Vacuum electron acceleration driven by a tightly focused radially polarized Gaussian beam

Lin Dai; Jian-Xing Li; Wei-Ping Zang; Jian-Guo Tian

doi:10.1364/OE.19.009303

1. Introduction

Due to the advance of Petawatt lasers [1] with repetition of ultrahigh intensity and ultrashort pulses, vacuum laser-driven electron acceleration has attracted widespread attentions [2–6]. For electrons to be accelerated and reach a net maximum energy gain from interaction with the fields of a laser beam, the beam must be focused onto small spatial dimensions [2]. Compared with linearly polarized light, the radially polarized laser light may be focused to a smaller spot size [7]. Therefore, electron acceleration driven by a radially polarized laser beam has been shown to be a more promising configuration [4].

Tightly focusing Gaussian laser beam would result in non-Gaussian field components in all three dimensions [8]. Therefore, a Gaussian profile is inadequate, and more accurate representations are required. By the use of Lax series approach [9,10], some researchers [11,12] obtained the high order correction terms with respect to the diffraction angle ε in the description of the tightly focused radially polarized Gaussian field. However, Lax series appears to be divergent for nonparaxial beams, and the higher order corrections of Lax series cannot always produce better results [12–15]. For the linearly polarized Gaussian field, Weniger transformation method has been used to effectively eliminate the divergence of Lax series [13–15].

In this paper, we study electron acceleration in vacuum driven by a tightly focused radially polarized Gaussian laser beam. By using Weniger transformation we eliminate the divergence of Lax series field (LSF), and use the plane wave spectrum approach [16,17] to validate the accuracy of Weniger transformation field (WTF). Meanwhile, if electrons are not accelerated along the longitudinal axis exactly, the transverse field components force electrons to deflect outside the intense field region. Due to the divergence of LSF, there are serious errors in electron dynamics simulated by using LSF. These problems can be avoided by using WTF. Furthermore, we have analyzed the important roles of the initial phase of the electromagnetic field and the injection angle, position and energy of electron in energy gain of electron.

2. Electromagnetic field of a radially polarized Gaussian beam beyond the paraxial approximation

Employing cylindrical coordinates (r, θ, z), where the z-axis is aligned with the laser beam propagation direction, the vector potential for a radially polarized Gaussian beam is of the form

A (r, θ, z, t) = \hat{z} A_{0} ψ (r, z) e^{i η},

where

η = ω t - k z

is the plane wave phase factor, and A ₀ is a constant amplitude. Using Lax series method [9,10], one obtains the Lax series solution for ψ(r, z) in Eq. (1) [11,12,18]

ψ (r, z) = \sum_{n = 0}^{\infty} ε^{2 n} ψ_{2 n} (r, z) .

With the Lorentz gauge condition the fields can be derived from

E = - i k A - \frac{i}{k} \nabla (\nabla \cdot A), B = \nabla \times A .

Substituting Eq. (1) into Eq. (3), the components of the fields accurate up to ε ^{2m + 2} can be represented as [12,18]

E_{r} = E \sum_{n = 0}^{m} ε^{2 n + 1} E_{2 n + 1}^{} (f, ρ),

E_{z} = - i E \sum_{n = 0}^{m} ε^{2 n + 2} E_{2 n + 2} (f, ρ),

B_{θ} = E \sum_{n = 0}^{m} ε^{2 n + 1} B_{2 n + 1} (f, ρ),

where

E = E_{0} \exp [i (φ_{0} + η) - f ρ^{2}]

,

E_{0}

= kA ₀ is field amplitude, φ ₀ is an initial phase, f = i/(i + ζ), ζ = z/z _r, z _r is the Rayleigh length, ρ = r/w ₀, and w ₀ is the beam waist radius. Other components E_θ, B_r, and B_z vanish identically. E ₂ _n, E_2n ₊₁, and B_2n ₊₁, as well as ψ ₂ _n _, are available in Refs. [12,18]. Note that Gaussian unit is used throughout this paper.

