dx dy? ?
? ?? ??? ? ? ?? ? ?? ?? ?? ???    (2.1) 
where we assume A(x,y) = const. at X > 0 and A(x,y) = 0  at X < 0. A similar expression 
can be derived for a Y scan. One can easily show that the resulting intensity scans are 
error functions. Simple fits of  the experimental horizontal and vertical scan data to the 
corresponding error functions and their respective Gaussians functions, which are the 
derivatives of these functions, are shown in Fig. 2.10 (b).  The widths of the horizontal 
and vertical Gaussians are 3.0 and 2.1 respectively. The width provides as estimate of the 
resolution of the entire optical system.   
0 10 20 30
1
1.5
2
2.5
3
3.5
4
x 104
Pixels
In
te
n
si
ty
 
Le
v
el
 
(ar
b.
 
u
n
its
)
 
 
Vertical
Vertical erf fit
Horizontal
Horizontal erf fit
(a)
ertical
E f fit
orizontal
Erf fit
In
te
n
sit
y 
Le
v
el
 
(ar
b.
 
u
n
its
)
Pixels
 
  22 
 
0 5 10 15 20 25 30 350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pixels
No
m
a
liz
ed
 
Co
u
n
t
 
 
Vertical
Horizontal
2? 6.04.3
(b) Vertical
H ri tal
N
or
m
a
liz
ed
 
co
u
n
ts
Pixels
 
Figure 2.10: (a) Horizontal and vertical scan of resolution target corner shown above in 
Fig. 2.9 and (b) Related Gaussian functions. 
 
  
 We compare these measured values of the resolution of the DMD optical system, 
to that of a ?standard? optical system in which the DMD is replaced by a simple mirror. 
The major difference in the optics when the mirror is used is that Scheimpflug 
compensation is not required.  So for the resolution test with the mirror we rotate the 
CCD camera back to its initial orientation, i.e. perpendicular to the optical axis of the 
second optical channel.  We then observe the knife edge resolution target and follow 
same procedure described above to measure the resolution. The measured width of the 
black-white transition region in both the horizontal and vertical directions is ? ~3 pixels, 
which is approximately the same value measured for the DMD optical system. This 
indicates that the optical resolution of the DMD optics is essentially the same as that of 
the ?standard? mirror system. 
 
  23 
 
2.5.3 Single Pixel Response 
We have also measured the single pixel response of the second optical channel. 
To accomplish this, a uniform white light source beam is directed onto the entire DMD 
but with only a single micro-mirror activated.  The light from this single illuminated pixel 
is imaged onto the CCD via lenses L3 and L4. Figure 2.11 shows the resulting image 
(shown as a negative) as well as horizontal and vertical line scans across the image.  The 
scans show that the width of a Gaussian fit to distribution of intensity in both the 
horizontal and vertical directions is ?  ~ 3 pixels, which, interestingly, is the same as the 
measured optical resolution of the entire optical system. This result means that the DMD 
does not significantly influence the resolution or PSF of the optics system.  Furthermore, 
since the contribution to the PSF of the second channel is comparable to that of the 
overall system, the width of the PSF of the first channel must be less than that of the 
second channel or very close to it. 
520510500490
490
480
470
460
76
26191
76
26
19
1
 
                 Figure 2.11: Response of a single DMD pixel to uniform white light 
illumination. 
  24 
 
2.5.4 High Dynamic Range Measurement of the point spread function (PSF) 
 As far as dynamic range of the optical imaging system is concerned, it is only 
necessary to measure the PSF of the first channel.  The reason for this is that the second 
optical channel merely reimages the first image from the DMD plane with or without a 
mask in place, with an inherently low dynamic range imaging system.   
 The optical masking technique we will describe below uses a series of images 
each of which are taken using a CCD camera with a (DR ~100-1000). Thus the wings of 
the PSF of second channel below 10-2 or 10-3 of the peak intensity of any masked or 
unmasked image of the beam are not visible to the CCD camera. In fact any 
contamination of the true beam halo due to the wings of the real PSF will be visible in the 
first channel and affect image on the DMD.  This means that if the wings of the PSF 
exceed the beam halo in first image they will also exceed the beam halo in the 
distribution, which is reconstructed using the DMD masking method.  Similarly, if the 
wings of PSF are below the true beam halo level, then this will be the case in the 
reconstructed image as well.  Thus, for all intensive purposes it is sufficient to measure 
the PSF with high dynamic range for the first channel only. In order to do this, we use a 
wide band (white), ?point like? source with a homogeneous angular distribution, which 
closely mimics that of the phosphor screen used at UMER. A schematic of measurement 
system is shown in Fig. 2.12. The source is a white thread illuminated by light produced 
by a light-emitting diode (LED) that is 4 mm in diameter. The LED light is focused onto 
the thread by a Nikon camera lens which has focal length f = 28 mm and is oriented so 
that the normal input aperture of the lens faces the LED. The thread is a very good 
diffuser and scatters the focused LED light uniformly into a wide angle. We verified this 
  25 
 
by imaging the angular distribution with a CCD camera placed in the Fourier plane of the 
first optical channel which showed a uniform the irradiance across the sensor. The image 
of the LED on the thread is 0.45 mm in length and 0.15 mm in width, which corresponds 
to the diameter of the thread. When the light scattered by the thread is focused onto the 
CCD sensor by the first optical channel (i.e. lenses L1 and L2) the size of geometrical 
spot is on the CCD sensor is 0.04 ? 0.12 mm, or about 1.5 ? 4 pixels. 
 
LED
Lens F=28mm
Vertical thread
L1
CCD 
sensor
L2
 
Figure 2.12: Sketch of the experimental setup for PSF measurement of the first optics 
channel. 
  
 We measured the PSF of the first channel by progressively shifting the bright 
central spot in the image of our source away from the active area of sensor of CCD 
camera via a mechanical linear actuator and successively applying neutral density filters 
to attenuate the light to avoid saturating the CCD. By means of this technique we were 
able to utilize the whole size of the CCD sensor and achieve a dynamic range DR~107. 
The results are shown in Fig. 2.13.   
  26 
 
Pixels
0 100 200 300 400 500
N
or
m
a
liz
ed
 
In
te
n
si
ty
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
-1
-2
-3
-4
-5
-6
-7No
m
al
ize
d
in
te
ns
ity
 (l
og
10
)
pixels
1 2 3 4 5
 
Figure 2.13: Measured PSF of the first optical channel. 
  
 We note that the measured PSF has a much wider FWHM than the PSF, 
calculated from diffraction theory, i.e. the Airy disk.  Furthermore, the intensity of the 
wings of the measured PSF in the interval 100-500 pixels is several orders of magnitude 
greater than the level calculated from diffraction theory.  This is due to the combined 
effects of scattering, aberration, non-uniformities and diffraction in all the elements in the 
optical transport. The importance of this measurement will become apparent later when 
we discuss the dynamic range measurements. 
 Also, because of the finite size of our source, the measured PSF differs from that 
of a ?true point source?, especially in the region close to the source.  However, at large 
distances from the source the intensity distribution will be close to that of the ?true PSF? 
because it is not affected by the size of the source at sufficiently large distances, i.e when 
ratio of distance over the source size is large. 
 
  27 
 
2.6 Simulation tool 
2.6.1 Trace3D 
 TRACE 3-D [34] is an interactive beam-dynamics program that calculates the 
envelopes of a bunched beam, including linear space-change forces, through a user-
defined transport system.  In TRACE 3-D, the basic assumption is that all forces are 
linear or can be linearized. To linearize the space charge, the distribution of the beam 
should be postulated as uniform in real space. In realistic sense, the beam distribution is 
never uniform, so the envelope in this code is in RMS sense. Several element-fitting and 
beam-matching options are available that determine values for the beam-ellipse 
parameters or for specified transport-system parameters (such as quadrupole gradients) to 
meet specified objectives.  In this dissertation, I use it for beam matching and mismatch 
mode generation for experiment.  
 
2.5.2 WARP 
 In UMER, we use a self-consistence PIC code called WARP to simulate the beam 
behaviors [35].  It allows nonlinear space charge, emittance growth, image charge forces 
and arbitrary distribution that most matrix codes or tracking codes don?t have. It has been 
successfully benchmarked against UMER experimental data in the past [36]. In the 
simulation, considering the computing power, we usually use macro particles. Each 
macro particle represents millions of particles in the realistic sense.  The macro particles 
are advanced in a transverse slice under the impact of external forces and self-consistent 
self-fields using the leap-frog algorithm. In the code, we calculate the self-filed on a mesh 
of sufficient resolution to capture the beam potential variations. In our case, it has been 
  28 
 
shown to be accurate to use the x-y 2.5 dimension model [37] since the beams in UMER I 
use is about long enough compared to the pipe diameter and we focus on the transverse 
dynamic mostly in the longitudinal center of the beam in a short longitudinal distance. 
For large longitudinal distance, we might need full 3D simulation or keep 2.5D version to 
compare with the experiment results when the longitudinal confinement is applied e.g. 
using the longitudinal focusing [38].   
 
2.7 Chapter conclusion 
  In this Chapter, I introduce the experimental environment of UMER. I discuss 
several basic diagnostics including the actuator and screen system, the imaging system 
and tomography method for phase space reconstruction. Followed by my master thesis, I 
discussed the optical properties of the novel adaptive masking method for halo imaging. 
Finally, I discuss the simulation tools that will be used later.  
  29 
 
Chapter 3: Transverse Matching of Space Charge Dominated Beam 
 
 For a space charge dominated beam, transverse envelope match plays an essential 
role in control of beam loss and halo formation. It is believed that a mismatched beam 
will drive a parametric resonance that force the particle moving back and forth around the 
center core and then leaving to large radius. This well-modeled mechanism will speed up 
other processes forcing the individual particle near the beam edge to create halo. 
Especially in a ring, integer, half integer or higher order resonance will be amplified by 
this mechanism, and fasten the process which degrades the beam quality and eventually 
results in beam loss. Here I present a general way to obtain an envelope match for space 
charge dominated beams, and tried to eliminate the source for halo generation and beam 
quality degradation. The ultimate goal is to achieve a halo free beam for further study of 
the mechanism of different halo formation sources. The organization of this chapter is as 
follow. We first address an Eddy current problem [39] that we realized and then solved. 
Then it follows with discussion of the concept of basic envelope match of the space 
charge dominated beam. After that, envelope matching will be addressed with beams of 
different space charge levels or intensities.  
3.1 Verify Eddy current of pulsed quadrupole and its compensation 
 YQ and QR1 are special type printed circuit quadrupoles due to its location and 
functionality that they provide confinement for both injection and recirculation (see Fig. 
2.1). The radius from the printed circuit to the center of the quadrupole is more than twice 
larger than the normal quadrupole used in the ring or injection for the pipe merging. 
  30 
 
Accordingly, the current applied to the circuit should be larger to generate the same 
magnetic confinement. The principle current value for the power supply of YQ and QR1 
should be around 4.6 A. Since we don?t have a water cooling system for the quadrupoles, 
there are concerns that YQ and QR1 could be overheated and burn if their currents are 
DC. Thus YQ and QR1 are pulsed for certain amount of time (~0.2 ms) each injection 
(60 Hz), and usually good for more than 1000 turn operation of the less space charge 
beam in UMER. In the past, YQ are usually set to over 6.5 A (instead of ~4.6 A) to allow 
most of the beam go through. This introduces a big error in the match in the first turn 
which deviates from the code prediction.  
 In order to study the difference of current setting, I performed a scan of YQ and 
QR1. By taking 6mA beam images at the first screen downstream (at RC1), I compared 
the results in experiment and simulation. Note that, before the comparison in RC1, I 
found good beam size agreements between experiment data and simulation in both IC1 
and IC2 of the injection with different solenoid current values as well as different values 
of Q1, Q2, and Q3. This indicated a good model of regular quadrupoles and solenoid in 
UMER. However, the model failed at RC1 when I did a scan of YQ. Fig. 3.1 shows a plot 
of beam sizes vs. YQ current. From the plots, I found that the curve of 2*Yrms from 
experiment (purple solid line) is stretched and offset from the simulation curve (green dot 
line).  
  31 
 
3 4 5 6 7
2
3
4
5
6
7
8
9
10
11
YQ current (A)
Si
ze
s 
(m
m
)
 
 
2*Xrms(Exp)
2*Yrms(Exp)
2*Xrms(Sim-full gradient of YQ)
2*Yrms(Sim-full gradient of YQ))
 
Figure 3.1: Comparison of beam sizes between experiment and simulation before 
correction. 
 
It is not hard to have a guess that the simulation model for YQ has a bigger peak 
gradient per ampere than experiment. I compared the difference of two curves, which 
gave us an empirical value of ~0.67, which I applied to reduce the peak gradient per 
ampere for YQ in simulation to match with experiment. After this modification, the 
comparison result is shown in figure 3.2. It is getting close although not perfect, and this 
could be due to the same error by QR1.  
 
  32 
 
2 3 4 5 6 7 82
2.5
3
3.5
4
4.5
5
5.5
6
6.5
YQ current (A)
Si
ze
s(m
m
)
 
 
2*Xrms(Exp)
2*Yrms(EXP)
2*Xrms(Sim-reduced gradient)
2*Yrms(Sim-reduced gradient)
 
Figure 3.2: Comparison of beam sizes between experiment and simulation with peak 
gradient per ampere for YQ model lowered by 0.67 
 
 From the previous experience, if we set YQ as 7.3 A, the beam is good for several 
turn but lost very quick. At the meantime, if we set YQ as about 4.6 A, the value from 
code for beam matching,  the beam is lost first turn but finally the loss is much less than 
previous one. This phenomenon indicates that the YQ and QR1 strengths might be 
changed during each time the beam go through Y-section. If we consider that YQ and 
QR1 are pulsed, it is straightforward that there is Eddy currents in both quadrupoles. 
Assuming the beam is injected into Y-section too soon, before the Eddy currents in YQ 
and QR1 have fully stabilized, the peak field will be weaker; this can only be partially 
compensated by increasing the YQ current (e.g. set YQ current 7.3 A, the actual value is 
about 4.6A for first turn). As the transients die down, the fields inside YQ will continue 
  33 
 
to increase, mismatching and kicking the recirculating beam during each turn. This can 
explain the 1%/turn beam loss seen during the first 20-30 turns. In other words, an YQ 
current of 7.3 A is initially seen as 5 A. As the Eddy currents die down, the field inside 
keeps increasing, sweeping the beam towards a wall downstream. Thus, by increasing the 
width of these pulses and pre-pulsing them earlier, we should be able to operate YQ, QR1 
at currents of 4.6-4.8A and obtain good transport and matching.   
 Following this assumption, we will test the effect of different pre-pulse of YQ and 
QR1 on beam dynamics. In order to do that, the TTL triggers to YQ and QR1 were 
separated so that both could be independently delayed while the other remained fixed. 
We first test the different pre-pulse for YQ. This measurement was performed using 
the PIMAX-II camera located at RC1 to image the beam on the 1st turn. The injection 
pulsed dipole, gun trigger and camera trigger remained fixed while the overall YQ pulse 
delay was adjusted such that it sampled different regions of the pulse in time. Fig. 3.3 
below illustrates the concept. 
 
  34 
 
-80 -60 -40 -20 0 20 40 60
-10
-5
0
5
10
Cu
rr
en
t (A
)
Time (us)
 YQ 18us
 YQ 28us
 YQ 38us
 YQ 48us
 YQ 58us
 YQ 68us
 
Figure 3.3:  YQ current profiles at different injection points. The injection pulse from the 
pulse dipole is illustrated as the rectangular box at time 0s. The horizontal axis is time 
and the vertical axis is current. 
 
 The YQ pulse delay was shifted in 10 ?s steps, beginning with an 18 ?s pre-pulse 
(pre-pulse is the period of time beginning from the turn on of the YQ pulse to the turn on 
of the injection dipole pulser). Screen images are illustrated below in Fig. 3.4 for the 
6mA beam.  
 
 
 
Injection 
  35 
 
     
     
Figure 3.4:  Screen images at RC1, measured at different injection points of the YQ pulse 
for the 6mA beam (as shown in Fig. 3.3).  
 
 The x and y centroids as well as the rms beam sizes, obtained from these images, 
are illustrated in Fig. 3.5 below. 
10 20 30 40 50 60 70
-4
-3
-2
-1
0
1
2
3
4
5
6
YQ Variation
Ce
n
tr
o
id
 
an
d 
Be
am
 
Si
ze
 
(m
m
)
Prepulse Time (us)
 2x
rms
 2y
rms
 x-centroid
 y-centroid
 
Figure 3.5:  X, Y centroids and beam sizes from screen images, measured at different 
injection points of the YQ pulse for the 6mA beam (as shown in Fig. 3.3). The horizontal 
axis is the prepulse time and the vertical axis is the x, y centroid or beam size. 
18?s 28?s 38?s 
48?s 58?s 68?s 
21.3 mm 
40.1 mm 
x 
y 
  36 
 
 The x-centroid and vertical beam size changes substantially over the limited pre-
pulse time explored in this experiment. The original machine parameters set a pre-pulse 
of 38 us, but as seen in Fig. 3.4 and 3.5; both are still changing beyond that period. 
Vertically, the beam reduces in size overall by 64% at the waste and horizontally, the 
centroid drifts by 7.5 mm over 50 us, and continues to drift but at a slower and slower 
rate.  
Then we test QR1 with different pre-pulse. This measurement was also performed 
using the PIMAX-II camera located at RC1 to image the beam on the 1st turn. Once 
again, the injection pulsed dipole, gun trigger and camera trigger remained fixed while 
the overall QR1 pulse delay was adjusted such that it sampled different regions of the 
pulse in time. The QR1 pulse delay was shifted in 10 us steps, beginning with an 18 us 
prepulse. Screen images are illustrated below in Fig. 3.6.  
      
      
Figure 3.6:  Screen images at RC1, measured at different injection points of the QR1 
pulse for the 6mA beam (as shown in Fig. 3.3).  
 
The x and y centroids as well as the rms beam sizes, obtained from these images, 
are illustrated in Fig. 3.7. 
18?s 28?s 38?s 
48?s 58?s 68?s 
21.3 mm 
40.1 mm 
x 
y 
  37 
 
10 20 30 40 50 60 70
-2
-1
0
1
2
3
4
5
6
7
QR1 Variation
Ce
n
tr
o
id
 
an
d 
Be
am
 
Si
ze
 
(m
m
)
Prepulse Time (us)
 2x
rms
 2y
rms
 x-centroid
 y-centroid
 
Figure 3.7:  X, Y centroids and beam sizes from screen images, measured at different 
injection points of the QR1 pulse for the 6mA beam (as shown in Fig. 3.3). The 
horizontal axis is the prepulse time and the vertical axis is the x, y centroid or beam size. 
 
For QR1, the centroid and beam size settled down sooner than YQ, as this magnet sits 
above a reducer and does not sit above a metal block that houses the injection and 
recirculation beam pipes as YQ does.  
We then lengthen the YQ and QR1 pulses close to the existing limits of the power 
supplies. Beam current each turn were then captured for various injection cases, where 
the delay of injection pulse and YQ current was varied. This experiment was performed 
by capturing beam current traces from the wall current monitor at RC10, beginning with 
the nominal operating case using a matched setting calculated from code. The QR1 pulse 
was fixed in amplitude for all cases presented below, but the pre-pulse time was varied 
along with YQ amplitude. Figs. 3.8 a-b and 3.9 a-b below illustrates the results. 
  38 
 
0 5 10 15 20 25 30 35
-30
-25
-20
-15
-10
-5
0
5
10
Un
co
rr
ec
te
d 
Vo
lta
ge
 
fro
m
 
RC
10
 
(m
V)
Time (us)
 
0 5 10 15 20 25 30 35
-30
-25
-20
-15
-10
-5
0
5
10
Un
co
rr
ec
te
d 
Vo
lta
ge
 
fro
m
 
RC
10
 
(m
V)
Time (us)
 
Figure 3.8:  Beam current measured at the RC10 wall current monitor, (a) -38 ?s pre-
pulse with an YQ current of 6.51A; (b) 208 ?s pre-pulse with an YQ current of 6.51A. 
The horizontal axis is time and the vertical axis is the uncorrected voltage from the 
oscilloscope. 
 
