The recent proposal to use Weinger transformation field (WTF) [Opt. Express 17, 4959-4969 (2009)] for describing tightly focused laser beams is investigated here in detail. In order to validate the accuracy of WTF, we derive the numerical field (NF) from the plane wave spectrum method. WTF is compared with NF and Lax series field (LSF). Results show that LSF is accurate close to the beam axis and divergent far from the beam axis, and WTF is always accurate. Moreover, electron dynamics in a tightly focused intense laser beam are simulated by LSF, WTF and NF, respectively. The results obtained by WTF are shown to be accurate.
©2009 Optical Society of America
Invention of the chirped pulse amplification (CPA) technique  continues to motivate research efforts into the issue of electron acceleration by a laser, which has been investigated theoretically [2–4]. The primary problem in the theoretical simulation of electron dynamics is to obtain the accurate field representation of intense laser beam. 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 focus region when the beam waist w 0 is much larger than the laser wavelength λ. If a laser beam is focused down to the order of the laser wavelength, a Gaussian beam description becomes insufficient. In 1975, Lax et al.  proposed a perturbative approach to derive the solution of the free space propagation of a monochromatic electromagnetic beam, starting from knowledge of the paraxial solution in whole space. For highly non-paraxial beams, Lax series appears to be divergent, but Weniger transformation can eliminate that divergence [6,7].
In this paper, the numerical field (NF) of a tightly focused beam is derived by the plane wave spectrum method [8,9]. Comparing Lax series field (LSF) and Weniger transformation field (WTF) with NF, we illustrate that LSF is accurate close to the beam axis and divergent far from beam axis, respectively. However, Weniger transformation is an effective method to eliminate the divergence of LSF, and WTF always keep convergent. We adopt NF to simulate electron acceleration by a tightly focused intense laser beam, and compare the results with those simulated by LSF and WTF. Results show electron dynamics simulated by WTF are accurate.
In the second section, we review the representations of LSF and WTF, and derive those of NF using the plane wave spectrum method. To ensure the accuracy of NF, we employ the accurate boundary field. In the third section, firstly, LSF and WTF are compared with NF. Secondly, the electron dynamics in an intense laser beam are simulated by LSF, WTF and NF. The simulated results of WTF are shown to be accurate. The conclusion is given in the last section.
2. Electromagnetic field of tightly focused laser beams
2.1 Lax series approach and Weniger transformation
The laser beam adopted here polarizes along the x direction and propagates along the z axis. The electromagnetic field can be described in form of the vector potential A=x̂A 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
We can define x=ξw 0, y=υw 0, z=ζzr, where w 0 and zr=kw 2 0/2 are the beam radius and Rayleigh length, respectively. Equation (2) can be rewritten as
where ∇2 ⊥=∂ 2/∂ξ 2+∂ 2/∂υ 2 and the diffraction angle ε=w o/zr. Since ε 2 is small, one can expand ψ as a sum of even power of ε ,
Equation (5) has an exact paraxial approximation solution:
To obtain the high order functions ψ2n and the purpose of gaining physical insight, we follow the work of Davis et al . 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≫zr the condition f→izr/z holds, this line of reasoning suggests that
where D 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. Such as,
After getting vector potential, one can obtain the scalar potential ϕ=i∇·A/k by using the Lorentz gauge. Then, the components of LSF can be obtained by substituting A and ϕ into Maxwell’s equations E=-ik A-∇ϕ and B=∇×A, accurate up to ε 2m+1, as [7,12]:
where m≥0, E=E oexp[i(ωt-kz+ϕ 0), E 0=kA 0 and ϕ 0 is the constant phase. For example, the components of electromagnetic field accurate up to ε 7 can be expressed as
It is well known that LSF shown above 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 , 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 [6,7].
The Weniger transformation, when applied to the partial sum of an infinite series, sm=∑mn=0an(m≥0), can convert them into the following sequence:
where (b)j denotes the Pochhammer symbol.
Therefore, the electric field z component of Eq. (16), accurate to ε 2m+1, can be rewritten by using Weniger transformation:
where sezz=∑n i=0 ε 2i+1 Ezi(f,ρ). Equation (27) gives the z component of electric field of WTF, and other components can be obtained by the same process.