For linearly polarized Gaussian laser beams, the divergence of LSF gets worsened as the beams are more tightly focused. Simply truncating the Lax series does not guarantee an accurate description of the beam fields. However, the divergence of Lax series can be eliminated by using Weniger transformation [13–15]. Therefore, we use Weniger transformation to process LSF for a radially polarized Gaussian beam. For example,

E_{z} = \frac{\sum_{j = 0}^{n} {(- 1)}^{j} (\begin{matrix} n \\ j \end{matrix}) {(1 + j)}_{n - 1} (S_{2 j}^{} / E_{2 j + 2}^{})}{\sum_{j = 0}^{n} {(- 1)}^{j} (\begin{matrix} n \\ j \end{matrix}) {(1 + j)}_{n - 1} (1 / E_{2 j + 2}^{})}

where

S_{2 n}^{} = \sum_{l = 0}^{n} ε^{2 l} E_{2 l}^{} (f, ρ)

, note that the series expression for any field component possesses only either odd or even order terms, which entails the forms of dummy variables in Eq. (7). The Weniger transformation is based on a nonlinear sequence transformation. The results obtained in this way are superior to those obtained using Pade approximants. Meanwhile, the Weniger transformation fulfills an accuracy-through-order relation [19–21]. Figure 1 illustrates the propagation dynamics of LSF and corresponding WTF. LSF is accurate in the focal plane, and the divergence of LSF in the far regions can be eliminated completely. w(z) = w ₀(1 + ζ ²)^1/2 is the beam radius for an arbitrary z.

Fig. 1 (Color online) (a) and (b) Propagation dynamics of E_z, accurate up to ε ¹⁰, of LSF and WTF, respectively. w ₀ = λ.

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To further confirm the accuracy of WTF, plane wave spectrum approach [16,17] is used to reach integral representations of fields, which are the rigorous solutions of Maxwell’s equations. We essentially follow the arguments in Refs. [16,17]. The vector potential A in Eq. (1) can be represented by its angular spectrum representation

A (r, z) = \frac{1}{2 π} {\int \int}_{- \infty}^{\infty} P (k_{x}, k_{y}) e^{- i (k_{x} x + k_{y} y + k_{z} z)} d k_{x} d k_{y},

where k_x ² + k_y ² + k_z ² = k ², and

P (k_{x}, k_{y}) = \frac{1}{2 π} {\int \int}_{- \infty}^{\infty} A (r, 0) e^{i (k_{x} x + k_{y} y)} d x d y .

To simplify Eq. (8), polar coordinates are introduced

k_{x} = κ \cos θ^{'}, k_{y} = κ \sin θ^{'}, x = r \cos θ, y = r \sin θ .

Note that the coordinates x and y are completely symmetrical, and so P(k_x, k_y) always degenerates into the form P(κ). Consequently, Eq. (8) can be represented as

A (r, z) = \frac{1}{2 π} \int_{0}^{\infty} P (κ) e^{- i z \sqrt{k^{2} - κ^{2}}} J_{0} (κ r) κ d κ,

where J ₀ is the zeroth Bessel function of the first kind. Substituting Eq. (11) into Eqs. (3), the angular spectrum representations of the fields are

E_{r} = \frac{1}{k} \int_{0}^{\infty} κ^{2} \sqrt{k^{2} - κ^{2}} P (κ) e^{- i z \sqrt{k^{2} - κ^{2}}} J_{1} (κ r) d κ,

E_{z} = - \frac{i}{k} \int_{0}^{\infty} κ^{3} P (κ) e^{- i z \sqrt{k^{2} - κ^{2}}} J_{0} (κ r) d κ,

B_{θ} = \int_{0}^{\infty} κ^{2} P (κ) e^{- i z \sqrt{k^{2} - κ^{2}}} J_{1} (κ r) d κ .

where J ₁ is the first Bessel function of the first kind. When exploring the integrals in Eqs. (12)–(14), the integration range can be decomposed into two parts:

\int_{0}^{\infty} ... = \int_{0}^{k} ... + \int_{k}^{\infty} ...

where

\int_{0}^{k} ...

indicates the propagation wave component, and

\int_{k}^{\infty} ...

the evanescent wave component [16,17]. The contribution of the evanescent wave turns out so small that it suffices to explore

\int_{k}^{n k} ...

instead of the infinite integration, with n = 10 for instance.

