Abstract

Simulation of vacuum laser acceleration, because of its scheme’s simplicity, attracts many people involved in. However, how to put the particle in the initial positions in the field has not been considered seriously in some such schemes. An inattentive choice of electron’s initial conditions may lead to misleading results. Here we show that arbitrarily placing the particle within the laser field leads to an overestimation of its energy gain, and offer suggestions for selecting appropriate initial conditions.

© 2010 OSA

1. Introduction

After the use of lasers to accelerate particles was first proposed by Shimoda in 1961 [1], especially after laser-plasma acceleration scheme was proposed by Tajima and Dawson in 1979 [2] and chirped pulse amplification (CPA) laser technique appeared in 1980’s [3], scientists began enthusiastically exploring the field of laser acceleration [4, 5]. Recent years have seen remarkable achievements in the development of laser-based particle accelerators [6]. Simulations are often employed to study vacuum laser acceleration (VLA) in the contexts of theory [7] and experiment [8].

In VLA simulations, the particle interacts with a laser beam in free space, where any optical components are at very large distances from the focus, compared to the Rayleigh length. The particles are usually given initial positions close to the focus, but may have zero or nonzero velocities. Some works begin simulating the particles at an early time, before the arrival of the pulse, while others begin with the pulse contained the particles or even use a continuous laser beam. In the latter cases, the particles are in a strong laser field region from the very start, so would have required a large injection energy to reach that point. Even in the first case, the initial distance between the particle and the laser pulse may also be too short for the field to be negligible. It is important for VLA simulations to take the injection energy into account when predicting the energy gain of the particle.

The energy gain of a particle in the laser field is always temporary, so discussions in the literature often focus on how best to retain or utilize this energy. However, the initial placement of a particle in the field is not considered seriously in some VLA schemes and some early particle acceleration scenarios [9]. This paper is neither a comment on a certain published paper nor for denying the validity of VLA schemes, nor to seek an alternative VLA scheme. Its goal is simply to quantify the errors induced in estimated energy gains by selecting improper initial conditions.

2. Simulation method

Without loss of generality, we may assume that the center of a laser pulse reaches the origin (x=y=z=0) at t=0 and we begin the calculation at t=tb (<0), as shown in Fig. 1 . A dimensionless parameter a0eE0/meωc measures the laser intensity. E0 denotes the electric field amplitude of the laser beam at its focus, and ω is the laser’s angular frequency. The electron’s charge and rest mass are e and me respectively. Throughout this paper, time is expressed in units of 1/ω, length in units of 1/k (k is the laser wave number), momentum in units of mec, and energy in units of mec2. The numerical simulation method is similar to those used previously [10, 11]. Radiation damping is neglected.

 

Fig. 1 Scheme for vacuum acceleration of an electron by a laser pulse.

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A fundamental-mode Gaussian laser beam polarized in the x-direction and propagating along the z-axis is described as follows [12]:

Ex=E0w0w(z)exp[ikziωt+iφ(r,z)+iϕ0r2w(z)2],
where w(z)=w01+z2/ZR2​ ​, w0 is the beam width at the focus (the beam waist), ZR=πw02/λ is the Rayleigh length, and φ(r,z)+ϕ0 is the phase. The other field components can be obtained from the relations Ez=(i/k)(Ex/x)​ ​ and B=(i/ω)×E​ ​. In the case of a finite pulse duration τ (the FWHM pulse length τ0=τ2ln2), the paraxial field components are multiplied by a Gaussian time envelope profile, f(η)=exp[(zct)2/(cτ)2]. All other corrections to the fields of order 1/(ωτ) or higher have been neglected [13]. It should be pointed out that the laser pulses adopted here are focused to about 10λ​ ​, and are longer than 60 cycles. Thus, high-order corrections to the paraxial expressions [13, 14] need not be taken into account. Every example presented below was optimizes the laser initial phase ϕ0​ ​ to obtain the maximum outgoing energy.

