Stable monoenergetic ion acceleration by a two color laser tweezer
In the past decades, the phenomenal progress in the development of ultraintense lasers has opened up many exciting new frontiers in laser matter physics, including laser plasma ion acceleration. Currently a major challenge in this frontier is to find simple methods to stably produce monoenergetic ion beams with sufficient charge for real applications. Here, we propose a novel scheme using a two color laser tweezer to fulfill this goal. In this scheme, two circularly polarized lasers with different wavelengths collide right on a thin nano-foil target containing mixed ion species. The radiation pressure of this laser pair acts like a tweezer to pinch and fully drag the electrons out, forming a stable uniform accelerating field for the ions. Scaling laws and three-dimensional particle-in-cell simulations confirm that high energy (10-1000 MeV) high charge () proton beams with narrow energy spread () can be obtained by commercially available lasers. Such a scheme may open up a new route for compact high quality ion sources for various applications.
With the rapid development of ultraintense lasers in the past decades, many exciting new frontiers in the field of laser matter physics have been opened up, such as fast ignition of inertial confinement fusion, laboratory astrophysics[2, 3], compact particle accelerators [4, 5, 6, 7, 8, 9, 10, 11] and light sources[12, 13, 14]. Among these topics, laser driven ion acceleration has attracted much attention worldwide for its potential to generate compact ultrafast high quality ion sources[15, 16]. For many applications, ion beams that are quasi-monoenergetic and have a sufficient charge are desired. Furthermore, such beams need to be produced stably[17, 18, 19]. Developing schemes to meet these requirements is the current focus of research in the field of laser ion acceleration. During more than a decade of intense research, several acceleration schemes have been proposed, such as target normal sheath acceleration (TNSA)[20, 21, 22], collisionless shock acceleration (CSA)[23, 10, 24], radiation pressure acceleration (RPA)[25, 26, 27, 28, 29, 30], breakout afterburner acceleration (BOA)[31, 32], magnetic vortex acceleration (MVA)[33, 34, 35], chirped standing wave acceleration (CSWA) etc. However, none of the above schemes provides a simple stable mechanism to produce a monoenergetic ion beam with sufficient charge.
In this paper, we propose a new scheme using a two color laser tweezer, which provides a possible solution to meet these demands. In this scheme, two circularly polarized (CP) intense laser pulses with different frequencies ( and , ) but nearly equal normalized vector potential () collide on a nanometer thick foil (nano-foil) with mixed atomic species (as shown in Fig.1(a) and (b)). The electrons in the foil quickly get squeezed into a thin sheet by the radiation pressure of this beat wave, i.e., the slowly moving ponderomotive potential well, and then are fully dragged out of the foil as a whole, as if they were grabbed and pulled out by a laser “tweezer”. This creates a nearly constant longitudinal electric field between these electrons and the heavy ions that lag behind (see Fig.1(c)). If the target contains a minority species of protons (or light ions), they then can be accelerated by such field to high energy in a very short distance. It is noted that these electrons are spatially separated from the ions. Therefore, instabilities caused by electron-ion streaming coupings, a frustrating problem for the RPA scheme, are fully avoided, thus the acceleration structure is much more stable. In this paper, we develop a simple theoretical model for this scheme and a scaling law for the dependence of the beam energy on the laser energy. Two and three dimensional Particle-in-Cell (PIC) simulations using OSIRIS  are systematically used to verify the theory and scaling formulas, and good agreement has been obtained. Based on these formulas and simulations, we predict that 10 MeV-GeV level proton beams with sufficient charge ( protons) and narrow energy spread () may be realizable using laser pulses containing 1-80 J energy through OPA or OPCPA technique in the near future[39, 40]. Since the moving ponderomotive bucket of the laser beat wave acts like a “light tweezer”, we call this scheme “Bi-Color Laser Tweezer Acceleration” (BCTA). Such a scheme may open up a new route for compact high quality laser based ion source for various applications.