Figure 2 presents the comparisons of electromagnetic field components among LSF, WTF and PWSF. The terms of high order correction of LSF can introduce better correction effect in the near-axial region. However, the divergence of LSF becomes more and more serious with the increasing of correction order. Weniger transformation not only maintains the advantaged correction effect of high order terms, but also eliminates the divergence of LSF. Furthermore, E_r and B_θ both vanish completely in the longitudinal axis and peaks at $r = w_{0} / \sqrt{2}$ [11,18].

Fig. 2 (Color online) (a)–(c) Cross sections of electromagnetic field components E_z, E_r, and B_θ, respectively. w ₀ = λ, and z = 5z_r. Red, blue and grey curves represent the components of LSF accurate up to ε ², ε ⁸, and ε ²⁴, respectively. Green solid and black dash-dot curves represent the components of WTF and PWSF, both accurate up to ε ²⁴, respectively.

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3. Electron dynamics in a tightly focused radially polarized Gaussian laser beam

Electron dynamics in vacuum driven by a laser beam is governed by the relativistic equations of motion d p/dt = -e(E + β × B), and dΞ/dt = -ec β·E, where the momentum p = γmc β, the energy Ξ = γmc ², the Lorentz factor γ = (1-β ²)^-1/2, and β is the velocity scaled by c, e is the charge of an electron. It is convenient to introduce a normalized factor q = eE ₀ /mωc indicating the intensity of fields. We define the lines x = ± w(z) as the beam boundaries where the field intensity falls to e ⁻² of its maximum value on the beam axis.

If an electron is injected along the longitudinal axis from an arbitrary point on the longitudinal axis, where E_r and B_θ both vanish completely, it only interacts with E_z and may be accelerated to high energy along the longitudinal axis [4]. However, in many cases, such as laser-driven electron beam acceleration, the dynamics of electrons which move outside the longitudinal axis should be considered. Figure 3 presents the dynamics of an electron injected with an angle θ ₀ = 25° relative to the longitudinal axis. As the electron moves outside the longitudinal axis, transverse electric field also makes significant contribution for its energy gain. Due to the intensity diffraction of Gaussian beam, the electron gains its energy mainly in the focal area. As the electron moves far away from the axis, where the field is very weak, it will gain little or no net energy, as simulated by WTF. The divergence of LSF causes the serious errors of electron dynamics simulation. Even though the injected angle of electron is very small, the transverse electric field still deflects the electron outside the beam axis, where LSF is divergent. Therefore, WTF should be used to simulate electron dynamics in a tightly focused radially polarized Gaussian beam instead of LSF.

Fig. 3 (Color online) (a)–(e) Electron trajectories, energy gains, and electromagnetic field components E_z, E_r, and B_θ, sensed by electron along its trajectory, respectively. An electron is initially set at the origin and injected in x-z plane, with injection angle θ ₀ = 25°, γ ₀ = 100, w ₀ = λ, q = 150, and φ ₀ = π. Integration limit takes η_f = 10³ π. Black curves present the beam boundaries.

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The roles of initial phase of field $ϕ_{0}$ and injection angle θ ₀, position, and energy γ ₀ of electron in energy gains of electron are studied in Fig. 4 . In Fig. 4(a), $ϕ_{0}$ plays a very important role in electron energy gain. For the given parameters, E_z can afford maximum accelerating force at $ϕ_{0} = π$ , and the electron can be quickly accelerated to its peak energy. Meanwhile, in the region $270 ° < ϕ_{0} < 450 °$ , E_z initially affords negative or few accelerating force, and the electron will experience accelerating and decelerating phase alternately. So, the electron gains few or no net energy. For electron acceleration by a radially polarized Gaussian beam, E_z affords mainly accelerating force for electron energy gain, and E_r forces electron to deflect outside the intense field region. Therefore, an electron injected with smaller angle can be accelerated to higher energy, as shown in Fig. 4(b). For the same reason, an electron injected at a closer distance from the axis can gain higher energy, as shown in Fig. 4(c). The phase velocity of field around the longitudinal axis is very fast, an electron with finite initial energy cannot be synchronously accelerated to much high energy, as shown in Fig. 4(d). Furthermore, the results simulated by LSF have obvious differences from those of WTF. Therefore, electron acceleration by an intense radially polarized Gaussian beam should be studied by using WTF.

