ABSTRACT 
 
 
 
 
Title of Document: STUDY OF LONGITUDINAL SPACE CHARGE 
WAVES IN SPACE-CHARGE DOMINATED BEAMS   
  
 Jayakar Charles Tobin Thangaraj,  
Doctor of Philosophy, 2009 
  
Directed By: Prof. Patrick O?Shea 
Prof. Rami A. Kishek 
Department of Electrical and Computer Engineering 
 
 
Future x-ray free electron lasers will probe matter at the atomic scale with 
femtosecond time resolution. Such x-ray sources require a high current electron beam 
with very low emittance and energy spread.  Any density fluctuation in an intense 
beam can launch space charge waves that lead to energy modulation. The energy 
modulations may cause further density modulations in any dispersive element and 
can, for example, excite the microbunching instability in x-ray free electron lasers. 
Hence, it is important to understand and control the evolution of density modulations 
on an intense beam. This dissertation focuses on long path-length experimental study 
of intense beams with density perturbations. The experimental results are compared 
with theory and computer simulation.  
 We took advantage of the multi-turn operation of the University of Maryland 
Electron Ring (UMER), to carry out long path-length (100 m) experimental studies of 
space-charge-dominated beams with density perturbations. First, a single density 
  
perturbation is introduced on a space-charge dominated electron beam using 
photoemission from a laser. The perturbation splits and propagates as a fast and a 
slow wave on the beam. The speed of the space charge waves is measured 
experimentally as a function of beam current and perturbation strength. The results 
are in good agreement with Particle-in-cell (PIC) simulation and 1-D cold fluid theory 
in the linear regime. We then show that linear space-charge waves can be used as 
non-interceptive transverse beam diagnostics in UMER. Using time-resolved imaging 
techniques, we report the transverse effects of a longitudinal perturbation in a circular 
machine. 
We introduce multiple perturbations on the beam and show that the fast and 
the slow waves superpose and cross each other. We then present experimental results 
on the beam response from introducing a controlled energy modulation on the density 
modulated beam and compare them with the theory.  In the non-linear regime, where 
the strength of the perturbation is large (>25% compared to the beam current), we 
report, for the first time, a wave train formation of the space charge waves. Finally, 
experimental observation of a photo-emitted beam pulse splitting into sub-pulses 
under high laser power is presented and compared with 1-D virtual cathode theory. 
From this work, we conclude that density modulations on an intense beam 
produce fast and slow waves, which, in the linear regime at least, can be controlled 
through energy modulation. Moreover, a large amplitude density modulation, when 
allowed to propagate, can break into sub-pulses, causing energy modulation. Hence, a 
density modulation should not be allowed to grow and must be controlled as soon as 
possible. 
  
 
 
 
 
 
 
 
 
 
 
 
STUDY OF LONGITUDINAL SPACE CHARGE WAVES IN SPACE-CHARGE 
DOMINATED BEAMS   
 
 
 
By 
 
 
Jayakar Charles Tobin Thangaraj 
 
 
 
 
 
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 
2009 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Patrick O?Shea, Chair/ Advisor 
Professor Rami  Kishek, Co-Advisor 
Professor Victor Granatstein 
Professor Kristine Rosfjord 
Professor Richard Ellis 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Jayakar Charles Tobin Thangaraj 
2009 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ii 
 
Dedication 
To my beloved family, 
Joshua Daniel Thangaraj, 
Beulah Thangaraj, 
Jeromie Wesley Vivian 
 
 iii 
 
Acknowledgements 
I thank the Lord God Almighty for providing me with the good fortune to 
meet many people who have inspired me with their creativity, their compassion and 
their courage. First and foremost is my advisor Prof. Patrick O?Shea, who leads one 
of the top engineering schools in the nation, but still took time for my research work. 
He has motivated me to relish the scientific spirit of exploring unknown challenges in 
research and to pursue excellence in my professional work. Prof. Kishek has been my 
mentor and a brilliant scientist. His style of problem-solving, presenting ideas and 
writing have influenced me heavily. He taught me persistence and motivated me to 
strive for clarity and to bring this thesis to fruition.  
This dissertation is an experimental work involving many people. First, I 
would like to thank Don Feldman for his guidance from day one. He was with me 
every single day of the experiment and was an inspiration. His patience coupled with 
his eye to experimental details has made him a joy to work with. I miss Renee 
Feldman who enquired about my family and helped me with data processing. I would 
like to thank Santiago and Brian for keeping the machine running. Santiago is a 
superb experimental physicist. He taught me that experimental physics is hard but 
rewarding. I will never forget that lesson. I enjoy his physics conversations peppered 
with jokes and stories. Brian has been a great friend and always wanted me to 
succeed. I went home happy after many challenging experiments because of him. I 
also thank Dave Sutter for helping with dispersion calculations among other things. 
Whenever I travel to accelerator conferences and physics meetings around the world, 
 
 iv 
 
there is always someone asking for Dave! A major achievement in this work is the 
experimental evidence of solitary waves in beams and I thank Haber for his 
simulation confirming my findings. I thank Prof. Reiser for the whole idea of UMER 
that was born from his lifetime of research. He is both an inspiring physicist and a 
leader in the field. His book is invaluable. 
The last year of my research work involved people in Linac Coherent Light 
Source in Stanford, California and FERMI@Elettra, Italy. I am greatly indebted to 
Max Cornacchia for this. Max joined the group, instantly recognized my work and 
involved me with these labs. His expertise along with his cheerful disposition and his 
enthusiasm for research is stimulating. His leadership was invaluable while we 
worked on the microbunching instability. I also thank Fiorito and Anatoly for their 
support for Coherent Optical Transition Radiation research while I was here and 
abroad.  
I also thank all the UMER graduate students, both past and present, especially 
Mike, Mark and Quinn. I thank Chris, Diktys, Chao and Kai for their support. Thanks 
to Eric Voorhies for painstakingly proofreading the draft. I greatly enjoyed his 
historical anecdotes. I also thank the IREAP and ECE staff for their administrative 
assistance. Last but not least, I would like to thank Prof. Eugenia Kalnay who 
provided me with complete support during my first year. I enjoyed working with all 
of them. 
I thank my church, Indians for Christ (IFC) for helping me throughout these 
years. I thank my Pastor Rev. Lite and his family for they took care of me as their 
 
 v 
 
own son. I thank Valentine and Sheba for their generosity and helping me in every 
single situation through all these years.  Fridays with Noble and Chamini have always 
been filled with love and affection. I thank Mano and Shanthi for they have shown 
what Christian love is all about with their example. I am greatly indebted to all of 
them for the rest of my life. I can only aspire to be of such an influence toward 
someone else. There are many others in the church who have helped me including the 
Edwins, Bhaskarans, Henrys, Prems and the James family. Regi mentored me during 
my tough times. His helping nature and positive attitude toward life are inspiring. I 
am grateful for Regi and Daniel Sumanth for their constant dose of encouragement.  
I thank my friend Deepak who taught me to think independently and who has 
been a great influence. Deepak and Jyoti have provided me with a positive and 
accepting environment that I have been searching for. I am very grateful to them and 
the only way to thank them would be to do what they have done for me for someone 
else.  I thank Rahul who has been an understanding friend. I thank Thangamani whom 
I will miss, for he always shared my success as his very own. I thank Ashwin and 
Mahesh for their help, especially during my initial years at Maryland. I thank my best 
friend Satish for motivating me throughout graduate school. I owe my interest in 
physics research to Dr. Murali from my alma mater Anna University. I thank my 
parents and teachers who always motivated me to excel in academics and pursue 
knowledge.  
I thank the Department of Energy and the Office of Naval Research and Joint 
Technology office for supporting our research. 
 
 vi 
 
Table of Contents 
 
 
Dedication..................................................................................................................... ii 
Acknowledgements......................................................................................................iii 
Table of Contents......................................................................................................... vi 
List of Tables .............................................................................................................viii 
List of Figures.............................................................................................................. ix 
Chapter 1 :  Introduction............................................................................................... 1 
1.1 Previous work on Space-Charge Waves ............................................................. 3 
1.2 One-dimensional cold fluid theory ..................................................................... 5 
1.3  Longitudinal Geometry factor (g-factor) ........................................................... 9 
1.4 Sound Speed of the fast and slow space-charge waves .................................... 11 
1.5 Limitations of the cold-fluid theory.................................................................. 13 
1.6 Nonlinear waves, Korteweg-de Vries equation and Burgers equation:............ 15 
1.7 Organization of the thesis ................................................................................. 16 
Chapter 2 : EXPERIMENTAL SETUP, DIAGNOSTICS AND THE UNIVERSITY 
OF MARYLAND ELECTRON RING....................................................................... 18 
2.1 University of Maryland Electron Ring (UMER) .............................................. 19 
2.2 UMER Gun ....................................................................................................... 21 
2.3 Laser Setup........................................................................................................ 27 
2.4 Beam Diagnostics: ............................................................................................ 30 
2.5 Summary:.......................................................................................................... 37 
Chapter 3 . Experimental investigation of a single perturbation on a space-charge 
dominated beam.......................................................................................................... 39 
3.1 Wave speed as a function of beam current ....................................................... 39 
The center- difference method to calculate Cs: ...................................................... 43 
3.2 Wave speed as a function of perturbation strength........................................... 48 
3.3 Simulation using WARP................................................................................... 50 
3.8 Beam End Erosion ............................................................................................ 55 
3.4 Space-charge waves as beam diagnostics for measuring transverse beam size 56 
3.5 Transverse effects of the space-charge waves .................................................. 60 
3.11 Summary......................................................................................................... 69 
Chapter 4 : Multiple perturbations: Propagation, Control and Nonlinearities............ 71 
4.1 Generating multiple perturbations on a beam:.................................................. 71 
4.1.1 Experiments on a beam with multiple perturbations: .................................... 73 
4.1.2 WARP Simulation of beam with multiple perturbations:.............................. 75 
4.2 Controlling density perturbations on a beam:................................................... 78 
4.2.1 Coherent pulse flattening and 1-D cold theory:............................................. 80 
4.2.2 Analytical Modeling of Coherent pulse flattening using MATLAB:............ 83 
4.2.3 Experimental cancellation of density modulation by controlled energy 
perturbation:............................................................................................................ 89 
4.2.4 Limitations of the coherent pulse flattening technique:................................. 94 
4.2.5 Experimental results of the perturbation-compression scheme: .................... 95 
4.2.6 Perturbation compression scheme: ................................................................ 97 
 
 vii 
 
4.3 Non-linear space charge waves......................................................................... 98 
4.3.1 Experiment showing wave train formation:................................................. 100 
4.3.2 KdV equation, Burgers equation and Dawson limit:................................... 102 
4. 4 Summary:....................................................................................................... 104 
Chapter 5 :  Longitudinal perturbations due to space charge effects in the gun....... 105 
5.1 Pulse splitting near the gun ............................................................................. 105 
5.2 Time-independent diode behavior .................................................................. 107 
5.3 Potential minimum oscillation ........................................................................ 115 
5.4 Classical Short ? Pulse Child-Langmuir Law (SCL)...................................... 117 
5.5 Explanation for the experimental observations............................................... 120 
5.6 Extending the results for a high gradient photoinjector.................................. 121 
5.7 Limitation of the theory .................................................................................. 122 
5.8 Conclusion ...................................................................................................... 123 
Chapter 6 :  Conclusion and Future work ................................................................. 124 
6.1 Summary and Conclusion............................................................................... 124 
6.2 Suggestion for future work ............................................................................. 125 
Appendix- A.............................................................................................................. 127 
Bibliography ............................................................................................................. 129 
 
 
 
 
 
 
 
 
 
 
 
 viii 
 
List of Tables 
 
Table 1-1: Table referring to the quantities used in 1-D cold fluid theory................... 7 
Table 2-1: Beam and lattice parameters of UMER..................................................... 20 
Table 2-2: Cathode aperture sizes used in longitudinal experiments and related 
parameters. Beam size values courtesy of [44]........................................................... 22 
Table 2-3: The table showing the important parameters of the UMER gun............... 24 
Table 2-4: Nd:YAG laser parameters ......................................................................... 29 
Table 3-1: Table showing the values of the time separation ?T of the space charge 
waves with distance and the corresponding values of Cs calculated using linear and 
center difference techniques. ...................................................................................... 47 
Table 3-2: Normalized Space-charge impedance values for different beam current as 
a function of distance as per eqn (3.26). ..................................................................... 60 
Table 4-1: Table comparing of the velocity modulation strength predicted by theory 
to cancel the perturbation and the experimental values.............................................. 90 
 
 
 
 ix 
 
List of Figures 
Figure 2.1: The layout of the University of Maryland Electron Ring (top view)....... 19 
Figure 2.2: Schematic of the UMER gun electronics showing the Cathode (K), Grid 
(G) and Anode (A). The Grid is biased and an accelerating voltage of 10kV is applied 
in the gap. Distances not to scale................................................................................ 23 
Figure 2.3: Schematic of the laser setup (left) with the experimental setup (right).... 27 
Figure 2.4: A block diagram of the laser setup in UMER along with the multiplying 
crystals. Measured energy output at each stage is indicated....................................... 28 
Figure 2.5: Schematic layout of Minilite II laser showing the internal optical setup. 
The red line is IR while the next harmonics are shown in green................................ 30 
Figure 2.6: Timing pulse sequences of the laser. The flash lamp is triggered externally 
with the gun trigger. On a 100ns beam, the perturbation is around 60ns from the head 
of the beam.................................................................................................................. 31 
Figure 2.7 :Beam current profile measured using Bergoz current monitor before 
injection into ring. A 100-ns rectangular beam pulse without the perturbation (Top) 
and with a 5-ns pure density perturbation is at the center of the main beam pulse 
(Bottom)...................................................................................................................... 32 
Figure 2.8: The four signals are from the top, bottom, left and right plates of the beam 
position monitor. The beam current measured was 23mA at 1.84 A quad current 
settings. As the beam propagates, the beam expands and fills the ring. ..................... 33 
Figure 2.9:The wall current monitor principle: The image current induced in the beam 
pipe is forced to flow through the resistors using the ferrite core and the voltage 
across the resistors measured is proportional to the beam current. Courtesy [30] ..... 35 
Figure 2.10:Fast imaging diagnostics set up in UMER. The beam hits the fast 
phosphor screen (ZnO:Ga) and the light emitted is captured by the camera (ICCD 
PIMAX2) for further processing via the quartz window............................................ 37 
Figure 3.1: Turn-by-Turn plot of the beam current obtained from wall current 
monitor. Turn 1 is on the bottom with subsequent turns plotted on top of each other. 
Injected beam current was 16 mA. For clarity, every turn of beam current is shifted 
with a constant value of 5mA. The tune of the machine was set at 0 6.1? = . ............. 41 
Figure 3.2: 3-D plot of beam current evolution with single density perturbation. The 
plot is generated by interpolating the turn by turn data obtained using the wall current 
monitor. The injected beam current is 16 mA at 10keV............................................. 42 
Figure 3.3: The top graph plots the time separation of space-charge waves with the 
distance. The bottom graph is the beam current as a function of distance. The beam 
current is changing as a function of distance which affects the linearity of the wave 
speed. Hence, we have to take into account the varying beam current in calculating 
the value of sound speed from the measurements. ..................................................... 44 
Figure 3.4 : The graph shows two ways of measuring the value of Cs from the 
experimental data. The black curve is the sound speed calculated from the theoretical 
formula. The two red curves correspond to the calculation of the sound speed from 
experimental values using the linear method eqn (3.21) and the center-difference 
 
 x 
 
method eqn(3.22) . As the distance increases, the center-difference method is closer 
to the theory as expected............................................................................................. 45 
Figure 3.5: Graph comparing the sound speeds for different main beam current for the 
same strength of the initial density modulation along with values predicted by theory.  
The higher the main beam current, the faster the waves depart from each other. A 5% 
error bar in the data is not shown in the graph for clarity........................................... 48 
Figure 3.6: Sound speed as a function of distance for different strengths of the initial 
density modulation. The higher the strength of the perturbation, faster the wave 
moves. ......................................................................................................................... 49 
Figure 3.7 : Turn by turn plot of beam current obtained from PIC simulation using 
WARP. Since there is no beam loss assumed in the simulation, the waves reach the 
ends of the beam faster than in the experiment. The injected beam current is 16mA. 
For the sake of clarity, the beam profile from every turn is shifted vertically. .......... 53 
Figure 3.8 : Sound speeds measured from the experiment, and theory along with those 
obtained from simulation with and without beam loss. Theory and WARP agrees with 
measured data when beam loss is included in the calculation. For the zero-th turn, the 
measured and the calculated values of the sound speed show a large discrepancy. This 
is because the fast and the slow waves have not yet separated at this distance. These 
points are indicated by gray boxes.  The black line with the black triangles uses the 
formula for Cs for a given beam current and hence does not change with the turn. But, 
in the experiment, the current changes and taking this account the Cs is calculated and 
this is the black line with the black boxes. The red line is the experimentally 
measured value of Cs. The blue lines correspond to the WARP simulation. The blue 
line with the triangle ignores beam loss while the blue line with the boxes includes 
beam loss..................................................................................................................... 54 
Figure 3.9: WARP simulation including beam loss is compared with the experimental 
data. The beam current is 16mA with a 20% perturbation. The time-evolution of the 
perturbation obtained from WARP is overlaid on the experimental measurement 
showing both the fast and the slow waves evolving at almost the same speed as seen 
in the experiment. The turns shown in this plot are turn 3 (Top: left), turn 4 (Top: 
right), turn 5 (Bottom: left) and turn 6 (Bottom: right)............................................... 55 
Figure 3.10: The average beam radius of various beams as a function of distance in 
the ring. The beam radius is calculated from measuring the wave-speed and beam 
current for each turn and applying equation(3.23). The beam size tends to decrease 
slightly as the distance increases................................................................................. 57 
Figure 3.11: The wall current monitor signal at RC10 showing the measured beam 
current of 45mA peak with a 12% perturbation. The strength of the perturbation was 
deliberately increased to capture the splitting before the fast screen at RC15. .......... 62 
Figure 3.12: Integrated beam image of the whole beam. Top: 45mA with a positive 
perturbation; Bottom; the same beam in color-coded format. .................................... 63 
Figure 3.13: (Landscape) Black and white montage of the images captured by the 
camera in the 51-ns window at RC15. The total intensity of each image is calculated 
expressed in terms of the peak current.  The peak beam current is 45mA. The 
montage is read from top to bottom starting from left to right. This is done to capture 
the x-motion of the beam. ........................................................................................... 64 
 
