tremaine 1999 0378
TRANSCRIPT
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UNIVERSITY OF CALIFORNIA
Los Angeles
Coherent Radiation Diagnosis of Self Amplified Spontaneous Emission Free Electron
Laser-Derived Electron Beam Microbunching
A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Physics
by
Aaron Matthew Tremaine
1999
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The dissertation of Aaron Matthew Tremaine is approved.
Steven Cowley
Harold R. Fetterman
Claudio Pellegrini
James B. Rosenzweig, Committee Chair
University of California, Los Angeles
1999
ii
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conviction is not a criterion for truth
-Nietzsche
Any enjoyment is weakened when shared
-The Marquis de Sade
You dont know me, but you dont like me.
You say you care less how I feel.
How many of you that sit and judge me,
Have every walked the streets of Bakersfield?
-Buck Owens
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Table of Contents
Chapter 1
Introduction to Coherent Transition Radiation Theory 1
1.1 Motivation for Experiment 1
1.2 Coherency of Radiation 4
1.3 Single Particle Transition Radiation 8
1.3.1 Overview of Transition Radiation 8
1.3.2 General Considerations of Transition Radiation 10
1.3.3 Transition Radiation using a Collision Model 13
Chapter 2
Coherent Transition Radiation from Multi-Particle Electron Beams 19
2.1 Overview of a SASE FEL 20
2.1.1 Electron Trajectories and Radiation Emission Inside a
Magnetic Undulator 20
2.1.2 Electron Beam Micro-bunching 24
2.1.3 Electron Beam Distribution at the Undulator Exit 28
2.1.4 Brief Review of the High Gain Regime 30
2.2 Calculation of Coherent Transition Radiation 32
2.2.1 Form Factor and Energy Density Spectrum 32
2.2.2 Angular Dependence of the CTR Spectrum 35
2.2.3 CTR Line Spectrum 38
2.2.4 Dependencies of CTR Energy 40
iv
2.2.4.1 Electron Beam Energy 41
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2.2.4.2 Micro-bunching Wavelength 43
2.2.5 Relating emitted UCTR to Incident Electron Beam field
Energy 45
2.2.5.1 Transversely Parabolic Beam 45
2.2.5.2 Transversely Gaussian Beam 47
2.2.6 Emitted CTR energy for a Transversly Asymmetric Beam 49
2.3 Foil Scattering Effecting Forward Emitted CTR 50
Chapter 3
Experimental Setup 57
3.1 AFEL Overview 57
3.2 Electron Beam and Radiation Measurements 63
3.2.1 Measuring N, , r, z ,UCTR ,USASE, , rCTR/SASE
63
3.3 Data Acquisition 70
Chapter 4
Experimental Results and Discussion 73
4.1 Simulations using the Code GINGER 73
4.2 Running the CTR/SASE Experiment 74
4.3 CTR Energy Measured 76
4.4 Measurement of CTR/SASE Line Spectra 78
4.5 Relevant Future Experiments and Discussion for New
Transition Radiation Modeling 824.5.1 Future Experiments Using CTR 82
v
4.5.2 Explaining the CTR Spectrum Shift: Improving the
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Standard Transition Radiation Model 85
Appendices 90
A. Micro-bunch Modeling 90
B. Calculation for the Electric fields of a Micro-bunched
Electron Beam 93
C. Graphed Radial and Longitudinal Fields from Appendix B 99
D. CTR Degradation due to Foil induced Scattering 112
E. FWHM of a Captured Image 116
F. Ginger Bunching Output 118
References 119
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less CTR will be emitted when the beam strikes a conducting foil. 44
Figure 2.8: Velocity and configuration space for electrons at interaction
point for scattering CTR. Electrons receive transverse momentum which
will increase the transverse spot size for the electron beam at the back surface
of the foil. 51
Figure 3.1: The AFEL Beamline. Courtesy of Dinh Nguyen. 58
Figure 3.2: Pulse train out of the drive laser. Individual pulses are 10 ps
FWHM and are 9.23 ns apart. The bunch train can have between 0 and
1000 individual pulses. 59
Figure 3.3: Aluminum CTR screen inserted. SASE FEL radiation is
reflected. Only forward emitted CTR propagates to the HgCdTe detector. 61
Figure 3.4: Diagnostics of electron beam including OTR blades for spot
size and BPMs for charge measurements. In addition, there are three extra
steering magnets along the undulator. See Figure 3.1 for reference. 64
Figure 3.5: Digital readout from oscilloscope. Top trace is BPM2 and the
bottom trace is HgCdTe. 65
Figure 3.6: Image of transition radiation from OTR3 caught by CCD
camera. This is the transverse size of the electron beam at the CTR foil. 66
Figure 3.7: Dipole spectrometer calibration. Energy of electron vs.
Pixel position at OTR4 after dipole and before beam dump. 67
Figure 3.8: Energy spread vs. RF phase for the AFEL accelerator. 68
Figure 3.9: Calibration of the Jerrell Ash monochromator. Relating
the Count Number to a wavelength. 70Figure 3.10: Schematic of the data acquisition implemented at the AFEL 71
Figure 4.1: SASE and CTR signals vs. RF phase. CTR signal has been
normalized to make it the same scale as the SASE signal. 75
Figure 4.2: SASE spectrum with narrow bandwidth on the Jerrell Ash
monochromator. 79
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Figure 4.3: CTR and SASE spectrums detected at the HgCdTe through
the Jerrell Ash monochromator with input slits removed. CTR normalized
to SASE to make it the same scale. 80
Figure 4.4: Formation Zone for transition radiation. Radiation wave and
electron beam must stay in phase for at leas a formation length, Zv . 88
Table 2.1: Electron beam and SASE FEL parameters. 19
Table 4.1: Expected CTR energy considering foil thickness and angular
acceptance of the optical beamline. 77
Table 4.2: Parameters expected for the VISA SASE FEL experiment
at BNL. 84
ACKNOWLEDGMENTS
Here I acknowledge the people who were instrumental to me throughout my graduatecareer:
My advisor, James Rosenzweig, gave me opportunities in the accelerator physics,
for which I am very appreciative. He not only was an excellent guide, but understood that
some days I just wasnt going to be very productive.... especially during the World
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Series.
A luxury of working in the UCLA group is the diversity of research interest
offered. Claudio Pellegrini, although not my advisor, was a major component in getting
my thesis experiment done at LANL. For this, I am very thankful.
While my stay at LANL, I worked closely with Dinh Nguyen. Without the talents
of Dinh, the UCLA experiments would not of had the success they did. And Dinh, Ive
learned to clean the wiggler.
Instead of writing some long, drawn out, thanks, I will list it:
Experiment:
Pedro Frigola: Helped mount the very thin CTR foils and with the construction of
the undulator.
Scott Anderson: Simulations, steering magnet assembly, and printing techniques.
Alex Murokh: Guidance in CTR calculations - scattering effects.
Pietro Musumeci: Simulations and beer drinking.
Travis Holden: Initially helped, then took over the Neptune Laser transport.
Xiaodong Ding: Three words -The Real Deal.
Alexander Varfolomeev: Led construction of undulator which performed excellently.
Gil Travish: Taught me laboratory fundamentals. Irritated by antics. Go to some Cubs
games.
Mark Hogan: By doing his experiment, was a major player in this experiment. See text.
Al, Harry, Ted, Ken, Dave, John: Machined parts needed in timely manner.
Administration:
Christine, Carol and Jim: Kept the orders coming.
Penny and Joyce: Answered questions regarding grad school.
Kurt: Answered questions regarding shipping.
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Outside of Physics:
Blake and Jason: Demonstrating ... too much of a good thing...
Bryant: Sharing his extensive knowledge on management skills.
Kara: Good friend during my graduate career. Now tinkering with mouse genetics in
Cleveland.
Mike: Met at the USPAS in Seattle and will continue to keep in contact. Good luck at
Stanford.
Sven Reiche: I do thank a Northern German from the Lower Saxon region that
1. I can beat in chess.
2. was infiltrated in his German home by a pizza commercial filmed at my
apartment.
3. owes me $100. Ich glaube, Zeigen Sie mir das Geld!
Soren Telfer: I dont thank vegetarians to whom I owe $80.
My Parents: Taught me some of life's lessons at an early age.... pointing out a bum,
Son, thats what happens when you dont go to college.
I would also like to thank Steve Cowley and Harold Fetterman for their time and being
signing members on this dissertation committee.
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VITA
May 16, 1970 Born in Bakersfield, CA.
May, 1991-September, 1991 Undergraduate Student Research Assistant
University of California, Santa Barbara
June, 1993 B.S in Physics at UCSB
June, 1993 B.S in Electrical Engineering at UCSB
June, 1995 US Particle Accelerator School-University of Washington
August, 1997September, 1998 Visiting Scientist-Los Alamos National Laboratory
September, 1997 Laser Technician-Lightwave Electronics
December, 1997 US Particle Accelerator School-University of Texas, Austin
February, 1998-March, 1998 Visiting Scientist-Los Alamos National Laboratory
Presently Research Assistant, Particle Beam Physics and NeptuneLaboratories- UCLA
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PUBLICATIONS AND PRESENTATIONS
M. Hogan, et al., Measurements for High Gain and Intensity Fluctuations in a SASE
Free-Electron Laser, Physical Review Letters, vol. 80, 1998:298.
