## Abstract

High-quality ultrashort electron beams have diverse applications in a variety of areas, such as 4D electron diffraction and microscopy, relativistic electron mirrors and ultrashort radiation sources. Direct laser acceleration (DLA) mechanism can produce electron beams with a large amount of charge (several to hundreds of nC), but the generated electron beams usually have large divergence and wide energy spread. Here, we propose a novel DLA scheme to generate high-quality ultrashort electron beams by irradiating a radially polarized laser pulse on a nanofiber. Since electrons are continuously squeezed transversely by the inward radial electric field force, the divergence angle gradually decreases as electrons transport stably with the laser pulse. The well-collimated electron bunches are effectively accelerated by the circularly-symmetric longitudinal electric field and the relative energy spread also gradually decreases. It is demonstrated by three-dimensional (3D) simulations that collimated monoenergetic electron bunches with 0.75° center divergence angle and 14% energy spread can be generated. An analytical model of electron acceleration is presented which interprets well by the 3D simulation results.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

With the rapid development of laser technologies, the peak intensity of ultrashort ultraintense laser pulses is already far exceeding the relativistic intensity [1–3], at which the interaction of laser and plasma is pushed to the relativistic regime. Ultraintense laser pulses have potential applications in inertial confinement fusion [4], charged particle acceleration [5,6], novel radiation sources [7], nuclear physics [8] and laboratory astrophysics [9]. Especially, high-quality ultrashort electron beams accelerated by ultrashort ultraintense laser pulses have attracted widespread attention [10–13], since they have diverse applications in a variety of areas such as 4D electron diffraction and microscopy [14], relativistic electron mirrors [15] and ultrashort radiation sources [16]. As one of the most extensively investigated schemes to generate ultrashort electron beams, the laser wakefield acceleration (LWFA) has an acceleration gradient of 100s GV/m. In the laser wakefield, electrons are accelerated to several GeV by plasma waves excited in underdense plasma [17,18]. High-quality electron beams with high energy [19], low divergence [20] and low energy spread [21] may be generated via this indirect laser acceleration mechanism. However, the beam charge is limited to tens of pC due to beam loading effects [22,23]. When irradiating laser pulses onto targets with a higher density, such as a near-critical-density target or a solid target, electron beams with several nC even hundreds of nC charge can be produced [24–26]. The acceleration process is dominated by the DLA [26–32], such as ponderomotive acceleration [27,31], resonance acceleration [5,33] and vacuum acceleration [32,34]. In the DLA mechanisms, massive electrons gain energy directly from the laser electromagnetic fields with acceleration gradient up to 10s TV/m [32], which is different from the energy gain from plasma waves in the ILA regime. When a laser pulse transports in a plasma channel, electrons will make transverse oscillations and can directly gain energy during betatron resonance. When injecting free electrons into a laser pulse in vacuum, electrons can be directly accelerated by the laser fields [33]. In addition, electrons can also gain energy directly from the laser ponderomotive force [27]. Electrons will obtain energy from the transverse electric field and the laser magnetic field will turn the transversal electron momentum into longitudinal momentum via the Lorentz force.

However, the generated electron beams via the DLA regime generally have large divergence and wide energy spread which prevent the stable acceleration of the electrons [32,35,36]. Hu $et~al.$ [35] obtained dense electron bunches by irradiating a Gaussian laser pulse onto a cone target. However, the generated electron beams have a usual exponential energy spectrum and a two-peak divergence angle distribution. When electrons leave the cone target, the bunches quickly spread out in space. Thèvenet $et~al.$ [32] reported experimental observations of vacuum laser acceleration using plasma mirror as electron injectors, where an electron beam with overall charge of 3 nC is obtained. However, the accelerated electrons have a broad peaked energy spectrum and the full width at half maximum (FWHM) of the divergence is about $9^{\circ }$. Ma $et~al.$ [36] proposed a scheme to produce quasi-monoenergetic sub-femtosecond electron bunches by using a transversely thin solid target, where electrons experience almost the same ponderomotive force, but the emission angle is larger than $15^{\circ }$. Overall, it is still challenging to generate electron beams simultaneously with narrow energy spread and low divergence angle in the DLA regime.

