Electron acceleration driven by sub-cycle and single-cycle focused optical pulse with radially polarized electromagnetic field

Xunming Cai; Jingyun Zhao; Qiang Lin; Hong Tong; Jiangtao Liu

doi:10.1364/OE.26.030030

1. Introduction

Electron acceleration by ultrashort optical pulse is an interesting and important topic of research. Many works have been done on this topic [1–8]. The electron acceleration by few-cycle and sub-cycle laser pulses are also investigated [9–15]. In the most cases, the radial polarized optical beams were focused to obtain the optical field with strong axial electric field. Two methods are used to describe the radial polarized optical beams. One method is based on Lax series where the electric fields are expanded as series of diffraction angle $ε {=w}_{0} {/z}_{r}$ , where $w_{0}$ is the beam waist and $z_{r}$ is the Rayleigh range [11]. The other method is based on plane wave angular spectrum representation [16–18]. But for the sub-cycle pulses, the temporal and spatial parameters are coupled, which causes the special properties of sub-cycle pulse [19]. The expressions of sub-cycle focused optical pulse with radially polarized electromagnetic (EM) field should be deduced directly from Maxwell’s equation. The expressions of sub-cycle pulsed focused vector beams can be derived from Sink-Source Model [14,19–21]. Many new properties of sub-cycle pulse, such as the self-induced blue-shift of the central frequency have been found and proved by the experiment [22]. In our previous works, the expressions of sub-cycle pulsed focused beams with the transverse polarization of electric field on the beam waist plane are obtained. The electron can be accelerated to sub-GeV energies by the sub-cycle pulses when the peak intensity is $I =1 .077 \times 10^{21} W {cm}^{- 2}$ [10]. The expression of Poisson-like ultrashort pulse based on Sink-Source Model is given in [14]. The spectrum of the pulse can be reduced to a Gaussian function if its duration is sufficiently long. In this paper, both of the expressions of Lorentz type and Poisson type sub-cycle and single-cycle focused optical pulses with radially polarized EM field are derived from the Sink-Source Model. The acceleration of electrons by the two types of pulses is studied after accounting for the radiation-reaction force.

The process of radiation is fundamentally quantum. Such both of the electrons and the pulses should be processed by the quantum methods [14]. In this paper, the cases of sub-cycle laser pulses are considered. The space and time scales of pulse are extremely limited. Such classical methods can be good approximations. The electronic trajectories can also be conveniently studied.

The effect of the spin force on the acceleration of electrons is also researched by some papers [23,24]. For the sub-cycle pulse cases, the effects of the spin force are a wide range of issues and need to be discussed carefully in the future. In this paper, the topics are focused on the effect of the time-space structure of sub-cycle pulse on the classical radiation reaction force and the radiation reaction force on the motion of atom.

2. Model and equations

The expressions of sub-cycle and single-cycle focused pulses propagating along a certain direction can be deduced from dipole model using the complex sink-source source theory [14]. The complex sink-source source theory is traditionally considered to overcome the singularity problem on the beam waist plane of the expressions of the complex point source theory. But recent research indicates that the discontinuities still arise in phasors generated from the complex source-sink model for all odd orbital angular momentum modes [25]. In order to study the effect of different pulse configurations on the electron acceleration, the Lorentz type and Poisson type pulses are studied. The temporal shape of Poisson pulse vanishes more quickly at large time than that of the Lorentz pulse. The space-time coupling property of ultrashort pulse beams are discussed in the Ref [26]. and the Lorentz sub-cycle pulse is discussed in the Ref [27].

The expressions of sub-cycle focused pulse with radially polarized EM field can be deduced by using the Sink-Source method and the dipole moment model [14]. The dipole moment can be described as an oscillating dipole source located at the origin of the coordinate. The upper arrow indicates the vectors and the upper dot indicates derivative.

P (r, t) = p_{0} f (t) \vec{e_{z}} δ (\vec{r})

_{$f (t)$}can be written as

T / (T - i t) \cdot e^{i ω_{0} t + i ϕ_{0}}

for the Lorentz type pulse and

1/(1 - i ω_{0} t {/s)}^{s+1} \cdot e^{i ϕ_{0}}

for the Poisson like pulse.

T

is the pulse width parameter,

ω_{0}

is the carrier frequency and

s

is a real positive parameter.