Results and Discussions
Basic concept and formulation
This scheme can be most easily understood in a moving frame with the velocity . In this frame, the two counter-propagating laser pulses have the same frequency of and a standing wave is formed as they overlap if they have nearly equal vector potentials (). An electron can be trapped in the ponderomotive potential well of the standing wave under proper conditions, and this condition can be obtained by examining the one dimensional (1D) single electron dynamics in a circularly polarized standing wave as follows. To simplify the derivation, we adopt the standard normalization in laser plasma physics, where , , and are the laser wave number in the moving frame, electron rest mass, light speed in vacuum and electron charge respectively. In reality the lasers each have finite longitudinal extent (a spectrum of frequencies). When the peaks of the lasers overlap and they are sufficiently long compared to the dominant wavelength, we can approximate the vector potential of CP standing wave as , where z represents the propagation axis of the laser pulse, and are the unit vectors of transverse directions. We assume the electrons initial transverse momentum was zero and that they started in a region where the vector potential also vanished, i.e., (). The equations of motion for an electron can be written as:
where and are the Lorentz factor and longitudinal velocity of the electrons respectively. Eq. 1c indicates that the electron experiences a spatially periodic ponderomotive force . This force is focusing in the direction at nodes and de-focusing at antinodes of the standing wave, making it possible to trap electrons at nodes. We can obtain a relation between and z by integrating Eq. 1c, leading to
where , , are the initial longitudinal velocity, Lorentz factor and phase position, respectively. From the assumption of and Eq. 1b, one gets (N=0,1,2,3…) and . For what follows we assume without loss of generality. It is then straightforward to obtain . If electrons are trapped, their trajectories in phase space should be closed in this standing wave frame which implies there must have at least two intersections with axis. This leads to a trapping condition of . Combined with initial conditions in the moving frame ( and ), one can get
The above analysis is simply based on single electron dynamics in one-dimension. For the standing wave to be formed in the first place, the laser pulses need to penetrate through the thin foil. For cases of interest the foil has a thickness greater than the the non-relativistic collisionless skin depth, . Therefore, it is necessary that the self-induced relativistic transparency condition be satisfied[41, 42], where , and are the intensities for both lasers, the foil’s initial density and thickness, respectively.
Upon reexamining the trapped electron dynamics in the lab frame, one finds that essentially all the electrons are first quickly squeezed into a very thin sheet, then this sheet is trapped in the moving ponderomotive potential well with a phase velocity of , and eventually gets dragged out as a whole from the ion background. A nearly constant capacitor-like large electric field () is induced between this stable electron sheet and the heavy ions left behind, and such a constant field then can be used to accelerate minority protons embedded in the foil to generate a quasi-monoenergetic beam in a very short distance.
The maximum energy gain of the protons will be reached when they move beyond the trapped electron sheet, where the accelerating field nearly vanishes. The maximum acceleration time for the protons can be easily estimated as:
where is acceleration rate of the protons, is the proton rest mass and is the Lorentz factor of the moving frame. By simply integrating the accelerating field over distance, one can get the expression of :
PIC simulation verification for BCTA scheme
To verify the validity and stability of the BCTA scheme, 3D PIC simulations have been systematically performed, and very good agreement between simulation and the idealized theoretical model has been achieved. In Fig. 2 we show the results from a typical 3D simulation. In this simulation, a flattop temporal envelope with equal rise and fall time of 2 fs and a super-gaussian transverse profile (, , ) are used for the laser pulses to make direct comparison with the theoretical prediction. A 27fs 800nm CP laser pulse and a 54fs 1.6m CP laser pulse are initialized to counter-propagate towards a 45nm thick target located at , and reach the front and back surface at the time . The target has a uniform density profile, and it is composed of electrons(), immobile ions () and small fraction of protons(), where cm is the critical plasma density for a 800nm laser. Sufficiently small cell sizes ( nm) and sufficient large particle numbers ( particles per species per cell) are used to ensure the fidelity of the simulations, where z represents the longitudinal direction and x, y represent the transverse directions.
Fig. 2(a)-(c) show the slices (at ) of the electron density, the electric field () and the proton momentum () at three different times, and (d)-(f) give lineouts of the electron density (green line), the transverse field (red line) and the longitudinal field (purple line) along the central axis () of (a)-(c).
