- Email: [email protected]
Oncological hadrontherapy with laser ion accelerators
1 July 2002
Physics Letters A 299 (2002) 240–247 www.elsevier.com/locate/pla
Oncological hadrontherapy with laser ion accelerators S.V. Bulanov a , T.Zh. Esirkepov b , V.S. Khoroshkov c , A.V. Kuznetsov b , F. Pegoraro d,∗ a General Physics Institute, RAS, Moscow, Russia b Moscow Institute of Physics and Technology, Dolgoprudny, Russia c Institute of Theoretical and Experimental Physics, Moscow, Russia d Department of Physics and INFM, Pisa University, Pisa, Italy
Received 20 April 2002; received in revised form 20 April 2002; accepted 23 April 2002 Communicated by F. Porcelli
Abstract The use of an intense collimated beam of protons produced by a high-intensity laser pulse interacting with a plasma for the proton treatment of oncological diseases is discussed. The fast proton beam is produced at the target by direct laser acceleration. An appropriately designed double-layer target scheme is proposed in order to achieve high-quality proton beams. The generation of high quality proton beams is proved with particle in cell simulations. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Effective ion acceleration during the interaction of an ultra short and ultra intense laser pulse with matter is one of the most important applications of the presently available compact laser systems with multi-terawatt and petawatt power [1,2]. Collimated beams of fast ions have been observed in experiments on the interaction of laser radiation with solid targets (see Refs. [3–7]) and the process of ion acceleration has been studied in detail with multi-dimensional Particle In Cell simulations [8–12]. In the recent experimental results presented in Refs. [3–7], electron energies were observed in the range of hundreds of MeV while the proton energy was about tens of MeV. The generation of fast ions becomes highly effective * Corresponding author.
E-mail address: [email protected] (F. Pegoraro).
when the laser radiation reaches the petawatt power limit [12]. In Refs. [9–12] it was shown that, by optimizing the laser–target parameters, it becomes possible to accelerate protons up to energies in the several hundred MeV range. In Ref. [13] the possibility of using the plasma produced by the ultra short and ultra intense laser pulses generated by such systems was proposed as a source of high energy ions for hadrontherapy in oncology. One of the main challenges in radiation therapy is to deliver a substantially high and homogeneous dose to a tumor, while sparing neighboring healthy tissues. In this Letter we show that the laser plasma interaction can provide proton beams of sufficiently high quality for radiation therapy. We show that such a beam can be obtained using a target made of a layer of heavy ions followed by a thin proton layer. The transverse size of the proton layer must be smaller than the size of the pulse waist since an inhomogeneity in the pulse causes an inhomogeneity of the accelerating electric
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 5 2 1 - 2
S.V. Bulanov et al. / Physics Letters A 299 (2002) 240–247
field and thus a degradation of the beam quality as seen in experiments. We notice that obtaining proton beams with high quality, i.e., with sufficiently small energy spread ∆E/E, is of fundamental importance also for the application of laser produced fast ion beams to the injectors of conventional accelerators (see Ref. [14]) and for using laser accelerated proton beams for the fast ignition of thermonuclear targets as discussed in Refs. [15,16].
2. Hadrontherapy with laser accelerated plasma ions Hadron radiotherapy [17–22] is a part of radiation therapy, which uses not only beams of high energy ions, but also π -mesons, neutrons, electron beams, X- and gamma rays to irradiate cancer tumors (see Ref. [17] and literature quoted therein). In cancer treatment surgery, chemotherapy and radiotherapy are applied together as they complement each other. The use of protons in radiotherapy has several advantages. First of all the proton beam scattering on the atomic electrons is weak and thus there is less irradiation of healthy tissues in the vicinity of the tumor. Second, the slowing down length for a proton with given energy is fixed, which avoids undesirable irradiation of healthy tissues at the rear side of the tumor. Third, the well localized maximum of the proton energy losses in matter (the Bragg peak) leads to a substantial increase of the irradiation dose in the vicinity of the proton stopping point (see, for example, [17]). After more than 40 years of experimental investigations different hospital based centers of proton radiotherapy are currently under construction. All these centers are located in large hospitals and have the capacity to admit up to 1000 patients per year. Already three centers are operating [21] and soon their number will grow to nine. These centers have accelerators specialized for medical applications, from which the proton beams are delivered into 3 to 5 procedure rooms equipped with treatment units. A necessary and most expensive treatment unit of the hospital based proton therapy centers is the GANTRY which is a device that provides multi-directional irradiation of a laying patient. By now, proton beams with the required parameters have been obtained using conventional accelerators of charged particles (synchrotrons, cyclotrons,
241
linacs). On the other hand, the use of a laser based accelerator is fairly attractive because of its compactness and of the additional possibility it offers of controlling the proton beam parameters. There are at least two possible ways of using a laser accelerator. In one, it simply replaces a conventional proton accelerator. In the other, which appears to be much more attractive, the laser radiation is delivered to the target where its energy is converted into the energy of fast ions. This can simplify the technical problems related to the generation and transport of the ion beam and reduce costs substantially. In this case the central accelerator, the channels through which the fast ions are transported and, finally, most part of the GANTRY are no longer required. The basic parameters required in a proton beam for medical applications can easily be reached with present accelerator technology. The proton beam intensity must be in the range 1010 to 5 × 1010 protons per second, the maximum proton energy must be in the range 230 to 250 MeV. At the same time, for practical realization, the following two conditions are the most demanding for the laser accelerator. First is the requirement of a highly monoenergetic proton beam with ∆E/E = 10−2 . Second is the system duty factor, i.e., the fraction of the time of during which the proton beam can be used that must not be smaller than 0.3. Actually, the most important factor is the economical feasibility of the use of a laser accelerator. Although it is not easy now to determine what the real price of a laser accelerator will be, its use is likely to represent a significant simplification of present equipment and to lower the price of construction of hadrontherapy centers.
