Acceleration of electrons by tightly focused azimuthally polarized ultrashort pulses in a vacuum

Yali Zheng; Xunming Cai; Xin Zhao; Wei Wang

doi:10.1364/OE.448464

1. Introduction

As the special focusing characteristics of the azimuthally polarized and radially polarized ultrashort pulses [1–5], the two polarized beams are widely used in the research of particle trapping and particle acceleration [6–12]. These beams are also widely used in the fields of medicine, biology and chemistry [13–16]. The radially polarized light has the high efficiency in capturing particles in the longitudinal direction and the azimuthally polarized light has the high capture efficiency in the transverse direction [17,18]. Many models of accelerating electrons by laser pulses in vacuum and plasma have been proposed [19–21]. High kinetic energy of electrons can be obtained by accelerating electrons with laser pulses in plasma, but the collision between electron and plasma, the plasma uniformity and other problems need to be dealt with [22]. Accelerating electrons by few-cycle or even sub-cycle pulses in vacuum is also a hot research issue [23–28].

The exact solution of the Helmholtz equation for non-paraxial beams can be obtained by CSSM [29–32]. Using CSSM and HPM, the space-time expression of the electromagnetic field that strictly satisfies Maxwell's equations can be derived, and the expression of azimuthally polarized pulse (TE) can be obtained [32]. The Richards-Wolf diffraction integral theory can also be used to derive the vector wave equations of tightly focused beams [1,33–34]. The time part and the space part of the ultrashort pulse expression derived from Maxwell's equations using CSSM are mutually coupled, which also reflects the physical characteristics of the spatiotemporal coupling of ultrashort laser pulses [35–39].

Radially polarized ultrashort laser pulses have been proven to have high efficiency in the electron acceleration [8,40]. The field of a focused radially-polarized light pulse is purely longitudinal on-axis and the off-axis fields can act to contain a pulse while it is accelerated. Some theoretical and experimental studies have shown that the focused radially-polarized light pulse is well-suited to accelerate a directed electron beam [9,11,41]. At present, as far as we know, the theoretical and experimental studies on the acceleration of electrons by the azimuthally polarized pulse to obtain a good directional electron beam are rare. Here we show that it is possible to obtain a low emittance and directedness high-energy electron beam using the azimuthally polarized pulses. There is a strong longitudinal magnetic field distribution in the focus center of the azimuthally polarized laser pulse, which makes the movement of electrons in the lateral direction strongly constrained when the electrons are accelerated by the azimuthally polarized pulse. So, the movement range of electrons is restricted. Thus, it is possible to obtain a fairly uniform directional electron beam by the azimuthally polarized pulsed beam. In this paper, the relativistic dynamic equations of electron in the laser field are solved numerically, and the interaction of tightly focused sub-cycle and few-cycle azimuthally polarized beams with electrons is analyzed. It is found that the electrons can obtain sub-$GeV$ kinetic energy gain by both the sub-cycle and the few-cycle azimuthally polarized pulses, and the electrons can be confined by the pulses in the lateral direction for hundreds of femtoseconds. The variation of the radiation spectrum of electron acceleration with the peak electric field parameter ${E_0}$ of the pulse is also analyzed. The results show that the characteristics of the radiation spectrum are affected by both of the laser intensity and the pulse time domain width.

2. Theoretical analysis

In order to derive the electromagnetic field expressions of azimuthally polarized tightly focused ultrashort pulses, a pulsed oscillating magnetic dipole is introduced [23–24,35] as

(1)$$\overrightarrow P (\overrightarrow r ,t) = {p_0}f(t)\delta (\overrightarrow r )\overrightarrow {{e_\textrm{z}}} .$$

In the formula, $f(t) = {(1 - i{\omega _0}t/s)^{ - (s + 1)}} \cdot {e^{i{\phi _0}}}$ is the Poisson pulse function [24,32], which is used to represent the pulse oscillation of the magnetic dipole in the time domain. ${\omega _0}$ is the carrier frequency, $s$ is a real parameter, ${\phi _0}$ is the initial phase of the oscillation, and the peak value of the dipole moment ${p_0}$ determines the peak power of the pulse. The pulse expressions can be derived from the CSSM [32,42]. The space-time transformation is introduced as

(2)$$z \to z^{\prime} = z + ia, t \to {t^{\prime}}\textrm{ = }t - {t_0}\textrm{ + }{{\textrm{i}a} / c}.$$

In the above formula, $a = 1\sqrt {{{(1 + {k^2}{w_0}^2/2)}^2} - 1} /k$ is the confocal parameter and is determined by the beam waist ${w_0}$. $k$ is the wave vector. The complex distance $R^{\prime} = \sqrt {{x^2} + {y^2} + {{(z + ia)}^2}}$ and the complex time $\tau ^{\prime} = t^{\prime} - R^{\prime}/c$, $\tau ^{\prime\prime} = t^{\prime} + R^{\prime}/c$ are introduced. The expressions of the electric and magnetic Hertz vectors in the CSSM are derived by folding the following $\delta$ function with the source,

(3)$$D(R^{\prime},t^{\prime}) = \frac{{{c^2}{p_0}{u_0}}}{{4\pi }}\frac{{\delta (t^{\prime} - R^{\prime}/c) - \delta (t^{\prime} + R^{\prime}/c)}}{{R^{\prime}}}.$$

The singularities in the field expressions are eliminated in this way. According to the method of using HPM to generate TE pulse in the literature [32], the following expressions for the electric hertz vector and the magnetic hertz vector of the azimuthally polarized pulse can be obtained as

