- Email: [email protected]
Acceleration of ions with a nonadiabatic linearly polarized laser pulse
Physics Letters A 375 (2011) 1135–1141
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Acceleration of ions with a nonadiabatic linearly polarized laser pulse Victor V. Kulagin a,∗ , Vladimir A. Cherepenin b , Vladimir N. Kornienko b , Hyyong Suk c,∗ a b c
Sternberg Astronomical Institute, Moscow State University, Universitetsky prosp. 13, 119899 Moscow, Russia Institute of Radioengineering and Electronics, RAS, Mohovaya 11, Moscow 125009, Russia Advanced Photonics Research Institute, GIST, Gwangju 500-712, Republic of Korea
a r t i c l e
i n f o
Article history: Received 8 June 2010 Received in revised form 10 January 2011 Accepted 11 January 2011 Available online 15 January 2011 Communicated by F. Porcelli Keywords: Laser acceleration of ions Ultraintense nonadiabatic laser pulses Relativistic electron mirror
a b s t r a c t Acceleration of ions during interaction of a nonadiabatic laser pulse (i.e., a pulse with a sharp front) with a nanofilm is considered. If the amplitude of such a pulse is large enough, all electrons are removed from the target at the beginning of interaction and an energy of the most energetic ions follows approximately a parabolic law with time. Two main physical mechanisms limiting the maximal ions energy are identified and investigated in detail with the help of two-dimensional (2D) particle-in-cell (PIC) simulations. The first effect is a compensation of the ions charge due to the longitudinal return of the electrons to their initial position. The second effect is the compensation of the ions charge in the laser spot due to the transverse motion of the electrons from the periphery of the target. The theory for both effects is developed and a good agreement with the 2D PIC results is established. This theory allows predicting the optimal parameters for ions acceleration. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Acceleration of ions with laser pulses became a hot topic recently due to unique properties of laser-driven ion beams such as a low transverse emittance, a good collimation, a short burst duration, and a high beam current, besides, the laser ion source can be made compact and relatively inexpensive because of a very high accelerating field. The laser-driven energetic ions can be used in radiography of transient processes in laser–plasma interactions, for fast ignition of fusion cores, in hadron therapy, for isohoric heating of matter, etc. For not an extremely high intensity, the physics of laser acceleration of ions is essentially different for the cases of linear and circular laser polarizations so all schemes of laser ion acceleration can be divided into two main classes. For the first class, the ions are accelerated with a circularly polarized laser pulse [1–12] and the “heating” of the electrons with the laser pulse is strongly suppressed, which allows, in principle, achieving higher efficiency of ions acceleration and smaller spread of their energy. However, in this case, the optimization of the process requires careful adjustment of laser and target parameters, namely, laser amplitude, thickness of the target and density of electrons in it. Besides, three-dimensional effects destroy the optimality of acceleration, especially for a small laser spot size. Also, high-power lasers generate usually the linearly polarized pulses so transfor-
*
Corresponding authors. E-mail addresses: [email protected] (V.V. Kulagin), [email protected] (H. Suk). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.01.022
mation to the circular polarization requires additional efforts and decreases the ultimate power and energy of the pulse. For the second class, the acceleration of ions with linearly polarized laser pulses is used [13–28]. Heating of the electrons is essential here. Historically, ions acceleration with a linearly polarized laser pulses has been realized in the first experiments [13, 14], and a target normal sheath acceleration model was used to explain the physics of interaction [18,19,21,24–26]. In this model, the time-dependent component of the ponderomotive force generates hot electrons, which then escape from the back and front surfaces of the target [18,19,21,25], and ions are accelerated by the electrostatic field induced at the target surfaces. Here, precise adjustment of the laser-target parameters and transformation of the laser pulse from linear to circular polarization are not necessary. Also, double layer [17] and complex [28] targets can be used for ions acceleration in this case, which allow better controlling the ions parameters and can be considered as an additional advantage of the linear polarization. The acceleration will be most effective when all electrons are removed with the laser pulse from the target. This can be possible when ponderomotive force of the laser pulse can overcome the Coulomb forces of attraction between the ions and the electrons in the target. Just this case is realized in the process of generation of relativistic electron mirrors from a nanofilm target [29,30], i.e., from the target having thickness in the nanometer range. In this Letter, we investigated the parameters of the ion beams accelerated during relativistic electron mirrors generation. For nonadiabatic laser pulses (i.e., for pulses having a sharp front), at the beginning of acceleration when all electrons are removed
1136
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
from the target, energy of ions follows approximately a parabolic law with time so this interval can be called an interval of effective acceleration. Then, the rate of acceleration decreases. Acceleration of ions in the regime of full target electron evacuation during interaction with a laser pulse was considered in a number of papers (see, e.g., Refs. [12,24]). However, physical mechanisms restricting the acceleration of ions in this regime were not studied enough. We identified and investigated in detail the mechanisms limiting the maximal energy of ions. The first mechanism is a longitudinal reverse motion and return of the electrons, which are near the central part of the relativistic electron mirror (near the laser beam axis), due to attraction of the ions. These returned electrons compensate the charge of the ions, which strongly reduces the longitudinal accelerating field. The second effect is a compensation of the ions charge in the laser spot by the electrons moving transversely from the periphery of the target, which means that the diameter of the positive window decreases with time contrary to the estimates in the previous papers. The theory for both effects was developed and a good correspondence with two-dimensional (2D) particle-in-cell (PIC) simulations was established. According to this theory, the time of effective acceleration determined by the return of the electrons from the central part of relativistic electron mirror depends primarily on the ratio of the dimensionless laser pulse amplitude a0 and the dimensionless value for the surface charge density of the target α , which is a measure of the Coulomb forces of attraction between the ions and the electrons in the target [a0 = |e | E 0 /(mc ω), α = π (ω2p /ω2 )(l/λ), where e and m are the charge and the mass of an electron, c is the speed of light, E 0 , ω, and λ are the amplitude, the frequency, and the wavelength of the laser field in a vacuum, ω p = 4π n0 e 2 /m is the characteristic plasma frequency, n0 and l are the density and the thickness of the nanofilm]. The time of effective acceleration defined by the transverse motion of the electrons depends strongly on the laser beam diameter. The maximal energy of ions at the end of the interval of effective acceleration will be achieved when both limiting effects come into play approximately simultaneously. The developed theory allowed to derive an equation for optimal values of a0 , α and laser beam diameter. Below, we limited our consideration to the case of moderate (for the problem of ions acceleration) values for dimensionless amplitude of the laser field, 12 a0 130, and laser pulse energy of few tens of joules. These values can be accessed in modern experiments. For these laser pulse parameters, the laser-piston model [20] is inadequate. The most energetic ions are accelerated in the combined Coulomb field of other ions and electrons so the motion of the electrons around the ions is very important (contrary to the case of pure Coulomb explosion model). The Letter is organized as follows: the results for the 2D PIC simulations of the ions acceleration with the nonadiabatic laser pulses are presented in Section 2. In Section 3, the analytical theory to estimate the time of effective acceleration is derived and comparison with the 2D PIC simulations is presented. Section 4 concludes the discussion. 2. Two-dimensional PIC simulations for acceleration of ions with nonadiabatic laser pulses To study the acceleration of ions with the laser pulses, we performed the 2D PIC simulations with the XOOPIC [31] code. In simulations, we used a linearly y-polarized laser pulse (λ = 1 μm), running along the x axis in the positive direction. In the simulations for this section, the laser pulse had a Gaussian transverse profile with the beam waist w 0 = 8λ at focus (laser spot size is 2w 0 = 16λ) and a step-like longitudinal envelope, the full duration of the laser pulse was 3λ. The acceleration of ions from two different nanofilms (positioned at the beam focus) with thickness 10 nm
Fig. 1. (Color online.) Dependence on time for the maximal energy of ions near the laser beam axis for the nanofilm with α = 3 and laser pulse amplitude a0 = 12 (blue circles), a0 = 18 (red triangles), a0 = 30 (cyan squares), and a0 = 40 (magenta diamonds). Green solid line corresponds to the energy of the ions when all electrons are removed from the nanofilm [cf. Eq. (1)].