3. Multi-layer targets for high quality ion beams In the simplest scheme of ion acceleration, the laser pulse interacts with a thin foil. The foil thickness ranges from less than a micron up to several microns. When multiterawatt laser radiation interacts with a target, matter is ionized in an interval shorter than a single optic oscillation period of the laser radiation, producing in such a way a collisionless plasma. Hence, we will approximate the foil as a narrow layer of overdense plasma. Under the action of the laser radiation the electrons are expelled from a region
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on the foil with transverse size of the order of the diameter of the focal spot. For femtosecond long laser pulses with multiterawatt power, the typical time scale of the hydrodynamic expansion of a micron plasma slab is much longer than the laser pulse duration. Under these conditions ions remain at rest, which results in the formation of a positively charged layer of ions. However, after a time interval equal to or longer than the inverse of the ion Langmuir frequency 1/ωpi = (mi /4πn0 Zi2 e2 )1/2 the ion layer explodes because of the Coulomb repulsion of electric charges with equal sign. In order to estimate the typical energy gain of the fast ions we assume that all free electrons produced by ionization in the irradiated region of the foil are expelled. In this case the electric field near the positively charged layer is equal to E0 = 2πn0 Zi el. Here n0 is the ion density in the foil, Zi e the ion electric charge, and l the foil thickness. The region of strong electric field has a transverse size of the order of the diameter 2R⊥ of the focal spot. Thus the longitudinal size of this region where the electric field remains essentially one-dimensional is also of order 2R⊥ and the typical energy of the ions with charge Za e accelerated by the electric field due to charge separation can be estimated as Emax = 4πn0 Za Zi e2 lR⊥ .
(1)
In deriving Eq. (1) we have assumed that the energy gained by the electrons in the laser field is well above the ion energy and larger than the energy they need to leave the irradiated region. The electron energy in the electromagnetic wave is given by Ee = me a 2 /2 (see Ref. [23]), where a = eE/me cω is the dimensionless amplitude of the laser pulse. From this condition we can find the required value of the laser pulse intensity, its power and, once the pulse duration is chosen, its energy. In Table 1 we present four set of parameters of the target and of the proton beam as well as of the parameters of the laser pulse that are required in order to accelerate the protons. In all four versions we consider a solid density plasma inside the target with transverse size determined by the radius of the focal spot, but change he thickness of the foil. In versions 2, 3, and 4 the size of the focal spot is approximately one wavelength of the laser radiation. The value of the product of the laser pulse duration τl times the
Table 1 Parameters of the fast proton beam Zi = 1 versus the parameters of the laser pulse and of the target. The proton energy is calculated on the basis of Eq. (1) Version Foil thickness (µ) Particle density (1023 cm−3 ) Focus radius (µ) Pulse amplitude Intensity (1020 W cm−2 ) Power (TW) Pulse energy (J) Pulse length (fs) τ1 ωpp Number of fast protons 1010 Proton energy (MeV)
1
2
3
4
0.5 1 5 130 220 1.7 × 104 8.5 × 103 480 210 400 4500
0.25 1 1 40 22 70 17 240 100 7.8 450
0.1 1 1 25 8.8 27 2.7 100 40 3 1810
0.05 1 2 26 8.8 110 5.4 48 200 6.3 180
proton Langmuir frequency ωpp = (4πn0 e2 /mp )1/2 (mp is the proton mass) shown in Table 1 characterizes whether the immobile ion approximation is valid during the laser–target interaction. We see that from this point of view version 4 with a very thin foil is the best for proton acceleration in hadrontherapy. Indeed, one of the main obstacles in obtaining such effective ion acceleration regimes is related to the condition that the laser pulse must be shorter than the ion response time. It is not yet clear whether the effect of the ion motion does deteriorate the efficiency of the ion acceleration and lower the ion beam quality. It is however clear that for long laser pulses with moderate intensities the assumption of immobile ions is not valid and that the ion acceleration evolves in a much less controlled way. In order to satisfy the condition τl ωpp < 1 we must either choose a short pulse, as the one corresponding to version 4 in Table 1, or we must use a target made of heavy ions. In the case of a heavy ion target with approximately the same ion density n0 , the ion response time is equal to 1/ωpi = µ1/2 /(Zi ωpp ), where µ = mi /mp . For µ1/2 /Zi 1 we have 1/ωpi 1/ωpp . We see that this is possible only for non fully stripped ions inside the heavy ion layer of the target. A version of such model will be considered below. If we compare the form of the energy spectra of the accelerated ions observed in the experiments or obtained in numerical simulations with the narrow energy spectrum, ∆E/E < 10−2 , that is required for medical applications in hadrontherapy, we see that the energy spectra of the laser accelerated ions are at
S.V. Bulanov et al. / Physics Letters A 299 (2002) 240–247
present quite far from the required ones. In Refs. [3, 11] it was found that the ion spectra below a maximum energy Emax (E < Emax ) can be approximated by a quasi-thermal distribution with an effective temperature T , which is several times smaller than Emax . Such spectra are not appropriate for our purposes because an ion beam with such an energy spread would not provide a sufficient increase of the irradiation dose in a well localized region and would lead to an unacceptably high damage of the healthy tissues. In order to improve the proton beam quality one can cut the beam into beamlets with a narrow spread in energy space. However, in this case the efficiency of the laser energy transformation into the energy of fast particles decreases significantly, and, what is more important, the number of fast particles decreases. A different approach which seems more promising, is connected with the use of multi-layer targets. In this scheme a thin foil is used as a target and its rear surface is coated with a thin hydrogen layer. When the ultra short laser pulse irradiates the target, the heavy atoms are partly ionized and the ionized electrons abandon the foil, generating an electric field due to charge separation. Because of the large value of the ratio (µ/Z) the heavy ions remain at rest, while the lighter protons are accelerated. The requirement that the number of fast protons in the beam must be approximately 1010 to 5 × 1010 protons per second can be satisfied relatively easily in a multi-layer target, extrapolating from the experimental results in Refs. [3–7] and the computer simulations presented in Refs. [9–12] where the number of fast protons accelerated per pulse varies from 1012 to 1013 particles. Thus, in order to achieve 1010 fast protons per pulse from a two-layer target it is enough to have a proton layer approximately 0.02 µm thick and a laser pulse focused onto a spot with diameter equal to two laser wavelengths. A sketch of a doublelayer target is shown in Fig. 1. The first layer is made of heavy ions, i.e., of ions with a sufficiently large value of µ1/2/Z and the target is sufficiently thick so as to give a large enough electric field due to charge separation. This electric field has opposite sign on the two different sides of the target, has a zero inside the target and vanishes at a finite distance from it, as illustrated in Fig. 1(a). Since the number of protons is assumed to be sufficiently small not to produce any significant effect on the electric field,
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Fig. 1. Scheme of a two-layer target. Density distribution of heavy ions, protons, and electrons and dependence of the electric field on the x coordinate (a). The form of two-layer target in the r, x plane (b). Shape of a curved two layer target in the plane r, x (c).