(a) 
(b) 
  39 
 
0 5 10 15 20 25 30 35
-30
-25
-20
-15
-10
-5
0
5
10
Un
co
rr
ec
te
d 
Vo
lta
ge
 
fro
m
 
RC
10
 
(m
V)
Time (us)
0 5 10 15 20 25 30 35
-30
-25
-20
-15
-10
-5
0
5
10
Un
co
rr
ec
te
d 
Vo
lta
ge
 
fro
m
 
RC
10
 
(m
V)
Time (us)
 
Figure 3.9:  Beam current measured at the RC10 wall current monitor. (a) -38 ?s pre-
pulse with an YQ current of 5.34 A, (b) -208 ?s pre-pulse with an YQ current of 5.34 A. 
The horizontal axis is time and the vertical axis is the uncorrected voltage from the 
oscilloscope. 
 
(a) 
(b) 
  40 
 
When we operate the magnets with the original 38 ?s pre-pulse, with YQ and QR1 at 
a current of 6.51A and 5.34A respectively, we obtain many turns, as shown in Fig. 3.8a. 
If we extend the prepulse to 208 ?s (as shown in Fig. 3.8b), then we lose those turns. If 
we then reduced the strength of the YQ magnet to 5.34A (as shown in Fig. 3.9b), we 
obtain multiple turns again, indicating that the transients are settling down and were 
approaching the actual effective strength of the magnet.    
Finally, the settings that are being used currently have a pre-pulse of 258 ?s. We are 
able to run YQ and QR1 at values of 4.638 A and 4.754 A, respectively and obtain 
multiple turns as shown in Fig. 3.10. 
 
0 5 10 15 20 25 30 35
-30
-25
-20
-15
-10
-5
0
5
10
Un
co
rr
ec
te
d 
Vo
lta
ge
 
fro
m
 
RC
10
 
(m
V)
Time (us)
 
Fig. 3.10:  Beam current measured at the RC10 wall current monitor with a 258 ?s pre-
pulse and YQ current of 4.638A and QR1 of 4.754A. The horizontal axis is time and the 
vertical axis is the uncorrected voltage from the oscilloscope. 
  41 
 
3.2 General concept of envelope match for UMER beam 
 As can be seen for UMER configuration, the envelope match includes three 
sections:  injection, Y section and ring section.  
There are 35 FODO cells in the ring section. Each cell is 32 cm long, with focusing 
and defocusing quadrupoles located in 8 cm and 24 cm, and dipole in 16 cm related to the 
start of each cell.(Note that the focusing or defocusing here means for x-direction, and 
they are opposite for y direction, similarly hereinafter.) The dipole bends the ideal path of 
the electron beam by 100 (see Fig. 2.1). In our model, we ignore the dispersion by assume 
there is no energy spread. By this simplification, we can ignore the dipole component in 
the ring. However, if considering the contribution of the edge focusing from dipole, we 
should modify the calculation model by adding a small focusing quadrupole both in X 
and Y direction in each cell as shown in Fig. 3.11. Notice that there is a small difference 
in the focusing strength by dipole will result in the slightly asymmetry of the envelope in 
horizontal and vertical axes.  
 
 
Figure 3.11: Diagram of UMER FODO lattice in one period  
  42 
 
Y section is a special UMER type FODO. The dipole is pulsed for both injection 
and recirculation. The radius of the pipe in Y section is larger than in the ring, to allow 
pipe merging and a bigger acceptance. As a result, the quadrupole YQ and QR1 is bigger 
than normal quadrupole in UMER and need more current to create the same field as in 
the ring which will brings more heat.  For overheat issue from that high current, YQ and 
QR1 are pulsed in order to reduce heat as discussed in Section 3.1. Moreover, YQ 
provides focusing for both injection and recirculation, each with its pipe coming in with 
an angle and an offset in x direction, which actually complicate the match geometrically. 
In a FODO focus system, envelope match indicates the beam envelope and slope 
have a period the same as the FODO cell period (see Eqn. 3.1). Practically, It simply 
means the two pair of parameters (both in x and y) are identical when entering and 
exiting the FODO or at the mid-plane of two adjacent cells. When the quadrupoles? 
strengths are fixed, for electron beams in UMER with certain current without considering 
transverse emittance growth, the match condition (the envelope size and slope) in the 
beginning of each FODO is determined.  It can be directly derived from the envelope 
equations (see Eqn.3.2) with the periodic boundary condition as shown in Eqn. 3.1.  
    
  (3.1) 
          
   (3.2) 
 
where s is ideal beam path coordinate, S is the period of the lattice, X, Y, X?,Y? represent 
the transverse beam RMS sizes and slopes, K is the generalized perveance which depends 
( ) ( ), ( ) ( )
'( ) '( ), '( ) '( )
X s S X s Y s S Y s
X s S X s Y s S Y s
+ = + =
+ = + =
2
2
3
( )
'' ( ) 0K IX k s X
X X
?+ + + =
  43 
 
on beam current I, ? is the effective emittance and k is the field strength which depends 
on the lattice setting.  
The injection is designed to match the initial beam parameter from aperture to the 
required match condition determined by the ring FODOs. Fig. 3.12 shows a diagram of 
the whole injection, Y-section and partial ring in UMER, where the SD, Q, PD, YQ, QR 
represent steering dipole, quadrupole, pulsed dipole, Y-shape quadrupole and ring 
quadrupole respectively. Note in Eqn. 3.2, k2 is also dependent on quadrupole currents. 
From a known initial size and slope from the aperture, through the whole injection to 
obtain an envelope match condition before entering the ring, we have six quadrupoles and 
one solenoid to vary. Since there are only four constraints, the envelope size and slope in 
both x and y direction, it is enough to use only four quadrupoles to satisfy them. We 
usually fix the solenoid field. Q1 is the quadrupole close to the solenoid, which will 
separate the beam from radial symmetry to asymmetry. To avoid the effect from 
unbalanced image charge forces from too much asymmetry between horizontal and 
vertical axes and reduce the overlap of the fringe field between solenoid and Q1, we fix 
Q1 with a small value with respect to the ring quadrupole.  Under all these considerations, 
the four quadrupoles used to optimize the matching solution in the injection are Q2-Q6. 
We usually only scan 4 of them, but we also add some flexibility when the solution cause 
too much asymmetry in the injection which could introduce a nonlinearity to the beam 
distribution.  
  44 
 
YQ
 
Figure 3.11: Magnet components in UMER injection for matching. 
 
3.3 Simulation code used for envelope matching 
I used a tracking code Trace3D to get a match solution. Trace3D assumes linear 
field and no emittance growth, and have variety types of matching procedure, which is 
suitable for our beam experiment to get a rough match solution. Here, I use the emittance 
value from solenoid tomography result to replace the emittance in the ring with the 
assumption of no emittance growth. This is well enough since I only use the code for a 
rough condition. Later results will show that when beam current increase, nonlinear force 
will become more and more obvious, and the linear assumption of Trace3D is no longer 
that valid. Thus, the solution from the Trace3D will be less accurate. However, the 
solution will still be a starting point to start with when I discuss the empirical method 
later.  
  45 
 
For beam match in Trace3D, I start with the ring section. From the start to the end 
of each FODO cell, I can use Type 1 match procedure in Trace3D to get a solution of 
Twiss parameters which give a match solution inside the ring. As indicated by envelope 
equation, this solution is determined by the assumed emittance and the predefined lattice 
setting, specially the ring quadrupole strength. In UMER, the ring quadrupoles are setting 
to 1.826A with positive and negative alternating. In this way, the tune will be 6.694 away 
from the integer and half integer resonances. This quadrupole setting is not a necessary 
requirement, and be varied to any value which will give a new lattice and result in new 
matching Twiss parameters. For example, UMER could work at 2.220A, 2.068A, 1.518A 
for ring quadrupole, which give a tune about 7.630, 2.068, and 1.518 respectively. Note 
that, Twiss parameter is used in Trace3D or other accelerator field generally instead of 
beam size and slope which can easily interchange to each other through Eqn. 3.3. 
     
     (3.3) 
 
After I got the matched Twiss parameters for ring section, particularly the initial 
parameters in the beginning, we need to use the same Twiss parameters to determine the 
quadrupole settings for Y-section, which will allow the beam have a multiple turn match. 
Since one matched the beam for four independent Twiss parameters, ?x, ?y, ?x, and ?y, 
typically one needs four quadrupoles to do this job as the same as the discussion in the 
injection section. Therefore, I include QR71 and QR2 into this section and use Type 8 
matching procedure in Trace3D to find the solution for QR71, YQ, QR1 and QR2. 
2
( )
2
( )
2
( ) ( ) ( )
( ) /
'( ') /
1
x y
x y
x y x y x y
X Y rms
X Y rms
? ?
? ?
? ? ?
=
=
= +
  46 
 
Next step is to find appropriate injection quadrupole settings with the YQ and 
QR1 strength determined from multiple turn requirements and a matching condition from 
the ring lattice requirement. The components here will be solenoid, Q1-Q6, YQ, PD and 
QR1. The lattice starts from the aperture and ends with the entrance of a regular FODO 
cell. I set YQ and QR1 value from previous Y-section match, and input the final Twiss 
parameters as the matched initial Twiss parameters in the ring section. I also set the initial 
Twiss parameter for the injector as the ones I got from Solenoid Tomography. Using a 
Type 8 matching procedure in Trace3D, I will finally get a quadrupole setting of Q2-Q5.  
 
3.4 Empirical method using response matrix 
 In practice, many factors, such as beam initial properties, lattices imperfection, 
vacuum condition and etc., will affect calculations and simulations, and make them 
deviate from experiments. The magnets strength obtained though calculation and 
simulation does not necessarily yield a good matching condition as showed previously. A 
further empirical method [40] will be used to achieve the final match. Here, I first assume 
the mismatch is small (linear approximation), so that a matrix implementation could be 
applied. Followed the previous discussion, I use the quadrupoles Q2 ? Q5 for online 
adjustment and the matrix form is as follows:   
 
 
 (3.4) 
 Where 
Xi and Yi are the 2?rms transverse beam sizes in the i?s screen, which is measured with 
2 3 4 5
2
2 3 4 5 3
2 3 4 5 4, , ,
5
xi xi xi xii m
yi yi yi yii mI I I I
I
R R R RX X I
R R R R IY Y
I
?
?+ ?
?
? ?
? ?? ?? ? ? ? ? ?= ? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ?
  47 
 
E R ?
current magnet strength setting of I2, I3, I4, I5. Xm and Ym is the matched beam size on 
each screen. ?I is the magnet strength change from current setting to a better setting to 
minimize the mismatch. Rwij is the beam size response in i?s screen when changing 
quadrupole j (w can be x or y), defined as  
    
    (3.2) 
 
and can be measured by perturbing the current quadrupole strength and observing the 
beam size in each screen. The full matrix form can be rewritten and extended as 
 
 
 
(3.3) 
 
 
 
 
 
or simplified as  
E R= ?      (3.4) 
This is standard linear equations, and the optimal solution can be obtained in a least 
square sense, i.e. ? = (RTR)-1RTE. A better injection setting is given by I1- ?I1, I2- ?I2, I3- 
?I3, I4- ?I4. Details for this method can be referred to [40].  
i
i
w j
j
WR
I
?= ?
  48 
 
 I test this method with the simulations in Warp by scanning the quadrupoles Q2-
Q5, and monitoring the envelope change in periodic point, e.g. the screen position in the 
ring. Fig. 3.13 shows the comparison of before empirical correction (red star points) and 
the one after the empirical correction (blue circle point). There is an obvious 
improvement since the deviation of the beam sizes is much smaller after the empirical 
correction both in X and Y directions. In real experiment, one might need several 
implementations of this method to let the solution converge. 
2 3 4 5 6 73.5
3.7
3.9
4.1
4.3
4.5
Z (m)
X 
(2*
RM
S,
 
m
m
)
 
 
After empirical method
Before empirical method
 
(a) 
  49 
 
2 3 4 5 6 72.7
2.9
3.1
3.3
3.5
3.7
Z (m)
Y 
(2*
RM
S,
 
m
m
)
 
 
After empirical method
Before empirical method
 
(b) 
Figure 3.13: Simulated beam sizes in screen location before and after empirical method: 
(a) 2*Xrms (b)2*Yrms 
 
3.5 Correction for Skewness 
 Due to fringe field of solenoid and rotation error of injection quadrupoles, there 
will be a beam rotation complicating beam match. Here I use two skew quadrupoles in 
Q3 and Q6 to correct this rotation. The skew quadrupole is a type of UMER quadrupole 
with a normal pair of printed-circuits [40] and a 45-degree rotated pair as in Fig. 3.14. 
Each pair is powered by different current supplies, so the normal and skew components 
can be independently adjust. By scanning the skew quadrupoles and comparing the 
rotation angle of beam images in each chamber, one can minimize the rotation in least 
square sense similarly to the empirical matching. 
 
  50 
 
  
Figure 3.14: schematic diagram of (a) skew quadrupole and (b) piece of printed circuit. 
  
 Next sections, I will use these basics to discuss three specific matching cases of 
UMER beams. 
 
3.6 Transverse beam matching for 6mA beam 
 As indicated by the name, the beam current for this case is 5.96 mA close to 6 mA. 
The initial conditions are obtained from the beam phase space plot which is reconstructed 
from solenoid tomography. The reconstructed plot in x direction is shown in Fig. 3.15. 
For symmetry of two transverse directions at aperture and through solenoid, I found an 
initial condition listed in Table 3.1.  
  51 
 
               
 
Figure 3.15: R-R? phase space plot of 6mA beam in aperture (a) and IC1 (b) 
 
Table 3.1: Initial parameters of 6 mA beam  
Initial envelope values X (mm) X? (mrad) Y(mm) Y?(mrad) Emittance(um rad) 
At aperture 0.88 -16.20 0.88 -16.20 13.0 
At IC1 2.52 -7.09 2.52 -7.09 13.0 
 
 For ordinary operation setting, the ring quadrupoles is fixed to 1.826 A, which 
corresponding to a hard edge model of 659.0 G/m in strength, 4.475 cm in effective 
length. This results in a tune of 6.694. Using this hard edge model in trace3D, I found an 
envelope match solution in the ring listed in Table 3.2. 
 
 
 
  52 
 
 
Table 3.2: Envelopes and slopes in 3 location for a matched 6 mA beam from Trace3D 
Parameter Mid-plane of ring dipoles Mid-plane of BPMs Screen 
X (mm) 3.14 3.13 3.49 
X? (mrad) 11.78 11.88 12.86 
Y (mm) 3.04 3.01 2.69 
Y? (mrad) -11.87 -11.44 -10.38 
  
 The hard edge models used for quadruples in Y section are listed in Table 3.3. 
Using the match Parameter, I first obtain the setting for YQ and QR1 for recirculation 
using the match solver in Trace3D, which is 4.646 A and 4.735 A.   
 
Table 3.3: Hard edge model of UMER quadrupole [41] 
Quadrupole 
type 
Effective 
lenghth (cm) 
Peak Gradient 
per A (G/cm/A) 
Hard edge 
factor 
Average Gradient 
per A (G/cm/A) 
Injection 4.475 3.609 0.8354 3.015 
YQ 5.833 1.110 0.8557 0.950 
QR1 5.999 1.010 0.8965 0.905 
Ring 4.475 3.609 0.8354 3.015 
 
 I input the initials and match solution into trace3D code, and use its match module 
to get the current settings for injection quadrupoles. I use the same lattice as discussed in 
previous section. I obtain a current setting for matching shown in Table 3.4. The beam 
envelope from aperture to the second FODO is plotted in Fig. 3.16.   
 
  53 
 
 
Table 3.4: Magnets location and matching solution for 6mA from Trace3D 
Magnet Sol Q1 Q2 Q3 Q4 Q5 Q6 
Position (cm) 17.13 40.53 53.73 72.73 92.33 106.63 122.63 
Gradient (G/cm) 98.13 3.93 6.98 6.64 6.29 6.56 6.67 
Current (A) 5.500 1.090 1.935 1.842 1.742 1.819 1.849 
 
YQ QR1 QR2 QR3 QR4 QR5SOL Q1 Q2 Q3 Q4 Q5 Q6
Injection Ring
22
Y
X
S
X
Y
 
Figure 3.16: Envelope in injector for a match solution from Trace3D simulation. 
 
 In Fig. 3.17, I compare the beam images in each chamber before and after I solve 
the Eddy current problem. Both setting is from a solution of Trace3D. Since the field in 
YQ and QR1 is not built up thoroughly, there will be a huge mismatch caused by the field 
error. In both raw (a) of Fig. 3.17 and the plot in Fig. 3.18, one can see the oscillation of 
the beam sizes, which latter will be a source of halo formation and cause beam loss. From 
row (b), the beam still experience a small mismatch, and an empirical match and rotation 
correction will be performed later. 
  54 
 
RC6RC2
x
y
4 mm RC12RC11RC1 RC5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1RC1 RC2 RC5 RC6 RC12 RC13
4 mm
(a)
(b)
 
Figure 3.17: Comparison of beam (a) before and (b) after compensate Eddy current in 
experiment. 
0 5 10 152
3
4
5
Chamber number
Si
ze
 
(m
m
)
 
 
Old 2*Xrms
New 2*Xrms
 
0 5 10 152
2.5
3
3.5
4
Chamber number
Si
ze
 
(m
m
)
 
 
Old 2*Yrms
New 2*Yrms
 
  55 
 
Figure 3.18: Comparison of beam sizes in each chamber before after beam match in 
experiment:  x axis (upper); y axis (lower). 
 
 
 In practice, many factors, such as beam initial properties, lattices imperfection, 
vacuum condition and etc., will affect calculations and simulations, and make them 
deviate from experiments. I input the setting from Trace3D into experiment, and the 
result is shown in row (b) of Fig. 17 as well as row (a) of Fig. 19.  Here, the beam is close 
to match, but not the perfect. I use empirical method to adjust the match, and then find a 
matched beam showed in the row (b) of Fig. 19.  The final setting for Q2 ? Q5 is 1.992 A, 
1.785 A, 1.858 A, and 1.940 A.  Comparing these two cases, I conclude that I get a more 
regular shape beam and the size difference is greatly reduced. The average size of beam 
in each chamber are 3.45 in x axis and 2.62 in y axis, which is very close to what I expect 
in Trace3D, 3.49 mm in x and 2.69 mm in y. The standard derivation of the beam sizes 
after empirical matching are 0.17 mm and 0.14 mm for x and y axis repeatedly. In 
general, by performing this beam envelope matching, the beam loss is reduced and the 
beam can propagate more turns without any loss.  
 
  56 
 
 
Figure 3.19: Beam images for 6 mA beam in each chamber for (a) solution from 
Trace3D, (b) after empirical match, (c)after empirical match and rotation correction. 
 
 
 
Table 3.5: Beam sizes of 6mA for 3 cases in Fig. 3.19  
Chamber (mm) RC1 RC2 RC5 RC6 RC11 RC12 
2*Xrms (Trace3D solution) 3.48 3.70 3.65 3.33 3.18 3.70 
2*Xrms(Empirical match) 3.41 3.46 3.54 3.43 3.17 3.67 
2*Yrms (Trace3D solution) 2.46 2.73 2.55 2.96 2.35 2.76 
2*Yrms(Empirical match) 2.43 2.54 2.63 2.70 2.64 2.55 
 
 When viewing case after empirical matching, I notice that the beam have a 
rotation angle in different chamber, the data can be found in Table 3.5. I first suspect this 
angle could come from following sources: 1) imperfection of ring quadrupole such as 
rotation error, 2) the residual magnetic field from induction cell in RC4 and wall current 
  57 
 
monitor in RC10, 3) asymmetry in YQ, and 4) the skewness of injection. I did a leveling 
of all quadrupoles in the ring and a demagnetization of induction cell and wall current 
monitor to reduce the rotation from source 1 and 2. I insert a new skew quadrupole in Q3 
in the injection to correct the rotation in the injection and YQ since I found one 
previously installed skew quadrupole in Q6 is not enough to correct this rotation angle. 
The corrected beam images are shown in row (c) of Fig. 3.19 and the derivation of 
rotation angles (also see Table 3.6) is reduced from 15.0 degree to 6.9 degree.  
Table 3.6:  Rotation angle of the near match case for 6mA 
Chamber RC1 RC2 RC5 RC6 RC11 RC12 
Rotation (degree) 8.63 -8.04 21.5 -16.8 8.85 -12.8 
After correction(degree) 0.8 6.7 11.3 -6.7 0.0 -5.5 
2 4 6 8 10 12
-50
-40
-30
-20
-10
0
10
20
30
40
50
Ring Chamber Index

Ro
ta
tio
n
 
A
n
gl
e
 
(d
eg
re
e)
 
Figure 3.20: Comparison of rotation angle (red) before and (blue) after rotation correction. 
  