3. Some typical examples

First, we consider an example with the following laser parameters: wavelength λ=1μm, peak field intensity a0=10 (I=1.37×1020W/cm2), focused spot size w0=60 (9.55μm), and pulse duration τ=200 (the FWHM pulse length is about 125 fs). The electron is initially at rest at the origin (x0=y0=z0=0), and we begin our calculation at tb=0. The electron energy γ=(1β2)1/2 as a function of time t is shown in Fig. 2(a) (solid line). Its outgoing energy is γf68 .02 ​, or about 34MeV. Acceleration to 34MeV from a state of rest would seem to be a very good result. However, how much energy would the electron have needed to move from a point outside the field to its rest state at the origin? To answer this question, we use the same initial conditions but run the simulation in reverse. The electron energy γ obtained by the time-reversed simulation is shown in Fig. 2(b) (dotted line). We find that the electron would have needed an identical amount of injection energy of γi68 .02 ​. Unfortunately, the net energy gain γfγi in this process is close to zero.

 

Fig. 2 Panels (a) and (b) show the result for tb=0, in normal time and reverse time respectively. Panel (d) shows the case for tb=1200 in normal time. Panel (c) plots the longitudinal momentum as a function of coordinate z, with tb=0. The solid line in these panels represents a simulation in normal time with the laser propagating along the z-axis, the dotted line represents a simulation in inverse time with the laser propagating along the z-axis, and the dot-dashed line represents a simulation in normal time with the laser propagating along the −z-axis. In all cases, the electron’s initial position is x0=y0=z0=0​ ​, and the laser parameters are a0=10​ ​, w0=60​ ​, and τ=200​ ​.

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Note that a time-reversed simulation is not equivalent to a simulation where the laser pulse propagates in the opposite direction. To distinguish these two cases, the longitudinal momentum of the electron as a function of its z-coordinate is shown in Fig. 2(c). The dotted line is a time-reversed simulation, while the dot-dashed line is for a laser propagating along the negative z-axis.

A case where we begin the calculation very early (tb=1200​ ​) is shown in Fig. 2(d). Using the same initial conditions, the time-reversed simulation tells us that the electron needed an injection energy γi​ ​ is equal to the initial energy γ0 of the normal calculation, i.e., γi=γ0=1​ ​, as expected. This case is a good simulation example.

Table 1 presents a variety of other cases. We begin with cases where the electron starts on the optical axis (x0=y0=0,β0x=β0y=0) with one of three initial velocities in the z-direction: at rest (β0z=0), at low speed (β0z=0.05, about 640eV), and at high speed (β0z=0.999, about 10.9MeV). The laser parameters are always a0=10, w0=60 and τ=200. We consider the following 5 kinds of cases of initial conditions: a) the electron and the center of the laser pulse are both at the origin (Nos. 1, 2, 3); b) the laser pulse is centred on the origin and the electron is one pulse duration ahead (Nos. 4, 5, 6); c) the electron is at the origin and the center of the laser pulse is one pulse duration behind (Nos. 7, 8, 9); d) the pulse center and electron coincide initially at a point two Rayleigh lengths behind the laser focus (Nos. 10, 11, 12); and e) the electron is at the origin and the center of the laser pulse is five pulse durations behind (Nos. 13, 14, 15). For each case we run the simulation forwards and backwards to determine the outgoing energy γf and needed injection energy γi.

Tables Icon

Table 1. The outgoing energy γf and needed injection energy γi in some typical cases. z0 is the initial position, γ0 the initial energy, Up0 the initial ponderomotive potential in units of mec2.

We find that γf overestimates the net energy gain in all listed cases except for the final three, where the initial electron positions are far from the pulse center. In the case No. 3, for example, where the electron has an initial kinetic energy of about 10.9MeV in the center of the laser pulse, the net energy gain γfγ0 seems to be more than 150MeV. However, when the energy needed to reach that initial state is taken into account, the actual net energy gain γfγi is no more than 50MeV. We would also like to point out that the initial conditions of part (a) are not very realistic. In the case No. 12, although γfγ0 is more than 90 MeV, the net energy gain is negative. Only the final three cases, underlined in Table 1, satisfy γi=γ0.

In views of the ponderomotive potential model (PPM) [15], the particle initially being set in the laser field corresponds to it initially being set in a high potential point. It is not strange that a particle can gain kinetic energy when it goes from a high potential point to a lower one. However, pushing a particle to a high potential point needs initial kinetic energy.

The ponderomotive potential of an electron in a fundamental-mode Gaussian laser beam in the paraxial approximation can be given by [7]

Up=A02w02w(z)2exp[2r2w(z)22(zct)2(cτ)2],
where A0=eE0/(2mωc) is the normalized time-averaged laser intensity parameter amplitude. Thus, the ponderomotive potential outside the laser field should be zero.