and α = 3 or α = 10 was investigated, which corresponds to the electron densities of n0 = 95.5ncr and n0 = 318.3ncr , where ncr is the critical density. The plasma was assumed to be preionized and collisionless with the ions mass of about mi = 1840m (hydrogen). The simulation box size was from 20λ to 50λ along x axis and from 40λ to 80λ along y axis. The number of grid points per laser wavelength was from 200 to 2000 along x axis and 40 along y axis with 250 particles per cell. These numbers were controlled to give the required accuracy of simulations. The boundaries of the simulation box were conducting for the field and absorbing for particles. The time dependences for the maximal energies of ions near the laser beam axis are shown in Figs. 1 and 2 (time is normalized with the period T 0 of the laser field). The physics of generation of relativistic electron mirrors from a nanofilm target depends primarily on the ratio a0 /α [29,30]. So for each value of parameter α , we used a set of the laser pulse amplitudes chosen as a0 = 4α , a0 = 6α , a0 = 10α , and a0 = 13α . Figs. 1 and 2 look very similar. The main difference is in a different scale for the y axis. This fact confirms that the physics for acceleration of ions from nanofilms with the nonadiabatic laser pulses depends basically on the ratio a0 /α as the physics for relativistic electron mirrors generation does [29,30]. Also, for the laser pulses with the amplitudes of about a0 ∼ 100, the energy of the accelerated ions can be on the level of several hundreds of MeV. All curves in Figs. 1 and 2 show very similar time behavior. At the beginning of acceleration, the relativistic electron mirror is formed and all electrons are removed from the nanofilm. The accelerating electric field is approximately constant in time and uniform in space since it is simply the field of a plane capacitor. The kinetic energy ε of the most energetic ions on the edge of the ion layer (i.e., just inside the plane capacitor) follows approximately the parabolic dependence on time:
ε=
1 + (2ζ t )2 − 1
ε0i 2ε0i (ζ t )2 ,
(1)
where
ζ = 2παm/ M ,
(2)
ε0i and M are the ions rest energy and mass, and t is normalized here (and in equations below) with the laser field period T 0 . Then, the rate of acceleration decreases. For a0 = 4α , a0 = 6α , and
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
1137
Fig. 2. (Color online.) Dependence on time for the maximal energy of ions near the laser beam axis for the nanofilm with α = 10 and laser pulse amplitude a0 = 40 (blue circles), a0 = 60 (red triangles), a0 = 100 (cyan squares), and a0 = 130 (magenta diamonds). Green solid line corresponds to the energy of the ions when all electrons are removed from the nanofilm [cf. Eq. (1)]. Note the different vertical scale here and in Fig. 1. Fig. 4. (Color online.) Phasespaces x– y for electrons (red) and ions (cyan) at time t = 6.05T 0 from the beginning of interaction (α = 10, a0 = 60, and w 0 = 8λ). The initial position for the nanofilm is at x = 6.5λ, the remainder of the relativistic electron mirror is situated from x = 11λ to x = 12λ.
Fig. 3. (Color online.) Spectrum of accelerated ions near the laser beam axis (within 6λ spot) at t = 20T 0 (α = 10, a0 = 100).
a0 = 10α , the time of effective acceleration (i.e., the time, at which there is a bend of the acceleration curve) increases with the laser pulse amplitude giving increasing ions energy at the “knee” point. For the largest laser pulse amplitude from the set (a0 = 13α ), the time of effective acceleration is about the same as for the smaller amplitude a0 = 10α and, correspondingly, the ions energy at the knee point does not increase when the laser pulse amplitude is changed from a0 = 10α to a0 = 13α . So obviously, there is a physical mechanism, which determines the appearance of the knee point on the acceleration curve for the ions, and there is another process, which limits the energy of the ions at the knee point when the laser pulse amplitude is increased. As we will see below, the first mechanism is the reverse longitudinal motion of the electrons of the relativistic electron mirror and their return to initial position due to attraction of the ions [29,30]. These returned electrons compensate the charge of the ions, which strongly reduces the longitudinal accelerating field and the rate of acceleration for the most energetic ions. The second effect is compensation of the ions charge in the laser spot by the electrons, which move transversely along the nanofilm from the periphery of the target. The spectrum of accelerated ions at t = 20T 0 is presented in Fig. 3 for α = 10 and a0 = 100 (only the ions within 6λ spot near the laser beam axis are considered). The spectra for the other
curves in Figs. 1 and 2 are very similar in shape to one presented in Fig. 3. This spectrum demonstrates almost flat-top feature for ions energy from 150 MeV to 280 MeV, which can be useful in some applications. Also, the relative number of high-energy ions is not exponentially small as for the TNSA mechanism. And at last, the clear cut-off for the maximal energy of ions is evident, which allows to characterize the acceleration process with the maximal (cut-off) energy. We performed also some additional simulations to reveal the ions energy dependence on thickness and electron density of the nanofilm when the parameter α is kept constant. These simulations show that the energy of the accelerated ions weakly depends on the nanofilm thickness provided it is considerably smaller than λ (cf. also [30]). 3. Analytical estimates for the time of effective acceleration of ions by the nonadiabatic laser pulses 3.1. Limitation of the ions maximal energy due to reverse longitudinal motion of the electrons The physics for turning back of the electrons on the left side of the relativistic electron mirror is considered in detail in Refs. [29, 30] (note that the laser pulse is going from the left to the right). This effect can be seen already in the one-dimensional model for interaction, and it can limit the energies of the ions when the ratio a0 /α is not very large. The phasespaces x– y for the electrons and the ions are presented in Fig. 4 for the time t = 6.05T 0 from the beginning of interaction when the electrons from the left side of relativistic electron mirror return back and touch with the most energetic ions on the right edge of the ion layer (α = 10, a0 = 60, and w 0 = 8λ). The time of effective acceleration t ea is about a return time t r for the electrons and can be approximately estimated as the doubled lifetime tl for the relativistic electron mirror (tl is the time when the electrons on the left side of relativistic electron mirror near the laser beam axis begin to turn back [29,30]): tr ≈ 2tl . This expression is a good approximation for the case of heavy ions or small laser pulse amplitudes when tl is also small with respect
1138
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
Fig. 5. (Color online.) Acceleration curve for the ions (blue circles) for α = 10, a0 = 60, and w 0 = 8λ. Two estimates for the time of effective acceleration t ea are shown: one is based on the direct estimation of the return time for the electrons from the 2D PIC simulations (vertical black line marked with squares) and the other is obtained from Eq. (4) with the lifetime tl derived from the 2D PIC simulations (vertical red line marked with diamonds). The time for the closing of the positive window in the nanofilm due to the transverse motion of the electrons along the nanofilm is represented with vertical magenta line marked with triangles. Green solid line corresponds to the dependence of ions energy on time according to Eq. (1).
to ζ −1 . The energy of the ions grows quadratically with tl in this case. The exact value for the time of effective acceleration in the 1D model with full electron evacuation from the nanofilm can be calculated from the following equation:
ζ tr2 + tr − 2tl = 0.
(3)
This equation takes into account a displacement of the ions at the right edge of the ion layer (i.e., the displacement of the most energetic ions) due to their acceleration under the action of the Coulomb force. In Eq. (3), we supposed that the time for removing of the electrons from the nanofilm is considerably smaller than tl . Also, the return time for the electrons to the initial position of the nanofilm is supposed equal to tl (i.e., electrons move forth and back during the same time tl ) and the return velocity of the electrons when they touch the ions is about c. Then, Eq. (3) gives for the time of effective acceleration of the ions the following expression
√ tr =
1 + 8ζ t l − 1 2ζ
.
(4)
In Fig. 5, the acceleration curve for the ions (blue circles) is presented for α = 10, w 0 = 8λ, and a0 = 60 (this curve is identical to the corresponding curve in Fig. 2). Also, two estimates for the return time t r of the electrons are shown: one is based on the direct estimation of the return time for the electrons from simulation results (vertical black line marked with squares), just for this time the phasespaces for the ions and the electrons are shown in Fig. 4. The other estimate is obtained from Eq. (4) with the lifetime tl derived from the PIC simulations (vertical red line marked with diamonds). These two estimates give almost coincident values for t r , which are also equal to the time when the acceleration curve of the ions declines from the curve described by Eq. (1), i.e., the multidimensional effects are unimportant in this case, and the 1D model for ions acceleration can be used effectively. The time for the closing of the positive window in the nanofilm due to the transverse motion of the electrons along the nanofilm is also presented in Fig. 5 with vertical magenta line marked with triangles. There is no any special feature on the acceleration curve near this time.
Fig. 6. (Color online.) Phasespaces x– y for electrons (red) and ions (cyan) at time t = 13.4T 0 from the beginning of interaction (α = 10, a0 = 130, and w 0 = 8λ). The initial position for the nanofilm is at x = 15λ, the relativistic electron mirror is situated at x = 28λ.