M. Hogan, et al., Measurements of Gain Larger than 105 at 12 m in a Self-Amplified
Spontaneous-Emission Free-Electron Laser, Physical Review Letters, vol.81, 1998:4867.
D.C. Nguyen, et al.,High-gain Self-Amplified Spontaneous Experiments in the Infrared,
20th International Free-Electron Laser Conference, Williamsburg, Va., 1998.
S. Reiche, J. B. Rosenzweig, et al. , Experimental Confirmation of Transverse Focusing
and Adiabatic Damping in a Standing Wave Linear Accelerator, Physical Review E, vol.
56, (No. 3), 1997:3572-7.
J. Rosenzweig, G. Travish, and A.Tremaine, Coherent Transition Radiation diagnosis
of Electron beam microbunching, Nuclear Instruments in Methods A 365 (1995), 255.
J. Rosenzweig et al. , The Neptune photoinjector, Nuclear Instruments and Methods A
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410 (1998), 437.
A. Tremaine, J.B. Rosenzweig, et al., Observation of Self-Amplified Spontaneous-
Emission-Induced Electron-Beam Microbunching Using Coherent Transition Radiation,
Physical Review Letters, Dec. 1998, vol. 81 (No. 26), 5816-5819.
A. Tremaine, J. Rosenzweig, and P. Schoessow, Electromagnetic Wake Fields and Beam
Stability in Slab-symmetric Dielectric Structures, Physical Review E, Dec. 1997, vol. 56
(No. 6), 7204-7216.
A. Tremaine, Wake-fields in Planar Dielectric-Loaded Structures, Presented at the
Advanced Accelerator Conference (AAC96) (Lake Tahoe, California: 1996).
A.Tremaine, Status of the UCLA Infrared Free-Electron Laser, Presented at the Advanced
Accelerator Conference (AAC96), Lake Tahoe, California: 1996.
A. Tremaine, Coherent Transition Radiation in Longitudinal Electron Beam Diagnostics,
Presented at the American Physical Society (APS) Spring Conference, Columbus,
Ohio, 1998.
A. Tremaine, Free-Electron Laser Micro-bunching Measured using Coherent Transition
Radiation, Presented at the 20th International Free-Electron Laser Conference and
published in the proceedings, Williamsburg, Virginia:1998.
A. Tremaine, Coherent Transition Radiation used in Free-Electron Laser-induced Micro-
bunching Measurements, Presented at the SPIE Conference: LASE 99, San Jose,
California:1999.
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ABSTRACT OF THE DISSERTATION
Coherent Radiation Diagnosis of Self Amplified Spontaneous Emission Free Electron
Laser-Derived Electron Beam Microbunching
by
Aaron Matthew Tremaine
Doctor of Philosophy in Physics
University of California, Los Angeles, 1999
Professor James B. Rosenzweig, Chair
This thesis presents an experiment in which the longitudinal profile of an electron
beam was measured by studying the coherent transition radiation (CTR) emitted when the
beam strikes a thin conducting foil. A high gain Self Amplified Spontaneous Emission
(SASE) Free Electron Laser (FEL) was implemented and the source for the longitudinal
beam profile modulation.
Diagnostics measuring very short periodic electron beam modulations will be
necessary for future experiments in which the modulating wavelength will be severalmicrons. Up to this point, there have been reliable tools used for such longitudinal beam
profile measurements. However, the limits of resolution in these devices are being
approached and new and less expensive methods are needed. Transition radiation from an
electron beam striking a metallic surface is an easily emplementable and inexpensive
diagnostic and is shown to be a reliable diagnostic for the future.
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This thesis presents the theoretical calculation of the expected CTR photon spectrums
and compares the analysis with an experiment recently done on an electron beam which
has been longitudinally modulated by a SASE FEL. The accelerator beamline and its
parameters important to the experiment are described. Also discussed, are the requirements
on the system needed for the best CTR emission possible and the importance in choosing
a good metallic radiating foil. Results from the data are compared with computer simulation
in which these issues are taken into account. Also, the experimental results point out
approximations used in traditional transition radiation modeling that will not be valid in
future CTR experiments and more rigorous theoretical analysis will be needed.
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Chapter 1
Introduction to Coherent TransitionRadiation
1.1 Motivation for Experiment
The ability to diagnose electron beam parameters reliably is an important necessity
in any accelerator lab. Accurate beam profile measurements must be done in order to
describe and use the beam in a variety of applications. New acceleration techniques
[1,2,29] and Free Electron Lasers (FELs) [3,6,13] will deliver electron beams in which
periodic longitudinal modulations will be several femtoseconds and a dependable means
of measurement will be needed.
Typical electronic measurements (Beam Position Monitors (BPM), Integrating
Current Transformers (ICT), etc.) on electron beams are limited to a few GHz (several
hundred picoseconds) resolution. With very longitudinally short electron beams on the
order of a picosecond and less becoming more available, better diagnostic resolution is
needed. Usually to study these very short beams, the electron beam will interact with a
device producing electromagnetic radiation with the same time and space profile as the
electron beam distribution. The radiation produced is either coherent or incoherent and
the degree of coherency determines what can be done and what type of diagnostic can be
used with this signal. This radiation is then directed into a diagnostic which can measure
the lights intensity profile, thus giving information on the electron beam.
1
Up to the present, the streak camera has been a reliable tool to measure longitudinal
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beam profiles of incoherent light. Figure 1.1 shows a schematic of a typical streak camera
design. The incident light to be measured by the streak camera hits a photocathode where
electrons are emitted and accelerated by an electric field and the electron beam distribution
created will have the same longitudinal profile as the incident light intensity. Once the
beam enters the deflecting plates, a triggered ramped high voltage is applied and electrons
at the head of the beam are deflected differently than those at the tail because of the
changing (ramping) transverse electric field between the plates. Here, the longitudinal
information of the beam is transformed into a dimension perpendicular to the original
axis of the incident light. After amplifying the number of electrons, the beam is sent to a
phosphor screen, fluorescing the image onto a CCD camera. Up to the resolution of the
streak camera, the longitudinal profile of the original light can be determined.
It can be seen the triggering of the high voltage and actual demands on the sweep
Acceleratingtube
Photocathode
ElectronMultiplier
PhosphorScreen
CCDCamera
electrons
Voltage RampSupply
DeflectingPlates
Outputlight
Light to bemeasured
Figure 1.1: Schematic of a Streak Camera for measuring longitudinal profiles of light.
2
voltage ramping to measure very short beam modulations are very critical. The shorter
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Happek [4] describes an interferometer technique using CTR in a time domain
autocorrelation to measure picosecond electron beam bunch lengths. A recent experiment
using this technique successfully measured a beam bunch length down to a few picoseconds
[5]. However, to measure modulations to wavelengths much lower than this we need to
use frequency domain techniques as discussed by Rosenzweig [6]. Proposals to measure
periodic longitudinal beam modulations in the infrared to visible wavelength range induced
by SASE FEL [6] interactions suggest that frequency domain CTR has the potential for a
much higher resolution than the usual streak cameras and autocorrelation techniques
described above.
1.2 Coherency of Radiation
It will be shown below that when many electrons radiate the same spectrum, an
enhancement of part of the intensity spectrum is possible. Furthermore, having coherency
in the output signal implies information about the radiator can be deduced. In this section,
the dependencies of the radiation spectrum on the electron bunch distribution as well as
the total number of electrons in a bunch which radiate is examined.
The next section will discuss in detail the actual transition radiation process.
There, transition radiation from an electron/aluminum foil interaction will be modeled by
a collision between an electron with its image charge where Jacksons [7] power spectrum
for an accelerated charge is used for the calculation.
To illustrate radiation enhancement by coherence, we can use this intensity spectrum
for a single accelerated electron [Jackson] derived from the Lienard-Wiechert fields for
4
moving charges
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d2U
d dsingle e-
=e2
42c
n [(n r
) r
]
(1r
n)2exp i t
n rr( t)
c
dt
+
2
(1.1)
wherer
andr
are the velocity and acceleration of the electron normalized to the speed
of light, c ,rr(t) is the electrons trajectory, n is the unit vector from origin to the
observation point, and e is the charge on an electron. More will be said about Eq. 1.1 in
Section 1.3, but to analyze coherency affects, we just need to concentrate on the phase
term shown above.