  58 
 
 I list the matched beam sizes, rotation angle and their average compared with the 
prediction from Trace3D in Table 3.7. The comparisons of beam sizes for the three cases 
are plotted in Fig. 3.21 and Fig. 3. 22. 
 
Table 3.7: Match beam parameters in each chamber and their statistics for 6mA beam 
Chamber RC1 RC2 RC5 RC6 RC11 RC12 Ave. Std. Trace3D 
2*Xrms(mm) 3.42 3.35 3.39 3.5 3.32 3.41 3.40 0.06 3.49 
2*Yrms(mm) 2.43 2.54 2.63 2.7 2.66 2.39 2.56 0.13 2.69 
Angle(o) 0.824 6.71 11.3 -6.7 -0.0 -5.5 1.0 6.9 0.00 
 
2 4 6 8 10 122.5
3
3.5
4
Ring Chamber Index
2*
X r
m
s 
(m
m
)

 
 
Prediction by Trace3D
Trace3D Solution
After Empirical Match
After Empirical Match and Rotation Correction
 
Figure 3.21: Plot of 2*Xrms of 6 mA beam in each chamber with 3 cases in Fig. 3.19 
compared with Trace3D prediction. 
  59 
 
2 4 6 8 10 12
1.8
2
2.2
2.4
2.6
2.8
3
3.2
Ring Chamber Index
2*
Y r
m
s 
(m
m
)

 
 
Prediction by Trace3D
Trace3D Solution
After Empirical Match
After Empirical Match and Rotation Correction
 
Figure 3.22: Plot of 2*Xrms of 6 mA beam in each chamber with 3 cases in Fig. 3.19 
compared with Trace3D prediction 
 
 Then I use tomography to reconstruct the beam phase space in x-x?. Since there 
will be four ring quadrupole involved for the phase space tomography in each screen, if 
one want to get phase space plot for RC1, he need to use QR1, which is not well modeled 
and might bring ambiguity to the reconstruction process. Therefore, I will not present any 
phase space plot in RC1. The phase space of other 5 chamber is shown in Fig. 23. 
Comparing the phase space images proceeding from RC2 to RC12, one can see there is 
minimum halo and the shape of beam in x-x? space is only slightly changed in these 
periodic locations. The beam mostly keeps a uniform distribution as indicated by the 
theory of space charge dominated beam.  
  60 
 
 
Figure 3.23: Phase space plot (x-x?) of matched beam in screen RC2-RC5. 
  
 The data are analyzed in a way that one can take a threshold to filter out the 
artifact that is introduced by the reconstruct algorithm. The threshold is chosen to match 
the beam sizes from phase space with previous results I measured from configuration 
space of the beam images in all the chambers. The results are listed in Table 3.8. The 
average and standard derivation of 2*X?rms is 13.54 mrad and 1.03 mrad, and those of the 
effective emittance in x direction is 13.74 ?m and 0.87 ?m, which is in the range of the 
prediction of a matched beam from Trace3D simulation.  In this sense, I obtain the match 
of the 6 mA beam for the first turn.  One can see that the beam still experience a small 
emittance growth which might due to other non-perfection in the ring. 
Table 3.8: Phase space measurement of beam parameters for 6mA beam 
 Trace3D RC2 RC5 RC6 RC11 RC12 
2*Xrms (mm) 3.49 3.35 3.39 3.50 3.32 3.41 
2*X?rms (mm) 12.86 14.60 13.00 15.10 12.8 12.9 
Effective emittance (?m) 13.00 13.90 12.91 13.42 13.893 15.29 
  61 
 
 
3.7 Transverse beam matching for 21mA beam and 80 mA 
 Followed by the discussion of 6mA beam, it is naturally to look at a beam with 
higher currents or higher space charge effects. The next apertures I use here are the 21 
mA beam and 80 mA. Again, I start with the initial condition I got from the solenoid 
tomography shown in Fig. 3. 24 (21mA) and Fig. 3.25 (80 mA), and the parameters are 
listed in table 3.9. 
 
Figure 3.24: x-x? phase space plot of 21 mA beam in aperture (a) and IC1 (b) 
  62 
 
 
Figure 3.25: x-x? phase space plot of 80 mA beam in aperture (a) and IC1 (b) 
 
Table 3.9: Initial parameters for 21mA and 80 mA beams  
Initial envelope values  X (mm) X? (mrad) Y(mm) Y?(mrad) Emittance(?m) 
 At aperture 
 21 mA 2.30 -15.90 2.30 -15.90 29.47 
80 mA 5.50 -40.10 5.50 -40.10 72.42 
At IC1 
21 mA 3.64 -11.50 3.64 -11.50 29.49 
80 mA 5.37 -21.30 5.37 -21.30 72.73 
 
 Trace3D gives the match parameters for both beam in three location of the ring 
including the mid-plane if ring dipoles, mid-plane of BPMs and the screen, which I listed 
in Table 3.10. Later, the parameters in screen will be used most.  
 
 
  63 
 
Table 3.10: Envelopes and slopes in 3 locations for a matched 21 mA and 80 mA beams 
from Trace3D 
Parameter Mid-plane of ring dipoles Mid-plane of BPMs Screen 
 21 mA 80 mA 21 mA 80 mA 21 mA 80 mA 
X (mm) 5.62 10.60 5.61 10.57 6.23 11.75 
X? (mrad) 20.88 39.21 -21.06 -39.55 -22.85 -42.94 
Y (mm) 5.40 10.12 5.33 10.01 4.79 8.98 
Y? (mrad) -20.87 38.94 20.09 37.48 18.23 34.02 
  
 Based on the initial parameters from Table 3.10 and matched condition from 
Table 3.9, I use the fitting module of Trace3D to obtain a match solution for the injection, 
which is shown in Table 3.11. Notice that the setting for YQ and QR1 are slightly 
different in both cases comparing with 6 mA beam, which I listed in Table 3.12.  
 
Table 3.11: Magnet location and matching solution for 21mA and 80 mA beams from 
Trace3D 
Magnet Sol Q1 Q2 Q3 Q4 Q5 Q6 
Position (cm) 17.13 40.53 53.73 72.73 92.33 106.63 122.63 
21 mA 
Gradient (G/cm) 98.10 3.93 7.17 7.51 6.97 5.43 4.51 
Current (A) 5.5 1.090 1.990 2.075 1.934 1.507 1.251 
80 mA 
Gradient (G/cm) 103.42 4.33 9.01 7.61 6.94 6.06 6.86 
Current (A) 5.8 1.200 2.499 2.108 1.923 1.680 1.900 
 
  64 
 
Table 3.12: YQ and QR1 settings for 21 mA and 80 mA beams 
Magnet YQ QR1 
21 mA 4.643 4.732 
80 mA 4.600 4.700 
  
 I apply this solution of 21 mA beam in experiment, and the beam images in all the 
chambers are shown in row (a) of Fig. 3.26. One can see that, with large beam current or 
higher intensity, there will be nonlinear effect because of space charge.  Moreover, since 
the high current beam have a larger beam size the image charge from the pipe will also 
bring nonlinearity to the beam when there is a large asymmetry. As a result, the linear 
space charge model in Trace3D might not be accurate and the experiment will deviate 
from the prediction from Trace3D more.  
 
 
Figure 3.26: Beam images for 21 mA beam in each chamber for (a) solution from 
Trace3D, (b) after empirical match and (c) after empirical match and rotation 
correction. 
  65 
 
 However, the empirical method is still valid. Following the procedure, I perform 
the empirical match with Q3-Q6 for 21 mA, so that the updated current setting for them 
will be modified. The result of current setting for Q3-Q6 is shown in Table 3.13. The 
beam images after empirical matching are shown in row (b) of Fig. 3.26. The row (c) of 
Fig. 3.26 shows a group of images after a rotation correction as before. 
Table 3.13: Magnet settings after empirical match 
Magnet Q3 Q4 Q5 Q6 
21 mA 1.690 1.560 1.750 1.670 
 
 I list the matched beam sizes, rotation angle and their average for 21 mA beam 
compared with the prediction from Trace3D in Table 3.13. The comparisons of beam 
sizes for the three cases are plotted in Fig. 3.27 and Fig. 3. 28. The comparison of rotation 
angle for latter two cases is plotted in Fig. 3.29 as well. 
2 4 6 8 10 120
2
4
6
8
10
Ring Chamber Index
2*
X r
m
s 
(m
m
)

 
 
Prediction by Trace3D
Trace3D Solution
After Empirical Match
After Empirical Match and Rotation Correction
 
Figure 3.27: Plot of 2*Xrms of 21 mA beam in each chamber with 3 cases in Fig. 3.26 
compared with Trace3D prediction  
 
  66 
 
2 4 6 8 10 121
2
3
4
5
6
7
8
Ring Chamber Index
2*
Y r
m
s 
(m
m
)

 
 
Prediction by Trace3D
Trace3D Solution
After Empirical Match
After Empirical Match and Rotation Correction
 
Figure 3.28: Plot of 2*Yrms of 21 mA beam in each chamber with 3 cases in Fig. 3.26 
compared with Trace3D prediction  
2 4 6 8 10 12
-30
-20
-10
0
10
20
30
Ring Chamber Index

Ro
ta
tio
n
 
A
n
gl
e
 
(d
eg
re
e)
 
 
Before Rotation Correction
After Rotation Correction
 
Figure 3.29: Comparison of rotation angle (red) before and (blue) after rotation correction. 
 
 
 
 
 
 
  67 
 
Table 3.13: Match beam parameters in each chamber and their statistics (21mA) 
Chamber RC1 RC2 RC5 RC6 RC11 RC12 STD. Ave. Trace3D 
2*Xrms(mm) 5.56 6.42 6.05 6.18 6.02 5.78 0.30 6.00 6.23 
2*Yrms(mm) 4.67 4.7 4.74 5.13 4.87 4.74 0.17 4.81 4.79 
Angle(o) -3.2 4.7 2.7 -4.1 5.9 -14.6 -1.42 -1.42 0.00 
 
 For even higher current, e.g. 80 mA beam, the beam match is not that simple. 
First, for such a higher current, the space charge will be much severer than previous two 
cases. The match solution from Trace3D will be less accuracy or even useless. Second, 
the beam is large, and the average rms size is about 10 mm, which is about half of the 
beam pipe. If one does not have a good steering solution or lack of steering dipoles, when 
he starts from a coarse empirical match, he will expect the beam loss which might affect 
the efficiency of the method. That usually means a tedious job of several round of 
empirical match. Third, even after I overcome the beam loss problems, I still face an 
image force issue because of the betatron motion and envelope oscillation.  Last, notice 
the screen radius is only 15.875 mm, which is less than 1.5 times of x rms size of the 
beam predicted by Trace3D, how to choose a scan range to do the empirical match is also 
challenging. 
 Since there is no good steering solution for this beam, I need to fine adjust the 
horizontal or vertical dipoles to allow most of the beam come through. Then, instead of a 
coarse empirical match, I adjust the injection from the Trace3D manually and check the 
images screen by screen until they are close to the predicted sizes. Finally, a fine 
empirical match is performed by scan the injection quadrupole Q2-Q5 in a small range 
about 0.03 A per step for 7 steps each quadrupole. In this way, the beam can be confined 
  68 
 
in the screen and beam loss will be ignored. The best match of 80 mA beam I can achieve 
for the first turn is shown in Fig. 3.30.  The beam is very bright for this case so that the 
fiducial mark can be seen very easily. Since the fiducial mark is transparent plastic foil 
set behind the screen, there might be gaps between them to form the diffraction pattern 
shown as the ring structure on the beam. As one can see, the beam is mostly occupied the 
whole screen, and a clear indication of image force is the irregular shape of the beam 
even though the RMS size is matched. The matched size is listed in Table 3.14. 
 
Figure 3.30: Matched beam images in screens for 80 mA beam 
 
Table 3.14: Match beam parameters in each chamber and their statistics (80mA) 
Chamber RC1 RC2 RC5 RC6 RC11 STD. Ave. Trace3D 
2*Xrms(mm) 10.5 11.1 10.9 11.2 11.4 0.34 11.02 11.75 
2*Yrms(mm) 8.46 8.58 8.48 8.31 8.05 0.21 8.38 8.98 
Angle(o) 2.7 9.0 -12.1 4.6 8.2 8.6 2.5 0.0 
 
3.8 Chapter conclusion 
 In this chapter, I start with the discussion of envelope match in space charge 
dominated beam. Then I solve an Eddy current problem of pulsed magnets which puzzled 
us for years to obtain a beam match. By using Trace3D as a match tool, I can find a 
  69 
 
coarse match solution of the beam in RMS sense.  Empirical method allows us do a fine 
adjustment of that match solution.  Later, rotation correction avoids the possibility of halo 
formation from beam rotation. I apply this method to three of UMER beams with 
different space charge level, and I find a good match for these cases. The halo is greatly 
reduced, which allow us to continue with the follow mismatch experiment to study the 
mechanism of halo formation.  
 
  70 
 
Chapter 4: Major sources for halo formation in UMER 
 
4. 1 Mismatch and halo formation 
 From the matched 6 mA beam (before I correct the beam rotation) described in 
chapter 3, I mismatch the beam simple by reduce one of the quadrupole in the injection, 
e.g., Q5 by 20%, which is a huge error in realistic sense. I compared the beam images 
between matched and mismatched case in row (a) and row (b) of Fig. 4.1. These images 
are taken with the Ethernet cameras (GigE Vision Flea3) with gain -5.5 and shutter time 
0.3 ms. Here I use a special pseudo-color, which is from low value (black) to medium 
value (color) and then to high value (black again). In this way, I can show the halos more 
clear since the quasi-uniform core is not that interesting.  
x
y 4 mm
RC6RC2
x
y
4 mm RC12RC11RC5
a
x
y
b
c
d
0
0.2
0.4
0.6
0.8
1
 
  71 
 
Figure 4.1: Images of 6mA beam in RC2, 5, 11 and 12: (a) matched beam, (b) 
mismatched beam, (c) simulation of mismatched beam without lattice rotation, and (d) 
simulation of mismatched beam with lattice rotation. Note the mismatch is generated by 
reduce one of the injection quadrupole (Q5).  
 Since the matched beam is not rotation corrected, one can see there is a wobbling 
of the ?matched? beam, which could also be a source for halo which I will discuss later. 
From the mismatch beam, the wobbling is also obvious such as the tilted image in RC2, 
an asymmetry outer ring structure in RC5, and two rotated poles in RC6. When 
comparing two cases, one can pay attention to the process of halo formation. In the early 
stage such as in RC2, although I see a bigger mismatch in the y axis than in the x axis, 
there is no obvious halo and the beam look still quasi-uniform. It can also be seen from 
the x and y profile in Fig. 4.2 and Fig. 4.3. In the log scale plot, the edge falls off 
smoothly in x and y axis for both cases, which indicate no halo in this early stage.  
0 50 100 150 200
10-2
10-1
100
Pixel
No
rm
al
iz
e
d 
in
te
n
si
ty
 
pr
o
fil
e 
in
 
x
 
a
x
is
 
Figure 4.2: Normalized beam profile in x axis of matched (blue) and mismatched (red) 6 
mA beam at RC1 
  72 
 
0 50 100 150 200 250 300
10-2
10-1
100
Pixel
No
rm
al
iz
e
d 
in
te
n
si
ty
 
pr
o
fil
e 
in
 
y 
a
x
is
 
Figure 4.3: Normalized beam profile in y axis of matched (blue) and mismatched (red) 6 
mA beam at RC1 
 
 When the beam propagates to RC11, after 5 mismatch oscillation (with period ?m 
about 1.1 m for breathing mode or 1.3 m for quadrupole mode), there are much more halo 
generated from mismatched case than the ?matched? case. The profile in x and y axis 
from central point is shown in Fig. 4.4 and Fig. 4.5. Compared with the smooth tail of the 
match case, the mismatched one has a second peak or even a third peak outside the beam 
core edge. This is strong indication of halo formation from the envelope mismatch 
instead of beam rotation because both cases are suffered from the beam rotation. If 
looking at the halo structure of the images in RC11 and RC12, the matched beam has a 
quasi-uniform tail; while the mismatched beam have a more irregular shape of beam halo. 
  73 
 
0 50 100 150 200 250 300
10-2
10-1
100
Pixel
No
rm
al
iz
e
d 
in
te
n
si
ty
 
pr
o
fil
e 
in
 
x
 
a
x
is
 
Figure 4.4: Normalized beam profile in x axis of matched (blue) and mismatched (red) 6 
mA beam at RC11 
 
0 50 100 150 200 250 300
10-2
10-1
100
Pixel
No
rm
al
iz
e
d 
in
te
n
si
ty
 
pr
o
fil
e 
in
 
y 
a
x
is
 
Figure 4.5: Normalized beam profile in y axis of matched (blue) and mismatched (red) 6 
mA beam at RC11 
 
  74 
 
 In WARP (PIC code) simulation, I see the similar behaviours too. In the 
simulation, I include the injection, Y-section and ring section. I use 40000 macro 
particles to simulation the beam. The grids is set to 256?256. Rows (c) and (d) in Fig. 4.1 
are simulation from WARP (PIC code). Both of them have the same setting of the 
mismatched case as in the experiment but without any rotation of the beam. The 
difference of these two simulations is I set the quadrupole in the ring with a rotation of 10 
degree for the latter one to approximately replace the injected beam angle. One can see 
that although the simulation don?t match the experiment perfectly, the characteristic 
behaviour of beam is reserved in simulation, such as the hot spot structure in RC5, 
several layers in RC11, and poles structure in RC12. Moreover, when the lattice is rotated, 
the symmetry is broken. In RC5, two of the hot spot rotate into core and the poles in 
RC12 not sit in up and down but rotated with an angle just as in the experiment.  
 I also compare the phase space x-x? plots in the same chambers in Fig. 4.6 from 
phase space tomography. From the configuration space in Fig. 4.1, it is clear that there 
are large amount of halo generated due to mismatch. Here, I address three points. First, 
we don?t have a perfect match beam because there is no rotation correction for this 
matched beam (this is done before I install the new skew quadrupole) and there are halos 
generated from the rotation, which can be seen from the phase space plot in row (a). 
Second, from the mismatch beam that I intentionally generate, the beam core first goes 
through large amplitude oscillation, which can be seen from the slope of the phase space 
plot from RC2, RC5 and RC6. Third, the associate free energy is transferred to halo 
particles later and the oscillation amplitude of the core is then reduced.  
  75 
 
RC6RC2 RC5
x
x?
(a)
(b)
RC11
Matched
Mismatched
0
0.2
0.4
0.6
0.8
1
4 mm
9 m
ra
d
RC12
 
Figure 4.6: Comparison of reconstructed phase space of (a) matched  and (b) mismatched 
6mA beam 
 
4.2 Injection Quadrupole scan 
 In order to see the halo clearly, I intentionally increase the camera gain from -5.5 
to 0 and shutter time from 0.3 ms to 1 ms. The view point is set at RC12 (about 6 
mismatch periods) to allow the halo particle to be exited by mismatch.  In this sitting, the 
beam core region reach beyond the CCD maximum readout and cause saturation of the 
pixels, and the intensity of beam halo increase dramatically in order for us to have a 
better view. The drawback for saturating the camera is the blooming and potential 
damage of the CCD. Based on the matched setting for 6 mA and 21 mA beam, I scan 
injection quadrupole Q4 and Q5 by steps of 0.1 A. The images are presented in Figs. 4.7- 
4.10, and the special pseudo-color is also applied here to emphasize the halo outside the 
beam core.  
  76 
 
0
0.2
0.4
0.6
0.8
1
10 mm
x
y
 
Figure 4.7: Images of 6 mA beam with injection quadrupole Q4 scan. 
 In Fig. 4.7, I first scan the injection quadrupole Q4 from 0.858 to 2.858 with a 
step of 0.1 A. As I increase the quadrupole strength, the halo region is first reduced till 
the beam match at 11th image and then begins to increase. The intensity of the halo is 
experiencing the same process. At the same time, the beam core goes through the 
opposite process. It increase first until beam match and then decreased. Notice that, for 
the first 5 images and last 3 images, there are small beam current losses monitored by an 
upstream beam current monitor. Even for the case where there is no beam loss, it might 
still face image forces from beam pipe either due to large asymmetry of beam shape in 
the injection or due to a non-perfect steering in the ring. Note that the halo distribution is 
again not regular for small or large Q4 current, again indicating an image force from the 
conducting pipe. 
 Similar behaviors can be also observed through Q5 scan. The scan range is from 
0.84 A to 2.84 A with a 0.1 A step. The images of Q5 scan are shown in Fig. 4.9. 
  77 
 
0
0.2
0.4
0.6
0.8
1
10 mm
x
y
 
Figure 4.8: Images of 6 mA beam with injection quadrupole Q5 scan 
 
4.3 Beam rotation and halo formation 
 Beam rotation is also a driving source for halo formation [15]. It is first 
discovered in simulation by Kishek, but has never been studied through experiment. In 
order to see the halo generated in first turn, I intentionally introduce a beam rotation by 
changing the skew quadrupole in the same location as Q6. The beam images of two 
typical cases compared with zero skew current case are shown in Fig. 7. For row (a), the 
skew current is set to be 0.4 A, or 144.4 G/m peak gradient. For row (c), the skew current 
is -0.5 A, or -180.5 G/m peak gradient.  
 