The focused pulse propagating in the z direction can be obtained by introducing the spatiotemporal translation

z \to z = z^{'} + i a, t \to t^{'} = t - t_{0} + i \frac{a}{c}

where $a = 1 \sqrt{{(1 + k^{2} w_{0}^{2} / 2)}^{2} - 1} / k$ . $k$ is the wave vector. $w_{0}$ is the waist spot size, and is a constant value in the simulation. The carrier frequency $ω_{0}$ , together with the waist spot size, determines the parameter $a$ . The complex distance is introduced as

R^{'} = \sqrt{x^{2} + y^{2} + {(z + i a)}^{2}}

According to the Sink-Source method, the optical field can be obtained by folding the source with the

δ

function,

D (R^{'}, t^{'}) = \frac{c^{2} μ_{0}}{4 π} \frac{δ (t^{'} - R^{'} / c) - δ (t^{'} + R^{'} / c)}{R^{'}}

The complex retarded times are introduced as

τ^{'} = t' - R' / c

and

τ^{''} = t' + R' / c

. Such time domain pulse functions

f (t)

can be written as

f_{1} (τ')

and

f_{2} (τ'')

.The detailed derivation process of the expressions of electromagnetic field can be found in the Ref [14]. and they can be written as

\begin{array}{l} \vec{E_{x_{1}}} = \frac{c^{2} μ_{0} p_{0}}{4 π} \cdot r e a l {\frac{z^{'}}{R^{'}^{2}} \cdot (\frac{{\overset{..}{f}}_{1} - {\overset{..}{f}}_{2}}{c^{2} R^{'}} + \frac{3 ({\dot{f}}_{1} + {\dot{f}}_{2})}{c R^{'}^{2}} + \frac{3 ({\overset{}{f}}_{1} - {\overset{}{f}}_{2})}{R^{'}^{3}})} \cdot x \cdot \vec{e_{x}} \\ \vec{E_{y_{1}}} = \frac{c^{2} μ_{0} p_{0}}{4 π} \cdot r e a l {\frac{z^{'}}{R^{'}^{2}} \cdot {\frac{{\overset{..}{f}}_{1} - {\overset{..}{f}}_{2}}{c^{2} R^{'}} + \frac{3 ({\dot{f}}_{1} + {\dot{f}}_{2})}{c R^{'}^{2}} + \frac{3 ({\overset{}{f}}_{1} - {\overset{}{f}}_{2})}{R^{'}^{3}}}} \cdot y \cdot \vec{e_{y}} \\ \vec{E_{z_{1}}} = \frac{c^{2} μ_{0} p_{0}}{4 π} \cdot r e a l {\frac{z^{'}^{2}}{R^{'}^{2}} \cdot [\frac{{\overset{..}{f}}_{1} - {\overset{..}{f}}_{2}}{c^{2} R^{'}} + \frac{3 ({\dot{f}}_{1} + {\dot{f}}_{2})}{c R^{'}^{2}} + \frac{3 ({\overset{}{f}}_{1} - {\overset{}{f}}_{2})}{R^{'}^{3}}] \\ - [\frac{{\overset{..}{f}}_{1} - {\overset{..}{f}}_{2}}{c^{2} R^{'}} + \frac{({\dot{f}}_{1} + {\dot{f}}_{2})}{c R^{'}^{2}} + \frac{({\overset{}{f}}_{1} - {\overset{}{f}}_{2})}{R^{'}^{3}}]} \cdot \vec{e_{z}} \\ \vec{B_{x}} = - \frac{c μ_{0} y p_{0}}{4 π} \cdot r e a l {(\frac{{\overset{..}{f}}_{1} + {\overset{..}{f}}_{2}}{c^{2} R^{'}^{2}} + \frac{({\dot{f}}_{1} - {\dot{f}}_{2})}{c R^{'}^{3}})} \vec{e_{x}} \\ \vec{B_{y}} = \frac{c μ_{0} x p_{0}}{4 π} \cdot r e a l {(\frac{{\overset{..}{f}}_{1} + {\overset{..}{f}}_{2}}{c^{2} R^{'}^{2}} + \frac{({\dot{f}}_{1} - {\dot{f}}_{2})}{c R^{'}^{3}})} \vec{e_{y}} \end{array}