their behavior can be described within the framework of the test particle approximation. A most important requirement is that the transverse size of the proton layer be smaller than the pulse waist so as to decrease the influence of the laser pulse inhomogeneity in the direction perpendicular to its direction of propagation. The pulse inhomogeneity causes an inhomogeneity of the accelerating electric field, which results in an additional energy spread of the ion beam, as seen in experiments. The effect of the finite waist of the laser pulse leads also to an undesirable defocusing of the fast ion beam. In order to compensate for this effect and to focus the ion beam we can use the properly
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deformed targets, as suggested in Ref. [9]. A sketch of a deformed target is shown in Fig. 1(b). The energy spectrum of the protons can be found by taking the electric field in the vicinity of the target to be non-zero inside the region 0 < x < R0 and to be of the form E(x) = E0 (1 −x/R0), where E0 = 2πn0 Zel and R0 is the scale-length of the electric field in the longitudinal and in the transverse directions (R0 = 2R⊥ ). The distribution function of the fast protons f (x, v, t) obeys the kinetic equation ∂f eE ∂f ∂f +v + = 0, ∂t ∂x mp ∂v
(2)
which gives f (x, v, t) = f0 (x0 , v0 ), where f0 (x0 , v0 ) is the distribution function at the initial time t = 0, i.e., f (x, v, t) is constant along the characteristics of Eq. (2) which can be written as x = R0 − (R0 − x0 ) cos Ωt +(v0 /Ω) sin Ωt and v = v0 cos Ωt +(R0 − x0 )Ω sin Ωt, where Ω = (eE0 /mp R0 )1/2 and x < R0 . If v0 = 0, all particles reach x = R0 at the same time t = π/2Ω independently of their initial position. After this time they exit the acceleration region. Thus, the linear dependence of the electric field on the x coordinate produces both the acceleration and bunching of the particles. The number of particles per unit volume in phase space, dx dv, is equal to dn = f dx dv = f v dv dt = f dE dt/mp . We assume that at t = 0 all particles are at rest and that their spatial distribution is given by n0 (x0 ) which corresponds to the distribution function f0 (x0 , v0 ) = n0 (x0 )δ(v0 ), with δ(v0 ) the Dirac delta function. For the applications discussed in this Letter we are interested in the particle distribution function integrated over time. Time integration of the distribution f v dv dt gives the energy spectrum of the beam dt N(E) dE = dE n0 (x0 )δ(v0 ) mp n0 (x0 ) dt = (3) dE. mp dv0 At the end of the acceleration region, from the equation of the characteristics with v0 = 0 we obtain |dt/dv0 | = |Ω(2E/mp )1/2|−1 , i.e., N(E) dE =
n0 (x0 ) dE . |Ω(2Emp )1/2|
(4)
Here x0 = x0 (E) is obtained by inverting the relationship E = mp [(R0 − x0 )Ω]2/2. The spread of the pro-
ton energy in the two-layer target model is determined by the width of the proton layer ∆x0 and is given approximately by ∆E = 2πn0 Zi e2 l∆x0 . The ratio between this energy spread and the maximum energy is ∆E/Emax = ∆x0 /R0 . For a proton layer 2 × 10−6 cm thick, with R0 = 2 × 10−4 cm we find ∆E/Emax = 10−2 with Emax = 200 MeV, when the product of the heavy ion charge times the foil thickness l is given by Zi l = 2 × 10−4 cm. An additional parameter for controlling the properties of the proton beams is provided by the use of deformed targets. As was demonstrated in Refs. [9–12] with 2D and 3D PIC simulations, the protons accelerated from the rear side of a deformed (concave) foil are focused into a point localized near the center of the foil curvature.