When describing a bunched beam with N electrons, Eq. 1.1 must be summed
over all particles keeping track of all phase factors which is the basis for coherency
enhancement. Figure 1.2 shows the multi-electron beam radiative process. Here, r(t) is
ObservationPoint
Origin
r(t)
n
Figure 1.2: Coordinates for bunch beam radiative process.
radiationpath
electron beam
rn
5
now the position of the bunch center and rn is the position of the n th electron with
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respect to the center of the beam. Noting r(t) r(t) + rn and the observation point is in
the far field, the total energy spectrum from all N electrons in the beam using Eq. 1.1 is
given by
d2U
d d=
e2
42c
n [( n r
) r
]
(1r
n)2exp i t
n rr( t)
c
expi
rrn nc
n=1
N
dt
+
2
. (1.2)
Equation 1.2 shows that there is a phase shift factor due to path length differences
between different parts of the electron bunch and the observation point. Assuming all
electrons in the bunch are traveling with the same velocity (
r
n = z ) and the observation
point is far away, the summation can be taken out of the integral and Eq. 1.2 can be
rewritten as a function of the single particle spectrum given by Eq. 1.1,
d2U
d d exp
irrn nc
n=1
N
2
d2U
d dsingle e-
. (1.3)
Most electron beams worked with in accelerator labs are described by continuous
distributions (Gaussian, parabolic, step functions, etc.) with many electrons and doing the
actual summation over discrete electron positions is computationally difficult (but used
often in numerical simulation codes). Equation 1.3 can be put into a continuous form
similar to the method outlined by Nodvick and Saxon [8] and Hirshmugl [9] and noting
expi
rrn nc
n=1
N
2
=N+ expi
rrj n
c
j =1
N
exp i
rrk nc
k=1
N
(j k) (1.4)
6
where the result of N on the right side of the equation comes from the part of the
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summation where j = k. In order to eliminate the summation on the right side of Eq. 1.4,
we assume a continuous normalized particle distribution of S(r) , where NS(r) is the
particle density at a given point, r, and the total number of electrons is large. The sums in
Eq. 1.4 can now be written as integrals,
expi
rrn nc
n=1
N
2
=N+N2 expi
rrj n
c
S(rj)d
3rj exp
irrk nc
S(rk)d
3rk
(1.5)
The integrals in Eq. 1.5 are just the Fourier transforms of the particle distribution function,
S(r) , and the energy spectrum of a multi-electron beam from Eq. 1.3 is simply
d2U
d d [N+ N2F( )]
d2U
d dsingle e-
(1.6)
where F( ) (termed the form factor) is the square of the Fourier transform of the
radiating electron particle distribution shown in Eq. 1.5. The spectrum proportional to
N2
is termed the coherent spectrum and is enhanced by a factor ofN over the incoherent
spectrum and as long as the Fourier transform of the electron distribution is not near zero.
This is case for typical particle distributions (Gaussian, parabolic, step functions , etc.)
where the wavelength, r, of the radiation spectrum emitted is larger than the longitudinal
size of the electron beam, z [9]. Stated another way, the spectrum is enhanced where
the wavenumber of the radiation, kr= /c , is smaller than the inverse of the beam
length, 1/ z . Otherwise, the spectrum is incoherent and its power density is proportional
to N, not N2
.
7
By using the coherent portion of the spectrum, the signal not only has a potentially
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much higher energy output, but also contains valuable information concerning the beam
bunch shape. The experiments described in References 4,5, 6 and 25 take advantage of
this effect with regards to coherent transition radiation. It should be noted the results for
the total energy spectrum in Eq. 1.6 can be applied more generally than to just the
radiated spectrum given by Eq. 1.2. As long as the length of a radiating bunch is smaller
than the radiated wavelength, this enhanced coherency effect occurs.
1.3 Single Particle Transition Radiation
Before discussing transition radiation from a bunched beam with an assumed
distribution, Eq. 1.6 shows that the single particle radiation spectrum must be known.
This section presents the single electron transition radiation spectrum in which an
approximated calculation is done in Section 1.3.3 and compared to more formal and
general results cited in Section 1.3.1. The next chapter will apply this single particle
theory to an electron beam distribution which has been modulated by a Self Amplified
Spontaneous Emission Free Electron Laser (SASE FEL).
1.3.1 Overview of Transition Radiation
Transition radiation is emitted when a particle travels from one medium with
permittivity 1( ) into a second medium with 2( ) and is shown in Fig. 1.3. In
medium one, the electron carries fields specific to 1( ) properties,
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Boundary
Figure 1.3: Transition radiation model. An electron passing through a
boundary between two mediums emits radiation to an observation point.
Observationpoint
Medium 1 Medium 2
1 2
e-
Electric fieldspropagatingwith particle
_
y
z
n
likewise in medium two. At the boundary, the fields must obey the continuity conditions
derived from the Maxwell equations. In traversing the boundary, the electron will reorient
its fields to suit the new medium, and in doing so, fields will be emitted as transition
radiation. This effect was first studied by Ginzburg and Frank [10] and a rigorous approach
is provided by Ter-Mikaelian [11] in which the radiation spectrum emitted to the observation
point is given by
d2I
d d=
e2 2 2 sin2 cos2
2c
( 1 2)(1
22 1 2 sin
2
(1 2 2 cos2 )(1 1 2 sin
2 )( 1 cos + 1 2 22 sin2
2 (1.7)
9
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where is the angle between the electrons velocity axis and the observation point in
medium two. Equation 1.7 is seen to be quite complicated. More will be said about this
very general result in Section 1.3.3 where Eq. 1.7 will be compared with a simpler
method for obtaining the transition radiation spectrum.
1.3.2 General Considerations of Transition Radiation
There are some important aspects that must be known about the single particle
transition radiation process in order to apply the results of this chapter accurately to
multi-particle electron beams. Two questions need to be answered: As the electron
approaches the boundary, which of the fields (parallel or perpendicular to the boundary
plane) the electron carries with it are responsible for the transition radiation? At what
angle can we expect the transition radiation to be emitted?
The dependence of emitted transition radiation on the incident electron fields can
be found by expanding on general arguments used in the discussion of this process given
by Jackson [7]. Here we assume an electron propagates (
r
= z ) in medium one (vacuum)
normal to a boundary with a medium 2 . Figure 1.3 shows the model being used where
n (n = cos( )z + sin( ) y ) is the unit vector to the observation point and is the angle
between the observation point and the particle axis ( z ). It is seen this process has azimuthal
symmetry and the observation point is chosen in the y-z plane for convenience. The
electric field in the first medium assumed to travel along with the particle is
rE1 =Ex x + Ey y +Ez z where the exact field dependencies have been omitted for brevity.
It should be noted these are not radiating fields since they travel along with the
electron in vacuum, but have been referred as virtual photons [7,26,33,34] with a phase
10
velocity of z . The virtual photon method is known more formally as the Weizsacker-
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Williams method where the fields of the moving charge are related to the fields emitted
as radiation in processes such as collisions or Bremsstrahlung.
As the electron approaches the boundary, the fields from the electron polarize
(
r
P(x ,y,z , ) ) medium two at a point (x,y,z ). It can be seen that as the electron moves
closer to this point, the polarization will be time dependent and therefore will radiate to
the observer. For there to be significant radiation, the phase difference between the
incident electron (and its fields) and the polarization radiation must be constant and will
be shown below to determine what constraints can be placed on .
From Maxwells equations the polarization in medium two is proportional to the
incident electrons fields,rP
rE1. The electric field emitted to the observation point due
to this polarization radiation is given by the electromagnetic relation [Jackson]
rEobs. ( n
rP) n[ ]dxdydz . (1.8)
Taking the cross products in Eq. 1.8, the dependence of the field emitted (transition
radiation) to the observer due the incident electrons fields in medium one has the form
rEobs. ( n
rP) n Ex cos
2( ) x +Ey cos
2( ) y . (1.9)
This important result states that the emitted transition radiation detected at the observation
point from Eq. 1.8 is primarily due to the incident electron fields (Ex,Ey ) which are
polarized in the x-y plane-the same plane as the boundary. (From Fig. 1.3 these are the
radial fields with respect to the propagation axis of the electron.) If the second medium is
metal, the results above show transition radiation being derived from the surface currents
11
on the metal driven by the fields of the incident electron beam which are in the conductor
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surface plane. This will be considerably helpful in understanding the results of transition
radiation from multi-particle micro-bunched electron beams discussed in the next chapter.
There we will see that it is possible for the radial fields of a longitudinally modulated
electron beam to be diminished, thus suppressing the surfaces currents on the metal and
reducing the amount of transition radiation emitted.
For completeness, it should be noted that when the full fields are used
(Ex(x ,y ,z),Ey(x,y,z )) the first term in Eq. 1.9 will integrate to zero when put into Eq.
1.8 and the transition radiation is seen to be polarized in the plane of observation ( y ). As
the observation point rotates around thez-axis, Eq. 1.9 can be easily modified and the
transition radiation field polarization is always in the plane of observation and seen to be
radially polarized.
An approximation of the radiation emission angle can also be made. The phase
between the incident electron(s) (and its fields) and the emitted radiation has to stay
essentially constant. This requires the component of the transition radiation wave velocity
along the particle direction of motion (z ) to be equal to the particle velocity (see Figure
1.3 or Fig. 4.4). Using the approximations that = 1 1/(2 2) and the speed of the
radiation along the particles axis is (normalized to the speed of light, c ) cos( ) = (1 2/2) ,
the angle for an invariant phase difference between the particle and radiation wave is
max 1
. (1.10)
Equation 1.10 is the angle that is commonly quoted for the incoherent transition radiation
angle of emission, but it will be shown below that this is not entirely accurate and the
correct angle for emission is a factor of 3 larger. Equation 1.10 shows the dependence
12
of emission angle on electron beam energy for incoherent radiation, and as the beam
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energy increases, the radiation will be emitted in a narrower cone. We will see in the next
chapter that coherent radiation is contained in a much narrower emission cone and is
more strongly dependent on beam energy than the incoherent radiation.
