Vacuum laser-driven acceleration by Airy beams

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

doi:10.1364/OE.18.007300

1. Introduction

Due to the invention of the chirped pulse amplification (CPA) technique [1], laser acceleration in vacuum has been an active research area in recent years [2–5]. Laser-driven electron acceleration relies on the large laser intensity that can be achieved by focusing laser beams down to spot sizes in the order of wavelength. However, a shortcoming of many of these schemes is that the interaction length over which the high intensity can be sustained is relatively short due to transverse spreading (diffraction). Therefore, electron acceleration by quasi-diffraction-free beams, Bessel beam [6], has attracted widespread attentions [7–9].

Another known diffraction-free beam is Airy wave packets, first predicted by Berry and Balazs within the context of quantum mechanics [10]. This intriguing class of beams was only recently predicted and realized in optical domain [11,12]. Its key features are transversely accelerating and diffraction-free during propagation. Unlike other-diffracting beams, the Airy beam does not result from conical superposition, is possible even in one-dimension, and is highly asymmetric. Airy beams have been used in optical micromanipulation [13]. In this paper we present the first use of the Airy beam in vacuum electron acceleration. The characteristics of acceleration and non-diffraction of Airy beam in one dimensional configuration lead to the formation of a long “asymmetric field channel” (AFC) along the propagation axis. Electron senses a continual acceleration phase in AFC, so 1D Airy beam is more appreciable to the accelerating of electron than other diffraction and diffraction-free beams. Moreover, the injection energy of electron plays an important role in determining the final energy gain. An initial fast electron can be captured by AFC, and a slow electron could be captured or reflected.

This paper is organized as follows. In Sec.2, we first present an analytical derivation of electromagnetic field of 1D Airy beam. Then, we analyze the characteristics of AFC in detail. In Sec.3, we study the electron dynamics in an intense 1D Airy beam, and the effect of the injection energy of electron on the final energy gain. Finally, conclusions are drawn in Sec.4.

2. The field components of 1D Airy beams

We start our analysis by considering the field components of 1D Airy beam. The laser beam adopted here polarizes along the x direction and propagates along the z axis. Its electromagnetic field can be derived from the vector potential $A = \hat{x} A_{0} ψ (x, z) \exp (i η),$ where $A_{0}$ is a constant amplitude, and $η = ω t - k z .$ The field envelope $ψ (x, z)$ is the solution of the paraxial equation of diffraction [11,12]

\frac{\partial^{2} ψ}{\partial s^{2}} + 4 i \frac{\partial ψ}{\partial ξ} = 0,

where,

s = x / x_{0}

represents a dimensionless transverse coordinate,

x_{0}

is an arbitrary transverse scale,

ξ = z / z_{r}

is a normalized propagation distance, the axis scale

z_{r} = k x_{0}^{2} / 2

is similar to Rayleigh range for Gaussian beam, and

k = 2 π / λ

is the wave number of optical wave. The evolution of a finite energy Airy beam, whose field profile at the origin is given by

ψ (s, ξ = 0) = A i (s) \exp (a s),

can be obtained by directly integrating Eq. (1) in closed form [11]:

ψ (s, ξ) = A i (A) \exp (B),

where,

A = s - {(ξ / 4)}^{2} + i a ξ / 2, B = a s - a ξ^{2} / 8 - i ξ^{3} / 96 + i a^{2} ξ / 4 + i s ξ / 4,

Ai represents the Airy function, and a is a positive parameter so as to ensure containment of the infinite Airy tail. Typically, for

a < < 1

the obtained wave packet closely resembles the Airy function, and, in the limit

a = 0

Eq. (2) reduces to the non-dispersive wave packet found in Ref. (10).