4. Simulations of a quasi-monoenergetic proton beam In order to take into account the numerous nonlinear and kinetic effects as well as to extend our consideration to a multi-dimensional geometry, we performed numerical simulations of the proton acceleration during the interaction of a short, high power laser pulse with a two-layer target. We used a twodimensional version of a code REMP (relativistic electro-magnetic particle-mesh code) [24]. The size of the computation box is 20λ × 30λ with cell size equal to 0.03λ × 0.03λ. The total number of particles is 106 . The plasma consists of three species: electrons, protons and heavy ions with mp /me = 1836 and mi /(me )Zi = 18360. Here we show the results corresponding to the case where the target consists of two neutral plasma layers with their front side at x = 5λ. The electron density in the heavy ion layer corresponds to the ratio ωpe /ω = 3.0 between the plasma and the laser frequencies. The second layer is localized at the rear side of the target and its electron density is smaller than the critical density ncr = me ω2 /4πe2 and corresponds to ωpe /ω = 0.7. The sizes of the first layer and of the proton layer are 0.25λ × 10λ and 0.1λ × 5λ, respectively, so that the number of electrons in the first layer is 50 times larger than in the proton layer. A circularly polarized laser pulse with dimensionless amplitude a = 20 is initiated at the left-hand side
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Fig. 3. Distribution of the z component of the electric field in the x, y plane at t = 12 × 2π/ω.
Fig. 2. The proton energy spectrum at t = 45 × 2π/ω (a), the heavy ion energy spectrum at t = 97.5 × 2π/ω (b).
boundary of the computation region. The pulse size is 20λ × 15λ, with a constant profile and with a gradual decrease of its amplitude down to zero on the scale length 2λ. The simulation results are shown in Figs. 2–5, where the coordinates are measured in wavelengths of the laser light and the time in laser periods. In Fig. 2 we present the spectrum of the proton energy at time t = 45 × 2π/ω (a) and of the energy per nucleon of the heavy ions at time t = 97.5 × 2π/ω (b). In Figs. 3 and 4 we present the distributions of the z and of the x components of the electric field in the x, y plane at t = 12 × 2π/ω which show the shape of the laser pulse and of the accelerating electric field. This field is localized in the vicinity of the first layer (the heavy ion layer) of the target. In Fig. 5 we show the electric
Fig. 4. The same as in Fig. 3 for the x component of the electric field.
charge distribution in the x, y plane at t = 18 × 2π/ω (a), at t = 27 × 2π/ω (b), t = 51 × 2π/ω (c). We see that the proton layer moves in the x direction and that the distance between the two layers increases. The heavy ion layer expands and tends to become rounded. The two black areas show the form of the clouds of the heavy ions (left) and of the proton layer (right) and the white cloud corresponds to the electrons blown out of the target by the laser light. We notice that for the adopted simulation parameters the electrons do not abandon the region irradiated by the laser light completely. Even if only a portion of the electrons is accelerated and heated by the laser pulse, the electric field they generate appears to be strong enough to accelerate the protons up 100 MeV in the first configuration, and up to 60 MeV in the second.
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The energy per nucleon acquired by the heavy ion component is approximately ten times smaller than the proton energy. As seen in Fig. 2, the heavy ions have a wide energy spectrum while the protons form a quasi-mono-energetic bunch with ∆E/E ≈ 3%. The proton beam remains localized in space for a long time due to the bunching effect of the linearly decreasing dependence of the electric field on the coordinate in the acceleration direction discussed above.
5. Conclusions (a)
(b)
The future of hadron radio therapy requires the construction of specialized centers, equipped with modern diagnostics and medical accelerators near oncological clinics. The construction of small centers would make it possible to provide cheaper facilities for hadrontherapy closer to the patients. A possible solution of this problem relies on producing a specialized laser proton accelerator. Such an accelerator can simplify the technical solutions adopted in the hospital centers for proton radio therapy and lower their construction costs significantly. The use of the multi-layer targets with different shapes and composition opens up additional opportunities for controlling the parameters of the fast proton beam, for optimizing its energy spectrum, the number of particles per bunch, the beam focusing and the size of the region where the beam deposits its energy. To increase the duty factor we must use a system of several high-power, 1 Hz repetition rate lasers or to use a multi-stage acceleration scheme in the case of moderate power, high repetition rate (1 kHz) lasers.
References
(c) Fig. 5. Distribution of the electric charge density (black corresponds to both heavy (thick shell) and light ions (thin shell) whereas white corresponds to electrons) in the x, y plane at different times: t = 18 × 2π/ω (a), t = 27 × 2π/ω (b), t = 51 × 2π/ω (c).
[1] G.A. Mourou, C.P.J. Barty, M.D. Perry, Phys. Today 51 (1998) 22. [2] G.A. Mourou, Zh. Chang, A. Maksimchuk et al., Plasma Phys. Rep. 28 (2002) 12. [3] A. Maksimchuk, S. Gu, K. Flippo, D. Umstadter et al., Phys. Rev. Lett. 84 (2000) 4108. [4] E.L. Clark, K. Krushelnick, M. Zepf et al., Phys. Rev. Lett. 85 (2000) 1654. [5] S.P. Hatchett, C.G. Brown, T.E. Cowan et al., Phys. Plasmas 7 (2000) 2076. [6] R. Snavely, M.H. Key, S.P. Hatchett et al., Phys. Rev. Lett. 85 (2000) 2945.
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[7] A.J. Mackinnon, M. Borghesi, S. Hatchett et al., Phys. Rev. Lett. 86 (2001) 1769. [8] T.Zh. Esirkepov, Y. Sentoku, K. Mima et al., JETP Lett. 70 (1999) 82. [9] S.V. Bulanov, N.M. Naumova, T.Zh. Esirkepov et al., JETP Lett. 71 (2000) 407. [10] Y. Sentoku, T.V. Lisseikina, T.Zh. Esirkepov et al., Phys. Rev. E 62 (2000) 7271. [11] H. Ruhl, S.V. Bulanov, T.E. Cowan et al., Plasma Phys. Rep. 27 (2001) 411. [12] S.V. Bulanov et al., in: V.D. Shafranov (Ed.), Reviews of Plasma Physics, Vol. 22, Plenum Publishers, New York, 2001, p. 227. [13] S.V. Bulanov, V.S. Khoroshkov, Plasma Phys. Rep. 28 (2002) 453. [14] K. Krushelnik, E.L. Clark, R. Allot et al., IEEE Trans. Plasma Sci. 28 (2000) 1184. [15] M. Roth, T.E. Cowan, M.H. Key et al., Phys. Rev. Lett. 86 (2001) 436.