The ponderomotive potential values Up0 of the initial positions for these examples are present in Table 1. In the parts (a)-(d), we can find that the outgoing energy of an electron initially at rest or at low speed is mainly contributed from its initial ponderomotive potential energy, i. e. γf~Up0+1.0. However, in these cases, the outgoing energy of an electron initially at high speed is far from its initial ponderomotive potential energy. The reason is that here the ponderomotive acceleration mechanism plays an important role for low energy electrons but not for high energy electrons [5], or the PPM should be corrected for relativistic electrons [16].

Assuming that the laser field is a one-dimensional (1D) plane wave, the electron orbits can be calculated exactly [17, 18]. Physically, as the laser pulse impinges upon the electron, the nonlinear ponderomotive force associated with the front (rise) of the laser pulse accelerates the electron. Eventually, the laser pulse outruns the electron and the electron is decelerated by ponderomotive force on the back of the pulse. Once the electron exits the back of the pulse, there is no net energy gain. Though the initial ponderomotive potential energy is almost zero in the part (e), a finite energy gain can result, as shown in Table 1, if the electron leaves the vicinity of the laser pulse before it has a chance to be decelerated by the back of the pulse. In 3D, this can occur by the pulse diffracting, or by transverse scattering of the electrons, as discussed in Refs [8, 16].

4. Identifying proper initial positions

In principle, the criterion γi=γ0​ ​ could also be used to identify proper initial positions, but this constraint is difficult to implement using simulations. Only a point at infinity can be considered completely outside the laser field. Normally, the electron’s initial position is considered to lie outside the laser beam if the electric field strength is no larger than a small parameter ε.

One of the optimum acceleration conditions is always that the moving electron interacts with the laser pulse near the focus, i.e. the laser’s strongest region [7, 11]. In order to estimate the optimal acceleration under these circumstances, we assume that an electron arrives at the x-y plane (z = 0) at time t = 0 in a state of free motion, uninfluenced by the laser field. The electron’s initial velocity is β0 (β0x, β0y, β0z). If it exactly encounters the pulse center at the focus, then z0=β0zctb. From Eq. (1), we conclude that the electric field strength experienced by an electron at the beginning of this scenario should satisfy (in normalized units):

|E|[a0ZRβ0z2c2tb2+ZR2]×exp[(tbβ0ztbτ)2(β02β0z2)c2tb2w02(1+β0z2c2tb2/ZR2)]ε,
where ε is the small threshold mentioned above. From Eq. (3), we find that in order to satisfy γi=γ0​ ​, the absolute value of tb should increase along with β0z, τ, a0 and w0. For a continuous laser beam (τ), however, tb is meaningless. We must select an initial position far from the focus (z0<<ZR) to guarantee that the electron is initially immune to the laser field, as was done in Ref [19]. It is worth noting that for a particle with a different charge-to-mass ratio, ε should be chosen differently under the same initial conditions.

Figure 3 presents the dependence of the electric field strength |E|​ ​ on tb​ ​ for electrons injected along the z-axis with various kinematic parameters. The solid circles in this plot correspond to the cases listed in Table 1. In Fig. 3 we see that the absolute value of tb​ ​is very large for electrons with high injected energy. Indeed, it can be more than two orders of magnitude larger than in the low-energy case. On the other hand, for low-energy injected electrons the field strength |E|​ ​ is very sensitive to tb​ ​. Normally, if |E|<106, this factor has a negligible impact on the final results. To ensure that this is the case, we set ε=109, corresponding to a maximum field of 32.1V/cm for λ=1μm. The motion of a high-energy electron is relatively unresponsive to the electric field, so the parameter ε could be larger in this case. The dotted and dot-dashed curves in Fig. 3 show that changing the waist size w0 has little effect on tb​ ​, while changing the pulse duration τ has a strong effect on tb​ ​.

 

Fig. 3 The electric field strength |E|​ ​ as a function of tb​ ​, where β0x=β0y=0​ ​ and a0=10​ ​. The solid lines are for w0=60​ ​, τ=200​ ​ and β0z=0,0.05,0.5,0.7,0.80.999; The dotted line is for w0=120​ ​, τ=200​ ​ and β0z=0.999; The dot-dashed line is for w0=60​ ​, τ=300​ ​ and β0z=0.999.