After getting vector potential, one can obtain the scalar potential $ϕ = i \nabla \cdot A / k$ by using the Lorentz gauge. Finally, the fields may be derived from $E = - i k A - \nabla ϕ$ and $B = \nabla \times A,$ and can be expressed as

E_{x} = - i E {C_{e x} A i (A) + D_{e x} A i' (A)} / (2 k^{2} x_{0}^{2}),

E_{y} = 0,

E_{z} = E {C_{e z} A i (A) + D_{e x} A i' (A)} / (4 k^{3} x_{0}^{3}),

B_{x} = 0,

B_{y} = i E {C_{b y} A i (A) + D_{b y} A i' (A)} / (4 k^{2} x_{0}^{2}),

B_{z} = 0,

where

A i' (A)

denotes the derivate of

A i (A) .

We define following parameters

E = E_{0} \exp (i (η + φ_{0})) \exp (B),

C_{e x} = 2 a^{2} + 2 i a ξ - ξ^{2} / 4 + 2 s + 2 k^{2} x_{0}^{2},

D_{e x} = 4 a + i ξ,

C_{e z} = 2 a^{3} + 9 i a^{2} ξ / 2 + a (- 3 ξ^{2} / 2 + 6 s - 4 k^{2} x_{0}^{2}) - i (2 i + ξ^{3} / 8 - 3 s ξ / 2 + ξ k^{2} x_{0}^{2}),

D_{e z} = 6 a^{2} + 4 i a ξ + 2 s - ξ^{2} / 2 - 4 k^{2} x_{0}^{2},

C_{b y} = 2 a^{2} + 2 i a ξ - ξ^{2} / 4 + 2 s - 4 k^{2} x_{0}^{2},

D_{b y} = 4 a + i ξ .

Furthermore, the field amplitude

E_{0} = k A_{0},

and

φ_{0}

is the constant phase.

In this section, we focus on the propagation characteristics of the electromagnetic fields of 1D Airy beam. It is clearly shown in Fig. 1 that the field components are quasi-diffraction-free and propagates along parabolic trajectories. The accelerating property deflects the main lobe of intensity toward the positive x direction, and reflection distance is well described by the theoretical relation $x_{d} ≅ z^{2} / 4 k^{2} x_{0}^{3}$ [12]. The corresponding cross-sections of the field profiles at various distances are shown in Fig. 2 . The field components are roughly divided into two regions. In the region near the propagation axis, about −10x ₀ to 10x ₀ in the transverse direction, the field components are asymmetric, and the oscillations of the asymmetric field languishes with the propagation distance. The region of asymmetric field forms a “channel” along the propagation axis, namely, “asymmetric field channel” (AFC). Outside AFC, the components of the field oscillate rapidly and become symmetrical gradually.

Fig. 1 (a)-(c) Propagation dynamics of the field components E _x, E _z, and B _y, respectively. Parameters used here are $a = 0.05,$ $λ = 1 μ m,$ and $x_{0} = 5 λ .$

Download Full Size | PDF

Fig. 2 (a)-(d) Cross sections of E _x at z_r, 10z_r, 30z_r, and 50z_r, respectively. (e)-(h) Cross sections of E _z at z_r, 10z_r, 30z_r, and 50z_r, respectively. And, (i)-(l) cross sections of B _y at z_r, 10z_r, 30z_r, and 50z_r, respectively. Other parameters are the same as those of Fig. 1.

Download Full Size | PDF

3. Electron dynamics in an intense Airy beam

Dynamics of the electron in a laser beam in vacuum is governed by the equations

\frac{d p}{d t} = - e (E + β \times B), \frac{d χ}{d t} = - e c β \cdot E,

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.