 xi 
 
Figure 3.14:  (Landscape) Color montage of the images captured by the camera 
showing transverse density waves at RC15. The beam size of the perturbed part of the 
beam is larger in x and the beam centroid moves transversely in the x-direction 
possibly to the dispersion effect of the ring................................................................ 65 
Figure 3.15: The plot of the slice beam current as a function of time. The total light 
intensity from the fast screen follows the fast and the slow waves. The measurements 
are at RC15 with a peak current of 45mA with a perturbation of 12%. The plot allows 
confirms the linearity of the screen with the charge................................................... 67 
Figure 3.16: The RMS beam radius measured from the images collected at RC15 
using the fast screen and camera as a function of time. There is a 10% increase in the 
x-radius with the perturbation. As predicted by the theory, the perturbation causes a 
radius change in a space charge dominated beam to keep the volume charge density a 
constant. ...................................................................................................................... 68 
Figure 3.17: The centroid shift, the distance between the beam center and the pipe 
center, as measured from the images as function of time. The x-position of the beam 
centroid oscillates while the y-position of the beam centroid is relatively unchanged. 
This is due to the effect of the velocity modulation of the perturbation causing a shift 
in beam position due to the dispersion in the ring. ..................................................... 69 
Figure 4.1 : Beam current profile measured using the Bergoz current monitor before 
injection into the ring. Two 5-ns pure density perturbations are introduced using two 
lasers. The time between A and B is 15ns. The beam current is 21 mA with 6 mA (A) 
and 4.5 mA (B)  perturbation...................................................................................... 72 
Figure 4.2: : 3-D plot of beam current evolution with two density perturbation. The 
plot is generated by interpolating the turn by turn data obtained using the wall current 
monitor. The waves are observed to superimpose and cross each other between turns 
2 and 4. The injected beam current is 24mA. ............................................................. 74 
Figure 4.3:Graph showing the value of sound speed of the perturbations A and B 
compared with the theoretically predicted values over multiple turns.  The sound 
speed is higher for A compared to B because of the higher strength of the perturbation 
A (see Chapter 3) . The injected beam current is 24 mA............................................ 75 
Figure 4.4: WARP Simulation results of a beam with multiple density perturbations. 
The slow wave of one perturbation and the fast wave from the other perturbation 
superimpose each other on the second turn and then cross. The beam used in 
simulation is the beam shown in Figure 4.1. The WARP simulation used one million 
particles....................................................................................................................... 76 
Figure 4.5: Experimental results of a UMER beam with multiple perturbation. The 
beam current is 12.6mA with a 1.7mA perturbation (A) and 1 mA perturbation (B). 
The waves approaching each other, superposing each other and then crossing each 
other is clearly observed. The lines connect the peaks of the fast and the slow waves 
in each perturbation..................................................................................................... 78 
Figure 4.6: Graph showing the strength of velocity modulation (? ) required to cancel 
density modulation of initial strength (?) for different UMER beam currents at 10 
keV. A high current beam with a larger perturbation needs a larger energy 
modulation. ................................................................................................................. 82 
 
 xii 
 
Figure 4.7: (Landscape) MATLAB Simulation showing the effect of velocity 
modulation on a beam. The beam current is 20mA and the velocity modulation is 
0.2%. In the current space, the fast and the slow wave have opposite polarity.......... 85 
Figure 4.8: (Landscape) ) MATLAB Simulation showing the effect of density 
modulation on a beam. The beam current is 20mA and the density modulation is 10%. 
In the current space, the fast and the slow wave have same polarity, while they have 
opposite polarity in the velocity space........................................................................ 86 
Figure 4.9:  MATLAB Simulation showing the cancellation of the fast wave. The 
beam current is 20mA and the density modulation is 10% with a velocity modulation 
of 0.2%. Though the fast wave is cancelled, the amplitude of the slow wave increases.
..................................................................................................................................... 87 
Figure 4.10: (Landscape) MATLAB Simulation showing the cancellation of the slow 
wave. The beam current is 20mA and the density modulation is 10% with a velocity 
modulation of 0.2%. Though the slow wave is cancelled, the amplitude of the fast 
wave doubles............................................................................................................... 88 
Figure 4.11: Turn-by-Turn plot of the UMER beam (7 mA) with a density 
perturbation (8-ns). The perturbation width is 5ns from a single laser. Therefore, two 
lasers were combined to get  an 8 ns density perturbation. The fast and the slow 
waves have the same polarity. .................................................................................... 91 
Figure 4.12: Turn-by-Turn plot of the UMER beam (7 mA) with a velocity 
perturbation (8-ns). The fast and the slow waves have opposite polarity. This property 
allows the velocity perturbation to cancel a density perturbation, which generates a 
fast and a slow wave but with the same polarity. ....................................................... 92 
Figure 4.13: Turn-by-Turn plot of the UMER beam (7 mA) with a velocity 
perturbation (8-ns) and a density perturbation (8 ns). Only the slow wave is 
propagating on the beam. The fast wave of the density modulation is cancelled by the 
fast wave of velocity modulation because of opposite polarity while the slow waves, 
which are of same polarity, adds up............................................................................ 93 
Figure 4.14: Experimental results of perturbation compression for a 23mA beam 
current with a 10% density perturbation. The sub-graphs show the effect of 
compression on every turn for different values of electric field strength. The 
perturbation, when compressed, takes a longer time to evolve into a slow and a fast 
wave. For the case shown, the perturbation was applied only once in the first turn at 
RC4. ............................................................................................................................ 97 
Figure 4.15: Turn-by-turn plot of the beam current (36 mA) with a 17.5 mA 
perturbation showing wave train formation in the third turn. As the turn increases, the 
number of waves increases. The perturbation is introduced near the tail of the beam to 
allow the fast wave to spend a longer time in the flat-top portion of the beam........ 101 
Figure 5.1: Single perturbation pulse splitting into multiple sub-pulses at (a) Laser 
generated electron beam (b) Pulse splitting into two sub-pulses (0.68mJ) (c) Three 
sub-pulses (1.3mJ) (d) Four sub-pulses observed at (2.7mJ). This measurement is 
taken at the Bergoz (placed 64 cm from the anode exit) in the injection line. ......... 106 
Figure 5.2: A short-circuited diode showing the virtual cathode and the electron beam 
direction. The virtual cathode separated the gap into Region I and Region II. ........ 107 
 
 xiii 
 
Figure 5.3: The plot shows the value of the normalized potential minimum as a 
function of the input current density. When the input current density exceeds 8, the 
potential minimum starts to reduce and then vanishes. ............................................ 109 
Figure 5.4: The plot shows the potential profile inside a diode as a function of 
distance and the input current density. As the input current density increases, the 
potential profile sags until it vanishes when ? >8..................................................... 110 
Figure 5.5: The plot shows the variation of the potential profile in the region before 
the virtual cathode (Region I). After the virtual cathode is formed, increasing the 
injected current moves the virtual cathode closer toward the cathode causing the 
output current to follow the Child-Langmuir limit................................................... 112 
Figure 5.6: The plot shows the transmitted current as a function of input current after 
the virtual cathode is formed. Once the virtual cathode is formed, complete 
transmission occurs only when the value of input current (?) is reduced to 4. ........ 113 
Figure 5.7: The diode output current as a function of the input current. The output 
current follows the input current up to a critical value of the input current after which 
there is a sudden drop in the output. The path CD constitutes the virtual cathode 
formation................................................................................................................... 114 
Figure 5.8: The plot shows that the virtual cathode after it forms is not stable. It 
oscillates in position as well as in amplitude. The blue curve plots the value of the 
normalized potential minimum and the red curve plots the normalized position of the 
potential minimum.................................................................................................... 116 
 
 1 
 
Chapter 1:  Introduction 
 
There is a growing demand for short wavelength free electron lasers (FELs), 
which have unprecedented brightness and temporal properties for probing biological 
and material structures [1-4]. Short wavelength FELs place very tight constraints on 
the quality of the electron beam required for operation. Advanced accelerators like 
high-luminosity colliders and high energy-recovery linacs also demand both high 
energy and superior beam quality [5, 6]. Space charge plays a major role in the 
generation and propagation of high-quality beams. Understanding the role of space-
charge in such beams will be pivotal in the design and operation of modern machines. 
Near the cathode, all beams of interest begin as space-charge dominated beams. As 
the beam is accelerated to relativistic energies, any beam quality degradation that 
occurred at low energy, where the collective effects dominate the dynamics, will 
remain at high energies. Hence, to maintain the beam quality of bright beams 
throughout the beam transport line, it is important to identify the factors degrading the 
beam quality at low energy and control them.  
As an example, X-ray free electron lasers (XFELs) require a high current 
beam with a low emittance and a low energy spread. Any density perturbations on the 
longitudinal distribution can induce energy modulations on the beam through space 
charge forces at low energies. Such induced energy modulations may lead to density 
modulations at higher energies when the beam passes through a dispersive section 
such as bunch compressor, possibly leading to microbunching instability [7, 8]. This 
instability can adversely affect the operation of the XFEL. Therefore, it is important 
 
 2 
 
to generate and transport a uniform and stable longitudinal current distribution. As 
another example, in ion machines, where the beam for the most part is space charge 
dominated, any fluctuations on the beam can cause resistive wall instability, leading 
to unacceptable energy spread.  
Upon their generation at the cathode, beams typically have density 
modulations, velocity modulations, or both. Some of the sources of these 
perturbations are the drive laser fluctuations in a photoinjector, RF modulations, 
cathode non-uniformity, shot noise or impedance mismatch in the transport lines. 
These modulations can be desirable or undesirable, depending on the application.  
By pre-modulating a beam, say at mm-wave or THz frequencies, an efficient 
radiation source can be implemented. For certain applications like XFEL, where the 
beam quality is critically important, such beam modulation can be undesirable. In a 
high brightness beam, initial density modulations on the beam get converted into 
energy modulations and hence distort the longitudinal phase space. This will cause 
emittance growth and thus reduce FEL output.  Moreover, the microstructures inside 
the beam can radiate coherently when passing through a bend, thereby introducing 
energy modulation. This microbunching instability, currently observed in the Linac 
Coherent Light Source (LCLS) as Coherent Optical Transition Radiation (COTR) [9], 
may potentially degrade the beam quality and is undesirable for seeded machines like 
FERMI@Elettra [10]. Bright COTR may swamp OTR radiation and therefore prevent 
the use of OTR as a diagnostic tool [11].  
 One of the goals of the University of Maryland Electron Ring (UMER) is to 
study beams in such a non-ideal state. In this work, we deliberately introduce a stable, 
 
 3 
 
localized, density perturbation on a space charge dominated beam and observe its 
evolution over a long path length. 
1.1 Previous work on Space-Charge Waves 
          Space-charge wave studies began with microwave generation during the 
1950?s. Research on space charge waves in beams continued with the Heavy Ion 
Fusion research [12, 13]. In the 1990?s, J.G. Wang [14, 15] and D.X. Wang [16] 
worked on several experiments on the Maryland Electron Beam Transport (MEBT) 
facility. In their experiments, they generated perturbations by modulating the grid-
voltage of an electron gun.  They measured the phase velocity of the space charge 
waves and found good agreement with 1-D cold fluid theory and experiment. They 
also observed the space charge waves reflecting off the beam edges [17]. Their work 
was done in a beam pipe with a conducting wall.  
Suk [18] found a good agreement of the growth rate of the slow space charge 
wave with cold fluid theory in the linear regime. In the linear regime, the strength of 
the perturbation is small compared to the main beam current. Similarly, Zou [19] 
observed the energy width of the slow wave growing and of the fast wave decaying. 
A theory on the possibility of launching solitary waves in a space charge dominated 
beam was also developed [20]. 
Further work was done by Neumann [21] who used a pre-modulated beam to 
generate radiation. On UMER, Huo [22] demonstrated generating perturbations using 
a UV laser and found good agreement between theory, simulation, and experiment for 
the wave velocity in the linear region under various initial conditions. Also, on 
UMER, Harris and Neumann [23, 24] investigated generating perturbations in the 
 
 4 
 
UMER gun both by modulating the cathode and by using photoemission from a laser. 
Tian [25] has observed initial density modulations getting converted into energy 
modulations through space-charge waves in current space as well as velocity space in 
the Long Solenoid Experiment (LSE) facility.    
Although these experiments revealed information about space charge wave 
propagation in intense beams, the experiments were limited in two aspects. First, the 
beam transport in all these experiments was limited to 10 meters or less. A longer 
transport distance has an advantage of increasing data resolution by probing the space 
charge waves for a longer time and also allows long path length experiments. 
Secondly, when introducing a perturbation on the beam by modulating the grid 
voltage of an electron gun, both the energy and the density of the beam are modulated 
and this makes it difficult to know the exact initial conditions. A pure density or a 
pure velocity modulation on the electron beam can simplify the initial conditions.  
Therefore, in this work we first extend the studies on a longer distance and 
secondly introduce a pure density modulation to simplify the initial conditions. We 
conduct our experiments on the University of Maryland Electron Ring [26, 27], which 
is a circular machine 11.52 m in circumference. Even modest multi-turn operation of 
UMER makes it is possible to conduct experiments over path lengths much longer 
than 100m compared to 5-m of previous experiments. Secondly, in our experiments 
we introduce a density modulation using photoemission from a laser. With the use of 
a laser, we can introduce a localized charge modulation without changing the velocity 
of the electron beam. Using multiple lasers, we extend this technique to introduce 
multiple perturbations on the beam. We also report the observation of space charge 
induced instabilities above a critical value of the laser power. Though such effects 
 
 5 
 
have been observed before in photoinjectors [28, 29], we compare our observations 
with theory and discuss the relevance of such effects for future photoinjectors. As of 
this writing, Beaudoin [30] introduced energy modulations on a space-charge 
dominated beam, using an induction cell, and found good agreement between theory, 
measurement and simulation over a distance of 60 meters in UMER. 
The outline of this chapter is as follows. We begin by a brief overview of 1-D 
cold fluid theory in Section 1.1. We then discuss the longitudinal geometry factor in 
Section 1.2 and sound speed in Section 1.3. We discuss the limitation of the 1-D cold-
fluid theory in Section 1.4 and extend it to nonlinear regime in Section 1.5 and we 
finally conclude in Section 1.6 with the organization of the thesis. 
 
1.2 One-dimensional cold fluid theory 
The cold fluid theory is derived by truncating the hierarchy of fluid equations. 
This truncation is done by assuming a cold beam without energy spread. Since the 
theory treats the beam as a fluid, the fluid model is suitable to explain phenomena 
which are inherently fluid in nature like wave propagation and wave growth.  The 
fluid equations are obtained by taking moments of the Vlasov equations. We refer to 
[31] for details of Vlasov equation and its moments. We begin with the continuity 
equation (1.1) and the momentum equation(1.2).   
( ) 0
z t
?? ?? ?+ =
? ?                                                                 (1.1) 
                   0 3
0
zqE
z t m
? ??
?
? ?+ =
? ?                                                                   (1.2) 
 
 6 
 
 
Let us assume that the initial perturbation is small compared to the beam. This allows    
the equations to be linearized later on. The symbols used in the equation are described 
in Table 1-1 .  
 
 
 
 
 
 7 
 
 
Table 1-1: Table referring to the quantities used in 1-D cold fluid theory 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
In our experiment, we introduce a pure charge density perturbation that can be 
expressed in the form:  
 ( ) ( ) ( )1 1 0, 0, ,z t z t p t? ? ??= = =  (1.3) 
Symbol used  Quantity referred  
q Charge of an electron (-1.6x10-19C) 
z Distance along the beam (m) 
t Time (s) 
? Line charge density (C/m) 
I Beam current (A) 
? Velocity of the beam (m/s) 
? Gamma factor of the beam 
m Mass of an electron (kg) 
Ez Longitudinal electric field (V/m) 
p(t) Function representing the perturbation 
K Perveance  
? Ratio of beam velocity to the velocity of 
light 
a Transverse Beam size (m) 
b Beam pipe radius (m) 
 
 8 
 
where ?  is the amplitude of the density perturbation relative to the main beam 
current, ?0 and  ?1 are the non-perturbed and perturbed values of line charge density. 
( )p t  is a smooth function with unit amplitude and it represents the shape of the 
perturbation. Since it is a pure density perturbation, 1(0, ) 0t? =  , where 1?  is the 
perturbed velocity. The initial perturbation current can then be expressed as: 
 
( )
0 1 1 0 0 0
0 0 0
0 0
1 0
                   = (0, )
                  ( )
                (0, ) ( ),
I I
p t I t
I p t I
I t I p t
?? ?? ? ?
?? ?
?
?
+ = = +
+
= +
? =
 (1.4) 
 ,where 1I  is the perturbed current. Equation (1.4) shows that a pure charge 
modulation converts to a beam current modulation of same strength. For a pure initial 
density modulation, the linearized continuity and momentum equations are reduced to 
the following set of equations.  
 
 1 1 10 0 0z t z
? ? ?? ?? ? ?+ + =
? ? ?  (1.5) 
 
 
1 1
0 3
0
1
5
0 0
( , )
( , )                     
4
zqE z t
t z m
z tqg
m z
? ??
?
?
pi? ?
? ?+
? ?
?= ?
?
asymptequal
 (1.6) 
 The geometry factor, g , given by 2ln bg a?= + ,  represents the dependence of the 
longitudinal electric field on the transverse beam properties with ?b? being the beam 
 
 9 
 
pipe radius and ?a? the transverse beam radius and ? is a constant ranging from 0 to 
1.  A complete derivation of these equations is shown in [31] . 
1.3 Longitudinal Geometry factor (g-factor) 
Initial work [32] used a uniform beam model with a constant radius leading to 
a value of ?=0.5. But this assumption breaks down for perturbations in a space-charge 
dominated beam because in a space-charge dominated beam it is the volume charge 
that remains constant while the radius changes with the variation of the line-charge 
density. To show this, let us calculate the line-charge density for a uniform beam of 
radius a and volume charge density ?. The line-charge density ? is 
 2( ) ( ) ( )z a z z? pi ?=  (1.7) 
In a machine with a periodic focusing lattice like UMER, the average matched 
radius of the beam, which depends on the wave constant k0 of the focusing lattice, the 
perveance K and the emittance? , is approximately given by 
 
1
2
2 00
Ka
kk
?? ?? ?? ?
? ?
? ?? ?
? ?
? +  (1.8) 
The equation (1.8) is obtained from equation 5.293 of [31]. The relation is exact only 
for either a zero emittance beam (? =0) or beam with negligible space charge (K=0). 
Combining (1.7)  and replacing perveance with beam current, we get: 
 
 2
2 0
2 3 2
0 0
( )
2
z
kmck
e
a ? ?
pi? ? ?
+? ?
? ?? ?
=            (1.9) 
 
 10 
 
If the beam is emittance-dominated such as beams in a high energy accelerator, then 
the beam size will be determined by the second term in equation(1.8), then  
                                            2
0
2( ) ( )
constantk
z z a
a ?
? pi?
=
? =
=                                         (1.10)                                           
  
In this case, modulating the charge density is essentially modulating the volume 
charge density in an emittance-dominated beam with a constant beam radius. 
But, if the beam is space-charge dominated, which is the situation in high-
brightness injectors, then the first term in equation (1.8) dominates leading to  
 
 
2
2
2 2
2 3 2 2 3 2
0 0 0 0
     
( ) ( ) ( )( )
2 2
( )
z a z zz
mc mck k
e e
a
z constant
? pi ?
pi? ? ? pi? ? ?
?
=? ? ? ?
? ? ? ?? ? ? ?
=
=?
 (1.11) 
 
In this case, the volume charge density remains constant. This means any 
variation in the line-charge density on the beam changes the beam size accordingly to 
keep the volume charge density a constant. This is a very important feature of space-
charge dominated beams causing a change in g-factor. The value of ? in the g-factor 
formula for a space-charge dominated beam was measured experimentally [15] 
showing it be equal to zero. Hence, the g-factor for a space-charge dominated beam 
of radius a inside a pipe of radius b, is given by 
 
 11 
 
 2ln
bg
a=  (1.12) 
Because the g-factor is different between the space-charge dominated beam 
and the emittance-dominated beam, there is a difference in the longitudinal electric 
field Ez produced by the waves in those beams. From equation(1.6), the longitudinal 
electric field Ez is  
 1
( , )             ( , )  
z
z tE z t g
z
???
?  (1.13) 
 
  In the case of an emittance dominated beam, the beam size a(z) is constant 
along the beam and so the Ez field is a function of radius r across the beam, so 
E ( , )=E ( )z zr z r . But when the space-charge forces dominate, the longitudinal field 
Ez is constant across the beam radius and changes along the beam, so   
E ( , )=E ( ( ))z zr z a z . In high brightness photoinjectors, density modulations can 
create longitudinal space charge fields in the beam leading to rapid transverse beam 
size variation across the beam. So, any efforts to model the physics of the beam inside 
modern photoinjectors should take the radial dependence of longitudinal space-charge 
forces into account [33]. 
 