To remove

p_{0}

from the expression of electromagnetic field, the peak value of the electric field is assumed to be

E_{0}

. The peak value of the electric field appears at the position

x, y, z = 0

and in the time

t - t_{0} = 0

can be written as

E_{0} = \max {E_{z}} |_{x, y, z = 0}

Such expressions of

E_{0}

can be deduced as

E_{0} = p_{0} \cdot \frac{c^{2} μ_{0}}{4 π} [\frac{2}{{(i a)}^{2}} (\frac{{\dot{f}}_{1} (0) + {\dot{f}}_{2} (2 i a / c)}{c} + \frac{(f_{1} (0) - f_{2} (2 i a / c)}{i a})]

Then

p_{0}

can be replaced by

E_{0}

.

The electron acceleration by the Lorentzian focused sub-cycle pulse can be studied by solving the relativistic Newton-Lorentz equation. After the radiation-reaction force of electrons is considered, the modified relativistic Newton-Lorentz equation is given as:

\frac{d \vec{β}}{d t} = \frac{1}{γ m c} [e (\vec{β} (\vec{β} \cdot \vec{E}) - (\vec{E} + c \vec{β} \times \vec{B})) + \vec{f}]

where

β

is the velocity of particle scaled by

c

. The classical radiation reaction model is discussed in Ref [28]. In this paper, the Landau-Lifshitz equation is selected because it is concise and clear, and is convenient for numerical calculations [29]. The component equations of Eq. (17) and the displacement equations can be written as

\begin{array}{l} \frac{d \vec{β_{x}}}{d t} = \frac{e}{γ m c} [\vec{β_{x}} (β_{x} E_{x} + β_{y} E_{y} + β_{z} E_{z}) - (\vec{E_{x}} + c (β_{y} B_{z} - β_{z} B_{y}) \vec{e_{x}})] + \frac{\vec{f_{x}}}{γ m c} \\ \frac{d \vec{β_{y}}}{d t} = \frac{e}{γ m c} [\vec{β_{y}} (β_{x} E_{x} + β_{y} E_{y} + β_{z} E_{z}) - (\vec{E_{y}} + c (β_{z} B_{x} - β_{x} B_{z}) \vec{e_{y}})] + \frac{\vec{f_{y}}}{γ m c} \\ \frac{d \vec{β_{z}}}{d t} = \frac{e}{γ m c} [\vec{β_{z}} (β_{x} E_{x} + β_{y} E_{y} + β_{z} E_{z}) - (\vec{E_{z}} + c (β_{x} B_{y} - β_{y} B_{x}) \vec{e_{z}})] + \frac{\vec{f_{z}}}{γ m c} \\ \frac{d \vec{x}}{d t} = c \vec{β x}, \frac{d \vec{y}}{d t} = c \vec{β y}, \frac{d \vec{z}}{d t} = c \vec{β z} \end{array}

In the SI system, the general form of the radiation reaction term can be written as:

\begin{array}{l} \frac{\vec{f}}{γ m c} = \frac{1}{3 \times 10^{7}} \cdot \frac{2 e^{3}}{m^{2}} {\frac{1}{c} [\frac{\partial}{c \partial t} + (\vec{β} \cdot \vec{\nabla})] \vec{E} + [\vec{β} \times (\frac{\partial}{c \partial t} + (\vec{β} \cdot \vec{\nabla})) \vec{B}]} \\ + \frac{1}{3 \times 10^{7}} \cdot \frac{2 e^{4}}{m^{3} γ c} {\frac{\vec{E} \times \vec{B}}{c} + [\vec{B} \times (\vec{B} \times \vec{β})] + \frac{\vec{E}}{c^{2}} (\vec{β} \cdot \vec{E})} \\ - \frac{1}{3 \times 10^{7}} \cdot \frac{2 e^{4} \vec{β} γ}{m^{3} c} {{(\frac{\vec{E}}{c} + \vec{β} \times \vec{B})}^{2} - \frac{1}{c^{2}} {(\vec{β} \cdot \vec{E})}^{2}} \end{array}

Equation (8) can be solved by some high precision numerical algorithms. The initial speed of the electron is

β_{0}

and the initial kinetic energy of the electron is

K_{0} = (γ_{0} - 1) m c^{2}

.

