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 $t=t_b$ (<0), as shown in Fig. 1 . A dimensionless parameter $a_0\equiv e{E}_0/m_e\omega c$ measures the laser intensity. $E_0$ 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 $m_e$ 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 $m_{e}c$, and energy in units of $m_{e}c{}^2\text{}$. 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]:

3. Some typical examples

First, we consider an example with the following laser parameters: wavelength $\lambda =1\mu m$, peak field intensity $a_0=10\text{}$ ($I=1.37\times {10}^{20}W/c{m}^2\text{}$), focused spot size $w_0=60\text{}$ ($9.55\mu m\text{}$), and pulse duration $\tau =200\text{}$ (the FWHM pulse length is about 125 fs). The electron is initially at rest at the origin ($x_0=y_0=z_0=0\text{}$), and we begin our calculation at $t_b=0\text{}$. The electron energy $\gamma ={\left(1-{\beta }^2\right)}^{-1/2}\text{}$ as a function of time t is shown in Fig. 2(a) (solid line). Its outgoing energy is ${\gamma }_f\approx \text{68 .02 }$, 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 $\gamma \text{}$ 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 ${\gamma }_i\approx \text{68 .02 }$. Unfortunately, the net energy gain ${\gamma }_f-{\gamma }_i\text{}$ in this process is close to zero.

 

Fig. 2 Panels (a) and (b) show the result for $t_b=0\text{}$, in normal time and reverse time respectively. Panel (d) shows the case for $t_b=-1200\text{}$ in normal time. Panel (c) plots the longitudinal momentum as a function of coordinate z, with $t_b=0\text{}$. 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 $x_0=y_0=z_0=0\text{ }$, and the laser parameters are $a_0=10\text{ }$, $w_0=60\text{ }$, and $\tau =200\text{ }$.

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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 ($t_b=-1200\text{ }$) is shown in Fig. 2(d). Using the same initial conditions, the time-reversed simulation tells us that the electron needed an injection energy ${\gamma }_i\text{ }$ is equal to the initial energy ${\gamma }_0$ of the normal calculation, i.e., ${\gamma }_i={\gamma }_0=1\text{ }$, 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 ($x_0=y_0=0\text{}$,${\beta }_{0x}={\beta }_{0y}=0\text{}$) with one of three initial velocities in the z-direction: at rest (${\beta }_{0z}=0\text{}$), at low speed (${\beta }_{0z}=0.05\text{}$, about 640eV), and at high speed (${\beta }_{0z}=0.999\text{}$, about 10.9MeV). The laser parameters are always $a_0=10\text{}$, $w_0=60\text{}$ and $\tau =200\text{}$. 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 ${\gamma }_f\text{}$ and needed injection energy ${\gamma }_i\text{}$.

Table 1. The outgoing energy ${\gamma }_f\text{}$ and needed injection energy ${\gamma }_i\text{}$ in some typical cases. $z_0\text{}$ is the initial position, ${\gamma }_0\text{}$ the initial energy, $U_{p0}$ the initial ponderomotive potential in units of $m_{e}c{}^2\text{}$.

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We find that ${\gamma }_f\text{}$ 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 ${\gamma }_f-{\gamma }_0\text{}$ 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 ${\gamma }_f-{\gamma }_i\text{}$ 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 ${\gamma }_f-{\gamma }_0\text{}$ is more than 90 MeV, the net energy gain is negative. Only the final three cases, underlined in Table 1, satisfy ${\gamma }_i={\gamma }_0\text{}$.

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]

The ponderomotive potential values $U_{p0}$ 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. ${\gamma }_f~U_{p0}+1.0$. 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 ${\gamma }_i={\gamma }_0\text{ }$ 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 ${\beta }_0\text{}$ (${\beta }_{0x}\text{}$, ${\beta }_{0y}\text{}$, ${\beta }_{0z}\text{}$). If it exactly encounters the pulse center at the focus, then $z_0={\beta }_{0z}c{t}_b\text{}$. 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):

Figure 3 presents the dependence of the electric field strength $\left|E\right|\text{ }$ on $t_b\text{ }$ 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 $t_b\text{ }$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 $\left|E\right|\text{ }$ is very sensitive to $t_b\text{ }$. Normally, if $\left|E\right|<{10}^{-6}\text{}$, this factor has a negligible impact on the final results. To ensure that this is the case, we set $\epsilon ={10}^{-9}\text{}$, corresponding to a maximum field of 32.1$V/cm\text{}$ for $\lambda =1\mu m\text{}$. 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 $w_0\text{}$ has little effect on $t_b\text{ }$, while changing the pulse duration $\tau \text{}$ has a strong effect on $t_b\text{ }$.

 

Fig. 3 The electric field strength $\left|E\right|\text{ }$ as a function of $t_b\text{ }$, where ${\beta }_{0x}={\beta }_{0y}=0\text{ }$ and $a_0=10\text{ }$. The solid lines are for $w_0=60\text{ }$, $\tau =200\text{ }$ and ${\beta }_{0z}=0,0.05,0.5,0.7,0.80.999\text{}$; The dotted line is for $w_0=120\text{ }$, $\tau =200\text{ }$ and ${\beta }_{0z}=0.999\text{}$; The dot-dashed line is for $w_0=60\text{ }$, $\tau =300\text{ }$ and ${\beta }_{0z}=0.999\text{}$.

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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 $\lambda =1\mu m\text{}$, $a_0=10\text{}0\text{}$ ($I=1.37\times {10}^{22}W/c{m}^2\text{}$), $w_0=60\text{}$ and $\tau =200\text{}$, we present an example of x-z plane sideways injection at ${\theta }_i=5^0\text{ }$ (the angle between the electron’s initial velocity and the z-axis) into the focus. The initial velocity is ${\beta }_0=0.9997\text{}$ (about 20.35MeV), and two initial positions are considered: $z_0=-Z_R\text{}$ and $z_0=-10Z_R\text{}$. Provided that the electron and pulse arrive at the origin at the same time, we get $t_b=z_0/({\beta }_0\cos {\theta }_i)\text{}$, $x_0=z_0\tan {\theta }_i\text{}$. Figure 4 presents simulation results for the full range of initial phase ${\varphi }_0\text{}$. Solid lines give the outgoing energy ${\gamma }_f\text{ }$, and dotted lines give the injection energy ${\gamma }_i\text{}$. For comparison, the initial energy ${\gamma }_0\text{}$ is also plotted with a dot-dashed line.

 

Fig. 4 The outgoing energy ${\gamma }_f\text{}$ (solid line), needed injection energy ${\gamma }_i$ (dotted line) and initial energy ${\gamma }_0$ (dot-dashed line) as functions of the laser’s initial phase ${\varphi }_0$. The laser pulse parameters are $\lambda =1\mu m$, $a_0=10\text{}0$, $w_0=60$ and $\tau =200$. Other parameters are ${\beta }_0=0.9997$, ${\theta }_i=5^0\text{}$, $t_b=z_0/({\beta }_0\cos {\theta }_i)$, $x_0=z_0\tan {\theta }_i$, (a) $z_0=-Z_R$, (b) $z_0=-10Z_R$.

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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 ($z_0=-Z_R\text{}$) 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), $\left|E\right|=\text{0 .2254214}$, 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 ($z_0=-10Z_R\text{}$) is ten Rayleigh lengths away from the focus. A brief calculation gives $\left|E\right|=\text{9 .46}\times {10}^{-4}\text{}$ 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.