1.3.3 Transition Radiation using a Collision Model
A simple method can be used to find the single electron transition radiation
spectrum when the two mediums are vacuum and metal. The interaction at the boundary
can now be modeled by a collision between the electron with its image charge. In this
annihilation process, both particles are accelerated from near the speed of light to zero,
r 0 , and Eq. 1.1 (the intensity spectrum for an accelerated charge) can be used.
Figure 1.4 shows the model being used for the collision where n is the unit vector from
13
Boundary
(Metal)
(Vacuum)
Figure 1.4: Transition radiation from an electron/metal interaction
modeled by an annihilation between an electron and its image charge.
e+
e-
Observationpoint
y^
z
n
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the collision point at the boundary to the observation point in the vacuum. For simplicity,
n is assumed to be in they-z plane and is the angle from the particle axis ( z ) to the
observation point. The next chapter will discuss more advanced topics which will include
non-normal incidence between an electron and metallic surface.
Equation 1.1 is the spectrum an accelerated charge radiates. It is derived from the
general energy spectrum formula [7]
d2U
d d= 2
rA( )
2(1.11)
wherer
A(t) = c/4( )1/2 [Rr
E]ret andr
E is the electric field from the Lienard-Wiechert
fields for moving charges. In its exact form Eq. 1.1 is quite complicated, but the integral
can be reduced by a couple of well known approximations. First we note
n [(n r
) r
]
(1 n r
)2 =
d
dt
n (n r
)
1 n r
. (1.12)
Second, because the time of collision is assumed very short, the radiated frequencies will
be relatively small over the course of the collision/annihilation. In other words, the
wavelengths of radiation are large ( 0 ) compared to the characteristic time of collision
and the exponential in Eq. 1.1 goes to unity. It should be noted this is the approximation
used in standard transition radiation theory, but the results found from the experiment in
this thesis imply this exponential term can not be neglected in future work and is discussed
more in Chapter 4.
Using Eq. 1.12 and assuming relatively low emitted frequencies, the integrand in
Eq. 1.1 is a perfect differential. Integrating from a velocity of ( 1) 0 results in a
14
single particle acceleration spectrum of
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d2U
d dparticle
e2
42c
n ( n r
)
1 n r
2
(1.13)
where, from Fig. 1.3, n = cos z + sin y andr
= z . In order to get the total spectrum
at the observation point of the collision model, the spectrums from the electron and image
charge must be added together. Looking at the denominator in Eq. 1.13, one sees that
spectrum to reach the observation point will be mainly due to the image charge and not
the electron. (If for relatively small angles ( ) , which is a good approximation discussed
below, and | |1 , the denominator for the image charge will be much less than the
denominator for the electron). Thus, the dominant portion of the energy spectrum is due
to the particle with its velocity most parallel to the observation point.
Multiplying out Eq. 1.13 and noting the acceleration of the particle is in the
opposite direction of its velocity, we get the familiar form for an electron/metal transition
radiation
d2U
d dsingle e-
e2
42
c
sin1
r
cosy
2
. (1.14)
The energy spectrum will be shown below to be confined to small angular cones, and Eq.
1.14 has been approximated by small angles. Also from Eq. 1.14, the radiation is seen to
be primarily polarized ( y ) in the plane of observation (see Fig. 1.4). As the observation
point rotates around the z-axis, the vector potential is seen to always point towards the
z-axis, meaning the transition radiation is radially polarized. This agrees with the previous
15
result obtained from general considerations accompanying Eq. 1.9 in Section 1.3.2.
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The photon dependence on angle is found by dividing Eq. 1.14 by h and
integrating from ( 1 2),
dN
d single e- 2sin3( )
1 cos( )( )2 ln 12
(1.15)
where is the wavelength ( = 2 c/ ) and is the fine structure constant. The
maximum number of photons for incoherent transition radiation from Eq. 1.15 is found to
be at
max =
3
(1.16)
which, as stated before, is different than the more commonly quoted angle given by Eq.
1.10.
Figure 1.5 graphs just the angular dependence of Eq. 1.15 for the transition
radiation spectrum assuming a particle energy of 17.5MeV ( = 35 ). The radiation is a
narrow cone centered around the particles propagation axis (z ) where the maximum of
the spectrum is shown to be around 2.7 degrees agreeing with the result given by Eq.
1.16. The next chapter will establish when micro-bunched multi-particle beams are
considered using Eq. 1.6 for coherent transition radiation, the spectrum has a much
narrower emission cone than the incoherent radiation shown in Fig. 1.5. This will be an
important factor in the experimental separation of these two signals.
One other point should be mentioned about Eq. 1.15. Since this is the spectrum
for one electron, it is found for typical beams (1010
electrons) there is a significant
amount of light in the visible wavelength range emitted. As will be discussed in Chapter
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0 5 10 15 20 250
0.005
0.01
0.015
.
Figure 1.5: Photon number angular distribution for incoherent transition radiation.
Angle ()
dNd
3, this light can be monitored by a CCD camera in which optical measurements on the
electron beam transverse spot size can be made.
It can be deduced from Fig. 1.4 that as the electron exits a metal foil, there will
also be transition radiation. Now the model is the creation of an electron/image charge
pair at the boundary in which the electron accelerates into the vacuum and the image
charge accelerates into the metal. It is easy to show that this energy spectrum would be
exactly the same as given by Eq. 1.14. In general, when an electron propagates through a
metal foil, there will be a back emitted transition radiation cone at the front surface of the
foil and a forward emitted transition radiation cone at the foils back surface (Fig. 1.4
shows the front surface).
These same results can be obtained following Wartski [12] using the more generalspectrum cited by Ter-Mikaelian in Eq. 1.7 for a metal/vacuum boundary with 1.
17
For the case of the electron exiting metal ( 1 >> 1) to vacuum ( 2 = 1) (the creation
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model discussed above) and using small angle approximations, Eq. 1.7 can be reduced to
the spectrum described by Eq. 1.14. When the electron goes from vacuum ( 1 = 1) to
metal ( 2 >> 1) the result from Eq. 1.7 is
d2U
d d=
e2
42c
sin2( )
1 cos( )( )22 1
2 + 1
2
. (1.17)
This is the spectrum emitted back into vacuum and the last ratio in the equation is the
Fresnel reflection term. When the second medium is a metal, no fields can propagate into
it and are reflected. Eq. 1.13 can be reduced to Eq. 1.11 ( 2 >> 1) showing that the
forward and backward emitted transition radiation spectrums are the same for an
electron/metal interaction as predicted by the simpler collision model discussed above.
Now the transition radiation spectrum can be written for a bunched electron
beam. Combining the effects of multi-particle radiation from Section 1.2 and the single
electron transition radiation spectrum from the present section, the transition radiation
spectrum for a multi-particle electron beam is found to be
d2U
d de- beam
N+N2F( )[ ]e2
42c
sin2
(1 cos )2(1.18)
where the second term is the coherent spectrum mentioned in Sections 1.1 and 1.2. The
next chapter will look into more detail about the form factor, F( ) , where a particle
beam profile will be assumed and the complete energy spectrum for this distribution will
be studied.
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Chapter 2
Coherent Transition Radiation fromMulti-particle Electron Beams
The present chapter expands on the transition radiation discussion from the previous
chapter and presents analysis for coherent transition radiation from multi-particle beams.
As the experiment for this thesis was performed on an electron beam exiting a Self
Amplified Spontaneous Emission (SASE) FEL, the beam modulation (micro-bunching)
from the SASE FEL interaction is examined. The expected CTR spectra arising from this
beam striking a radiating foil is found using the methods discussed in Chapter 1 and the
19
Beam Energy
Charge/bunch
Bunch length
(FWHM)
Wiggler
period
On axis field
FEL Wavelength
RMS beam size
17.5 MeV
1.2 nC
9.2 ps
2 cm
7.4 kG
13 m
180 m
E
Q
u
B0
r
r
Table 2.1: Electron beam and SASE FEL parameters.
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models shown here for SASE FELs. Also, scattering affects in the radiating foil are
analyzed and shown to degrade the forward emitted CTR signal. Table 2.1 is a list of
typical experimental parameters for this thesis and is used to give quantitative results
illustrating the theoretical analysis discussed in this chapter.
2.1 Overview of a SASE FEL
2.1.1 Electron Trajectories and Radiation Emission Inside a Magnetic
Undulator
The SASE FEL process is described in great detail in a number of places [13,15]
and is discussed briefly here reviewing the important results critical to this thesis. Figure
2.1 shows the SASE FEL operation in which there is an energy exchange between an
electron beam and electromagnetic wave. An electron beam with an assumed velocity
r
= z enters a planar magnetic undulator with a magnetic field perpendicular to the
beam axis and sinusoidally varying along the electron axis of propagation,
r
B =B0 cos(kuz) y (2.1)
where B0 is the peak undulator magnetic field and u = 2 /ku is the magnetic period of
the undulator.As the electrons propagate through the undulator, they start to oscillate in
the x dimension due to the Lorentz force and emit spontaneous radiation. In order for the
electromagnetic wave to amplify (gain energy at the expense of the electron beam), there
20
has to be a constant exchange of energy from the electron beam to the electromagnetic
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wave given by
d
dt ebeam=
qr
r
Ewave
mc. (2.2)
where q is the particle charge, is the particle velocity, and Ewave is the electric field
of the co-propagating EM wave shown in Fig. 2.1.