At a time slightly before the lasers reach and overlap each other on the target (), as shown in Fig. 2 (a) and (d), the target has not been perturbed by the lasers. Soon after the laser collides (), a high density electron sheet is formed by the ponderomotive potential well of the laser beat wave (a “laser tweezer” ) and moving towards the +z direction, as can be clearly seen in Fig. 2 (b) and (c). The sheet moves at a speed of , in very good agreement with the estimated value of 1/3c (). A nearly constant longitudinal electric field of 64 TV/m () is formed between the electron sheet and the immobile ion background. Due to the fact that electrons and ions are spatially separated, there are no transverse ripples growing up in the central region of the electron density as shown in Fig. 2 (b) and (c), and therefore the longitudinal electric field is not perturbed and proceeds almost unchanged during the whole process (see Fig. 2 (e) and (f)). Meanwhile, Protons with a doped rate of 10% almost have no influence on this constant field, and are accelerated together to high energy in only 40fs (see Fig. 2 (b)(c)).
Fig. 3 (a) gives the proton () phase space at the time . One can see protons that obtain the maximum energy show good 1D acceleration feature with very low transverse momentum. For simplicity, here we take the angle mrad as a critical point to check the proton energy spectra of different divergence angles in Fig. 3 (b). It is shown that protons with small divergence angles mrad (blue curve) eventually reach the energy around 214 MeV with a narrow energy spread (FWHM) 4.5, in good agreement with the estimated value of 230 MeV from 1D theory (Eq. 5). The total particle number of this bunch is , which is already sufficient for many applications. Interestingly, protons of relatively large angles mrad (orange curve) have a nearly flattop spectra profile in the range of 30-200 MeV with a charge number of . These two beams can be easily separated by using a narrow-size aperture at a proper distance behind the foil. We note that the proton charge can be further increased for wider laser focal spot (e.g.m) or by increasing the proton doping fraction (e.g. 20%) for the scheme.
Proton energy scaling
It is of great interest to derive a scaling law for the proton energy of the BCTA scheme. For simplicity, we adopt instead of in the following derivations, where is the laser wave number of shorter wavelength. As mentioned above, idealized 1D theory indicates that the acceleration time (Eq. 4) does not depend on the transverse profile of the laser. However, in a real multi-dimensional geometry, laser pulses with finite transverse width lead to trapped electron sheets whose widths decrease with propagation distance, as can be seen from previous 3D simulation in Fig. 2(b) and (c). This phenomenon caused by the diffraction of the electron sheet and of the lasers cause the sheet to dissipate even before for tightly focused lasers, leading to a termination of the BCTA process. This process is difficult to incorporate into the scaling law so we rely on simulations to find the effective acceleration time (). Note that this will depend on the transverse profile of the laser.
Combining the relativistic transparency condition and the expression of leads to the following formula for the maximum energy of the proton beam:
It is also important to estimate requiured laser energy needed to obtain . Firstly, the minimal pulse duration of the laser with shorter/longer wavelength is . Then, for a gaussian temporal envelope and super-gaussian transverse profile () with a FWHM intensity duration , only laser energy within the FWHM region will effectively take part in the “laser tweezer” process, and thus the required energy can be roughly estimated as .
If an 800nm laser is chosen as the one with shorter wavelength, the total energy of the laser pair can be written as:
In order to verify the above predictions of Eq. 6 and 7, five different 2D PIC simulations have been performed using realistic parameters. Laser pulses in all cases had a gaussian temporal envelope and super-gaussian transverse profile with . The target was composed of carbon and hydrogen with a thickness of 45 nm. All the simulation parameters and their results are presented in Table 1, and the spectra of protons with mrad are plotted in Fig. 4(a).