Fig. 4 (Color online) (a)-(d) Variation of the energy gain with $ϕ_{0}$ , θ ₀, γ ₀, and r. q = 150. In (a) γ ₀ = 100, θ ₀ = 15°, and η_f = 35π. In (b) γ ₀ = 100, φ ₀ = π, and η_f = 2.2×10³ π. In (c) φ ₀ = π, γ ₀ = 10, θ ₀ = 0° and η_f = 40π. In (d) $ϕ_{0}$ = π, θ ₀ = 15°, and η_f = 10³ π. Other parameters are same as those of Fig. 3.

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4. Conclusion

Vacuum laser-driven electron acceleration by an intense radially polarized Gaussian beam is studied in detail. Field representation derived from Lax series method is divergent. And, the terms of high order correction in LSF do not always produce better effect. Weniger transformation can be used to eliminate completely the divergence of LSF. In study of electron dynamics in an intense radially polarized Gaussian beam, WTF should be employed instead of LSF. Meanwhile, the initial phase of field and the injection angle, energy and position of electron play very important roles in energy gain of electron. Since the longitudinal electric field affords mainly accelerating force, electrons should be accelerated along the longitudinal axis to gain high energy.

Acknowledgment

We acknowledge financial supports from the National Natural Science Foundation of China (grant 11074130), Chinese National Key Basic Research Special Fund (2011CB922003), Program for New Century Excellent Talents in University, and 111 Project (B07013).

References and links

1. M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H. T. Powell, M. Vergino, and V. Yanovsky, “Petawatt laser pulses,” Opt. Lett. 24(3), 160–162 (1999). [CrossRef]

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

3. N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204(1-6), 7–15 (2002). [CrossRef]

4. Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006). [CrossRef]

5. J. X. Li, W. P. Zang, and J. G. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. 96(3), 031103 (2010). [CrossRef]

6. J. X. Li, W. P. Zang, and J. G. Tian, “Analysis of electron capture acceleration channel in an Airy beam,” Opt. Lett. 35(19), 3258–3260 (2010). [CrossRef] [PubMed]

7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

8. S. X. Hu and A. F. Starace, “Laser acceleration of electrons to giga-electron-volt energies using highly charged ions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6), 066502 (2006). [CrossRef] [PubMed]

9. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975). [CrossRef]

10. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]

11. Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31(17), 2619–2621 (2006). [CrossRef] [PubMed]

12. H. Luo, S. Y. Liu, Z. F. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 32(12), 1692–1694 (2007). [CrossRef] [PubMed]

13. R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28(10), 774–776 (2003). [CrossRef] [PubMed]

14. J. X. Li, W. P. Zang, and J. G. Tian, “Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method,” Opt. Express 17(7), 4959–4969 (2009). [CrossRef] [PubMed]

15. J. X. Li, W. P. Zang, Y.-D. Li, and J. G. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express 17(14), 11850–11859 (2009). [CrossRef] [PubMed]

16. A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136(1-2), 114–124 (1997). [CrossRef]

17. P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152(1-3), 108–118 (1998). [CrossRef]

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

19. E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10(5-6), 189–371 (1989). [CrossRef]

20. U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, “Resummation of QED perturbation series by sequence transformations and the prediction of perturbative coefficients,” Phys. Rev. Lett. 85(12), 2446–2449 (2000). [CrossRef] [PubMed]

21. E. J. Weniger, “Summation of divergent power series by means of factorial series,” Appl. Numer. Math. 60(12), 1429–1441 (2010). [CrossRef]

Vacuum electron acceleration driven by a tightly focused radially polarized Gaussian beam

Abstract

1. Introduction

2. Electromagnetic field of a radially polarized Gaussian beam beyond the paraxial approximation

3. Electron dynamics in a tightly focused radially polarized Gaussian laser beam

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Equations (15)

Optics Express