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5. Sideways injection examples

Finally, we would like to point out that a high-energy electron injected along the optical axis and satisfying Eq. (3) will not interact with the laser pulse in the focus area. The electron will leave the laser beam long before arriving at the focus due to the transverse ponderomotive force. Thus, high-energy electrons always use a sideways injection scheme (also called a capture acceleration scheme) [10, 11]. Using a laser pulse with λ=1μm, a0=100 (I=1.37×1022W/cm2), w0=60 and τ=200, we present an example of x-z plane sideways injection at θi=50​ ​ (the angle between the electron’s initial velocity and the z-axis) into the focus. The initial velocity is β0=0.9997 (about 20.35MeV), and two initial positions are considered: z0=ZR and z0=10ZR. Provided that the electron and pulse arrive at the origin at the same time, we get tb=z0/(β0cosθi), x0=z0tanθi. Figure 4 presents simulation results for the full range of initial phase ϕ0. Solid lines give the outgoing energy γf​ ​, and dotted lines give the injection energy γi. For comparison, the initial energy γ0 is also plotted with a dot-dashed line.

 

Fig. 4 The outgoing energy γf (solid line), needed injection energy γi (dotted line) and initial energy γ0 (dot-dashed line) as functions of the laser’s initial phase ϕ0. The laser pulse parameters are λ=1μm, a0=100, w0=60 and τ=200. Other parameters are β0=0.9997, θi=50, tb=z0/(β0cosθi), x0=z0tanθi, (a) z0=ZR, (b) z0=10ZR.

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In Fig. 4(a), we see that the initial position has a very strong impact on the final energy when the initial position (z0=ZR) is only one Rayleigh length from the focus. Figure 4(a) also reveals that the best acceleration case may not be the maximum impact case. For some phases, the net energy gain is negative (shown with an arrow in Fig. 4(a)). In the simulations of Fig. 4(a), |E|=0 .2254214, so that the initial position is not far enough from the focus. Figure 4(b) shows that the initial position’s impact is slight when the initial position (z0=10ZR) is ten Rayleigh lengths away from the focus. A brief calculation gives |E|=9 .46×104 in this case of Fig. 4(b).

6. Summary

In summary, the particle’s initial position in the laser field at the very start may have an important impact on the final simulation results of vacuum laser acceleration. Some “apparently safe” initial conditions such as those listed in parts (b)-(e) of Table 1 lead to misleading results. In order to obtain accurate results, the electron has to begin rather far from the pulse. Special cases of an electron “born” at rest may come from an ionization of an atom or from an electron-positron production in an intense laser beam. That is another story and we will not discuss it here.

Acknowledgments

This work was supported by CORE (Center for Optical Research and Education). P. X. Wang was also partly supported by Program for NCET (New Century Excellent Talents in University), Chinese Ministry of Education.

References and links

1. K. Shimoda, “Proposal for an electron accelerator using an optical maser,” Appl. Opt. 1(1), 33–36 (1962). [CrossRef]  

2. T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979). [CrossRef]  

3. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985). [CrossRef]  

4. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006). [CrossRef]  

5. Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006). [CrossRef]  

6. V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008). [CrossRef]  

7. G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001). [CrossRef]   [PubMed]  

8. G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997). [CrossRef]  

9. S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991). [CrossRef]   [PubMed]  

10. P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001). [CrossRef]  

11. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002). [CrossRef]   [PubMed]  

12. E. Siegman Lasers, (University Science Books, Mill Valley, California, 1986).

13. J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004). [CrossRef]  

14. Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. 8(8), 133 (2006). [CrossRef]  

15. T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. 16(23), 1054–1056 (1966). [CrossRef]  

16. B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3719–3732 (1998). [CrossRef]  

17. P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids 16(2), 321–328 (1973). [CrossRef]  

18. E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993). [CrossRef]   [PubMed]  

19. C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B 74, s83–s88 (2002). [CrossRef]  

References

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  1. K. Shimoda, “Proposal for an electron accelerator using an optical maser,” Appl. Opt. 1(1), 33–36 (1962).
    [CrossRef]
  2. T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979).
    [CrossRef]
  3. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
    [CrossRef]
  4. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
    [CrossRef]
  5. Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
    [CrossRef]
  6. V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
    [CrossRef]
  7. G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001).
    [CrossRef]
  8. G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
    [CrossRef]
  9. S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991).
    [CrossRef]
  10. P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
    [CrossRef]
  11. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
    [CrossRef]
  12. E. Siegman Lasers, (University Science Books, Mill Valley, California, 1986).
  13. J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
    [CrossRef]
  14. Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. 8(8), 133 (2006).
    [CrossRef]
  15. T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. 16(23), 1054–1056 (1966).
    [CrossRef]
  16. B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3719–3732 (1998).
    [CrossRef]
  17. P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids 16(2), 321–328 (1973).
    [CrossRef]
  18. E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993).
    [CrossRef]
  19. C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B 74, s83–s88 (2002).
    [CrossRef]