The radiation losses in the process of electron acceleration can be calculated from the Lienard-Wiechert potential. The radiation power of an electron in the acceleration process can be written as

\begin{array}{l} P (t^{'}) = \frac{e^{2}}{16 π^{2} ε_{0} c^{3}} \oint \frac{| \vec{e_{r}} \times [(\vec{e_{r}} - \vec{v} / c) \times \dot{\vec{v}}] |^{2}}{{(1 - \vec{e_{r}} \cdot \vec{v} / c)}^{5}} d Ω \\ = \frac{e^{2}}{16 π^{2} ε_{0} c^{3}} \oint \frac{{[a_{θ} - (a_{θ} \cdot v_{r} - a_{r} \cdot v_{θ}) / c]}^{2} + {[a_{ϕ} - (a_{ϕ} \cdot v_{r} - a_{r} \cdot v_{ϕ}) / c]}^{2}}{{(1 - v_{r} / c)}^{5}} \sin θ d θ d ϕ \end{array}

the radiation energy loss can be written as

W = \int_{t_{0}}^{t_{e n d}} P (t^{'}) d t^{'}

. The velocity and the acceleration of an electron in the spherical coordinate system can be written as

\begin{array}{l} \vec{a_{r}} = \vec{a_{x}} \sin θ \cos ϕ + \vec{a_{y}} \sin θ \sin ϕ + \vec{a_{z}} \cos θ \\ \vec{a_{θ}} = \vec{a_{x}} \cos θ \cos ϕ + \vec{a_{y}} \cos θ \sin ϕ - \vec{a_{z}} \sin θ \\ \vec{a_{ϕ}} = - \vec{a_{x}} \sin ϕ + \vec{a_{y}} \cos ϕ \end{array}

where

\vec{v_{x}} = c \vec{β_{x}}, \vec{v_{y}} = c \vec{β_{y}}, \vec{v_{z}} = c \vec{β_{z}}, \vec{a_{x}} = d \vec{v_{x}} / d t, \vec{a_{y}} = d \vec{v_{y}} / d t, \vec{a_{z}} = d \vec{v_{z}} / d t

. The expressions of

\vec{v_{r}}

are the same forms.

In the numerical computations, we use the fifth-order prediction-correction Runge-Kutta algorithm. In order to avoid too long calculation time, the algorithm must be adaptive variable step. We use the matlab function ode45 to do the numerical computations. In order to ensure the accuracy of the result, the relative error parameter ‘RelTol’ is set to $2 .25 \times 10^{-14}$ and the absolute error parameter ‘AbsTol’ is set to $1 \times 10^{-21}$ . The numerical results can be confirmed by our self-written fifth-order Runge-Kutta Cash-Karp algorithm. The computing speed of ode45 is faster.

3. Space-time characteristics of sub-cycle and single-cycle pulses

The space-time characteristics in the center of the non-paraxial pulse field can be analyzed by setting $x, y, z = 0$ . The temporal waveform, power and the frequency spectrum of a 0.45-cycle Poisson-like pulse are shown in Fig. 1. The frequency $ω_{0}$ is $2 .355 \times 10^{15} H z$ and the period $T_{0}$ is $2 .6685 \times 10^{-15} s$ . The parameter $s$ is 4.12.

Fig. 1 (a) The 0.45-cycle waveforms versus time. (b) The frequency spectrum of the pulse.

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From Fig. 1(a), we can define the number of cycles of the pulse by the ratio of the full width at half maximum(FWHM) of the power to the period $T_{0}$ . Figure 1(b) shows the self-induced blueshift of the center frequency of spectrum in the center of the pulse field. The center frequency of spectrum is $2 .95 \times 10^{15} H z$ in Fig. 1(b), which is much higher than $ω_{0}$ . The reason is that the sub-cycle pulse has an ultra-wide spectrum. For the tightly focused sub-cycle radially polarized EM field, there are strong axial electric field distributions on the waist plane. According to the principle of the Rayleigh diffraction limit, the focus spot of the high frequency component is much smaller than that of the low frequency component on the waist plane. Figure 2 is given to clarify this question.

Fig. 2 The power distribution of the focus spots of different wavelength components of the tightly focused sub-cycle radially polarized EM field on the waist plane.