1.4 Sound Speed of the fast and slow space-charge waves 
Solving the linearized fluid equations (1.5) and (1.6) along with Maxwell?s 
equation [31]  in the beam frame, we obtain a wave solution for the perturbation 
 
 12 
 
current ,velocity and charge density. The solution in the beam frame can be expressed 
as: 
  
0 0
1
0 0
( , ) 1 ( ) 1 ( )2 2s s
s s
I c I cz zI z t p t p t
v c v c
? ?? ? ? ?= ? + + + ?? ? ? ?
? ? ? ?  (1.14)  
1( , ) ( ) ( )2 2
s s
s s
c cz zv z t p t p t
c c
? ??= + + ?
 (1.15) 
00
1( , ) ( ) ( )2 2s s
z zz t p t p t
c c
?? ??? = + + ?  (1.16) 
 
The solution shows that the density perturbation propagates along the beam in 
the form of waves, one of which has a phase velocity greater than the beam velocity 
and  hence is called the fast wave, while the other one has a phase velocity smaller 
than the beam velocity and hence is called the slow wave. Here 
 0 5
04
s
qgC
mpi? ?
?=
 (1.17)  
is the velocity of the space charge waves in the beam frame, in analogy to the 
propagation of sound in a gas and hence is called the ?sound speed?. Equations from 
(1.14) through (1.16) reveal some of the properties of the space-charge waves. First, 
the fast wave has higher amplitude than the slow wave in the perturbed beam current 
as seen in equation(1.14). The polarity of the fast and the slow waves are both 
positive in the density space (equation(1.16)) while they are opposite in velocity 
space (equation(1.15)). This is a fundamental difference between a density 
 
 13 
 
perturbation and an energy perturbation. In the latter, the fast and the slow waves 
have positive polarity in the velocity space and have opposite polarity in the density 
space [30]. Finally, the space charge waves have the same shape as that of the initial 
perturbation. The same conclusion can also be arrived by constructing the dispersion 
relation for the space charge waves 
 0
 
0
)(
group velocity
s
k
Ck
? ?
? ? ?
?? ?? =
? ??? ?
=  (1.18) 
where k is the wave constant of the perturbation. The dispersion relation shows that 
the group velocity is equal to the phase velocity of the waves meaning the space 
charge waves are dispersion free.  
1.5 Limitations of the cold-fluid theory 
Both the cold-fluid theory and the g-factor model have some limitations in 
their validity. First, the cold-fluid theory assumes a cold beam with zero energy 
spread. Also, the linearization of the continuity and momentum equation means the 
amplitude of the perturbations is assumed to be small compared to the beam current: 
i.e. 1 0? ?<< . Additional terms must be taken into account if the perturbation is 
sufficiently large compared to the small amplitude perturbation approximation.  
The g-factor model has at least two limiting cases. First, the g-factor model 
cannot be applied near beam ends where the line-charge density changes rapidly in a 
short distance thereby violating the assumption of uniform beam density. Secondly, 
the g-factor calculation changes significantly for highly compressed beams. This is 
also called the short-wavelength limit - when the wavelength of the perturbation is 
 
 14 
 
comparable to the dimension of the beam radius or pipe radius (  pert beamsizea? ? ). 
For an ultra-short beam bunch the beam length is comparable to the pipe radius. So 
any perturbation on such a short beam affects the overall Ez-field of the beam causing 
the g-factor to be a function of the wavelength of the perturbation.  
Only in the long-wavelength limit of  pert beamsizea? greatermuch   does the g-factor become 
independent of the wavelength of the perturbation. 
              In UMER, we introduce a 5-ns perturbation (28 cm long) on a 100-ns (5.6 m 
long) coasting beam with a beam pipe radius of one inch (2.54 cm) and hence we 
operate in the long-wavelength limit. The density perturbation will be introduced near 
the beam center where the charge density is uniform. Hence the g-factor model is a 
valid model for calculating the space-charge wave velocity in our experiments. 
There are also other longitudinal space charge wave models beside cold-fluid 
theory that take a gas-dynamic approach with an adiabatic equation and an index [34]. 
As seen from eqn(1.15), a pure density modulation introduces a modulation in 
velocity. The sound speed varies as 2.5? ? . As the beam is accelerated, Cs decreases 
very fast according to eqn(1.17). This implies that in an accelerated beam, the waves 
slow down and the modulations are essentially frozen in the beam. Such frozen 
energy modulations can still get converted into further density modulation when the 
beam passes through a dispersive section. In proton machines, where the beam for the 
most part is space-charge dominated, such perturbations can lead to large energy 
spread and wave-growth leading to instabilities. 
 
 15 
 
1.6 Nonlinear waves, Korteweg-de Vries equation and Burgers 
equation: 
Previous studies predicted that a space-charge dominated electron beam is 
capable of supporting solitary waves and wavebreaking[35, 36]. Detailed 
mathematical analysis shows that the 1-D cold fluid theory can be reduced to a more 
general form of the Korteweg-de Vries Equation (KdV) equation[37]. The KdV 
equation describes wave propagation in physical phenomena ranging from water 
waves to solitary wave transport.  
We extend Suk?s analysis to the specific case of non-dispersive, non-linear 
wave transport and show that it reduces to the inviscid Burgers? equation. The 
Burgers? equation is a partial differential equation that exhibits shock waves solutions 
for smooth sinusoidal initial conditions. 
We begin with the classical Korteweg-de Vries equation for the line-charge 
density [35]: 
 
3
1 1 1
1 1 2 3 0,c ct z z
? ? ??? ? ?+ + =
? ? ?
             (1.19) 
Where ?1 is the line-charge density of the perturbation and the arbitrary coefficients 
c1 and c2 describe the nonlinear effect and the dispersion term respectively. In our 
case, as mentioned before we are in the long-wavelength limit and hence we assume a 
non-dispersive or weakly dispersive medium. Thus equation (1.19) reduces to  
 1 11 0,
t z
? ??? ?+ =
? ?
                       (1.20)  
 
 16 
 
where we have made c1=1 for convenience.  Equation (1.20) is the classical inviscid 
Burger equation. We will discuss this equation and its relevance for wave train 
formation in Chapter 4. 
 
1.7 Organization of the thesis 
The outline of the thesis is as follows: 
In Chapter 2, we discuss the experimental setup, and the beam diagnostics that 
are used in this work.  
In Chapter 3, we begin by presenting experimental results on the evolution of 
a single density perturbation under various initial conditions. We then compare the 
results with PIC simulation and 1-D theory. We show how space-charge waves can be 
used as a non-interceptive, transverse beam diagnostic. We also present time-resolved 
measurement of the transverse effects of longitudinal perturbation.  
In Chapter 4, we present experimental study on the evolution of multiple 
perturbations on the beam and compare with PIC simulation. We then demonstrate 
experimentally how energy modulation can be used to cancel a fast or a slow wave 
from a density modulation. Experimental results of a large amplitude perturbation and 
wave breaking are discussed with a simple theory to explain the nonlinear behavior. 
 In Chapter 5, we investigate sub-pulse formation in the photo-emitted beam 
pulse when the laser power is increased above a certain value. We briefly discuss the 
1-D virtual cathode theory and the short-pulse Child-Langmuir limit and conclude by 
extending our results to a high gradient gun. 
 
 17 
 
In Chapter 6, we summarize the dissertation work and then suggest possible 
ideas for future work in understanding the behavior of longitudinal dynamics of 
space-charge dominated beams. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 18 
 
Chapter 2: EXPERIMENTAL SETUP, DIAGNOSTICS 
AND THE UNIVERSITY OF MARYLAND ELECTRON 
RING 
In this chapter we briefly discuss, the University of Maryland Electron Ring 
(UMER) [26, 27, 38, 39], which forms the centerpiece of the apparatus used for this 
work. UMER is a circular machine with a circumference of 11.52 m. It uses a 10-keV 
electron beam with other scaled beam parameters. The primary scientific goal of the 
machine, among other things, is to investigate the effects of space charge forces on 
intense beams among other things.  By using a scaled, low-energy electron beam, 
UMER deliberately enhances space charge effects and is able to model high-
brightness injectors but at a lower cost. Most modern and proposed future machines, 
as described earlier, use bright beams, therefore, understanding space charge effects is 
important for their successful commission and operation. UMER parameters of 
interest are listed in Table 2-1. 
After the first multi-turn operation of UMER [40] and subsequent 
developments in the steering algorithms [41], UMER is capable of transporting the 
low current beam (0.55mA)  to more than two kilometers and the high current beam 
(?=0.9)  to 100-m. In this thesis, we focus on longitudinal dynamics and hence will 
only describe those parts of the machine that are directly relevant to the work.  The 
history and the details of the machine progress over the years are provided in [31]. 
In this chapter, after providing a brief overview of UMER in section 2.1, we discuss 
the UMER gun in section 2.2, the laser setup in section 2.3, the UMER beam 
diagnostics in section 2.4 and finally summarize in section 2.5. 
 
 19 
 
2.1 University of Maryland Electron Ring (UMER) 
The layout of the University of Maryland Electron Ring (UMER) is shown in Figure 
2.1 and some important parameters of the machine is listed in Table 2-1. 
 
 
Figure 2.1: The layout of the University of Maryland Electron Ring (top view).  
 
 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 
Table 2-1: Beam and lattice parameters of UMER 
 
 
 
UMER employs a thermionic gun, the details of which are given in the next section. 
The UMER gun is followed by a 1.4 m injection line containing a solenoid and six 
matching quadrupoles. A Bergoz fast current transformer, located 64 cm from the 
anode plane of the gun section, is used to measure the beam current before injection 
into the ring. The UMER ring section consists of 36 FODO cells, with a periodicity of 
32 cm, distributed over the 11.52 m circumference. As shown in the figure, there are 
three glass gaps for induction modules, which can provide electric fields for 
longitudinally focusing the beam. Out of these three glass gaps, two house a wall 
current monitor. Beam position monitors, one per section except the three glass gaps, 
are used to measure and steer the beam centroid through a computer-controlled 
LabView interface.   
 
Beam Energy  10 keV (?=1.02) 
? (= beam velocity/speed of light) 0.2 
FODO Period 0.32 m 
Circulation time 197 ns 
Pulse length 120-5 ns 
Beam current 0.55 - 100 mA 
Circumference 11.52 m 
 
 21 
 
2.2 UMER Gun 
 
The UMER gun has a thermionic dispenser cathode, which is made of a porous 
tungsten (W) matrix impregnated with barium oxide and calcium aluminate. While 
the cathode is typically operated at a temperature of 9000C as a thermionic gun, it can 
also be used as a photocathode [42]. The dispenser photocathode provides the 
flexibility to simultaneously generate electrons using both thermionic emission and 
photoemission. The gun has a Pierce geometry cathode and anode with a grid placed 
between them. A general feature of these electrodes is: they make an angle of 67.5 to 
the beam edge [43]. By shaping the electrodes to this angle, a suitable potential can 
be produced to generate a uniform, laminar beam flow through the gap. Normally, a 
negative bias voltage (15 V) is applied to the grid relative to the cathode to suppress 
the emission of the electrons. To generate a beam, a positive pulse (36 V) is applied at 
the grid at 60 Hz, thus allowing the electrons to escape towards the anode. At the exit 
of the gun, there is an aperture wheel which consists of six apertures, each in different 
sizes. By selecting an aperture (shown in Table 2-2 ) on the wheel, the desired beam 
current can be chosen.   
 
 
 
 
 
 
 
 22 
 
 
Table 2-2: Cathode aperture sizes used in longitudinal experiments and related parameters. Beam 
size values courtesy of [44]. 
Aperture 
size 
Cathode current 
in the space-
charge limited 
mode (mA) 
Cathode current range in  
the temperature limited 
mode (mA)? 
Beam radius# 
0.875 7 2-5  3.19 
1.5  23 7-18  4.89 
2.85 78 10-50  8.7 
3.2* 104 20-80 9.92 
*full beam aperture: used for parabolic beam studies and alignment only. 
# Matched beam radius at ?0=760 
? Though the temperature limited current can be lowered by reducing the cathode 
temperature, the current range provided in the table corresponds to those values 
that will remain stable for a sufficient period of time before the cathode cools 
down. 
 
 
 
 
 
 
 
 
 23 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 2.2: Schematic of the UMER gun electronics showing the Cathode (K), Grid (G) and Anode 
(A). The Grid is biased and an accelerating voltage of 10kV is applied in the gap. Distances not to 
scale. 
 
 
 
 
 
 
 
BIAS 10 kV 
K 
G 
A 
 
 24 
 
Table 2-3: The table showing the important parameters of the UMER gun 
 
Instead of holding the cathode current constant and changing the grid as done 
in a typical triode, UMER pulses the cathode while keeping the grid at a fixed 
voltage.  The anode is gridded to provide uniform potential across the extracted beam. 
A schematic of the UMER gun is shown in Figure 2.2 . 
Type Triode, gridded 
Geometry Pierce 
Cathode type Barium dispenser 
Cathode Eimac Y-646B 
Cathode radius 4 mm 
Cathode grid voltage Variable from 0 to 50 V 
Accelerating voltage 10 kV 
Cathode Temperature (normal) 11000 C 
Space-charge limited mode > 10000 C 
Temperature-limited mode  650-8500 C 
Heater voltage (V) 4.0 ? 7.5  
 
 25 
 
Various transverse effects have been observed by changing the grid bias. A 
detailed description of the gun is given in [23].The Table 2-3 lists the important gun 
parameters of interest. 
The UMER gun can be operated in two different modes: temperature-limited 
and space-charge limited. In the temperature-limited mode, the current across the gap 
is controlled by the cathode temperature. In this region, anomalous behavior such as a 
decrease in the beam current when the temperature is increased is observed owing to 
the triode nature of the gun and the transverse effects. In the space-charge limited 
emission mode, the gun follows the Child-Langmuir law [45, 46]. The space-charge 
limited mode is the normal mode of UMER operation. We use photoemission from a 
laser to enhance the electron density above the surface of the cathode. To be able to 
use that as a method for perturbing the beam, the gun needs to be operated in the 
temperature-limited mode. There are two reasons for operating the gun in the 
temperature-limited mode. By controlling the laser power and changing the cathode 
temperature, the total beam current- from both the thermionic emission and the 
photoemission- can be adjusted. This helps us to introduce controlled perturbation for 
beam experiments. Secondly, by turning the laser on when the gun is in the space 
charge limited mode, the current density can exceed the Child-Langmuir limit and 
hence lead to instabilities.  So, unless otherwise noted, all the experiments in this 
dissertation involve operation in the temperature limited mode. 
In the UMER gun, we can produce either a positive or a negative perturbation 
using the laser. We discuss here the physics behind a negative perturbation. By 
design, when the UMER gun is operated in the space-charge limited mode, the 
 
 26 
 
current density is held uniform across the gap for the full beam current. When the 
laser is turned on, the current density inside the gap increases for a short time leading 
to increase in self-field forces. Due to the defocusing effects of space charge, the 
beam radius at the aperture increases and becomes larger than the aperture size and is 
cut-off causing a gap in the current flow i.e. a negative perturbation [31]. So, the 
negative perturbation is due to the transverse effects caused by sudden increase in 
current density by the photo-emitted electrons. Negative perturbations can also be 
produced by changing the grid bias and overfocusing the beam [23]. Since the 
primary focus of this dissertation is longitudinal effects of density perturbations, we 
will consider only positive perturbation on the beam. 
 
 
 
 
 
 
 
 
 
 27 
 
2.3 Laser Setup 
 
 
Figure 2.3: Schematic of the laser setup (left) with the experimental setup (right) 
Figure 2.3 shows the laser setup. A Minilite [47] Q-switched Nd: YAG laser 
from Continuum Inc. forms the source of optical power. The wavelength of this laser 
is 1064 nm. The specified energy at this wavelength is 50 mJ. The full-width half-
maximum of the laser (FWHM) is around 5 ns. The pulse repetition frequency can be 
varied from 1 Hz to 15 Hz. During the experiment, the laser is operated at 15 Hz. A 
second harmonic of this frequency 532 nm is generated using Potassium Titanium 
Oxide Phosphate (KTP) crystal. A BBO crystal is used to generate a third harmonic, 
ultraviolet light at 355 nm. The maximum energy measured at this wavelength was 3 
mJ. The UV light is reflected by two dielectric mirrors whose wavelength of 
reflection is 355 nm. The dielectric mirrors are optional and can be changed to 266nm 
 
 28 
 
by adding additional BBO crystals. The crystals can be either inside the laser or can 
be external as shown in Figure 2.4. The laser power is controlled by the knob which 
rotates the polarizer inside the laser providing the needed attenuation. 
By allowing the laser light to pass through another multiplying crystal, a 
fourth harmonic (266 nm) can be produced. After the UV light is reflected by these 
mirrors, the light passes through the quartz window and is reflected by another mirror 
installed inside the chamber (IC1) then hits the cathode. The final energy deposited 
on the cathode is approximately 3 mJ.       
Figure 2.4: A block diagram of the laser setup in UMER along with the multiplying crystals. 
Measured energy output at each stage is indicated 
 
 
 
 
 
 
 
 
 
 
 
 
 
?=532 nm 
E= 12 mJ 
KNOB 
 
 KTP 
 
BBO 
 
Dielectric 
 Mirrors  
(355nm) 
 
 
     LASER 
TO CATHODE 
?=1064 nm 
E= 35 mJ 
?=355 nm 
E= 3 mJ 
 
 29 
 
 
Table 2-4: Nd:YAG laser parameters 
Repetition Frequency (Hz) 1-15 
Pulse width (ns) 5 
Diameter (mm) 3 
Divergence (mrad) < 3 
Jitter (ns) ?0.5 
Energy (@1064nm,@355) 35,3 
Polarization(@1064nm,@355nm) Horizontal, Horizontal 
  The flash lamp of the laser source is triggered at 15 Hz. Around 150 s?  after 
the flash lamp triggers, the Q-switch triggers. The laser pulse is produced 60 ns later 
than the Q-switch trigger. So, by triggering the electron gun and the Q-switch at the 
same time, the perturbation is generated at around the middle of the electron beam 
pulse which is 100 ns long.  A schematic optical layout is shown in Figure 2.5 and the 
timing is shown in Figure 2.6. 
 