A symmetrical laser field structure contributes to improve the quality of the generated electron bunches [37,38]. As a kind of symmetrical polarized laser, radially polarized laser pulses have circularly-symmetric longitudinal and transverse laser electric field, which can be conducive to the generation of monoenergetic electron bunches with low divergence. Besides, the hollow electromagnetic distribution can change the manner of electron beam generation, and the strong longitudinal electric field components can affect the electron acceleration mechanism. Therefore, many works concerning electron acceleration using radially polarized laser pulses via numerical simulations and experiments have been reported [39–44]. Since high-quality radially polarized laser pulses with up to $\sim 6 \times 10^{19}$ $\textrm{W}/cm^2$ intensity and exceeding $93\%$ polarization purity have been experimentally generated in 2014 and 2019 [45,46], they may provide new possibilities for the DLA regime. In this paper, a novel all-optical scheme which generates high-quality sub-femtosecond electron bunches by irradiating a radially polarized laser pulse onto a hydrogen nanofiber has been proposed. The generation, acceleration and confinement mechanisms of electron bunches are discussed in detail through three-dimensional (3D) particle-in-cell (PIC) simulations. It is found that every generated electron slice is monoenergetic with small divergence angle. The energy spread, the center divergence angle and the FWHM of the divergence angle of the selected electron slice at $t=50T_0$ are 14%, $0.75^{\circ }$ and $1.75^{\circ }$, respectively, which is conducive to the stable and continuous acceleration of electrons in the DLA regime.

## 2. Model

The fully 3D PIC code Virtual Laser-Plasma Laboratory (VLPL) [47,48] is employed to carry out numerical simulations. In the simulations, a radially polarized laser pulse is incident along the $+x$ direction and focused at the left tip of a nanofiber target. The transverse electric field component can be written as

## 3. Results

Figure 1(a, b) show the density distribution of the sub-femtosecond electron bunches. It is shown that several dense electron bunches with a density of about $3n_{c}$ and total charge of 2.2 nC are generated at $t=10T_0$, as the laser pulse with hollow radial electric fields sweeps the nanofiber target. Different from the sawtooth-like or helical structure in other schemes [36,49,50], each electron bunch here is isolated and doughnut-like. The electron bunches will be compressed and converged into pie-like disks, after they are detached from the nanofiber. Afterwards, electrons are continuously accelerated and can transport stably along the laser pulse at a speed close to the speed of light in vacuum. The structure of the bunches is still intact at $t = 50T_{0}$ with an average density of about $1n_{c}$, as can be seen from Fig. 1(b). The interval between the electron bunches at this time is about $1\lambda _0$, and the bunch durations are 498 as, 400 as (the selected slice), 180 as, 133 as and 103 as, respectively, with the average value of 260 as. The dense sub-femtosecond electron bunches with high beam charge are similar to those generated by using circularly-polarized LG laser pulses [51,52], but our scheme is more feasible experimentally. In addition to the high beam density and charge, the generated electron bunches are also characterized by low energy spread and small divergence angle. Figure 1(c, d) show the evolution of energy spectrum and angular distribution of the selected electron slice as marked by the black dashed curve. We can see that the generated electron bunch is monoenergetic, which is different from the Maxwellian distribution [35] or a broad peaked energy spectrum [32] in the normal DLA regime. Fortunately, as electrons are continuously accelerated, the relative energy spread gradually decreases. The relative energy spreads are $30\%$, $20\%$ and $14\%$ at $t=20T_0$, $30T_0$ and $50T_0$, respectively.

The divergence angle $\theta$ of the produced electron bunches is defined as $\theta = \arctan (p_\perp /p_x)$, with $p_\perp = \sqrt {p_y^2+p_z^2}$. The corresponding center divergence angle and FWHM of the divergence angle are $2.5^{\circ }$, $1^{\circ }$, $0.75^{\circ }$ and $3.25^{\circ }$, $2.25^{\circ }$, $1.75^{\circ }$ at $t=20T_0$, $30T_0$ and $50T_0$, respectively, which is much better compared with the two-peak distribution with the emission angle larger than $15^{\circ }$ in the previous work [36]. The divergence angle is very low and gradually decreases as electrons transport stably along the laser pulse, which is totally different from the DLA regime driven by the Gaussian laser pulse [35,36]. In this novel scheme, monoenergetic and collimated electron bunches can be generated with the simultaneously improved relative energy spread and divergence angle.

#### 3.1 Generation of the sub-femtosecond electron bunches

When a laser pulse sweeps a nanofiber target, electrons within the skin depth will be thrown toward the bulk of the plasma by the laser ponderomotive force or electric field force. Then attractive force from the ions left behind and the repulsive force from electrons themselves will lead to a relativistic electron counterstream toward the skin layer. When the counterstream encounters the inward electrons, an electron bunch are formed inside the plasma. When these electron bunches arrive at the plasma surface, electrons with relativistic velocities escape the nanofiber from the locations where the electromagnetic field is weak, such as nodes of the laser field at every half cycle. The process of electron bunching and ejection were first discussed by N. Naumova $et~al.$ [49,53]. As we all know, the radially polarized laser pulse has different electromagnetic energy distribution (can be measured by $E^2$ here) compared with the normal Gaussian laser pulse. The $E^2$ distribution of the radially polarized laser pulse is given by