Figure 4.9: Images of 6mA beam with three different initial rotation   
  78 
 
 It is obvious from images in RC1 to RC5 that there is a wobbling mode related to 
beam rotation. Since we lack the data between screens, it is hard to say the frequency of 
the wobbling mode. Essentially, it is a coupled mismatch mode from both transverse 
directions. When the beam finally gets close to equilibrium state (see in RC11 and RC12), 
the wobbling energy will be transfer to particles to form halo. The beam rotation angles 
are calculated using Eqn. 4.1.  
      
    (4.1) 
where ? is the rotation angle, and ? ab = <ab> - <a> <b>. Here operator < > means to 
average and a (b) could be x or y.  The beam rotation angles are also listed in Table 4.1.  
Table 4.1: Beam rotation angles at each screen for three ISkew6 settings (6mA) 
Chamber RC1 RC2 RC5 RC6 RC11 RC12 
(a) -11.3 44.5 -22.8 34.6 1.2 1.1 
(b) 14.5 -17.9 12.7 -15.9 18.8 -20.8 
(c) 22.3 -32.8 23.6 -26.6 7.7 -13.5 
 
 As before, I plot the beam profile from the centroid point in x axis and y axis for 
three cases. The profiles of x and y axis in RC1 are plotted in Fig. 4.10 and Fig. 4.11 and 
those in RC12 are plotted in Fig. 4.12 and Fig. 4.13. Notice in the beam edge in both x 
axis and y axis, it is similar to what I discussed in the mismatch section that the beam 
initially almost don?t have any halo and the edge falls smoothly even in the logarithmic 
scale.  
1
2 2
2
tan
xy
x y
? ? ? ??= ? ?? + ?? ?
  79 
 
0 50 100 150 200 250 300
10-2
10-1
100
pixel
No
m
a
liz
ed
 
In
te
n
s
ity
 
 
I
skew = 0.4 A
I
skew = 0.0 A
I
skew = -0.5 A
 
Figure 4.10: Normalized beam profiles in x axis with three setting of Iskew6 at RC1 
 
0 50 100 150 200 250 300
10-2
10-1
100
Pixel
No
rm
al
iz
e
d 
In
te
n
si
ty
 
 
I
skew = 0.4 A
I
skew = 0.0 A
I
skew = -0.5 A
 
Figure 4.11: Normalized beam profiles in y axis with three setting of Iskew6 at RC1 
 
 However, when the beams propagate to RC12, there is halo formation for three 
cases. The zero skew current is not the optimal but close to it, so one can see a small 
amount of halo (red curve) generated in both x and y.  For other two cases, since I 
  80 
 
introduce large rotation amplitudes initially, the associate energy will transfer to beam 
particle to form halo as the blue and green curve shows especially in the x axis.   
50 100 150 200 250
10-2
10-1
100
Pixel
No
m
a
liz
ed
 
In
te
n
s
ity
 
 
I
skew = 0.4 A
I
skew = 0.0 A
I
skew = -0.5 A
 
Figure 4.12: Normalized beam profiles in x axis with three setting of Iskew6 at RC12 
40 60 80 100 120 140 160 180 200
10-2
10-1
100
Pixel
No
rm
al
iz
e
d 
In
te
n
si
ty
 
 
I
skew = 0.4 A
I
skew = 0.0 A
I
skew = -0.5 A
 
Figure 4.13: Normalized beam profiles in y axis with three setting of Iskew6 at RC12 
 
  81 
 
4.4 Chapter conclusion 
 In this Chapter, I find out two major sources of halo formation for intense beam in 
UMER, envelope mismatch and beam rotation. By performing a quadrupole scan, large 
amount of halo particle can be driven out when the quadrupole strength is far from the 
one for the matched beam. For the beam rotation case, initial beam rotation will couple 
both the focus in x and y. I observe a wobbling of the beam rotation, and the energy of 
beam wobbling will finally relax to the formation of halo. This is the first time in 
experiment to find out the skewness is also a driving source for halo formation. 
  82 
 
Chapter 5:  Halo formation in mismatch modes 
 
 In the particle core model, mismatch is the key factor to cause halo. To systematic 
study the halo formation from mismatch, one needs to quantify the mismatch strength by 
specific envelope mismatch mode and mismatch parameters. In this chapter, I will 
discuss the basic envelope mismatch mode, particle core model for halo formation and 
the free energy model for emittance growth. Simulation and experiment are used latter to 
testify the theories. 
5.1 Envelope Mismatch and Mismatch modes 
5.1.1 Envelope mismatch mode 
 In a quadrupole focus channel, the system will have two planes of symmetry as 
well as the beam. Thus, one needs two equations to describe the beam envelopes and the 
particle trajectories in each plane. The trajectory equation will be:  
 
          (5.1) 
 
   (5.2) 
Where x, y is the single particle trajectory and ?x0(z), ?y0(z)  are the external focusing 
functions, and K is the generalized perveance as before, and X, Y is the beam envelope. 
The beam envelope can be determined as  
    (5.3) 
 
0
2
'' 0( )y
Ky y y
Y X Y
?+ ? =+
2
0 3
2
'' 0xx
KX X
X Y X
??+ ? ? =+
0
2
'' 0( )x
K
x x x
X X Y
?+ ? =+
  83 
 
 
   (5.4) 
From the envelope equations, in a quadrupole channel, even the beam is perfect matched, 
there will be an envelope oscillation, and the oscillations are transversely coupled. Here, 
one can assume that ?x = ?y = ? and take a mean value of the envelope equation over a 
period, which is  
      (5.5) 
In this way, the mean value of an matched envelope will be X  = Y  = R, where the mean 
radius satisfy  
     (5.6) 
Where k0 is the wave number of the lattice, and k02 replaces the ?x0 and ?y0 in smooth 
approximation assumption. Here a space-charge depressed wave number k will also be 
used, which is defined as 
     (5.7) 
Usually, the ratio between the space charge depressed wave number and the lattice wave 
number is called tune depression ratio, which is a very important parameter in high 
intensity beam to describe how much space charge forces are involved in the beam 
dynamics.   
 When the mismatch is small, the mean value of envelopes X  and Y  will slightly 
away from the mean radius R with perturbations ? (s) and ? (s)  separately. 
The perturbations in envelopes will result a set of coupled equations as 
    (5.8) 
 
2
0 3
2
'' 0xy
KX X
X Y X
??+ ? ? =+
1 ( )s S
s
A A s ds
S
+= ?
2
2
0 3 0
Kk R
R R
?? ? =
2 2
0 2
Kk k
R
= ?
2 2 2 2
0 03 5
'' 0
2 2
k k k k? ? ?+ ?+ + =
2 2 2 2
0 03 5
'' 0
2 2
k k k k? ? ?+ ?+ + =
  84 
 
    (5.9) 
To solve this equation one can get two basic modes. Each mode associate with a wave 
number k1 and k2.  
      (5.10)  
       (5.11) 
Where k1 is the in phase mode or breathing mode and k2 is the out of phase mode or 
quadrupole mode.   
5.1.2 Particle core model 
 From [13, 14], the particle core model depicts the interaction of a simple central 
charge distribution, or beam core and a single test particle. The core could have an initial 
mismatch and then introduce an envelope oscillation. The simplest case will be a round 
continuous beam propagating in a uniform beam transport system with azimuthal 
symmetry and a linear radial focusing force. As discussed in [7], this could also describe 
the smoothed or average behavior of a beam in a quadrupole focusing channel. In this 
case, the envelope equation will be  
       
    (5.12) 
 
For a test particle, the trajectory function will be 
     (5.13) 
where Fsc is the force due to space charge. The particle will face linear space charge force 
inside the beam envelope while nonlinear space charge force outside as in Eqn. 5.14. 
 
2 2 2
1 02 2k k k= +
2 2 2
1 0 3k k k= +
2 2
2
02 3 0
d X Kk X
ds R R
?+ ? ? =
2
2
02 0sc
d x k x F
ds
+ ? =
  85 
 
      
       (5.14) 
 If a test particle transit from the edge of the core and pass away, it will be 
decelerated by the space charge force. Otherwise, it will be accelerated. The net energy 
gain or loss depends on the contributions from entering core and exiting core process. 
Usually, for a match beam here, the core radius does not change, so there will be no 
energy gain or loss. However, with envelope mismatch, there will be an envelope 
oscillation. Thus, when the test particle enters core with larger radius than the matched 
beam size and exits core with smaller radius, there will be an energy gain to that particle. 
This motion is described by Gluckstern as a nonlinear parametric resonance. The 
resonance occurs when the particle wave number ? is less than half of the wave number 
of the core mismatch oscillation km. Notice that ? = k (space charge depressed wave 
number) for particles inside the beam core. When the particles are outside the core, the 
wave number ? increases as the space charge force decreased. If ? >  0.5 kc, the resonant 
condition cannot be maintained, which will limit further growth of the amplitude. One 
description of this maximum particle amplitude can be obtained from the Poincar? 
surface of section plot.  Each curve of the plot is taken by following a random picked test 
particle in phase space trajectory when the beam core radius reaches its minimum. For 
example, for beam with mismatch parameter ? = 6.2, and tune depression ratio ? = 0.5, 
the Poincar? surface of section plot is shown as Fig. 5.1. There are three different regions 
in this plot including a central core in the center, two islands symmetrically located on the 
x axis representing the parametric resonance trajectories, and a group of quasi-elliptical 
trajectories outside the core and islands, which hardly play any role with beam core. The 
2/ ,
/ ,sc
KX R X R
F
K X X R
? <= ? ??
  86 
 
maximum halo radius can be obtained by examining the maximum radius of the particle 
which is located on the separatrix of the 2:1 resonance island. Wangler describes this 
maximum particle amplitude by an approximate empirical formula [14] as  
     (5.15) 
where a is matched core RMS size, A and B are weak functions of the tune depression 
ratio, and approximated by A = B = 4 and ? is the mismatch parameter, which is the ratio 
of initial mismatch beam size and matched beam size. 
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
x
x'
 
Figure 5.1:  Poincar? plot of mismatched beam (? = 0.62, ? = 0.5) 
 
5.1.3 Free energy model 
 In [17], the equilibrium state of a continuous beam in a linear periodic focusing 
system is best described by a transverse Maxwell-Boltzmann distribution with the smooth 
approximation. Its profile will tend to be uniform in the space-charge dominated regime.  
Linked to UMER beams, when they are injected into the ring, but not satisfy the 
stationary state requirement (non-stationary), they will have a higher energy per particle 
max / lnX a A B ?= +
  87 
 
2
2
0
1 1 lnf fi
i i
a ak h
a k a
? ? ? ?? ? ? =? ? ? ?? ? ? ?
than that of the corresponding stationary beams. A free energy model is used to describe 
the result that the energy difference ?E (free energy) between the non-stationary and the 
stationary beams can be thermalized to cause beams relax to a stationary state with higher 
energy per particle. This process will cause emittance growth and it can be analytically 
calculated.    
 In this chapter I focus on the case of non-stationary initial beam due to mismatch. 
The free energy will be in the form  
     (5.16) 
where h is the dimensionless free-energy parameter. In the case of mismatch, it can be 
calculated by  
   (5.17) 
where i denotes the initial stationary state, ai is the initial beam envelope in stationary 
state or matched beam size, a0 is the initial mismatched beam envelope and ki is initial (or 
matched beam) space charge depressed wave number.  
 Then the final beam envelope in final stationary state af can be calculated using 
    (5.18) 
where f denotes the final stationary state.  Later, the emittance in final stationary state can 
be calculated using 
    (5.19) 
 
5.2 Procedure for generating the pure-mode mismatch[42] 
 I start from the matched quadrupole setting discussed in previous chapter. I 
assume that the setting results in a good envelope match at screens in Chamber RC1, RC2, 
2 2 2
0
1
2 i
E mv k a h?? =
2 2 2 2
0
2 2 2 2
0 0 0 0
1 11 1 1 ln
2 2
i i i i
i
k a a k ah
k a a k a
? ? ? ? ? ?= ? ? ? + ?? ? ? ? ? ?? ? ? ? ? ?
1/222
0
21 1
f f f
i i i i
a ak
a k a
?
?
? ?? ?? ?? ?= + ?? ?? ?? ?? ?? ?? ?? ?? ?
  88 
 
RC5, RC6, so that the four Xrms values are nearly equal and the four Yrms values are nearly 
equal. Using these values, I obtain the averages of Xrms and Yrms, as <Xrms> and <Yrms>. 
These two values can be a good estimation of the matched RMS sizes at these screens.  
 Then, I use the Match Type 1 procedure in Trace3D to find the matched x and y 
alphas and betas at the screen position that give a envelope match over the period from 
screen in RC1 to that in RC2. Note the emittance I use initially is the value used in 
Chapter 3, which is from the Solenoid tomography.  In this step, the matched sizes from 
Trace3D will not in general be equal to the measured average values <Xrms> and <Yrms>. I 
will use the matched beta values from Trace3D with Eqn. 5.20 to recalculate the 
transverse emittance ?x,rms and ?y,rms, 
        
    (5.20) 
 
 
With this new emittance, I can rematch the beam with the Match Type 1 procedure in 
Trace3D to get advanced beta values. After several iterations, I can get a self-consistent 
solution with the right average RMS beam size in the screen in RC1 and RC2 while 
obtaining a match. 
 The next step is to find the self-consistent initial conditions in aperture with the 
assumption of no emittance growth. From the self-consistent match solution from 
previous step, I must run the Trace3D backwards from RC1 to Aperture to get the initial 
size and slope values. The quadrupoles I used for matching should be set to the match 
values in Chapter 3 after empirical method. Although the RMS size and slope values in 
aperture from this procedure in general are different from the ones from solenoid 
tomography (which is used as the initial condition for matching in Chapter 3), we expect 
2
,
/x rms rms xX? ?=< >
2
,
/y rms rms yY? ?=< >
  89 
 
that the differences are small. The new RMS size and slope values will be used to 
generate the quadrupole solution for mismatch.  
 To generate quadrupole solutions that produce a pure-mode mismatch beam, I 
need to go back to the matched ellipse parameters at a little downstream of the screen in 
RC1, where beta functions (or beam size in the case of same emittance in both transverse 
axis) are equal. A breathing mode mismatch solution is obtained by scaling ?x, ?x, ?y and 
?y at that point by a common factor ?2, which is the mismatch parameter mentioned 
previous. From previous aperture condition and lattice settings in Trace3D, I change the 
setting of the final condition as the parameters modified by mismatch parameter. By 
using a Type 8 matching procedure, I can find the settings of quadrupole Q2, Q3, Q4, and 
Q5. In this process, sometimes if I cannot find a setting of this group of quadrupoles with 
mismatch parameter ?2 both less than (smaller beam than matched at the screen in RC1) 
or greater than (lager beam than matched) one, I switch to another group of quadrupoles 
Q3, Q4, Q5, and Q6. To get the quadrupole mode solutions I must scale ?x and ?x with 
the mismatch parameter ?2, and scale ?y and ?y with a different parameter ?12 given by the 
approximate formula in Eqn. 5.21. The scaling should be performed in the same location 
as the breathing mode. The quadrupole setting is also found by Trace3D using Type 8 
match procedure. 
?12=3.29067 ? 4.27792 ?2+2.83012 ?4-1.01391 ?6+0.18356 ?8-0.0130688 ?10     (5.21) 
 
5.3 Verification of pure mismatch mode using simulation 
 In order to test the procedure for generating pure mismatch mode, I first use 
WAPR code to focus on the envelope mismatch mode. To make the case simple, I only 
use the ring lattice (see Chapter 3). The currents and emittance are the same as measured 
  90 
 
by experiment in previous section (5.96 mA, 12.9 mm mrad for 6mA beam and 20.56 
mA, 29.5 mm mrad for 21mA beam). In this way, I can avoid the problematic injection 
and Y-section, and directly obtain a match beam by using a match module called 
?Rami_match? embedded in WARP. The idea here is to test if we generate a pure mode 
envelope mismatch using the scaling procedure at the point when the beta functions of 
both transverse directions are equal. From the matched solution, this point is located at 
284 mm ahead of the first screen for 6 mA beam and 282 mm for 21 mA beam. The 
matched beam parameters here are shown in Table 5.1. Later I will start our simulation 
from this point.  
Table 5.1 Initial condition for the matched beam (6mA and 21mA) for Warp simulation 
Parameter 6 mA 21 mA 
2*Xrms (m) 0.002976 0.5261 
2*Yrms (m) 0.02982 0.5263 
2*X?rms (rad) -0.011369 -0.019859 
2*Y?rms (rad) 0.011400 0.019810 
 
 Since the initial mismatch procedure is applied to the twiss parameters, one needs 
to use Eqn. 5.22 to transform the beam sizes and slopes to twiss parameters. After 
multiplying the twiss parameters by the according mismatch parameters as indicated in 
section 5.2, I then transform them back to the sizes and slopes. The initial beam sizes and 
slopes associated with according mismatch parameters are list in Table 5.2 (5.4) for 
breathing mode and Table 5.3 (5.5) for quadrupole mode for 6 mA (21 mA) . 
 