(4)$$\overrightarrow {{\Pi _e}} = 0,\textrm{ }\overrightarrow {{\Pi _m}} = \eta _0^{ - 1}D(R^{\prime},t^{\prime})\overrightarrow {{\textrm{e}_z}} .$$

In the above formula, ${\eta _0} = {({\mu _0}/{\varepsilon _0})^{1/2}}$ is the inherent impedance in vacuum. ${\mu _0}$ and ${\varepsilon _0}$ represent the permeability and the permittivity in vacuum. Substituting Eq. (4) into the following equation, the expressions of TE pulse can be obtained as

(5)$$\left\{ \begin{array}{l} \overrightarrow E = \nabla \times \nabla \times \overrightarrow {{\Pi _e}} - {\mu_0}\frac{\partial }{{\partial \textrm{t}}}\nabla \times \overrightarrow {{\Pi _m}} \\ \overrightarrow H = \nabla \times \nabla \times \overrightarrow {{\Pi _m}} + {\varepsilon_0}\frac{\partial }{{\partial \textrm{t}}}\nabla \times \overrightarrow {{\Pi _e}} \end{array} \right..$$

By solving the above equations, the expressions of TE pulse can be written as:

(6)$$\begin{array}{l} \overrightarrow {{E_x}} = \frac{{c{u_0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{y}{{R^{\prime}}} \cdot \left[ {\frac{{({{\dot{f}}_1} - {{\dot{f}}_2})}}{{{{R^{\prime}}^2}}} + \frac{{({{\ddot{f}}_1} + {{\ddot{f}}_2})}}{{cR^{\prime}}}} \right]} \right\}\overrightarrow {{e_x}} ,\\ \overrightarrow {{E_y}} ={-} \frac{{c{u_0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{x}{{R^{\prime}}} \cdot \left[ {\frac{{({{\dot{f}}_1} - {{\dot{f}}_2})}}{{{{R^{\prime}}^2}}} + \frac{{({{\ddot{f}}_1} + {{\ddot{f}}_2})}}{{cR^{\prime}}}} \right]} \right\}\overrightarrow {{e_y}} ,\\ \overrightarrow {{B_x}} = \frac{{c{u_0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{{xz^{\prime}}}{{{{R^{\prime}}^2}}} \cdot \left[ {\frac{{3({f_1} - {f_2})}}{{{{R^{\prime}}^3}}} + \frac{{3({{\dot{f}}_1} + {{\dot{f}}_2})}}{{c{{R^{\prime}}^2}}} + \frac{{({{\ddot{f}}_1} - {{\ddot{f}}_2})}}{{{c^2}R^{\prime}}}} \right]} \right\}\overrightarrow {{e_x}} ,\\ \overrightarrow {{B_y}} = \frac{{c{u_0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{{yz^{\prime}}}{{{{R^{\prime}}^2}}} \cdot \left[ {\frac{{3({f_1} - {f_2})}}{{{{R^{\prime}}^3}}} + \frac{{3({{\dot{f}}_1} + {{\dot{f}}_2})}}{{c{{R^{\prime}}^2}}} + \frac{{({{\ddot{f}}_1} - {{\ddot{f}}_2})}}{{{c^2}R^{\prime}}}} \right]} \right\}\overrightarrow {{e_y}} ,\\ \overrightarrow {{B_z}} = \frac{{c{u_0}{p_0}}}{{4\pi }} \cdot real\left\{ {\frac{{{{z^{\prime}}^2}}}{{{{R^{\prime}}^2}}} \cdot \left[ {\frac{{3({f_1} - {f_2})}}{{{{R^{\prime}}^3}}} + \frac{{3({{\dot{f}}_1} + {{\dot{f}}_2})}}{{c{{R^{\prime}}^2}}} + \frac{{({{\ddot{f}}_1} - {{\ddot{f}}_2})}}{{{c^2}R^{\prime}}}} \right] - \left[ {\frac{{({f_1} - {f_2})}}{{{{R^{\prime}}^3}}} + \frac{{({{\dot{f}}_1} + {{\dot{f}}_2})}}{{c{{R^{\prime}}^2}}} + \frac{{({{\ddot{f}}_1} - {{\ddot{f}}_2})}}{{{c^2}R^{\prime}}}} \right]} \right\}\overrightarrow {{e_z}} . \end{array}$$

The space-time expressions of the tightly focused azimuthally polarized pulse are the same as the results in the literatures [1,32]. There is no longitudinal electric field component and the longitudinal magnetic field is the strongest in the center of the pulse. ${f_1}$ and ${f_2}$ in Eq.(6) represent $f(\tau ^{\prime})$ and $f(\tau ^{\prime\prime})$ respectively. $\dot{f}$ and $\ddot{f}$ represent the first derivative and the second derivative respectively. By calculating the expression ${B_0} = \max \{{{B_x} + {B_y} + {B_z}} \}|{_{x,y,z = 0}} $ for the magnetic field at the focus, and using the conversion ${E_0} = {B_0}/\sqrt {{\mu _0}{\varepsilon _0}}$, ${p_0}$ and ${E_0}$ can be related by the following equation

(7)$${E_0} = \frac{{{c^2}{u_0}{p_0}}}{{2\pi {\textrm{a}^2}}} \cdot \left\{ {\frac{{(s + 1){\omega_0}}}{{cs}} \cdot \left[ {1 + {{\left( {1 + \frac{{2a{\omega_0}}}{{sc}}} \right)}^{ - (s + 2)}}} \right] - \frac{1}{a} \cdot \left[ {1 - {{\left( {1 + \frac{{2a{\omega_0}}}{{sc}}} \right)}^{ - (s + 1)}}} \right]} \right\}.$$

It should be noted that the parameter ${E_0}$ here does not represent the peak value of the electric field. The peak value of the electric field should be calculated by Eq. (6).