|parameter and results111 Laser pulses adopted in all cases have a gaussian temporal envelop and super-gaussian transverse profile with the expression of . The spot size is chosen as 4m.||A||B||C||D||E|
|laser wavelength [m]:,||0.8, 1.6||0.8, 1.6||0.8, 1.6||0.8, 2.4||0.8, 3|
|laser pulse duration (FWHM) [fs]222These are demanded minimal pulse durations for effective acceleration time . Lasers with longer pulses can still work, only wasting the redundant parts: ,||9, 18||9, 18||9, 18||7, 21||6, 22.5|
|laser energy [J]: ,||0.82, 0.41||6.0, 3.0||13.1, 6.6||32.1, 10.7||65.5, 17.5|
|plasma density [cm]:||21||52||77||112||175|
|proton density [cm]:||2||5||7||10||17|
|proton final peak energy [MeV]:||18||110||190||490||820|
|particle number: |
As shown in Table. 1 and Fig. 4(a), proton beams with the energy 20 MeV to 1 GeV and narrow spread ( to ) are obtained for a laser energy in the range of 1-80 J. The charge in the proton beam is estimated by using the proton density per unit length and then assuming a round beam for cylindrical symmetry.
Two useful conclusions also can be obtained in Fig. 4(a). First, a laser pulse with a gaussian temporal envelope (the purple solid curve) is able to capture the monoenergetic feature of BCTA, although is slightly lower than that obtained from a reference simulation which used ideal flattop profile (the black dashed curve); second, the averaged energy of proton beam in the 3D simulation (the pink dashed curve) is very close to that from 2D (only 5% lower). Both these two observations indicate the validity and robustness of BCTA for real experiments.
Fig. 4(b) gives the relation between proton maximum energy and laser energy, and one can see that the scaling from simulations has good agreements with the earlier estimations. From the scaling laws and the PIC simulations it is predicted that less than 20 J laser energy is required to produce a 200 MeV proton beam. Furthermore, if the laser energy is increased to 100 J, the scaling laws predict that a 1 GeV monoenergetic proton beam can be obtained, indicating BCTA is very promising for generating 100-1000 MeV high quality ion beams with commercially available lasers.
In summary, we propose a simple new scheme named Bi-Color Laser Tweezer Acceleration (BCTA) that has the potential to generate a stable proton beam with a narrow energy spread and a significant charge. In this scheme, the ponderomotive force of the beat wave (laser tweezer) formed by two CP colliding laser pulses with different wavelengths can fully drag the electrons out of a thin nano-target with mixed species, creating a stable uniform longitudinal electric field for the monoenergetic acceleration of doped protons (or any light ion species). Scaling laws and companying PIC simulations show that 10-1000 MeV level proton beams with sufficient particles () and narrow energy spread () may be obtained by using a bi-color femtosecond laser with J energy, making BCTA a potential approach for future compact high quality ion sources.
The mechanism for BCTA is explored using the fully relativistic particle-in-cell code OSIRIS in two and three dimensional Cartesian coordinates with a fixed window. For all simulations, sufficiently small cell sizes ( nm for 2D, nm for 3D) and sufficiently large particle numbers ( particles per species per cell for 2D, particles per species per cell for 3D) are used to ensure fidelity, where z represents the longitudinal direction and x, y represent the transverse directions. The simulation parameters and relevant results are presented in the section of results and discussions.
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This work was supported by the National Basic Research Program of China No. 2013CBA01501, NFSC Grant No. 11425521, No. 11535006, No. 11175102, No. 11005063, No. 11375006 and No. 11475101, the Foundation of CAEP No. 2014A0102003, Tsinghua University Initiative Scientific Research Program, the Thousand Young Talents Program, the U.S. DOE Grant No. DE-SC0010064, No. DE-SC0014260, the U.S. NSF Grant PHY-1415386, No. ACI-1339893 and No. 1500630, and the Ministry of Science and Technology of Taiwan under Contracts No. 104-2112-M-001 -030 -MY3. Simulations are performed on the HOFFMAN2 clusters at UCLA and Hopper, Edison cluster at National Energy Research Scientific Computing Center (NERSC).
Y.W. and W.L. proposed the concept. Y.W. developed the theoretical model and performed the simulations. Y.W. and W.L. wrote the paper. C.J.Z., J.F.H., F.L., C.H.P., Y.P.W., Y.Q.G, W.B.M., and C.J., contributed to refining the details of the paper. All authors reviewed the manuscript.