21
x xxx x
x xxx x
B=B0cos(kuz) y^
u
Magnets
x
z
e - beam
EM Wave
r
Figure 2.1: SASE FEL process showing the electron beam
trajectory and the copropagating EM wave. Amplification requires
a net exchange of energy to the EM wave from the electron beam.
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It can be shown for there to be a net energy exchange, the electron beam must slip
approximately one wavelength of the electromagnetic field for every period of the magnetic
undulator travelled. This ensures the transverse velocity of the electrons and electric field
of the wave will remain in phase through the undulating motion, resulting in a net energy
exchange given by Eq. 2.2. The wavelength of radiation which can sustain this process is
given by the well known FEL relation
r =u
22 1+
K2
2
(2.3)
where K= (e2
/mc
2
)B0 is the normalized undulator field and a measure of the transverseelectron oscillation amplitude, u = 2 /ku is the undulator magnetic period, and is the
energy of the electron beam. Using the parameters in Table 2.1, Eq. 2.3 gives an FEL
wavelength of r = 13 m for this system.
Figure 2.1 shows the electrons velocity and trajectory inside the undulator being
modulated by the undulator magnetic field. The longitudinal and transverse trajectories of
the electrons in the undulator are found by solving the Lorentz force equations,
dr
p/dt= q(r
r
B) , giving [13]
z( t) = c 0t+K
2
82ku 0
2sin(2kuc 0t) +z0
x(t) =K
ku 0[cos(kuc 0t) 1]+x0
y(t) = y0
0 = 11 + 1
2K2
2
1
2
(2.4)
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where 0 is the average axial velocity (normalized to the speed of light) of the electrons
in the undulator and x0,y0,z0 are the initial positions of the electrons at the entrance of
the undulator. It can be seen from Fig. 2.1 and Eqs. 2.4 the electron transverse ( x )
velocity through the undulator oscillates with a period u and the longitudinal velocity
(z ) is modulated with a period one half of an undulator wavelength, u/2 .
Since the electrons are being accelerated, they will emit spontaneous radiation and
Eq. 1.1 can be used. The radiation spectrum emitted from electrons traversing an undulator
with the trajectories given by Eqs. 2.4 (and its derivatives) is found using Eq. 1.1, the
spectrum for accelerated charges, to be
d2U
d d=
Nu2e2 2K2
c(1+ K2/2)2Fn(K)
sin2 (xn/2)
(xn/2)2
where
Fn(K) = J(n+1)/2nK2
4(1+ K2/2)
J(n1)/2
nK2
4(1+ K2/2)
2
n2
, (2.5)
xn = 2 Nu( n
n ) , n =
2ncku2
(1+ K2/2)
for n = 1,3,5... (odd).
Here Jn is the Bessel function of the first kind with order n . Each harmonic (n ) of the
energy spectrum given by Eqs. 2.5 will emit radiation with a harmonic wavelength given
by
23
r,n =r
n(2.6)
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which is the same radiation wavelength needed for amplification of an EM wave in this
system (see Eq. 2.3). Only the odd harmonics in radiation are emitted forward (co-
propagating with the electron beam) and as we will see below, these harmonic wavelengths
will correspond to a longitudinal density modulation experienced by the electron beam.
Also, the electric field in Fig. 2.1 is seen to be polarized ( x ) perpendicular to the
undulator magnetic field ( y ) and the axis of propagation ( z ).
The total field the electrons see in propagating through the undulator is the
combination of the radiation fields given by Eqs. 2.5 and the undulator magnetic field
given by Eq. 2.1. In the next section it will be shown that the combination of the these
two fields acting on the electron beam will create a pondermotive force causing the
electron beam to micro-bunch.
2.1.2 Electron Beam Micro-Bunching
We know from Section 1.2 that when many electrons radiate within a bunch that
is shorter than the emitted radiation wavelength, z < r, an enhancement of signal is
possible. It will be shown below that in a Self Amplified Spontaneous Emission (SASE)
FEL the electrons will bunch at certain positions in the radiated EM field shown in Fig.
2.1. As the electrons become more micro-bunched, the radiation fields emitted from the
electrons will intensify due to the coherency effects discussed in Section 1.2. An increase
in the radiation amplitude will then cause the micro-bunching to intensify, and an instability
is formed. As the process continues, it is seen that not only does the radiation field
amplitude grow, but the degree of electron beam micro-bunching increases and is directly
24
responsible for this radiation amplification process.
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Taking into account the effect of the undulator (Eq. 2.1) and radiation (Eqs. 2.5)
fields on the electron beam and using the energy relation given by Eq. 2.2, the micro-
bunching of the electron beam can be shown. The change in electron energy through the
undulator is found to be [13]
d
dz=
eKErFn(K)
2msin( ) (2.7)
where = (ku + kr)z krct and Er is the amplitude of the radiation field. Figure 2.2a
graphs the change in electron energy vs. longitudinal phase position in the EM wave
given by Eq 2.7 at a specific time. The pondermotive force (the right side of Eq. 2.7) is
shown to be the cause the of the electron beam micro-bunching. Electrons that have a
phase in which the pondermotive force is positive will gain energy and thus will be
pushed forward in phase and electrons which see a negative pondermotive force will be
pushed back in phase. Figure 2.2a shows that there will be regions of high and low
electron density where the pondermotive potential is zero. It is also seen that the micro-
bunching periodicity is the same as the radiation wavelength, r, since ku + kr kr for
most SASE FEL configurations (see Table 2.1)
In order to model the micro-bunching induced from the pondermotive potential, it
is seen (Fig. 2.2) that peaks in the electron distribution are needed every r with a
deficiency of electrons between each peak. This is done using a Fourier series of harmonics,
bn cos(nkrz)n=1
(2.8)
where bn are the harmonic bunching amplitudes. Shown in Fig. 2.2b is an example of
25
such a Fourier series modeling a step function using Eq. 2.8 for the first forty harmonics
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Figure 2.2:a. Pondermotive Potential for the first harmonic, n=1. Electrons are pushed toward andaway from the zeros in the pondermotive potential.b. Simulation model for micro-bunching using a step function for first forty harmonics in a Fourier
Cosine Series.
0 1 2-1-2
-.75
.75
0
sin[(ku+kr)z]
electrons
Pondermotive Force
r
Phasezr
( )
(a)
0 1 2-1
Phasezr
( )
-2Bunch
ingprofileusing
thefirst40harmonics
deficiency
of e-
r
(b)
dz
d__
(
)
,
surplus
of e-
-10
1
2
3
4
26
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with a bunching factor
bn =sin(nkr)
nkr. (bunching factor for a step function)
This and other examples of longitudinal micro-bunching are shown in Appendix A.
Figure 2.2b shows the electrons have moved from the deficiency to the surplus regions
separated by r. This step function properly models the micro-bunching formed by the
pondermotive force shown in Fig. 2.2a where the electrons have moved toward and away
from the zero-crossings of the pondermotive potential. Appendix A also shows an example
of more intense bunching where the model approaches a delta function every r. This
scenario for a SASE FEL is termed saturation and further energy exchange from the
electron beam to the EM wave is not possible. For the SASE FEL in this experiment the
bunching factors will greatly diminish with increasing harmonic number and only the
fundamental harmonic in micro-bunching, b1 , can be measured with all other higher
harmonics being negligible. Once again, as the micro-bunching amplitude, bn , increases,
the radiated field, Er, emitted from the micro-bunched electrons will also increase. This
increase in field amplitude will cause the pondermotive force in Eq. 2.7 to induce more
micro-bunching, thus an instability is formed and is the amplification process of the
SASE FEL.
It should be noted that Eq. 2.7 is the pondermotive force due to the radiation
fundamental harmonic (n = 1 ) in Eqs. 2.5 from the electron trajectories given by Eqs. 2.4
and shown in Fig. 2.2. In the above section, it was shown electrons undergoing this
undulating motion will also emit radiation at higher harmonic wavelengths, r,n (Eq.
2.6). These higher harmonics will induce a pondermotive potential in Eq. 2.7 with
27
(ku nkr)z nkrct, thus micro-bunching the electrons with a periodicity of r/n .
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Each one of these higher harmonic micro-bunching distributions can also be modeled by
a harmonic Fourier series similar to the one given in Eq. 2.8 and shown in Fig. 2.2b.
2.1.3 Electron Beam Distribution at the Undulator Exit
At the exit of the undulator, the electron beam has been longitudinally modulated
with wavelengths equal to the electromagnetic wave harmonics, r,n , given by Eq. 2.6.
The longitudinal electron beam profile from this process is shown in Fig. 2.3 for
28
RelativeAmplitude
z (mm)
10 5 0 5 100
200
350
Figure 2.3: Modulation of the longitudinal electron beam
distribution from a SASE FEL with bunching factor, b=.1.