The parameter $s = 4.12$ corresponds to the full width at half maximum (FWHM) of the pulse time-domain envelope is 0.45-cycle and $s = 100$ corresponds to the FWHM is 2.6 cycles. The center wavelength of pulses is $\lambda = 800nm$ in the paper. On the focal plane, the peak electric field power distribution and the amplitude distribution of the longitudinal magnetic strength $|{B_z}|$ of the pulse are shown in Fig. 1. The parameters are ${E_0} = 9 \times {10^{13}}V/m$, $s = 4.12$, ${w_0} = 1.5\mu m$ and $|{B_z}|= \sqrt {{B_z} \cdot B_z^ \ast }$.

Fig. 1. (a) The peak electric field power distribution on the focal plane. (b) the amplitude distribution of the longitudinal magnetic strength $|{B_z}|$ on the focal plane.

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We can see that the electric field is distributed around the central optical axis. The longitudinal magnetic field is distributed in the center of the pulse. Therefore, the electrons can be confined in the center of the pulse by the magnetic field force and obtain acceleration energy from the electric field.

In order to study the dynamic behavior of electrons in the electromagnetic field, the relativity Newton-Lorentz equation [43–45] and the energy equation are used:

(8)$$\frac{{d\overrightarrow p }}{{dt}} = e({\overrightarrow E + \overrightarrow V \times \overrightarrow B } ),\textrm{ }{W^2} = {p^2}{c^2} + {m^2}{c^4}.$$

In the above equations, $\overrightarrow p = \gamma m\overrightarrow v$ is the electron momentum, $m$ is the rest mass, $\gamma = {({1 - {v^2}/{c^2}} )^{ - 1/2}}$ is the relativistic factor, and v is the electron velocity. After calculating Eq.(8) and considering the radiation reaction force f, the following equation is obtained:

(9)$$\frac{{d\overrightarrow \beta }}{{dt}} = \frac{1}{{\gamma mc}}[{e({\overrightarrow \beta (\overrightarrow \beta \cdot \overrightarrow E ) - (\overrightarrow E + c\overrightarrow \beta \times \overrightarrow B )} )+ \overrightarrow f } ],$$

where $\overrightarrow \beta = {{\overrightarrow v } / c}$ is the velocity of particle scaled by c. In order to facilitate the numerical calculation, the Eq.(9) is rewritten as the following first-order differential equations:

(10)$$\left\{ \begin{array}{l} \frac{{d\overrightarrow {{\beta_x}} }}{{dt}} = \frac{e}{{\gamma mc}}[{\overrightarrow {{\beta_x}} ({\beta_x}{E_x} + {\beta_y}{E_y} + {\beta_z}{E_z}) - (\overrightarrow {{E_x}} + c({\beta_y}{B_z} - {\beta_z}{B_y})\overrightarrow {{e_x}} )} ]+ \frac{{\overrightarrow {{f_x}} }}{{\gamma mc}}\\ \frac{{d\overrightarrow {{\beta_y}} }}{{dt}} = \frac{e}{{\gamma mc}}[{\overrightarrow {{\beta_y}} ({\beta_x}{E_x} + {\beta_y}{E_y} + {\beta_z}{E_z}) - (\overrightarrow {{E_y}} + c({\beta_z}{B_x} - {\beta_x}{B_z})\overrightarrow {{e_y}} )} ]+ \frac{{\overrightarrow {{f_y}} }}{{\gamma mc}}\\ \frac{{d\overrightarrow {{\beta_z}} }}{{dt}} = \frac{e}{{\gamma mc}}[{\overrightarrow {{\beta_z}} ({\beta_x}{E_x} + {\beta_y}{E_y} + {\beta_z}{E_z}) - (\overrightarrow {{E_z}} + c({\beta_x}{B_y} - {\beta_y}{B_x})\overrightarrow {{e_z}} )} ]+ \frac{{\overrightarrow {{f_z}} }}{{\gamma mc}}\\ \frac{{d\overrightarrow x }}{{dt}} = c\overrightarrow {{\beta_x}} \\ \frac{{d\overrightarrow y }}{{dt}} = c\overrightarrow {{\beta_y}} \\ \frac{{d\overrightarrow z }}{{dt}} = c\overrightarrow {{\beta_z}} \end{array} \right..$$

The specific form of the radiation reaction force f is:

(11)$$\begin{array}{l} \frac{{\overrightarrow f }}{{\gamma mc}} = \frac{1}{{3 \times {{10}^7}}} \cdot \frac{{2{e^3}}}{{{m^2}}}\left\{ {\frac{1}{c}\left[ {\frac{\partial }{{c\partial t}} + (\overrightarrow \beta \cdot \overrightarrow \nabla )} \right]\overrightarrow E + \left[ {\overrightarrow \beta \times \left( {\frac{\partial }{{c\partial t}} + (\overrightarrow \beta \cdot \overrightarrow \nabla )} \right)\overrightarrow B } \right]} \right\}\\ + \frac{1}{{3 \times {{10}^7}}} \cdot \frac{{2{e^4}}}{{{m^3}\gamma c}}\left\{ {\frac{{\overrightarrow E \times \overrightarrow B }}{c} + [{\overrightarrow B \times (\overrightarrow B \times \overrightarrow \beta )} ]+ \frac{{\overrightarrow E }}{{{c^2}}}(\overrightarrow \beta \cdot \overrightarrow E )} \right\} - \frac{1}{{3 \times {{10}^7}}} \cdot \frac{{2{e^4}\overrightarrow \beta \gamma }}{{{m^3}c}}\left[ {{{\left( {\frac{{\overrightarrow E }}{c} + \overrightarrow \beta \times \overrightarrow B } \right)}^2} - \frac{1}{{{c^2}}}{{(\overrightarrow \beta \cdot \overrightarrow E )}^2}} \right]. \end{array}$$