 30 
 
 
Figure 2.5: Schematic layout of Minilite II laser showing the internal optical setup. The red line is 
IR while the next harmonics are shown in green. 
2.4 Beam Diagnostics: 
UMER is equipped with several diagnostics for measuring the beam along the 
injection line as well as the ring section. We will briefly discuss some of them with a 
focus on longitudinal dynamics. 
2.4.1 Beam Current Diagnostics 
The Bergoz fast current transformer, model number FCT-082-20:1, is placed 
at 64 cm from the anode plane in the injection line. It is basically a transformer with 
the beam as the primary. The output voltage of the transformer is proportional to the 
beam current. The measured rise time for a 23 mA rectangular beam is ~2 ns. An 
oscilloscope trace from the Bergoz current transformer is shown in Figure 2.7 
 At low- current operation, the beam current at the Bergoz shows some ringing 
noise. The noise is associated with the injection pulsar and is suppressed appreciably 
when the injection and recirculation pulsars are turned off.  
 
 
 
 31 
 
 
Figure 2.6: Timing pulse sequences of the laser. The flash lamp is triggered externally with the 
gun trigger. On a 100ns beam, the perturbation is around 60ns from the head of the beam. 
 
 
EXT 
INT 
EXT 
INT 
EXT 
FLASHLAMP 
TRIGGER 
Laser Pulse 
Width ~ 5ns 
 
+5V 
Q-switch delay ? 150 ?s yields highest energy laser pulse 
Q-SWITCH TRIGGER 
FLASHLAMP SYNC 
Q-SWITCH SYNC OUT 0V 
Envelope of flashlamp output 
is ~ 250?secs 
Build up time 
~ 60nsec 
 
 32 
 
 
 
Figure 2.7 :Beam current profile measured using Bergoz current monitor before injection into ring. 
A 100-ns rectangular beam pulse without the perturbation (Top) and with a 5-ns pure density 
perturbation is at the center of the main beam pulse (Bottom).  
 
 
 33 
 
Beam position monitors (BPM) are standard diagnostics used in storage rings 
for measurement and correction of beam position. They measure beam position by 
coupling to the electric field of the beam. They consist of four electrodes placed 
symmetrically around the beam axis. The signal at the individual electrode is 
proportional to the distance of the beam from the electrode.  UMER BPM [48] are 
designed to achieve good spatial and temporal resolution of the order of 2 ns. Recent 
work [39, 49]  increased the signal-to-noise ratio of the BPM signal and measured its 
linearity. An experimental BPM trace tracking a UMER beam is shown in Figure 2.8. 
All the beam centroid measurements in UMER use the BPM signals to track the 
beam.  Beam-based steering solutions obtained from simulation are used to make 
corrections to the orbit. 
 
Figure 2.8: The four signals are from the top, bottom, left and right plates of the beam position 
monitor. The beam current measured was 23mA at 1.84 A quad current settings. As the beam 
propagates, the beam expands and fills the ring. 
 
 
 
 
 34 
 
2.4.2. Wall Current Monitor 
In the closed path around the vacuum chamber, there is no magnetic field. 
i.e. . 0externalB dr =?integralloop ; whereas inside the chamber .inside beamB dr I=?integralloop . So there must be 
an equal and opposite current, called the image current, flowing on the beam pipe. A 
wall current monitor measures this image current. In order to measure the image 
current, a ceramic (glass in UMER) gap is created in the beam pipe and a resistor is 
added parallel to the gap. By measuring the voltage drop across the resistor, the beam 
profile is captured. In UMER there is an added complexity. The image current 
actually flows through a ground loop created by the plates that hold the FODO 
sections. In order to choke off the image current from flowing through the ground 
loop created by the plates, a high ferrite material is added and the image current is 
forced to flow through the resistor. The details of the wall current monitor and the 
equivalent circuit model are given in [30]. A schematic of the wall current monitor is 
shown in Figure 2.9 . The wall current monitor is AC coupled and hence when the 
beam expands and fills the ring (especially near the ends), the DC component 
registers as an apparent beam loss.  
 
 
 35 
 
 
Figure 2.9:The wall current monitor principle: The image current induced in the beam pipe is forced 
to flow through the resistors using the ferrite core and the voltage across the resistors measured is 
proportional to the beam current. Courtesy [30] 
 
2.4.3 Fast Imaging Diagnostics 
Conventional fluorescent screens such as P-43 have a long response time on 
the order of microseconds. Therefore, they are not capable of resolving nanosecond 
time structures inside the beam. Lexel Imaging Systems, Inc [50] manufactures fast 
phosphor screens that have a time response around 3 ns. The screen consists of 
ZnO:Ga formulation on a quartz plate. When an electron beam hits the screen, it 
emits a bluish light in the near UV (around 390 nm), visible to the eye. A thin layer 
 
 36 
 
(1?m) of aluminum is coated on the screen to conduct the electrons away from the 
screen and prevent charge buildup.  In UMER, the fast phosphor screens are installed 
at IC1 (31.1 cm from the aperture), RC 8 and at RC 15, while the conventional P-43 
screens are installed at all the other locations of the beam position monitors. The light 
from the screen is reflected by a mirror, placed 45? the beam line, and seen through 
the viewport.  
The light from the fast phosphor screen is captured by an ICCD (Intensified 
charge coupled device) PIMAX-2 camera manufactured by Princeton 
Instruments[51]. The bandwidth of the camera is between UV and Near-IR making it 
suitable for the experiments. The high-speed camera has an internal pulse generator 
for adjusting the gate/delay time and can do time-gated imaging at nanoseconds (3-
ns). A wide dynamic range (16-bit) makes the camera sensitive to halos and internal 
structures in the beam. A general setup of the image acquisition system used in 
UMER is shown in Figure 2.10. 
 
 37 
 
 
Figure 2.10: Fast imaging diagnostics set up in UMER. The beam hits the fast phosphor screen 
(ZnO:Ga) and the light emitted is captured by the camera (ICCD PIMAX2) for further processing via 
the quartz window.  
2.5 Summary: 
                             We have briefly discussed the University of Maryland Electron 
Ring and the beam current diagnostics. In this thesis, the primary diagnostic for all 
the experiment will be the wall current monitor and the fast current transformer 
(Bergoz). The fast imaging diagnostics will be used to study the transverse effect of 
 
 38 
 
perturbation in the following chapter. In addition to all these diagnostics, we are 
currently testing the energy analyzer on UMER and have not used it in this work. The 
measurements from the energy analyzer will complement the work done in this thesis.  
 
 39 
 
Chapter 3. Experimental investigation of a single 
perturbation on a space-charge dominated beam  
 
Laboratory beams have either density or velocity modulations. These 
modulations are due to cathode non-uniformities, laser fluctuations and various other 
sources. In order to study the effect of such modulations, we begin by introducing a 
single, localized and controlled density perturbation on a beam using a laser.  
In this chapter, we present various experiments with a single perturbation on 
the beam. We know that a perturbation emits a pair of space charge waves that travel 
at the sound speed. We begin by measuring the sound speed as a function of the beam 
current in Section 3.1. In Section 3.2, we measure the sound speed as a function of the 
perturbation strength by keeping the beam current constant. In Sec 3.3, we compare 
the experimental results with theory and WARP simulation. In Section 3.4, we show 
how space charge waves can be used as a beam diagnostics. Finally, in section 3.5, 
we present time-resolved measurements of the perturbation and conclude. 
3.1 Wave speed as a function of beam current 
              In order to measure the variation of the wave speed as a function of beam 
current, the following experiment was performed in UMER. We use the gun aperture 
(2.85mm) and then adjust the cathode temperature to generate a 100ns rectangular 
beam pulse with 16mA peak beam current.  An UV laser (355nm) is switched on to 
generate a 5ns perturbation on top of the main beam current. The current settings on 
the focusing and steering magnets were set for a phase-advance of 00 61? =  . The 
 
 40 
 
quadrupoles current was set at 1.83 A which corresponds to a peak gradient of 6.63 
G/cm. The beam pulse is measured at the Bergoz coil which is shown in the  
Figure 2.7.  
After injecting the beam into the ring, the beam current is measured using the 
wall current monitor at RC10. Figure 3.1 shows the turn-by-turn beam current profile 
from the wall current monitor. The perturbation splits into a fast and a slow wave 
with the waves travelling toward the head and the tail of the beam respectively with 
sound speed.  Since there is no longitudinal focusing applied to confine the beam 
longitudinally, the beam itself is expanding and the edges of the beam move inwards. 
By applying ear-fields using the induction modules, longitudinal focusing and 
acceleration can be provided for the UMER beam [30].  
 
 
 41 
 
 
Figure 3.1: Turn-by-Turn plot of the beam current obtained from wall current monitor. Turn 1 is on 
the bottom with subsequent turns plotted on top of each other. Injected beam current was 16 mA. 
For clarity, every turn of beam current is shifted with a constant value of 5mA. The tune of the 
machine was set at 0 6.1? = . 
 
The wall current monitor is AC-coupled and hence measures only time-
varying component of the beam current. When the beam ends meet, the beam 
becomes quasi-DC and so the AC component is lost and appears as the beam loss at 
the wall current monitor. Hence, the actual beam loss in the ring is smaller than the 
measured beam loss. 
After recording the beam profile from the wall current monitor, the main 
beam current is changed by changing the temperature of the cathode, keeping the 
 
 42 
 
relative perturbation strength (?=0.2) constant. In this experiment, the beam currents 
used were 8mA, 16mA and 24mA.  A 3-D waterfall plot is shown in Figure 3.2. 
 
 
 
Figure 3.2: 3-D plot of beam current evolution with single density perturbation. The plot is 
generated by interpolating the turn by turn data obtained using the wall current monitor. The 
injected beam current is 16 mA at 10keV. 
 
We use the following method to calculate the sound speed from the experimental 
data. Let us assume the fast and the slow waves travel a distance of z?  in a given 
time t? . The velocity of the fast and the slow waves are 0 sC? +  and 0 sC? ?  
respectively in the lab frame. Let us take the fast-wave as a reference for calculation 
 
 43 
 
purposes. The fast and the slow waves move away from each other at twice the sound 
speed. So,  
 
 
( )
0
0
2 2
0
2 2 2 2
0
( )2
/( )
0.5   
0.5 since 
s
s
s
s s
s s
C tDistanceC
Time z C
tC C
z
tC c C
z
?
?
?
? ?
+ ?= =
? ?
?? = ?
?
??
?asymptequal greatermuch
 (3.21) 
? is the ratio of the beam velocity to the velocity of light  0 c?? = .  z?  is equal to 
7.67 m+ (n-1)11.52 m where n is the number of turn, with n=0,1,2..    
The center- difference method to calculate Cs: 
The eqn (3.21) has few limitations. It assumes that the beam current is not changing 
between the measurements. This is a good assumption for a short distance beam 
transport like Long Solenoid Experiment [52] or within the first turn of UMER [22]. 
But, in our case, we have many turns and the beam current is changing between the 
turns. So, we have to include the effect of varying beam current into the formula for 
Cs. 
             Let us begin by plotting t? , the time separation between the peaks of the fast 
and the slow waves against the distance. We then plot the beam current as a function 
of distance to show how the current varies with distance. This is shown in Figure  
 
 44 
 
               
Figure 3.3: The top graph plots the time separation of space-charge waves with the distance. The 
bottom graph is the beam current as a function of distance. The beam current is changing as a 
function of distance which affects the linearity of the wave speed. Hence, we have to take into 
account the varying beam current in calculating the value of sound speed from the measurements. 
 
 45 
 
 
Figure 3.4 : The graph shows two ways of measuring the value of Cs from the experimental data. 
The black curve is the sound speed calculated from the theoretical formula. The two red curves 
correspond to the calculation of the sound speed from experimental values using the linear method 
eqn (3.21) and the center-difference method eqn(3.22) . As the distance increases, the center-
difference method is closer to the theory as expected. 
Since our quantity of interest is ( )
( )
d t
d z
?
?
 ,we use the center-difference method, a form 
of finite difference method, to calculate Cs. Though an ideal method would fit a 
function and calculate slope, we use difference equations as follows: Let us assume 
we want to calculate the sound speed at turn 5 say Cs5. To do this, we measure the 
time separation of the space charge waves in turn 5 and record it as 5t? . Then Cs5 is 
calculated using the formula:  6 45 2 11.52mt tCs ? ??= ? .  For calculating Cs for the first and 
 
 46 
 
the last turn we use 2 2 2 21 210.5  , 0.5 ,11.52m 11.52m 7.67mmm t tCs c Cs c? ?+? ?? ? ? ?= = ? ?? ? +
? ?? ?
 where 
m is the last turn number. This reduces the uncertainty due to the variation in beam 
current.  So, in general we can write, Cs from the center-difference method as  
 2 21 10.5 2 11.52mm mm
t tCs c?+ ?? ??? ?=
? ??? ?  (3.22) 
 
In order to calculate the value of sound speed from theory, the following 
method is used. For every turn of data, the beam current I  is measured. Now, we 
need to know the beam radius every turn. Assuming smooth approximation in a 
periodic focusing lattice, the beam size a can be calculated from 
2 1/ 2( 1 )a R u u= + + [31] , where 
0
SR ??=  is the matched beam radius in a 
smooth focusing lattice with emittance ? = 30 microns and S=32cm is the lattice 
periodicity in UMER and 0? is the zero current phase advance and 
02
KSu
??=
 where 
3
0
2
( )
IK
I ??=
 is the beam perveance and 0I = 17 kA for electrons. For this experiment, 
as mentioned before 0?  is set at 610. 
 
 
 
 
 
 
 
 47 
 
 
Table 3-1: Table showing the values of the time separation ?T of the space charge waves with 
distance and the corresponding values of Cs calculated using linear and center difference 
techniques.  
Distance 
(m) 
?T (ns) Cs_Theory 
(106m/s) 
Cs_linear_technique 
(106m/s) 
Cs_center_difference 
(106m/s) 
7.67 45 1.17 1.05 1.09 
19.19 117 1.05 1.09 1.06 
30.71 181 0.97 1.06 0.98 
42.23 243 0.94 1.03 0.90 
53.75 297 0.91 0.99 0.86 
65.27 353 0.88 0.97 0.83 
76.79 404 0.83 0.94 0.81 
88.31 457 0.80 0.93 0.82 
 
The calculated values of the sound speed versus the values predicted by theory are 
shown in Figure 3.5. As predicted by the 1-D theory, the wave speed increases as the 
beam current increases. This is because as the beam current increases, the line-charge 
density increases, leading to the increase of longitudinal space charge field, which 
drives the waves faster. This is similar to sound waves, which travel faster in denser 
media.  
 
 48 
 
 
Figure 3.5: Graph comparing the sound speeds for different main beam current for the same 
strength of the initial density modulation along with values predicted by theory.  The higher the 
main beam current, the faster the waves depart from each other. A 5% error bar in the data is not 
shown in the graph for clarity.  
3.2 Wave speed as a function of perturbation strength 
Further experiments were carried out to study the effect of the perturbation 
strength (?) on the wave speed of the space-charge waves. In these experiments, we 
used three beams: 15.5 mA beam with 1mA perturbation, 16.4 mA beam with 2 mA 
perturbation and 17.4 mA beam with 4 mA perturbation. The perturbation strength 
was 6%, 10% and 20% of the main beam current for each of these cases. The sound 
 
 49 
 
speed is calculated from the theory and the experimental data as described in the 
previous section. The results are plotted in Figure 3.6 . 
 
Figure 3.6: Sound speed as a function of distance for different strengths of the initial density 
modulation. The higher the strength of the perturbation, faster the wave moves.  
 
We see that, as the perturbation strength increases, the wave speed increases 
indicating a non-linear trend. One of the important properties of a non-linear wave is 
that the speed of the wave will depend on its amplitude. Such nonlinearity can lead to 
wave breaking. Moreover, there is a discrepancy between the theory and the 
experiment for the value of the sound speed in the case of ?=0.2. This is because the 
 
 50 
 
1-D cold-fluid model assumed a small initial perturbation to linearize the variables 
( 1 0? ?<< ) while ? is not negligible in this case. In conclusion, these experiments 
illustrate the range of perturbation strengths for which 1-D cold fluid theory holds 
true. 
3.3 Simulation using WARP 
In order to simulate the evolution of perturbations in the beam, we use the 
particle-in- cell code (PIC) WARP. WARP is a self-consistent code, meaning that 
once the density and velocity distribution of the particles is given, WARP correctly 
includes the effect of the evolving self-fields. It does so by creating grids and by 
calculating self-fields at every grid point for the given distribution and the given 
boundary conditions. The calculated self-field and the external fields are then applied 
to move the particles to a new position resulting in a new distribution, whose self-
field is calculated again, and the procedure is repeated until the end of the simulation. 
The initial particle density distribution and velocity distribution as well as other 
boundary conditions (such as the pipe radius, pulse length) must be specified in 
WARP.  
In our case, since the laser introduces a pure density modulation, we have no 
velocity modulation at the cathode.  Therefore, we import the Bergoz current 
transformer signal directly into WARP for the initial current distribution. We assume 
that the velocity modulation is still zero at the Bergoz coil. We also initialize the code 
with the values of matched beam radius, current, energy and emittance. We made the 
following approximations. We assumed a smooth uniform field (solenoid field) to 
represent the UMER FODO lattice. Since we use a long solenoid field for focusing, 
 
 51 
 
we have an external field which is symmetric in the x and y-axis. Hence we used 
WARP-RZ, a version of WARP-3D, for all our simulations.  
 
The initial transverse distribution used is a semi-Gaussian (uniform in density 
and Gaussian in velocity). Two important quantities of interest in our simulation were 
the number of particles and the longitudinal thermal spread. When we attempted to 
increase the accuracy of the simulation by increasing the number of particles, we ran 
into numerical instabilities. We avoided instabilities by increasing the longitudinal 
thermal speed. The values we used are: the number of particles = 1 million and the 
thermal speed was 51.5 10? m/s, which corresponds to 0.25%p p? = .  The transverse 
grid size was 64x1 cells (R-Z) and the z-grid was 256x256. Before running every 
simulation, the appropriate dE dr  was calculated for matching the beam with the 
assumed solenoidal field. All simulations are done in the beam frame without any 
longitudinal focusing. 
 
For simulating the experiment discussed in the previous section, we loaded the 
16 mA beam current with a density perturbation strength of 0.2? = . The beam 
current profile obtained from the current monitor in the injection line is used as the 
initial longitudinal distribution. Figure 3.7 shows the turn by turn plot of the beam 
current profile obtained from WARP. The space charge wave velocity is calculated 
from the simulation for every turn and is in good agreement with the theory.  But the 
experimental values of wave speed differ from the values obtained from WARP. This 
is because we assume there is no beam loss in the WARP simulation.   In other 
 
 52 
 
words, WARP agrees with the predicted theory. But both differ from the experimental 
results possibly due to exclusion of beam loss from the simulation and the theory. 
So we deliberately introduced beam loss into the simulation and compared it 
with the experiment. We used the WARP variable (top.pgroup.sw) to change the 
particle weights every turn. The variable effectively changes the number of real 
particles per macro particle. We changed the particle weight such that the beam 
current in the simulation equals the beam current in the experiment after every turn. 
Figure 3.8 shows the experimentally measured wave speed values and compares them 
with the prediction for theory and simulation, with and without beam loss. This 
improves the agreement between WARP and the experiment for many turns as shown 
in Figure 3.9. In essence, we have predicted, measured and numerically verified the 
wave speed with a good accuracy. The discrepancy in the zero-th turn is because the 
waves have not separated yet and hence those points should be ignored. We can still 
improve the accuracy by doing the following: the simulation can be done in the lab 
frame rather than the beam frame. This will compensate for the slight asymmetry 
between the fast wave and the slow wave in time separation from the beam centre. 
Also, we have ignored the energy profile at the Bergoz and have taken it to be zero. 
This could be measured and the velocity profile loaded into the simulation to improve 
the accuracy. 
  We have also neglected transverse effects of space charge. The transverse 
effect can play an important role in space-charge dominated beams. We discuss in the 
next section how transverse effects of space charge waves can be used to measure 
beam size. 
 