0
280
320
z(mm)
13mr=
-5 10-2. 5 10-2.
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the fundamental harmonic, n = 1 , where z = 0 is taken to be the longitudinal center of the
beam. It is assumed before entry into the undulator, the electron beam has a Gaussian
longitudinal profile with a bunch length (FWHM) of about 10 ps. Upon travel through
the undulator, the micro-bunching structure is superimposed on this distribution and the
resulting profile is shown in Fig. 2.3 for a 10% micro-bunching amplitude, b = .1 . For the
experiment described in this thesis, the SASE FEL radiation and thus the electron micro-
bunch modulation wavelength for the fundamental harmonic is r,1 = 13 m using Eq.
2.6 and the experimental parameters given in Table 2.1. It can be seen from Fig. 2.3 that
at the exit of the undulator the electron beam longitudinal profile with micro-bunching
(Eq. 2.8) can be written as a modulated Gaussian distribution,
h(z) =
exp z2
2 z2
2 z bn cos nkrz( )
n=1
, (2.9)
where bn are the micro-bunching factors and the amplitude of the beam micro-bunching
and n are the harmonic numbers discussed in Section 2.2.2 by Eq. 2.8. Noting ifn = 0
and b0 =1 , Eq. 2.9 will show no longitudinal harmonic micro-bunching and the longitudinal
profile will be a simple Gaussian.
The transverse electron beam profile is also assumed to be Gaussian and is not
modulated by the SASE FEL. At the exit of the undulator, the normalized three dimensional
electron beam distribution can be described as
(r,z) = S(r
x) = g(r)h(z) =exp
r2
2 r2
2 r
2
exp
z2
2 z2
2 z
bn cos nkrz( )n=1
. (2.10)
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It should be noted that it is possible for other transverse modes to be present in the
undulator, but as we will see in the experiment for this thesis, the transverse Gaussian
profiles given by Eq. 2.10 will accurately describe the electron beam distribution.
When the electron distribution given by Eq. 2.10 interacts with a conducting foil,
the CTR spectrum produced is found by taking the Fourier transform of Eq. 2.10 to find
the form factor, F( ) , and using the result given by Eq. 1.18. It should be noted again
when n = 0 and b0 =1 the above distribution gives purely Gaussian distribution functions
in all three dimensions allowing for the calculation of CTR due to an unmodulated
electron beam necessary for the interferometer technique described in Reference 5 for
electron beam bunch lengths measurements.
2.1.4 Brief Review of the High Gain Regime
For completeness, the high amplification regime of the EM wave co-propagating
with an electron beam through an undulator is briefly discussed here. Just results important
in illustrating the exponential gain process are presented without derivation. For further
and more detailed study of SASE FEL, one should refer to References 3, 13, 15, 27.
As stated above, the EM wave gains energy from the electron beam through an
instability in the system. In order to get a full understanding of this process, detailed
calculation of the electron beam coupled with the EM field is required. In other words,
Maxwells Equations are needed in which the electron beam and EM wave are in constant
energy exchange with each other. Kroll et al. [15] went through this derivation and
derived the well known FEL equations termed as the KMR equations. These equations
describe the changes in field amplitude, electron beam phase with respect to the
30
pondermotive potential, and electron beam energy.
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The Pierce or FEL parameter, , important in describing the gain and requirements
necessary for high gain in an FEL, is defined by
=K
0p4 R
2
2/ 3
(2.11)
where p =4 nee
2
m 0
1/2
is the modified plasma frequency of the electron beam and R is the resonant energy of
the system derived from Eq. 2.3. This parameter is important in describing the FEL
radiation bandwidth and angular effects and in setting limits on the initial electron beam
parameters [3,13,15,27]. It also gives a rough estimate for the over efficiency of the
system to extract energy from the electron beam to the EM wave. For the experiment in
this thesis, the experimental parameters in Table 2.1 give a Pierce parameter of .01.
A commonly quoted relation in the references for the power exponential gain in
these systems is given by
P = P0 expz
Lg
(2.12)
where the gain length is Lg =u
4 3, z is the distance the down the undulator and P0 is
the start up power derived from the spontaneous fields given by Eqs. 2.5. It is usually
estimated it takes about one gain length before exponential power gain starts and just
over ten gain lengths to saturate a SASE FEL (b 1). Using the result above and
31
experimental parameters, we have a power gain length ofLg 10cm . Also, the amount
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of power expected in the EM wave can be related to the electron beam power and the
Pierce parameter and is given by the approximation, PEM Pbeam .
GINGER, a 3D code, models the SASE FEL gain and bunching using the KMR
equations [15] and is used to predict these results for the SASE FEL in this thesis
experiment. More will be said about GINGER in Chapter 4 where experimental results
will be compared with the theoretical modeling.
2.2 Calculation of Coherent Transition Radiation
This section concentrates on the calculation of the coherent transition radiation
spectrum following the methods outlined by Shibata et al. [16] and Rosenzweig et al. [6].
The importance of the line and angular spectra are emphasized and the expected CTR
energy output is calculated. Also, dependencies of the CTR energy on many experimental
parameters critical to this thesis are discussed.
2.2.1 Form Factor and CTR Power Density Spectrum
To study the radiation spectra, the Fourier transform of the electron beam distribution,
S(r
x ) , from Eq. 2.10, needs to be calculated and the form factor found as discussed in
Chapter 1. Substituting the relationr
k = ( /c)n = kn ( n is the unit vector to the observation
point from Chapter 1) into Eq. 1.5 gives a form factor
F(k) = eir
krx S(r
x)d3x
2
(2.13)
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where the three dimensional coordinate vector,r
x , has been introduced to for clarity. The
coordinate system from Fig. 1.3 for a bunched beam instead of a single electron is used,
where z is the electron bunch axis of propagation normal to the foil surface, is the
azimuthal angle around the z -axis and
r
k r
x = kz cos + krsin cos . Combining Eqs.2.10 and 2.13, the Fourier transform can be separated into a longitudinal and a transverse
component giving a form factor
F(k, ) = fl(k, )ft(k, )2. (2.14)
First, the transverse component is found,
ft(k, ) =1
2 r2 exp
r2
2 r2
exp ikrsin cos( )rdrd (2.15)
When the integration over is done, the result is a Bessel function of the 0th order and
final integration over the radial coordinate, r, results in
ft(k, ) = exp (k rsin )2
2
. (2.16)
Now the longitudinal Fourier transform must be calculated,
fl(k, ) =1
2 zexp
z2
2 z2
bn cos n
2
r
z
exp(ikz cos )dz
n . (2.17)
Putting the cos(nkrz) term into its exponential equivalent and integrating, Eq. 2.17 can be
33
written as
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fl(k, ) =bn
2iexp
(kcos nkr)2
z2
2
n . (2.18)
Once again it should be mentioned when n = 0 and b0 =1 , this is the Fourier transform
of an unmodulated longitudinal Gaussian profile and Eq. 2.18 can be used for the calculation
of bunched beam CTR described in References 4 and 5. To find the total form factor,
Eqs. 2.16 and 2.18 need to be put into Eq. 2.14. Squaring the summation in Eq. 2.18
gives
fl(k, )
2
=
1
4 bn exp
(kcos nkr)2
z2
2
n bm exp (kcos mkr)
2z2
2
m .
It can be seen the only terms remaining in the above relation are where m = n. Each
Gaussian in the series above is centered around an integer multiple of kr and has a width
much less than the spacing between adjacent Gaussians. Results from multiplying two of
these Gaussians not centered at the same wavelength gives
exp (kcos nkr)
2z2
2
exp
(kcos mkr)2
z2
2
0 when m n
for typical experiments kr z >>1 (see Table 2.1) and the form factor from Eq. 2.14 looks
like
F(k, ) =b12
4exp (k rsin )
2( ) exp (kcos nkr)2 z2( ) . (2.19)
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Now the CTR energy density spectrum from Eq. 1.18 for small angles (a good approximation
which is discussed below) and using = ck is found to be
d2U
dkde - beam
N2e2b1
2
162
sin2
(1 cos )2exp (k rsin )
2[ ]exp (k nkr)2 z2[ ]n
(2.20)
The CTR spectrum above shows sharply peaked Gaussians at the Fourier components of
the electron beam modulation wavelength, r,n (the SASE FEL radiation wavelength).
This result is expected since CTR has the same Fourier spectrum as the electron beam
distribution as seen in Eq. 1.6 and any periodic modulations in the electron beam distribution
will be seen in the emitted CTR spectrum.
2.2.2 Angular Dependence of the CTR Spectrum
Integrating Eq. 2.18 over the wavenumber, k, and using the solid angle
d = 2 sin d , the angular spectrum for the first harmonic (n = 1 ) is found to be
dU
d e- beam
N2b12e2
8 z
sin3
(1 cos )2exp (kr rsin )
2[ ]. (2.21)
Figure 2.4 graphs the coherent angular energy dependence from Eq. 2.21 using the
harmonic wavelength, r = 13 m , and assuming a micro-bunching amplitude of 1%,
b1 = .01 . Also graphed is the incoherent spectrum found from Eq. 1.15 and Fig. 1.5 for a
narrow frequency bandwidth of 1% around r. Both plots use the experimental parameters
35
given in Table 2.1. The incoherent spectrum had to be enhanced by an additional factor
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CoherentIncoherent x 6*103
(rad)
6x10-10
3x10-10
00 .02 .04 .06 .08
dUd
(J/rad)
Figure 2.4: Angular spectrum for coherent transition radiation
and incoherent transition radiation for a 1% bandwidth around r.
of 6*103
to make it the same scale as the coherent radiation in this plot. It is seen not
only does the coherent radiation at this micro-bunching level have significantly more
signal than the incoherent, but is also contained in a much narrower angular spectrum.