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[16] V.Yu. Bychenkov, V. Rozmus, A. Maksimchuk, D. Umstadter, Plasma Phys. Rep. 27 (2001) 1076. [17] V.S. Khoroshkov, E.I. Minakova, Eur. J. Phys. 19 (1998) 523. [18] U. Amaldi, M. Silari, The TERA Project and the Center for Oncological Hadrontherapy INFN-LNF Divisione Ricerca, Frascati, 1995. [19] G. Kraft, Physica Medica XVII (Suppl. 1) (2001) 13. [20] U. Amaldi, Physica Medica XVII, (Suppl 1) (2001) 33. [21] M. Goitein, in: Advances in Hadron Therapy, Elsevier, Amsterdam, 1997, p. 141; J.M. Slater, J.D. Slater, D.W. Miller et al., in: Advances in Hadron Therapy, Amsterdam, Elsevier, 1997, p. 181. [22] Report of the Advisory Group Meeting on the utilization of particle accelerators for proton therapy, IAEA, UN, F1-AG1010, July, 1998. [23] L.D. Landau, E.M. Lifshits, The Classical Theory of Fields, Pergamon, Oxford, 1984. [24] T.Zh. Esirkepov, Comput. Phys. Commun. 135 (2001) 144.
Physics Letters A 299 (2002) 240–247 www.elsevier.com/locate/pla
Oncological hadrontherapy with laser ion accelerators S.V. Bulanov a , T.Zh. Esirkepov b , V.S. Khoroshkov c , A.V. Kuznetsov b , F. Pegoraro d,∗ a General Physics Institute, RAS, Moscow, Russia b Moscow Institute of Physics and Technology, Dolgoprudny, Russia c Institute of Theoretical and Experimental Physics, Moscow, Russia d Department of Physics and INFM, Pisa University, Pisa, Italy
Received 20 April 2002; received in revised form 20 April 2002; accepted 23 April 2002 Communicated by F. Porcelli
Abstract The use of an intense collimated beam of protons produced by a high-intensity laser pulse interacting with a plasma for the proton treatment of oncological diseases is discussed. The fast proton beam is produced at the target by direct laser acceleration. An appropriately designed double-layer target scheme is proposed in order to achieve high-quality proton beams. The generation of high quality proton beams is proved with particle in cell simulations. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Effective ion acceleration during the interaction of an ultra short and ultra intense laser pulse with matter is one of the most important applications of the presently available compact laser systems with multi-terawatt and petawatt power [1,2]. Collimated beams of fast ions have been observed in experiments on the interaction of laser radiation with solid targets (see Refs. [3–7]) and the process of ion acceleration has been studied in detail with multi-dimensional Particle In Cell simulations [8–12]. In the recent experimental results presented in Refs. [3–7], electron energies were observed in the range of hundreds of MeV while the proton energy was about tens of MeV. The generation of fast ions becomes highly effective * Corresponding author.
E-mail address: [email protected] (F. Pegoraro).
when the laser radiation reaches the petawatt power limit [12]. In Refs. [9–12] it was shown that, by optimizing the laser–target parameters, it becomes possible to accelerate protons up to energies in the several hundred MeV range. In Ref. [13] the possibility of using the plasma produced by the ultra short and ultra intense laser pulses generated by such systems was proposed as a source of high energy ions for hadrontherapy in oncology. One of the main challenges in radiation therapy is to deliver a substantially high and homogeneous dose to a tumor, while sparing neighboring healthy tissues. In this Letter we show that the laser plasma interaction can provide proton beams of sufficiently high quality for radiation therapy. We show that such a beam can be obtained using a target made of a layer of heavy ions followed by a thin proton layer. The transverse size of the proton layer must be smaller than the size of the pulse waist since an inhomogeneity in the pulse causes an inhomogeneity of the accelerating electric
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 5 2 1 - 2
S.V. Bulanov et al. / Physics Letters A 299 (2002) 240–247
field and thus a degradation of the beam quality as seen in experiments. We notice that obtaining proton beams with high quality, i.e., with sufficiently small energy spread ∆E/E, is of fundamental importance also for the application of laser produced fast ion beams to the injectors of conventional accelerators (see Ref. [14]) and for using laser accelerated proton beams for the fast ignition of thermonuclear targets as discussed in Refs. [15,16].