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From Fig. 2, one can see that the focus spots of the short wavelength components are far smaller than that of the long wavelength components. Therefore, the high-frequency components are more concentrated in the center of the pulse than that of the low frequency component. That is called as the central frequency blue shift of pulse. This has a significant impact on the acceleration of electrons due to the large electric field gradient in the center of the pulse.

4. Results and discussions

The initial electron velocity components in the $y$ direction are $0 .01c$ and $0 .1c$ . The initial velocities can be easily prepared by a simple electron accelerator. The choice of the initial velocities has no effect on the main conclusions in the paper. These cases that the electrons initially enter the light beam at a wide range angle are considered. If the time of the pulse peak is zero, the initial interaction time $t_{0}$ is negative $15fs$ for the 0.45-cycle pulse and negative $40fs$ for the 1.37-cycle pulse. Zero time is the time when the pulse center arrives. The initial positions of the electron are $x = 0, y = v_{0} \cdot t_{0}, z = y / \tan θ_{0}$ , where $v_{0}$ is the initial velocity component in the $y$ direction and $θ_{0}$ is the initial incidence angle. The initial electron velocity is $v_{0} /sin θ_{0}$ .The frequency $ω_{0}$ is $2 .355 \times 10^{15} H z$ . The peak intensity of pulse is $I =1 .077 \times 10^{21} W {cm}^{- 2}$ . For reading, the different input parameters of most figures are summarized in the Table 1.

Table 1. Parameters of Figures

View Table

In Table 1, c indicates the velocity of light and fs indicates femtosecond.

4.1 The 0.45-cycle pulse case

For the Poisson type 0.45-cycle pulse, the exit kinetic energy of the electron versus the carrier-envelope phase (CEP) and the beam waist is shown in Fig. 3. The initial positions of an electron are $x = 0, y = v_{0} \cdot t_{0}, z = y / \tan θ_{0}$ and the velocity $v_{0}$ is $0.01 c$ . The incidence angle $θ_{0}$ is 10 degrees in Fig. 3.

Fig. 3 (a), (b) The exit kinetic energy of the electron versus the CEP and the beam waist. The radiation-reaction force is considered in (a) and is neglected in (b). (c) The kinetic energy of the electron versus time, the CEP and beam waist of the pulse are $0. 62 π$ and $15 .1 μ m$ .(d) Trajectory of the electron, the parameters are the same as (c).

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Figure 3 shows that there is a big difference for the kinetic energy of electrons when the radiation-reaction effect is considered in Fig. 3(a) and is neglected in Fig. 3(b). The ideal working point in the waist-CEP plane is modified when the radiation reaction force is considered. In Fig. 3(a), the maximal exit kinetic energy gain of an electron is $9.269 G e V$ when the CEP is $0.62 π$ and the beam waist is $15.1 μ m$ . In Fig. 3(b), the maximal exit kinetic energy gain of an electron is $37 .36 G e V$ when the CEP is $1 .7 π$ and the beam waist is $15.1 μ m$ . This can be explained from Fig. 3(c)-3(d). In order to clearly demonstrate the effects of radiation reaction, only the case of the initial acceleration phase is shown in Fig. 3(d). The significant differences of kinetic energy of electrons occur from the moment of 600 femtoseconds. When the electrons enter the center of the laser pulse, the strong electric field gradients cause significant acceleration of electrons, the radiation reaction force also increases significantly at the same time. The cumulative effect of the radiation reaction force causes that the electron can’t always stay in the acceleration phase [30]. From the embedded graph of Fig. 3(c) and Fig. 3(d), one can see that the electron experiences more acceleration and deceleration processes when the radiation reaction force is considered than that the radiation reaction force is neglected. The self-induced blue shift of the frequency in the center of the sub-cycle pulse field significantly increases the effect of the radiation reaction force on the electron’s motion. From Fig. 3(d), one can see that the electron leaves the beam with a larger ejection angle when the radiation reaction force is considered than that is neglected. This indicates that the electron leaves the beam in a relatively short period of time. The electron can obtain more than $7 G e V$ exit kinetic energy gain in the ranges of CEP varying from $0.3 8 π$ to $0.7 6 π$ and beam waist varying from $12 .1 μ m$ to $15.1 μ m$ when the radiation reaction force is considered. This indicates that the electron can obtain the high kinetic energy gain in the wide range of the beam waist and CEP parameters. The parameter ranges in which the electron obtains high exit kinetic energy are far smaller in the CEP interval of $π$ to $2 π$ than that of 0 to $π$ . This indicates that the initial direction of the electromagnetic field plays an important role in the electron acceleration. The radiation loss of the acceleration electron in the case of Fig. 3(c) is $2 .25 \times 10^{6} e V$ .