The initial kinetic energy of the electron is ${K_0} = ({\gamma _0} - 1)m{c^2}$, where ${\gamma _0} = {({1 - ({\beta_{x0}}^2 + {\beta_{y0}}^2 + {\beta_{z0}}^2)} )^{ - 1/2}}$, ${\beta _0}$ is the initial reduced velocity. By the Fourier transform of the radiated electromagnetic field, the electron radiation spectrum can be obtained. The radiation energy corresponding to the frequency $\omega$ can be written as:

(12)$${W_\omega } = 4\pi {\varepsilon _0}c{\oint {|{{E_\omega }} |} ^2}{R^2}d\varOmega .$$

${E_\omega }$ represents the Fourier transform of the time-domain expression of the electron radiation field, where $d\varOmega = \sin \theta d\theta d\varphi$.$\theta$ and $\varphi$ represent the polar angle and the azimuth angle in the spherical coordinates. The following expression can be obtained by the further calculation of the Eq.(12) :

(13)$${W_\omega } = \frac{{{e^2}}}{{16{\pi ^3}{\varepsilon _0}{c^3}}}\int\limits_0^{2\pi } {\int\limits_0^\pi {{{\left|{\int_{ - \infty }^{ + \infty } {\left( {{{\overrightarrow {{e_r}} \times \left[ {\left( {\overrightarrow {{e_r}} - \frac{{\overrightarrow v }}{c}} \right) \times \overrightarrow {\dot{v}} } \right]} / {{{\left( {1 - \frac{{\overrightarrow v \cdot {{\overrightarrow e }_r}}}{c}} \right)}^3}}}} \right) \cdot {e^{i\omega t}}dt} } \right|}^2}} } \sin \theta d\theta d\varphi .$$

$\overrightarrow {{e_r}}$ is the unit vector from the electron position to the field point. $\omega$ is the frequency of the radiated electromagnetic field. The radiation spectrum of electrons can be obtained by Eq.(13).

The adaptive variable step-size fifth-order prediction modified Runge-Kutta method is used to solve the electron motion equation in the electromagnetic field. The Matlab ode45 function is used in the calculation. The relative error parameter RelTol is set to 2.25×10⁻¹⁴ and the absolute error parameter AbsTol is 1×10⁻²¹ in the softwave.

3. Numerical analysis

The relativity Newton-Lorentz equation is solved numerically to study the electron acceleration by the tightly focused azimuthally polarized pulse. In order to analyze the conditions for obtaining the maximum exit kinetic energy of electrons, the acceleration effects of electrons emitted in four different initial directions are analyzed. The schematic diagram is shown in Fig. 2. The black arrows a, b, c, and d indicate that the initial electron velocity directions are parallel to the optical axis, pointing to the focus, perpendicular to the optical axis, and pointing to the ${z_d}$ respectively, where the optical axis refers to the z axis. The distance from ${z_d}$ to the focus is $5.1\mu m$. Thus the coordinates of the point ${z_d}$ is $\{ 0,0, - 5.1\mu m\}$ and are optimized values selected from the numerical calculation results. The laser pulse propagates along the z-axis, and the symbol e represents an electron. The cases a, b and c are shown in Fig. 2(a). The case d is shown in Fig. 2(b) for the sub-cycle pulse and in Fig. 2(c) for the few-cycle pulse. For the cases a, b and c, the coordinates of the initial position of the electron are $\{ {x_0},{y_0},{z_0}\} = \{ 0,h \cdot {v_{y0}} \cdot {t_0},h \cdot {v_{y0}} \cdot {t_0}/tan{\theta _0}\}$, where ${y_0}$ is $h \cdot {v_{y0}} \cdot {t_0}$ for the cases b and c and ${y_0}$ is a selected value for the case a. ${v_{y0}}$ is the y component of the initial velocity ${v_0}$ and h is a coefficient used to adjust the position of the electron to obtain the optimal acceleration position. For the case d, the coordinates are $\{ {x_0},{y_0},{z_0}\} = \{ 0,h \cdot {v_{y0}} \cdot {t_0},{z_d} + (h \cdot {v_{y0}} \cdot {t_0}/tan\theta )\}$. ${\theta _0}$ is the position angle of the electron. $\theta$ is the direction angle of the velocity and is shown in Fig. 2(b) and Fig. 2(c).

Fig. 2. Schematic diagram of the interaction between electrons and ultrashort pulses. (a) The cases a, b and c. (b)The case d for the sub-cycle pulse. (c)The case d for the few-cycle pulse.

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If the time when the pulse peak reaches the focus is set to zero, the initial interaction time ${t_0}$ of the electron with the sub-cycle pulse and the few-cycle pulse is set to $- 15fs$ and $- 60fs$, respectively. $fs$ means femtosecond.

The electric field peak parameter ${E_0}$ is $9 \times {10^{13}}V/m$ in the CSSM. ${E_0}$ is not the actual electric field peak and is converted from ${B_0}$ to characterize the intensity of the magnetic dipole ${p_0}$. The actual peak value of the electric field can be calculated from Eq.(6) and is $2.787 \times {10^8}V/m$. The peak power of the electric field is $I = 1.0314 \times {10^{10}}Wc{m^{ - 2}}$. The pulse beam waist parameter is ${w_0} = 1.5\mu m$.