Electrons with different initial energies are injected and interact with an intense 1D Airy beam. The results are shown in Fig. 3 . The initial fast electron with the scaled injection energy $γ_{0} = 30$ is captured by the asymmetric field in AFC. The captured electron can remain in the acceleration phase of the laser field for a sufficiently long time, thereby obtaining considerable energy from the field. In the region z < 15z_r, the asymmetric field is intense and oscillates rapidly. The captured electron senses complicated accelerating fields and gains some energies. In the region 15z_r < z < 500z_r, the asymmetric field oscillates slowly. The captured electron senses continually accelerating fields and gains a majority of its final energy. And, in the region z > 500z_r, the field in AFC is very weak due to the dispersion of the beam intensity, and the electron can’t gain appreciable energy any more. The initial slow electron with $γ_{0} = 15$ is reflected by the asymmetric field in AFC, it senses intense asymmetric fields and gains a majority of its energy. Once the reflected electron moves outside AFC, it senses symmetric fields, where every accelerating half cycle is followed by an equally decelerating one, and gains no energy. Therefore, for the given parameters, the available accelerating distance of 1D Airy beam reaches about 500z_r. It is well known that, for a tightly focused Gaussian beam with the same width and peak intensity, the interaction of electron with laser beam is located near the focal region, 2-3 z_r, due to the serious diffraction of the beam [4]. Moreover, due to the existence of the continual accelerating phase in AFC, the 1D Airy beams are more beneficial to the accelerating of electron than other diffraction or quasi-diffraction-free beams.

Fig. 3 (Color online) (a) The trajectory, and (b) the energy gain of electron in an intense Airy laser beam. (c)-(e) The field components E _x, E _z, and B _y, respectively, sensed by the electron along its trajectory. (f)-(j) are small portions of (a)-(e), respectively. And, (k)-(o) are small portions of (f)-(j), respectively. Parameters used here are a = 0.05, λ = 1μm, x ₀ = 5λ, q = 20, φ ₀ = 0, the full interaction time ωt = 1 × 10⁶, the initial location of electron (x, z) = (0.1x ₀, 0), the injection angle of electron θ ₀ = 0, the black curves and the red curves are calculated with the initial injection energy γ ₀ = 15 and 30, respectively. Energy gain = (γ -γ ₀)mc ².

Download Full Size | PDF

The effect of the injection energy $γ_{0}$ of electron on the final energy gain is shown in Fig. 4 . For the given parameters, the initial fast electron with $γ_{0} \geq 19$ is always captured by AFC, and gains high energy. The initial fast electron can quickly escape the effects of the transverse field, and be accelerated by the longitudinal field. However, the initial slow electron with $γ_{0} < 19$ is sensitively affected by the transverse field, so that the trajectory of the slow electron is indeterminate, and it could be captured or reflected. Thus, the choice of the injection energy of electron is important for achieving high energy electrons.

Fig. 4 Variation of the energy gain with the scaled injection energy γ₀. Parameters used here are the same as those of Fig. 3.

Download Full Size | PDF

4. Conclusion

In conclusion, vacuum electron acceleration by 1D Airy beams is studied. The acceleration and quasi-diffraction-free characteristics of Airy beam lead to the formation of a long AFC along the propagation axis, which is beneficial to the accelerating of electron. Moreover, the injection energy of electron plays an important role in determining the final energy gain. For the given parameters, the initial fast electron with $γ_{0} \geq 19$ is always captured by AFC, and gains high energy.

Acknowledgement

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).

References and links

1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 55(6), 447–449 (1985). [CrossRef]

2. Y. Cheng and Z. Xu, “Vacuum laser acceleration by an ultrashort, high-intensity laser pulse with a sharp rising edge,” Appl. Phys. Lett. 74(15), 2116–2118 (1999). [CrossRef]

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

5. 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]

6. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

7. B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3539–3545 (1997). [CrossRef]

8. S. Liu, H. Guo, H. Tang, and M. Liu, “Direct acceleration of electrons using single Hermite-Gaussian beam and Bessel beam in vacuum,” Phys. Lett. A 324(2-3), 104–113 (2004). [CrossRef]

9. D. Li and K. Imasaki, “Vacuum laser-driven acceleration by a slits-truncated Bessel beam,” Appl. Phys. Lett. 86(3), 031110 (2005). [CrossRef]

10. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

11. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

12. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]

13. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]

Vacuum laser-driven acceleration by Airy beams

Abstract

1. Introduction

2. The field components of 1D Airy beams

3. Electron dynamics in an intense Airy beam

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (4)

Equations (17)

Optics Express