2. Hadrontherapy with laser accelerated plasma ions Hadron radiotherapy [17–22] is a part of radiation therapy, which uses not only beams of high energy ions, but also π -mesons, neutrons, electron beams, X- and gamma rays to irradiate cancer tumors (see Ref. [17] and literature quoted therein). In cancer treatment surgery, chemotherapy and radiotherapy are applied together as they complement each other. The use of protons in radiotherapy has several advantages. First of all the proton beam scattering on the atomic electrons is weak and thus there is less irradiation of healthy tissues in the vicinity of the tumor. Second, the slowing down length for a proton with given energy is fixed, which avoids undesirable irradiation of healthy tissues at the rear side of the tumor. Third, the well localized maximum of the proton energy losses in matter (the Bragg peak) leads to a substantial increase of the irradiation dose in the vicinity of the proton stopping point (see, for example, [17]). After more than 40 years of experimental investigations different hospital based centers of proton radiotherapy are currently under construction. All these centers are located in large hospitals and have the capacity to admit up to 1000 patients per year. Already three centers are operating [21] and soon their number will grow to nine. These centers have accelerators specialized for medical applications, from which the proton beams are delivered into 3 to 5 procedure rooms equipped with treatment units. A necessary and most expensive treatment unit of the hospital based proton therapy centers is the GANTRY which is a device that provides multi-directional irradiation of a laying patient. By now, proton beams with the required parameters have been obtained using conventional accelerators of charged particles (synchrotrons, cyclotrons,
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linacs). On the other hand, the use of a laser based accelerator is fairly attractive because of its compactness and of the additional possibility it offers of controlling the proton beam parameters. There are at least two possible ways of using a laser accelerator. In one, it simply replaces a conventional proton accelerator. In the other, which appears to be much more attractive, the laser radiation is delivered to the target where its energy is converted into the energy of fast ions. This can simplify the technical problems related to the generation and transport of the ion beam and reduce costs substantially. In this case the central accelerator, the channels through which the fast ions are transported and, finally, most part of the GANTRY are no longer required. The basic parameters required in a proton beam for medical applications can easily be reached with present accelerator technology. The proton beam intensity must be in the range 1010 to 5 × 1010 protons per second, the maximum proton energy must be in the range 230 to 250 MeV. At the same time, for practical realization, the following two conditions are the most demanding for the laser accelerator. First is the requirement of a highly monoenergetic proton beam with ∆E/E = 10−2 . Second is the system duty factor, i.e., the fraction of the time of during which the proton beam can be used that must not be smaller than 0.3. Actually, the most important factor is the economical feasibility of the use of a laser accelerator. Although it is not easy now to determine what the real price of a laser accelerator will be, its use is likely to represent a significant simplification of present equipment and to lower the price of construction of hadrontherapy centers.
3. Multi-layer targets for high quality ion beams In the simplest scheme of ion acceleration, the laser pulse interacts with a thin foil. The foil thickness ranges from less than a micron up to several microns. When multiterawatt laser radiation interacts with a target, matter is ionized in an interval shorter than a single optic oscillation period of the laser radiation, producing in such a way a collisionless plasma. Hence, we will approximate the foil as a narrow layer of overdense plasma. Under the action of the laser radiation the electrons are expelled from a region
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on the foil with transverse size of the order of the diameter of the focal spot. For femtosecond long laser pulses with multiterawatt power, the typical time scale of the hydrodynamic expansion of a micron plasma slab is much longer than the laser pulse duration. Under these conditions ions remain at rest, which results in the formation of a positively charged layer of ions. However, after a time interval equal to or longer than the inverse of the ion Langmuir frequency 1/ωpi = (mi /4πn0 Zi2 e2 )1/2 the ion layer explodes because of the Coulomb repulsion of electric charges with equal sign. In order to estimate the typical energy gain of the fast ions we assume that all free electrons produced by ionization in the irradiated region of the foil are expelled. In this case the electric field near the positively charged layer is equal to E0 = 2πn0 Zi el. Here n0 is the ion density in the foil, Zi e the ion electric charge, and l the foil thickness. The region of strong electric field has a transverse size of the order of the diameter 2R⊥ of the focal spot. Thus the longitudinal size of this region where the electric field remains essentially one-dimensional is also of order 2R⊥ and the typical energy of the ions with charge Za e accelerated by the electric field due to charge separation can be estimated as Emax = 4πn0 Za Zi e2 lR⊥ .
(1)
In deriving Eq. (1) we have assumed that the energy gained by the electrons in the laser field is well above the ion energy and larger than the energy they need to leave the irradiated region. The electron energy in the electromagnetic wave is given by Ee = me a 2 /2 (see Ref. [23]), where a = eE/me cω is the dimensionless amplitude of the laser pulse. From this condition we can find the required value of the laser pulse intensity, its power and, once the pulse duration is chosen, its energy. In Table 1 we present four set of parameters of the target and of the proton beam as well as of the parameters of the laser pulse that are required in order to accelerate the protons. In all four versions we consider a solid density plasma inside the target with transverse size determined by the radius of the focal spot, but change he thickness of the foil. In versions 2, 3, and 4 the size of the focal spot is approximately one wavelength of the laser radiation. The value of the product of the laser pulse duration τl times the
Table 1 Parameters of the fast proton beam Zi = 1 versus the parameters of the laser pulse and of the target. The proton energy is calculated on the basis of Eq. (1) Version Foil thickness (µ) Particle density (1023 cm−3 ) Focus radius (µ) Pulse amplitude Intensity (1020 W cm−2 ) Power (TW) Pulse energy (J) Pulse length (fs) τ1 ωpp Number of fast protons 1010 Proton energy (MeV)
1
2
3
4
0.5 1 5 130 220 1.7 × 104 8.5 × 103 480 210 400 4500
0.25 1 1 40 22 70 17 240 100 7.8 450
0.1 1 1 25 8.8 27 2.7 100 40 3 1810
0.05 1 2 26 8.8 110 5.4 48 200 6.3 180