 
 53 
 
 
Figure 3.7 : Turn by turn plot of beam current obtained from PIC simulation using WARP. Since 
there is no beam loss assumed in the simulation, the waves reach the ends of the beam faster than 
in the experiment. The injected beam current is 16mA. For the sake of clarity, the beam profile 
from every turn is shifted vertically. 
 
 
 
 54 
 
 
Figure 3.8 : Sound speeds measured from the experiment, and theory along with those obtained 
from simulation with and without beam loss. Theory and WARP agrees with measured data when 
beam loss is included in the calculation. For the zero-th turn, the measured and the calculated 
values of the sound speed show a large discrepancy. This is because the fast and the slow waves 
have not yet separated at this distance. These points are indicated by gray boxes.  The black line 
with the black triangles uses the formula for Cs for a given beam current and hence does not 
change with the turn. But, in the experiment, the current changes and taking this account the Cs is 
calculated and this is the black line with the black boxes. The red line is the experimentally 
measured value of Cs. The blue lines correspond to the WARP simulation. The blue line with the 
triangle ignores beam loss while the blue line with the boxes includes beam loss. 
 
 
 
 
 55 
 
 
Figure 3.9: WARP simulation including beam loss is compared with the experimental data. The 
beam current is 16mA with a 20% perturbation. The time-evolution of the perturbation obtained 
from WARP is overlaid on the experimental measurement showing both the fast and the slow 
waves evolving at almost the same speed as seen in the experiment. The turns shown in this plot 
are turn 3 (Top: left), turn 4 (Top: right), turn 5 (Bottom: left) and turn 6 (Bottom: right). 
3.8 Beam End Erosion 
As seen from the experimental data, the beam ends erode inward as the beam 
propagates. There is no longitudinal focusing applied to the beam causing the beam to 
expand and fill the ring.  At the edge, the charge density varies sharply leading to a 
large electric field. Appropriate external fields, called ?ear-fields?, have to be applied 
to focus the beam.  
Currently, they are applied at RC4 through the induction module. For the low-
current beam (0.55 mA), this increases the number of turns to 200 turns. Space-
 
 56 
 
charge wave experiments have shown that the waves reflect off the beam ends[17]. 
With the induction module, UMER is capable of repeating such experiment over a 
longer distance and track the propagation of the waves after reflection from beam 
ends. In theory, the fast wave after reflection from the beam ends will become a slow 
wave and vice-versa. Experiments conducted in UMER, without the longitudinal 
focusing, show that the waves walk off the edge of the beam without any reflection. 
3.4 Space-charge waves as beam diagnostics for measuring transverse 
beam size 
In order to measure the beam size over multiple turns, we need a non-
intercepting beam diagnostic. As an alternative to fluorescent gas monitors or 
diagnosis through an extraction line, space-charge waves can be used to measure 
transverse beam size turn-by-turn. The principles behind the technique are Equation 
(1.17) and (1.12). Combining those two equations, we get: 
 
5 2
0 0 02ln sc mb
a qI
pi? ? ?=
 (3.23) 
After measuring Cs and I for every turn, Equation (3.23) can be used to calculate the 
values of beam radius. As the flat-top erodes, the beam current decreases leading to a 
decrease in beam size. Other sources of particle loss like beam scraping can also lead 
to decrease in beam current. Therefore, we expect a decrease in beam radius as the 
distance increases. Figure 3.10 shows the turn by turn plot of the beam size as a 
function of distance for three different beam currents. As the beam propagates a 
longer distance, the beam size decreases and then settles to a final value. The beam 
 
 57 
 
self-matches itself into the ring. The error percent is calculated by assuming 2% error 
in the measurement of Cs which causes a 4% change in the beam size. There are 
some limitations on this diagnostic. Firstly, Cs becomes ill-defined at the beam ends.  
The higher the beam current, the faster the space-charge waves reach the beam ends 
and so this diagnostic is unreliable after some turns depending on the beam current.   
 
 
Figure 3.10: The average beam radius of various beams as a function of distance in the ring. The 
beam radius is calculated from measuring the wave-speed and beam current for each turn and 
applying equation(3.23). The beam size tends to decrease slightly as the distance increases.  
 
 
 58 
 
Hence this diagnostic is useful only in that range of beam currents where the waves 
spend a longer time in the flat-top portion of the beam. Secondly, since the formula 
depends on the g-factor, this diagnostic cannot be applied for compressed beams or 
very short bunches. Finally, the perturbation applied to the beam should be very 
small. A large perturbation produces a non-linear wave which causes a larger error in 
the measurement of Cs and hence the beam size. 
Another important longitudinal parameter in accelerators and storage rings is 
the longitudinal impedance. Though there are several types of impedances that can 
occur in a machine, we focus on space-charge impedance. When a charged particle 
beam travels through vacuum it does not experience a resistance directly like 
electrons moving inside a wire. Rather, the beam encounters impedance due to image 
space-charge fields that can interact back with the beam. The space-charge impedance 
per unit length is given by 
 *
1
( , )( , )
( , )
z
s
E kZ k
I k
??
?= ?
 (3.24) 
where Ez and I1 are the perturbed longitudinal electric field and the perturbed current. 
The impedance is a function of frequency and hence is expressed in the Fourier 
transformed coordinates of space and time. Since low-amplitude space charge waves 
are dispersionless, we can assume 0k? ??  and the space charge impedance can be 
calculated as 
 * * 0
0 0 02 2( ) ( ) 4s s
gZ iX i Z
c
?? ?
pi? ?= ? ? ?
 (3.25) 
As can be seen from eqn (3.25) , the space-charge impedance is a function of the g-
factor and the perturbation frequency for a give beam energy. Z0 is the free space 
 
 59 
 
impedance and equals 377 ?. For a given perturbation wavelength, the space-charge 
impedance can be normalized as 
 * 0
2
0
( )
2
s p
s
X gX
Z
? ?
??= =
 (3.26) 
By measuring the g-factor, the space charge impedance can be calculated 
experimentally. We can extend this technique to calculate the growth rate of the slow 
wave or the decay of the fast wave [53].  We list the space-charge impedance for 
three different beam currents in the Table. As expected the table shows that when the 
beam current increases the product of beam current and the normalized space-charge 
impedance increases- a high current beam produces a larger self-field. The 20 mA 
beam with a perturbation of frequency 200 MHz (5 ns) would have a space-charge 
impedance Xs*=12k?/m. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 60 
 
Table 3-2: Normalized Space-charge impedance values for different beam current as a function of 
distance as per eqn (3.26).  
Distance (m) I=9mA I=20mA I=30mA 
7.67 
12 9.6 8.3 
19.19 
16.6  11.7 9.6 
30.71 
13.0  12.2 9.4 
42.23 
14.5  10.9 8.5 
53.75 
19.2  10.5 9.5 
65.27 
19.6 10.81 9.8 
76.79 
16.7 10.8 9.3 
88.31 
17.4 11.6 9.9 
 
 
3.5 Transverse effects of the space-charge waves 
In a space charge dominated beam, as mentioned earlier, any line charge 
modulation manifests as a change in the beam radius to make the volume charge 
density a constant. We are also interested in the transverse effects because they may 
be responsible for nonlinear effects. To study the transverse effects of longitudinal 
modulation, we need diagnostics that can resolve the perturbation. In our case, the 
perturbation is 5 ns and hence we need a high resolution optical diagnostic. The fast 
phosphor screen emits UV light when a charged particle hits the screen and has a 
good temporal response making it suitable for our experiment. In our experiment, the 
 
 61 
 
light from the phosphor screen is captured by a PIMAX camera and the image is 
stored in a computer. The fast phosphor screen and the camera were discussed in 
Chapter 2. 
A 5-ns density perturbation is introduced on a 100-ns long beam using the 
laser. After injecting into the ring, the beam hits the fast screen at RC15 and emits a 
UV light. The light is captured by a PIMAX camera in the auto focus mode. After 
focusing the camera onto the beam, the camera is set to gated mode in which the 
camera is triggered by an internal pulse generator. The timing sequence is adjusted 
with the BPM trigger (RC14) and camera runs at 15Hz. Once the timing sequence is 
checked, a gate is set for 3-ns width. Finally, the internal sequence generator is turned 
on with the appropriate start time and end time. A window of 51ns is used to generate 
17 images of 3-ns each. The camera collected images are stored in a single file which 
is then sliced into TIFF images for processing. 
The wall current monitor output is shown in the Figure 3.11 indicating the width of 
the gate and the scanning range of the gate within the pulse. The integrated beam 
image when on the camera is in the auto-focus mode is shown in the Figure 3.12. 
 
 
 62 
 
               
 
Figure 3.11: The wall current monitor signal at RC10 showing the measured beam current of 45mA 
peak with a 12% perturbation. The strength of the perturbation was deliberately increased to 
capture the splitting before the fast screen at RC15. 
 
 
 
 
 63 
 
                    
                     
Figure 3.12: Integrated beam image of the whole beam. Top: 45mA with a positive perturbation; 
Bottom; the same beam in color-coded format. 
The sliced images are processed and then calibrated according to the intensity of the 
light collected per gating. The montage is shown in the  Figure 3.13 and Figure 3.14 
below along with the calculated beam current.                
 
 64 
 
 
Figure 3.13: (Landscape) Black and white montage of the images captured by the camera in the 51-
ns window at RC15. The total intensity of each image is calculated expressed in terms of the peak 
current.  The peak beam current is 45mA. The montage is read from top to bottom starting from 
left to right. This is done to capture the x-motion of the beam. 
  
 
 65 
 
                     
Figure 3.14:  (Landscape) Color montage of the images captured by the camera showing transverse 
density waves at RC15. The beam size of the perturbed part of the beam is larger in x and the beam 
centroid moves transversely in the x-direction possibly to the dispersion effect of the ring. 
 
 66 
 
We follow the following algorithm [54] to calculate the beam current and the centroid 
from each image of the montage. First, the total intensity of the image is calculated by 
adding intensity at each point of the image. 
 ( , ),
N N
i j
I G i j= ??  (3.27) 
where G( i, j) represent the intensity of the i-th row and j-th column of the image.  
The x and y centroid is then calculated by taking the first moment as following: 
 
( , ) /
( , ) / ,
N N
c
i j
N N
c
i j
x iG i j I
y jG i j I
=
=
??
??  (3.28) 
Where xc and yc are the x and y centroid respectively. The code similarly calculated 
the beam size by taking the second moment. Once the intensity is calculated for all 
the images, the image with the maximum intensity corresponds to the peak beam 
current. The rest of the images are then scaled to calculate the beam current for each 
image. The calculated sliced current for each gating is plotted as a function of time. 
This is shown in Figure 3.15.  
  The intensity pattern follows the current profile seen with the current monitor 
and hence confirms the linearity of the fast-screen with respect to charge. 
 
 67 
 
 
Figure 3.15: The plot of the slice beam current as a function of time. The total light intensity from 
the fast screen follows the fast and the slow waves. The measurements are at RC15 with a peak 
current of 45mA with a perturbation of 12%. The plot allows confirms the linearity of the screen 
with the charge. 
The radius and the centroid of the beam are affected by the perturbation and 
are shown in Figure 3.16 and Figure 3.17  shows a displacement of centroid in the x-
direction only. This is likely due to the velocity modulation introduced by the space-
charge waves.  The fast and the slow waves create an energy modulation within the 
beam which causes the perturbed part of the beam to move transversely in x-direction 
due to dispersion. This experiment indicates that a pure density modulation evolves 
into an energy modulation over time. The beam size increase is more pronounced in 
the x-direction because of the placement of the screen with respect to the UMER 
 
 68 
 
FODO lattice. The plane of the phosphor screen is 2.7 cm from the lattice centre, so 
the UMER beam looks elliptical with major axis along x-direction [55]. Hence, 
though the beam radius increases inside the perturbation, the radius appears to 
increase only in the x-plane. In order to cross check these result from WARP, we 
need to perform a WARP 3-D simulation that takes into account the quads and the 
bends. A WARP R-Z simulation will not be sufficient. 
       
 
Figure 3.16: The RMS beam radius measured from the images collected at RC15 using the fast 
screen and camera as a function of time. There is a 10% increase in the x-radius with the 
perturbation. As predicted by the theory, the perturbation causes a radius change in a space charge 
dominated beam to keep the volume charge density a constant. 
 
 69 
 
 
Figure 3.17: The centroid shift, the distance between the beam center and the pipe center, as 
measured from the images as function of time. The x-position of the beam centroid oscillates while 
the y-position of the beam centroid is relatively unchanged. This is due to the effect of the velocity 
modulation of the perturbation causing a shift in beam position due to the dispersion in the ring. 
3.11 Summary 
  In this chapter, we derived the wave speed formula from 1-D cold fluid 
theory. We compared the measured values of the sound speed with theory and WARP 
simulation. The agreement is good within the linear regime and there is an evidence 
of non-linear behavior at larger amplitudes. We showed that space charge waves can 
also be used as diagnostics to measure beam size in UMER and finally we performed 
 
 70 
 
progressive, time-sliced imaging of the fast and the slow waves using the imaging 
diagnostics to show the transverse effects of longitudinal perturbation due to space-
charge. The experiment concluded that a pure density modulation evolves into an 
energy modulation over time. In free electron lasers, this mechanism can lead to an 
increase in the energy spread of the beam as follows: In a high brightness electron 
beam, density modulations at low-energies can evolve into energy modulations due to 
space-charge. Once, the beam gets accelerated and goes through a bunch compressor 
or a bend, these energy modulations can get converted into density modulations 
through dispersion. Now, due to these density modulations, the longitudinal charge 
density of the beam is not smooth anymore but is modulated at wavelengths much 
smaller than the bunch length. This can lead to coherent synchrotron radiation (CSR) 
in the bunch compressor and thus introduce additional energy modulation, which 
distorts the transverse and longitudinal phase-space of the beam and affects the FEL 
performance. 
 
 
 
 71 
 
Chapter 4: Multiple perturbations: Propagation, 
Control and Nonlinearities  
 
In this chapter we look at how multiple perturbations propagate in a space-
charge dominated beam and ways to control them. We also show how a large 
perturbation can launch nonlinear wave that can lead to pulse train formation. 
  In real laboratory beams, often the beam has more than one perturbation. It 
may have multiple modulations resulting from cathode non-uniformities, RF 
fluctuations, varying impedances along the beam pipe, etc. Such modulations occur 
throughout the beam and can vary in their frequency and amplitude. Hence, by 
understanding how multiple perturbations evolve in a space-charge dominated beam, 
we can control them in real laboratory beams. In Section 4.1, we discuss generating 
and propagating multiple perturbations on a beam. In Section 4.2, we then present 
experimental results on how to control the density perturbation by modulating the 
beam energy. Finally in Section 4.3, we show how a large perturbations can lead to 
non-linear wave propagation and thus to pulse train formation.   
4.1 Generating multiple perturbations on a beam: 
  In the previous chapter, we showed how a single perturbation can be 
generated using a laser. Here, we take advantage of the photoemission technique to 
add a second perturbation. For this, we installed another Minilite laser system that 
used the same optics as the first one to introduce a second density modulation on the 
beam. By adjusting the timing of the pulses that trigger the laser systems, the distance 
between the two perturbations can be adjusted. This allows us to introduce 
independently controlled modulations (by adjusting the laser power) with controlled 
 
 72 
 
separations (by changing the trigger timing). Figure 4.1 shows a beam current profile 
with two density perturbations (A and B) measured at the current monitor in the 
injection line.  
 
 
Figure 4.1 : Beam current profile measured using the Bergoz current monitor before injection into 
the ring. Two 5-ns pure density perturbations are introduced using two lasers. The time between A 
and B is 15ns. The beam current is 21 mA with 6 mA (A) and 4.5 mA (B)  perturbation. 
 
 
 73 
 
4.1.1 Experiments on a beam with multiple perturbations: 
The beam shown in Figure 4.1 was injected into the UMER ring and the beam 
current profile was measured using the wall current monitor at RC10 for nine turns. 
Figure 4.2 shows a 3-D turn by turn plot of the measured beam current profiles. After 
the perturbation splits into a fast and a slow wave, the fast and slow waves from each 
perturbation travel toward the head and the tail of the beam, respectively, at the sound 
speed as discussed in Chapter 3. The slow wave from one of the perturbations (A-
closer to the head) and fast wave from the other perturbation (B-closer to the tail) 
superimpose on each other in the second turn and then cross each other. Hence, in the 
second turn, the amplitude of the perturbation near the center is higher compared with 
the other peaks. The wave velocity of each of the perturbation (A and B) is measured 
and is shown along with 1-D cold fluid theory predictions in Figure 4.3. The black 
curve is the value of wave speed calculated from theory using the measured peak 
current. The red line corresponds to the experimentally measured value of Cs using 
the center-difference method described in Chapter 2. The blue line values are from 
the WARP simulation which we discuss in the following section. 
 
 
 74 
 
 
Figure 4.2: : 3-D plot of beam current evolution with two density perturbation. The plot is 
generated by interpolating the turn by turn data obtained using the wall current monitor. The 
waves are observed to superimpose and cross each other between turns 2 and 4. The injected 
beam current is 24mA. 
 
 
 
 
 
 
 
 
 
 75 
 
 
Figure 4.3:Graph showing the value of sound speed of the perturbations A and B compared with 
the theoretically predicted values over multiple turns.  The sound speed is higher for A compared 
to B because of the higher strength of the perturbation A (see Chapter 3) . The injected beam 
current is 24 mA. 
4.1.2 WARP Simulation of beam with multiple perturbations: 
  The beam profile obtained from the current monitor shown in Figure 4.1 is 
used to initialize the current distribution in WARP. The profile is then smoothed 
using a weighted average filter (of width 3) in the simulation to filter noisy spikes in 
the current distribution due to the injection and re-circulation pulsing circuits. The 
initial transverse distribution used is a semi-Gaussian (uniform in density and 
Gaussian in velocity). We used one million particles in the R-Z simulation and took 
measured beam loss into account. We used 64 cells in R-direction and 256 cells in Z-
direction with a thermal spread of 1.5e5. After calculating the necessary smooth 
 
 76 
 
focusing for matching the beam, the beam is transported to a distance equivalent to 8 
turns.  
 
Figure 4.4: WARP Simulation results of a beam with multiple density perturbations. The slow wave 
of one perturbation and the fast wave from the other perturbation superimpose each other on the 
second turn and then cross. The beam used in simulation is the beam shown in Figure 4.1. The 
WARP simulation used one million particles. 
 
