Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Dose properties of a laser accelerated electron beam and
prospects for clinical application
K. K. Kainza), K. R. Hogstrom, J. A. Antolak, P. R. Almond, and C. D. Bloch,
Department of Radiation Physics, The University of Texas M. D. Anderson Cancer
Center, Houston, Texas 77030.
C. Chiu, M. Fomytskyi, F. Raischel+, M. Downer, and T. Tajima++, Department of
Physics, The University of Texas at Austin, Austin, Texas 78712.
a) Correspondence to: Kristofer K. Kainz, Ph.D., Department of Radiation Physics, Unit 0094,
The University of Texas M.D. Anderson Cancer Center, Houston, TX 77030. Phone No.: (713)-563-
2587. Fax No.: (713)-563-2482. E-mail: [email protected].
+ Present address: Institut fur Theoretische Physik, Julius-Maximilians-Universitat at Wurzburg,
Am Hubland, 97074 Wurzburg, Germany
++ Present address: Kansai Research Establishment, JAERI, 8-1 Umemidai, Kizu, Kyoto, 619-0215, Japan
(Received _______________)
2
Abstract:
Laser Wakefield Acceleration (LWFA) technology has evolved to where it should be evaluated for its
potential as a future competitor to existing technology that produces electron and x-ray beams. The
purpose of the present work is to investigate the dosimetric properties of an electron beam that should be
achievable using existing LWFA technology, and to document the necessary improvements to make
radiotherapy application for LWFA viable. This paper first qualitatively reviews the fundamental
principles of LWFA and describes a potential design for a 30-cm accelerator chamber containing a gas
target. Electron beam energy spectra, upon which our dose calculations are based, were obtained from a
uniform energy distribution and from two-dimensional particle-in-cell (2D PIC) simulations. The 2D PIC
simulation parameters are consistent with those reported by a previous LWFA experiment. According to
the 2D PIC simulations, only approximately 0.3 % of the LWFA electrons are emitted with an energy
greater than 1 MeV. We studied only the high-energy electrons to determine their potential for clinical
electron beams of central energy from 9 − 21 MeV. Each electron beam was broadened and flattened by
designing a dual scattering foil system to produce a uniform beam (103% > off-axis ratio > 95%) over a
25×25 cm2 field. An energy window ( E) ranging from 0.5 − 6.5 MeV was selected to study central-axis
depth dose, beam flatness, and dose rate. Dose was calculated in water at a 100-cm source-to-surface
distance using the EGS/BEAM Monte Carlo algorithm. Calculations showed that the beam flatness was
fairly insensitive to E. However, since the falloff of the depth-dose curve (R10−R90) and the dose rate both
increase with E, a tradeoff between minimizing (R10−R90) and maximizing dose rate is implied. If E is
constrained so that R10−R90 is within 0.5 cm of its value for a mono-energetic beam, the maximum practical
dose rate based on 2D PIC is approximately 0.1 Gy⋅min−1 for a 9-MeV beam and 0.03 Gy⋅min−1 for a 15-
MeV beam. It was concluded that current LWFA technology should allow a table-top terawatt (T3) laser to
produce therapeutic electron beams that have acceptable flatness, penetration, and falloff of depth dose;
however, the dose rate is still 1 − 3 % of that which would be acceptable, especially for higher-energy
electron beams. Further progress in laser technology, e.g., increasing the pulse repetition rate or number of
high energy electrons generated per pulse, is necessary to give dose rates acceptable for electron beams.
Future measurements confirming dosimetric calculations are required to substantiate our results. In
3
addition to achieving adequate dose rate, significant engineering developments are needed for this
technology to compete with current electron acceleration technology. Also, the functional benefits of
LWFA electron beams require further study and evaluation.
Keywords: electron radiotherapy, laser-electron acceleration, LWFA, dosimetry
4
I. INTRODUCTION
In 1979, Tajima and Dawson1 originally proposed a method by which the wake generated
in a plasma by high-intensity laser pulses could accelerate charged particles to ultra-
relativistic energies. This process, which has since been established experimentally, is
referred to as Laser Wakefield Acceleration (LWFA). While the original theoretical and
experimental efforts regarding the acceleration of charged particles using lasers have
concentrated on electron beam formation, the acceleration of protons2−4 and heavy ions5
is also feasible.
LWFA technology has evolved to the point where its potential as a future alternative to
conventional electron radiotherapy technology should be evaluated. In fact, given the
experimental success over the past decade in accelerating electrons to energies of tens of
MeV, radiotherapy could potentially be one of the first practical applications of LWFA
technology. Conventional technology typically uses a linear accelerator that is powered
by a radio frequency (RF) amplifier system; by contrast, LWFA technology will utilize a
small gas-cell accelerating medium and a pulsed terawatt laser. Conventional technology
is mature, relatively inexpensive, dependable, safe, easily maintained, and clinically
useful. Although LWFA for radiation therapy has yet to be proven in any of these areas,
it has the potential to compete with conventional technology in all of them. Furthermore,
the potential of laser acceleration for proton beams might drive improvements to this
technology in the future; should that occur, then the utility of electron and x-ray therapy
using laser-accelerated electron beams will undoubtedly be reasonable to assess. One
5
purpose of the present work is to provide, for the benefit of medical physicists, an
overview of the underlying physics mechanisms of LWFA and the key hardware
elements utilized in the existing LWFA technology.
To date, there have been few if any reports that show the potential benefits of LWFA
from a functional perspective. Although the purpose of this paper is not to speculate on
potential benefits, a few points can illustrate some benefits that might be apparent. For
example, LWFA technology should offer opportunities for higher electron energies. This
would allow treatments to midline head and neck tumors and abdominal tumors, those
sites that motivated the development of the 50-MeV racetrack microtron.6 Second,
discontinuous segments of the LWFA energy spectrum could be selected, which would
allow a more uniform depth dose. This might be useful in chest wall irradiation,
particularly if arc therapy is used.7 Third, a LWFA device could allow continuously
available energies without retuning the accelerator. This should make more practical the
delivery of energy-modulated radiotherapy in continuous energy steps.8 Fourth, because
the acceleration region is so small, it might be possible to deliver scanned beam therapy
by mechanical scanning of the laser. LWFA technology might enable the development of
compact, inexpensive accelerators for use as dedicated electron machines for
intraoperative electron therapy or total skin irradiation. Also, a LWFA device might be
used to generate higher-resolution radiographs, given that a small laser focal spot size
could yield a small focal spot size for x-rays produced immediately beyond the plasma
target. Although the above ideas seem radical, they and others should not be discounted
until studied further.
6
However, for the potential clinical application of LWFA to be regarded seriously, it must
be demonstrated that a LWFA device can be used to create electron (and photon) beams
of a quality that at least matches that from existing RF linacs. Thus, the electron beam
extracted from laser-plasma interactions must exhibit certain characteristics. First,
LWFA must be capable of accelerating electrons to energies of at least 25 MeV, since the
beam energies routinely used for conventional electron therapy range from approximately
6 − 20 MeV. Second, the beam current achievable using LWFA must be sufficiently
large that an eventual LWFA device can deliver dose at a rate of at least 4 Gy⋅min−1.
Third, the width of the electron energy distribution must be sufficiently narrow so that the
depth-dose curves from these beams remain clinically useful. Therefore, another purpose
of this article is to provide dosimetry data useful to laser-plasma theoreticians and
experimentalists with regard to these three items above. Such feedback would
benchmark the magnitude of improvements in LWFA technology necessary to make the
modality adequate from a dosimetric perspective, and thus motivate further research of
LWFA.
The primary goal of the study performed by the group at The University of Texas M. D.
Anderson Cancer Center is to determine whether a laser accelerator system can meet the
above conditions for direct electron and x-ray therapies. Thus, a detailed analysis of the
dose capabilities of this system, for energies in the range E ∈ (6, 20) MeV, is required.
Part of this study includes calculations for the rate of total absorbed dose and the
dependence of dose versus depth for electron beams. The present investigation is based
7
on particle-in-cell simulation work, conducted by the group at The University of Texas at
Austin (UTA), which predicts the phase space of the electron beam from LWFA.9
An overview of LWFA physics and an introduction to some of the key hardware
elements of a laser accelerator system are given in Section II. Section III discusses the
conditions for which the particle-in-cell simulations were conducted, and describes how a
therapeutic beam might be produced from the LWFA beam. This includes energy-
window selection and beam broadening. There is then a description of the methods used
by investigators at M. D. Anderson Cancer Center to design the dual foil scattering
system and to calculate key properties of the dose distribution (e.g. off-axis profiles,
central-axis percent depth dose, and dose rate). Section IV presents the results for the
angular and energy distributions of these simulated electron beams, along with
calculations of dosimetric properties of the therapeutic beams. The prospects for a
clinical electron or x-ray therapy device, given these results, are discussed in Section V,
along with a proposed experimental program related to this LWFA system.