     (5.22) 
*
' *
* 1
x x
x x
x x x
Xrms
X rms
? ?
? ?
? ? ?
=
=
= +
  91 
 
 
Table 5.2: Initial conditions for Warp input of the breathing mode (6 mA) 
? 2*Xrms (m) 2*Yrms (m) 2*X?rms (m) 2*Y?rms (m) 
0.6 0.0017856 0.0017892 -0.00958004 0.009583811 
0.7 0.0020832 0.0020874 -0.00961021 0.009622128 
0.8 0.0023808 0.0023856 -0.00999931 0.010018405 
0.9 0.0026784 0.0026838 -0.01061325 0.010638652 
1 0.002976 0.002982 -0.011369 0.0114 
1.1 0.0032736 0.0032802 -0.01221556 0.012251642 
1.2 0.0035712 0.0035784 -0.0131215 0.013162307 
1.3 0.0038688 0.0038766 -0.01406721 0.014112477 
1.4 0.0041664 0.0041748 -0.01504018 0.015089716 
 
Table 5.3: Initial conditions for Warp input of the quadrupole mode (6 mA) 
? ?1 2*Xrms (m) 2*Yrms (m) 2*X?rms (m) 2*Y?rms (m) 
0.6 1.439827 0.0017856 0.004293564 -0.00958004 0.015484552 
0.7 1.328506 0.0020832 0.003961605 -0.00961021 0.014388707 
0.8 1.214751 0.0023808 0.003622389 -0.00999931 0.013300348 
0.9 1.103641 0.0026784 0.003291057 -0.01061325 0.012283908 
1 0.999726 0.002976 0.002981182 -0.011369 0.011397769 
1.1 0.906279 0.0032736 0.002702525 -0.01221556 0.010682967 
1.2 0.824523 0.0035712 0.002458729 -0.0131215 0.010153658 
1.3 0.753214 0.0038688 0.002246085 -0.01406721 0.009798427 
1.4 0.689058 0.0041664 0.002054772 -0.01504018 0.009597448 
 
 
 
 
 
  92 
 
Table 5.4: Initial conditions for Warp input of the breathing mode (21 mA) 
? 2*Xrms (m) 2*Yrms (m) 2*X?rms (m) 2*Y?rms (m) 
0.6 0.0031566 0.0031578 -0.01476469 0.014739014 
0.7 0.0036827 0.0036841 -0.01555656 0.015524725 
0.8 0.0042088 0.0042104 -0.01677523 0.016737447 
0.9 0.0047349 0.0047367 -0.01824272 0.018199238 
1 0.005261 0.005263 -0.019859 0.01981 
1.1 0.0057871 0.0057893 -0.02156711 0.021512722 
1.2 0.0063132 0.0063156 -0.02333378 0.02327411 
1.3 0.0068393 0.0068419 -0.02513901 0.025074115 
1.4 0.0073654 0.0073682 -0.02697036 0.026900282 
 
 
 
 
Table 5.5: Initial conditions for Warp input of the quadrupole mode (21 mA) 
? ?1 2*Xrms (m) 2*Yrms (m) 2*X?rms (m) 2*Y?rms (m) 
0.6 1.439827 0.0031566 0.007577809 -0.01476469 0.027632994 
0.7 1.328506 0.0036827 0.006991927 -0.01555656 0.025592429 
0.8 1.214751 0.0042088 0.006393237 -0.01677523 0.023537572 
0.9 1.103641 0.0047349 0.005808461 -0.01824272 0.021575973 
1 0.999726 0.005261 0.005261556 -0.019859 0.019805434 
1.1 0.906279 0.0057871 0.004769748 -0.02156711 0.018296758 
1.2 0.824523 0.0063132 0.004339467 -0.02333378 0.017077703 
1.3 0.753214 0.0068393 0.003964167 -0.02513901 0.016130576 
1.4 0.689058 0.0073654 0.003626513 -0.02697036 0.015413713 
 
  
  93 
 
 In the simulation, I use a 256?256 grid for the Poisson solver, a step size of 4 mm 
along z, and 5,000 particles. The initial distribution is semi-Gaussian based on RMS 
value mention above. The reason for using fewer particles and large step size here is to 
quickly access the envelope mismatch while including the nonlinear space charge force, 
image charge force and emittance growth. Running a large number of test simulations 
with more particles or higher resolution resulted in no perceptible difference in the final 
result of envelope oscillations. In order to get enough sampling point for FFT analysis, I 
run the particles through 400 lattice period. To get the FFT analysis of the mismatch 
mode and avoid the interference from the dominated oscillation from FODO lattice, I 
choose the sampling frequency as one sample per lattice period. The final results of FFT 
analysis plots for each case of 6 mA beam are shown in Fig. 5.2 (breathing mode) and 
Fig. 5.3 (quadrupole mode). 
0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5 x 10
-4 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
? = 0.6
? = 0.7
? = 0.8
? = 0.9
? = 1.0
? = 1.1
? = 1.2
?
 = 1.3
? = 1.4
0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5 x 10
-4 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
? = 0.6
? = 0.7
? = 0.8
? = 0.9
? = 1.0
? = 1.1
? = 1.2
?
 = 1.3
? = 1.4
 
   (a)      (b) 
Figure 5.2: FFT analysis of envelope from the breathing mode for 6 mA (left: x-axis; 
right: y-axis). 
  94 
 
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6 x 10
-4 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
? = 0.6
? = 0.7
? = 0.8
? = 0.9
? = 1.0
? = 1.1
? = 1.2
?
 = 1.3
? = 1.4
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6 x 10
-4 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
? = 0.6
? = 0.7
? = 0.8
? = 0.9
? = 1.0
? = 1.1
? = 1.2
?
 = 1.3
? = 1.4
 
   (a)      (b) 
Figure 5.3: FFT analysis of envelope from the quadrupole mode for 6 mA (left: x-axis; 
right: y-axis). 
 From Fig. 5.2, there is a strong indication of a dominated breathing mode both in 
x or y direction because of the high peaks shown in the figure. The wave number of the 
high peak is 5.522 m-1, which is close to the calculation value 5.649 m-1 using analysis 
discussed in section 5.1.1. Note that, although there is another peak with smaller wave 
number shown in the plots, the amplitude is small and did not vary much among different 
mismatch parameters. Later, the wave number is found close to the calculated value of 
quadrupole mismatch mode, which means that the mode I generate is not that pure and 
the mismatch have a small portion of quadrupole mismatch. Meanwhile in the quadrupole 
mismatch case (see Fig. 5.3), the peaks indicate the mismatch wave number is 4.449 m-1 
(estimated 4.605 by calculation), also testify the much lower frequency of the quadrupole 
mode than breathing mode. Note that a small portion of breathing mismatch is also 
presented here. I plot the amplitudes of breathing mode peak from the breathing mode 
case and that of the quadrupole mode peak in Fig. 5.4. From the figure, the curve shape is 
similar to the maximum radius of the halo particles shown in [8], which indicate that the 
  95 
 
amplitudes of the peaks are associated with the potential energy in that mismatch mode 
which could be given to particles to form halos. With the same mismatch parameter, the 
amplitude of the quadrupole mismatch is smaller than that of the breathing mode, which 
points out that the breathing mode has larger potential energy which can be transfer to 
halo particles.  
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
-1
-0.5
0
0.5
1
1.5
2
2.5
3
x 10-4
Mismatch parameter
Am
pi
ltu
de
2  
[A
rb
.
 
Un
its
]
 
 
x (Breathing mode)
y (Breathing mode)
y (Quadrupole mode)
y (Quadrupole mode)
 
Figure 5.4: Simulated amplitude of the mismatch oscillation versus mismatch parameters 
for 6mA 
  
 For higher current like 21 mA beam, I do the similar PIC simulation with higher 
beam current and use the same sampling method and FFT analysis to the mismatched 
envelope for different mismatch parameters and mismatch modes. The plots of the FFT 
analysis are shown in Fig. 5.5 (breathing mode) and Fig. 5.6 (quadrupole mode). I plot 
the amplitudes of breathing mode peak from the breathing mode case and that of the 
quadrupole mode peak in Fig. 5.7. 
  96 
 
 In the breathing mode case, one can see that the dominated peak with obvious 
higher amplitude than the 6 mA beam case, but with a peak wave number at 5.369 m-1 
close to calculated value 5.434 m-1, smaller than that of 6 mA case. This is because that 
higher beam current has a higher tune depression ratio and the wave number of the 
mismatch mode is monotone increasing with tune depression ratio. This can be also seen 
in the case of quadrupole mode mismatch where the peak wave number is 4.180 m-1 
(estimated 4.199 by calculation). Note that there is also another small peak in the 
breathing mode case indicating a small portion of quadrupole mode while another small 
peak in the quadrupole mode case indicating a small portion of breathing mode. 
Comparing with 6 mA cases, the amplitudes of the secondary peaks in 21 mA cases are 
much smaller than the dominated mismatch mode. Another point is when the mismatch 
parameter is as small as 0.6, there is a peak in Fig. 5.3 (b) and Fig. 5.5 (b) at the wave 
number bigger than characteristic wave number of the breathing mode. It is either due to 
the strong mismatch, or some numerical artifact which is not been investigated further.  
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 x 10
-3 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
?  = 0.6
?  = 0.7
?  = 0.8
?  = 0.9
?  = 1.0
?  = 1.1
?  = 1.2
?  = 1.3
?  = 1.4
0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 x 10
-3 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
?  = 0.6
?  = 0.7
?  = 0.8
?  = 0.9
?  = 1.0
?  = 1.1
?  = 1.2
?  = 1.3
?  = 1.4
 
   (a)      (b) 
Figure 5.5: FFT analysis of envelope from the breathing mode for 21 mA (left: x-axis; 
right: y-axis). 
  97 
 
0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5 x 10
-4 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
?  = 0.6
?  = 0.7
?  = 0.8
?  = 0.9
?  = 1.0
?  = 1.1
?  = 1.2
?
  = 1.3
?  = 1.4
0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5 x 10
-4 Spectrum of Envelop Oscillation
wave number (m-1)
|Y(
k)|
 
 
?  = 0.6
?  = 0.7
?  = 0.8
?
  = 0.9
?  = 1.0
?  = 1.1
?  = 1.2
?  = 1.3
?
  = 1.4
 
   (a)      (b) 
Figure 5.6: FFT analysis of envelope from the quadrupole mode for 21 mA (left: x-axis; 
right: y-axis). 
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
-2
0
2
4
6
8
10 x 10
-4
Mismatch parameter
Am
pi
ltu
de
2  
[A
rb
.
 
Un
its
]
 
 
x (Breathing mode)
y (Breathing mode)
y (Quadrupole mode)
y (Quadrupole mode)
 
Figure 5.7: Simulated amplitude of the mismatch oscillation versus mismatch parameters 
for 21 mA 
 
5.4 Maximum extent of halo radius and Emittance growth in simulation 
 In a real accelerator with large current or intensity, each particle will carry large 
amount of kinetic energy, especially for ion machine. The loss of beam particle will hit 
  98 
 
the pipe and cause the machine to be radio-active. In this section, I will focus on the 
maximum extent of halo radius, which will help us to better guide the geometric design 
of future high intensity accelerators to reduce the beam loss due to pipe-particle 
interaction.  
 In order to study the maximum extent of halo radius, one must run similar 
simulation as studying the mismatch mode but with large number of particle to access 
faint halo. In the simulation, considering the computing power, I use one million particles 
to best address the halo issue. Further increasing the particle number may improve the 
accuracy, but the improvement is not so obvious. We showed the comparison of 
maximum particle radius in x direction with different simulation macro particle number 
in the Fig. 5.8.  This beam is under breathing mode mismatch with mismatch parameter 
1.3. There is no difference between the blue (1 million simulation particle) and green 
curve (4 million), while the purple curve (100 k) is not far away, but the red curve (10 k) 
is way off. 
0 2 4 6 8 10
2
4
6
8
10
12 x 10
-3
Distance (m)
M
a
x
im
u
m
 
pa
rt
ic
le
 
ra
di
u
s 
in
 
x
 
di
re
ct
io
n
 
 
Macro Particle number = 10000
Macro Particle number = 100000
Macro Particle number = 1000000
Macro Particle number = 4000000
 
Figure 5.8: Comparison of simulations using different number of macro particles (6 mA). 
  99 
 
 I also increase the number of grid to 512?512 for field solver and use 0.001 m as 
the time step. This gives us more accuracy when dealing with the field halo particles face 
and their trajectories. In each time step, I record the number outside the ellipse with its 
major and minor radius equal to 2, 3 or 4 time of the RMS beam sizes.  This will allow us 
to estimate the halo particle number in specific range. I am also interested in the 
maximum radius of a particle can reach, so I trace the boundary of individual particle 
trajectory and record the largest radius in each step.  The beam image in the start, RC1 
and RC2 is shown in Fig. 5.9 for breathing mode and Fig. 5.10 for quadrupole mode. It is 
clear that the beam sizes increase equally with radial symmetry with mismatch parameter 
increased for the breathing mode in the start while for the quadrupole mode, the beam 
sizes increase in x axis and decrease in y axis. This behavior is not obvious in the screen 
location, since the screens are not in the periodic location of the mismatch oscillation.  
 
Figure 5.9: Simulated 6mA beam images at start (upper row), RC1 (middle row) and 
RC12 (lower row) with mismatch parameter from 0.6-1.4 for breathing mode. 
 
  100 
 
 
Figure 5.10: Simulated 6mA beam images at start (upper row), RC1 (middle row) and 
RC12 (lower row) with mismatch parameter from 0.6-1.4 for quadrupole mode. 
  
 In fig. 5.11, I illustrate a typical case of halo formation. The mismatch parameter 
is 0.6, and it plots the ratio of particles outside the 3*RMS beam ellipse. Initially, since I 
start from a semi-Gaussian distribution. There will be no particle reach beyond the 
3*RMS ellipse. But as the beam size continues decrease in x axis, a small portion of halo 
particles are first driven out. This process continues every time when the beam core 
reaches its minimum. As one can see in the plot, there is a periodicity for the halo particle 
driven out, which is close to the mismatch wave period. Since particles feel nonlinear 
space charge force outside the beam core, they will have different frequency with the 
particle inside. As the beam propagate through several mismatch period, although halo 
particles can still oscillate back to core region, but there are more newly formed halo will 
come out of the beam core. The net number of particles outside 3*RMS ellipse region 
will stay above zero.  
  101 
 
0 2 4 6 8 10
0
1
2
3
4
5
6
7
8
distance (m)
Pa
rt
ic
le
 
R
at
io
 
(%
)
 
 
Particle outside 3*RMS ellipse
 
Figure 5.11: Simulated number ratio of particles outside 3*RMS region of 6 mA beam 
with mismatch parameter ? =0.6 in breathing mode 
 In Fig. 5.12, I plot the number ratio of particles outside 3*RMS region at screen 
location of RC12 with different mismatch mode and mismatch parameters. The red solid 
curve is for the breathing mode and blue dashed curve is for quadrupole mode. For both 
mode, the minimum of number ration outside the 3*RMS region occurs when the 
mismatch parameter is unity. When we increase or decrease the mismatch parameter, 
there is always an increase of particle number in the region outside 3*RMS size. In large 
value of mismatch parameter, there is more particles outside the 3*RMS region in 
breathing mode than in quadrupole mode. When close to unity, the difference is not so 
obvious.  
  102 
 
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0
1
2
3
4
5
6
Mismatch Parameter
Pa
rt
ic
le
 
R
at
io
 
(%
)
 
 
Breathing Mode
Quadrupole Mode
 
Figure 5.12: Simulated number ratio of particles outside the 3*RMS region of 6 mA 
beam with different mismatch parameter and different mismatch mode at screen in RC12 
 
 In Fig. 5.13, I plot the maximum particle radius divided by the matched RMS size 
with mismatch parameters from 0.6 to 1.4 in two mismatch mode. The blue solid line is 
for breathing mode, and the red dashed line is for quadrupole mode. The maximum 
particle radius is the largest size of individual particles along the way when beam 
propagate from the ring entrance to RC12, but averaged between x and y axis. In each 
axis, the largest size is the distance from maximum extent to the minimum extent divided 
by two. A theory prediction curve from Eqn. 5.15 of maximum particle radius divided by 
the matched RMS beam size with different mismatch parameter is plot as purple curve in 
Fig. 5.13. One can see that the shape of the simulation curve is consistent with the curve 
predicted by particle core model. We see the data for breathing mode mismatch is always 
larger than that of the quadrupole mode. This could be explained as previous discussion 
that for the same mismatch parameter, the beams in breathing mode mismatch carry more 
  103 
 
free energy than the beams in quadrupole mode mismatch. From the figure, the points of 
the breathing mode lie below the theory prediction when mismatch is small (close to 
unity) while for the other large mismatch, points exceed the prediction from the core 
particle model for larger mismatch. The possible explanation is that, for the small 
mismatch the particles only go through the mismatch oscillation about 6 periods when 
reaching to RC12, so the halo particles are not being exited to maximum energy level in 
order to reach the maximum radius predicted by the theory. For the large mismatch case, 
there could be other mechanisms which fasten this process, for example, the free energy 
stored in the initial distribution.  
 
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
Mismatch Parameter
M
a
x
im
u
m
 
Pa
rt
ic
le
 
Ra
di
u
s/
M
a
tc
he
d 
RM
S 
si
ze
 
 
Simulated breathing mismatch mode
Simulated quadrupole mismatch mode 
Particle core model
 
Figure 5.13: Simulated maximum particle radius along the way propagating to RC12 
versus mismatch parameter compared with prediction from particle core model.  
 For a mismatched beam, the beam is not in an equilibrium state, and the associate 
free energy will cause emittance growth. In Fig. 5.14, the blue curve plot the ratio of the 
final (at RC12) and initial emittance versus mismatch parameter calculated from the free 
  104 
 
energy model with the assumption that the ratio of final radius and initial radius is small 
(see Eqn. 5.17-5.19).  The red, purple, green and light blue dashed curve represent the 
emittance growth versus the mismatch parameters in x axis, in y axis for breathing mode 
and for quadrupole mode separately.  One can see that the emittance growth in y axis for 
breathing mode is close to the theory curve for most of the mismatch parameters while 
the emittance growth in x axis for quadrupole mode approaches the theory in mismatch 
parameter higher than 1.0. Otherwise, the curves are below the maximum emittance 
predicted by the theory. In the simulation, for the mismatched case, we always observe 
anisotropy (x-y differences) for emittance growth, especially for breathing mode, which 
can be seen also from Fig. 5.14. This result was also found out by Franchetti, Hofmann, 
and Jeon [43], and they think it is related to the initial x-y tune differences as small as 1%.  
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0.5
1
1.5
2
2.5
3
3.5
Mismatch parameter
RM
S 
Em
itt
an
c
e 
G
ro
w
th
 
 
Breathing mode (x-axis)
Breathing mode (y-axis)
Quadrupole mode (x-axis)
Quadrupole mode (y-axis)
Therory Max
 
Figure 5.14: Emittance growth verses mismatch parameter at RC12 for 6mA beam. The 
solid curve shows maximum growth from the free energy model. 
  105 
 
 Later, I test the halo formation with higher current beam (21 mA) in simulation. 
The simulation setting is the same except the beam current, initial sizes, slopes and 
emittance increase accordingly. The beam images at the start, RC1 and RC12 are shown 
in Fig. 5.15 for breathing mode and Fig. 5.16 for quadrupole mode. With higher currents, 
the beam size is larger and then the halo from mismatch will oscillate to even bigger 
radius. Comparing with the curve predicted in particle core model in Fig. 5.17, which is 
not related to beam current (or tune depression ratio), one can see the similar behaviors as 
6mA beam case that the maximum radius for quadrupole mode are smaller than the 
breathing mode. Most case of maximum radius lies below the prediction curve from 
particle core model. For larger mismatch (far away from unity), the maximum radius of 
breathing mode exceed the prediction from the particle core model, which indicate that 
when we apply this simplified model in design of high intensity accelerators, some 
margin should be taken into consideration. From the free energy model, the emittance 
growth is dependent on both the mismatch parameter and the tune depression ratio. With 
lower tune depression value (higher current), the theory prediction value is higher than 
the 6 mA case.  There is still a large anisotropy for emittance growth. Since the simulated 
emittance growth is still below the theory curve, the free energy model is still valid in this 
range of mismatch parameter at least for such a length of beam propagation.   
  106 
 
 
Figure 5.15: Simulated 21mA beam images at start (upper row), RC1 (middle row) and 
RC12 (lower row) with mismatch parameter from 0.6-1.4 for breathing mode. 
 
 
Figure 5.16: Simulated 21mA beam images at start (upper row), RC1 (middle row) and 
RC12 (lower row) with mismatch parameter from 0.6-1.4 for quadrupole mode. 
 
  107 
 
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.43
3.5
4
4.5
5
5.5
6
6.5
7
7.5
Mismatch Parameter
M
ax
im
u
m
 
pa
rt
ic
le
 
ra
di
u
s/
M
at
ch
ed
 
R
M
S 
si
ze
 
 
Particle core model
Simulated breathing mode mismatch
Simulated quaduopole mode mismatch
 
Figure 5.17: Simulated 21mA maximum particle radius along the way propagating to 
RC12 versus mismatch parameter 
 
 
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Mismatch parameter
RM
S 
Em
itt
an
c
e 
G
ro
w
th
 
 
Breathing mode (x-axis)
Breathing mode (y-axis)
Quadrupole mode (x-axis)
Quadrupole mode (y-axis)
Therory Max
 
Figure 5.18: Emittance growth verses mismatch parameter at RC12 for 21mA beam. The 
solid curve shows maximum growth from the free energy model. 
  108 
 
5.5 Experiment study of the mismatch mode 
 In experiment, I follow the procedure in section 5.2 to generate the pure mode 
envelope mismatch. The settings of the injection quadrupoles Q2-Q5 for 6 mA beam are 
listed in Table 5.6 for breathing mode and in Table 5.7 for quadrupole mode.  
 