Seven turns from the WARP simulation in Figure 4.4 shows how the fast and the 
slow waves from each perturbation superimpose and cross each other. We calculated 
Cs from WARP and compared with the experimental and the theoretical values. This 
is plotted in Figure 4.3 as blue lines. The same procedure to calculate Cs using the 
center-difference method mentioned in Chapter 3 is also followed here. As the graph 
shows, the waves from perturbation A travel faster than the perturbation B.  This is 
 
 77 
 
because of the large amplitude perturbation (A) travels faster compared to low 
amplitude perturbation (B) as discussed in Chapter 3.  The slightly larger discrepancy 
with the theory may be possibly due to the multiple perturbations on the beam 
changing the beam size along the beam causing a variation in g-factor, which the 1-D 
theory ignores. In other words, transverse effects like variation in g-factor due to 
beam loss must be taken into account in a lab-frame, WARP 3-D simulation.  
 
A long pulse length experiment with multiple perturbations: 
To track the evolution of the waves over a long timescale, we need the waves 
to remain in the flat top portion of the beam for a longer time. So, we increased the 
beam pulse length to 150 ns and performed the two laser experiment. The beam 
current was 12.6 mA with a 1.7 mA perturbation (A) and 1.0 mA perturbation (B). 
We show the result of the experiment in Figure 4.5 . A straight line fit of the wave 
peaks of the respective perturbation is also shown. It shows the linear variation of the 
distance between the waves (with a slight change due to change in current) as the turn 
increases. In conclusion, within the linear regime, the presence of another density 
perturbation does not affect the velocity of the waves and each perturbation behaves 
as if it were the only perturbation present on the beam.  
 
 78 
 
 
Figure 4.5: Experimental results of a UMER beam with multiple perturbation. The beam current is 
12.6mA with a 1.7mA perturbation (A) and 1 mA perturbation (B). The waves approaching each 
other, superposing each other and then crossing each other is clearly observed. The lines connect 
the peaks of the fast and the slow waves in each perturbation. 
 
 
4.2 Controlling density perturbations on a beam: 
In most applications density or velocity perturbations on a beam is 
undesirable. So any method to eliminate or at least control the perturbation will be 
useful in preserving the beam quality. As an example, to clean a beam pulse before it 
is injected into a bend can prevent unwanted radiation. In this section, we discuss 
techniques for controlling a density perturbation using a energy modulation. 
 
 79 
 
 Several beam experiments were conducted at UMER, either by changing the 
perturbation strength while keeping the beam current constant or by changing the 
beam current while keeping the perturbation current constant. All these experiments 
showed a trend: space-charge waves superpose and cross each other just like linear 
waves would. This is because of the dispersionless property of the space-charge 
waves. The waves behave as if they are independent of each other and hence 
superimpose and cross each other without a change in shape. Since linear waves 
superimpose each other constructively, they can also superpose destructively. Thus, if 
a space charge wave interacts with another space charge wave of the same amplitude 
and pulse width but different polarity, then the waves will cancel each other. This 
principle can be used to cancel the propagation of a fast or a slow wave.  
In order to test this principle, introduced a density modulation to create a 
positive slow wave and then we used an energy modulation to create a fast wave with 
negative polarity. By adjusting the timing, amplitude, and the pulse width of the 
modulation carefully, we were able to suppress the propagation of the fast wave.  
We show how energy modulations can be used to control density modulations 
in a beam in the following section. The basic idea behind the technique can be 
described as follows: the electrons in the fast or slow waves are travelling with a 
velocity v0+Cs or v0-Cs in a density modulated beam, respectively, where v0 is the 
beam velocity. Adding energy modulation to such a beam creates velocity waves that 
interfere constructively (depending on the phase) with the velocity waves produced 
by the density modulation. By appropriately choosing the amplitude and the phase of 
the energy modulation, we can cancel the fast or the slow wave. The perturbation 
beam current becomes zero. We call this coherent pulse flattening. This technique 
 
 80 
 
assumes that the density perturbation is small. In other words, the perturbation is in 
the linear regime and the waves are not dispersing. Also, the time scale for applying 
the correction should be small compared to the time scale for the waves to phase mix. 
In other words, the beam should have low incoherent energy spread. 
We show in the next section how an energy modulation can produce current 
waves of opposite polarity [30] and how waves in a density modulated beam can be 
controlled using energy modulation. 
4.2.1 Coherent pulse flattening and 1-D cold theory: 
From cold-fluid theory, an initial velocity perturbation of the form: 
1 0( 0, ) ( )v z t v h t?= =  and a density perturbation of the form: 
( ) ( ) ( )1 1 0, 0,i z t i t i h t?= = , where the h(t) is the shape of the velocity perturbation 
with unit amplitude ,? is the strength of the velocity perturbation and ? the strength of 
the density perturbation, evolves according to the following linearized equations [56]. 
 
   
0 0
1
0
0 0
0
( , ) ( )2
( )2
s
s s
s
s s
i v c zi z t h t
c v c
i v c zh t
c v c
? ? ? ?
? ? ? ?
? ? ? ?= ? ? + ? +
? ?? ?? ? ? ?
? ? ? ?+ + + ? ?
? ?? ?? ? ? ?
 (4.1) 
 
   0 01
0 0
( , ) ( )2
( )2
s s
s s
v zz t h t
c c
v zh t
c c
?? ? ? ?
? ? ? ?
? ? ? ?= ? ? ? +
? ?? ?? ? ? ?
? ? ? ?+ + ? ?
? ?? ?? ? ? ?
 (4.2) 
 
 
 81 
 
      
0
1
0
0
0
( , ) ( )2
( )2
s
s
s
s
v c zv z t h t
v c
v c zh t
v c
? ? ?
? ? ?
? ? ? ?= ? ? +
? ?? ?? ? ? ?
? ? ? ?+ + ? ?
? ?? ?? ? ? ?
 (4.3) 
 
 
First of all, we notice that if ?=0, then the current exhibits a pair of space 
charge waves with opposite polarity. This shows that a pure energy modulation 
creates a pair of space charge waves in beam current which has opposite polarity. 
Let us suppose we want to cancel the slow wave (or fast wave) through energy 
modulation. By setting the first or second term, respectively, to zero in eqn. (4.1), we 
get: 
    
 
0
0
0
( ) 0s
s
s
s
v c
c v
c
v c
? ? ? ?
? ?
+ ? =
? ?? =
? ?? ?
?
?
 (4.4) 
Where the + sing is for the fast wave and ? sign is for the slow waves. So, for a given 
strength of the density modulation (?), a particular value for the strength of the 
velocity modulation (?) calculated from the eqn (4.4) can cancel the fast wave. To 
cancel the slow wave we set the second term of the eqn (4.1) to zero and again we get 
eqn (4.4) but with a sign change in the denominator.  
 By modulating the charge, we get waves in velocity and current. Similarly by 
modulating velocity we get waves both in charge and current. Therefore, by 
simultaneously introducing a density and a velocity wave, we can control the current 
 
 82 
 
perturbation on a beam. The value of  ?  to cancel a given density modulation 
depends on both  ?  and the sound speed (Cs), and therefore depends on the beam 
current (I) and beam energy as well. We plot the values of ? as a function of ? and 
Cs for a 10-keV beam. This is shown in the Figure 4.6 . As expected, a larger current 
with a stronger density modulation requires a stronger energy modulation to cancel 
the density perturbation.  
 
 
Figure 4.6: Graph showing the strength of velocity modulation (? ) required to cancel density 
modulation of initial strength (?) for different UMER beam currents at 10 keV. A high current 
beam with a larger perturbation needs a larger energy modulation.  
 
 
 83 
 
4.2.2 Analytical Modeling of Coherent pulse flattening using 
MATLAB: 
We used a MATLAB to numerically solve eqns (4.1), (4.2) and (4.3) to 
observe how a density modulated beam behaves when it is energy modulated. We 
assumed a Gaussian shaped perturbation with a full-width half maximum of 5ns. 
First, we show what happens when a beam starts with a pure density modulation. 
Next, we show what happens when a beam starts with a pure energy perturbation and 
show that a fast and a slow wave of opposite polarity are generated. To get a clearer 
view, we show the evolution of all the quantities, charge, velocity and current with 
respect to time. Finally, we superimpose both the density and the energy modulation 
to show how, by choosing the correct strength (? ) for the energy modulation from 
equation (4.4) , one of the waves could be cancelled. We note that the energy 
modulation is applied at a distance of RC4 (z =3.8 m) downstream. 
In the simulation, we use a 20 mA beam with a 10% density modulation 
(?=0.1). We begin the simulation with no energy modulation (? =0). The results are 
shown in the Figure 4.7. We then set ?=0 and set ? =0.0022, a pure velocity 
modulation. The results are shown in Figure 4.8 . Finally, we set ?=0.1 and 
? =0.0022 , so that the beam starts with a density modulation and then a velocity 
modulation is applied. The results under this condition are shown in Figure 4.9 
illustrating the cancellation of the fast wave but the amplitude of the slow wave is 
doubled. By reversing the polarity of ? , we can also cancel the slow wave. This is 
shown in Figure 4.10. We used three colors in the simulation to indicate the 
following:  the blue color indicates a beam with just a density modulation, the red 
 
 84 
 
color indicates a beam with just a velocity modulation and the black color sums the 
two beams to bring out the effect of the modulation. We add them due to the linearity 
of fluid equations in the 1-D cold fluid theory.
 
 85 
 
 
 
 
Figure 4.7: (Landscape) MATLAB Simulation showing the effect of velocity modulation on a beam. 
The beam current is 20mA and the velocity modulation is 0.2%. In the current space, the fast and 
the slow wave have opposite polarity. 
 
 86 
 
 
Figure 4.8: (Landscape) ) MATLAB Simulation showing the effect of density modulation on a beam. 
The beam current is 20mA and the density modulation is 10%. In the current space, the fast and the 
slow wave have same polarity, while they have opposite polarity in the velocity space. 
 
 87 
 
 
Figure 4.9:  MATLAB Simulation showing the cancellation of the fast wave. The beam current is 
20mA and the density modulation is 10% with a velocity modulation of 0.2%. Though the fast wave 
is cancelled, the amplitude of the slow wave increases. 
 
 88 
 
 
Figure 4.10: (Landscape) MATLAB Simulation showing the cancellation of the slow wave. The beam 
current is 20mA and the density modulation is 10% with a velocity modulation of 0.2%. Though the 
slow wave is cancelled, the amplitude of the fast wave doubles. 
 
 
 89 
 
4.2.3 Experimental cancellation of density modulation by controlled 
energy perturbation: 
In order to test the prediction from analytical calculation, we injected a 7 mA 
beam with a density perturbation of duration 8 ns. Because the width of a single laser 
pulse is 5 ns, we combined two laser pulses (from two similar laser sources) and 
created a single 8ns density perturbation. We injected the beam into the UMER ring, 
as shown in Figure 4.11. Next, the lasers were switched off and we introduced an 
energy modulation on the beam. The beam after the energy modulation is shown in 
the Figure 4.12.  After making the timing adjustment to make the energy modulated 
waves occur at the same time as the fast (or slow wave) from the density perturbation, 
the density perturbation was turned on. The amplitude of the energy modulation was 
increased to cancel the fast wave as shown in the Figure 4.13. It also shows that the 
slow wave is not propagating as predicted by the simulation. We then repeated the 
experiment by changing the polarity of the energy modulation, to cancel the slow 
wave. We repeated the experiment for the 23 mA and 1 mA beam. In each case, the 
fast wave was cancelled and the beam carried only the slow wave. 
These experiments show that when a controlled energy modulation is 
introduced on a density modulated beam and when the shape and the timing of the 
modulation is matched with the original perturbation, coherent pulse flattening 
occurs. In the experiment, we apply an external voltage to the beam ( appliedE ) to the 
beam. In order to compare the experimental values   appliedE  with the theoretical 
 
 90 
 
values of ?  in equation (4.4) , we use the following conversion in the non-relativistic 
limit: 
 
 
2 2
1 0 0 1 1 0
2
21 1 0 01
2 2
10 0
2
0.5 ( ) 0.5 0.5 ( 2 )
0.5 ( 2 ) 1 2 2
0.5
2 0
2 4
applied
E m m m
mE
E m
E
E
EE
E E
? ? ? ? ? ?
? ? ? ?? ? ?
?? ?
? ?
?
? = + ? ? +
? ?+? = = + = +
? ?? ?
?? + ? =
?? = =
 (4.5) 
E?  is the coherent energy spread for a single wave and E is the beam energy. We 
note that the factor of two arises because when the wave splits, the strength of the 
energy modulation is halved. 
 
Table 4-1: Table comparing of the velocity modulation strength predicted by theory to cancel the 
perturbation and the experimental values. 
Beam 
current 
 
? ?  (theory) 
2
appliedEE? =    (V) ?  (Experiment) 
1 mA 0.06 4.6e-4 20 5e-4 
7 mA 0.05 8.2e-4 40 1.0e-3 
23 mA 0.1 0.026 160 0.040 
 
 91 
 
 
 
 
 
Figure 4.11: Turn-by-Turn plot of the UMER beam (7 mA) with a density perturbation (8-ns). The 
perturbation width is 5ns from a single laser. Therefore, two lasers were combined to get an 8 ns 
density perturbation. The fast and the slow waves have the same polarity. 
 
 
 92 
 
              
Figure 4.12: Turn-by-Turn plot of the UMER beam (7 mA) with a velocity perturbation (8-ns). The 
fast and the slow waves have opposite polarity. This property allows the velocity perturbation to 
cancel a density perturbation, which generates a fast and a slow wave but with the same polarity. 
 
 93 
 
 
Figure 4.13: Turn-by-Turn plot of the UMER beam (7 mA) with a velocity perturbation (8-ns) and a 
density perturbation (8 ns). Only the slow wave is propagating on the beam. The fast wave of the 
density modulation is cancelled by the fast wave of velocity modulation because of opposite 
polarity while the slow waves, which are of same polarity, adds up. 
 
The experimentally determined value ?  for different beam current for 
different perturbation strength is listed in Table 4-1. As the Table 4-1 shows the value 
of the voltage need to cancel a perturbation increase with the strength of the initial 
density perturbation. The discrepancy between the theory and the experiment is 
possibly due to variation of beam size inside the perturbation. 
In our experiment we used a coasting UMER beam with 10-keV with no 
acceleration. Accelerating the beam would change the beam energy and hence reduce 
the required value of ?   for cancelling the density modulation. When a low energy 
 
 94 
 
beam is accelerated, Cs decreases and v0 increases, causing the ratio of 0sC ?   
decrease substantially. Therefore if a density-modulated beam is accelerated quickly, 
the energy modulation required to cancel the density modulation will be reduced from 
eqn (4.4). Ongoing work on acceleration [57] will clarify the timescale needed to 
perform the cancellation experiment in UMER. 
4.2.4 Limitations of the coherent pulse flattening technique: 
There are some limitations in the coherent pulse flattening experiment 
demonstrated above. In principle, either the fast or the slow wave can be cancelled 
but not both of them. This is because UMER, as of writing, has only one energy 
modulator to cancel either the fast or the slow wave. By adding another energy 
modulator we can, in principle, cancel the other wave. Also, if we start with a pure 
fast or a pure slow density wave, we can use the coherent pulse flattening technique 
to cancel the wave. Methods to generate a pure fast or slow density wave in a beam 
has been investigated before [25]. In UMER, out of the three glass gaps, currently one 
of the glass gaps is used as an induction module to introduce the energy modulation 
on the beam while the second glass gap is used as a current monitor. We still have 
one glass gap left which could be used to cancel the remaining wave in the beam. 
Also, in our experiment when the density modulation was in the linear regime and the 
energy modulation was applied in a time scale of nanoseconds. This assumes that the 
waves do not disperse appreciably and phase mix with other particles. In reality, the 
timescales involved are short and the energy modulation might have to be applied 
before the wave becomes nonlinear and dispersive.  
 
 95 
 
In this technique, the coherent energy spread of the beam increases 
substantially because while one of the waves is cancelled, the amplitude of the 
remaining wave is increased. A good technique would be to measure the energy 
profile created by the density perturbation and then apply an energy modulation that 
exactly negates the profile. However, this assumes we can measure the energy profile 
of the beam on an appropriate timescale/resolution which can be a limitation in bigger 
machines. We therefore discuss another technique for controlling the space-charge 
effect of perturbations on the beam.      
4.2.5 Experimental results of the perturbation-compression scheme: 
When the electron beam is injected into the UMER ring without longitudinal 
focusing, the beam ends expand at a speed of 2Cs until the beam fills the ring. This is 
primarily due to longitudinal space-charge forces, which defocus the beam 
longitudinally. Hence, in order to contain the beam in the longitudinal direction, 
appropriate longitudinal electric fields must be applied repeatedly. This is done in 
UMER presently through the induction cell. Like the beam ends, the perturbation also 
expands and splits into a fast and a slow wave when the beam is allowed to freely 
expand. So, by longitudinally compressing the perturbation part of the beam 
periodically, we can contain the perturbation from splitting into a fast and a slow 
wave. We show experimental results of the perturbation compression in the following 
section.  
We injected a 23 mA beam with a 10% density perturbation into a ring and 
applied no longitudinal focusing. This is shown in Figure 4.14 with a dark blue curve. 
We then applied longitudinal electric field at RC4 with varying strength, to contain or 
 
 96 
 
?compress? the perturbation. We tracked the beam current for each turn in all theses 
cases at RC10. The results of the experiment are shown for up to 4 turns in Figure 
4.14. Every color within a sub-plot shows the beam current for a specific value of 
compression voltage. As the applied voltage is increased, the waves become closer or 
in other words, the wave splitting is delayed.  Due to the limitation in the apparatus, 
we applied the longitudinal focusing only for a single turn. Because of the circular 
layout of UMER, we can in principle apply the same longitudinal focusing field 
periodically for every turn, and thus ?hold? the perturbation without splitting. This 
technique is very similar to the previously discussed coherent pulse flattening 
experiment except here we introduce the energy waves at the very beginning before 
the wave splits. This allows us to delay the perturbation from splitting. We 
summarize this technique in the next section. 
 
 
 97 
 
 
Figure 4.14: Experimental results of perturbation compression for a 23mA beam current with a 10% 
density perturbation. The sub-graphs show the effect of compression on every turn for different 
values of electric field strength. The perturbation, when compressed, takes a longer time to evolve 
into a slow and a fast wave. For the case shown, the perturbation was applied only once in the first 
turn at RC4. 
4.2.6 Perturbation compression scheme:  
We start with the observation that it is because of space-charge that the 
density perturbations are converted into energy modulation, which can then create 
problems when the beam goes through a bunch compressor. So, we want to prevent 
the density perturbation from splitting into a fast and a slow wave. We do this by 
compressing the density perturbation near the gun/injector by applying a linear 
electric field which prevents the perturbation from splitting. This is similar to ear-
 
 98 
 
fields applied at the beam ends for longitudinal focusing, but we apply them for the 
perturbation alone.  
Initial experiments in UMER using this approach show that by applying ear-
fields to a perturbation, we can control the wave speed. If a density modulation is 
compressed before it splits into fast and slow waves, we have in effect compensated 
for the space-charge mechanism. We are then left with a space-charge dominated 
beam with a density modulation that remains as a single density modulation and does 
not split into fast and slow waves as long as the compression is applied periodically. 
 If the beam is now accelerated to relativistic energies, particles within the 
perturbation are travelling with approximately the same velocity as that of the other 
particles. The space-charge effects were compensated for at the beginning. So, when 
such a ?space-charge compensated? beam goes through a dispersive section we have 
no additional density modulation because the effect of dispersion on a beam with 
uniform energy is same for all particles. There are some limitations in this technique. 
First, the beam will still have a density bump which could lead to coherent radiation 
and thus start an energy modulation again. Secondly, it is not clear how does the 
perturbation evolve under compression and acceleration combined. Future 
experiments in UMER involving acceleration will give us some clues to understand 
these limitations. 
 