8
II. OVERVIEW OF LASER WAKEFIELD ACCELERATION FOR
ELECTRON BEAMS
II. A. LWFA physics
A comprehensive overview of the LWFA and other plasma-based accelerator concepts
has been given by Esarey et al.10 Here, we briefly summarize the main features of the
LWFA. Fig. 1 illustrates the basic interactions between a laser pulse and the plasma that
the pulse creates. When a laser pulse of sufficiently high intensity travels through a
neutral gas, the leading edge of the pulse ionizes the gas. Because this ionization process
occurs rapidly, the remainder of the laser pulse does not interact with neutral particles,
but rather with a plasma. The remainder of the laser pulse, in propagating through the
plasma, applies a force upon the electrons that is proportional to the gradient of the pulse
intensity. This force, referred to as the “ponderomotive force,” has a significant
component in the direction of propagation of the laser pulse. Thus, the electrons in the
plasma will be longitudinally displaced relative to the heavier positive ions. This space
charge displacement pulls the electrons back and forth, setting up a longitudinally
oscillating plasma wave.
The oscillation frequency ωp of the plasma wave, in a plasma with electron density ne, is
as follows11 (given in meter-kilogram-second units):
2
0
e ep
e
n qm
ωε
= , (1)
9
where me is the electron mass, qe is the electron charge, and ε0 = 8.85×10−12 C2⋅N−1⋅m−2 is
the permittivity of free space. The phase velocity vp of the plasma wave matches the
group velocity vgEM of the laser pulse:
cvcv EMg
pp <=−= 2
2
1ωω
, (2)
where ω is the oscillation frequency of the photons in the laser pulse.
Under certain circumstances, the travelling plasma wave, which is oscillating
longitudinally, will trap and accelerate charged particles, such as electrons, to velocities
that match and then exceed that of the travelling wave. The charged particles to be
accelerated could be injected into the plasma wave from an external source. If the plasma
wave amplitude is sufficiently high, they could also be drawn from the thermal
background charged particles that are trapped in the plasma wave. For high plasma wave
amplitude, there could also be a supply of electrons due to so-called wave breaking
phenomena.
To bring about plasma wave formation using a single laser source, a high-intensity pulse
(on the order of 1018 W⋅cm−2 or higher) is necessary. In the “standard” LWFA scheme, a
pulse length on the order of the plasma wavelength λp provides the most efficient
wakefield excitation:
2 ppulse p
p
vL
πλ
ω≈ = . (3)
10
An optimum resonance condition occurs when the plasma-wave oscillation period τp is
twice the temporal length τ of the laser pulse (or, equivalently, when the plasma
wavelength λp is approximately twice the laser pulse length Lpulse).10 For a hydrogen or
helium gas at standard temperature and pressure, the electron density ne is about 6×1019
cm−3 and ωp is about 4.4×1014 s−1, making 2 /p pτ π ω= about 14 fs. As the lengths of
pulses typically generated by terawatt laser systems are on the order of 100 fs, the
optimum resonance condition would be difficult to meet unless the plasma density was
below that for gas at STP. Thus, lower-density plasmas on the order ne ~ 1017 cm−3 are
used for this “standard” LWFA scheme.
Fig. 2 illustrates an inelastic scattering mechanism called Raman scattering, which is used
for generating plasma waves at a much higher plasma density, when Lpulse >> λp. When
Raman scattering occurs in atoms or molecules, incident photons at ω0 excite a bound
level of intermediate frequency ω1 < ω0, such as a molecular vibration, then emerge with
reduced frequency ωscatter = ω0 − ω1 (see [A]). When Raman scattering occurs in
plasmas, incident photons at ω0 excite a plasma wave at ωp, and emerge at the Stokes-
shifted frequency ωscatter = ω0 − ωp. The scattered light can emerge in any direction, but
forward Raman scattering12 is of greatest interest for acceleration because it excites a
plasma wave that travels at nearly the speed of light. In this case, the Raman-scattered
light co-propagates and beats with the incident light, as shown in [B] and [C]. The
beating modulates the incident light pulse, breaking it up into a train of shorter pulses, as
shown in [D]. Because the beat frequency is at ωp, the pulse train reinforces the growth
11
of the plasma wave at ωp, which in turn deepens the modulation of the incident pulse. In
the simple one-dimensional (1D) limit shown in Fig. 2, this positive feedback mechanism
is often called the “forward Raman instability.”12 When its 2D or 3D aspects are
included, it is called the “self-modulation instability.”13 The use of this laser-plasma
instability to create an accelerating structure for charged particles has come to be known
as the self-modulated LWFA (SM-LWFA).14
Of the various schemes for accelerating charged particles via laser-plasma interactions,
the SM-LWFA so far yielded electron bunches of the highest energy, charge (> 1 nC per
bunch), and collimation (transverse emittance ε < 0.1π mm⋅mrad). In an experiment
conducted at the University of Michigan, the generation of electron beams with energy E
up to 20 MeV from a plasma of density ne = 3.6×1019cm−3 was reported.15 Another group
at the Naval Research Laboratory reported an electron beam with energy Emax = 30 MeV
from a plasma of density ne = 1.4×1019cm−3.16 A maximum electron beam energy Emax =
90 MeV, achieved using a plasma density ne = 1.4×1019cm−3, was reported by a
collaboration at the Rutherford Appleton Laboratory.17 Leemans et al.18 at the Lawrence
Berkeley Laboratory have demonstrated SM-LWFA that produces nC bunches with Emax
= 30 − 50 MeV at repetition rates as high as 10 Hz. This represents about the upper limit
of repetition rate achievable with the SM-LWFA with current laser technology because of
the high peak power required to drive the modulation instability. Clearly, laser-plasma
acceleration can produce electron beam energies and charge per bunch adequate for
therapy, although repetition rate (and therefore average current) needs to be improved.
12
Recent simulations19, 20 have suggested that the pulse energy needed to drive the forward
Raman scattering process can be reduced significantly by the method of “Raman
seeding,” in which the primary pulse co-propagates into the plasma with a much lower
intensity secondary pulse that is offset in frequency by ωp from the primary pulse. The
Raman-seeded LWFA superficially resembles another laser-plasma acceleration scheme,
the plasma beat-wave accelerator (PBWA),21 in which the primary and Raman-shifted
secondary pulses are comparable in energy. However, according to simulations, it
requires less laser energy, is more compatible with plasmas of nonuniform density, and
can generate multiple MeV electrons as efficiently as the SM-LWFA. Although not yet
demonstrated in the laboratory, the Raman-seeded LWFA is a promising approach for
increasing the repetition rate produced by laser-plasma accelerators.
II. B. LWFA accelerator design
In this section, we describe the key hardware elements that comprise a prospective
electron LWFA system. Included are the laser and pulse amplification systems that are
common to most LWFA experiments as well as a proposed option for a plasma-gas
cylinder.9, 24
High intensities are achieved by amplifying the laser pulse and focusing the laser pulse
within the gas target. The laser source must be capable of generating pulses with
intensity (irradiance) on the order of 1018 W⋅cm−2. The components of such a “table-top
terawatt” (T3) system include a pump laser (such as a Nd:YAG laser) to provide an initial
13
laser pulse and a series of typically 3 or 4 amplifiers (such as Ti:sapphire crystals) to
provide the additional gain in pulse energy. The output terawatt laser pulse typically has
a pulse length of 30 − 100 fs and a wavelength of 0.8 − 1 µm.
To achieve the required pulse intensity without damaging the amplifier elements within
the T3 laser system, a technique known as “chirped pulse amplification” (CPA) is used.22
The CPA scheme is illustrated in Fig. 3. First, an optical configuration consisting of a
pair of diffraction gratings and a telescope is used to broaden the pulse, in time, into its
constituent frequencies. This “stretched” pulse is amplified, and the amplified stretched
pulse is then cast upon another pair of diffraction gratings that recompresses the pulse in
time. Initial experimental tests of CPA in 198522 increased the intensity of laser pulses
by a factor of 106; since then, gain factors of 1011 have been achieved.23
CPA pulses with energy approaching 1 joule and repetition rates as high as 10 Hz have
been used for all SM-LWFA experiments to date. Details of an experiment in progress to
implement the Raman-seeded LWFA have been illustrated by Downer et al.24 The
scheme being investigated generates a Raman-shifted seed pulse 1% as intense as the
primary pulse in a barium nitrate crystal internal to a CPA system. Simulations suggest
that this has the potential for acceleration equivalent to the SM-LWFA with lower pulse
energy and higher repetition rate.