3.2. Limitation of the ions maximal energy due to transverse motion of the electrons along the nanofilm Phasespaces x– y for the ions and the electrons at time t = 13.4T 0 are presented in Fig. 6 (α = 10, a0 = 130, and w 0 = 8λ). At this time, the closing of the positive window formed by the laser pulse in the nanofilm occurs, i.e., t c = 13.4T 0 (simulations in the window with larger dimensions along y axis give the same value for t c ). The electrons from the central part of the nanofilm still move forward within the relativistic electron mirror. However, the electrons from the outlying regions of the nanofilm move transversely along the nanofilm due to attraction of the ions near the laser beam axis and partially compensate the ions charge. Then the accelerating longitudinal field decreases and the rate of acceleration for the most energetic ions on the right side of the ion layer also decreases. This effect cannot be seen in the one-dimensional model so this is a multidimensional effect. In Fig. 7, the acceleration curve for the ions (blue circles) is presented for α = 10 and a0 = 130 (this curve is identical to the corresponding curve in Fig. 2), and the time t c for the closing of the positive window in the nanofilm due to the transverse motion of the electrons along the nanofilm is shown with the vertical magenta line marked with triangles (two estimates for the return time t r of the electrons for this case are also indicated). The time t c is close to the point, where the acceleration curve of the ions declines from the law described by Eq. (1). Two estimates for the return time t r of the electrons are not equal in this case, besides, they are far from the knee point of the acceleration curve. So the time of effective acceleration t ea is about t c , and it is defined mainly by the multidimensional effects. To estimate the time t c for the closing of the positive window, one needs to estimate a transverse field due to the ions charge at the edge of the window in the nanofilm. We suppose during this estimation that the short laser pulse already left the nanofilm and ions don’t move. The positive window in the 2D model is a long strip, which has infinite length (along the z axis), its thickness is equal to the thickness of the nanofilm (short edge of the strip is situated along the x axis) and its width is equal to the diameter of
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
the window (long edge of the strip is situated along the y axis). To calculate the field at the middle of the short edge of the strip (i.e., at the median plane of the nanofilm), it is necessary to divide the strip along the y axis into substrips, which will have the infinite length along the z axis, thickness dy along the y axis, and width l along the x axis. The field of some substrip with coordinate y can be calculated supposing that this substrip consists of charged wires of infinite length with linear charge density n0 |e | dy dx (note that only ions are present in the window). After integration along x from −l/2 to l/2, one has (we suppose that x = 0 for the median plane of the nanofilm)
dE str = 4n0 |e | dy arctan
l 2y
(5)
.
Further integration along y from 0 to 2R, where R is the radius of the positive window (the width of the window along the y axis is 2R), gives for the transverse field at the edge of the window in the median plane of the nanofilm:
E w = 2n0 |e |l
4R l
arctan
l 4R
+
1 2
ln 1 +
4R l
2 .
(6)
When the thickness of the nanofilm l is considerably smaller than the value for R, the last equation can be simplified considerably:
E w 2n0 |e |l 1 + ln
4R l
.
(7)
The motion of the electrons at the edge of the window can be approximately described then with the following set of equations (momentum p R , R and t are normalized with mc, λ, and T 0 correspondingly)
dp R dt dR dt
= −2α =
4R l
pR
arctan
1139
l 4R
+
1 2
ln 1 +
4R l
2 ,
.
(8)
1 + p 2R
The dimensionless force from the first equation of the system (8) is shown as a function of R in Fig. 8 for α = 3 and l = 10 nm. Since the focal diameter of the laser beam cannot be smaller than λ, the normalized force acting on the electrons at the edge of the positive window in the median plane of the nanofilm just after beginning of interaction is considerably greater than 1 (and directly proportional to α ). Then, p R becomes greater than 1 in a time, which is considerably smaller than the laser field period T 0 , i.e., the transverse motion of the electrons in the positive window becomes relativistic almost immediately after beginning of interaction. So the time for the closing of the window can be estimated approximately as
tc R 0 ,
(9)
where R 0 is the initial diameter for the positive window, normalized with λ. Numerical solution for Eqs. (8) entirely confirms this result. To use Eq. (9) for the practical estimation of the closing time for the positive window, one needs to define the initial diameter R 0 for this window. It is reasonable to define R 0 as the value, for which the laser pulse amplitude a0 is about the parameter α for the nanofilm since only the electrons inside this spot move longitudinally with noticeable amplitude [32]. Then (w 0 is normalized with λ here)
R 0 w 0 ln
a0
α
(10)
and for the closing time of the positive window one has
t c w 0 ln
a0
α
.
(11)
Comparison of the time t c derived from the 2D PIC simulations with Eq. (11) for different laser spot sizes w 0 is shown in Fig. 8b (α = 3, a0 = 30, l = 10 nm). It occurs that all simulation points belong to the same straight line, a best linear fit to which has the following parameters
t c fit = 1.28w 0 + 2.52.
Fig. 7. (Color online.) The same as in Fig. 5 but for a0 = 130.
(12)
For α = 3, a0 = 30, the coefficient before w 0 in the analytical equation for t c according to Eq. (11) is equal to 1.52, i.e., t c an = R 0 1.52w 0 .
Fig. 8. (a) Dependence for the absolute value of the normalized transverse force [cf. the first equation of the system (8)], which acts on the electrons at the edge of the positive window in the median plane of the nanofilm, on the radius of the window (α = 3, l = 10 nm, R is normalized with λ). (b) Time for the closing of the positive window as a function of the laser beam waist w 0 (α = 3, a0 = 30, l = 10 nm): red stars are simulation points, red solid line is the best linear fit to the simulation points, and blue line is the analytical estimation for the closing time from Eq. (11).
1140
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
One can infer a good coincidence of the simulation and the analytical lines from Fig. 8b so the derived theory for t c can be used for analytical estimation of the time for effective acceleration of the ions when w 0 is from 2λ to 20λ. However, there is a small regular difference (about 13%) in the tangents for the inclination angles of the lines with respect to the x axis and correspondent nonzero value for t c at w 0 = 0 for the data derived from the 2D PIC simulations. This means that the real value for R 0 differs a little from the value determined by Eq. (10). Knowing the dynamics for the positive window closing, one can improve the estimate given above for the time of effective acceleration, t ea . Actually, the effective longitudinal field accelerating the ions starts to decrease earlier than the positive window fully closes because, due to the longitudinal motion, the most energetic ions move away from the plane of the window. For example, one can suppose that the accelerating field decreases noticeably when the positive window diameter becomes about the displacement of the most energetic ions from the initial position of the nanofilm. In general case, the following equation can be used to estimate the time of effective acceleration t ea
R 0 − t ea χ S i = χ
2 ai t ea
2
,
(13)
where S i is the distance, which the most energetic ions travel from the beginning of interaction, ai is the acceleration for these ions, and χ is a coefficient, which defines the relation between the dimension of the positive window at the moment of time t ea and the displacement of the most energetic ions (for χ = 0, the estimate t c is used for t ea ; χ = 0.5, if at the time t ea , the diameter of the positive window is about S i ; and χ = 1, if at the time t ea , the diameter of the positive window is two times larger than S i ). Then, for t ea , one has from Eq. (13)
√
t ea =
1 + 4ζ χ R 0 − 1 2ζ χ
.
Fig. 9. (Color online.) Maximal energy of ions versus α for a0 = 100, w 0 = 8λ: red curve is for the theoretical values, green curve marked with squares is for the energy from the 2D PIC simulations for time t = t ea (this time depends on α so it is different for each point), blue curve marked with circles is for the energy from the 2D PIC simulations for t = 10T 0 .
tr is about the closing time t c for the positive window (or a corrected value for this time if χ is nonzero). The estimate for the lifetime tl of the relativistic electron mirror is derived in [29,30]
tl
a20 8α 2
T 0.