4.3 Non-linear space charge waves 
Previous experiments by using a localized, single density perturbation showed 
that the wave speed depends on the strength of the perturbation. The higher the 
 
 99 
 
relative strength of the perturbation compared to the main beam, the faster the waves 
propagate. This is a general property of non-linear wave propagation. We also know 
from the physics of nonlinear waves that non-linear waves are prone to wave-
breaking [58]. Wave-breaking occurs when the crest of the wave travels faster 
compared to the other part of the wave leading to steepening and eventually breaking. 
It is a phenomenon most commonly observed in ocean waves. Though the problem of 
wave-breaking is mathematically involved, we extend the fluid equation presented in 
the previous chapter to understand some elementary ideas in non-linear wave 
propagation. The original dispersionless, linear equation for the space-charge waves 
can be written in the form, 
                                          
1 1 0
sC z t
? ?? ?+ =
? ?                                                                                 (4.6)  
The solution of equation  (4.6) can be written as 1 ( , ) ( )sz t f z C t? = ?  where Cs is 
the wave velocity and ?1 is the line-charge density of the perturbation. From 
experiments, we know that the velocity of the waves is dependent on the amplitude 
i.e. 0 1( ) ( )sC G G?? ?= = , where  ? is the strength of the perturbation and ?0 is 
the line-charge density and let G be a corresponding function between  the 
perturbation strength and the wave speed. Keeping the dispersionless transport 
assumption but including the nonlinear behavior, the solution changes to 
1 1( , ) ( ( ) )sz t f z C t? ?= ? . As shown in the appendix A, as the waves of the form 
1 1( , ) ( ( ) )sz t f z C t? ?= ?  propagate, they steepen and then break. An intuitive way 
 
 100 
 
to understand this would be to think of the apex of the wave travelling faster than the 
base of the wave causing the apex to overtake. At this point, the wave curls like a 
large ocean wave. Now, the wave becomes multi-valued at a certain time due to 
steepening and we say that the wave breaks.  
4.3.1 Experiment showing wave train formation: 
                         We introduced a relatively large perturbation (?=0.5) compared to the 
main beam to test wave-breaking in space-charge waves. The beam current was 36 
mA and the perturbation current was 17.5 mA. For clarity, the perturbation was 
introduced near the edge so that either the slow wave or the fast wave alone 
propagates along the whole length of the beam. The turn-by-turn profile of the beam 
current is shown in Figure 4.15.  
 
 101 
 
 
Figure 4.15: Turn-by-turn plot of the beam current (36 mA) with a 17.5 mA perturbation showing 
wave train formation in the third turn. As the turn increases, the number of waves increases. The 
perturbation is introduced near the tail of the beam to allow the fast wave to spend a longer time 
in the flat-top portion of the beam. 
As seen from Figure 4.15, the perturbation introduced near the tail splits into a 
fast and a slow wave. The fast wave propagates along the beam toward the beam head 
while the slow wave rolls off the edge of the beam tail. The fast wave starts to steepen 
and then starts to break into sub-pulses after second turn and continues to break in the 
subsequent turns as well. The distance between each peak is 2 ns (on average). The 
plasma frequency of the beam is approximately 400 MHz confirming that the sub-
pulses oscillate close to the plasma frequency. We show in the next few sections that 
these sub-pulses are due to the excitation of non-linear waves and wavebreaking. 
 
 102 
 
4.3.2 KdV equation, Burgers equation and Dawson limit: 
 We take a simplified approach to present how an initial perturbation can 
become steep and break [59]. We begin with eqn (1.20) discussed in Chapter 1. Let 
the initial condition be expressed as 
 0( , 0) ( )z t? ? ?= =  
where ? is the line charge density. We do a change of variable as z t? ?= ?  . 
Now, we take the temporal and the spatial derivatives to get, 
 
0
0 0
0
0 0
( )
[1 ]
( )
[1 ]
t t t
tt
z z z
tz
?? ? ? ?
? ?
? ?? ?
? ?
?? ? ? ?
? ?
? ??
? ?
?? ? ? ?= +
? ? ? ? ?
? ??? + = ?
? ? ?
?? ? ? ?= +
? ? ? ? ?
? ??? + =
? ? ?
 (4.7) 
Combining the two parts of equation (4.7) into equation, we get the following 
condition 
 0(1 ) ( ) 0t
t z
? ? ??
?
? ? ?+ + =
? ? ?
 (4.8) 
So if,  
 0(1 ) 0,t ?
?
?+ ?
?
 (4.9) 
then ( )z t? ??  is a solution to the Burgers equation. 
Let us take a concrete example to demonstrate that the Burgers equation with 
a smooth initial profile can lead to wave steepening and then wave breaking over 
 
 103 
 
time. Let    0 1( ) cos( ); ( , ) cos( ( ))z z z t z t? pi ? pi ?= = ? , then according to 
equation (4.9) we get, 
 1
sin( )t zpi pi?
 (4.10) 
This equation is satisfied for all  1 Bt t
pi< =
 , where Bt  is the breaking time. So, 
until Bt t< , the wave just propagates and steepens.  But at  Bt t= , the initial 
perturbation becomes steep and breaks.  
The summary of the previous discussion is that we have shown that once the 
1-D cold fluid theory is reduced to KdV equation, the KdV equation can then be 
simplified into the inviscid Burgers equation to explain the observation of wave 
breaking at large amplitude perturbation. We conclude this section by applying 
equation (4.9) to a parabolic shaped perturbation of the form: 
2
20 0( ) (1 )
m
zL
zz? ?= ? , where 0L?  is the peak line-charge density 
(amplitude) of the perturbation and zm is the maximum length of the pulse. 
Differentiating with respect to z and substituting we get the breaking time to be, 
 
0
0
constant2 mB L B
L
zt t?
?? ? =
 (4.11) 
This implies that for a perturbation of given width (5ns in our case) a larger line-
charge density perturbation can cause the wavebreaking to occur sooner than with a 
lower charge density one. This behavior is observed regularly in UMER beam 
perturbation experiments. In the non-linear regime, a larger perturbation causes the 
 
 104 
 
space charge waves to break earlier compared to a perturbation with relatively smaller 
amplitude. The phenomenon of wavebreaking was first predicted in cold plasmas by 
Dawson [60] and has been demonstrated to accelerate particles [61]. Even nonlinear 
longitudinal plasma oscillations are harmonic with the plasma frequency [62] , as can 
be seen from  Figure 4.15.   
 We note that we have ignored dispersion in all our calculation in trying to 
explain wavebreaking. Dispersion can become a critical factor, once the wave 
steepens. For example, when the wave breaks, the pulse width of the secondary waves 
are of the order of 2ns, which is 11.5cm long while the pipe diameter is 5.08 cm. 
When the ratio of the pipe diameter to the beam length becomes comparable, 
dispersion effect must also be included [31, 36]. 
 
4. 4 Summary: 
                      In this chapter we have experimentally shown how multiple 
perturbations propagate in a space-charge dominated beam. Space-charge waves 
behave like linear waves and so superpose and cross each other. We applied this 
principle to demonstrate the cancellation of a space charge wave through energy 
modulation. We also showed that by compressing the perturbation at the start we can 
prevent the perturbation from splitting into a fast and slow wave. Finally, we showed 
that when there is a large perturbation, space-charge waves behave like non-linear 
waves and can undergo wavebreaking.  
 
 105 
 
Chapter 5:  Longitudinal perturbations due to space 
charge effects in the gun 
 
Up until now, we have discussed the effects of space charge on perturbations 
over a long distance under different initial conditions.  In this chapter, we focus on 
studying space charge effects near the gun. We begin with the experimental 
observation of pulse splitting near the gun at higher laser power. Then we briefly 
discuss the 1-D virtual cathode theory (VC) and short-pulse Child-Langmuir limit 
(short pulse space-charge limit) to explain the sub-pulse formation. We conclude this 
chapter by discussing the relevance of such short pulse effects for photoinjectors. 
5.1 Pulse splitting near the gun 
  In UMER, it is possible to photoemit a short electron beam pulse using a laser, 
without any thermionic emission. Under such conditions, the beam current profile 
follows the laser profile up to a critical value of laser power. Above the critical value, 
the beam pulse starts splitting into multiple sub-pulses within the gun as shown in 
Figure 5.1.  
 
 
 106 
 
 
Figure 5.1: Single perturbation pulse splitting into multiple sub-pulses at (a) Laser generated 
electron beam (b) Pulse splitting into two sub-pulses (0.68mJ) (c) Three sub-pulses (1.3mJ) (d) Four 
sub-pulses observed at (2.7mJ). This measurement is taken at the Bergoz (placed 64 cm from the 
anode exit) in the injection line.  
 
Similar effects have been observed before in high current RF photoinjectors 
and other low energy electron guns [28, 29, 42]. We compare our observations with 
those theories and discuss the relevance of such effects for future photoinjectors. This 
effect is likely due to the formation of a virtual cathode. A virtual cathode is formed 
when the total injected current in the cathode exceeds the space charge limit of the 
gun.  We discuss the 1-D virtual cathode theory in the following section. 
 
 107 
 
5.2 Time-independent diode behavior 
The space-charge fields inside a diode can become large to form a virtual 
cathode.  In order to explain the formation of virtual cathode, we do a simple DC 
analysis of the diode under the absence of the applied field. This problem is well-
known and is treated in several papers on vacuum tubes [63, 64].  
Our goal in this section is to perform a DC analysis of a 1-D diode to see how 
and at what critical input current to a diode does the virtual cathode forms. We follow 
Birdsall and Bridges[65] in the subsequent derivation. 
 
Figure 5.2: A short-circuited diode showing the virtual cathode and the electron beam direction. 
The virtual cathode separated the gap into Region I and Region II. 
x 
z 
y 
X=0 X=d 
V0 
V0 
e- beam 
I II 
Virtual 
Cathode 
X= Xm 
 
 108 
 
    Let us assume two infinite (in y and z) parallel plates held at potential V0 and 
separated by a distance d.  This is shown in Figure 5.2. By writing the Poisson 
equation and continuity equation, and simplifying the equations give, 
                      1/2 ,1/2 1/20
0
2 4
( 2 / ) m
dV
dz
J V V
e m?
? ? ? ?
? ? ? ?? ?
? ?? ?
??=                         (5.1) 
where Vm is a constant of integration. Vm is the minimum potential inside the space 
between the cathode and the anode.  In order to proceed from here, we have to solve 
the above equation for two boundary conditions. They are (i) Vm >0 and (ii) Vm =0. 
(i) Vm > 0: 
                                   
The parameter? is defined as the following: 
 60 1/2
0
22.33 10
J
V d? ? ?= ? ?  (5.2) 
When ? =1, we simply get the Child-Langmuir limit. The solution of these equations 
for Vm >0 are reported elsewhere [65]. We used MATLAB to plot the potential 
minimum as a function of the input current ?  as shown in Figure 5.3.  
 
 
 109 
 
 
Figure 5.3: The plot shows the value of the normalized potential minimum as a function of the 
input current density. When the input current density exceeds 8, the potential minimum starts to 
reduce and then vanishes. 
 
We also plot potential minimum as a function of distance for a few values of the input 
current.  This is shown in Figure 5.4. 
 
 110 
 
 
Figure 5.4: The plot shows the potential profile inside a diode as a function of distance and the 
input current density. As the input current density increases, the potential profile sags until it 
vanishes when ? >8. 
 
From these plots, we can conclude that between 0<? <8, the potential minimum 
is symmetric about the center. For ? >8, Vm approaches zero, causing reflection of 
charges towards the cathode. At this point, the potential minimum becomes a virtual 
cathode. At the virtual cathode, both the potential and the electric field are zero, while 
at a potential minimum only the electric field is zero. 
 
Now we solve the equation (5.1) after the virtual cathode is formed. i.e. Vm =0.   
 
 
 111 
 
(ii) Vm =0: 
 
               When Vm =0, some of the electrons will be transmitted through the 
virtual cathode while others will be reflected back toward the cathode. Hence 
though the velocity of the electrons may change, the net current will remain 
unchanged. Let us assume that a fraction f of the injection current is transmitted 
and (1-f) is reflected. We can write, 
 
 
0 0 0
0 ,
(1 ) (2 )
I
II
J f J f J
v v v
fJ
v
?
?
? ? ?= + =
?
=
 (5.3) 
where I and II refers to the regions before (0< x< xm) and after (xm < x< d) the 
virtual cathode respectively.  This is shown by a red dotted line in Figure 5.2. Figure 
5.6 plots the fraction of the input current that is transmitted as a function of the input 
current.   
 
 112 
 
 
Figure 5.5: The plot shows the variation of the potential profile in the region before the virtual 
cathode (Region I). After the virtual cathode is formed, increasing the injected current moves the 
virtual cathode closer toward the cathode causing the output current to follow the Child-Langmuir 
limit. 
 
 
 113 
 
 
Figure 5.6: The plot shows the transmitted current as a function of input current after the virtual 
cathode is formed. Once the virtual cathode is formed, complete transmission occurs only when 
the value of input current (?) is reduced to 4. 
 
The minimum value of  ?  for which all the current is transmitted without 
reflection is ? =4. When ? >4, the virtual cathode moves towards the cathode and 
for large values of ? , the m? =0 and we get the Child-Langmuir limit. This is shown 
in Figure 5.5 for some potential profiles as a function of the input current? .  
 
  So, to summarize, we have the range of currents 4< ? <? for Vm=0 and the 
range 0<? <8 for Vm >0. Both of these solutions can cover the entire range of input 
currents. Next, we will see how the classical behavior of the diode can be explained 
using these two solutions. Figure 5.7 shows the transmitted current as a function of 
the input injected current. 
 
 114 
 
    
 
Figure 5.7: The diode output current as a function of the input current. The output current follows 
the input current up to a critical value of the input current after which there is a sudden drop in the 
output. The path CD constitutes the virtual cathode formation. 
As ?  is increased from 0 to 8, there is complete transmission of the injected 
current shown by the points A, B and C.   If the input current is increased beyond 
? >8, there is a drop in the transmitted current to the point D. Any further increase in 
the input current causes the transmitted current to decrease and it asymptotically 
reaches the Child-Langmuir limit (point E). Now, if we then decrease the input 
current, the transmitted current increases through the point (E, D and F). After 
reaching the point F, it follows the original curve through B and C. Note that, though 
 
 115 
 
the point B and F are transmitting the same current, the point F has a virtual cathode 
in the potential profile while point B doesn?t. (Refer Figure 5.5).  
The jump from the point C to point D cannot be captured by the classical DC 
analysis. We have to do a transient analysis to understand the changes in the diode 
region between those points. Since a complete AC analysis is beyond the scope of 
this work, we refer time-dependent analysis to works by Birdsall and Bridges[65] and 
references therein. 
5.3 Potential minimum oscillation 
When the current diode is increased beyond the ? =8, a steady oscillatory 
solution is found for both the value and the position of the potential minimum [66, 
67]. A schematic of this behavior is shown in Figure 5.8. 
 
 116 
 
 
Figure 5.8: The plot shows that the virtual cathode after it forms is not stable. It oscillates in 
position as well as in amplitude. The blue curve plots the value of the normalized potential 
minimum and the red curve plots the normalized position of the potential minimum. 
 
Figure 5.8 shows that not only does the value of the potential minimum 
oscillates but the position oscillates as well. This means that the number of electrons 
crossing the virtual cathode to reach the anode will also vary with respect to time. 
This cycle can be explained as follows: Initially, all the electrons injected from the 
cathode reach the anode. Once the charge density becomes high inside the diode, a 
virtual cathode is formed. When a virtual cathode is formed, some of the electrons get 
reflected back to the cathode while others pass through the virtual cathode to the 
anode. Due to this reflection, the number of electrons near the virtual cathode 
decreases, causing the potential to increase and hence the potential minimum shifts 
 
 117 
 
again. This causes more electrons to pass through the virtual cathode and the cycle 
repeats again. In our experiment, the laser creates a virtual cathode inside the diode 
causing the sub-pulses to form. Since, the virtual cathode formation depends on the 
electron density which increases with laser power, the larger the laser power, the 
greater the number of sub-pulse formation. 
                      We have discussed why the virtual cathode forms when the Child-
Langmuir limit is exceeded. Now, we will look into how we can exceed the Child-
Langmuir limit when the pulse duration is short compared to the transit time in the 
diode. 
5.4 Classical Short ? Pulse Child-Langmuir Law (SCL) 
         We follow Valfells [28] to derive the short-pulse Child-Langmuir limit using a 
single sheet model. A much more accurate approximation is the equivalent diode 
approximation which was also done by Valfells [28]. We assume a planar 1D diode 
with a transit time of TCL in the Child-Langmuir limit. 
Single Sheet Model 
 
In this model, we treat the electron pulse as a sheet of charge with charge 
density ?. When there are no electrons in the gap, the electric field in the gap is gVD , 
where Vg is the gap voltage and D is the separation of the electrodes. Once the sheet 
is injected, the electric field changes by 
0
E ??? =  and at the virtual cathode, the 
effective electric field becomes zero. So, at the virtual cathode, the charge density is 
0 gV
D
?? = . Since we are treating the pulse as a single sheet of charge, we can write 
 
 118 
 
pJ? ?= , where J is the injection current density and p?  is the pulse length. An 
important condition in this is that the pulse length is short compared to the transit-
time i.e. p?  < TCL.  So, combining these equations, we can write JSCL the maximum 
current density in the short-pulse limit, before the virtual cathode is formed as: 
 
0
SCLJ
g
p
V
D
?
?=  (5.4) 
From the classical Child-Langmuir law we know that, 
 
3/2
0
2
4 2
9
3 / 2 /CL
g
CL
g
VeJ
m D
T D eV m
? ? ?? ?
? ?? ?
? ?
=
=
 (5.5) 
So, eqn (5.4) can be rewritten as, 
   
 ,34 CLSCL
CL
JJ
X=  (5.6) 
where pCL
CL
X T?= is the ratio of the beam pulse duration to the transit time of the 1-D 
planar diode in the Child-Langmuir limit. Equation (5.6) indicates that when XCL 
<<1, the current density (JSCL) can exceed the 1-D Child-Langmuir limit JCL.  
We note that the planar 1-D Child-Langmuir was extended to 2-D by 
Lunginsland [68] using PIC simulation and shown to be monotonically decreasing 
function of W/D where W is the width of the emitting strip and D is the gap 
separation according to, 2 1D CL
W
D
CJ J? ?? ?= +
? ?? ? , where C is a constant depending on the 
shape of the emitter and W is the characteristic length of the emitter. Valfells 
 
 119 
 
experimentally estimated the dependence of the current density on the emitter area to 
be 1.81critJ R?? . From both Valfells? estimation and the single-sheet model, we 
can conclude that the total charge emitted (without virtual cathode formation) is 
weakly dependent on the pulse-length and depends on the charge-density of the 
emitter material and the surface electric field. i.e., pQ J E? ?= = .  From (5.6) we 
can write the total charge in the short-pulse limit as a function of the total charge in 
the Child-Langmuir limit as  
 ,3 4CLSCL QQ =  (5.7) 
where QCL is the charge emitted in the Child-Langmuir limit and QSCL is the charge 
emitted in the short-pulse limit before virtual cathode formation.  
In summary, in the short-pulse limit with no virtual cathode formation, the 
current density can exceed the Child-Langmuir limit, but we cannot exceed the limit 
on total charge in the pulse. Therefore, future photoinjectors where XCL<<1 should be 
operated with bunch charges well below QSCL to prevent virtual cathode oscillations 
on the electron beam. 
 