A typical accelerator chamber consists of an evacuated vessel enclosing a gas jet or cell
1−2 mm thick and focusing mirrors, as shown in Fig. 4. A curved mirror on the input
14
side focuses the incident laser pulse, or laser pulse combination, typically at f/35 to a spot
radius on the order of 18 µm. To reach a peak intensity on the order of 1018 W⋅cm−2
needed for LWFA, the incident pulse should therefore have a peak power of 1 TW, e.g.,
an energy of 0.1 J for a 100 fs pulse. Somewhat higher intensity and energy is typically
needed for SM-LWFA, and potentially somewhat less for the Raman-seeded scheme. At
10 Hz repetition rate, the average power of this beam is on the order of 1 W. The laser
spot on the incident mirror should be 0.46 cm in diameter to keep the fluence below a
safe damage threshold of approximately 1 J⋅cm−2 for dielectric mirrors. Thus, a typical
focal length of about 15 cm, or total chamber length of about 30 cm, is needed.
The cylinder would be evacuated except for the gas cell. This is necessary because the
pulse would not focus well if the entire cylinder were filled with the plasma gas. Small
orifices would be placed on each side of the cell, along the chamber axis. This is
required because the laser pulses would eventually damage the gas-cell walls. Because
some of the gas within the cell will stream out of the orifices, the flow of gas into the cell
must be maintained. The pressure difference between the gas cell and the vacuum would
be maintained using differential pumping.
To produce the plasma wave in this accelerator via the seeded-pulse scheme, the helium
gas in the plasma cell would have an electron density of at least ne = 3.3×1019 cm−3. This
is slightly below the electron density of 5.4×1019 cm−3 for hydrogen or helium gas at STP.
Using a higher-density plasma gas leads to a larger number of thermal electrons that are
trapped and accelerated. This in turn allows a sufficiently large dose delivery without the
15
need for an injected external beam. A low-Z gas such as hydrogen or helium is used
because a higher-Z gas attenuates the laser pulse and increases the distortion of the pulse
within the plasma, thus inhibiting the ability to focus the pulse.
III. METHODOLOGY
III. A. Particle-in-cell simulations of LWFA
Given the promise of medical application for LWFA, the UTA investigators conducted
particle-in-cell (PIC) simulations of a laser accelerator apparatus similar to that described
above that might fulfill the requirements for direct electron beam therapy. The output
electron phase space distributions that are predicted by these simulations are used to
model a therapeutic electron beam and to assess the dosimetric properties of such a beam.
An iteration of the PIC-simulation algorithm proceeds as follows. Temporal and spatial
grids are defined over the region of interaction between a collection of charged particles
and the electric and magnetic fields with which they interact. These electric and
magnetic fields arise internally from interactions among the charged particles and
externally from the incident laser pulse. From the positions and velocities of the charged
particles at a given moment in time, the charge density and current density at each spatial
grid point is obtained. Maxwell’s equations are then used to determine, from these
charge and current densities, the electric and magnetic fields, and thus the Lorentz force,
at the grid points. Incorporating these Lorentz forces into equations of motion, and then
16
integrating these equations, gives the updated positions and velocities of the charges (i.e.,
the updated charge and current densities) to use in the next iteration. After a sufficient
number of iterations, the phase space distribution of an output electron beam is
determined.26
In a 2D PIC simulation, a two-dimensional grid (the simulation window) is defined;
associated with each grid point is the charge and current density of the plasma. At the
onset of the PIC simulation, the frame of reference of the simulation window is that of
the plasma, and the laser pulse is propagated into the window from one boundary. Once
the pulse has reached the center of the window, the window’s frame of reference
becomes that of the pulse. For the remainder of the simulation, in each iteration the grid
is shifted one cell width in the direction of the motion of the pulse. The values for EM
fields and charge and current densities for plasma that falls out of the window are
removed from the simulation, and field and density values for new, “quiet” plasma are
added to the window’s front edge. Changing the frame of reference in this manner
reduces the computation time for an arbitrary interaction length.
In a 1D PIC simulation, the simulation window encompasses a one-dimensional rather
than a two-dimensional grid. It is expected that results from 1D PIC are equivalent to
those from 2D PIC using an arbitrarily large spot size, although a careful study of
intermediate spot sizes (and the dependence of other laser-plasma parameters upon spot
size) is still in progress. Chiu et al.9 determined the electron energy distributions for two
“extreme” cases of the laser spot size, using 2D PIC to model the LWFA for a tight spot
17
size (i.e. one that is consistent with previous LWFA experiments) and 1D PIC to model a
large spot size. They found that the resulting electron energy distribution appears thermal
(exponentially decreasing with increasing energy) for 2D PIC simulations, but tends to
have an enhanced yield of higher-energy electrons (a flatter distribution) for 1D PIC
simulations.
The UTA PIC simulations modeled the evolution of a plasma generated by a seeded laser
pulse described in Section II.B. The UTA group had optimized several key parameters of
the laser-plasma system, such as primary pulse and seed pulse intensity and plasma
density. The guiding principle of this optimization is to maximize the electron current for
therapeutic electron energies (6−20 MeV) while keeping the laser intensity to a
minimum. Although adjusting the pulse intensity should increase the current (and thus
the dose rate) from the system, keeping the intensity as low as possible yields reductions
in the size, complexity, and operating cost of the LWFA system. A list of the key
parameters for the proposed accelerator system is given in Table I. Among these
parameters are the density and dimensions of the plasma cell, as well as the power, cross-
sectional size, and repetition rate of the oscillation-generating laser pulse; the parameter
set is consistent with the experimental setup of Leemans et al.18 For the PIC results upon
which our analyses are based, the intensity of the seed pulse was 1/100 that of the
primary laser pulse. The report by Chiu et al.9 discusses in greater detail the optimization
of this ratio.
18
The maximum energy of accelerated electrons, for the 2D PIC simulations, is also shown
in Table I, along with the quantity Ntotal per pulse. This is the total number of electrons
irradiated by the pulse at the focal point and along an effective interaction length of about
200 µm, and is thus the number of electrons that could potentially comprise the output
beam following each laser pulse. The quantity NE > 1 MeV is the number of electrons
accelerated per pulse with energy E > 1 MeV. Of the total number of plasma-cell
electrons irradiated by a laser pulse, only about 0.3% of them are accelerated to energies
suitable for therapy.
III. B. Producing a therapeutic beam from simulated data
As will be observed from the UTA 2D PIC simulation results presented in Section IV, the
energy distributions for LWFA electron beams are quite broad, on the order of tens of
MeV. Thus, tasks to perform within the M. D. Anderson Cancer Center analyses of the
UTA PIC simulation output include the selection of subsets of the output beam that have
narrower energy spreads, and for each subset, the design of hardware to broaden this
beam into the desired field area.
The beam exiting a conventional linac has a diameter on the order of 1 mm, which must
be transformed into a beam with a uniform region of at least 25×25 cm2 to treat the
patient. Two standard techniques are used to broaden beams from conventional linacs.
In one method, the beam is scanned across the treatment field using orthogonal, time-
varying magnetic fields;27 in the other method, a dual scattering foil system is utilized.
19
The former method has become less popular because of the risk of a malfunction of the
scanning magnet that can be hazardous to the patient.28 Presently, broadening the beam
using a dual scattering foil system is almost exclusively used; therefore, we will restrict
our analyses to that method for beam broadening. In the dual scattering foil system, a
uniformly thick primary foil is used to provide an initial Gaussian spread for the beam
profile. A Gaussian-shaped secondary foil is then used to scatter the central part of the
primary-foil output farther from the beam axis. With appropriate foil dimensions, the
resulting beam profile at the phantom surface is reasonably flat over the treatment field
area, and falls off outside the edges of the field area.
To select narrow ranges in energy from the raw beam distribution, one can implement an
achromatic bending magnet to spatially separate the beam’s component energies and a
variable collimating slit that transmits only those electrons within the selected energy
range. At this stage in the M. D. Anderson studies, details of energy window selection
are not being modeled, but this will eventually be done using a particle transport code,
and any additional loss of the beam due to energy selection and transport will be
assessed. It may also be possible to perform some degree of beam energy selection
without a magnet and collimators. High-energy components might be removed from the
output beam by appropriately adjusting the plasma density.
Raischel20 has investigated an alternative method that transports the broad beam within a
defined energy spread through achromatic magnet systems. His results produced beams
20
too narrow for most clinical situations, but which might be useful in abutting multiple
fields. This offers the potential for intensity and energy-modulated electron therapy.8
III. C. Dose simulation and calculation
From the electron-beam phase-space distributions predicted by the UTA PIC simulations,
various subsets of the beam energy distribution were used to assess the following
attributes of an eventual therapeutic beam: (1) beam flatness at the phantom surface; (2)
percent depth dose; and (3) peak dose rate. Calculating these three quantities was done
using the EGS/BEAM Monte Carlo program,29, 30 which models the transport of electrons
and photons through a given beamline geometry. The user provides the composition and
dimensions of the beamline elements (for these analyses, the scattering foils and the
water phantom), as well as the position, momentum, and energy spectrum of the incident
electron beam. For the dose-property analyses to be presented below, we drew our
energy-distribution subsets from two different shapes of the overall beam energy spectra.