The exit kinetic energy of the electron versus the incidence angle for the Poisson type 0.45-cycle pulse is shown in Fig. 4. The exit kinetic energy of the electron for the case of the Lorentz type 0.45-cycle pulse is also shown in Fig. 4. The peak intensities of the two types of pulses are $I =1 .077 \times 10^{21} W {cm}^{- 2}$ .

Fig. 4 The exit kinetic energy of electrons for the 0.45-cycle pulse. (a), (b)The case of the Poisson type pulse. The incidence angle of electrons is 90 degrees in (a). (c), (d) The case of the Lorentz type pulse. The incidence angle of electrons is 10 degrees in (c). (b), (d)The maximal kinetic energy of an electron versus the incidence angle.

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Figure 4(a) shows the exit kinetic energy of the electron versus the beam waist and CEP for the initial incident angle of 90 degrees. Comparing Fig. 3(a) and Fig. 4(a), one can see that there are only some minor changes for the ideal working point when the incident angle changes. From Fig. 4(b), we can see that there are only some little changes for the maximal kinetic energy of an electron when the incident angle varies from 10 degrees to 90 degrees. This shows that the maximal kinetic energy gain of electrons is robust against the variation of the incident angles. The reason is that the Poisson type sub-cycle pulses have large time domain electric field gradients and the fringe electric fields disappear quickly. Such an electron can enter the center of the pulse field and interacts with the peak field. In Fig. 4(c), the maximal exit kinetic energy gain of an electron is $4 .596 G e V$ when the CEP is $0. 98 π$ and the beam waist is $15.1 μ m$ . The parameter ranges to win a high kinetic gain are also much smaller in Fig. 4(c) than that in Fig. 4(a). Figure 4(d) also shows that the exit kinetic energy of electrons will decrease as the incidence angle increases. The reason is that the electric field amplitude of the Lorentz sub-cycle pulse decreases slowly with time. Such electrons are easily driven away from the beam by the fringe electric field of the pulse. The exit kinetic energy of electrons is also sensitive to the Lorentz sub-cycle pulse parameters.

Figure 4 also shows that the electron can obtain more than 9 GeV energy gain in a very wide range of incident angles by the Poisson sub-cycle pulse when the peak intensity is $I =1 .077 \times 10^{21} W {cm}^{- 2}$ , which is far higher than that of the Lorentz sub-cycle pulse in the same peak intensity of light. Compared with the case of few-cycle laser pulse accelerating electrons, the gain of kinetic energy of electrons is a few times higher and the corresponding peak optical power is one order of magnitude lower in the case of the sub-cycle laser pulses accelerating electrons [13]. Such a Poisson type sub-cycle pulse is more suitable for accelerating electrons than a Lorentz type sub-cycle pulse.

The case that $v_{0} = 0.1 c$ is shown in Fig. 5. The temporal configuration of the pulse is Poisson type.

Fig. 5 (a), (b) The exit kinetic energy of the electron versus the CEP and the beam waist. The incidence angle of electrons is 10 degrees in (a) and is 90 degrees in (b). (c) The maximal kinetic energy of the electron versus the incidence angle.

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The maximal exit kinetic energy gain of electrons is $9.152 G e V$ in Fig. 5(a). Figure 5 shows that the exit kinetic energy of the electron is robust against the variation of the incident angles. From Fig. 5(a), one can see that the electron can obtain more than $7 G e V$ exit kinetic energy gain in the wide ranges of CEP varying from $0. 41 π$ to $0.7 1 π$ and beam waist varying from $12 .8 μ m$ to $15.1 μ m$ . Figure 5(c) shows that the maximal kinetic energy of an electron remains approximately constant when the incident angle varies from 10 degrees to 90 degrees.

4.2 The 1.37-cycle pulse case

The exit kinetic energy of the electron versus the CEP and the beam waist is shown in Fig. 6 for the Poisson type 1.37-cycle pulse. $v_{0}$ is $0.01 c$ and the parameter $s$ is 28.4.

