Dose properties of a laser accelerated electron beam and ...itiq/chiu/kainz.pdf · beam energies...

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 _______________)

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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

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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

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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

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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

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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.

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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

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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”

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)


radius (cm)

-40 -20 0 20 40

rela

tive

dose

(%)

0

20

40

60

80

100

120

prediction from analytical codeEGS4 prediction


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


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


distance from central axis (cm)

-40 -20 0 20 40


Ecent=9 MeV Ecent=15 MeV[C] [D]2D PIC, λp=4.8 µm 2D PIC, λp=4.8 µm


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



depth (cm)

0 2 4 6 8 10 12


[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


R10

−R90

(cm

)

0

1

2

3

4


R90 (cm)

0 1 2 3 4 5 6 7

R10

−R90

(cm

)

0

1

2

3

4


[A] uniform energy distribution

[B] 2D PIC, λp=4.8 µm


∆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


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


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