Table 5.6: Setting of injection quadrupoles Q2-Q5 for breathing mode mismatch (6mA) 
? Q2(A) Q3(A) Q4(A) Q5(A) 
0.6 1.819 1.265 1.128 1.794 
0.7 1.875 1.427 1.302 1.832 
0.8 1.922 1.562 1.475 1.865 
0.9 1.961 1.681 1.660 1.900 
1 1.992 1.785 1.858 1.940 
1.1 2.015 1.876 2.059 1.983 
1.2 2.028 1.952 2.243 2.029 
1.3 2.030 2.013 2.391 2.076 
1.4 2.023 2.062 2.493 2.128 
 
 
Table 5.7: Setting of injection quadrupoles Q2-Q5 for quadrupole mode mismatch (6mA) 
? ?1 Q2(A) Q3(A) Q4(A) Q5(A) 
0.6 1.439827 1.930 1.858 1.526 1.561 
0.7 1.328506 1.938 1.837 1.622 1.682 
0.8 1.214751 1.952 1.816 1.694 1.781 
0.9 1.103641 1.970 1.797 1.766 1.865 
1 0.999726 1.992 1.785 1.858 1.940 
1.1 0.906279 2.015 1.783 1.989 2.012 
1.2 0.824523 2.037 1.792 2.170 2.083 
1.3 0.753214 2.054 1.806 2.382 2.149 
1.4 0.689058 2.212 1.966 3.340 2.315 
  109 
 
 The beam images are shown in Fig. 5.19 for breathing mode and in Fig. 5.20 for 
quadrupole mode. For the breathing mode, one can observe a large beam rotation in RC1 
when the mismatch parameter is small. As discussed before, the rotation will contribute 
to the halo formation and cause more halo generation. For the cases that mismatch 
parameter close to 1.0 (0.9-1.1), the halo formation is relatively slow, which shows the 
intense beam have certain allowance for envelope mismatch. I compare the envelopes of 
three typical cases each mode with simulation and plot them in Fig. 5.21 ? Fig. 5.25. 
Note that the red color is for x axis and blue is for y axis. The dashed curves are from the 
simulation and the markers represent the experiment result. The error comes from the 
resolution of one pixel. Except few experimental data points are not on the simulation 
curve, most agree with simulation quite well. Since I already showed the simulation 
curve is a pure mode mismatch and here I have the agreement, I can claim a close 
situation of pure mode mismatch in the experiment, which is of course not perfect, but 
give us a frame to discuss the halo formation in experiment quantitatively.  
  110 
 
 
Fi
gu
re
 
5.
19
: 
Im
ag
es
 
o
f 6
 
m
A
 
be
am
 
at
 
R
C1
,
 
2,
 
5,
 
6,
 
11
,
 
12
 
w
ith
 
m
ism
at
ch
 
pa
ra
m
et
er
 
fro
m
 
0.
6 
to
 
1.
4 
o
f b
re
at
hi
n
g 
m
o
de
 
  111 
 
 
 
Fi
gu
re
 
5.
20
: 
Im
ag
es
 
o
f 6
 
m
A
 
be
am
 
at
 
R
C1
,
 
2,
 
5,
 
6,
 
11
,
 
12
 
w
ith
 
m
ism
at
ch
 
pa
ra
m
et
er
 
fro
m
 
0.
6 
to
 
1.
4 
o
f q
u
ad
ru
po
le
 
m
o
de
 
  112 
 
0 1 2 3 4 5 6 7 8
1.5
2
2.5
3
3.5
4
4.5
5
5.5
z(m)
En
v
e
lo
pe
(m
)
 
 
Simulated 2*Xrms
Simulated 2*Yrms
Experimental 2*Xrms
Experimental 2*Yrms
 
Figure 5.21: Comparison of Envelope between simulation and experimental data for 6mA 
beam in breathing mode (mismatch parameter ? = 0.8) 
 
0 1 2 3 4 5 6 7 8
2
2.5
3
3.5
4
4.5
5
z(m)
En
v
e
lo
pe
(m
)
 
 
Simulated 2*Xrms
Simulated 2*Yrms
Experimental 2*Xrms
Experimental 2*Yrms
 
Figure 5.22: Comparison of Envelope between simulation and experimental data for 6mA 
matched beam (mismatch parameter ? = 1.0) 
 
  113 
 
 
0 1 2 3 4 5 6 7 8
1.5
2
2.5
3
3.5
4
4.5
5
z(m)
En
v
e
lo
pe
(m
)
 
 
Simulated 2*Xrms
Simulated 2*Yrms
Experimental 2*Xrms
Experimental 2*Yrms
 
Figure 5.23: Comparison of Envelope between simulation and experimental data for 6mA 
beam in breathing mode (mismatch parameter ? = 1.2) 
 
0 1 2 3 4 5 6 7 8
2
2.5
3
3.5
4
4.5
5
z(m)
En
v
e
lo
pe
(m
)
 
 
Simulated 2*Xrms
Simulated 2*Yrms
Experimental 2*Xrms
Experimental 2*Yrms
 
Figure 5.24: Comparison of Envelope between simulation and experimental data for 6mA 
beam in quadrupole mode (mismatch parameter ? = 0.8) 
 
  114 
 
0 1 2 3 4 5 6 7 8
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
z(m)
En
v
e
lo
pe
(m
)
 
 
Simulated 2*Xrms
Simulated 2*Yrms
Experimental 2*Xrms
Experimental 2*Yrms
 
Figure 5.25: Comparison of Envelope between simulation and experimental data for 6mA 
beam in quadrupole mode (mismatch parameter ? = 1.3) 
 
 In order to measure the maximum halo radius and compare with the core particle 
theory and previous simulation, I have to use the optical mask method as discussed in 
section 2.5. Here, I use a PIMAX camera [44], which features a gated intensifier CCD 
with 1000*1000 pixels. The optics design is based on Fig. 2.4 and the whole setup is 
shown in Fig. 5.26. Before I take any image, two calibrations are made as shown in Fig. 
5.27 and Fig. 5.28. For the first calibration, I apply a checker board mask on the DMD 
and use a flash light to illuminate the DMD. It checks the focus of the second focus 
channel and identifies the DMD edges in the CCD coordinates, which allows us to 
generate the mask because in the mask generation process, I need to calculate distances 
of the points I want to mask out to the four edge of DMD in camera coordinate, and then 
I can transform them to the DMD coordinate. The second calibration gives the real 
resolution of each pixel for size calculation.  
  115 
 
Screen at RC12
DMD
L1
f = 150 mm
L2
f = 150 mm
PIMAX 
ICCD camera
Alignment 
Laser
 
Figure 5.26: Setup of the optical mask method for halo measurement 
 
Figure 5.27: Calibration image of DMD indicating the edges of the DMD in CCD 
coordinate. The DMD is illuminated by a flush light. Camera is in shutter mode with 5 ?s 
exposure time. 
  116 
 
 
Figure 5.28: Calibration image of screen. The 80 mA beam is defocused to fill full of the 
screen for this calibration. There is straight light from background since we did not shut 
down all the light for the calibration. Camera is in gate mode with gate time 500 ?s for 
100 integration frames. 
 When taking data, I cover all the optics with black cloth and shut down all the 
lights to prevent the any straight light from outside of the system.  I use the beam master 
trigger to trigger the camera externally and set the gate to 500 ?s to further cut down the 
noise light level while keeping enough light for image acquisition. For acquiring the 
image, I first apply a black mask to the DMD which let all the images go to the camera. 
To take the beam image, I set the integration frame to 20 to collect 20 pulses of the beam 
in order to let the image close to camera saturation to fully use the camera dynamic range. 
The peak intensity is about 25000. Based on the beam image, I generate a threshold mask 
with a 5000 threshold. I apply this mask on the DMD to block out the unwanted central 
beam core and retake another image with increased integration frame to 200. Note here, 
the UMER beams are quite stable and the difference between pulses is very small. The 
  117 
 
images of the full beam and the halos are shown in Fig. 5.29 for breathing mode and Fig. 
5.30 for quadrupole mode.  
 
10 mm x
y
25000
0
? = 0.6 ? = 0.8 ? = 1.0 ? = 1.2 ? = 1.4
Full 
beam
Masked 
beam
 
Figure 5.29: Beam images of full beam (upper row) and after core being masked out 
(lower row) for breathing mode (6mA beam). 
10 mm x
y
25000
0
? = 0.6 ? = 0.8 ? = 1.0 ? = 1.2 ? = 1.4
Full 
beam
Masked 
beam
 
Figure 5.30: Beam images of full beam (upper row) and after core being masked out 
(lower row) for quadrupole mode (6mA beam). 
  
 In the particle core model, it predicts the maximum radius of halo particles as a 
function of mismatch parameter as indicated by Eqn. 5.15. There are limitations in the 
experiment to determine the maximum radius for very faint halo due to the dynamic 
range of the measuring method. (Previously in my master thesis, I showed a dynamic 
  118 
 
range about 104 of the optical mask method in UMER. The limitation is due to the screen 
size and the dynamic range and efficiency of the phosphor screen.) Instead, I plot the 
measured radius (x-y averaged half-width of the beam) at 1% of peak intensity versus the 
mismatch parameter. The radiuses are determined in this way. From the full beam images, 
I obtain the peak intensity and according point for each mismatch case. From the peak 
intensity, I can calculate the 1% intensity level. This level should be then multiplied by 
factor of 10 for the difference of the integration frames. In the halo images, I do a 
horizontal or vertical scan from the peak point and determine the pixels at this level. The 
final radius is determined by half width each direction and then averaged by both 
directions. The error comes from the similar value of the adjacent point. I generally apply 
a ?15 pixels (about 0.53 mm) error, which implies a background noise. A plot of the half 
width of 1% of maximum intensity over matched rms size versus mismatch parameter is 
shown in Fig. 5.31 at RC12 for both mismatch modes. We compared the experimental 
results with simulation in same condition with Fig. 5.13 (In Fig. 5.13, the maximum 
radius is along the whole propagating channel, but here it is the local maximum at 
location of RC12). Here, we choose to plot the 1% of the maximum intensity averaged 
between x and y direction from the maximum point instead of the maximum extent. The 
experimental points lie on the simulation curve within the error bar except the largest 
mismatch case when ? =0.6, which is below the simulation curve. If compared with 
prediction curve from particle core model in Fig. 5.13, both experimental and simulation 
curves in Fig. 5.31 lies way below the theory curve (that is why it does not plotted here). 
Thus we can claim that the particle core model gives us an reasonable upper bound for 
maximum radius of halo particles.  
  119 
 
 
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.42
2.5
3
3.5
4
4.5
Mismatch parameterHa
lf 
w
id
th
 
o
f 1
%
 
m
ax
im
u
n
 
in
te
n
si
ty
 
/ m
at
ch
ed
 
rm
s 
si
ze
 
 
Breathing mode simulation
Quadrupole mode simulation
Brething mode exp
Quadrupole mode exp
 
Figure 5.31: Beam half widths at 1% level of the maximum intensity versus mismatch 
parameter for both breathing and quadrupole mode mismatch. The comparison is between 
measured beam widths and simulated widths at RC12 (6 mA beam). 
 I also use the Tomography method to measure the emittance of the mismatched 
beam in x axis in RC12. The experimental data are plotted in Fig. 5.32 as brown square 
marker for breathing mode and pink triangle marker for quadrupole mode. The blue solid 
curve, red and green dashed curves are from Fig. 5.14 for comparison. The experimental 
data agree with the simulation curve well for breathing mode, but all lies below the 
simulation curve for quadrupole mode. All the data lies below curve predicted by the 
theory, which states that either at RC12 there is still free energy stored in the beam or 
other mechanism could absorb the free energy. As said in previous section, the emittance 
could increase anisotropically. Thus the average emittance of x and y will be more 
effective to compare with the theory but we lack of the emittance in y axis due to tight 
experiment schedule, which could be a future work. 
  120 
 
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0.5
1
1.5
2
2.5
3
3.5
Mismatch parameter
RM
S 
Em
itt
an
c
e 
G
ro
w
th
 
 
Breathing mode (Simulation)
Quadrupole mode (Simulation)
Therory Max
Breathing mode (Experiment)
Quadrupole mode (Experiment)
 
Figure 5.32: Measured RMS emittance growth for 6mA at RC12 for both breathing and 
quadrupole mode mismatch. The solid curve show maximum growth from the free 
energy model and the dashed curve is from simulation 
 
5.6 Chapter conclusion 
 In this Chapter, I discuss the halo formation systematically with the mismatch 
parameter range from 0.6 to 1.4 in two envelope mismatch modes both in simulation and 
experiment. The agreement between the experiment and simulation is mostly satisfactory. 
The results support the prediction from free energy model as an upper limit for emittance 
growth with mismatch parameter in range of 0.6-1.4. The simulation results show the 
halo radius prediction from the particle core model is valid for mismatch parameter close 
to unity. For larger mismatch, when we apply this theory, especially for large intensity, it 
may be wise to consider a margin of a range about 20% to estimate the mismatch 
allowance and aperture requirement in the design of new high current, high intensity 
accelerators. 
  121 
 
Chapter 6:  DMD based application in accelerators 
 
 In chapter 2, I introduce an adaptive masking method for halo measurement. Later 
on, we found this diagnostic is quite useful in most of accelerators, not only to measure 
the beam halo using synchrotron radiation (in JLab FEL facility), but also to detect 
injected beam with high intensity stored beam presented in storage ring (in SLAC 
SPEAR3). In this chapter, I will discuss the application of this diagnostic. 
6.1 Halo Experiments at Jefferson National Lab FEL Facility 
6.1.1 Experiment Setup 
 The energy of the electron beam at the JLAB FEL accelerator [44] is 135 MeV.  
Various experiments such as the recently proposed Dark Light Project [45] require high 
current (10 mA) and high beam quality. Thus, it is important to know the spatial 
distribution of the beam with a high dynamic range.  To fulfil the experimental need to 
measure the halo non-interceptively under high current operation of the FEL, we have 
developed a halo imaging system using optics similar to those described above, which 
images the beam in optical synchrotron radiation (OSR) as it passes through a bending 
magnet.  
 The differences in the optical system used for JLAB are as follows. First, slits are 
used to restrict the horizontal extent of the synchrotron light. Second, we need to 
transport the OSR beam image from downstairs in the shielded accelerator vault to an 
upstairs gallery, which houses the DMD and secondary optics.  To do this we use an extra 
lens with a long focal length (1.5 m) to form an intermediate image in the transport path. 
  122 
 
There are also two pico-motor controlled mirrors in the path to steer the synchrotron light 
through the optics and two flipper controlled screens with associated cameras to monitor 
the OSR transport. Third, a separate target set at the same distance as the source is used 
in order to measure the magnification and to achieve the best focus of the source at the 
DMD. The total magnification measured by using this target is 0.71.  
 
6.1.2 Mask generation 
 The mask we applied to DMD is a 1024*768 pixels bitmap image with only 0 or 1 
value indicating whether the pixel is reflecting the partial image toward camera or not. 
We apply this bitmap image and load it onto the DMD by the Texas instrument software 
called Discovery 4100 Explorer.  To generate the bitmap mask image, a special Matlab 
GUI is developed as shown in Fig. 6.1.  
 
Figure 6.1: Mask generation GUI interface. 
 
 We first apply a check board mask (software included) onto DMD and take a 
calibration of the DMD. The image shown in Fig. 6.2 is the DMD with the check board 
  123 
 
mask. Since the DMD is rotated, we need 8 points (as indicated in the image) to define 
the DMD edges in camera coordinate. For generating a single point mask, we calculate its 
distance to each of the DMD edge in the camera coordinate in that image and then scale 
up to the 1024?768 DMD coordinate.  A general mask is a set of these single points. In 
the realistic operation, we first apply an all black mask (value 0 for all pixels) to obtain a 
full beam image. Based the full beam image, we have two methods to generate the mask. 
One is to choose a certain threshold value and mask out all the points higher than that 
value. This is applied in the point spread function measurement with gradually increased 
mask sizes (or decreased threshold) using certain turn of the stored beam which will be 
discussed later. Another one is to choose a region of interest (ROI) either by a polygon or 
a circle. For injected beam imaging, we choose a smaller threshold to generate a large 
mask, or choose a large ROI mask in order to prevent the leakage of the intense 
synchrotron radiation from the stored beam caused by jittering. A small correction is 
applied by shifting the mask in x or y axis in case that the mask is not perfect aligned 
with the region we want to mask out.  
 
Figure 6.2: Image of illuminated DMD with a check board mask for calibration 
  124 
 
6.1.3 Preliminary Results from JLAB 
 We used an electron beam with 1 ms macro pulse width, 60 Hz repetitive rate and 
4.68 MHz micro-pulse repetition rate. Each micro-pulse contains a charge of 60 pc. We 
generated several masks based on threshold level. For each mask, we selected the 
appropriate integration time to bring the peak intensity of the image close to the 
saturation level of the camera. To obtain a background image for each mask setting, we 
covered the PVC tube which connected downstairs and upstairs, and integrated for the 
same time period used to obtain the corresponding masked beam image. Background-
subtracted images are shown in Fig. 6.3, where the number in the lower left indicates the 
integration time used to obtain the particular image and the number in the lower right 
gives the threshold level for the generated mask. As shown in Fig. 6.3, the core masking 
method reveals an irregular spatial distribution of the beam halo. 
 
Figure 6.3:  Unmasked and masked OSR images of the JLAB FEL beam. 
  125 
 
 When analyzing line scans across images 1, 3, 5, 7, 9, and 11 (note for reference 
the horizontal red line in Fig. 6.3), we normalized them by the integration time taken for 
each image. The normalized line scans are plotted in Fig. 6.4. As is observed in the tails 
of the line scans, the longer the integration time is, the smaller the intensity fluctuations 
are. In addition, the noise level is decreased to below 10-4, which indicates a good 
dynamic range for the measurements obtained so far. This is expected to improve as we 
develop the optical system and focus the beam to a smaller spot size. 
0 200 400 600 800 100010
-6
10-4
10-2
100
Pixel
N
o
rm
a
liz
e
d 
Co
u
n
ts
 
Figure 6.4: Normalized horizontal scans of beam profiles 
 
6.2 Injected beam measurement in SLAC SPEAR3 
6.2.1 Experiment Setup 
 The visible diagnostic beam line at SPEAR3 [47, 48] contains a rectangular 
aperture mask about 7 m from the OSR source point with an angular acceptance of 3.5 ? 
6 mrad2. As showed as black dot in the right of Fig. 6.5, a ?0.47 mrad, horizontally 
extended ?cold figure? is located right after the primary aperture in order to absorb the 
  126 
 
intense x-ray component of the light. This complex compound aperture creates a 
window-like shape at the first beam line lens and diffraction from the edges of the 
aperture contributes significantly to the overall shape of the point spread function (PSF) 
of the optics that is discussed later.  
 
 
Figure 6.5: Layout of the optical system in the first synchrotron beam line of SPEAR3 
 A schematic of the beam line optics including the 2-channel DMD system is 
shown in Fig. 6.5. The image from the source is first imaged to a plane by f = 2 m 
objective lens and then re-imaged onto the DMD surface by a single acromat, f = 125 mm. 
At rest the micro-mirrors on the DMD surface are perpendicular to the incident light. The 
net magnification between the OSR source and the DMD camera is M1*M2=0.49. This 
allows us to easily image the ?8 mm betatron oscillations of the injected beam onto the 1 
cm2 surface of the DMD. When all the DMD micro-mirrors flip 12? toward the PiMax 
camera, the incident light will be reflected into the second optical channel which contains 
a third f = 100 mm achromat creating a 1:1 image relay system. Note that the camera is 
rotated 24? about the vertical axis to compensate for the Scheimplug effect [reference]. 
 