Fig. 6 (a), (c) The exit kinetic energy of the electron versus the CEP and the beam waist. The incidence angle of electrons is 10 degrees in (a) and is 90 degrees in (c).(b) The kinetic energy of the electron versus time, the CEP and beam waist of the pulse are $0. 32 π$ and $15 .1 μ m$ .(d) The maximal kinetic energy of the electron versus the incidence angle.

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The maximal exit kinetic energy gain of electrons is $8 .107 G e V$ in Fig. 6(a), which is lower than the maximal kinetic energy in Fig. 5(a). This shows that the sub-cycle pulses are more suitable for accelerating electrons than the single cycle pulses. The reason is that the electrons are easier to be driven away from the beam by the fringe electric field of the pulse if the pulse duration is shorter. The electron can obtain more than $7 G e V$ exit kinetic energy gain in the wide ranges of CEP varying from $0. 31 π$ to $π$ and beam waist varying from $14 .1 μ m$ to $15.1 μ m$ .The maximal kinetic energy gain of electrons is also robust against the variation of the incident angles. The radiation loss of the acceleration electron in the case of Fig. 6(b) is $1 .42 \times 10^{6} e V$ .

In Fig. 7, $v_{0}$ is $0.1 c$ .The temporal configuration of the pulse is Poisson type.

Fig. 7 (a), (b) The exit kinetic energy of the electron versus the CEP and the beam waist. The incidence angle of electrons is 10 degrees in (a) and is 90 degrees in (b). (c) The maximal kinetic energy of the electron versus the incidence angle.

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Compared with Fig. 6, the initial velocity of electrons is higher in Fig. 7. We can see that the parameter ranges for obtaining the high exit kinetic energy gain is wider in Fig. 7(b) than that in Fig. 7(a). If there is a large angle between the direction of the velocity of electrons and the propagation velocity of the beam, electrons easily enter the center of the pulse and are not easily driven away by the fringe electric field of the pulse. The maximal exit kinetic energy gain of electrons is $8 .187 G e V$ in Fig. 7(b). Figure 7(c) shows that the maximal kinetic energy gain of electrons is also robust against the variation of the incident angles. Figure 5 and Fig. 7 show that the electron can obtain more net exit kinetic energy gain with shorter FWHM of pulses.

4.3 The different initial positions and misaligned cases

In this section, these cases that the electrons are initially located at different locations are studied. Two cases are considered. The first case is that the initial interaction time $t_{0}$ is changed to negative $60fs$ for the 0.45-cycle pulse and negative $80fs$ for the1.37-cycle pulse. The second case is that the initial positions of electrons are changed to $x = 0, y = 2 \cdot v_{0} \cdot t_{0}, z = y / (2 \cdot \tan θ_{0})$ for both the 0.45-cycle and 1.37-cycle pulses. The incidence angle $θ_{0}$ is 10 degrees. The first case is shown in Fig. 8.

Fig. 8 The exit kinetic energy of the electron versus the CEP and the beam waist. (a), (b) The 0.45-cycle pulse cases. (c), (d)The 1.37-cycle pulse cases. (a), (c) $v_{0} = 0.01 c$ . (b), (d) $v_{0} = 0.1 c$ _{$v_{0}$} is $0.01 c$ in (a), (c) and is $0.1 c$ in (b), (d). Figure 8 shows that the electron acceleration cases are almost the same as Fig. 3-7. This illustrates that the initial different interaction times $t_{0}$ have little effect on the electron acceleration. The reason can be attributed to the ultra-short duration of the sub-cycle and single-cycle pulses and the fringe electric fields of the pulses disappear quickly. The second case is shown in Fig. 9. The incidence angle $θ_{0}$ is 80 degrees.

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The maximal exit kinetic energy gain of electrons is $9.41 G e V$ in Fig. 9(a) and $8.378 G e V$ in Fig. 9(b). Figure 9 shows that the electron can obtain the high exit kinetic energy in the wide CEP and beam waist parameter ranges. This indicates that with some specific CEP and beam waist parameters, the electron can enter the capture acceleration channel when interacting with the fringe electric fields. Such an electron can meet the center of the pulse field and obtain a high exit kinetic energy gain.