It can be seen that $E_{rad}^2=0$ and $E_{gauss}^2=0$ are both satisfied at $\psi = (1/2+n)\pi$, with $n= \pm 1,\pm 2,\pm 3,\ldots$, but $E_{rad}^2=0$ is also satisfied at $r=0$, which means that electrons generated by the Gaussian laser pulse can only be ejected from the half cycle nodes of the lateral surface but those generated by the radially polarized laser pulse can also be ejected from the left tip of the nanofiber. As can be seen from Fig. 2(a), the electrons in the Gaussian laser fields will be ejected from the half cycle nodes of the lateral surface of the nanofiber target. However, the hollow distribution of the radially polarized laser electromagnetic energy leads to a weak laser intensity near the axis on the left tip of the nanofiber. Copious electrons are ejected from the left tip of the nanofiber target, as shown in Fig. 2(b). It can also be seen from the electron trajectories that the electron bunches generated by the radially polarized laser pulse move leftwards after leaving the left tip and then be accelerated rightwards. The generation process of the sub-femtosecond electron bunches is illustrated in more details by Visualization 1.

#### 3.2 Acceleration of the sub-femtosecond electron bunches

After electrons are dragged out of the nanofiber, they are captured by and propagate along with the radially polarized laser pulse. The electron motion equations are $\mathrm {d} \vec {p}_{\perp , x}/\mathrm {d} t=-q_0\vec {E}_{\perp , x} -q_0( \vec {v} \times \vec {B})_{\perp , x}$, where $\gamma = \sqrt {1+\vec {p}^{2}/m_{e0}^2c^2}$ is the Lorentz factor. The radially polarized laser pulse is characterized by circularly symmetric radial electric field $E_r$ and azimuthal magnetic field $B_{\varphi }$ components. Electrons will obtain transversal momentum from the radial electric field component $E_r$ after they eject from the nanofiber. The azimuthal magnetic field component $B_{\varphi }$ will turn the transversal electron momentum into longitudinal momentum via the Lorentz force $\vec {v} \times \vec {B}$ [54]. Furthermore, the longitudinal electric field component $E_x$ is also rather important for tightly focused radially polarized laser pulse [41]. The expression of $\vec {E_{x}}$ can be given by $E_x=i/k\left (\partial E_y/\partial y+\partial E_z/\partial z\right )$, which can be obtained from Gauss law, i.e., $\nabla \cdot \vec {E} = 0$, by using the paraxial approximation [55–58], namely

The electron energy equation in the radially polarized laser pulse can be described by

The first term on the right side of Eq. (5) represents the power of the longitudinal electric field force $P_{\textrm{E}, x}$, and the second term represents the power of the transverse electric field force $P_{\textrm{E},\perp}$. In order to intuitively compare the magnitude of $P_{\textrm{E}, x}$ and $P_{E,\perp}$, the evolutions of $P_{\textrm{E}, x}$ and $P_{\textrm{E},\perp}$ of the selected electron slice are shown in Fig. 3(a,b). Almost all the electrons are located within the region ($P_{\textrm{E}, x} \gg 0$, $\lvert P_{\textrm{E}, \perp} \rvert \sim 0$), which means that $P_{\textrm{E}, x}$ is much larger than $P_{\textrm{E},\perp}$. To be more persuasive, the trajectories of a typical electron in the $(P_E,t)$ space are also shown in Fig. 3(a,b). It is shown that $P_{\textrm{E},\perp}$ is close to zero and sometimes negative. However, $P_{\textrm{E}, x}$ remains positive and one magnitude larger than $P_{\textrm{E},\perp}$. As is known, the radially polarized laser pulse is characterized by hollow electromagnetic fields. Therefore the transverse electric field $E_{\perp }$ is very weak near the optical axis. For example, the transverse electric field $E_{\perp }$ at $r = 0.5 \lambda _0$ is about one-third of its maximum value. However, the longitudinal electric field $E_{x}$ reaches its maximum at the optical axis and decreases with $r$ according to Eq. (4). The value of $E_{x}$ at $r = 0.5 \lambda _0$ is comparable with that of $E_{\perp }$. Furthermore, $v_{\perp }<0.1c$ and $v_x \approx c$ for many captured electrons are satisfied since electrons are well collimated in the radially polarized laser field. Therefore, $P_{\textrm{E},\perp}$ can be much smaller than $P_{\textrm{E}, x}$, which is consistent with the simulation results in Fig. 3(a, b).