3.1 Interaction between sub-cycle pulses and electrons

The electron emissions in the four different directions are analyzed. The parameters corresponding to the maximum kinetic energy gain of the electrons are shown in Table 1. In Table 1, $fs$ is femtosecond, c is the velocity of light, $E{k_0}$ represents the initial kinetic energy of electrons and $Ek$ represents the exit kinetic energy of electrons. W represents the radiation loss of the accelerated electrons. The y₀ coordinate of the electron is $h \cdot {v_{y0}} \cdot {t_0}$. The coefficient h is adjusted to obtain the optimal electron acceleration position. For the case a that the initial velocity direction of the electron parallel to the optical axis, y₀ is a selected value. The case d that the initial velocity direction of the electron pointing to the ${z_d}$ is found to be optimal. In case d, the exit kinetic energy of the electron is the largest and the initial velocity of the electron is the smallest.

Table 1. The parameters of the 0.45-cycle pulses and electron for obtaining the maximum kinetic energy gain of electrons^a

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The maximum kinetic energy gain of electrons versus time and angles is shown in Fig. 3. The parameters in Fig. 3 are shown in Table 1. In Table 1, the maximum kinetic energy gain can reach $201MeV$ when the electrons are initial emitted toward the ${z_d}$ point. The radiation loss of accelerated electrons is $KeV$. Numerical simulation shows that the maximum kinetic energy gain of electrons increases with the increase of the parameters ${E_0}$ and ${w_0}$. As the beam waist parameter ${w_0}$ increases, the FWHM of the beam focus spot will increase. Electrons will have the opportunity to stay in the acceleration phase for a longer acceleration time. Therefore, the maximum exit kinetic energy of electrons increases as the beam size ${w_0}$ increases. The maximum kinetic energy gain can exceed $1GeV$. The maximum electron kinetic energy gain corresponding to the parameter ${E_0} = 2.85 \times {10^{14}}V/m$ is twice that when ${E_0} = 9 \times {10^{13}}V/m$.

Fig. 3. (a) The kinetic energy gain versus time. The sub-figure is an enlarged view of the dotted line. (b) The maximum kinetic energy gain versus the position angle of the electron. The position angles of the electrons corresponding to the black, red, blue and green curves represent the initial movement directions of the electrons pointing to the focus, perpendicular to the optical axis, parallel to the optical axis and pointing to the point ${z_d}$.

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It can be seen from Fig. 3(a) that the electrons emitted in the four different directions will undergo the acceleration and deceleration during the initial time. The electrons get a small kinetic energy gain. The electrons emitted toward the focus and parallel to the optical axis are accelerated faster, and the time for the electrons to be accelerated to the maximum kinetic energy is about $250fs$. The maximum kinetic energy gain of the electrons emitted towards ${z_d}$ is higher than that of the other three directions. In Fig. 3(b), the optimal position angle of the electrons emitted toward the focus, perpendicular to the optical axis, and parallel to the optical axis is 10°. The best angle for electron emitted toward the ${z_d}$ point is 140° and the optimal position angle of the electron is 10.3° in the case. When the electronic position angle changes from 10° to 90°, the maximum kinetic energy gain of electrons varies greatly with the position angle of electron, which is different from the case of radially polarized pulses [24]. The reason is that there is a strong longitudinal electric field force along the optical axis of the radially polarized pulse. The electrons move under the action of the electric field force in the same direction. The magnetic field force along the optical axis of the azimuthally polarized pulse is very strong. Therefore, the motion direction of the electron changes constantly under the action of the magnetic field force. This results in a large change in the corresponding pulse electric field value when the electron is incident from different position angles. The calculation shows that the requirements for the position angle of electron to obtain the maximum kinetic energy gain by the interaction of the electron with the azimuthally polarized tightly focused sub-cycle pulse are relatively strict.

Carrier envelope phase (CEP) is the initial phase of the light field. CEP will affect the initial motion direction of the electron and play an important role in the electron trajectory. Thus, CEP will affect whether the electron can meet the pulse peak. The maximum kinetic energy gain of electrons emitted from four different directions versus CEP is shown in Fig. 4. It can be seen from Fig. 4 that the values of CEP corresponding to the maximum kinetic energy gain of electrons emitted from the four different directions are different and change periodically.

Fig. 4. The relationship between the maximum kinetic energy gain and CEP. The electron emission directions are (a) toward the focus, (b) perpendicular to the optical axis, (c) parallel to the optical axis, and (d) toward the point ${z_d}$.

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The data of the curve in Fig. 4 shows that the acceleration of electrons is very sensitive to changes in CEP.

3.2 Interaction between few-cycle pulses and electrons

The parameters of the maximum kinetic energy gain of electrons obtained by the interaction between the electron and the azimuthally polarized tightly focused few-cycle pulses (corresponds to the 2.6-cycle pulse) are shown in Table 2. The symbols in Table 2 have the same meaning as in Table 1.

Table 2. The parameters of the 2.6-cycle pulses and electron for obtaining the maximum kinetic energy gain of electrons^a

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It can be seen from Table 1 and Table 2 that the radiation loss is greater if the electron energy is higher. When the kinetic energy of electrons is accelerated to more than $200MeV$, the radiation loss of accelerated electrons is $KeV$ for the two cases of sub-cycle pulses and few-cycle pulses. The maximum kinetic energy gain of electrons versus time and angles is shown in Fig. 5.

Fig. 5. (a) The kinetic energy gain versus time. The sub-figure is an enlarged view of the dotted line. (b) The maximum kinetic energy gain versus the position angle of the electron. The position angles of the electrons corresponding to the black, red, blue and green curves represent the initial movement directions of the electrons pointing to the focus, perpendicular to the optical axis, parallel to the optical axis and pointing to the point ${z_d}$.