So the optimal parameters for acceleration of the ions with nonadiabatic laser pulses should satisfy the following equation
√
1 + 8ζ t l − 1 =
(14)
The numerical values for the case of Figs. 6 and 7 (α = 10, a0 = 130, and w 0 = 8λ) look as follows. The estimate for the initial dimensions of the positive window from Eq. (10) is R 0 12.8λ, whereas this value from the 2D PIC simulations (calculated from the time for the closing of the positive window) is R 0 13.4λ (cf. Fig. 6). For t = 12.8T 0 , which is the estimate for t ea according to Eq. (13) with χ = 0 and R 0 from Eq. (10), the energy of the most energetic ions from the 2D PIC simulations is about εpic 275 MeV and the prediction for the ions energy from Eq. (1) is εpar 325 MeV so the relative difference is (εpar − εpic )/εpar 15%. For χ = 0.5, the estimate for the time of effective acceleration from Eq. (14) is t ea 10.8T 0 , the energy of the most energetic ions from the 2D PIC simulations is about 220 MeV, and the prediction for the ions energy from Eq. (1) is 225 MeV so the relative difference is about 2%. And at last for χ = 1, the estimate for the time of effective acceleration is t ea 9.6T 0 , εpic 177 MeV, and εpar 179 MeV so the relative difference is about 1%. So the estimates for χ = 0 and χ = 0.5 can be considered as acceptable for the prediction of the energy of the most energetic ions at the end of the interval of effective acceleration (the last one gives enhanced accuracy), while the estimate for χ = 1 underestimates the value for t ea and, correspondingly, the maximal energy of the ions at this moment of time. 3.3. Optimal parameters for acceleration of the ions with nonadiabatic laser pulses At the end of the interval of effective acceleration, the energy achieved by the ions will be maximal when both limiting effects come into play approximately simultaneously, i.e., the return time
(15)
1 + 4ζ χ R 0 − 1
χ
,
(16)
where ζ and R 0 are defined by Eqs. (2) and (10), correspondingly. Then, for the time t ea of effective acceleration, one has the estimate according to Eq. (4) [or Eq. (14)], and the energy of the most energetic ions can be estimated from Eq. (1). If both square roots in Eq. (16) can be approximated with only two terms from the serial decomposition, Eq. (16) becomes
2tl R 0 ,
(17)
i.e., the electrons from the relativistic electron mirror should return to their initial position just at the time when the positive window will be closed by the electrons, which move transversely from the periphery of the target. Using Eqs. (10) and (15), one can conclude from Eq. (16) that if solutions for the optimal parameters exist then, for some value of w 0 , there should be two roots for the optimal ratio a0 /α . Generally speaking, only one of this root provides for a maximum ions energy, the other root can be just a breaking point for the energy curve. In Fig. 9, the maximal energy of ions versus the parameter α is presented. The red curve is calculated for the theoretical value of energy from Eq. (1) with the time of acceleration equal to the minimum of the values from Eqs. (4) and (14) (with χ = 0.5 in the last). Here, α = 12 (the maximal value for energy) corresponds to one root of Eq. (16), the other root near α = 100 is the breaking point. The green curve marked with squares is derived from the 2D PIC simulations for time t = t ea , which is different for each point (from 12T 0 for the point with maximal energy to 0 for the point at α = 110, where the electrons are not evacuated from the target). The shapes of the red and the green curves are similar. The difference in the maximal energy of ions originates from inaccuracy of the estimate for tl from Eq. (15) [30]. The other source for
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
the error is not high enough accuracy for the 2D PIC simulations, which generates Fig. 9. Actually, the lifetime of the relativistic electron mirror (and the maximal energy also) increases with decreasing the grid cell size. At least 10 000 grid points per λ along the longitudinal direction are necessary to reproduce accurately the coherent stage of the relativistic electron mirror evolution [30]. For the PIC simulations above, the error in tl is about 20%. From the experimental point of view, the maximal energy of ions versus α calculated at different time for each point is not very informative. More customary is a dependence where the maximal energy at each point is calculated for the same time. This curve is also presented in Fig. 9 with blue line marked with circles (t = 10T 0 ). For each value of α , the maximal energy here consists of two terms. One is just the energy from the green curve. The other term corresponds to the acceleration of ions during time interval from t ea to t = 10T 0 . This acceleration resembles the TNSA acceleration, however, the distribution of electrons in space is different here. The blue curve demonstrates two peaks, which means that there are two optimal values for the parameter α . The heights of the peaks depend on time, for which the energy is calculated. Detailed study of this result will be presented in the forthcoming paper. 4. Discussion of results and conclusions Conditions for effective acceleration of the ions with the laser pulses having a smooth front is similar to the conditions of relativistic electron mirrors generation [29,30]. The electrons of the target remain motionless in the longitudinal direction until the laser amplitude reaches the value a0 ∼ α . After that, the electrons will be displaced longitudinally by the first half-cycle, which has the amplitude a0 f larger than α . Then, the relativistic electron mirror will evolve until the turn of the left electrons, and the lifetime (the return time also) will be greater for larger values of a0 f /α . This picture is confirmed by our 2D PIC simulations. So for effective acceleration of the ions by a smooth laser pulse, the high peak-to-peak ratios ∼4 to 5 are necessary at the front of the laser pulse. For a Gaussian longitudinal profile, it corresponds to a full duration at half maximum of about (1.2–1.4)λ. Such pulses can be produced by shaping the incoming laser pulse with a thin plasma layer [33–35]. In conclusion, we studied the process of the ions acceleration during relativistic electron mirrors generation from the nanofilms with nonadiabatic laser pulses. Two main physical effects limiting the maximal energy of ions were considered in detail using numerical and analytical approaches: the longitudinal reverse motion and return of the electrons, which are near the central part of the relativistic electron mirror (near the laser beam axis), and the transverse motion of the electrons from the periphery of the target due to attraction of the ions. These electrons compensate the
1141
charge of the ions and strongly reduce the longitudinal accelerating field. The maximal energy of the ions at the end of the interval of effective acceleration will be achieved when both limiting effects come into play approximately simultaneously. Optimal values for a0 , α and the laser beam diameter were defined. Acknowledgements This work was supported by the RFBR projects 09-02-01483-a and 09-02-12322-ofi-m, by the Challenge Research Project of the NRF of Korea (grant number 2010-0000847), and by the Asian Laser Center Project of GIST. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
A. Henig, S. Steinke, M. Schnurer, et al., Phys. Rev. Lett. 103 (2009) 245003. A. Macchi, F. Cattani, T.V. Liseykina, et al., Phys. Rev. Lett. 94 (2005) 165003. A.V. Korzhimanov, A.A. Gonoskov, A.V. Kim, et al., JETP Lett. 0786 (2007) 577. X. Zhang, B. Shen, X. Li, et al., Phys. Plasmas 14 (2007) 073101. O. Klimo, J. Psikal, J. Limpouch, et al., Phys. Rev. ST Accel. Beams 11 (2008) 031301. X.Q. Yan, C. Lin, Z.M. Sheng, et al., Phys. Rev. Lett. 100 (2008) 135003. A.P.L. Robinson, M. Zepf, S. Kar, et al., New J. Phys. 10 (2008) 013021. A.A. Gonoskov, A.V. Korzhimanov, V.I. Eremin, et al., Phys. Rev. Lett. 102 (2009) 184801. B. Qiao, M. Zepf, M. Borghesi, et al., Phys. Rev. Lett. 102 (2009) 145002. T. Schlegel, N. Naumova, V.T. Tikhonchuk, et al., Phys. Plasmas 16 (2009) 083103. A. Macchi, S. Veghini, F. Pegoraro, Phys. Rev. Lett. 103 (2009) 085003. M. Grech, S. Skupin, R. Nuter, et al., New J. Phys. 11 (2009) 093035. R.A. Snavely, M.H. Key, S.P. Hatchett, et al., Phys. Rev. Lett. 85 (2000) 2945. A. Maksimchuk, S. Gu, K. Flippo, et al., Phys. Rev. Lett. 84 (2000) 4108. A. Andreev, A. Levy, T. Ceccotti, et al., Phys. Rev. Lett. 101 (2008) 155002. A. Henig, D. Kiefer, K. Markey, et al., Phys. Rev. Lett. 103 (2009) 045002. T.Zh. Esirkepov, S.V. Bulanov, K. Nishihara, et al., Phys. Rev. Lett. 89 (2002) 175003. Y. Sentoku, T.E. Cowan, A. Kemp, et al., Phys. Plasmas 10 (2003) 2009. S.V. Bulanov, T.Zh. Esirkepov, J. Koga, et al., Plasma Phys. Rep. 30 (2004) 18. T. Esirkepov, M. Borghesi, S.V. Bulanov, et al., Phys. Rev. Lett. 92 (2004) 175003. E. Fourkal, I. Velchev, C.-M. Ma, Phys. Rev. E 71 (2005) 036412. L. Yin, B.J. Albright, B.M. Hegelich, et al., Laser Part. Beams 24 (2006) 291. L. Yin, B.J. Albright, B.M. Hegelich, et al., Phys. Plasmas 14 (2007) 056706. S.S. Bulanov, A. Brantov, V.Yu. Bychenkov, et al., Phys. Rev. E 78 (2008) 026412. M. Passoni, M. Lontano, Phys. Rev. Lett. 101 (2008) 115001. A.V. Brantov, V.T. Tikhonchuk, V.Yu. Bychenkov, et al., Phys. Plasmas 16 (2009) 043107. J. Davis, G.M. Petrov, Phys. Plasmas 16 (2009) 023105. F. Wang, B. Shen, X. Zhang, et al., Phys. Plasmas 16 (2009) 093112. V.V. Kulagin, V.A. Cherepenin, M.S. Hur, et al., Phys. Rev. Lett. 99 (2007) 124801. V.V. Kulagin, V.A. Cherepenin, Y.V. Gulyaev, et al., Phys. Rev. E 80 (2009) 016404. J.P. Verboncoeur, A.B. Langdon, N.T. Gladd, Comp. Phys. Comm. 87 (1995) 199. V.V. Kulagin, V.A. Cherepenin, M.S. Hur, et al., Phys. Plasmas 14 (2007) 113101. V.A. Vshivkov, N.M. Naumova, F. Pegoraro, et al., Phys. Plasmas 5 (1998) 2727. V.V. Kulagin, V.A. Cherepenin, M.S. Hur, et al., Phys. Plasmas 14 (2007) 113102. L.L. Ji, B.F. Shen, X.M. Zhang, et al., Phys. Rev. Lett. 103 (2009) 215005.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Acceleration of ions with a nonadiabatic linearly polarized laser pulse Victor V. Kulagin a,∗ , Vladimir A. Cherepenin b , Vladimir N. Kornienko b , Hyyong Suk c,∗ a b c