A uniform beam energy distribution (akin to the UTA 1D PIC predictions) was
propagated through our software, to establish the “baseline” dependencies of the flatness
and depth dose upon the beam energy width. The energy spectrum predicted by the UTA
2D PIC simulations were put through the software chain to determine the flatness, depth
dose, and dose rate that are achievable using existing LWFA technology.
21
III. C. 1. Beam Flatness
Prior to running the BEAM simulations, the appropriate dimensions and material
composition for the scattering foils were determined for each canonical beam energy
under consideration. For a given beam energy, the dimensions of the foils were
determined by calculating the phantom-surface planar fluence for a given foil
configuration, and adjusting the foil dimensions within the calculation until the desired
shape for the planar fluence distribution was achieved.31 We briefly describe the
underlying theory and methodology for the analytical calculations of the phantom-surface
off-axis profile. We first set the distances along the beam axis (z-axis), between the
primary foil, secondary foil, and phantom surface, to be z=0, z1, and z2, respectively. We
then assume that any scattering medium (foil or air) will spread a collimated electron
“pencil beamlet” into a Gaussian-shaped profile, as is consistent with Multiple Coulomb
Scattering theory. Thus, a pencil beam incident upon the uniformly thick primary foil is
spread into a planar-fluence distribution Φ0(z1, ρ) at the secondary foil (z1), and is
centered about the beam axis with width σ0. A “pencil beamlet” originating at the
secondary-foil surface, at a distance ρ from the original beam axis, is spread into a
planar-fluence distribution Φ1(z2, r; z1, ρ) at the phantom surface (z2), centered at a
distance (z2/z1)ρ with width σ1. Both σ0 and σ1 depend upon the foil thickness and the
scattering power of each foil material; σ1 also varies with ρ because the secondary-foil
thickness varies with ρ. We calculate the scattering power using a thickness-dependent
expression presented in Huizenga and Storchi32 rather than the thickness-independent
ICRU35 formalism33; the latter expression gave scattering powers that were too large.
22
Since the probability of an electron being scattered at z=0 onto an area element ρ dρ dφ at
the location (z1, ρ) is Φ0(z1, ρ) ρ dρ dφ, the overall planar fluence Φp(z2, r) at the phantom
surface is calculated via a (numerical) convolution integral:
max2
2 0 1 1 2 10 0
( , ) ( , ) ( , ; , )p z r z z r z d dρπ
ρ ρ ρ ρ φΦ = Φ Φ . (4)
The initial design parameters for the foil system presumed a smooth-Gaussian shape for
the secondary foil. A corresponding stepped-foil approximation to the Gaussian shape
was then obtained for the secondary foil; this consisted of 3 beveled-edged disks of equal
thickness stacked on top of each other. This was done to more easily represent the
secondary foil geometry within the EGS simulation. The separation along the beam axis
between the foils was set to 10 cm, and the distance between the primary-foil plane and
the phantom surface was taken to be 100 cm (standard source-to-axis distance SAD).
The foils were designed to scatter the beam into a circular treatment field of radius
12.5√2 cm, to achieve flatness within a 25 × 25-cm2 square field. At the edges of the
circular field, the goal for the relative planar fluence of the scattered beam was 95% of
the central-axis planar fluence, with no value greater than 103% of the central-axis planar
fluence. The materials used for the scattering foils were gold for the primary and
aluminum for the secondary.
23
III. C. 2. Percent depth dose
Having determined an acceptable dual scattering foil system, EGS was then used to
propagate subsets of the LWFA electron-beam energy spectrum through the foils and
onto a water phantom. BEAM29 was used to model the beamline and a water phantom.
The “CHAMBER” component module was used to model a water phantom and to score
the central-axis depth dose. The following parameters were set within BEAM to halt
further transport of electrons and photons whose kinetic energy fell below 10 keV: AE =
0.521 MeV; ECUT = 0.521 MeV; AP = 0.010 MeV; and PCUT = 0.010 MeV.
The following quantities were extracted from each of the depth-dose curves generated
using EGS/BEAM. The “therapeutic depth” R90 is the penetration depth along the beam
axis at which the distal end of the 90% relative-dose contour resides. R10 is the depth
where the central-axis relative dose is reduced to 10%. The difference R10−R90 roughly
characterizes the minimum distance between the target volume and critical structures for
which the target receives the full-prescribed dose and the critical structures receive
minimal dose. A larger R10−R90 indicates a higher unwanted dose to regions beyond R90.
III. C. 3. Dose rate
A determination of the rate at which dose may be absorbed from a LWFA-generated
therapy electron beam is also of interest, given that such a device should be capable of
delivering dose at a rate of at least 4 Gy⋅min−1. EGS-based Monte Carlo simulations
24
were used to determine the dose rate from a “realistic” beam, i.e. one that emerges from
the dual scattering foil system such that the electrons may lose energy and/or be scattered
outside the treatment field. Of interest is the dose rate evaluated at R100, the point on the
central axis where the maximum dose is deposited, and this was calculated in the
following manner. First, the depth-dose curves obtained via EGS/BEAM, in the manner
described above and for the same canonical values for Ecent and ∆E, were output in units
of dose per incident electron. (This is in fact the default output format for central-axis
dose results from a BEAM simulation.) This quantity was then multiplied by the number
of electrons per minute (assuming a 10-Hz pulse repetition rate) predicted by the UTA
PIC simulation for the selected Ecent and ∆E values. This prescription may be
summarized by the equation:
100from100EGS4/BEAM
fromcentUTA PIC
dose per incident electronat on central axis
number of electrons per 600 pulsespulse within ( , ) minute
R
dDRdt
E E
= ×
×∆
. (5)
The objective of these calculations of absorbed dose rate is to study the dependence of
100( / )RdD dt on the central beam energy Ecent and the energy spread ∆E for beams
predicted using the UTA system with the current 2D PIC simulation parameters.
25
IV. RESULTS AND DISCUSSION
IV. A. Phase-space properties of LWFA electron beams
Before presenting the energy distributions for the E > 1-MeV electrons (that comprise
0.3% of the output beam) predicted by the UTA PIC simulations, some comments
regarding the beam angle distribution from 2D PIC should be made. In a full 3D
geometry, the polar angle θ is defined by Fig. 5[A] as the angle between the momentum
vector of the electron and the z-axis; the azimuthal angle φ denotes the direction of the
momentum’s transverse component in the plane perpendicular to the z-axis. The z-axis is
defined by the direction of the incident laser pulse, and coincides with the axis of the
accelerator cylinder. However, when modeling the interactions among the charged
particles within the plasma, the 2D PIC simulations account for only the longitudinal
component (in the direction of the incident laser pulse) and one transverse component
rather than two. Thus, from the output of the 2D PIC simulation, only two momentum
coordinates are given for each electron, a longitudinal (“z”) coordinate and a transverse
(“y”) coordinate. As illustrated in Fig. 5[B], we define an estimate of the polar angle
(“θy”) in terms of these two electron-momentum coordinates. Such an estimate should be
viewed as qualitative, though, as in the 3D case the self-focusing effects and evolution of
the plasma wave will differ from the 2D simulation.
The scatter plot in Fig. 5[C] shows the correlation between estimated polar angle and the
kinetic energy for the higher-energy (i.e. E > 1 MeV) electrons that result from the 2D
26
PIC simulation. For electrons with therapeutically useful energies (i.e. above 5 MeV),
the polar angle spread is relatively constant with respect to energy, and the FWHM of our
polar angle estimate is about 20°. This is actually somewhat greater than the “3D” polar
angle distributions with FWHM ~ 10° that have been observed in previous LWFA
experiments, for example by Wagner et al.15 Given that the 2D PIC simulations can only
qualitatively describe the trends regarding the polar angle, the approach for our
subsequent analyses is to use the distribution predicted by 2D PIC, but to assume a mono-
directional, point beam rather than an initial angular spread.
In a prospective clinical LWFA device, it should be possible to accommodate an inherent
angular spread in the electron beam in the following manner. First, the beam optics could
be designed to focus the beam to a point at the location of the primary scattering foil.
Second, it would most likely be necessary to decrease or eliminate the thickness of the
primary scattering foil derived for an incident pencil beam, to flatten the beam to the
desired profile. If the profile of the beam directly from LWFA is Gaussian in shape,
placing a uniform primary foil within the beam would merely increase the width of that
Gaussian distribution. Assuming a 10° FWHM in polar angle, at 100-cm SSD and with
no profile-flattening hardware to be in place, the FWHM of the beam profile at the
phantom surface would be about 17.5 cm. This is less than the 25√2 cm width that we
require for a clinical beam. Although incorporating an initial angular spread into our
scattering-foil design calculation is a focus of future studies, for our current analyses we
assume the initial angular spread of the electrons to be negligible. We then design the
primary and secondary foils accordingly, keeping in mind that for a beam with an initial
27
angular spread the primary foil thickness would be reduced in order to maintain the beam
profile at the SSD.