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From Fig. 3 and Fig. 5, one can see that when electrons interact with sub-cycle and few-cycle pulses, the electrons obtain the maximum kinetic energy gain corresponding to the initial emission direction of the electrons pointing to ${z_d}$. The initial kinetic energy of the electron is $K\textrm{e}V$. When electrons are initial emitted towards ${z_d}$, the electrons will experience the deceleration and acceleration when they interact with the pulse front, so the interaction time between the electrons and the pulse becomes longer. Thus, the electron may meet the strong pulse peak electric field in the area close to the focus, and obtain a large kinetic energy gain. The maximum kinetic energy gain obtained by the electrons in Fig. 5 is $217MeV$, which is numerically slightly higher than that in Fig. 3. The results in [24] show that that the maximum kinetic energy gain of electrons accelerated by the radially polarized sub-cycle pulses is several times higher than that of electrons accelerated by the radially polarized few-cycle pulses. However, the results of the studies of the azimuthally polarized ultrashort laser pulses are different. The reason for the difference is due to the different focusing characteristics of the radially polarized pulse and the azimuthally polarized pulse after passing through the high numerical aperture lens. There is a strong longitudinal magnetic field distribution and the electric field is zero in the focus center of the azimuthally polarized laser pulse. This ensures that the electron is confined in the pulse center for a long time by the action of the magnetic field force. Therefore, whether it is for the sub-cycle pulses or few cycle pulses, the electrons may meet the peak electric field to obtain the maximum kinetic energy gain, and then leave the beam. This is proved in the study of the Section 3.3.

According to Fig. 5(b), the maximum kinetic energy gain of electrons also varies greatly with the changes of the position angles of electrons. The optimal position angle corresponding to the maximum kinetic energy gain of the electrons emitted from the four directions does not exceed 10°. When the parameter ${E_0}$ is $2.85 \times {10^{14}}V/m$, the maximum kinetic energy gain of electrons in the cases of the few-cycle pulse is the same as that in the cases of the sub-cycle pulses, and it is also twice that when ${E_0}$ is $9 \times {10^{13}}V/m$.

The relationship between the maximum kinetic energy gain of electrons and CEP is shown in Fig. 6. The parameters are shown in Table 2. The results in Fig. 6 are the same as those in Fig. 4. The CEP parameters corresponding to the maximum kinetic energy gain of electrons emitted in the four different directions are different and change periodically. The difference from Fig. 4 is that the electronic kinetic energy gain is more sensitive to the changes of CEP when the electronic kinetic energy gain value is larger.

Fig. 6. The relationship between the maximum kinetic energy gain and CEP. The electron emission directions are (a) toward the focus, (b) perpendicular to the optical axis, (c) parallel to the optical axis, and (d) toward the point ${z_d}$.

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3.3 Electron trajectory and radiation spectrum

As the special focusing characteristics of the azimuthally polarized pulses, there have been a lot of studies on their application in the particle capture [46–47].

Figure 7 shows the trajectory of electrons when they interact with the pulse. The electrons are laterally confined near the optical axis of the pulse. The black curve in Fig. 7(a) represents the trajectory of the transverse constrained electrons in the pulse field with $CEP = 0.71\pi$ and the red curve corresponds to the trajectory of the electrons in the pulse field with $CEP = 0.11\pi$. In the calculation of the two electron trajectories, the parameters are consistent except for the CEP parameters. Figure 7(b) shows the case of the few-cycle pulses. From Fig. 7, one can see that the transverse motions of the electrons are constrained for the two cases of the sub-cycle pulses and the few-cycle pulses. The movement of electrons in the lateral direction is constrained within a certain range and the electron trajectory is spiral. The electron moves in a straight line after leaving the pulse. When the electron interacts with the pulse, the electrons are not only subjected to the Lorentz force, but also to the electric field force in the lateral direction. The kinetic energy of electron is accumulated under the action of the electric field force and the direction of electron velocity is changed under the action of the magnetic field force. Finally, the velocity of electron tends to a stable value, that is, the electron leaves the pulsed beam after obtaining the maximum kinetic energy. The corresponding parameters of Fig. 7(a) and 7(b) are shown in Table 3.

Fig. 7. The electron trajectory during the interaction between the electron and the pulse when it is laterally constrained near the optical axis of the pulse. (a) Sub-cycle pulses interact with electrons, (b) Few-cycle pulses interact with electrons

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Table 3. The corresponding parameters of electron trajectory when electron interacts with the sub-cycle and few-cycle azimuthally polarized pulses

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In Table 3, The parameter C_time represents the time when the movement of the electron is constrained in the lateral direction. $Ek$ is the exit kinetic energy of the electron. The Rayleigh range of the pulses is about $8.8\mu m$. The longest time that electrons are confined in the few-cycle pulsed field is $520fs$, the z coordinate of the electron is $149\mu m$, this is approximately 17 times the Rayleigh range. The longest time that electrons are confined in the sub-cycle pulsed field is $164fs$, the z coordinate of the electron is $45\mu m$ and is 5 times the Rayleigh range.

From the confinement time, it can be seen that the maximum time that the lateral movement of the electron is confined in the case of the few-cycle pulse is about three times longer than that in the case of the sub-cycle pulse. The central wavelength of the pulsed beam corresponds to a light cycle of $2.67fs$. The longest confinement time of the electron is about 61 times the light cycle for the sub-cycle pulse and is about 195 times the light cycle for the few-cycle pulse. The data in Fig. 7 and Table 3 show that the electrons are horizontally confined by the magnetic field force near the optical axis of the pulse and keep moving near the optical axis for a long time. Therefore, the electrons undergo a relatively long period of accelerated motion, and accumulate high kinetic energy and exit the pulsed beam. The initial position and initial velocity direction of the electron close to the optical axis will be conducive to the constraint of the transverse motion of the electron, and the initial position of the electron slightly away from the optical axis is conducive to the electron gaining high kinetic energy. When electrons are placed on the optical axis and emitted along the optical axis, the kinetic energy of the electrons remains unchanged. The changes of the electron position and the electric field peak position with time in Fig. 7 are shown in Table 4 and Table 5.