Sternberg Astronomical Institute, Moscow State University, Universitetsky prosp. 13, 119899 Moscow, Russia Institute of Radioengineering and Electronics, RAS, Mohovaya 11, Moscow 125009, Russia Advanced Photonics Research Institute, GIST, Gwangju 500-712, Republic of Korea
a r t i c l e
i n f o
Article history: Received 8 June 2010 Received in revised form 10 January 2011 Accepted 11 January 2011 Available online 15 January 2011 Communicated by F. Porcelli Keywords: Laser acceleration of ions Ultraintense nonadiabatic laser pulses Relativistic electron mirror
a b s t r a c t Acceleration of ions during interaction of a nonadiabatic laser pulse (i.e., a pulse with a sharp front) with a nanofilm is considered. If the amplitude of such a pulse is large enough, all electrons are removed from the target at the beginning of interaction and an energy of the most energetic ions follows approximately a parabolic law with time. Two main physical mechanisms limiting the maximal ions energy are identified and investigated in detail with the help of two-dimensional (2D) particle-in-cell (PIC) simulations. The first effect is a compensation of the ions charge due to the longitudinal return of the electrons to their initial position. The second effect is the compensation of the ions charge in the laser spot due to the transverse motion of the electrons from the periphery of the target. The theory for both effects is developed and a good agreement with the 2D PIC results is established. This theory allows predicting the optimal parameters for ions acceleration. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Acceleration of ions with laser pulses became a hot topic recently due to unique properties of laser-driven ion beams such as a low transverse emittance, a good collimation, a short burst duration, and a high beam current, besides, the laser ion source can be made compact and relatively inexpensive because of a very high accelerating field. The laser-driven energetic ions can be used in radiography of transient processes in laser–plasma interactions, for fast ignition of fusion cores, in hadron therapy, for isohoric heating of matter, etc. For not an extremely high intensity, the physics of laser acceleration of ions is essentially different for the cases of linear and circular laser polarizations so all schemes of laser ion acceleration can be divided into two main classes. For the first class, the ions are accelerated with a circularly polarized laser pulse [1–12] and the “heating” of the electrons with the laser pulse is strongly suppressed, which allows, in principle, achieving higher efficiency of ions acceleration and smaller spread of their energy. However, in this case, the optimization of the process requires careful adjustment of laser and target parameters, namely, laser amplitude, thickness of the target and density of electrons in it. Besides, three-dimensional effects destroy the optimality of acceleration, especially for a small laser spot size. Also, high-power lasers generate usually the linearly polarized pulses so transfor-
*
Corresponding authors. E-mail addresses: [email protected] (V.V. Kulagin), [email protected] (H. Suk). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.01.022
mation to the circular polarization requires additional efforts and decreases the ultimate power and energy of the pulse. For the second class, the acceleration of ions with linearly polarized laser pulses is used [13–28]. Heating of the electrons is essential here. Historically, ions acceleration with a linearly polarized laser pulses has been realized in the first experiments [13, 14], and a target normal sheath acceleration model was used to explain the physics of interaction [18,19,21,24–26]. In this model, the time-dependent component of the ponderomotive force generates hot electrons, which then escape from the back and front surfaces of the target [18,19,21,25], and ions are accelerated by the electrostatic field induced at the target surfaces. Here, precise adjustment of the laser-target parameters and transformation of the laser pulse from linear to circular polarization are not necessary. Also, double layer [17] and complex [28] targets can be used for ions acceleration in this case, which allow better controlling the ions parameters and can be considered as an additional advantage of the linear polarization. The acceleration will be most effective when all electrons are removed with the laser pulse from the target. This can be possible when ponderomotive force of the laser pulse can overcome the Coulomb forces of attraction between the ions and the electrons in the target. Just this case is realized in the process of generation of relativistic electron mirrors from a nanofilm target [29,30], i.e., from the target having thickness in the nanometer range. In this Letter, we investigated the parameters of the ion beams accelerated during relativistic electron mirrors generation. For nonadiabatic laser pulses (i.e., for pulses having a sharp front), at the beginning of acceleration when all electrons are removed
1136
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
from the target, energy of ions follows approximately a parabolic law with time so this interval can be called an interval of effective acceleration. Then, the rate of acceleration decreases. Acceleration of ions in the regime of full target electron evacuation during interaction with a laser pulse was considered in a number of papers (see, e.g., Refs. [12,24]). However, physical mechanisms restricting the acceleration of ions in this regime were not studied enough. We identified and investigated in detail the mechanisms limiting the maximal energy of ions. The first mechanism is a longitudinal reverse motion and return of the electrons, which are near the central part of the relativistic electron mirror (near the laser beam axis), due to attraction of the ions. These returned electrons compensate the charge of the ions, which strongly reduces the longitudinal accelerating field. The second effect is a compensation of the ions charge in the laser spot by the electrons moving transversely from the periphery of the target, which means that the diameter of the positive window decreases with time contrary to the estimates in the previous papers. The theory for both effects was developed and a good correspondence with two-dimensional (2D) particle-in-cell (PIC) simulations was established. According to this theory, the time of effective acceleration determined by the return of the electrons from the central part of relativistic electron mirror depends primarily on the ratio of the dimensionless laser pulse amplitude a0 and the dimensionless value for the surface charge density of the target α , which is a measure of the Coulomb forces of attraction between the ions and the electrons in the target [a0 = |e | E 0 /(mc ω), α = π (ω2p /ω2 )(l/λ), where e and m are the charge and the mass of an electron, c is the speed of light, E 0 , ω, and λ are the amplitude, the frequency, and the wavelength of the laser field in a vacuum, ω p = 4π n0 e 2 /m is the characteristic plasma frequency, n0 and l are the density and the thickness of the nanofilm]. The time of effective acceleration defined by the transverse motion of the electrons depends strongly on the laser beam diameter. The maximal energy of ions at the end of the interval of effective acceleration will be achieved when both limiting effects come into play approximately simultaneously. The developed theory allowed to derive an equation for optimal values of a0 , α and laser beam diameter. Below, we limited our consideration to the case of moderate (for the problem of ions acceleration) values for dimensionless amplitude of the laser field, 12 a0 130, and laser pulse energy of few tens of joules. These values can be accessed in modern experiments. For these laser pulse parameters, the laser-piston model [20] is inadequate. The most energetic ions are accelerated in the combined Coulomb field of other ions and electrons so the motion of the electrons around the ions is very important (contrary to the case of pure Coulomb explosion model). The Letter is organized as follows: the results for the 2D PIC simulations of the ions acceleration with the nonadiabatic laser pulses are presented in Section 2. In Section 3, the analytical theory to estimate the time of effective acceleration is derived and comparison with the 2D PIC simulations is presented. Section 4 concludes the discussion. 2. Two-dimensional PIC simulations for acceleration of ions with nonadiabatic laser pulses To study the acceleration of ions with the laser pulses, we performed the 2D PIC simulations with the XOOPIC [31] code. In simulations, we used a linearly y-polarized laser pulse (λ = 1 μm), running along the x axis in the positive direction. In the simulations for this section, the laser pulse had a Gaussian transverse profile with the beam waist w 0 = 8λ at focus (laser spot size is 2w 0 = 16λ) and a step-like longitudinal envelope, the full duration of the laser pulse was 3λ. The acceleration of ions from two different nanofilms (positioned at the beam focus) with thickness 10 nm
Fig. 1. (Color online.) Dependence on time for the maximal energy of ions near the laser beam axis for the nanofilm with α = 3 and laser pulse amplitude a0 = 12 (blue circles), a0 = 18 (red triangles), a0 = 30 (cyan squares), and a0 = 40 (magenta diamonds). Green solid line corresponds to the energy of the ions when all electrons are removed from the nanofilm [cf. Eq. (1)].
