## Abstract

For an electron accelerated by a tightly focused Gaussian laser beam, its dynamics are usually simulated through the field obtained by Lax approach [Phys. Rev. A 11, 1365 (1975)]. However, as Lax series field (LSF) is not always convergent, the obtained results are usually inaccurate and even illogical. Here we report that the divergence of LSF can be eliminated by using Weniger transformation, and the electron dynamics simulated by this new field are logical and accurate.

© 2009 Optical Society of America

## 1. Introduction

Advances in laser technology have resulted in the development of petawatt lasers [1], which can deliver super strong laser pulses with intensities as high as 10^{21} W/cm^{2}. Laser-based accelerators [2] are capable of producing high-energy electrons in much shorter distance than conventional accelerators due to their strong electric fields. The physics of the interaction of electron with such strong field is therefore attracting much attention [3, 4]. Acceleration of electrons by a laser in vacuum has been investigated theoretically. The problems inherent in laser-plasma interaction, such as instabilities, are absent in vacuum, and electrons can gain higher energy due to an increase in duration of the interaction between laser and the electron. Furthermore, preaccelerated electrons can be injected into vacuum and the peak energy attained by the electron can be increased [5].

It is well known that the field components of a laser beam must satisfy Maxwell’s equations. When focused by a lens or by a reflecting mirror, a laser beam is well described by a Gaussian profile function in region of the focus provided that the beam waist *w*
_{0} is much larger than the laser wavelength *λ*. However, if a laser beam is focused down to the order of the laser wavelength, a Gaussian beam description becomes inaccurate. Even for a laser beam that is linearly polarized outside the focal region, tight focusing results in non-Gaussian field components in all three dimensions [6]. By the use of Lax series approach [7], some researchers [8–10] analyzed this effect and got the high order correction terms of the diffraction angle e in the description of the associated fields. However, Lax series field (LSF) appears to be divergent for highly nonparaxial beams, and the higher order corrections of Lax series can not always produce better results. Moreover, it is shown that LSF is effectively only in the region near the beam axis and its divergence is serious in the region outside beam boundary. Meanwhile, Weniger transformation has been verified to be an effective method for eliminating the divergence of LSF, and Weniger transformation field (WTF) can describe accurately tightly focused beam [11].

When an electron is injected into a beam, for different initial parameters such as laser intensity, initial energy, injection aim point on the axis, initial phase and so on, it may be reflected, transmitted or captured. It is well known that for a tightly focused beam the field near focus is very intense and the field far off the focus is relatively weak. Therefore, the electron can gain much energy near the focus. For the cases of reflection and transmission, as moving far off the focus, the electron should be affected weakly by the field and can not obtain remarkable energy. However, by employing LSF to simulate electron acceleration, the electron energy gain still increases sharply even though the electron is off the beam boundary. By analyzing the field sensed by electron, it can be found that this illogical result is caused by the divergence of LSF. For the most interesting case, capture, as the electron is injected from a point outside beam boundary, the electron dynamics calculated by LSF are also inaccurate.

In this paper, Weniger transformation is used to eliminate the divergence of LSF. It is verified that WTF is an accurate analytic field by comparing with a plane wave spectrum solution. LSF and WTF are used to simulate electron dynamics in a tightly focused laser for all cases of reflection, transmission and capture. Results show that the results calculated by WTF are more logical and accurate. WTF is more feasible for the simulation of electron acceleration.

## 2. Laser field descriptions: LSF and WTF

The laser beam adopted here polarizes along the *x* direction and propagates along the *z* axis, and its electromagnetic field can be described in form of the vector potential **A** = *x*̂**A**
_{0}
*ψ*(**r**) exp(*iη*), where *A*
_{0} is a constant amplitude, and *η* = *ωt*-*kz*. The vector potential satisfies the following wave equation:

Direct substitution leads to

By defining *x* = *ξw*
_{0}, *y* = *υw*
_{0}, *z* = *ζz _{r}*, where

*w*

_{0}is the beam radius, and

*z*= k

_{r}*w*

_{0}

^{2}/2, Eq. (2) can be rewritten as

where ∇_{⊥}
^{2}= *∂*
^{2}/*∂ξ*
^{2} + *∂*
^{2}/*∂υ*
^{2} and *ε* = *w*
_{0}/ *z _{r}* is the diffraction angle. Since

*ε*

^{2}is small, one can expand [8]

*ψ*as a sum of even power of

*ε*,

by substituting it into Eq. (3), one can obtain

Equation (5) has an exact paraxial approximation solution:

For obtaining the high order functions *ψ*
_{2n} and the purpose of gaining physical insight, we follow the work of Davis *et al* [8] and for the moment consider a diverging spherical wave propagating along the *z* axis from the origin. Such a wave has an exponential factor, which can be expanded as

For *z* ≫ *z _{r}* the condition

*f*→

*iz*/

_{r}*z*holds, this line of reasoning suggests that

The coefficient *C*
_{2n} is determined by inserting this function into Eq. (6), and this process is complicated. However, *ψ*
_{2n} can be easily derived by simple recurrence relations [12]:

where *C*
_{0} = 1, and *a*
_{2n} (*ρ,f*) is given by the Eq. (8). These recurrence relations can be used to obtain accurate results with arbitrary order of *ε*. After getting vector potential, one can obtain the scalar potential *ϕ* = *i*∇ · **A**/*k* by using the Lorentz gauge. And then, the electromagnetic field components can be obtained with **E** = -*ik*
**A** - ∇*ϕ* and **B** = ∇ × **A**, as [10]:

where, *E*
_{0} = *kA*
_{0} and *ε*
_{0} is the initial phase.

It is well known that LSF is valid for weakly focused fields. The convergence properties of the Lax series will get worse and worse as the beam is focused gradually tightly. Simply truncating the Lax series does not guarantee accurate description of the beam fields. Higher-order correction terms do not necessarily lead to a better approximation. Nevertheless, a resummation scheme, introduced by Weniger [13], has been demonstrated recently to be an effective method for overcoming the divergence of Lax series. The accurate beam field can be obtained by resuming the Lax series of field [11].

The Weniger transformation, when applied to the partial sum of an infinite series, *s _{n}* = Σ

^{n}

_{j=0}

*a*(

_{j}*n*≥0), can convert them into the following sequence:

where (*b*)_{m} denotes the Pochhammer symbol.

Therefore, the electric field *x* component of Eq. (11), accurate to *ε*
^{2n}, can be rewritten by using Weniger transformation:

where *s _{j}* = Σ

^{j}

_{i=0}

*ε*

^{2i}

*E*(

^{x}_{i}*f*,

*ρ*,

*ξ*).

To validate that the tightly focused laser beam can be represented correctly by WTF, we compared the numerical results of LSF and WTF with the exact numerical fields obtained by a plane wave spectrum method [14]. Figure 1 gives the amplitude of *x* and *z* components of electric field in a line of *x* = 0, perpendicular to the beam axis at *z* = 19 *z _{r}* . Electric field

*x*and

*z*components of LSF with different order corrections are shown in Figs. 1(a) and 1(c). The exact numerical field is obtained by the plane wave spectrum method, whose boundary value is defined by

where *ψ*
_{2n} is the 2*n*-order correction to the *ψ*
_{0}, derived by Eq. (10). For a waist radius of the order of wavelength, it is enough to set *m* = 4 to get a high accuracy. Results show that LSF is accurate inside beam boundary and divergent outside beam boundary. Moreover, the divergence of LSF becomes more serious as the order of *ε* increases. Consequently, higher-order correction terms of Lax series do not always bring better results. To eliminate the divergence of LSF, we employ Weniger transformation to handle LSF, and the results are given in Figs. 1(b) and 1(d) respectively. It can be seen that the divergence of LSF has been eliminated. Moreover, WTF is more accurate when the higher order terms of *ε* are included. In the following sections we will use WTF to discuss electron acceleration and compare the results with those calculated by LSF.

## 3. Electron acceleration and numerical simulations

Electron acceleration can be simulated by solving the electron dynamics equations:

where the momentum **p** = *γmc*
**β**, the energy *χ* = *γmc*
^{2}, the Lorentz factor *γ* = (1-*β*
^{2})^{-1/2}, and **β** is the velocity normalized by the speed of light *c* in vacuum. The peak field intensity I_{0} can be expressed in terms of I_{0}
*λ*
^{2} ≈ 1.375×10^{18}
*q*
^{2} (W/cm^{2})(*μ*m)^{2}, where *q* = *eE*
_{0}/*mcω*. The boundary of the beam is described by the curves in the *xz* plane defined by *x* = ±*w*(*z*), *w*(*z*) = *w*
_{0}[1+(*z*/*z _{r}*)

^{2}]

^{1/2}. An electron will be transmitted if its trajectory crosses the line

*x*=

*w*(

*z*), and it will be reflected if its trajectory crosses the line

*x*= -

*w*(

*z*) twice or never crosses it. Otherwise, it will be captured by the beam. Strictly speaking, the field intensity on the curves

*x*= ±

*w*(

*z*) falls down to 1/

*e*

^{2}of its maximum value on axis. Thus a transmitted or reflected electron will be weakly affected by the laser field beyond the beam boundary, and its energy should keep constant or vary slightly when it moves far away from the focus.