Table 4. The z coordinate of electron position and electric field peak position for the sub-cycle pulse

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Table 5. The z coordinate of electron position and electric field peak position for the few-cycle pulse

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Table 4 and Table 5 show that in the interaction between the electron and the pulse, the peak of the pulse reaches the position of the electron and then gradually separates from the electron.

Electrons will radiate electromagnetic fields when they are accelerated. The electromagnetic field radiated by electrons is related to the peak power of the pulsed laser field. When electrons interact with azimuthally polarized pulsed lasers, there is a long-time spiral movement in the center of the pulses. Thus, the spectrum of electron radiation is worthy to be studied. The radiation spectrum generated by the interaction of electrons with the sub-cycle and few-cycle azimuthally polarized pulses and the characteristics of the angular frequency corresponding to the peak of the radiation spectrum as a function of the parameter ${E_0}$ are shown in Fig. 8. The cases of the sub-cycle pulses are shown in Fig. 8(a). ${E_0}$ is $9 \times {10^{13}}V/m$ for the purple curve in Fig. 8(a). The frequency $\omega$ is $1.3492 \times {10^{14}}rad/s$ for the peak of the radiation spectrum curve and is $\omega = 0.0573{\omega _0}$, where ${\omega _0}$ is the center frequency of the pulse. The cases of the few-cycle pulses are shown in Fig. 8(b). ${E_0}$ is $9 \times {10^{13}}V/m$ for the blue curve in Fig. 8(b). The frequency $\omega$ is $3.3 \times {10^{14}}rad/s$ for the peak of the radiation spectrum curve and $\omega$ is $0.142{\omega _0}$.When ${E_0} = 9 \times {10^{13}}V/m$, the peak frequency $\omega$ of Fig. 8(b) is about 2.5 times that of Fig. 8(a). The reason is that the time-domain width of the few-cycle pulse is larger than that of the sub-cycle pulse. There are more velocity changes when electrons are in the few-cycle pulsed field than in the sub-cycle pulsed field. Therefore, the radiation spectrum of electrons is widened and the center frequency of the spectrum is blue-shifted. From Fig. 8(a), it can be seen that the angular frequency $\omega$ for the peak of the radiation spectrum decreases with the increase of ${E_0}$.

Fig. 8. The curves of the radiation spectrum for the parameter ${E_0}$, (a) represents the case of sub-cycle pulse, (b) represents the case of few-cycle pulse.

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With the increase of ${E_0}$, the movement of electrons becomes simpler when the electrons interact with the pulses, the time that electrons are constrained laterally at the center of the pulse becomes shorter, and the electron undergoes less acceleration, deceleration and spiral motion, and are directly accelerated away from the beam under the action of the larger electric field of pulses. This results in a redshift in the radiation spectrum of electron, and the angular frequency $\omega$ for the peak of the radiation spectrum becomes smaller. In Fig. 8(a), when ${E_0}$ is less than $9 \times {10^{11}}V/m$, the electron radiation energy is very small and the angular frequency $\omega$ corresponding to the peak of the radiation spectrum is basically unchanged, so the radiation spectrum curves are also basically coincident. The result of Fig. 8(b) is opposite to that of Fig. 8(a). The angular frequency $\omega$ corresponding to the peak of the radiation spectrum increases with the increase of ${E_0}$, and the electron radiation spectrum is blue-shifted. The reason is that the time domain width of the few-cycle pulse is larger than that of the sub-cycle pulse. The electric field intensity increases with the increase of the parameter ${E_0}$. The electrons may be decelerated by the subsequent electric fields after being accelerated by the peak electric field when the electrons interact with the few-cycle pulse. There is a greater change of the electron acceleration in the case of the few-cycle pulse compared with the case of the sub-cycle pulse. Therefore, the radiation spectrum is widened, and the peak frequency of the radiation spectrum is blue-shifted. The results in Fig. 8 show that the laser intensities and the time-domain widths of pulses have a great influence on the radiation spectrum characteristics in the interaction between electrons and ultrashort pulses.