The circles in Fig. 6 indicate the distribution of dN/dE versus E for the electrons that
were generated via UTA 2D PIC simulations. These simulations used a laser pulse
power of 1.7 TW, a laser pulse intensity of 7.6×1017 W⋅cm−2, a spot radius of 3√2 µm,
and an electron density of 4.85×1019 cm−3 (corresponding to λp = 4.8 µm). For
comparison, the energy spectrum as reported by Leemans et al.18 (the experimental
conditions upon which the 2D PIC simulation parameters are based) is indicated by the
dotted line. The energy distribution of LWFA-generated electrons is quite broad, and a
means to select a narrower subset of beam energies from this distribution will be required
for a therapy device. The UTA LWFA system is capable of generating electrons with
therapeutically useful energies, up to about 21 MeV. The number of beam electrons per
pulse with E > 1 MeV is about 9.3×109, corresponding to a charge of 1.5 nC. This
prediction agrees with the measured values of about 1.7 nC per pulse reported by
Leemans et al.18 and is consistent with the 0.5 nC per pulse reported by Wagner et al.15
To smooth this distribution, in preparation for the calculations of central-axis depth-dose
curves and total dose rate, a fit of a fifth-degree polynomial to ln(dN/dE) vs. E has been
made. The solid line in Fig. 6 indicates the fit result. The dose calculations to follow
will presume the ability to select narrower portions of this distribution for therapy.
Energy spreads (∆E) of 0.5, 2.5, 4.5, and 6.5 MeV will be examined; these spreads will
28
be centered about the nominal “central” beam energies Ecent under evaluation: 9, 12, 15,
18, and 21 MeV.
IV. B. Dosimetric properties of LWFA beams
IV. B. 1. Flatness
A dual-foil system was designed for each of the canonical central energies Ecent. The foil
parameters were optimized for a mono-energetic beam (∆E = 0), and then the same foil
geometry was used to simulate the scatter of beams with a nonzero ∆E centered about
Ecent. The resulting dimensions of the primary and secondary foils, for each beam energy,
are summarized in Table II. If the shape of the secondary foil is described by the
Gaussian equation t=Texp(−r2/R2), where t is the foil thickness at a distance r from the
central axis, then the parameter T denotes the central-axis thickness of the foil and R
relates to the foil’s width. For each 3-layer secondary foil configuration, this parameter R
was (0.95)⋅(1.4) cm.
Before presenting results for the depth-dose curves from EGS, we established that a beam
profile distribution predicted by EGS agrees with the analytically calculated profile
distribution for the same scattering foil geometry. Fig. 7 compares the calculated profile
distribution for a 9-MeV beam with the prediction from EGS. The foil configuration was
optimized to scatter a 9-MeV electron beam into a circular field of diameter 25√2 cm.
29
Over the radial extent of the maximum treatment field (r ≤ 12.5√2 cm), the calculation
and the EGS prediction agree sufficiently well.
It is necessary to evaluate the effect of increasing the beam energy width upon the
flatness of the beam at the phantom surface. This effect has been estimated using the
analytically calculated beam profile distributions, given that they are consistent with the
predictions by EGS according to Fig. 7. In the top two frames, plots of the calculated
profile distributions, for uniform beam energy distributions and for beam energy widths
up to 6.5 MeV, are shown in Fig. 8 for [A] Ecent = 9 MeV and [B] Ecent = 15 MeV. In the
bottom two frames, the profile distributions are determined from beam energy
distributions that are subsets of the smoothed UTA 2D PIC distribution shown in Fig. 6;
profile distributions for [C] Ecent = 9 MeV and [D] Ecent = 15 MeV are shown. At each
energy, the same dual-foil geometry was used for each of the beam energy widths.
In general, the beam profile changes a small amount over the 12.5√2-cm field radius as
the energy spread is increased; thus, a broad central beam energy has only a small effect
on the flatness of the beam. The quantity ∆E/Ecent is larger for the lower-energy beams;
thus, as the profiles for the 9-MeV beams in Fig. 8[A] illustrate, keeping the beam energy
width below ~4.5 MeV will preserve the flatness of the profile. For higher-energy
beams, ∆E/Ecent changes little with increasing ∆E; as Fig. 8[B] shows, beam energy
widths up to 6.5 MeV are acceptable to maintain the profile flatness.
30
For the 9-MeV beams sampled from the 2D PIC distribution, the profile shown in Fig.
8[C] changes little with increasing ∆E. This is because, for a wider energy window about
Ecent = 9 MeV, the lower-energy component is enhanced relative to the corresponding
energy window from the uniform beam; thus, the decrease at the edges of the profile seen
in Fig. 8[A] is less apparent in Fig. 8[C]. The behavior seen near the edges of the profile
in Fig. 8[D], for the 15-MeV beams sampled from the 2D PIC distribution, is due to those
beam-energy subsets being biased toward lower energies as ∆E increases. Lower-energy
electrons are scattered farther from the central axis by the same foil configuration. If
unacceptable, the primary foil thickness can be re-optimized. The “shoulders” that
appear in the vicinity of approximately 22 cm from the central axis in Fig. 8[B] and [D]
are due to the radius of the bottom layer of the stepped-foil configuration being limited to
2.3 cm (projected to 23 cm at isocenter).
IV. B. 2. Percent depth dose
Central-axis depth-dose curves were calculated by EGS/BEAM using the previously
described scattering-foil geometries. Results are shown in Fig. 9 for ∆E values, ranging
from 0.5−6.5 MeV, serving as the input energy spectra for EGS/BEAM. Results are
shown in Figs. 9[A] and [C] for Ecent = 9 MeV and Figs. 9[B] and [D] for Ecent = 15 MeV.
Figs. 9[A] and [B] used a uniform energy spectrum, and Figs. 9[C] and [D] used the
energy spectrum generated by the 2D PIC simulations (cf. Fig. 6). The effect of
increasing the width of the energy distribution is demonstrated in these plots.
31
For uniform energy distributions, as ∆E increases, R90 decreases and R10 increases. For
both the 9-MeV and 15-MeV cases in Figs. 9[A] and 9[B], respectively, the increase of
R10 (as much as 1 cm) with increasing ∆E is significantly greater than is the
corresponding decrease in R90 (no greater than 1 mm). For the beam energy distributions
extracted from the 2D PIC simulation results, the behavior of the Ecent = 9 MeV depth-
dose curves with increasing ∆E (Fig. 9[C]) is similar to that observed for the uniform
distribution. However, the decreases in R90 are greater and the increases in R10 are less;
both effects are attributable to more lower-energy electrons in the ∆E window. For the
Ecent = 15 MeV beams (Fig. 9[D]) from 2D PIC, R90 decreases significantly (as much as 5
mm) whereas R10 decreases minimally (< 2 mm) with increasing ∆E. This difference
from the results of the uniform energy spectrum again is attributed to the sharply
decreasing beam-energy distribution in 2D PIC, beyond E ≈ 15 MeV (c.f. Fig. 6).
Fig. 10 plots the quantity R10−R90 from the EGS depth-dose curves against R90 for each of
the central energies (9, 12, 15, 18, and 21 MeV) and energy spreads under consideration.
Fig. 10[A] shows the curves from a uniform beam distribution, and Fig. 10[B] shows the
curves from the 2D PIC distribution of Fig. 6. In general, as the energy (and hence
therapeutic depth) increases, so also does R10−R90. At a lower value of R90, e.g. 3 cm,
R10−R90 ranges from approximately 1.5−2.5 cm. At a high value of R90, e.g. 5 cm,
R10−R90 ranges from approximately 2.7−3.6 cm. For the same central energy, although
R10−R90 for the 2D PIC distribution is slightly less than that for the uniform distribution,
R90 is also less for the former. When R10−R90 is plotted versus R90 (the key clinical
32
parameter), the resulting curves for the uniform distribution and the 2D PIC distribution
are comparable.
IV. B. 3. Dose rate
Dose rate at R100, calculated using Equation (5), requires the dose per incident electron
from the Monte Carlo calculation. At the depth R100 and for a given central energy Ecent,
the dose per incident particle tends to change little with increases in ∆E. For Ecent = 9
MeV beams from the UTA 2D PIC distribution, the dose per particle decreases slightly
with increasing ∆E, ranging from 1.2×10−13 Gy for mono-energetic beams to about
1.1×10−13 Gy at ∆E = 6.5 MeV. For the Ecent = 15 MeV beams, the dose per incident
particle at R100 again decreases with increasing ∆E, ranging from about 1.1×10−13 Gy for
∆E = 0.5 MeV to about 9.3×10−14 Gy at ∆E = 6.5 MeV. The product of the dose per
incident electron at R100 is multiplied by the number of electrons produced per minute
(obtained from Fig. 6) to obtain the dose rate (Gy⋅min−1) at R100 for the associated values
of Ecent and ∆E.