  127 
 
6.2.2 Point spread function Measurement 
 Point spread function (PSF) is the image of a point source through the whole 
optics. The image of any source can be considered as a convolution of the source 
distribution and the PSF. In our measurement of the PSF, we make the beam as small as 
possible to replace the point source. We use this PSF to test to which order our high 
dynamic range image is valid. It means that after normalization of both the PSF and the 
high dynamic image we measure later and lay them together in one plot, anything above 
the PSF is true; otherwise, it is un-trusted. 
 For the previous PSF measurement in UMER, we use a similar optics system with 
an actual line thread illuminated by a red laser as a point source. By progressively 
shifting the source away from the active area of CCD sensor of camera, and applying the 
well-calibrated neutral density filter to attenuate the light to avoid saturating the CCD, we 
achieved a dynamic range of 107 for halo measurement. In SPEAR3, however, it is not 
easy to find such a point source. Moreover, since our interest right now is the injected 
beam other than beam halos, and the injected beam is, in first several turns, away from 
the bright central stored beam, we want to prove the stored beam would not affect the 
measurement of the injected beam in this optics system. Therefore, the stored beam is 
sufficient approximation to simulate the point source. Strictly, the PSF mentioned here is 
a convolution of real PSF and the stored beam distribution. 
 To estimate the PSF, we were taking high dynamic ranges images of the stored 
beam alone. The stored beam was first imaged with an ND=2 filter and low MCP gain. 
Then we applied successive threshold level masks to the DMD in order to observe ?halo? 
or tails of the PSF with increasing detail. For each intensity mask we increased the 
  128 
 
integration time (shutter mode of this camera was used [refer to camera]) to bring the 
peak intensity up to near the saturation level of the CCD sensor. To use long integration, 
we assumed that the visible light generated by the stored beam did not change during that 
time. Then by obtaining a number of images, each of which examined a segment of the 
total intensity profile, we were able to reconstruct a high dynamic range (~ 106) picture of 
the PSF. Figures 6.6 (a) and (b) show the first 3 and last 4 decades of the PSF, 
respectively. The inserted image on the top right corner of (b) is the light distribution 
incident on the objective lens, i.e. the aperture function (AF) of the optical system that is 
produced as the OSR passes through the rectangular beam line aperture and the 
horizontal extended cold finger. The cross like structure observed in (b) is the Fourier 
transform of the AF that is visible in the image plane. It is due to the horizontal and 
vertical structures of the AF. Note the additional slanted ray seen on the upper right 
quadrant of (b), which is caused by the tilted edge seen in the upper left side of the AF.  
  (a)          (b)  
Figure 6.6: Log normalized intensity profile of the SPEAR3 stored beam; (a) first 3 
decades; (b) last 4 decades and insert showing aperture function of the visible beam line. 
  129 
 
6.2.3 Injected beam imaging in off-duty test 
 In SPEAR, the design of the injection is to inject 3 GeV electron beams with 
certain intensity to fill SPEAR to several hundred mini-amperes with-in 5 minutes [49]. 
The electron beam starts from a 2.5 MeV rf-gun and is accelerated to 120 MeV in a linear 
accelerator.  Then, it is injected into a booster synchrotron to reach the SPEAR injection 
energy, before being kicked into a beam transport line (BTS) to SPAER. A final orbit 
correction and envelope matching are performed inside the BTS to minimize beam loss 
due to injection. Since our diagnostic is located beside the first SPEAR synchrotron beam 
line, we can use it to record ordinary synchrotron radiation (OSR) of newer injected beam 
and determine how effective of the injection with certain magnet setting.  
 The OSR intensity from the injected beam is not strong enough, even with full 
MCP gain on the PiMax camera to image on a single pass with only one exposure. Thus, 
we utilized the synchronously-triggered gate mode of the PiMax to integrate several 
images per exposure (typically 15, which is the number of beams when each injection in 
a time interval of 5 mins during normal operation) prior to image readout.  With stored 
beam present, the problem of rejecting the stored beam light intensity is therefore 
compounded 15 folds. Two methods are used to reject the intense light in the core of the 
stored beam: (1) masking the image of the stored beam using the DMD and (2) gating in 
time on the injected charge. In order to image the injected beam in the presence of the 
stored beam, an intensity threshold mask or a fixed sized mask is applied on the DMD. 
To generate the threshold mask we integrated the stored beam intensity over a long-
exposure time, i.e. during the injection kicker bump excitation, and then numerically 
calculated the intensity threshold from the resulting image to define the mask. By doing 
  130 
 
so, we can prevent the damage of the camera from the bright stored beam due to its 
transverse oscillation. 
 The main storage ring of SPEAR3 can be divide into 372 (not sure about this 
number) time buckets, with each bucket filled with 1mA to its saturation. For the beam 
advance one term with 781 ns, each bucket is roughly 2.1 ns. For the first set of injected-
beam measurements we first filled 15 target buckets with the same amount of stored-
beam charge (~1nC), and then progressively injected a single ~50pC shot into each of 
them advancing from bucket to bucket each shot. Simultaneously, the camera gate was 
synchronized with the injected beam pulse and automatically advanced from bucket to 
bucket. We call this process ?clock mode? imaging. This has the advantage that the 
current of the stored beam in each bunch, which creates the background of each gated 
image, is kept nearly constant and the each bucket will not be saturated.  In the alternative 
?stacking mode?, in which injected pulses progressively increase the stored charge of a 
single bunch, we concerned that the increasing current of the stored beam might influence 
the injected beam dynamics. Moreover, the ?clock mode? is the only way to make online 
measurements when machine is dedicated to synchrotron users. However, the 
disadvantage of clock mode is time-consuming, because it is necessary to dump the beam 
or advance to another group of target buckets after the charge in the target buckets 
becomes excessively high. 
 For synchronizing the beam with camera and cutting out the background light, we 
use four digital delay/pulse generators (or trigger box, Stanford research system, Inc) to 
give external trigger to the PIMAX camera and control a mechanical shutter. The 
mechanical shutter is used to further cut out the background light even when the camera 
  131 
 
gate is closed or during the camera readout, because we see an effects of background 
light bleeding in these two events. The connections of all boxes are illustrated in Fig. 6.7 
and the timing table is shown in Fig. 6.8.    
  
Figure 6.7: Schematic plot of the triggering system. 
 
 Box 1, triggered by the master trigger of SPEAR3, gives external trigger to the 
PIMAX camera. The master trigger is 0.1 s, indicating the injection rate is 10 Hz. The 
camera gate is delayed 320.232 ns (delay 1 in the timing table) to synchronize with the 
first turn. For nth turn, the delay time will be 320.232 + (n-1)*780 ns. Note that the beam 
circulates the SPEAR3 main ring by 780 ns.  The gate opens 20 ns to cover the beam 
bucket. In the injection event, usually 15 new pulses are injected, so the number of gates 
is set to be 15. Box 2, triggered by the camera gate monitor signal, will trigger Box 3 and 
  132 
 
Box 4. The signal generated by Box 3 controls the mechanical shutter open time with the 
positive trigger value. This signal is inhibited while an inhibited signal generated by Box 
4 is zero. As shown in the timing table, the mechanical shutter inhibit signal is delayed by 
25 ms compared with the camera first gate signal and its length is (pulse number -1) *100 
ms. The mechanical shutter signal is delayed by 95 ms with a length of 4 ms. In this 
setting, the first beam pulse will not be collected as indicated by the dashed arrow, and 
the noise from readout will be reduced completely.  
100 ms
Delay 1
10 ns
Master trigger
Camera gate
Mechanical 
shutter (unibit) 4ms
Mechanical 
shutter inhibit
25 ms
Beam
Camera Readout
(Beam pulse number-1) *100 ms
96 ms
 
Figure 6.8: Timing table for injected beam imaging 
 
 We used the ?clock mode? technique to image the injected beam shown at turn 6 
for 15 shots with different single-bunch stored beam currents in Fig. 6.9. Here the outer 
crossing shape of the PSF clearly becomes more visible as the stored beam current is 
increased. Intensity profiles formed by integrating along the vertical axis are shown in 
Fig. 6.10 (a). The plots clearly show that with increasing stored beam current the 
illumination of the PSF (left peak) is enhanced while the intensity profile of the injected 
  133 
 
beam (right peak) remains nearly constant. The constant profile of the injected beam also 
demonstrates that at these current levels the injected beam dynamics is not affected by 
space charge amplitude in the stored beam. This result is further confirmed by comparing 
the 2?rms x- and y- beam size profile plotted in Fig. 6.10 (b), which shows a nearly 
constant injected beam profile at turn 6, with values about x=2.6 mm and y=5.5 mm. 
 
Figure 6.9: Masked with time-gated images of injected beam turn 6 with increasing 
single-bunch current. 
  134 
 
0 250 500 750 10001
1.5
2
2.5
x 105
Pixel
In
te
n
s
ity
 
le
v
el
 
 
0.42mA
1.52mA
3.05 mA
6.11mA(a)
 
0 2 4 62
3
4
5
6
Current per bucket (mA)
In
jec
te
d 
be
a
m
 
s
iz
e 
(m
m
)
 
 
X 2*RMS
Y 2*RMS
(b)
 
Figure 6.10: a) Horizontal intensity profile of the injected beam including PSF 
contribution from stored beam; b) 2?rms injected beam size vs. stored beam current. 
6.2.4 Beam Mismatch Experiment 
 By altering the strength of the 9th defocusing quadrupole in the Booster-to-
Storage ring (BTS) transport beam line, it is possible to change the phase-space matching 
of the injected beam with respect to the storage ring and impact charge capture efficiency. 
  135 
 
For these tests, we used the same ?clock mode? data acquisition process as before but 
generated a larger ?rectangular? DMD mask instead of an intensity mask. The reason is 
when the injection kicker fires, it also give the stored beam a kick, which increase the 
amplitude of the betatron oscillation. Fig. 6.11 shows images of the 20 even turns of the 
injected beam for three different BTS matching conditions. For each condition, the plots 
are delineated by injection rate.  
Turn 4 Turn 6Turn 2 Turn 8 Turn 10 Turn 12 Turn 14 Turn 16 Turn 18 Turn 20
61.0 mA/min
33.0 mA/min
15.3 mA/minfinal Injection Rate
0
6000
9 mm
default
17.5 A
14.9 A
Quadrupole current
 
Figure 6.11: First 20 turns of the injected beam for three cases of matching condition. 
 Since the stored beam also has a small betatron oscillation, the cross-like PSF 
created by the stored beam moves picture to picture. It is not simple to take only one 
background to do background subtraction for all the beam images. In order to measure 
precisely the beam parameters, such as beam centroid and rms size, we need to take a 
background for each beam image using the same condition only without the injection. In 
reality, especially when the machine is dedicated for users, this measure will be tedious 
and even impossible. However, over the observation of the PSF, we found the shape of 
PSF almost does not change although jittering always. This allows us only taking one 
background and moves this background PSF to overlap with the PSF in each beam 
picture to do subtraction. The only drawback is that we still do not know the distribution 
  136 
 
inside the mask where all the information is masked out by DMD. For this reason, we 
will not present the beam parameters for turn 12, 16 and 20.  
 The x and y centroid motions are plotted in Fig. 6.12. Comparing the oscillation in 
x and y direction, the amplitude is much larger in x (about 5 mm) than in y (about 2 mm), 
which consists with the fact that we inject the beam horizontally. Clearly the defocusing 
quadrupole strength has a limited effect on the horizontal betatron motion of the injected 
beam (dominated by the injection oscillation), but has a large impact in the vertical 
direction indicating a vertical beam offset in the 9thdefocusing BTS quadrupole.  
2 4 6 8 10 12 14 16 18 20
-5
0
5
10
15
Turn 
B
e
am
 
X-
ce
n
tr
o
id
 
(m
m
)
 
 
15.3 mA/min
33.0 mA/min
61.0 mA/min
 
(a) 
  137 
 
2 4 6 8 10 12 14 16 18 20
-2
0
2
4
Turn
B
e
am
 
Y-
ce
n
tr
o
id
 
(m
m
)
 
 
15.3 mA/min
33.0 mA/min
61.0 mA/min
 
(b) 
Figure 6.12: Beam centroid motion of the injected beam for the three different BTS 
matching condition. (a) x axis; (b) y axis. 
 
 We also plot the 2 ? rms beam size for each turn in Fig. 6.13. For the x and y 
coordinates, the beam sizes are initially similar yet undergo different turn-by-turn 
evolutions due to different phase-space dynamics in each axis.  The 2?rms size is not 
shown for turns 12, 16 and 20, since the images of the injected beam were partially 
blocked by the stored beam mask and light from the beam line PSF.  
  138 
 
2 4 6 8 10 12 14 16 18 201
2
3
4
Turn
2*
rm
s
 
be
a
m
 
X-
s
iz
e
 
(m
m
)
 
 
15.3 mA/min
33.0 mA/min
61.0 mA/min
 
(a) 
2 4 6 8 10 12 14 16 182
4
6
8
Turn
2*
rm
s
 
be
am
 
Y-
s
iz
e
 
(m
m
)
 
 
15.3 mA/min
33.0 mA/min
61.0 mA/min
 
(b) 
Figure 6.13: Beam size evolution of the injected beam with three different BTS 
quadrupole matching conditions. (a) x axis; (b) y axis.  
 
 For turns that can be clearly imaged, such as turns 6, 10 and 18, there is a linear 
relation between the integrated image intensity of the injected beam and the injection rate, 
  139 
 
which is shown in Fig. 6.14. This indicates that most of the beam loss occurs in the BTS 
transport line or in the injection septum prior to entering the storage ring when we 
compare the two lower injection rate cases with the higher one. 
20 30 40 50 60
0.5
1
1.5
2 x 10
7
Injected rate (mA/min)
In
te
n
si
ty
 
 
 
Turn 6
Turn 10
Turn 18
 
Figure 6.14: Total intensity of the injected beam versus injected rate for turn 6, 10 and 18. 
6.3.5 Injected beam image in on-line test 
 Since the SPEAR3 is a user facility, most of its running time was dedicated to the 
user experiments. While it operate, the stored beam will get loss so that we need fill the 
ring with about 15 new beam pulses in a time interval of 5 minutes. The transportation of 
this injected beam pulses and their acceptance into the storage ring is very important, 
because it decides the capture efficiency and the filling frequency. To study the injection 
without perturbing the existing user experiments, the method discussed previously is right 
on the target.  
 In the experiment, we will still use the so-called ?clock mode?. The filling time of 
this storage ring is about every 5 minutes, which means we can only get an injected beam 
image for a certain turn in that time interval. The gain is set as the maximum to increase 
  140 
 
the visibility, and the gate width is set as 10 ns to avoid unnecessary noise as well. Figure 
6.15 shows a series of turns of the injected beam while normal operation of SPEAR3. We 
can see the same structure of the stored beam is located in the center. Since the mask 
position is fixed according to the camera, it is obvious that the leakage of the cross-like 
shape indicates a jittering of the stored beam both in x and y direction. Thus in some case, 
the jittering is so big that we need make a larger mask (turn 22 and 26) to protect the 
camera from saturation. In order to show the injected beam in the same scale, the display 
range of these images is the same from 80 ? 3000 as indicated by color bar.  
 
Figure 6.15: Beam images in different turns during normal operation. 
 After subtracting the background and the cross-like point spread function of the 
stored beam using the same method mentioned in previous section, we can analyse beam 
centroid and 2*RMS beam size from the injected beam image. (It is still hard to 
reconstruct the information hidden in the mask, so when the centroid of the injection 
beam is close to that of the stored beam, the analysis is not applicable. After the beam 
  141 
 
smears, the measurement will have big error because we lack the information inside the 
mask). The beam centroid motions and beam sizes changes are plotted in Fig. 6.16.  
5 10 15 20 25 30-8
-6
-4
-2
0
2
4
6
Turn
Po
si
tio
n
 
(m
m
)
 
 
X centroid
Y centroid
 
(a) 
5 10 15 20 25 301
1.5
2
2.5
3
3.5
4
Turn
2*
R
m
s 
be
am
 
s
iz
e 
(m
m
)
 
 
X
Y
 
(b) 
Figure 6.16: Turn by turn evolution of beam centroids and beam sizes during normal 
operation 
  142 
 
 Comparing with the off-line matched case, which has a largest injection rate of 
61.0 mA/min, both the beams? centroid oscillation have a amplitude in the range of [-2,2] 
in y and [-5,5] in x. The different in oscillation detail could due to varied injection 
position and angle.  For beam sizes and shapes, we can see some similarity in the sixth 
turn, but in general, the beams are even not close in shape for other turns. This could be a 
result of different injection phase and match condition are slightly off.  
 
6.3 Chapter Conclusion 
 In this chapter, we discuss the application of the adaptive masking method used in 
halo diagnostic in JLab FEL facility and injected beam imaging in SLAC SPEAR3. We 
have succeeded in synchrotron radiation imaging and achieve a high dynamic range of 
104 again in JLAB FEL facility. In SLAC SPEAR3, the novel diagnostic help solving the 
problem of the dynamic range issue when one images the synchrotron radiation from the 
injected beam with a strong intensity synchrotron radiation of stored beam existed.  
 
 
  143 
 
Chapter 7:  Conclusion 
 
7.1 Summary 
 The goal of this dissertation is to better understand the halo formation in intense 
charged particle beams and figure out the dominated mechanism in beam dynamics for 
halo particles occurrence. As discussed in Chapter 1, the halo generation in accelerators 
can significantly degrade the beam quality and cause beam loss. The latter one might 
bring severe consequence for the maintenance of the accelerators and result in serious 
safety issues. The halo formation research gradually becomes an active research topic 
especially in high current and high intensity beams. The emphasis of this issue is very 
important and will give guidance for the design of the next generation of high intensity, 
high current accelerators.  
 As suggested by the particle core model, mismatch is major source for halo 
generation. I took a great effort to match the UMER beams in Chapter 3. First, I solved 
the magnet field error issue by studying the discrepancy between the experiments and 
simulations. This greatly reduces the beam loss for multi-turn operation and benefits not 
only my halo experiment but also other ongoing experiment in UMER. This is a more 
than important item to be checked before any sophisticated experiment. Moreover, I 
developed a matching procedure to do beam match by using tracking code as a coarse 
step and empirical method as a fine adjustment. Following by a rotation correction, I 
achieved an envelope match for three beams with increasing space charge intensities. The 
agreement of matched beam sizes for three cases is satisfactory between experiments and 
  144 
 
Trace3D code prediction. For extreme space charge case, since the beam sizes are large, 
the beam shape is not regular due to image force from beam pipe. This could be 
improved by a better steering, which could be address further in the future. This 
procedure is quite general that it can be used to other beams in alternate lattice in UMER 
or even other accelerators with space charge.  
 I found out in Chapter 4 experimentally that the envelope mismatch is not the 
only source for halo generation, but also the beam rotation or skewness could contribute 
in the space charge dominated beam. The latter has been study in simulation before, but 
this is the first time we confirm it in experiment. From an initial beam rotation, I 
observed that the beam goes through a wobbling oscillation, and the associate energy 
from the oscillation will transfer to the beam particles to form halo. Either the envelope 
mismatch or beam rotation could result in a quick halo formation if the initial mismatch 
or rotation is large.  
 In Chapter 5, I systematically studied the halo formation with respect to different 
level of mismatch in the frame of pure mismatch modes. I developed a method for pure 
mode mismatch and testified it in warp code. The FFT analysis of the simulated envelope 
sampled with the lattice period shows a dominated peak of the wave number. The value 
of the peak is close to the mismatch wave number calculated from a smooth theory of 
periodic FODO lattice. I compare the maximum halo radius and emittance growth 
between theory, simulation and experiments. Both simulations and experiments agree 
well and confirm that free energy model gives an upper limit for maximum emittance 
growth and the particle core model is good approximation of the maximum halo radius. 
This suggests that both the models can be used to guide design of the high intensity and 
  145 
 
high current accelerators. However, as said in the content, some margin should be taken 
into consideration. 
 I also include the application of the novel adaptive masking method in JLab FEL 
facility and SLAC SPEAR3. Followed from my master thesis, several properties are 
address in Chapter 2 to better understand the optics and compensations. In Chapter 6, I 
first applied the method for synchrotron radiation imaging and achieved a high dynamic 
range in JLab FEL facility. In SLAC SPEAR3, the implementation of the adaptive 
method helps to solve the injected beam imaging with present of the high intensity 
synchrotron radiation from stored beam.  
7.2 Future work 
 
 As mention in Chapter 5, in the experiment, due to the limitation of screen size, I 
can only measure the halo particle to 1% level of the maximum intensity. One idea is to 
use the steering dipole to kick the beam centroid to the edge of the screen to full-use the 
screen size. Another one is to install wire scanner to measure the beam profile. Moreover, 
the emittance growth in y axis could be measured to further verify the free energy model.   
 Due to the geometry and design of current YQ section, although I solve the Eddy 
current problem, there is still misunderstanding of the fringe fields and image forces in 
this section. New design in UMER has been proposed for the extraction section [50], 
which could also be applied to the YQ section. I hope this will eventually help match the 
beam in multi-turn.  
 Meanwhile, the knock-out method [51] or the installation of an extraction section 
in the future will help with multi-turn diagnostic which means the halo study in this 
  146 
 
dissertation could be extended to multi-turn and include infinity FODO lattice or 
mismatch periods if not consider the longitudinal erosion. The topic of halo formation 
rate could be studied for a long distance in the frame of small or large mismatch 
parameters or other possible parameter which could affect the halo formation process. 
Neither particle core model nor free energy model addresses this problem but it is of 
importance for new accelerator design and operation.  
   