and α = 3 or α = 10 was investigated, which corresponds to the electron densities of n0 = 95.5ncr and n0 = 318.3ncr , where ncr is the critical density. The plasma was assumed to be preionized and collisionless with the ions mass of about mi = 1840m (hydrogen). The simulation box size was from 20λ to 50λ along x axis and from 40λ to 80λ along y axis. The number of grid points per laser wavelength was from 200 to 2000 along x axis and 40 along y axis with 250 particles per cell. These numbers were controlled to give the required accuracy of simulations. The boundaries of the simulation box were conducting for the field and absorbing for particles. The time dependences for the maximal energies of ions near the laser beam axis are shown in Figs. 1 and 2 (time is normalized with the period T 0 of the laser field). The physics of generation of relativistic electron mirrors from a nanofilm target depends primarily on the ratio a0 /α [29,30]. So for each value of parameter α , we used a set of the laser pulse amplitudes chosen as a0 = 4α , a0 = 6α , a0 = 10α , and a0 = 13α . Figs. 1 and 2 look very similar. The main difference is in a different scale for the y axis. This fact confirms that the physics for acceleration of ions from nanofilms with the nonadiabatic laser pulses depends basically on the ratio a0 /α as the physics for relativistic electron mirrors generation does [29,30]. Also, for the laser pulses with the amplitudes of about a0 ∼ 100, the energy of the accelerated ions can be on the level of several hundreds of MeV. All curves in Figs. 1 and 2 show very similar time behavior. At the beginning of acceleration, the relativistic electron mirror is formed and all electrons are removed from the nanofilm. The accelerating electric field is approximately constant in time and uniform in space since it is simply the field of a plane capacitor. The kinetic energy ε of the most energetic ions on the edge of the ion layer (i.e., just inside the plane capacitor) follows approximately the parabolic dependence on time:
ε=
1 + (2ζ t )2 − 1
ε0i 2ε0i (ζ t )2 ,
(1)
where
ζ = 2παm/ M ,
(2)
ε0i and M are the ions rest energy and mass, and t is normalized here (and in equations below) with the laser field period T 0 . Then, the rate of acceleration decreases. For a0 = 4α , a0 = 6α , and
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
1137
Fig. 2. (Color online.) Dependence on time for the maximal energy of ions near the laser beam axis for the nanofilm with α = 10 and laser pulse amplitude a0 = 40 (blue circles), a0 = 60 (red triangles), a0 = 100 (cyan squares), and a0 = 130 (magenta diamonds). Green solid line corresponds to the energy of the ions when all electrons are removed from the nanofilm [cf. Eq. (1)]. Note the different vertical scale here and in Fig. 1. Fig. 4. (Color online.) Phasespaces x– y for electrons (red) and ions (cyan) at time t = 6.05T 0 from the beginning of interaction (α = 10, a0 = 60, and w 0 = 8λ). The initial position for the nanofilm is at x = 6.5λ, the remainder of the relativistic electron mirror is situated from x = 11λ to x = 12λ.
Fig. 3. (Color online.) Spectrum of accelerated ions near the laser beam axis (within 6λ spot) at t = 20T 0 (α = 10, a0 = 100).
a0 = 10α , the time of effective acceleration (i.e., the time, at which there is a bend of the acceleration curve) increases with the laser pulse amplitude giving increasing ions energy at the “knee” point. For the largest laser pulse amplitude from the set (a0 = 13α ), the time of effective acceleration is about the same as for the smaller amplitude a0 = 10α and, correspondingly, the ions energy at the knee point does not increase when the laser pulse amplitude is changed from a0 = 10α to a0 = 13α . So obviously, there is a physical mechanism, which determines the appearance of the knee point on the acceleration curve for the ions, and there is another process, which limits the energy of the ions at the knee point when the laser pulse amplitude is increased. As we will see below, the first mechanism is the reverse longitudinal motion of the electrons of the relativistic electron mirror and their return to initial position due to attraction of the ions [29,30]. These returned electrons compensate the charge of the ions, which strongly reduces the longitudinal accelerating field and the rate of acceleration for the most energetic ions. The second effect is compensation of the ions charge in the laser spot by the electrons, which move transversely along the nanofilm from the periphery of the target. The spectrum of accelerated ions at t = 20T 0 is presented in Fig. 3 for α = 10 and a0 = 100 (only the ions within 6λ spot near the laser beam axis are considered). The spectra for the other
curves in Figs. 1 and 2 are very similar in shape to one presented in Fig. 3. This spectrum demonstrates almost flat-top feature for ions energy from 150 MeV to 280 MeV, which can be useful in some applications. Also, the relative number of high-energy ions is not exponentially small as for the TNSA mechanism. And at last, the clear cut-off for the maximal energy of ions is evident, which allows to characterize the acceleration process with the maximal (cut-off) energy. We performed also some additional simulations to reveal the ions energy dependence on thickness and electron density of the nanofilm when the parameter α is kept constant. These simulations show that the energy of the accelerated ions weakly depends on the nanofilm thickness provided it is considerably smaller than λ (cf. also [30]). 3. Analytical estimates for the time of effective acceleration of ions by the nonadiabatic laser pulses 3.1. Limitation of the ions maximal energy due to reverse longitudinal motion of the electrons The physics for turning back of the electrons on the left side of the relativistic electron mirror is considered in detail in Refs. [29, 30] (note that the laser pulse is going from the left to the right). This effect can be seen already in the one-dimensional model for interaction, and it can limit the energies of the ions when the ratio a0 /α is not very large. The phasespaces x– y for the electrons and the ions are presented in Fig. 4 for the time t = 6.05T 0 from the beginning of interaction when the electrons from the left side of relativistic electron mirror return back and touch with the most energetic ions on the right edge of the ion layer (α = 10, a0 = 60, and w 0 = 8λ). The time of effective acceleration t ea is about a return time t r for the electrons and can be approximately estimated as the doubled lifetime tl for the relativistic electron mirror (tl is the time when the electrons on the left side of relativistic electron mirror near the laser beam axis begin to turn back [29,30]): tr ≈ 2tl . This expression is a good approximation for the case of heavy ions or small laser pulse amplitudes when tl is also small with respect
1138
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
Fig. 5. (Color online.) Acceleration curve for the ions (blue circles) for α = 10, a0 = 60, and w 0 = 8λ. Two estimates for the time of effective acceleration t ea are shown: one is based on the direct estimation of the return time for the electrons from the 2D PIC simulations (vertical black line marked with squares) and the other is obtained from Eq. (4) with the lifetime tl derived from the 2D PIC simulations (vertical red line marked with diamonds). The time for the closing of the positive window in the nanofilm due to the transverse motion of the electrons along the nanofilm is represented with vertical magenta line marked with triangles. Green solid line corresponds to the dependence of ions energy on time according to Eq. (1).
to ζ −1 . The energy of the ions grows quadratically with tl in this case. The exact value for the time of effective acceleration in the 1D model with full electron evacuation from the nanofilm can be calculated from the following equation:
ζ tr2 + tr − 2tl = 0.
(3)
This equation takes into account a displacement of the ions at the right edge of the ion layer (i.e., the displacement of the most energetic ions) due to their acceleration under the action of the Coulomb force. In Eq. (3), we supposed that the time for removing of the electrons from the nanofilm is considerably smaller than tl . Also, the return time for the electrons to the initial position of the nanofilm is supposed equal to tl (i.e., electrons move forth and back during the same time tl ) and the return velocity of the electrons when they touch the ions is about c. Then, Eq. (3) gives for the time of effective acceleration of the ions the following expression
√ tr =
1 + 8ζ t l − 1 2ζ
.
(4)
In Fig. 5, the acceleration curve for the ions (blue circles) is presented for α = 10, w 0 = 8λ, and a0 = 60 (this curve is identical to the corresponding curve in Fig. 2). Also, two estimates for the return time t r of the electrons are shown: one is based on the direct estimation of the return time for the electrons from simulation results (vertical black line marked with squares), just for this time the phasespaces for the ions and the electrons are shown in Fig. 4. The other estimate is obtained from Eq. (4) with the lifetime tl derived from the PIC simulations (vertical red line marked with diamonds). These two estimates give almost coincident values for t r , which are also equal to the time when the acceleration curve of the ions declines from the curve described by Eq. (1), i.e., the multidimensional effects are unimportant in this case, and the 1D model for ions acceleration can be used effectively. The time for the closing of the positive window in the nanofilm due to the transverse motion of the electrons along the nanofilm is also presented in Fig. 5 with vertical magenta line marked with triangles. There is no any special feature on the acceleration curve near this time.
Fig. 6. (Color online.) Phasespaces x– y for electrons (red) and ions (cyan) at time t = 13.4T 0 from the beginning of interaction (α = 10, a0 = 130, and w 0 = 8λ). The initial position for the nanofilm is at x = 15λ, the relativistic electron mirror is situated at x = 28λ.