The absorbed dose rate at R100 versus energy spread, for beams selected from the UTA
2D PIC distribution of Fig. 6, is shown in Fig. 11; the dose rates assume a pulse repetition
rate of 10 Hz. Since the 2D PIC energy distribution falls off with increasing Ecent, so also
does the dose rate. For even the widest beam energy widths, the dose rate for a 12.5√2-
cm radius treatment field is, at best, over ten times less than the 4 Gy⋅min−1 necessary for
a LWFA-based electron therapy to compete with conventional linacs.
33
Plots of the dose at R100 versus the R10−R90 for the depth-dose curve are featured in Fig.
12, for beams sampled from the 2D PIC distribution. In general, as ∆E increases, the
dose rate increases, but so does the depth-dose falloff distance R10−R90. For a given Ecent,
one could consider a value of R10−R90 that exceeds the minimum (for a mono-energetic
beam) by 0.5 cm to be acceptable. In that instance, the energy width for the 9-MeV beam
may be taken as high as approximately 4.5 MeV; for the 15-MeV beam, ∆E as large as
approximately 6.5 MeV is acceptable. However, the dose rate from both beams is still
well below 4 Gy⋅min−1.
V. CONCLUSIONS
V. A. Electron beams
According to the output from the 2D PIC simulations, the LWFA-generated electrons that
are potentially clinically useful (i.e. with E > 1 MeV) exhibit energy spectra that are quite
broad, on the order of tens of MeV. Whereas these spectra show the proposed LWFA
system to be capable of producing a clinically useful range of electron beam energies, the
widths of the spectra require a means to select subsets with narrower beam energy
widths. For example, a standard 270-degree achromatic magnet with collimating blocks
inside could be used to select the range of energies for the therapy beam.
34
Subsets of the beam energy distributions, from both a uniform distribution and the
spectrum predicted by the 2D PIC simulations, were implemented into EGS, which
propagated these subsets through a dual-foil beam scattering system and into a water
phantom. Results from these simulations demonstrated the flatness of the beam, the
shape of the central-axis depth-dose curve, and the absorbed dose rate at R100 on the
central axis. The dependence of these quantities on the central beam energy and beam
energy width was investigated. For beams from a uniform energy distribution, the
“gradient” of the depth-dose curves (R10−R90) increases as beam energy and beam energy
width increase. From near-monochromatic beams (∆E ~ 0.5 MeV) to beams with widths
of approximately 6.5 MeV, R10−R90 ranges from about 1.5 cm to about 5 cm for uniform-
distribution beams, and from about 1.5 cm to about 4 cm for 2D PIC-predicted beams.
The absorbed dose rate increases with increasing beam energy width, although the dose
rate is well below 4 Gy⋅min−1 even for energy widths as great as 6.5 MeV. The results
for depth-dose shape and dose per particle for uniform-distribution beams suggest that a
trade-off is involved when adjusting the beam energy width; whereas greater widths
result in high dose rates, smaller widths result in sharper falloff of the depth-dose curves.
For the prospective electron LWFA device with the above-proposed operating
parameters, the maximum practical dose rate (for the highest beam energy widths and for
central energies up to about 15 MeV) ranges from approximately 0.03 Gy⋅min−1 for 15-
MeV beams to 0.1 Gy⋅min−1 for 9-MeV beams. To achieve electron dose rates of at least
4 Gy⋅min−1, the beam current will need to be increased by about two orders of magnitude
assuming the 2D PIC-predicted beam currents. We should reiterate that the above dose
35
rates account only for the beam loss due to the foils’ scatter of the beam outside the
defined field, and not for any loss due to inefficiency in beam energy selection and
transport.
V. B. Potential for x-ray beams
For LWFA to be useful in a clinical setting, it is likely that this technology must also be
capable of producing suitable x-ray beams. Although not the subject of the present work,
studies of x-ray beams produced using LWFA electrons are important and should be the
focus of future studies. The present study showed that, for beam energies approximately
15 MeV, dose rates of about 4 Gy⋅min−1 for electrons could be achievable with an
appropriate increase in beam current (up to 100 times greater) using the current set of
UTA 2D PIC simulation parameters. However, the beam currents will need to be an
additional 2−3 orders of magnitude greater than that to achieve similar dose rates for
6−20 MV x-ray therapy.27
A key shortcoming of LWFA at this stage is the low pulse repetition rate of the current
terawatt laser systems. Increasing the pulse repetition rate would help to achieve the
higher beam currents necessary for x-ray therapy. The damage thresholds of the optical
elements in the laser cavity restrict the pulse repetition rate to the 10-Hz value currently
used. However, advances are being made that would considerably increase these damage
thresholds. A recent development34 suggests that pulse rates up to 1 kHz should be
achievable for pumped-laser systems. Additionally, one could envision increasing the
36
intensity gain of the laser pulse amplification system. Also, for x-ray beams, it should be
possible to relax the requirement for a narrow spread in the energy of the electron beam
upon the x-ray target, given that the energy distribution is already broad for x-rays
utilized in conventional treatments. We are planning studies similar to the present one in
the near future to evaluate dose rates and properties of the dose distribution for x-ray
beams produced using LWFA-generated electron beams.
V. C. Future work
In this work, our analysis is based on the UTA group’s 2D PIC simulation results for one
parameter set, and this approach is justified to some extent. Our analysis needs to be
repeated once the dependence of electron beam quantities (such as current, maximum
energy, and angular divergence) upon various PIC-simulation parameters (such as the
primary and seed pulse intensity, the dimensions of the laser-plasma interaction region,
the beam-spot radius, and the plasma density) are better understood. Eventually,
experimental verification is intended to be carried out by two experiments. One
experiment would construct the hardware to generate the seeded laser pulse, and establish
that the plasma wave amplitude and the subsequent trapping and acceleration of electrons
within the plasma are indeed enhanced relative to the SM-LWFA regime. Also, it must
be demonstrated that the accelerator cylinder and gas cell apparatus are viable.
Measurements verifying our calculations of dose rate, depth-dose, and the ability to
flatten the beam would follow.
37
Along with beam broadening, other design issues for an eventual clinical LWFA device
must be examined. For example, it is necessary to adequately shield the substantial
number of electrons (> 99.8 %) produced by LWFA that have energies below those
suitable for therapy. The heating of an eventual low-energy electron shield must be
estimated along with any possible leakage dose from such a shield.
For the use of LWFA-generated electrons in conventional radiotherapy (for electron and
x-ray beams) to ever become likely, it is necessary to increase the beam current from
laser-plasma interactions by approximately 105 times what is presently achievable. At
the same time, efforts to reduce the spread of the electron energy distribution should be
explored as well. An energy width up to about 4 MeV may be tolerable, in that it would
keep the profile flatness within 5 percent and the depth-dose falloff distance to within 0.5
cm of its minimum value for a given central beam energy.
38
References:
1 T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267-270
(1979).
2 E. L. Clark, K. Krushelnick, J. R. Davies, M. Zepf, M. Tatarkis, F. N. Beg, A.
Machacek, P. A. Norreys, M. I. K. Santala, I. Watts, and A. E. Dangor, “Measurements
of energetic proton transport through magnetized plasma from intense laser interactions
with solids,” Phys. Rev. Lett. 84, 670-673 (2000).
3 A. Maksimchuk, S. Gu, K. Flippo, D. Umstadter, and V. Yu. Bychenkov, “Forward ion
acceleration in thin films driven by a high-intensity laser,” Phys. Rev. Lett. 84, 4108-
4111 (2000).
4 T. E. Cowan, M. Roth, J. Johnson, C. Brown, M. Cristl, W. Fountain, S. Hatchett, E. A.
Henry, A. W. Hunt, M. H. Key, A. MacKinnon, T. Parnell, D. M. Pennington, M. D.
Perry, T. W. Phillips, T. C. Sangster, M. Singh, R. Snavely, M. Stoyer, Y. Takahashi, S.
C. Wilks, and K. Yasuike, “Intense electron and proton beams from PetaWatt laser-
matter interactions,” Nucl. Instrum. Meth. A 455, 130-139 (2000).
5 M. Hegelich, S. Karsch, G. Pretzler, D. Habs, K. Witte, W. Guenther, M. Allen, A.
Blazevic, J. Fuchs, J. C. Gauthier, M. Geissel, P. Audebert, T. Cowan, and M. Roth,
39
“MeV ion jets from short-pulse-laser interaction with thin foils,” Phys. Rev. Lett. 89,
085002-1-085002-4 (2002).