  
 
 
 
 
 
 
  
 
 120 
 
5.5 Explanation for the experimental observations 
In this section, we explain the observation of pulse splitting in the UMER gun. 
The laser generated electron beam has a rise time of 1 ns which is comparable to the 
0.75 ns transit time of the anode-cathode gap under the Child-Langmuir limit in 
UMER. When the laser is switched on, the measured value of charge extracted from 
the gun before the splitting (the critical point) is, 56crit peak FWHMQ I T pC= = , where 
peakI and FWHMT are the peak beam current and FWHM (Full Width Half-Maximum) 
of the beam pulse obtained from Figure 5.1. critQ  is the value of charge extracted 
before the pulse splits further. 
The maximum current density when the UMER gun is in the Child-Langmuir 
limit is 23.73 kACL mJ = . The corresponding charge 2 84.4CL CL laser laserQ J r t pCpi= = , where 
lasert = 5 ns is the FWHM of the laser and laserr =1.2 mm is the laser spot radius    The 
ratio of 0.66crit
CL
Q
Q = . From theory, at very short pulse-duration injection, the 
maximum charge that can be obtained before the onset of virtual cathode oscillations 
is 
max
0.75critQQ = , which is in good agreement with the measured value of 0.66.  
The oscillation    frequency    of    the    virtual    cathode    is    expressed    in 
terms    of    a    planar    diode    peak    current    density    as  2( )( ) 10.2 kAcmJf GHz
??=
 
[69]. For  a 10 keV electron beam  in  UMER ( ?? =0.2)  with a  photo-emitted beam  
current  of  7 mA   and  a radius of  1.2 mm, 1splitt f= , where splitt  is the time period 
of the virtual cathode oscillation. The time period for a 7 mA beam of radius 1.2 mm 
 
 121 
 
is in agreement with the observed experimental value of 2.5 ns. Since ??  is varying 
inside the gun, the value of ?? = 0.1 is taken for calculating the frequency of 
oscillation.  
The above experiment indicates that though it is possible to extract more 
current from a cathode using a short pulse, there is a critical current density above 
which the pulse starts to develop unwanted temporal structure on it. Thus, it is 
important to operate below the critical current density in a short-pulse injection 
scenario.  
5.6 Extending the results for a high gradient photoinjector 
 
          In the case of photoinjectors, which remain the primary choice for electron 
sources for next generation machines, it is important to consider effects introduced 
due to transient behavior owing to the very short pulse length compared to the anode-
cathode gap.  
Let us take an example of a high-gradient gun (e.g. LCLS gun [9]) and 
calculate the maximum charge the gun can handle: maxmax
0
EQ
A?= , where A is the 
cathode emission area with maxE  = 120 MV/m. Let the emission radius be 1.2mm, 
so max 4.8Q nC= . This is the maximum charge that can be extracted from the cathode. 
For the beam inside the RF-cell, f? = 6 for 2.5 MeV and AE =60 MV/m. Assuming a 
10ps laser pulse, the transit time 211 0.16 f
A f
mcT ns
eE
?
?= ? = , where f?  is the 
 
 122 
 
relativistic factor at the gun exit and AE is the average applied field. The transit time 
is much longer than the 10ps. Hence, short pulse effects are important in such a gun.  
From 1-D virtual cathode theory, at very short pulse-length injection, the 
maximum charge that can be obtained before the onset of cathode oscillations is 
max0.75 3.6critQ Q nC= = , which is much less  than the maximum obtainable value of 
4.8 nC. A typical gun (e.g. LCLS gun) uses a Cu cathode with a 255nm laser at 
500 J?  and a pulse length of 10 ps. So, ( ) ( ) (%)( ) 108.2 A124nm Power W QEI A ?= =  
which amounts to a charge of 1 nC, which is well below the critical value.  This is a 
good limit to operate for a generating a high quality beam. Any efforts to operate the 
gun at a much higher charge should then take the space charge effects into 
consideration. 
 
5.7 Limitation of the theory 
There are some limitations in the above derivation of the single sheet theory. 
Firstly, it ignores the transverse effects such as beam size, which depends on the 
critical current density and hence on the total charge. When the beam does not fill the 
diode region, the beam is all-ends and it is essential to take into account the complete 
3-D effects inside the gun to calculate the charge limit. Secondly, our calculations for 
the high-gradient gun overlooked acceleration that occurs inside the gun. 
Acceleration reduces the effects of space-charge. So, a 3-D simulation taking both 
space charge and acceleration into account will give a better understanding of the 
physics near the gun.  
 
 123 
 
5.8 Conclusion 
In summary, we have seen how space charge effects in the gun can lead to 
undesirable modulation on the beam pulse through space charge. The self-fields of a 
bright beam can initiate density modulations as well as energy modulations by 
splitting the beam pulse. In the case of modern photoinjector generating bright and 
ultra-short beams, the critical current density for such effects to start is much lower 
than the maximum current density achievable. Hence to prevent such behavior and 
maintain beam quality, the gun should be operated well below the space charge limit. 
This also indicates that there might be a space-charge imposed limit to the beam 
brightness that can be obtained from photoinjectors. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 124 
 
Chapter 6:  Conclusion and Future work 
 
In this chapter, we summarize the dissertation work and conclude by 
suggesting some ideas for possible future work. 
6.1 Summary and Conclusion 
  We started by using photoemission from a laser to produce a controlled, 
localized density perturbation on an intense beam. We showed that the density 
perturbation evolves as fast and slow space charge waves. We then calculated the 
wave speed for different beam currents at different perturbation strengths. We showed 
that we have good agreement with both the 1-D theory and PIC simulation (WARP) 
in the linear regime. We then proceeded to investigate experimentally the effect of 
multiple perturbations on a beam. We did this by adding another laser to generate a 
density perturbation. We found that multiple perturbations behave just like linear 
waves ? superposing and crossing each other. We exploited this property to control a 
density modulation by applying an energy modulation to cancel either the fast or the 
slow wave, thus showing that modulations can be controlled.  
In the non-linear regime, where the strength of the perturbation is large (>25% 
compared to the beam current), we observed wave breaking of the fast and slow 
waves. We also showed that linear space-charge waves can be used as non-
interceptive transverse beam diagnostics in UMER. We did a time-sliced imaging of 
the perturbation and showed that a density perturbation can become an energy 
modulation and, through dispersion, affect the beam centroid.  
Finally, we explained the observation of multiple sub-pulse formation near the 
cathode above a critical value. We extended the analysis for a high-gradient gun to 
 
 125 
 
show that there might be a practical limit on beam brightness that can be obtained 
from photoinjectors determined by space-charge.  
From this work, we conclude that density modulation on an intense beam can 
produce fast and slow waves, which can be controlled through energy modulation in 
the linear regime. Moreover, a large amplitude density modulation, when allowed to 
propagate, can break into sub-pulses, causing energy modulation. Hence, a density 
modulation should not be allowed to grow and must be controlled at the very 
beginning. Such effects can also occur near the gun, where high space charge can 
break a density modulation into multiple sub-pulses. This can be prevented by 
operating the gun below the space-charge limit. 
6.2 Suggestion for future work 
This dissertation focused on the effect of a density perturbation on a coasting 
beam without any longitudinal focusing. This work can be extended in three possible 
ways: long distance experiments, short-pulse experiments and parabolic pulse 
transport. 
In the long distance experiments, the first step would be to investigate the 
effects of a perturbation with the longitudinal focusing turned on. This will answer 
the question on what happens to the fast and slow waves as they reach the beam ends. 
Secondly, while this thesis demonstrated cancellation of either a fast or a slow wave, 
by adding one more induction module in UMER it should be possible to cancel both 
the fast and the slow wave. Moreover, by loading the induction gaps, the effect of 
impedances on the waves can be studied.  
 
 126 
 
In addition, there are still unresolved issues in the evolution of a large 
perturbation due to non-linear behavior. In the large amplitude limit, PIC simulation 
of pulse train formation in the non-linear regime should be investigated. 
Understanding the dispersive behavior of the space charge waves can give clues about 
the collective effects in a space-charge dominated beam. Also, extending the 1-D cold 
fluid theory to include large amplitude perturbation will answer some unresolved 
issues in the large amplitude limit to get better agreement between experiment and 
theory. By varying the wave speed such as by beam acceleration or beam 
compression, further understanding might be achieved about the wave breaking 
phenomenon. The velocity space of a density perturbation can be measured by 
installing an energy analyzer in UMER.  
In the short-pulse limit, the effect of transverse beam size on the critical 
current density needed to start virtual cathode oscillations can be studied.  This can be 
done by focusing the laser beam to different spot sizes and measuring the critical 
current density that triggers virtual cathode oscillations.  
A rich set of experiments can be done by using the 5-ns, parabolic beam pulse 
generated by photoemission from the laser. Since such a beam is ?all ends?, without 
any flat-top, it is realistic and is similar to the ultra-short beams generated in modern 
accelerators. Thus, studying the evolution of a space-charge dominated parabolic 
pulse with focusing and compression is of direct relevance to other machines. 
 
 
 127 
 
Appendix- A 
In this appendix, we show that equation of the form ( ) 0u uc u
t x
? ?+ =
? ?
 with the 
initial form 0 ( ) ( ,0) ( )u x u x f x= =  and the solution ( , ) ( ( ) )u x t f x c u t= ?  is 
prone to wave breaking using a simple example of a triangular wave. The piecewise 
linear initial data (a triangle)  is shown in the figure below can be expressed as: 
0
0
         0 1
( ) (2 ) 1 2
0 0; 2
u x x
f x u x x
x x
< <?
?= ? < <?
? ? ??
 
By changing the frame of reference, by substituting x t? = ?  we get: 
0
0
( - )         0 1
( , )= (2 ) 1 2
0 0; 2
u ut ut
u t u ut ut
x x
? ?
? ? ?
< ? <?
? ? + < ? <?
? ? ??
 
Now, we have an equation which is intrinsic in u , which can be solved to obtain: 
0
0
0
0
       0 11
(2 )( , )= 1 2
1
0 0; 2
u ut
u t
uu t ut
u t
x x
? ?
?? ?
? < ? <
? +
?
?? < ? <?
??
? ? ?
??
 
We differentiate the above equation with respect to ?  and calculate the gradient of 
the wave as time progresses as:  
 
 128 
 
0
0
0
0
       0 11
= 1 21
0 0; 2
u ut
u t
uu ut
u t
x x
?
??
? < ? <
? +
?
??? < ? <?
? ??
? ? ?
??
 
The gradient of the right side of the triangle (the part of the wave between 
( 1 2ut?< ? < ) changes from negative to positive i.e. it becomes vertical at  
01/t u= . The gradient of left side of the triangle slowly decreases over time. This is 
illustrated by the following diagrams.  
 
   
 
 
 
 
 
 
 
 
 
 
 
 
Illustrating the wavebreaking of a triangular waveform through nonlinear steepening. The x-
axis is time and y-axis is amplitude. The plot shows at various time (A,B,C,D), how the apex 
of the wave overtakes the base of the wave and lead to steepening. The dotted line is what 
happens if  the steepening continues without breaking 
 
 
Time 
A 
B 
C D 
Amplitude 
 
 129 
 
Bibliography 
[1] H. Wabnitz et al., Nature 420, 482 (2002). 
[2] P. G. O'Shea, and H. P. Freund, Science 292, 1853 (2001). 
[3] AckermannW et al., Nat Photon 1, 336 (2007). 
[4] H. N. Chapman et al., Nat Phys 2, 839 (2006). 
[5] J. E. Clendenin, and others, AIP Conf. Proc. 915, 1067 (2007). 
[6] L. Merminga, in Particle Accelerator Conference, 2007. PAC. IEEE2007), pp. 
22. 
[7] T. Shaftan, and Z. Huang, Physical Review Special Topics - Accelerators and 
Beams 7, 080702 (2004). 
[8] Z. Huang et al., Physical Review Special Topics - Accelerators and Beams 7, 
074401 (2004). 
[9] R. Akre et al., Physical Review Special Topics - Accelerators and Beams 11, 
030703 (2008). 
[10] C. J. Bocchetta, ICFA Beam Dyn. Newslett. 42, 58 (2007). 
[11] H. Loos, and others, in 30th International Free Electron Laser Conference 
(FEL 2008),Gyeongju,Korea, 2008). 
[12] C. M. Celata et al., Physics of Plasmas 10, 2064 (2003). 
[13] D. Callahan et al., Journal of Applied Physics 81, 3398 (1997). 
[14] J. Wang, and M. Reiser, Physics of Plasmas 5, 2064 (1998). 
[15] J. G. Wang et al., Physical Review Letters 72, 2029 (1994). 
[16] J. Wang, D. Wang, and M. Reiser, Physical Review Letters 71, 1836 (1993). 
[17] J. Wang et al., Physical Review Letters 74, 3153 (1995). 
[18] H. Suk et al., Journal of Applied Physics 86, 1699 (1999). 
[19] Y. Zou et al., Physical Review Letters 84, 5138 (2000). 
[20] H. Suk, J. G. Wang, and M. Reiser, Physics of Plasmas 3, 669 (1996). 
[21] J. G. Neumann et al., Journal of Applied Physics 105, 053304 (2009). 
[22] Y. Huo, in Electrical and Computer Engineering (University of Maryland 
College Park, 2004). 
[23] J. Harris, in Electrical and Computer Engineering (University of Maryland 
College Park, 2005). 
[24] J. G. Neumann et al., Review of Scientific Instruments 76, 033303 (2005). 
[25] K. Tian, in Department of Electrical and Computer Engineering (University 
of Maryland, College Park, 2008). 
[26] M. Reiser et al., Fusion Engineering and Design 32/33, 293 (1996). 
[27] P. O'Shea et al., Nuclear Inst. and Methods in Physics Research, A 464, 646 
(2001). 
[28] Valfells et al., Physics of Plasmas 9, 2377 (2002). 
[29] D. Dowell et al., Physics of Plasmas 4, 3369 (1997). 
[30] B. Beaudoin, in Electrical and Computer Engineering (University of 
Maryland, College Park, 2008). 
[31] M. Reiser, Theory and Design of Charged Particle Beams (WILEY-VCH 
Verlag GmbH & Co. KGaA, Weinheim, 2008). 
[32] V. K. Neil, and A. M. Sessler, Review of Scientific Instruments 36, 429 
(1965). 
 
 130 
 
[33] J. Wu, Z. Huang, and P. Emma, Physical Review Special Topics - 
Accelerators and Beams 11, 040701 (2008). 
[34] A. Dubinov, Plasma Physics Reports 33, 210 (2007). 
[35] H. Suk, in Department of Electrical and Computer Engineering (University of 
Maryland, College Park, 1996). 
[36] J. Bisognano et al., Nuclear Science, IEEE Transactions on 28, 2513 (1981). 
[37] H. Washimi, and T. Taniuti, Physical Review Letters 17, 996 (1966). 
[38] R. A. Kishek et al., Nuclear Instruments and Methods in Physics Research 
Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 544, 
179 (2005). 
[39] S. Bernal et al., in 13th Workshop on Advanced Accelerator Concepts (AAC) 
(New York: AIP Press, Santa Cruz, CA, 2008). 
[40] J. Thangaraj, in Electrical and Computer Engineerin (University of Maryland 
College Park, 2006). 
[41] C. Wu, in Electrical and Computer Engineering (University of Maryland 
College Park, 2008). 
[42] D. Feldman et al., Proceedings of the Particle Accelerator Conference, 2001. . 
[43] J. R. Pierce, Journal of Applied Physics 11, 548 (1940). 
[44] D. Stratakis, in Department of Electrical and Computer Engineering 
(University of Maryland, College Park, 2008). 
[45] C. D. Child, Physical Review (Series I) 32, 492 (1911). 
[46] I. Langmuir, Physical Review 2, 450 (1913). 
[47] ContinuumLasers, Inc. Minilite II Catalog, Santa Clara,CA. 
[48] B. Quinn et al., in Particle Accelerator Conference, 2003. PAC 2003. 
Proceedings of the2003), pp. 2571. 
[49] K. Fiuza et al., in 11th European Particle Accelerator ConferenceGenoa, Italy, 
(2008). 
[50] L. Imaging, Lexel Imaging (http://www.lexelimaging.com/). 
[51] P. Instruments, Princeton  Instruments Inc.(PIMAX). 
[52] K. Tian et al., Physical Review Special Topics - Accelerators and Beams 9, 
014201 (2006). 
[53] J. G. Wang, and M. Reiser, Review of Scientific Instruments 65, 3444 (1994). 
[54] R. Kishek, Program to calculate centroid,beam size from camera images 
(Private communication). 
[55] S. Bernal et al., in 45th Annual Meeting of the APS Division of Plasma 
Physics (AIP, Albuquerque, New Mexico (USA), 2004), pp. 2907. 
[56] J. Harris et al., Physical Review E 76, 26402 (2007). 
[57] R. A. Kishek, and K. Fiuza, Acceleraton in UMER, Technical note (2009)). 
[58] G. Whitham, Linear and Nonlinear waves (John Wiley & Sons, Inc., 1974). 
[59] Murawski.K., Analytical and Numerical Methods for Wave Propagation in 
Fluid Media (World Scientific Publishing Company, 2003). 
[60] J. M. Dawson, Physical Review 113, 383 (1959). 
[61] S. P. D. Mangles et al., Nature 431, 535 (2004). 
[62] W. B. Mori, and T. Katsouleas, Physica Scripta, 127 (1990). 
[63] B. Salzberg, and A. Haeff, RCA Review 2 (1938). 
[64] C. Fay, A. Samuel, and W. Shockley, Bell System Technical Journal 17, 49 
(1938). 
 
 131 
 
[65] C. K. Birdsall, and W. B. Bridges, Electron Dynamics of Diode Regions 
(Academic Press, New York, 1966). 
[66] J. Lomax, Proc. I.E.E. (London) 108, 119 (1981). 
[67] C. K. Birdsall, and W. B. Bridges, Journal of Applied Physics 32, 2611 
(1961). 
[68] J. W. Luginsland, Y. Y. Lau, and R. M. Gilgenbach, Physical Review Letters 
77, 4668 (1996). 
[69] D. J. Sullivan, J. E. Walsh, and E. A. Coutsias, in High-Power Microwave 
Sources, edited by V. L. Granatstein, and I. Alexeff (Artech House, Inc., Norwood, 
MA, 1987).