The injection energy plays an important role in electron acceleration. A slow electron will not be able to penetrate the high intensity region of the beam and may be reflected with little or no energy gain. But a fast electron may pass through the beam and gain or lose little energy. So, only the electron with an appropriate initial energy may be captured, accelerated and gain much energy.

## 3.1 Reflection and transmission

For a tightly focused laser beam whose parameters are given in the caption of Fig. 2, the electron with a scaled initial energy *γ*
_{0} = 16.03 is reflected by the beam, as shown in Fig. 2(a). Note that LSF, which we used in the simulations, is accurate to *ε*
^{7} and WTF is accurate up to *ε*
^{39} due to the conclusion given by Fig. 1. It can be seen in Fig. 2(b) that the energy gain calculated by WTF almost keeps constant as the electron moves outside the beam boundary since the real field is very weak and can not affect significantly the electron there. But the energy gain calculated by LSF increases absurdly, which is illogical and implies that the electron still senses intense field in the region outside the beam boundary. To explain these phenomena, we study the electromagnetic field and force sensed by the electron along its trajectories in Fig. 3. Obviously, LSF sensed by the reflected electron is divergent in the region outside the beam boundary. The force, caused by the divergent field according to **F** = -*e*(**E** + **β** × **B**), is also divergent and results in the illogical results as shown in Fig. 2(b). However, WTF sensed by the reflected electron is convergent and accords with the real field.

For a tightly focused laser beam whose parameters are given in the caption of Fig. 4, the electron with a scaled initial energy *λ*
_{0} = 200 transmits the beam. Be similar to the case of reflection above, as the electron moves beyond the boundary, its energy gain calculated by WTF almost keeps constant and that calculated by LSF still increases absurdly. It is shown in Fig. 5 that LSF sensed by the transmitted electron is divergent in the region outside the beam boundary, but WTF sensed is convergent and accords with the real field. Thus, LSF is inaccurate and illogical, and WTF is logical and accurate in the region outside the beam boundaries.

## 3.2 Capture

In electron acceleration, electron capture and violent acceleration to GeV are paid much attention. For a tightly focused laser beam whose parameters are given in the caption of Fig. 6, the electron, injected from a point outside the beam boundary with a scaled initial energy *λ*
_{0} =16 , is captured by the beam as shown in Fig. 6. Results show that when the electron moves inside the beam boundary after passing trough the focal region, the gains calculated by LSF and WTF almost keep constant, but there is a difference between them. The difference is due to the difference between WTF and LSF sensed by the electron as shown in Fig. 7. When the captured electron moves inside the beam boundaries, LSF and WTF sensed by it are convergent. However, before the electron enters into the beam, LSF is divergent, but WTF is convergent. Even though the field near the injection point is weak, the velocity of the electron is still sensitive to the field, as shown in Fig. 8. Therefore, the electron dynamics calculated by LSF are different from those calculated by WTF and the latter are more accurate.

For a laser beam focused down to a small dimension of the order of wavelength, LSF can effectively describe the field near the beam axis, but its divergence is obvious in the region outside the beam boundary. However, WTF is convergent and accurate in all regions. Therefore, if a captured electron is injected from a point inside the boundary, the electron will moves inside the boundary all along. The electron dynamics calculated by LSF should be the same as those calculated by WTF. Our simulations also validate their coincidence, as shown in Fig. 9. The coincident field components sensed by the electron are shown in Fig. 10. Therefore, for simulations of electron acceleration in a tightly focused laser, WTF are more suitable and accurate than LSF in all regions inside /outside beam boundary.

## 4. Conclusion

In conclusion, the divergence of LSF can be eliminated by Weniger transformation. WTF is an accurate analytic field for descriptions of a tightly focused laser beam. LSF only can describe accurately the field near the beam axis, or roughly speaking, inside the beam boundary. However, in simulation of dynamics of an electron injected outside beam boundary, LSF is invalid, but WTF is valid and accurate. Therefore, in simulation of electron acceleration or other applications of nonparaxial beam, WTF should be used to obtain accurate results.

## Acknowledgment

We acknowledge support from the Natural Science Foundation of China (grant 60678025), Chinese National Key Basic Research Special Fund (2006CB921703), Program for New Century Excellent Talents in University, and the Program for Changjiang Scholars and Innovative Research Team in University.

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