6 M. Karlsson and B. Zackrisson, “Exploration of new treatment modalities offered by
high energy (up to 50 MeV) electrons and photons,” Radiother. Oncol. 43, 303-309
(1997).
7 D. D. Leavitt, J. R. Stewart, and L. Earley, “Improved dose homogeneity in electron arc
therapy achieved by a multiple-energy technique,” Int. J. Radiat. Oncol. 19, 159-165
(1990).
8 K. R. Hogstrom, J. A. Antolak, R. J. Kudchadker, C.-M. Ma, and D. D. Leavitt,
“Modulated Electron Therapy” in Intensity Modulated Radiotherapy, edited by T. R.
Mackie and J. R. Palta (Proceedings of the AAPM 2003 Summer School, Medical
Physics Publishing, Madison, Wisconsin, 2003), pp. 749-786.
9 C. Chiu, M. Fomytskyi, F. Grigsby, F. Raischel, M. C. Downer, and T. Tajima, “Laser
Accelerators for Radiation Medicine: a Feasibility Study,” submitted to Med. Phys.
(2003).
10 E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Overview of plasma-based accelerator
concepts,” IEEE Trans. Plasma Sci. 24, 252-288 (1996).
40
11 J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley & Sons, New York,
New York 1975), p. 492.
12 C. D. Decker, W. B. Mori, T. Katsouleas, and D. E. Hinkel, “Spatial temporal theory of
Raman forward scattering,” Phys. Plasmas 3, 1360-1372 (1996).
13 T. M. Antonsen, Jr. and P. Mora, “Self-focusing and Raman scattering of laser pulses
in tenuous plasmas,” Phys. Fluids B 5, 1440-1452 (1993).
14 J. Krall, A. Ting, E. Esarey, P. Sprangle, and G. Joyce, “Enhanced acceleration in a
self-modulated laser wakefield accelerator,” Phys. Rev. E 48, 2157-2161 (1993).
15 R. Wagner, S.-Y. Chen, A. Maksimchuk and D. Umstadter, “Electron acceleration by a
laser wakefield in a relativistically self-guided channel,” Phys. Rev. Lett. 78, 3125-3128
(1997).
16 C. I. Moore, A. Ting, K. Krushelnick, E. Esarey, R. F. Hubbard, B. Hafizi, H. R.
Burris, C. Manka, and P. Sprangle, “Electron trapping in self-modulated laser wakefields
by Raman backscatter,” Phys. Rev. Lett. 79, 3909-3912 (1997).
17 D. Gordon, K. C. Tzeng, C. E. Clayton, A. E. Dangor, V. Malka, K. A. Marsh, A.
Modena, W. B. Mori, P. Muggli, Z. Najmudin, D. Neely, C. Danson, and C. Joshi,
41
“Observation of electron energies beyond the linear dephasing limit from a laser-excited
relativistic plasma wave,” Phys. Rev. Lett. 80, 2133-2136 (1998).
18 W. P. Leemans, D. Rodgers, P. E. Catravas, C. G. R. Geddes, G. Fubiani, E. Esarey, B.
A. Shadwick, R. Donahue, and A. Smith, “Gamma-neutron activation experiments using
laser wakefield accelerators,” Phys. Plasmas 8, 2510-2516 (2001).
19 D. L. Fisher and T. Tajima, “Enhanced Raman forward scattering,” Phys. Rev. E 53,
1844-1851 (1996).
20 F. Raischel, A Laser Electron Accelerator System for Radiation Therapy, M.A. Thesis,
The University of Texas at Austin, 2001 (unpublished).
21 C. E. Clayton, K. A. Marsh, A. Dyson, M. Everett, A. Lala, W. P. Leemans, R.
Williams, and C. Joshi, “Ultrahigh-gradient acceleration of injected electrons by laser-
excited relativistic electron plasma waves,” Phys. Rev. Lett. 70, 37-40 (1993).
22 D. Strickland and G. Mourou, “Comparison of amplified chirped optical pulses,” Opt.
Commun. 56, 219-221 (1985).
23 G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-intensity lasers: physics of
the extreme on a tabletop,” Phys. Today 51, 22-28 (January 1998).
42
24 M. C. Downer, C. Chiu, M. Fomyts’kyi, E. W. Gaul, F. Grigsby, N. H. Matlis, B.
Shim, P. J. Smith, and R. Zgadzaj, “Plasma channels and laser pulse tailoring for GeV
laser-plasma accelerators,” in Advanced Accelerator Concepts X, edited by C. E. Clayton
and P. Muggli (AIP Conference Proceedings Vol. 647, 2002), pp. 654-663.
25 R. L. Jones, III, Compact Laser-Driven Medical Electron Accelerators, M.A. Thesis,
The University of Texas at Austin, 1999 (unpublished).
26 E. Fourkal, B. Shahine, M. Ding, J. S. Li, T. Tajima, and C.-M. Ma, “Particle in cell
simulation of laser-accelerated proton beams for radiation therapy,” Med. Phys. 29, 2788-
2798 (2002).
27 C. J. Karzmark, “Advances in linear accelerator design for radiotherapy,” Med. Phys.
11, 105-128 (1984).
28 N. Leveson and C. Turner, “An Investigation of the Therac-25 Accidents,” in Ethics
and Computing: Living Responsibly in a Computerized World, K. W. Bowyer (IEEE
Computer Society Press, Los Alamitos, California 1996), pp. 18-41.
29 D. W. O. Rogers, B. A. Faddegon, G. X. Ding, C.-M. Ma, J. We, and T. R. Mackie,
“BEAM: A Monte Carlo code to simulate radiotherapy treatment units,” Med. Phys. 22,
503-524 (1995).
43
30 W. R. Nelson, H. Hirayama, and D. W. O. Rogers, The EGS4 Code System (Stanford
Linear Accelerator Center Report No. SLAC-265, Stanford, California, 1985).
31 A. D. Green, Modeling of Dual Foil Scattering Systems for Clinical Electron Beams,
M.S. Thesis, The University of Texas Health Science Center at Houston, 1991
(unpublished).
32 H. Huizenga and P. R. M. Storchi, “Numerical calculation of energy deposition by
broad high-energy electron beams,” Phys. Med. Biol. 34, 1371-1396 (1989).
33 ICRU, Radiation Dosimetry: Electron Beams with Energies Between 1 and 50 MeV,
ICRU Report 35, Bethesda, Maryland (1984).
34 V. Bagnoud and F. Salin, “Amplifying laser pulses to the terawatt level at a 1-kilohertz
repetition rate,” Appl. Phys. B−Lasers O. 70, S165-S170 (2000).
44
Tables:
TABLE I. Summary of the set of simulation parameters used by the UTA group in their simulations of the
proposed laser accelerator system considered. The list includes the plasma density, total number of
electrons irradiated by the laser pulse, and the number of beam electrons per pulse with total energy E > 1
MeV.
Laser power(instantaneous /average)
1.7 TW / 1.7 W
Laser pulse intensity 7.6×1017 W⋅cm−2
Repetition rate 10 HzPulselength 100 fsWavelength 0.8 µmBeam diameter atmirror
5 mm
Focal length of mirror 17 cmSpot radius 3√2 µmRayleigh length 1 mmElectron density (ne) 4.85×1019 cm−3
Plasma wavelength(λp)
4.8 µm
Maximum kineticenergy of acceleratedelectrons
21 MeV
Ntotal per pulse 2.9×1012 electronsNE>1 MeV per pulse 9.3×109 electrons (1.5 nC)
45
TABLE II. Optimum thicknesses of the scattering foils for each of the canonical central beam energies
under consideration. The shape of the secondary foil is a stepped approximation to a Gaussian function
t=Texp(−r2/R2), where t is the foil thickness at a distance r from the central axis, T is the central-axis
thickness of the foil, and R relates to the foil’s width. For each aluminum secondary foil, the parameter R
was (0.95)⋅(1.4) cm.
central beam energy
Ecent (MeV)
gold primary foilthickness (cm)
aluminum secondary foilcentral-axis thickness (cm)
9 0.00640 0.070312 0.01058 0.105515 0.01567 0.149418 0.02165 0.201321 0.02858 0.2594
46
Figure captions:
FIG. 1. Diagrams illustrating the processes by which a laser pulse generates a plasma wave in a gas and by
which the plasma wave traps low-energy electrons and accelerates them. [A] The leading edge of a high-
intensity laser pulse ionizes the molecules in the gas. [B] The remainder of the laser pulse exerts a
ponderomotive force upon the electrons in the plasma. This force has a significant longitudinal component
that displaces the electrons relative to the positive ions. Because the ponderomotive force is proportional to
the gradient of the pulse intensity (which is proportional to the square of the pulse’s electric field
amplitude), some electrons will experience a greater ponderomotive force than others. Those electrons that
encounter a maximal longitudinal force comprise a crest that moves with a velocity equal to that of the
laser pulse. The laser pulse proceeds to further ionize gas molecules and form electron crests. The space
charge separation between the crests and the positive ions triggers a longitudinal oscillation with period
2π/ωp (illustrated at the upper right of [B]) of the crests about the positive ions. In this way, a longitudinal
plasma wave is formed. Note that the wakefield has a much longer wave length, and its wave is travelling
with the pulse speed. Electrons may be trapped and accelerated by the wave. [C] The illustration at left
shows a thermal electron trapped within the electric field formed by the plasma wave crests. As shown in
the right-hand illustration, these trapped electrons are then accelerated by the plasma wave to ultra-
relativistic energies.