proton Langmuir frequency ωpp = (4πn0 e2 /mp )1/2 (mp is the proton mass) shown in Table 1 characterizes whether the immobile ion approximation is valid during the laser–target interaction. We see that from this point of view version 4 with a very thin foil is the best for proton acceleration in hadrontherapy. Indeed, one of the main obstacles in obtaining such effective ion acceleration regimes is related to the condition that the laser pulse must be shorter than the ion response time. It is not yet clear whether the effect of the ion motion does deteriorate the efficiency of the ion acceleration and lower the ion beam quality. It is however clear that for long laser pulses with moderate intensities the assumption of immobile ions is not valid and that the ion acceleration evolves in a much less controlled way. In order to satisfy the condition τl ωpp < 1 we must either choose a short pulse, as the one corresponding to version 4 in Table 1, or we must use a target made of heavy ions. In the case of a heavy ion target with approximately the same ion density n0 , the ion response time is equal to 1/ωpi = µ1/2 /(Zi ωpp ), where µ = mi /mp . For µ1/2 /Zi 1 we have 1/ωpi 1/ωpp . We see that this is possible only for non fully stripped ions inside the heavy ion layer of the target. A version of such model will be considered below. If we compare the form of the energy spectra of the accelerated ions observed in the experiments or obtained in numerical simulations with the narrow energy spectrum, ∆E/E < 10−2 , that is required for medical applications in hadrontherapy, we see that the energy spectra of the laser accelerated ions are at
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present quite far from the required ones. In Refs. [3, 11] it was found that the ion spectra below a maximum energy Emax (E < Emax ) can be approximated by a quasi-thermal distribution with an effective temperature T , which is several times smaller than Emax . Such spectra are not appropriate for our purposes because an ion beam with such an energy spread would not provide a sufficient increase of the irradiation dose in a well localized region and would lead to an unacceptably high damage of the healthy tissues. In order to improve the proton beam quality one can cut the beam into beamlets with a narrow spread in energy space. However, in this case the efficiency of the laser energy transformation into the energy of fast particles decreases significantly, and, what is more important, the number of fast particles decreases. A different approach which seems more promising, is connected with the use of multi-layer targets. In this scheme a thin foil is used as a target and its rear surface is coated with a thin hydrogen layer. When the ultra short laser pulse irradiates the target, the heavy atoms are partly ionized and the ionized electrons abandon the foil, generating an electric field due to charge separation. Because of the large value of the ratio (µ/Z) the heavy ions remain at rest, while the lighter protons are accelerated. The requirement that the number of fast protons in the beam must be approximately 1010 to 5 × 1010 protons per second can be satisfied relatively easily in a multi-layer target, extrapolating from the experimental results in Refs. [3–7] and the computer simulations presented in Refs. [9–12] where the number of fast protons accelerated per pulse varies from 1012 to 1013 particles. Thus, in order to achieve 1010 fast protons per pulse from a two-layer target it is enough to have a proton layer approximately 0.02 µm thick and a laser pulse focused onto a spot with diameter equal to two laser wavelengths. A sketch of a doublelayer target is shown in Fig. 1. The first layer is made of heavy ions, i.e., of ions with a sufficiently large value of µ1/2/Z and the target is sufficiently thick so as to give a large enough electric field due to charge separation. This electric field has opposite sign on the two different sides of the target, has a zero inside the target and vanishes at a finite distance from it, as illustrated in Fig. 1(a). Since the number of protons is assumed to be sufficiently small not to produce any significant effect on the electric field,
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Fig. 1. Scheme of a two-layer target. Density distribution of heavy ions, protons, and electrons and dependence of the electric field on the x coordinate (a). The form of two-layer target in the r, x plane (b). Shape of a curved two layer target in the plane r, x (c).
their behavior can be described within the framework of the test particle approximation. A most important requirement is that the transverse size of the proton layer be smaller than the pulse waist so as to decrease the influence of the laser pulse inhomogeneity in the direction perpendicular to its direction of propagation. The pulse inhomogeneity causes an inhomogeneity of the accelerating electric field, which results in an additional energy spread of the ion beam, as seen in experiments. The effect of the finite waist of the laser pulse leads also to an undesirable defocusing of the fast ion beam. In order to compensate for this effect and to focus the ion beam we can use the properly
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deformed targets, as suggested in Ref. [9]. A sketch of a deformed target is shown in Fig. 1(b). The energy spectrum of the protons can be found by taking the electric field in the vicinity of the target to be non-zero inside the region 0 < x < R0 and to be of the form E(x) = E0 (1 −x/R0), where E0 = 2πn0 Zel and R0 is the scale-length of the electric field in the longitudinal and in the transverse directions (R0 = 2R⊥ ). The distribution function of the fast protons f (x, v, t) obeys the kinetic equation ∂f eE ∂f ∂f +v + = 0, ∂t ∂x mp ∂v
(2)
which gives f (x, v, t) = f0 (x0 , v0 ), where f0 (x0 , v0 ) is the distribution function at the initial time t = 0, i.e., f (x, v, t) is constant along the characteristics of Eq. (2) which can be written as x = R0 − (R0 − x0 ) cos Ωt +(v0 /Ω) sin Ωt and v = v0 cos Ωt +(R0 − x0 )Ω sin Ωt, where Ω = (eE0 /mp R0 )1/2 and x < R0 . If v0 = 0, all particles reach x = R0 at the same time t = π/2Ω independently of their initial position. After this time they exit the acceleration region. Thus, the linear dependence of the electric field on the x coordinate produces both the acceleration and bunching of the particles. The number of particles per unit volume in phase space, dx dv, is equal to dn = f dx dv = f v dv dt = f dE dt/mp . We assume that at t = 0 all particles are at rest and that their spatial distribution is given by n0 (x0 ) which corresponds to the distribution function f0 (x0 , v0 ) = n0 (x0 )δ(v0 ), with δ(v0 ) the Dirac delta function. For the applications discussed in this Letter we are interested in the particle distribution function integrated over time. Time integration of the distribution f v dv dt gives the energy spectrum of the beam dt N(E) dE = dE n0 (x0 )δ(v0 ) mp n0 (x0 ) dt = (3) dE. mp dv0 At the end of the acceleration region, from the equation of the characteristics with v0 = 0 we obtain |dt/dv0 | = |Ω(2E/mp )1/2|−1 , i.e., N(E) dE =
n0 (x0 ) dE . |Ω(2Emp )1/2|
(4)
Here x0 = x0 (E) is obtained by inverting the relationship E = mp [(R0 − x0 )Ω]2/2. The spread of the pro-
ton energy in the two-layer target model is determined by the width of the proton layer ∆x0 and is given approximately by ∆E = 2πn0 Zi e2 l∆x0 . The ratio between this energy spread and the maximum energy is ∆E/Emax = ∆x0 /R0 . For a proton layer 2 × 10−6 cm thick, with R0 = 2 × 10−4 cm we find ∆E/Emax = 10−2 with Emax = 200 MeV, when the product of the heavy ion charge times the foil thickness l is given by Zi l = 2 × 10−4 cm. An additional parameter for controlling the properties of the proton beams is provided by the use of deformed targets. As was demonstrated in Refs. [9–12] with 2D and 3D PIC simulations, the protons accelerated from the rear side of a deformed (concave) foil are focused into a point localized near the center of the foil curvature.