4. Conclusion

Using the tightly focused azimuthally polarized sub-cycle and few-cycle Poisson pulses, the acceleration, lateral constraint and radiation spectrum characteristics of electrons are studied. The study found that the maximum kinetic energy gain of electrons varies greatly with the position angle of electron when the sub-cycle and few-cycle pulses interact with electrons. The electrons can obtain the largest exit kinetic energy when they are initial emitted toward the point before the focus on the optical axis. The maximum kinetic energy gain of electrons can reach $GeV$. In the case of the few-cycle pulses, the time that the lateral movement of the electrons is confined by the laser pulses can reach $520fs$, which is 3 times the time that the lateral movement of electrons is confined by the sub-cycle pulses. The few-cycle pulse is more suitable for the lateral movement confinement of electron than the sub-cycle pulse. At the same time, the physical mechanism of accelerating electrons to obtain large exit kinetic energy by the tightly focused azimuthally polarized ultrashort pulses is analyzed. The magnetic force of the strong longitudinal magnetic field at the center of the pulse restrains the transverse movement of the electron, which keeps the electron moving near the optical axis of the pulse center for a long time. During this period, the electrons accumulate kinetic energy by interacting with the pulsed electric field, until the peak electric field of the pulse interacts with the electrons and the electrons gain a large kinetic energy gain and leave the beam. The kinetic energy of electron can be accelerated to the $sub - GeV$ by both tightly focused azimuthally polarized sub-cycle and few-cycle laser pulses. The radiation spectrum generated by the interaction between electrons and the tightly focused azimuthally polarized ultrashort pulse and the variation of the angular frequency corresponding to the peak of the radiation spectrum with the electric field amplitude parameter of the pulse are also studied. It is found that the angular frequency corresponding to the peak of the radiation spectrum under the action of the sub-cycle pulse decreases with the increase of the electric field amplitude parameter of the pulse. The reason is that the electrons are directly accelerated away from the beam under the action of the large electric field amplitude of the pulse and the movement of electrons becomes simple. This leads to the red shift of the electron radiation spectrum. The angular frequency corresponding to the peak of the radiation spectrum under the action of the few-cycle pulse increases with the increase of the electric field amplitude parameter of the pulse. The reason is that the time-domain width of the few-cycle pulse is larger than that of the sub-cycle pulse, and the electron velocity undergoes more changes with the increase of the parameter ${E_0}$. This leads to the blue shift of the electron radiation spectrum.

Funding

National Natural Science Foundation of China (11964007); The key laboratory of Guizhou Minzu University (GZMUSYS[2021]03).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Emission direction	Φ₀	θ₀	y₀(m)	z₀(m)	v_y0 (m/s)	v_z0(m/s)	Ek₀(KeV)	Ek(MeV)	W(KeV)
Point to focus	0.60π	10°	-2.4cƒs	y₀/tanθ₀	0.1c	v_y0/tanθ₀	114	184	0.759
Parallel to optical axis	1.72π	10°	-2.4cƒs	y₀/tanθ₀	0	0.9c	661	176	0.571
Perpendicular to optical axis	0.44π	10°	-1.4cƒs	y₀/tanθ₀	0.7c	0	204	179	0.867
Point to $z_{d}$	0.16π	10.3°	-2.6cƒs	z_d+y₀/tanθ	0.1c	v_y0/tan(θ)	6.30	201	1.682

Emission direction	Φ₀	θ₀	y₀(m)	z₀(m)	v_y0 (m/s)	v_z0(m/s)	Ek₀(KeV)	Ek(MeV)	W(KeV)
Point to focus	0.84π	10°	-10.2cƒs	y₀/tanθ₀	0.1c	v_y0/tanθ₀	114	157	0.154
Parallel to optical axis	1.00π	10°	-10.2cƒs	y₀/tanθ₀	0	0.9c	661	187	0.247
Perpendicular to optical axis	0.16π	10°	-9cƒs	y₀/tanθ₀	0.9c	0	661	205	2.866
Point to $z_{d}$	0.16π	7.1°	-7.2cƒs	z_d+y₀/tanθ	0.05c	v_y0/tan(θ)	22.6	217	2.591

Pulse	Electron trajectory	Φ₀	θ₀	y₀(m)	z₀(m)	v_y0 (m/s)	v_z0(m/s)	C_time(ƒs)	Ek(MeV)
s = 4.12	6(a)Black curve	0.71π	10°	-0.15cƒs	y/tanθ₀	0.06c	v_y0/tanθ₀	164	5.32
s = 4.12	6(a)Red curve	0.11π	10°	-0.15cƒs	y/tanθ₀	0.06c	v_y0/tanθ₀	146	4.61
s = 100	6(b)Black curve	0.82π	10°	-1.2 cƒs	y/tanθ₀	0.03c	v_y0/tanθ₀	520	16
s = 100	6(b)Red curve	0.72π	10°	-1.2cƒs	y/tanθ₀	0.03c	v_y0/tanθ₀	388	14.6

Time (ƒs)( $Φ_{0} = 0.82 π$ )	65	200	336
Electron position (µm)	20	60	100
Electric field peak position (µm)	20	60.01	100.01
Time (ƒs)( $Φ_{0} = 0.72 π$ )	68	202	335
Electron position (µm)	20	60	100
Electric field peak position (µm)	20.46	60.75	100.8

Emission direction	Φ₀	θ₀	y₀(m)	z₀(m)	v_y0 (m/s)	v_z0(m/s)	Ek₀(KeV)	Ek(MeV)	W(KeV)
Point to focus	0.60π	10°	-2.4cƒs	y₀/tanθ₀	0.1c	v_y0/tanθ₀	114	184	0.759
Parallel to optical axis	1.72π	10°	-2.4cƒs	y₀/tanθ₀	0	0.9c	661	176	0.571
Perpendicular to optical axis	0.44π	10°	-1.4cƒs	y₀/tanθ₀	0.7c	0	204	179	0.867
Point to $z_{d}$	0.16π	10.3°	-2.6cƒs	z_d+y₀/tanθ	0.1c	v_y0/tan(θ)	6.30	201	1.682

Acceleration of electrons by tightly focused azimuthally polarized ultrashort pulses in a vacuum

Abstract

1. Introduction

2. Theoretical analysis

3. Numerical analysis

3.1 Interaction between sub-cycle pulses and electrons

3.2 Interaction between few-cycle pulses and electrons

3.3 Electron trajectory and radiation spectrum

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (5)

Equations (13)

Optics Express

Time (ƒs)( $Φ_{0} = 0.11 π$ )	33	67	101
Electron position (µm)	10	20	30
Electric field peak position (µm)	10.02	20.23	30.34
Time (ƒs)( $Φ_{0} = 0.71 π$ )	33	67	101
Electron position (µm)	10	20	30
Electric field peak position (µm)	9.93	20.01	30.19