Also, the maximum of the CTR signal in Eq. 2.14 is found to be
coh ( 2kr r)1
(2.22)
or about 8 mrad for the experimental parameters given in Table 2.1. This angle is much
smaller than the incoherent given by Eq. 1.16 where the maximum is near 50 mrad using
36
these parameters.
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Since the transverse dimension of the electron beam is usually larger than the
micro-bunch spacing, r > r, it is easy to understand why the angle for maximum CTR
would have to be quite small. This narrowing of the radiation cone comes from the fact
that radiation from different transverse points must add coherently together at the observation
point as discussed in Section 1.2. This states that the difference in distances from any two
points in the bunch to the observation point must be no more than about r/2 . Only at
small observation angles is this condition satisfied for beams that are transversely wide
compared to r (see Fig. 1.2).
Study of Fig. 2.4 introduces some experimental considerations concerning coherent
( coh. ) and incoherent ( i.c . ) radiation angular collection. From Fig. 2.4, it is possible for
the angular acceptance of a photon diagnostic beamline, acc , to collect the coherent
radiation and block most of the incoherent radiation when acc satisfies the criteria
1
2 rkr< acc a
where kr is the micro-bunching wavenumber, n is the harmonic number, bn is the
bunching factor, 0 is the density of the beam, and the radial distribution is modeled by a
parabola with radius, r= a , closely resembling a transverse Gaussian profile. The electric
fields are found by solving Poissons equation for the electric potential and implementing
the continuity conditions for the electric potential and radial electric field at r= a . The
radial electric fields of the micro-bunched electron beam are found to be (Appendix B)
Er(r,z) = 4 0bn cos nkrz( ) EnkrI1(n krr) 2A4
a2 n
kr
2
r
0 r a (B.21)
45
Er =Dnkrcos(nkrz)K1(nkr r) r a (B.22)
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2.2.5.2 Transversely Gaussian Beam
Now a comparison can be made between the electron beam radial field energy
and emitted CTR energy for the limit of
kr r
< .1
discussed with Eq. 2.24. From Eq. B.21 and B.22, we find the fields in this limit are
radially linear inside the beam and nonexistent outside the beam,
Er(r,z) = 4 0bn cos nkrz( )2A4
a2 n
kr
2
r 0 r a
Er = 0 a r
which agrees with the graphs in the Appendix for the 1.3 nm case.
We can use the results above in an analysis for the more realistic radial Gaussian
profile to find the fields of the micro-bunching in this limit (Eq. 2.24b) and compare the
results with Eq. 2.24a. In the limit where the micro-bunch spacing, r, is much smaller
than the radial beam size, r, the problem becomes one dimensional for finding the
longitudinal electric field (see longitudinal fields described above and in Appendix B,C
in this limit) and we can write Gauss law as
dEzdz = 4 (r,z) kr r 0 is vacuum,
z < 0 is metal, n is the unit vector to the observation point assumed to be in they-z plane
with angle from the z-axis and is the azimuthal angle around thez-axis.
The magnetic field emitted per particle (recalling transition radiation is modeled
by an electron interacting with its image charge as in Fig. 1.4) for transition radiation can
be found by noting the electromagnetic relations Jackson [7] for the magnetic field in
terms of the vector potential,
r
H(k,R) =1
c
r
A n , (2.37)
and the radiation power density,
d2U
dkd
=c2R2
4
2 H(k,R)2
. (2.38)
Using these and the power density/field relations in Chapter 1 for transition radiation
given by Eqs. 1.11 and 1.13, the total magnetic field to the observation point due to the
electron and its image charge is found to be
H(k,R) =e
cR
r
Vimage n
1 r
Vimage n
r
Ve
n
1 r
V
e
n
. (2.39)
For convenience and consistency with Chapter 1, the observation point is assumed to be
52
in they-z plane as shown in Fig 1.3. Noting n = cos z + sin y and the denominator in
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Eq. 2.39 for the field due to the electron is much smaller than the denominator for the
image charge (similar to the discussion accompanying Eq. 1.13 stating that the total field
at the observation point is primarily due to the particle with its velocity most parallel to
the observation vector, n ) results in a single electron transition radiation magnetic field
of
He_
(k,R) = e
cR
sin r'cos cos
1 + r' 2 cos + r' cos sin( )
x (2.40)
where a symmetric phase space distribution is assumed and any asymmetric functions in
can be ignored. This omission can be seen by examining Fig. 2.8 and Eqs 2.36 and
2.39. Here, the field at the observation point (in the y-z plane) due to an electron at a
point (x,y,z) with velocity vx( ) will cancel with the field due to an electron at (-x,y,z)
with velocity vx( ) = vx( ) making the last term in Eq. 2.36 negligible for this
model. It should be mentioned that if there is no radial velocity, r' = 0 , then the field
from Eq. 2.40 put into the energy spectrum of Eq. 2.37 results in the correct single
electron transition radiation energy spectrum obtained in Section 1.3.2 .
The total field at the observation point is the single particle field given in Eq. 2.40
summed over all particles with their differing angular momentum, r' , values. It is assumed
that the total phase space distribution is similar to Eq. 2.10 but with an additional normalized
radial velocity distribution,
(r,z,r' ) = S(r
x, r' ) =exp
r2
2 r2
2 r
2
exp
z2
2 z2
2 z
bn cos nkrz( ) exp
r' 2
2 'r2
2 'r
2n .
53
(2.41)
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The total field is found by integrating the field given in Eq. 2.40 over this distribution,
H(k,R) =N He_ (k,R) S(
r
x, r' )ei
r
krxd
3
xd2
r' (2.42)
where the additional phase factor accounts for the difference in distances from individual
electrons in the bunch to the observation point and is similar to the phase factor discussed
with Eq. 1.2 in Section 1.2. Many of the integrals are the same as in Section 2.2.1 and the
total field can be written as
H( ,R) = e
cRfl (k, )ft(k, )
sin r' cos cos
1+ r'2 cos + r'cos sin( )
exp r' 2
2 ' r2
2 'r2 r' dr' d
(2.43)
where the Fourier transforms from Eqs 2.16 and 2.18 have been used. Integrating over
the velocity space angle, , gives
H( ,R) = e
cRft(k, )fl(k, )
1
' r2 sin
a a2 b2
tan
exp r' 2
2 'r2
a2 b2r' dr' (2.44)
where the variables a and b are defined by
a =1 + r' 2
cos and b = r' sin . (2.45)
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The integration of Eq. 2.44 is quite complicated and must be done numerically. A Mathcad
document in Appendix D shows the method for this calculation. Defining
D( ) =1
' r2 sin
a a2 b2
tan
exp r' 2
2 ' r2
a2 b2r' dr'
0
10 scat
(2.46)
and assuming scat r' (not to be confused with the configuration space angle, ), the
energy density can be written as
d2U
dkd=
e2
4 2f
l(k, )f
t(k, )
2D
2( ) . (2.47)
It should be noted that the integral ofD( ) over the radial momentum out to 10 scat is
found to be sufficient for this calculation because the distribution in r' goes to zero after
about 5 scat. Also, the scattering angle for electrons depends on foil thickness and
material as defined in the Mathcad document. All integrations in Eq. 2.46 can be done
except for over . The total energy can now be written as
U=N2e2bn
2
8
1
z
D2
( )exp (kr rsin )2[ ]sin d
0
acc
. (2.48)
Equation 2.48 is very useful because it allows for the angular acceptance, acc , of the
optical beamline to limit the angular integration. The integral in Eq. 2.48 is done numerically
in the Appendix and a suitable angular step size chosen is seen to be the maximum of the
signal given by Eq. 2.22 divided by 200, d step ( 2kr r)1/ 200. It should be mentioned
the result of Eq. 2.25 for the undegraded CTR energy is done under the assumption that
55
the detector accepts the full angle, = 0 /2 which, as we will see in the description
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of the experiment for this thesis, is not the case and Eq. 2.48 will have to be used. Also,
the result of Eq. 2.25 assumes .1 rkr < . Using Eq. 2.48 no assumptions have to be
made relating the transverse beam size to the micro-bunch spacing.
As an example for this scattering induced degradation calculation, we us the
parameters in Table 2.1 and assume a 6um thick aluminum foil resulting in a scattering
angle of 9.7 mrad. The signal is found to be reduced by
Ud
U0 .60 (2.49)
where Ud is the forward emitted CTR from Eq. 2.41 and U0 is the forward emitted CTR
assuming a foil thickness of0 m . By using a very thin foil, it is seen around 40% of the
forward emitted CTR signal is lost due to scattering. In fact, if the thickness of the
aluminum is increased to 50 m , the degradation factor of Eq. 2.49 will be .11-a loss of
almost 90%. An experiment similar to the one described in this thesis was performed at
Brookhaven National Laboratory [19] in which a 50 m copper foil was used. In the BNL
experiment, the degradation factor in Eq. 2.49 turns out to be ~.005 explaining the
extremely low signal measured.