4. Simulations of a quasi-monoenergetic proton beam In order to take into account the numerous nonlinear and kinetic effects as well as to extend our consideration to a multi-dimensional geometry, we performed numerical simulations of the proton acceleration during the interaction of a short, high power laser pulse with a two-layer target. We used a twodimensional version of a code REMP (relativistic electro-magnetic particle-mesh code) [24]. The size of the computation box is 20λ × 30λ with cell size equal to 0.03λ × 0.03λ. The total number of particles is 106 . The plasma consists of three species: electrons, protons and heavy ions with mp /me = 1836 and mi /(me )Zi = 18360. Here we show the results corresponding to the case where the target consists of two neutral plasma layers with their front side at x = 5λ. The electron density in the heavy ion layer corresponds to the ratio ωpe /ω = 3.0 between the plasma and the laser frequencies. The second layer is localized at the rear side of the target and its electron density is smaller than the critical density ncr = me ω2 /4πe2 and corresponds to ωpe /ω = 0.7. The sizes of the first layer and of the proton layer are 0.25λ × 10λ and 0.1λ × 5λ, respectively, so that the number of electrons in the first layer is 50 times larger than in the proton layer. A circularly polarized laser pulse with dimensionless amplitude a = 20 is initiated at the left-hand side
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Fig. 3. Distribution of the z component of the electric field in the x, y plane at t = 12 × 2π/ω.
Fig. 2. The proton energy spectrum at t = 45 × 2π/ω (a), the heavy ion energy spectrum at t = 97.5 × 2π/ω (b).
boundary of the computation region. The pulse size is 20λ × 15λ, with a constant profile and with a gradual decrease of its amplitude down to zero on the scale length 2λ. The simulation results are shown in Figs. 2–5, where the coordinates are measured in wavelengths of the laser light and the time in laser periods. In Fig. 2 we present the spectrum of the proton energy at time t = 45 × 2π/ω (a) and of the energy per nucleon of the heavy ions at time t = 97.5 × 2π/ω (b). In Figs. 3 and 4 we present the distributions of the z and of the x components of the electric field in the x, y plane at t = 12 × 2π/ω which show the shape of the laser pulse and of the accelerating electric field. This field is localized in the vicinity of the first layer (the heavy ion layer) of the target. In Fig. 5 we show the electric
Fig. 4. The same as in Fig. 3 for the x component of the electric field.
charge distribution in the x, y plane at t = 18 × 2π/ω (a), at t = 27 × 2π/ω (b), t = 51 × 2π/ω (c). We see that the proton layer moves in the x direction and that the distance between the two layers increases. The heavy ion layer expands and tends to become rounded. The two black areas show the form of the clouds of the heavy ions (left) and of the proton layer (right) and the white cloud corresponds to the electrons blown out of the target by the laser light. We notice that for the adopted simulation parameters the electrons do not abandon the region irradiated by the laser light completely. Even if only a portion of the electrons is accelerated and heated by the laser pulse, the electric field they generate appears to be strong enough to accelerate the protons up 100 MeV in the first configuration, and up to 60 MeV in the second.
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The energy per nucleon acquired by the heavy ion component is approximately ten times smaller than the proton energy. As seen in Fig. 2, the heavy ions have a wide energy spectrum while the protons form a quasi-mono-energetic bunch with ∆E/E ≈ 3%. The proton beam remains localized in space for a long time due to the bunching effect of the linearly decreasing dependence of the electric field on the coordinate in the acceleration direction discussed above.
5. Conclusions (a)
(b)
The future of hadron radio therapy requires the construction of specialized centers, equipped with modern diagnostics and medical accelerators near oncological clinics. The construction of small centers would make it possible to provide cheaper facilities for hadrontherapy closer to the patients. A possible solution of this problem relies on producing a specialized laser proton accelerator. Such an accelerator can simplify the technical solutions adopted in the hospital centers for proton radio therapy and lower their construction costs significantly. The use of the multi-layer targets with different shapes and composition opens up additional opportunities for controlling the parameters of the fast proton beam, for optimizing its energy spectrum, the number of particles per bunch, the beam focusing and the size of the region where the beam deposits its energy. To increase the duty factor we must use a system of several high-power, 1 Hz repetition rate lasers or to use a multi-stage acceleration scheme in the case of moderate power, high repetition rate (1 kHz) lasers.
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(c) Fig. 5. Distribution of the electric charge density (black corresponds to both heavy (thick shell) and light ions (thin shell) whereas white corresponds to electrons) in the x, y plane at different times: t = 18 × 2π/ω (a), t = 27 × 2π/ω (b), t = 51 × 2π/ω (c).
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