2.2 Plane wave spectrum method
We use i, j, and k to denote unit vectors of in the positive x, y and z directions. Equations (28) and (29) are the exact solutions of Maxwell’s equations when the boundary value ψ(x, y, 0) of one of the components of the Hertz vector is given in the z=0 plane. The Gaussian function exp(-ρ 2) is usually chosen as the boundary value. However, for a non-paraxial laser beam, the terms of high order corrections should be included due to the Lax series theory . Thus, we define the boundary field according to Eq. (4) as
the electric and magnetic fields are given by
Then by using the transformation rules  of Bessel functions, the field components are given as
where Jn is the Bessel function, n order, first kind.
The accuracy of NF depends on the function P(κ), which is derived from the boundary field ψ(x, y, 0). Therefore, the accurate boundary field should be taken into account.
To illustrate the effects of high order correction in boundary field, we study the electric field z component of NF for different beam waist sizes in Fig. 1. For the beam waist w 0=5λ, the propagating field derived by the boundary field of ε 0 model is same as that obtained by high order model, as shown in Fig. 1(a). On the contrary, in the case of w 0=λ, the propagating field based on the ε 0 model deviates from that of the high order model, as shown in Fig. 1(b). Therefore, the terms of ε 2 should be included in the boundary field. In Fig. 1(c), we can see that the boundary field of ε 2 model is also inaccurate for w 0=0.5λ. The terms of ε 4 should be included into boundary field. Thus, we should consider the high order correction for a tightly focused laser beam.
3. Results and discussion
3.1 Comparison of LSF, WTF and NF
To validate the accuracy of WTF, we compare LSF and WTF with NF. Figure 2 gives the amplitude of the electric field z and x components for a tightly focused beam with waist w 0=λ. The boundary field of NF is accurate to ε 2. Results of Figs. 2(a) and 2(c) show that LSF is accurate close to the beam axis, and divergent far from the beam axis. The divergence of Lax series becomes more serious as increasing of the order of ε. We employ Weniger transformation to eliminate the divergence of LSF and obtain WTF, which are given in Figs. 2(b) and 2(d). It can be seen that the divergence of LSF has been eliminated. Moreover, WTF is accurate more and more as increasing of the order of ε.
3.2 Simulation of electron dynamics in a tightly focused laser beam
The final goal of our study on the accurate field description is to exactly simulate acceleration of electron by a tightly focused intense laser beam in a vacuum. Electron dynamics in an intense laser are always investigated by LSF [2–4]. However, due to the divergence of LSF, the simulation of LSF cannot be always accurate.
Dynamics of the electron in a laser beam in a vacuum is governed by the 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 scaled by c, the speed of light in vacuum. The peak field intensity I0, will be given in terms of the dimensionless parameter q=eE 0/mcω, where I0 λ 2≈1.375×1018 q 2 (W/cm2)(µm)2. The boundaries of the beam are described by the curves x=±w(z), where w(z)=w 0[1+(z/zr)2]1/2. An electron will be transmitted if its trajectory crosses the line x=w(z), and will be reflected if its trajectory crosses the line x=-w(z) twice or never. Otherwise, it will be captured by the beam.
An electron is injected from a point outside beam boundary toward to the laser beam focus and interacts with the laser. The trajectories and energy gains of electron for the cases of reflection, transmission and capture have been studied. The results showed that electron dynamics simulated by LSF were not always accurate due to the divergence of LSF, and those simulated by WTF are valid qualitatively . Here, we use NF to simulate electron dynamics, and compare the results with those simulated by LSF and WTF in the focal area, as shown in Figs. 3–5. In the simulations, the correction order of LSF and WTF is chosen to be ε 7 and ε 39, respectively, due to that as the correction order of ε increases, the divergence of LSF is more serious and WTF is more accurate. It is clearly shown that for all cases electron dynamics of WTF are consistent with those of NF, but, electron dynamics of LSF deviate from those of NF. Hence, electron dynamics of WTF are accurate.
In conclusion, to validate the accuracy of WTF, we derive NF by the plane wave spectrum method. By comparing LSF and WTF with NF, we illustrate that LSF is divergent and WTF is accurate. As the correction order increases, the divergence of LSF is more serious, and WTF is accurate more and more. Moreover, the electron dynamics are simulated by LSF, WTF and NF, respectively. The results of WTF are shown to be accurate.
We acknowledge financial supports 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 111 Project (B07013).
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