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Chapter 3
Experimental Setup
The CTR experiment described in this thesis was performed at the Advanced Free
Electron Laser (AFEL) at Los Alamos National Laboratory (LANL). The experimental
setup at this facility has been extensively described [20,21] but is reviewed in this chapter
emphasizing important and new aspects for this experiment. Section 3.1 describes the
general experimental setup and Sections 3.2 and 3.3 explore parameter measurements and
details of the data acquisition system.
3.1 AFEL Overview
Figure 3.1 shows the experimental setup for the thesis experiment. The AFEL
photo-injector uses a 10.5 cell L-band standing wave accelerator running at 1300MHz.
The drive laser uses a mode locked (at 108MHz frequency) diode pumped Nd:YLF
oscillator emitting infrared (IR) light pulses that are compressed using a fiber/diffraction
grating pair. The pulses are then amplified with a pair of flashlamp pumped Nd:YLF
rods. Next, the amplified IR pulses are frequency doubled to green with an LBO crystal
and frequency doubled again to ultra-violet (UV) with a BBO crystal. The output from
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the drive laser is shown in Fig. 3.2. Individual pulses are separated by 9.23 ns (108 MHz
rep rate) and have pulsewidths (FWHM) of 10ps and an energy of about 10 nJ. In
addition, the total length of the bunch train can be varied from 0-1000 individual pulses.
Before the UV light illuminates the cesium-telluride (Cs2Te) cathode creating an
electron train (photo-electric effect), it travels through an 8-14 mm optical iris insuring
transverse uniformity of the beam. In the experiment for this thesis, the iris was set at 12
mm and the laser pulse train length was around 3000 ns making a 325 bunch electron
train with about the same width pulse per pulse, 10 ps, similar to Fig. 3.2. Because of the
high quantum efficiency of the Cs2Te cathode, each laser pulse (10 nJ) could create an
electron bunch with up to 3 nC charge and after acceleration, each bunch would have an
energy around 18 Mev and a current of 180A. As the electron bunch train enters and
59
...
9.23 ns
10 ps(FWHM)
Bunch Train: 0-1000 Pulses
Figure 3.2: Pulse train out of the drive laser. Individual pulses are 10 ps FWHM and
are 9.23 ns apart.The bunch train can have between 0 and 1000 individual pulses.
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propagates through the undulator, each bunch will effectively be micro-bunched as was
discussed in Sections 2.1.2 and 2.1.3 and shown in Fig. 2.3. For the experiment in this
thesis, the system was run a frequency of 1 Hz, meaning every second 325 electron
bunches were accelerated and sent through the magnetic undulator.
When the laser pulse train hits the Cs2Te cathode, free electrons are emitted from
the surface and then accelerated by the 10.5 cell standing wave accelerator. At and near
the cathode, each electron bunch has a very low kinetic energy (0 3MeV) and is
highly space charge dominated and the transverse divergence of the electron bunch during
this period must be controlled [22]. Emitttance compensating solenoids are placed close
to the cathode and are used to focus the electron beam thus, controlling the transverse
divergence of each electron bunch. Downstream before the undulator, another solenoid is
used for matching the electron beam to the proper SASE FEL conditions mentioned in
Section 2.1.4. Between the photo-injector and the undulator, there are three steering
magnets used also for matching the trajectory of the electron beam to the undulator.
Three additional steering magnets were placed at .5, 1, and 1.5 m along the undulator for
additional steering of the electron beam.
The planar undulator used in this experiment was built from a collaboration between
UCLA and the Kurchatov Institute and is extensively described by Osmanov [23] and
Hogan [21]. Table 2.1 and Fig. 2.1 shows some of the parameters of this undulator. The
magnetic period along the electron propagation axis ( z ) is u = 2.06 cm , the peak magnetic
field, B0 , on axis is 7.4 kG, the normalized undulator field discussed with Eq. 2.3 is
K 1 and the vertical gap between the bottom and top sets of magnets is 5 mm. Since the
total length of the undulator is 2 m (98 periods) and the vertical gap is quite small, it is
easy to see why the addition of extra steering magnets to the undulator section of the
60
beamline was needed.
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Approximately 1 cm behind the last undulator period, an insertable CTR screen
was mounted. The diameter of the screen and holder is 2.54 cm and is much larger than
the exit aperture of the undulator, 5 mm, insuring that when inserted, all of the SASE
FEL light would be reflected back (see Fig. 3.3). At first, a 12 m thick aluminum foil
was used for the CTR screen. After a lower than expected CTR signal was measured, a
study of scattering effects showed we could increase the forward emitted CTR signal by
60% using a 6 m thick foil as discussed in Section 2.3. Results and a comparison of the
two signals will be discussed in the next Chapter. Three OTR blades are placed along the
undulator allowing for transverse electron beam spot size measurements. More will be
said about spot size measurements in the next section.
61
Forward emitted
CTR lobes
Figure 3.3: Aluminum CTR screen inserted. SASE FEL radiation isreflected. Only forward emitted CTR propagates to the HgCdTE detector.
electron beam
Backward
emitted
CTR lobes
Al Foil
Reflected
SASE
Radiation
Cathode
Undulator
HgCdTe
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When the screen is not inserted, the experimental setup in Fig. 3.1 shows the
SASE FEL radiation will propagate down the optical beamline. After insertion of the
CTR foil, all of the SASE radiation is reflected off the front foil surface and directed back
towards the cathode as shown in Fig. 3.3. The skin depth for 13 m light in aluminum is
around 50 nm insuring all the SASE radiation is reflected from the 6 m thick foil. Also,
the back emitted CTR propagates simultaneously with the reflected SASE radiation towards
the cathode and the only light to propagate towards the HgCdTe detector is the forward
emitted CTR as shown in Fig. 3.3. Using this setup, it is seen that the source points for
the CTR and SASE FEL radiation are at nearly the same location, the end of the undulator.
In addition, we know the CTR and SASE radiation will be at the same wavelength and
the collecting optics in the optical beamline can be the same for both CTR and SASE
radiation.
A1200
dipole spectrometer is located downstream from the undulator. The magnetic
field separates the electron beam from the co-propagating radiation (SASE or CTR) by
steering the electron beam towards the beam dump while the radiation continues propagating
down the optical beamline. The dipole spectrometer can be used to make energy and
energy spread measurements on the electron beam and will be discussed further in the
next section. As seen in Fig. 3.1, radiation traveling down the optical beamline will exit
from vacuum at the KBr window, be reflected by several 3 flat mirrors and is finally
focused by a ZnSe lens into the HgCdTe detector. The detector is about 3.5 m away from
the point source which limits the angular acceptance of the optical beamline for radiation
(SASE or CTR) collection to about acc 15 mrad . This important experimental parameter
allows for the separation of the forward emitted coherent radiation from the incoherent
radiation, as was discussed in Section 2.2.2 and Eq. 2.23. Also, to make spectral line
62
measurements of the CTR and SASE signals, a Jerrell Ash monchromator can be easily
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installed and removed from in front of the HgCdTe detector as shown in Fig. 3.1. When
installed, the measured CTR line spectrum through the monochromator can be compared
to the theoretical prediction shown in Fig. 2.5.
The experimental setup was briefly discussed above giving an overview of the
system. The next section will go into more detail about the measurements made and
needed to accurately describe the expected CTR spectra and energy.
3.2 Electron Beam and Radiation Measurements
We know from Eq. 2.25 the CTR energy is dependent on several of the electron
beam parameters,
UCTR N2b1
2e2
4 z kr r
4
. (2.25)
The present section discusses the methods and equipment used in measuring these electron
beam parameters necessary in understanding the emitted CTR/SASE radiation. All relevant
beamline diagnostic equipment is shown in Fig. 3.4. The interpretation of the resulting
data with regards to CTR radiation will be left for the next chapter where the experimental
results are discussed.
3.2.1 Measuring N, , r, z ,Uctr,USASE, , rCTR/SASE
The number of electrons in a bunch, N, or the charge, Q = eN, of the electron
63
bunch was measured using Beam Position Monitors (BPMs). (For a good discussion on
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Figure 3.4: Diagnostics of electron beam including OTR blades for spotsize and BPMs for charge measurements. In addition are the three extrasteering magnets along the undulator. See Figure 3.1 for reference.
To HgCdTe
Detector
BeamDump
End of
Linac
OTR Blades
Steering Magnets
BPMs
CTR Screen
Wall
these devices the reader should refer to Reference 27). These BPMs have a calibrated
output voltage of 33nC/V terminated into 50 . BPM1 is placed before the undulator
measuring the initial charge and BPM2 is after measuring the final charge exiting the
undulator. By measuring the ratio of the two BPM signals, BPM2/BPM1, the efficiency
(how much charge is being lost) of electron beam propagation through the undulator can
be determined. For the CTR experimental results, the charge read on