3.2. Limitation of the ions maximal energy due to transverse motion of the electrons along the nanofilm Phasespaces x– y for the ions and the electrons at time t = 13.4T 0 are presented in Fig. 6 (α = 10, a0 = 130, and w 0 = 8λ). At this time, the closing of the positive window formed by the laser pulse in the nanofilm occurs, i.e., t c = 13.4T 0 (simulations in the window with larger dimensions along y axis give the same value for t c ). The electrons from the central part of the nanofilm still move forward within the relativistic electron mirror. However, the electrons from the outlying regions of the nanofilm move transversely along the nanofilm due to attraction of the ions near the laser beam axis and partially compensate the ions charge. Then the accelerating longitudinal field decreases and the rate of acceleration for the most energetic ions on the right side of the ion layer also decreases. This effect cannot be seen in the one-dimensional model so this is a multidimensional effect. In Fig. 7, the acceleration curve for the ions (blue circles) is presented for α = 10 and a0 = 130 (this curve is identical to the corresponding curve in Fig. 2), and the time t c for the closing of the positive window in the nanofilm due to the transverse motion of the electrons along the nanofilm is shown with the vertical magenta line marked with triangles (two estimates for the return time t r of the electrons for this case are also indicated). The time t c is close to the point, where the acceleration curve of the ions declines from the law described by Eq. (1). Two estimates for the return time t r of the electrons are not equal in this case, besides, they are far from the knee point of the acceleration curve. So the time of effective acceleration t ea is about t c , and it is defined mainly by the multidimensional effects. To estimate the time t c for the closing of the positive window, one needs to estimate a transverse field due to the ions charge at the edge of the window in the nanofilm. We suppose during this estimation that the short laser pulse already left the nanofilm and ions don’t move. The positive window in the 2D model is a long strip, which has infinite length (along the z axis), its thickness is equal to the thickness of the nanofilm (short edge of the strip is situated along the x axis) and its width is equal to the diameter of
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
the window (long edge of the strip is situated along the y axis). To calculate the field at the middle of the short edge of the strip (i.e., at the median plane of the nanofilm), it is necessary to divide the strip along the y axis into substrips, which will have the infinite length along the z axis, thickness dy along the y axis, and width l along the x axis. The field of some substrip with coordinate y can be calculated supposing that this substrip consists of charged wires of infinite length with linear charge density n0 |e | dy dx (note that only ions are present in the window). After integration along x from −l/2 to l/2, one has (we suppose that x = 0 for the median plane of the nanofilm)
dE str = 4n0 |e | dy arctan
l 2y
(5)
.
Further integration along y from 0 to 2R, where R is the radius of the positive window (the width of the window along the y axis is 2R), gives for the transverse field at the edge of the window in the median plane of the nanofilm:
E w = 2n0 |e |l
4R l
arctan
l 4R
+
1 2
ln 1 +
4R l
2 .
(6)
When the thickness of the nanofilm l is considerably smaller than the value for R, the last equation can be simplified considerably:
E w 2n0 |e |l 1 + ln
4R l
.
(7)
The motion of the electrons at the edge of the window can be approximately described then with the following set of equations (momentum p R , R and t are normalized with mc, λ, and T 0 correspondingly)
dp R dt dR dt
= −2α =
4R l
pR
arctan
1139
l 4R
+
1 2
ln 1 +
4R l
2 ,
.
(8)
1 + p 2R
The dimensionless force from the first equation of the system (8) is shown as a function of R in Fig. 8 for α = 3 and l = 10 nm. Since the focal diameter of the laser beam cannot be smaller than λ, the normalized force acting on the electrons at the edge of the positive window in the median plane of the nanofilm just after beginning of interaction is considerably greater than 1 (and directly proportional to α ). Then, p R becomes greater than 1 in a time, which is considerably smaller than the laser field period T 0 , i.e., the transverse motion of the electrons in the positive window becomes relativistic almost immediately after beginning of interaction. So the time for the closing of the window can be estimated approximately as
tc R 0 ,
(9)
where R 0 is the initial diameter for the positive window, normalized with λ. Numerical solution for Eqs. (8) entirely confirms this result. To use Eq. (9) for the practical estimation of the closing time for the positive window, one needs to define the initial diameter R 0 for this window. It is reasonable to define R 0 as the value, for which the laser pulse amplitude a0 is about the parameter α for the nanofilm since only the electrons inside this spot move longitudinally with noticeable amplitude [32]. Then (w 0 is normalized with λ here)
R 0 w 0 ln
a0
α
(10)
and for the closing time of the positive window one has
t c w 0 ln
a0
α
.
(11)
Comparison of the time t c derived from the 2D PIC simulations with Eq. (11) for different laser spot sizes w 0 is shown in Fig. 8b (α = 3, a0 = 30, l = 10 nm). It occurs that all simulation points belong to the same straight line, a best linear fit to which has the following parameters
t c fit = 1.28w 0 + 2.52.
Fig. 7. (Color online.) The same as in Fig. 5 but for a0 = 130.
(12)
For α = 3, a0 = 30, the coefficient before w 0 in the analytical equation for t c according to Eq. (11) is equal to 1.52, i.e., t c an = R 0 1.52w 0 .
Fig. 8. (a) Dependence for the absolute value of the normalized transverse force [cf. the first equation of the system (8)], which acts on the electrons at the edge of the positive window in the median plane of the nanofilm, on the radius of the window (α = 3, l = 10 nm, R is normalized with λ). (b) Time for the closing of the positive window as a function of the laser beam waist w 0 (α = 3, a0 = 30, l = 10 nm): red stars are simulation points, red solid line is the best linear fit to the simulation points, and blue line is the analytical estimation for the closing time from Eq. (11).
1140
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
One can infer a good coincidence of the simulation and the analytical lines from Fig. 8b so the derived theory for t c can be used for analytical estimation of the time for effective acceleration of the ions when w 0 is from 2λ to 20λ. However, there is a small regular difference (about 13%) in the tangents for the inclination angles of the lines with respect to the x axis and correspondent nonzero value for t c at w 0 = 0 for the data derived from the 2D PIC simulations. This means that the real value for R 0 differs a little from the value determined by Eq. (10). Knowing the dynamics for the positive window closing, one can improve the estimate given above for the time of effective acceleration, t ea . Actually, the effective longitudinal field accelerating the ions starts to decrease earlier than the positive window fully closes because, due to the longitudinal motion, the most energetic ions move away from the plane of the window. For example, one can suppose that the accelerating field decreases noticeably when the positive window diameter becomes about the displacement of the most energetic ions from the initial position of the nanofilm. In general case, the following equation can be used to estimate the time of effective acceleration t ea
R 0 − t ea χ S i = χ
2 ai t ea
2
,
(13)
where S i is the distance, which the most energetic ions travel from the beginning of interaction, ai is the acceleration for these ions, and χ is a coefficient, which defines the relation between the dimension of the positive window at the moment of time t ea and the displacement of the most energetic ions (for χ = 0, the estimate t c is used for t ea ; χ = 0.5, if at the time t ea , the diameter of the positive window is about S i ; and χ = 1, if at the time t ea , the diameter of the positive window is two times larger than S i ). Then, for t ea , one has from Eq. (13)
√
t ea =
1 + 4ζ χ R 0 − 1 2ζ χ
.
Fig. 9. (Color online.) Maximal energy of ions versus α for a0 = 100, w 0 = 8λ: red curve is for the theoretical values, green curve marked with squares is for the energy from the 2D PIC simulations for time t = t ea (this time depends on α so it is different for each point), blue curve marked with circles is for the energy from the 2D PIC simulations for t = 10T 0 .
tr is about the closing time t c for the positive window (or a corrected value for this time if χ is nonzero). The estimate for the lifetime tl of the relativistic electron mirror is derived in [29,30]
tl
a20 8α 2
T 0.