FIG. 2. Illustration of the breakup of a long laser pulse into a train of shorter pulses that are better suited to
drive the plasma wave oscillation. As shown in [A], photons from the incident primary pulse are absorbed
by atoms in the plasma gas and re-emitted at a lower frequency. The wave comprised of the scattered
photons may beat together with the remainder of the primary pulse, as shown in [B] and [C]. If the
difference between the frequencies of the scattered photons and primary laser pulse is the plasma
frequency, then the widths of each subpulse in [D] will each be about half the plasma wavelength; this is
ideal for building up the longitudinal plasma wave.
47
FIG. 3. Diagram of the Chirped Pulse Amplification (CPA) method. Rather than directly amplify the low-
intensity pump laser pulse (represented at the upper far left of the figure), the pulse is stretched in time
prior to amplification using 1 pair of diffraction gratings and recompressed after amplification with another
pair of gratings. This figure was composed from illustrations given in Mourou et al.23
FIG. 4. A schematic illustration of an accelerator chamber under consideration. The 30-cm long
evacuated cylinder features mirrors at the ends to focus the laser pulse and a gas cell at the beam focal point
within which the plasma wave will be generated. An aperture is provided at one end of the cylinder to
accommodate the accelerated electron beam. This design is based on the schematic in Figure 4.1 in the
thesis by Jones.25
FIG. 5. [A] Illustration used to define the coordinate system and the angles of the electron momentum
relative to the coordinate system. The z-axis is taken to be the direction of propagation of the laser pulse;
the y-axis is the transverse axis along which the laser pulse is linearly polarized. In this 3D coordinate
system, the polar angle θ is that which the electron momentum vector ep makes with the z-axis. The
projection of ep into the plane defined by the x- and y-axes determines the azimuthal angle φ. [B]
Illustration to define an approximate “polar angle” using the two electron momentum components
(longitudinal and transverse) available from the output of the 2D PIC simulations. [C] Scatter plot of the
approximate polar angle versus the kinetic energy, for all of the E > 1 MeV electrons generated from the
2D PIC simulation.
FIG. 6. Electron production rate per MeV (d2N / dE dt) versus electron kinetic energy. The circles
represent the spectrum predicted by 2D PIC simulations; the solid line is a fifth-degree polynomial fit to the
natural logarithm of the 2D PIC spectrum. The dotted line is the energy spectrum from the LWFA
48
experiment reported by Leemans et al.;18 the parameters for the 2D PIC simulations (e.g. for the laser pulse
wavelength and spot size, and the plasma density) are consistent with the conditions of the Leemans et al.
experiment. In all cases, a 10-Hz laser pulse repetition rate is assumed.
FIG. 7. Profile distributions for a 9-MeV mono-energetic electron beam scattered through a gold primary
foil and aluminum secondary foil (stepped approximation) configuration optimized for a 9-MeV beam (see
Table II). The circles represent the EGS Monte Carlo calculation; the solid line indicates the analytical
calculation from the dual-foil design code.
FIG. 8. Distributions of the relative planar fluence (calculated using Green’s prescription31) of electrons
propagated through a dual-foil beam scattering system with the primary foil 100 cm from the phantom
surface. Calculations are shown for a central beam energy and foil configuration corresponding to 9 MeV
(frames [A] and [C]) and 15 MeV (frames [B] and [D]). For the top two frames, the beam energy
distribution (of width ∆E centered about Ecent) was assumed to be uniform; for the bottom two frames, the
beam energy distribution was extracted as slices from the spectrum predicted by the 2D PIC simulations.
FIG. 9. EGS/BEAM calculations of the depth-dose curves for [A] uniform energy distribution, Ecent = 9
MeV, [B] uniform energy distribution, Ecent = 15 MeV, [C] UTA 2D PIC distribution, Ecent = 9 MeV, and
[D] UTA 2D PIC distribution, Ecent = 15 MeV. In each frame, the effect of increasing the energy spread ∆E
about the central energy Ecent is illustrated. The vertical axes are normalized to percent depth dose.
FIG. 10. Plot of “inverse gradient” R10−R90 versus therapeutic depth (R90) for LWFA electron beams of
varying energy spreads ∆E, calculated using EGS/BEAM. Each data point corresponds to a particular
canonical central beam energy Ecent and beam energy spread ∆E, assuming a uniform energy distribution
(frame [A]) and extracted from the energy distribution predicted by UTA 2D PIC (frame [B]).
49
FIG. 11. Plot of the dose rate at R100 versus the energy spread (∆E) of the beam for different central beam
energies Ecent. The quantities ∆E and Ecent were extracted from the energy distribution predicted by UTA
2D PIC. The dose rates are normalized to dose per minute, assuming a 10-Hz pulse repetition rate.
FIG. 12. Plot of the dose rate (dose per minute) at R100 versus R10−R90, to demonstrate the dependence of
both quantities with change in beam energy spread. Each curve represents beams with a specific central
energy. The beams are sampled from the UTA 2D PIC distribution; a 10-Hz pulse repetition rate is
assumed.
FIG. 1. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
FIG. 2. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
FIG. 3. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
FIG. 4. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
kinetic energy (MeV)
0 5 10 15 20 25
pola
r ang
le (d
egre
es)
-80
-60
-40
-20
0
20
40
60
80[C]
FIG. 5. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical appliation”
electron kinetic energy (MeV)
0 5 10 15 20 25
d2 N/d
Edt
(min
− 1 M
eV− 1
)
109
1010
1011
1012
1013
UTA 2D PIC, λp = 4.8 µm5th-degree polynomial fitLeemans et al. (2001)
FIG. 6. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
radius (cm)
-40 -20 0 20 40
rela
tive
dose
(%)
0
20
40
60
80
100
120
prediction from analytical codeEGS4 prediction
FIG. 7. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
rela
tive
plan
ar fl
uenc
e
0
20
40
60
80
100
120
∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
Ecent=9 MeV Ecent=15 MeV[A] [B]
uniform E dist uniform E dist
distance from central axis (cm)
-40 -20 0 20 40
rela
tive
plan
ar fl
uenc
e
0
20
40
60
80
100
120
∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
distance from central axis (cm)
-40 -20 0 20 40
∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
Ecent=9 MeV Ecent=15 MeV[C] [D]2D PIC, λp=4.8 µm 2D PIC, λp=4.8 µm
FIG. 8. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
perc
ent d
epth
dos
e
0
20
40
60
80
100
120∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
depth (cm)
0 2 4 6 8
perc
ent d
epth
dos
e
0
20
40
60
80
100
120∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
depth (cm)
0 2 4 6 8 10 12
∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
[A]
Ecent = 9 MeV
Ecent = 9 MeV
[C]
[B]
[D]
Ecent = 15 MeV
Ecent = 15 MeV
uniform E dist uniform E dist
2D PIC, λp=4.8 µm 2D PIC, λp=4.8 µm
FIG. 9. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
R10
−R90
(cm
)
0
1
2
3
4
5∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
R90 (cm)
0 1 2 3 4 5 6 7
R10
−R90
(cm
)
0
1
2
3
4
5∆E = 0.5 MeV∆E = 2.5 MeV∆E = 4.5 MeV∆E = 6.5 MeV
[A] uniform energy distribution
[B] 2D PIC, λp=4.8 µm
FIG. 10. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
∆E (MeV)
0 1 2 3 4 5 6 7
dose
rate
at R
100 (
Gy
min
−1)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14 Ecent = 9 MeVEcent = 12 MeVEcent = 15 MeVEcent = 18 MeVEcent = 21 MeV
2D PIC, λp=4.8 µm
FIG. 11. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”
R10−R90 (cm)
0 1 2 3 4 5
dose
rate
at R
100 (
Gy
min
− 1)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14 Ecent = 9 MeVEcent = 12 MeVEcent = 15 MeVEcent = 18 MeV
2D PIC, λp=4.8 µm
FIG. 12. K. K. Kainz et al., “Dose properties of a laser accelerated electron beam andprospects for clinical application”