Fig. 9 The exit kinetic energy of the electron versus the CEP and the beam waist, $v_{0} = 0.1 c$ , $θ_{0} {=80}^{o}$ . (a) The 0.45-cycle pulse cases. (b) The 1.37-cycle pulse cases.

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The low velocity electrons can be easily prepared by a simple, commonly used electron accelerator. In the experiment, the most concern is the alignment of electrons and laser pulses. The alignment means that the electrons and the center of pulse will arrive at the same point at the same time if the interaction of electrons with pulses isn’t considered. In this paper, the analysis points out that this is not necessarily required. The arrival time can be staggered by dozens of femtoseconds. If the electron is captured by the edge field of pulse, the electron has the opportunity to interact with the center field of pulse and obtain a high exit kinetic energy.

Sub-cycle laser pulses can already be produced in the laboratory by the synthetic methods [31]. With the development of technology, sub-cycle laser pulses will be more convenient to produced and used for the electron acceleration experiments.

5. Conclusions

Using the Poisson-like and Lorentz type sub-cycle and single cycle pulse models, the electrons acceleration problem is studied. The space-time characteristics of sub-cycle and single-cycle pulses are found to have an important impact on the electrons acceleration. Compared with the Lorentz type pulse, the Poisson-like pulses can accelerate electrons to the high exit kinetic energy states in the wide CEP and beam waist parameter ranges. The maximal kinetic energy gain of electrons is robust against the variation of the incident angles for the Poisson-like pulse cases. The electron can obtain more net exit kinetic energy gain with the shorter FWHM of pulses. Compared with the few-cycle laser pulse accelerating electrons case, the gain of kinetic energy obtained by electrons is a few times higher and the corresponding peak optical power is one order of magnitude lower in the sub-cycle laser pulses accelerating electrons case. The radiation-reaction effect is also important in the sub-cycle and single cycle laser pulses accelerating electrons case.

Funding

National Natural Science Foundation of China (61727821, 61475139, 11564005, 11764008); Key Project for Innovation Research Groups of Guizhou Provincial Department of Education, China (KY[2016]030 and 028, KY[2017]035). The Thousand Levels of Innovative Talents of Guizhou Province ([2016]016).

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Figure	3(a)	3(b)	4(a)	4(c)	5(a)	5(b)	6(a)	6(c)	7(a)	7(b)	8(a)	8(b)	8(c)	8(d)	9(a)	9(b)
Initialy	−0.15·c·fs	−0.15·c·fs	−0.15·c·fs	−0.15·c·fs	−1.5·c·fs	−1.5·c·fs	−0.4·c·fs	−0.4·c·fs	−4·c·fs	−4·c·fs	−0.6·c·fs	−6·c·fs	−0.8·c·fs	−8·c·fs	−3·c·fs	−8·c·fs
Initial z	y/tan(10°)	0	0	y/tan(10°)	y/tan(10°)	0	y/tan(10°)	0	y/tan(10°)	0	y/tan(10°)	y/tan(10°)	y/tan(10°)	y/tan(10°)	y/(2·tan(80°))	y/(2·tan(80°))
angle	10°	90°	90°	10°	10°	90°	10°	90°	10°	90°	10°	10°	10°	10°	80°	80°
Initial velocity v_y	0.01c	0.01c	0.01c	0.01c	0.1c	0.1c	0.01c	0.01c	0.1c	0.1c	0.01c	0.1c	0.01c	0.1c	0.1c	0.1c
Initial velocity v_z	v_y/tan(10°)	0	0	v_y/tan(10°)	v_y/tan(10°)	0	v_y/tan(10°)	0	v_y/tan(10°)	0	v_y/tan(10°)	v_y/tan(10°)	v_y/tan(10°)	v_y/tan(10°)	v_y/tan(80°)	v_y/tan(80°)

Electron acceleration driven by sub-cycle and single-cycle focused optical pulse with radially polarized electromagnetic field

Abstract

1. Introduction

2. Model and equations

3. Space-time characteristics of sub-cycle and single-cycle pulses

4. Results and discussions

4.1 The 0.45-cycle pulse case

4.2 The 1.37-cycle pulse case

4.3 The different initial positions and misaligned cases

5. Conclusions

Funding

References

Cited By

Figures (9)

Tables (1)

Equations (12)

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