So the optimal parameters for acceleration of the ions with nonadiabatic laser pulses should satisfy the following equation
√
1 + 8ζ t l − 1 =
(14)
The numerical values for the case of Figs. 6 and 7 (α = 10, a0 = 130, and w 0 = 8λ) look as follows. The estimate for the initial dimensions of the positive window from Eq. (10) is R 0 12.8λ, whereas this value from the 2D PIC simulations (calculated from the time for the closing of the positive window) is R 0 13.4λ (cf. Fig. 6). For t = 12.8T 0 , which is the estimate for t ea according to Eq. (13) with χ = 0 and R 0 from Eq. (10), the energy of the most energetic ions from the 2D PIC simulations is about εpic 275 MeV and the prediction for the ions energy from Eq. (1) is εpar 325 MeV so the relative difference is (εpar − εpic )/εpar 15%. For χ = 0.5, the estimate for the time of effective acceleration from Eq. (14) is t ea 10.8T 0 , the energy of the most energetic ions from the 2D PIC simulations is about 220 MeV, and the prediction for the ions energy from Eq. (1) is 225 MeV so the relative difference is about 2%. And at last for χ = 1, the estimate for the time of effective acceleration is t ea 9.6T 0 , εpic 177 MeV, and εpar 179 MeV so the relative difference is about 1%. So the estimates for χ = 0 and χ = 0.5 can be considered as acceptable for the prediction of the energy of the most energetic ions at the end of the interval of effective acceleration (the last one gives enhanced accuracy), while the estimate for χ = 1 underestimates the value for t ea and, correspondingly, the maximal energy of the ions at this moment of time. 3.3. Optimal parameters for acceleration of the ions with nonadiabatic laser pulses At the end of the interval of effective acceleration, the energy achieved by the ions will be maximal when both limiting effects come into play approximately simultaneously, i.e., the return time
(15)
1 + 4ζ χ R 0 − 1
χ
,
(16)
where ζ and R 0 are defined by Eqs. (2) and (10), correspondingly. Then, for the time t ea of effective acceleration, one has the estimate according to Eq. (4) [or Eq. (14)], and the energy of the most energetic ions can be estimated from Eq. (1). If both square roots in Eq. (16) can be approximated with only two terms from the serial decomposition, Eq. (16) becomes
2tl R 0 ,
(17)
i.e., the electrons from the relativistic electron mirror should return to their initial position just at the time when the positive window will be closed by the electrons, which move transversely from the periphery of the target. Using Eqs. (10) and (15), one can conclude from Eq. (16) that if solutions for the optimal parameters exist then, for some value of w 0 , there should be two roots for the optimal ratio a0 /α . Generally speaking, only one of this root provides for a maximum ions energy, the other root can be just a breaking point for the energy curve. In Fig. 9, the maximal energy of ions versus the parameter α is presented. The red curve is calculated for the theoretical value of energy from Eq. (1) with the time of acceleration equal to the minimum of the values from Eqs. (4) and (14) (with χ = 0.5 in the last). Here, α = 12 (the maximal value for energy) corresponds to one root of Eq. (16), the other root near α = 100 is the breaking point. The green curve marked with squares is derived from the 2D PIC simulations for time t = t ea , which is different for each point (from 12T 0 for the point with maximal energy to 0 for the point at α = 110, where the electrons are not evacuated from the target). The shapes of the red and the green curves are similar. The difference in the maximal energy of ions originates from inaccuracy of the estimate for tl from Eq. (15) [30]. The other source for
V.V. Kulagin et al. / Physics Letters A 375 (2011) 1135–1141
the error is not high enough accuracy for the 2D PIC simulations, which generates Fig. 9. Actually, the lifetime of the relativistic electron mirror (and the maximal energy also) increases with decreasing the grid cell size. At least 10 000 grid points per λ along the longitudinal direction are necessary to reproduce accurately the coherent stage of the relativistic electron mirror evolution [30]. For the PIC simulations above, the error in tl is about 20%. From the experimental point of view, the maximal energy of ions versus α calculated at different time for each point is not very informative. More customary is a dependence where the maximal energy at each point is calculated for the same time. This curve is also presented in Fig. 9 with blue line marked with circles (t = 10T 0 ). For each value of α , the maximal energy here consists of two terms. One is just the energy from the green curve. The other term corresponds to the acceleration of ions during time interval from t ea to t = 10T 0 . This acceleration resembles the TNSA acceleration, however, the distribution of electrons in space is different here. The blue curve demonstrates two peaks, which means that there are two optimal values for the parameter α . The heights of the peaks depend on time, for which the energy is calculated. Detailed study of this result will be presented in the forthcoming paper. 4. Discussion of results and conclusions Conditions for effective acceleration of the ions with the laser pulses having a smooth front is similar to the conditions of relativistic electron mirrors generation [29,30]. The electrons of the target remain motionless in the longitudinal direction until the laser amplitude reaches the value a0 ∼ α . After that, the electrons will be displaced longitudinally by the first half-cycle, which has the amplitude a0 f larger than α . Then, the relativistic electron mirror will evolve until the turn of the left electrons, and the lifetime (the return time also) will be greater for larger values of a0 f /α . This picture is confirmed by our 2D PIC simulations. So for effective acceleration of the ions by a smooth laser pulse, the high peak-to-peak ratios ∼4 to 5 are necessary at the front of the laser pulse. For a Gaussian longitudinal profile, it corresponds to a full duration at half maximum of about (1.2–1.4)λ. Such pulses can be produced by shaping the incoming laser pulse with a thin plasma layer [33–35]. In conclusion, we studied the process of the ions acceleration during relativistic electron mirrors generation from the nanofilms with nonadiabatic laser pulses. Two main physical effects limiting the maximal energy of ions were considered in detail using numerical and analytical approaches: the longitudinal reverse motion and return of the electrons, which are near the central part of the relativistic electron mirror (near the laser beam axis), and the transverse motion of the electrons from the periphery of the target due to attraction of the ions. These electrons compensate the
1141
charge of the ions and strongly reduce the longitudinal accelerating field. The maximal energy of the ions at the end of the interval of effective acceleration will be achieved when both limiting effects come into play approximately simultaneously. Optimal values for a0 , α and the laser beam diameter were defined. Acknowledgements This work was supported by the RFBR projects 09-02-01483-a and 09-02-12322-ofi-m, by the Challenge Research Project of the NRF of Korea (grant number 2010-0000847), and by the Asian Laser Center Project of GIST. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
A. Henig, S. Steinke, M. Schnurer, et al., Phys. Rev. Lett. 103 (2009) 245003. A. Macchi, F. Cattani, T.V. Liseykina, et al., Phys. Rev. Lett. 94 (2005) 165003. A.V. Korzhimanov, A.A. Gonoskov, A.V. Kim, et al., JETP Lett. 0786 (2007) 577. X. Zhang, B. Shen, X. Li, et al., Phys. Plasmas 14 (2007) 073101. O. Klimo, J. Psikal, J. Limpouch, et al., Phys. Rev. ST Accel. Beams 11 (2008) 031301. X.Q. Yan, C. Lin, Z.M. Sheng, et al., Phys. Rev. Lett. 100 (2008) 135003. A.P.L. Robinson, M. Zepf, S. Kar, et al., New J. Phys. 10 (2008) 013021. A.A. Gonoskov, A.V. Korzhimanov, V.I. Eremin, et al., Phys. Rev. Lett. 102 (2009) 184801. B. Qiao, M. Zepf, M. Borghesi, et al., Phys. Rev. Lett. 102 (2009) 145002. T. Schlegel, N. Naumova, V.T. Tikhonchuk, et al., Phys. Plasmas 16 (2009) 083103. A. Macchi, S. Veghini, F. Pegoraro, Phys. Rev. Lett. 103 (2009) 085003. M. Grech, S. Skupin, R. Nuter, et al., New J. Phys. 11 (2009) 093035. R.A. Snavely, M.H. Key, S.P. Hatchett, et al., Phys. Rev. Lett. 85 (2000) 2945. A. Maksimchuk, S. Gu, K. Flippo, et al., Phys. Rev. Lett. 84 (2000) 4108. A. Andreev, A. Levy, T. Ceccotti, et al., Phys. Rev. Lett. 101 (2008) 155002. A. Henig, D. Kiefer, K. Markey, et al., Phys. Rev. Lett. 103 (2009) 045002. T.Zh. Esirkepov, S.V. Bulanov, K. Nishihara, et al., Phys. Rev. Lett. 89 (2002) 175003. Y. Sentoku, T.E. Cowan, A. Kemp, et al., Phys. Plasmas 10 (2003) 2009. S.V. Bulanov, T.Zh. Esirkepov, J. Koga, et al., Plasma Phys. Rep. 30 (2004) 18. T. Esirkepov, M. Borghesi, S.V. Bulanov, et al., Phys. Rev. Lett. 92 (2004) 175003. E. Fourkal, I. Velchev, C.-M. Ma, Phys. Rev. E 71 (2005) 036412. L. Yin, B.J. Albright, B.M. Hegelich, et al., Laser Part. Beams 24 (2006) 291. L. Yin, B.J. Albright, B.M. Hegelich, et al., Phys. Plasmas 14 (2007) 056706. S.S. Bulanov, A. Brantov, V.Yu. Bychenkov, et al., Phys. Rev. E 78 (2008) 026412. M. Passoni, M. Lontano, Phys. Rev. Lett. 101 (2008) 115001. A.V. Brantov, V.T. Tikhonchuk, V.Yu. Bychenkov, et al., Phys. Plasmas 16 (2009) 043107. J. Davis, G.M. Petrov, Phys. Plasmas 16 (2009) 023105. F. Wang, B. Shen, X. Zhang, et al., Phys. Plasmas 16 (2009) 093112. V.V. Kulagin, V.A. Cherepenin, M.S. Hur, et al., Phys. Rev. Lett. 99 (2007) 124801. V.V. Kulagin, V.A. Cherepenin, Y.V. Gulyaev, et al., Phys. Rev. E 80 (2009) 016404. J.P. Verboncoeur, A.B. Langdon, N.T. Gladd, Comp. Phys. Comm. 87 (1995) 199. V.V. Kulagin, V.A. Cherepenin, M.S. Hur, et al., Phys. Plasmas 14 (2007) 113101. V.A. Vshivkov, N.M. Naumova, F. Pegoraro, et al., Phys. Plasmas 5 (1998) 2727. V.V. Kulagin, V.A. Cherepenin, M.S. Hur, et al., Phys. Plasmas 14 (2007) 113102. L.L. Ji, B.F. Shen, X.M. Zhang, et al., Phys. Rev. Lett. 103 (2009) 215005.