2008 (1)

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

2006 (3)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[CrossRef]

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. 8(8), 133 (2006).
[CrossRef]

2004 (1)

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

2002 (2)

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B 74, s83–s88 (2002).
[CrossRef]

2001 (2)

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001).
[CrossRef]

1998 (1)

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3719–3732 (1998).
[CrossRef]

1997 (1)

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
[CrossRef]

1993 (1)

E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993).
[CrossRef]

1991 (1)

S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991).
[CrossRef]

1985 (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[CrossRef]

1979 (1)

T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979).
[CrossRef]

1973 (1)

P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids 16(2), 321–328 (1973).
[CrossRef]

1966 (1)

T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. 16(23), 1054–1056 (1966).
[CrossRef]

1962 (1)

Bulanov, S. V.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[CrossRef]

Cao, N.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

Chen, Z.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

Dawson, J. M.

T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979).
[CrossRef]

Esarey, E.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993).
[CrossRef]

Faure, J.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Gauduel, Y. A.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Hatsagortsyan, K. Z.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

Ho, Y. K.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

Hu, S. X.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

Hua, J. F.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

Kaw, P. K.

P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids 16(2), 321–328 (1973).
[CrossRef]

Kawata, S.

S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991).
[CrossRef]

Keitel, C. H.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

Kibble, T. W.

T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. 16(23), 1054–1056 (1966).
[CrossRef]

Kong, Q.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

Kulsrud, R. M.

P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids 16(2), 321–328 (1973).
[CrossRef]

Lefebvre, E.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
[CrossRef]

Lin, Y. Z.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

Malka, G.

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
[CrossRef]

Malka, V.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Maruyama, T.

S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991).
[CrossRef]

Miquel, J. L.

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
[CrossRef]

Mora, P.

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3719–3732 (1998).
[CrossRef]

Mourou, G.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[CrossRef]

Mourou, G. A.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[CrossRef]

Nishida, Y.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

Phuoc, K. T.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Piché, M.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B 74, s83–s88 (2002).
[CrossRef]

Quesnel, B.

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3719–3732 (1998).
[CrossRef]

Ride, S. K.

E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993).
[CrossRef]

Rousse, A.

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. 8(8), 133 (2006).
[CrossRef]

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

Sessler, A. M.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

Shimoda, K.

Sprangle, P.

E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993).
[CrossRef]

Strickland, D.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[CrossRef]

Stupakov, G. V.

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001).
[CrossRef]

Tajima, T.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[CrossRef]

T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979).
[CrossRef]

Takahashi, I.

S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991).
[CrossRef]

Varin, C.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B 74, s83–s88 (2002).
[CrossRef]

Wang, P. X.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

Watanabe, H.

S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991).
[CrossRef]

Xie, Y. J.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

Xu, J. J.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

Yan, Z.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

Yuan, X. Q.

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

Zhang, S. Y.

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

Zolotorev, M. S.

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B 74, s83–s88 (2002).
[CrossRef]

Appl. Phys. Lett. (2)

P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001).
[CrossRef]

J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004).
[CrossRef]

N. J. Phys. (1)

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. 8(8), 133 (2006).
[CrossRef]

Nat. Phys. (1)

V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008).
[CrossRef]

Opt. Commun. (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[CrossRef]

Phys. Fluids (1)

P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids 16(2), 321–328 (1973).
[CrossRef]

Phys. Rep. (1)

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (2)

E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993).
[CrossRef]

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3719–3732 (1998).
[CrossRef]

Phys. Rev. Lett. (6)

T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. 16(23), 1054–1056 (1966).
[CrossRef]

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979).
[CrossRef]

G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001).
[CrossRef]

G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997).
[CrossRef]

S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991).
[CrossRef]

Rev. Mod. Phys. (1)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[CrossRef]

Other (1)

E. Siegman Lasers, (University Science Books, Mill Valley, California, 1986).

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