     
  147 
 
Appendices A: Sample matching code in Trace3D 
 
&DATA 
ER=  0.510999  Q= -1.  W=     0.01000  XI=  6.000000 
EMITI =     17.856250       17.856250    0.1800000 
BEAMI =     0.180800       0.053600      0.180800       0.053600       
-1.15      250000.0 
BEAMF =     -2.670300       0.710700      2.614600       0.667800        
1.15      250000.0 
BEAMCI=       0.00000       0.00000       0.00000       0.00000       
0.00000       0.00000 
FREQ=    5.000  ICHROM= 0  IBS= 0  XC=   0.0000 
XM=  27.95  XPM=  50.0000  YM=  27.95  DPM=  30.00  DWM=  100.00  DPP=  
30.00 
 
N1=  1  N2= 37  SMAX=   0.1   NEL1=   1   NEL2=  37   NP1=   1   NP2=  
37 
MT=  8 
MP = 1,11,1,13,1,7,1,9 
CMT(001)='Gun-Sol '  NT(001)=  1  A(1,001)=  138.700000 
CMT(002)='Sol-smoo'  NT(002)=  5  A(1,002)=  81.807703      65.200000 
CMT(003)='Sol-Strt'  NT(003)=  1  A(1,003)=  116.100000 
CMT(004)='Strt-Q1 '  NT(004)=  1  A(1,004)=  62.925000 
CMT(005)='Q1-DeFoc'  NT(005)=  3  A(1,005)=  0.032860      44.750000 
CMT(006)='Q1-Q2   '  NT(006)=  1  A(1,006)=  87.250000 
CMT(007)='Q2-Focus'  NT(007)=  3  A(1,007)= -0.059070      44.750000 
CMT(008)='Q2-Q3   '  NT(008)=  1  A(1,008)=  145.250000 
CMT(009)='Q3-DeFoc'  NT(009)=  3  A(1,009)=  0.058410      44.750000 
CMT(010)='Q3-Q4   '  NT(010)=  1  A(1,010)=  151.250000 
CMT(011)='Q4-Focu '  NT(011)=  3  A(1,011)= -0.056370      44.750000 
CMT(012)='Q4-Q5   '  NT(012)=  1  A(1,012)=  98.250000 
CMT(013)='Q5-DeFoc'  NT(013)=  3  A(1,013)=  0.057520      44.750000 
CMT(014)='Q5-Q6   '  NT(014)=  1  A(1,014)=  115.250000 
CMT(015)='Q6-Focu '  NT(015)=  3  A(1,015)= -0.055740      44.750000 
CMT(016)='Q6-YQ   '  NT(016)=  1  A(1,016)=  106.459939 
CMT(017)='YQ-DeFo '  NT(017)=  3  A(1,017)=  0.044140      58.330123 
 
CMT(018)='        '  NT(018)=  1  A(1,018)=  50.834939 
CMT(019)='PD      '  NT(019)=  2  A(1,019)=  2754949.06      2754.95 
CMT(020)='        '  NT(020)=  1  A(1,020)=  50.005000 
CMT(021)='QR1-Foc '  NT(021)=  3  A(1,021)= -0.042870      59.990000 
CMT(022)='        '  NT(022)=  1  A(1,022)=  107.630000 
CMT(023)='QR2-DeFo'  NT(023)=  3  A(1,023)=  0.055840      44.750000 
CMT(024)='        '  NT(024)=  1  A(1,024)=  57.625000 
CMT(025)='Dipole  '  NT(025)=  2  A(1,025)=  15680.00      3500.00 
 
CMT(026)='Drift   '  NT(026)=  1  A(1,026)=  57.625000 
CMT(027)='Foc.Quad'  NT(027)=  3  A(1,027)= -0.055038      44.750000 
CMT(028)='Drift   '  NT(028)=  1  A(1,028)=  115.250000 
CMT(029)='Def Quad'  NT(029)=  3  A(1,029)=  0.055038      44.750000 
CMT(030)='Drift   '  NT(030)=  1  A(1,030)=  57.625000 
CMT(031)='Dipole  '  NT(031)=  2  A(1,031)=  15680.00      3500.00 
 
CMT(032)='Drift   '  NT(032)=  1  A(1,032)=  57.625000 
CMT(033)='Foc.Quad'  NT(033)=  3  A(1,033)= -0.055038      44.750000 
CMT(034)='Drift   '  NT(034)=  1  A(1,034)=  115.250000 
  148 
 
CMT(035)='Def Quad'  NT(035)=  3  A(1,035)=  0.055038      44.750000 
CMT(036)='Drift   '  NT(036)=  1  A(1,036)=  57.625000 
CMT(037)='Dipole  '  NT(037)=  2  A(1,037)=  15680.00      3500.00 
 
COMENT='UMER Y-matching Step 3' 
&END 
 
 
  149 
 
Appendices B: Sample simulation code in python 
 
from warp import * 
 
sys.path.insert(1, "/ebte/pywarp/rscripts/scripts_a") 
sys.path.insert(1, "/humer/shared/Universal") 
sys.path.insert(1, "humer/shared/WARP/Warp32_2011-2-22/scripts") 
 
if 'GISTPATH' in os.environ.keys(): os.environ["GISTPATH"] += 
":/ebte/pywarp/rscripts/scripts_a" 
else:   os.environ["GISTPATH"] = "/ebte/pywarp/rscripts/scripts_a" 
from rami_scripts import * 
from UMERGeometry import * 
from ParaKV import * 
from rami_match import * 
 
################ G L O B A L   P A R A M S ##################### 
 
ltest = 0 
lmatch = 0 
lscreen = 0 
lquadcenter = 0 
nperds = 30 
 
pm_initial = 0.0 #degree 
pm = pm_initial 
 
 
if lmatch: 
   nperds = 2 
 
a0_initial       = 0.0038688 
b0_initial      = 0.0038766 
ap0_initial     = -0.014067214 
bp0_initial     = 0.014112477 
offset_hao = -0.945 
 
I_initial = -7.45*0.8e-3 
emit_initial_x  = 12.874e-6 
emit_initial_y = 12.874e-6 
 
 
if len(sys.argv) == 1:  # Running single simulation 
    # quad current in Amps (positive: focus on x (horiz), de-focus on y (verical))  
 
  150 
 
    runnum = 501 
else:                   # Running one of a sequence of simulations 
    runnum = int(sys.argv[1]) 
    inputdata = sys.argv[2] 
    f1 = open(inputdata, 'r') 
    lines = f1.readlines() 
    f1.close() 
    data = lines[runnum] 
    items = data.split() 
    [a0_initial,b0_initial,ap0_initial,bp0_initial] = [float(i) for i in items] 
    print "a0 = %-.3f, ap0 = %-.3f, b0 = %-.3f, bp0 = %-.3f," 
%(a0_initial,ap0_initial,b0_initial,bp0_initial) 
 
file1_name = "halo_particle_MaxRadius"+str(runnum)+".txt" 
 
if lscreen: 
 a0_initial      = a0_initial 
 ap0_initial     = ap0_initial 
 b0_initial      = b0_initial 
 bp0_initial     = bp0_initial 
 offset_hao = -29.1425 
 
if lquadcenter: 
 a0_initial       = 0.003566#0.002435 
 b0_initial      = 0.002731#0.003947 
 ap0_initial     = -0.013296#-0.0001175 
 bp0_initial     = 0.010747#0.000453 
 offset_hao = -8.0 
 
##############  Injector Quadrupole scan############ 
# --- Set four-character run id, comment lines, user's name. 
 
if runnum < 10: 
   crun = "0"+str(runnum)             # Run number if more than one simu with same runid 
else: 
   crun = str(runnum) 
runid += crun 
top.runid[0] = runid 
 
 
top.pline2 = "TRACE Matching - 6 mA, fixed POS, after SOL, BEarth, Dipole f from R-
MAT" 
# top.pline1 is automatically filled later 
top.runmaker = "Hao Zhang" 
 
# --- run control 
  151 
 
ltest      = ltest          # set to 1 to speed up simulation for testing 
if lmatch: 
    ltest      = 1  
lpicts     = 0          # Take density pictures? 
lcalc_mom  = 1          # Calculate additional moments? (needed for quad rotation or 
dispersion) 
if lmatch: 
    ltest = 1  
# --- Comparison Plots against other simulation 
 
lcompare = 0 
dot1 = "FixedMatch_6mA_AS_BE_NewDip_exp"       # Runid of old simulation (to 
compare against) 
old_crun = "1"          # Run number of old simulation 
cf_title = "Change Q1" 
 
 
# --- Invoke setup routine (THIS IS MANDATORY) 
setup() 
 
 
############ B E A M   P A R A M E T E R S   1 ############### 
 
# +++ energy [eV] and species (current and zion -ve for electrons) 
top.ekin     = 10.0e03 
top.aion     = top.emass/top.amu 
top.zion     = -1. 
top.lrelativ = yes                        # relativistic particle push 
 
derivqty()  # calculate additional beam parameters, eg. top.vbeam from ekin. 
 
# +++ For dispersion and chromaticity: add a longitudinal thermal velocity 
e_spread     = 10.0                     #  Longitudinal energy spread in eV 
top.vthz     = top.vbeam*e_spread/(2.*top.ekin) 
 
 
#################### INITIAL DISTRIBUTION ###################### 
 
w3d.distrbtn = "semigaus"  # load  semi-Gaussian distribution 
#w3d.distrbtn = "K-V"       # load KV distribution 
 
 
 
#################### GEOMETRY & GRIDDING ###################### 
 
# +++ General 
  152 
 
top.npmax = 1000000           # Number of simulation macro-particles (40,000 typical, 
more if tracking 
stepsize  = 0.001           # step size in m 
ncells    = 512             # number of cells in each direction (power of 2) 
 
if lpicts:  top.npmax = max(top.npmax, 320000) 
if ltest:                   # DO NOT CHANGE THESE.  Use only for fast testing 
    top.npmax = 5000 
    ncells = 256 
    stepsize = 0.004 
 
# +++ Boundaries and Gridding 
top.prwall = pr1 = 2.45e-2  # Remove particles outside of Pipe radius 
boxrad = 2.5e-2          # Box 1/2 side > pipe radius (pr1) 
 
w3d.xmmin = w3d.ymmin = -boxrad 
w3d.xmmax = w3d.ymmax =  boxrad 
w3d.nx = w3d.ny = nint(ncells) 
 
 
# +++ Particles 
top.stickyxy = 1            # remove particles when they hit pipe. 
top.ibpush   = 1            # set type of pusher to  vXB push without tan corrections 
 
# +++ Stepsize 
top.dt = stepsize/top.vbeam                  # OK, 0 injection 
#wxy.lvzchang = 1    # If 1, fancy algorithm is used to find top.dt 
 
###################  SPECIFY LATTICE  ########################## 
 
# --- do a doc() on each method to determine optional arguments 
#ring = UMER('iyr', oppoint="83%", crun=crun, nturns=1, nperds=2, lEarth=1, 
verbose=1, lpf=0, offset = offset_hao, diags='a',pdiags='s') 
ring = UMER('r',oppoint="83%", crun=crun, nperds=nperds, nturns=4, lEarth=0, 
verbose=1, lpf=0,stopatscreen=0, diags='a',pdiags='s', funcs=[pptrace], 
pltype='density',offset = offset_hao ) 
 
 
 
############ B E A M   P A R A M E T E R S   2 ############### 
 
# --- ring.select_beam(aperture, operting point) automatically loads a matched beam 
# --- can override by commenting out and/or directly editing top.* parameters 
#beam = ring.select_beam("6mA", "83%") 
 
# +++ envelope and slope [m, rad], specific number for the starting point 
  153 
 
top.a0       = a0_initial   #beam.a0  
top.b0       = b0_initial   #beam.b0 
top.ap0      = ap0_initial    #beam.ap0 
top.bp0      = bp0_initial    #beam.bp0 
 
# +++ centroid and angle [m, rad] 
top.xcent_s  = 0.0#beam.xcent 
top.xpcent_s = 0.0#beam.xpcent 
top.ycent_s  = 0.0 
top.ypcent_s = 0.0 
 
# +++ current [A], unnormalized 4*rms emittance [m-rad] 
top.ibeam    = I_initial#beam.ibeam 
top.emitx    = emit_initial_x#beam.emitx 
top.emity    = emit_initial_y#beam.emity 
top.emit     = emit_initial_x#top.emitx 
 
# --- Hollow or peaked initial distribution, if desired 
w3d.hollow = 0          # 0 = flat; 2 = parabolic density 
w3d.hollow_h = 0.5      # n(w3d.rho)  not  1 + ((1-h)/h)*w3d.rho**2 
if (w3d.hollow == 2):  # WHY IS THIS COMMENTED OUT? 
    # --- form factor for parabolic density profile 
    Stf = sqrt( (2.*w3d.hollow_h+4.)/(3.*w3d.hollow_h+3.) ) 
    # --- scale loading parameters so the correct rms values are loaded 
    top.a0 = top.a0*Stf 
    top.b0 = top.b0*Stf 
    top.ap0 = top.ap0*Stf 
    top.bp0 = top.bp0*Stf 
    top.emit = top.emit*Stf**2 
 
 
# !!! 5-beamlet: create module with choices? 
# !!! Hollow-vel: must load after generate() 
 
if top.lrelativ: top.ibeam = top.ibeam/(top.gammabar**2)    # Correct field solve 
 
top.pline1 = ring.calc_pline() 
 
 
########### F I E L D    S O L V E R #################### 
 
top.fstype = 1 
#      -1       To Turn off field solver 
#       1       FFT Solver w/ Capacity Matrix (inifinite) 
 
# --- Set up Capacity Matrix for WARPxy 
  154 
 
fxy.ncxy    = fxy.ncxymax = w3d.nx+w3d.ny   # set number of points 
gallot("CapMatxy",0)                        # Allocate the arrays 
# -- Coordinates of boundary points in meters 
fxy.xcond[0:fxy.ncxy] = pr1*cos(2.0*pi*arange(fxy.ncxy)/fxy.ncxy) 
fxy.ycond[0:fxy.ncxy] = pr1*sin(2.0*pi*arange(fxy.ncxy)/fxy.ncxy) 
fxy.vcond[0:fxy.ncxy] = 0.e0                # Put pipe at ground 
 
f3d.lcndbndy = true     # enable subgrid interpolation 
 
############### D I A G N O S T I C S ################# 
 
top.nhist = 1       # at what frequency of steps to save moment histories 
top.itmomnts[0:4] = [0,1000000,abs(top.nhist),0] 
 
top.xpplmin = top.ypplmin = -0.015    # limits to use on particle plots 
top.xpplmax = top.ypplmax = 0.015 
 
top.verbosity = 1   # Turn off built-in oneliner output 
top.npplot = [top.npmax/10, 5000, 2500]      # Number of particles to save 
top.ncolor = 10 
#palette("gray.gp") 
 
 
###############  G E N E R A T E   ################## 
 
# --- Generate the PIC code (allocate storage, load ptcls, t=0 plots, etc.) 
package("wxy"); generate() 
 
w3d.zmmax = 0.1; 
# +++ Hollow velocity here, if needed 
# +++ 
file1 = open(file1_name,"w") 
 
def check_halo(): 
 Ellipse_ratio = array([1,1.5,2]) 
 Length_ratio=len(Ellipse_ratio) 
 num1 = array([0.0]*(3*Length_ratio+7)) 
 
 for iii in range(0,3):  
  out = 
where((top.pgroup.xp/2/top.xrms)**2+(top.pgroup.uxp/top.vbeam/2/top.xprms)**2>Elli
pse_ratio[iii]**2,1,0) 
  out1 = 
where((top.pgroup.yp/2/top.yrms)**2+(top.pgroup.uyp/top.vbeam/2/top.yprms)**2>Elli
pse_ratio[iii]**2,1,0) 
  155 
 
  out2 = 
where((top.pgroup.xp/2/top.xrms)**2+(top.pgroup.yp/2/top.yrms)**2>Ellipse_ratio[iii]*
*2,1,0) 
  num1[iii]=sum(out) 
  num1[3+iii]=sum(out1) 
  num1[6+iii]=sum(out2) 
  
 num1[3*Length_ratio] = max(top.pgroup.xp) 
 num1[3*Length_ratio+1] = min(top.pgroup.xp) 
 num1[3*Length_ratio+2] = max(top.pgroup.yp) 
 num1[3*Length_ratio+3] = min(top.pgroup.yp) 
 
 ix_cen = sum(where(w3d.xmesh < 0.,1,0)) 
 iy_cen = sum(where(w3d.ymesh < 0.,1,0)) 
 rho_x = getrho(iy=iy_cen) 
 rho_y = getrho(ix=ix_cen) 
 rho_min = max(minnd(rho_x),minnd(rho_y)) #electron, negative charge density 
  
 xx = sum(where(rho_x/rho_min>=0.01,1,0))*(max(w3d.xmesh)-
min(w3d.xmesh))/(size(w3d.xmesh)-1)/2 #1% maximun intensity location 
 yy = sum(where(rho_y/rho_min>=0.01,1,0))*(max(w3d.ymesh)-
min(w3d.ymesh))/(size(w3d.ymesh)-1)/2 #1% maximun intensity location 
 xxyy = (xx+yy)/2; 
 num1[3*Length_ratio+4] =xx 
 num1[3*Length_ratio+5] =yy 
 num1[3*Length_ratio+6] =xxyy 
 
 output = str(top.it*stepsize)+"\t" 
 for aaa in num1: 
  output += str(aaa)+"\t" 
 output += "\n" 
 file1.write(output) 
 
 
# +++ output parameters 
if lpicts: installafterstep(ring.snapshot) 
installafterstep(ring.oneliner) 
installafterstep(ring.pplots) 
installafterstep(check_halo) 
 
 
#################hollow velocity################# 
#para_temp(delta_hp=-0.3,delta_h=None) 
#loadvels(func=dualgauss, vm=0.01, vs=0.008, cut=0.038, npts=1000) 
 
 
  156 
 
 
############### Particle SIMULATION  ################## 
 
pplot(0, [pptrace], "density")    # Plot at beginning 
ring.oneliner() 
 
stime = wtime() 
 
 
# ===  PHASE III:  RING 
top.zlatperi = ring.rperi 
top.zlatstrt = ring.roffset 
 
ring.initialize() 
bgidq=ring.load_ring_quads() 
bgidd=ring.load_ring_dipos() 
ring.load_ring_steerers() 
ring.load_B_Earth() 
 
ring.finalize() 
 
 
 
step(ring.nstr) 
 
etime = wtime() 
print "Time Running = ", etime-stime 
 
pplot(0, [pptrace], "density")    # Plot at end 
 
if lmatch: 
    s = 0.32/stepsize 
    match2(imtch=5,s=s) 
 
################### S A V E  D A T A ##################### 
 
top.lenhist = top.jhist     # get rid of the zeros at the end of moment history arrays 
gchange('Hist') 
 
if lcalc_mom:   # see rami_scripts.py for following functions 
    calc_mom()              # calculates rotation and dispersion moments 
    save_long(crun=crun)    # appends runid to selected moments and saves to pdb file 
else: 
    save_data(crun=crun)    # appends runid to selected moments and saves to pdb file 
 
ring.dump_moments()         # generate text files of moment data at diagnostic locs 
  157 
 
 
####################### P L O T S ######################## 
 
begin  = swhere(ring.nperds>48, ring.nsti+ring.nsty, 0) 
strobe = swhere(ring.nperds>48, ring.nstpp, 1) 
 
# see rami_scripts.py for following plotting routines 
plot_cent(begin=0,strobe=1, width=4.0) 
(xi,xf,yi,yf, dum)=limits(); xpd = ring.pd_loc+ring.yoffset 
if ring.rperds == 33: 
    rpd = ring.prd_loc+ring.roffset 
    plg(array([yi,yf]), array([rpd,rpd]), color='red',marker='PD') 
plg(array([yi,yf]), array([xpd,xpd]), color='red',marker='PD');   fma() 
 
if ring.rperds == 33:   # This zooms in on the recirculation part (optional) 
    plot_cent(begin=0,strobe=1, width=4.0) 
    (xi,xf,yi,yf, dum)=limits(); xpd = ring.pd_loc+ring.yoffset 
    rpd = ring.prd_loc+ring.roffset 
    plg(array([yi,yf]), array([rpd,rpd]), color='red',marker='PD') 
    limits(11.0,15.0,yi,yf); fma() 
 
if ring.nperds>48: 
    plot_env( begin=0,strobe=1, end=begin, width=4.0, ymin=0.0) 
    plot_env( begin=begin,strobe=ring.nstpp, width=4.0, ymin=0.0, type="dot"); fma() 
else: plot_env( begin=0,strobe=1, width=4.0, ymin=0.0); fma() 
 
plot_emit(begin=0,strobe=1, width=4.0, ymin=10, ymax=40);   fma() 
plot_np(ymin=0.0);     fma() 
 
if lcalc_mom: 
    plot_remit(begin=0,strobe=1, width=4.0, ymin=10, ymax=40);   fma() 
    plot_rot(begin=begin+0.25*ring.nstpp,strobe=ring.nstpp, width=4.0, type="none", 
marker='x', color='red') 
    plot_rot(begin=begin+0.75*ring.nstpp,strobe=ring.nstpp, width=4.0, xmin=0.0, 
type="none", marker='o', color='blue'); fma() 
####################### TOMO: FINAL P L O T S ######################## 
 
#pplot(0, [ppxy, ppxxp, ppyyp], "cellarray", lphoto=1)    # Plot at end 
 
#################### C O M P A R E ####################### 
file1.close() 
 
if (lcompare): 
    plot_comp(runs={dot1+old_crun: {'type': "dot"}}, crun=crun, titlet=cf_title, 
                    begin=begin, strobe=strobe) 
  158 
 
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