In order to investigate whether the longitudinal or the transverse electric force is dominant in the acceleration process, the integrals of $P_{\textrm{E}, x}$ and $P_{\textrm{E}, \perp}$ over time $t$ are calculated respectively. $\eta _{x}\left (t\right )=-q_0 \int _{0}^{t} v_{x} E_{x} \mathrm {d} t /\left (m_{e,0} c^{2}\right )$ represents the energy gain from longitudinal electric fields, while $\eta _{\perp }\left (t\right ) = -q_0 \int _{0}^{t} \vec {v_{\perp }} \cdot \vec {E_{\perp }} d t /\left (m_{e 0} c^{2}\right )$ represents that from the transverse electric fields, with $\gamma (t)=\gamma (0)+\eta _{\textrm{x}}+\eta _{\perp }$. The evolution of the energy gain of a typical electron over time is shown in Fig. 3(c). It is found that the magnitude of $\eta _{\perp }\left (t\right )$ is much less than that of $\eta _{x}\left (t\right )$, while the curve of $\eta _{x}\left (t\right )$ almost coincides with that of $\gamma (t)$, which infers that the energy gain from the transverse electric fields is much less than that from the longitudinal electric fields. The energy distribution of the selected electron slice in $(\eta _{\textrm{x}}, \eta _{\perp })$ space at $t = 50 T_{0}$ is also shown in Fig. 3(d). It is shown that most electrons are located in the region ($350< \eta _{\textrm{x}} <450$, $-50< \eta _{\perp } <50$), which means that the longitudinal electric force is dominant in the acceleration process. In addition, it also implies that most electrons are concentrated in a small area in $(\eta _{\textrm{x}}, \eta _{\perp })$ space, which is beneficial to mono-energetic electron bunch generation. According to Eq. (4), the longitudinal electric field of the radially polarized laser pulse is circularly-symmetric. Since electrons are concentrated near the optical axis and are well collimated, electrons will experience almost the same longitudinal electric field force. Although electrons are continuously accelerated and the peak energy gradually increases, the FWHM of the energy spectrum remains almost unchanged. Therefore, the relative energy spread can keep decreasing over time, as shown in Fig. 1(c).

We define the phase experienced by an electron as $\psi _e = \omega t - k x_{e} - k r_{e}^2/(2 R_{ce}) +2 \arctan (x_{e}/x_{Re})$, which is similar to the laser phase except all the coordinates are replaced by that of the electron. Assuming that electrons are accelerated near the optical axis ( $r \approx 0$) at the speed of light in vacuum, we have $\psi _e = \Delta \psi _e+\psi _{e,0}$ and $\Delta \psi _e = 2\arctan (x_{e}/x_{Re}) - 2\arctan (x_{0}/x_{Re})$, where $\psi _{e,0} = - k x_{0} +2 \arctan (x_{0}/x_{Re})$ with $x_{0}$ the initial position relative to the laser focus plane when electrons are injected into the laser [59]. Taking the partial derivative of $\psi$ with respect to $x$, the phase velocity of the laser can be given by $v_{ph} = \omega \left [k -2/\left (x_R+x^2/x^2_R\right )\right ]^{-1}$. It is easy to see that the phase speed $v_{ph}$ near the axis is superluminal at $x=0$, but it decreases to $c$ at $x \gg x_R$. Therefore, the phase slip reaches its maximum at $x \to +\infty$, which is $\Delta \psi _{e,m} = \pi - 2\arctan (x_{0}/x_{Re}) \approx \pi$. As is known that an electron can be continuously accelerated as long as it is located in the half cycle of the laser with $E_x <0$. Therefore, electrons are well trapped in the acceleration phase of the laser pulse, as long as they are initially injected into the proper acceleration phase and are rapidly accelerated close to the speed of light in vacuum.

The electron acceleration process in the radially polarized laser pulse can be divided into two stages. In stage I, after electrons are pulled out of the nanofiber, they will stay near the target surface with a doughnut-like shape and are accelerated by $E_x$. In stage II, electrons converge into dense disks after leaving the nanofiber, and then are continuously accelerated near the optical axis. We can theoretically estimate the electron energy at $x$ with the assumption that $x \approx x_0+ct$ by integrating $E_x$ over the electron path in stage I and stage II

In order to get an analytical solution, we assume that the electrons transport near the optical axis ( $R_1 = R_2 = 0$), and the $E_x$ experienced by the electron can be simplified as $E_x = {2 \sqrt {2 e} E_{L,0} f\left (x_{0}\right ) }/{\left [ k \sigma _0\left (1+x_e^{2} / x^2_{R}\right ) \right ]} \sin \left (\Delta \psi _e +\psi _{e,0}\right )$. Then, the electron energy at $x$ can be given by

The maximum electron energy can be given by $\varepsilon \left (\infty \right )={ -\sqrt {2 e} q_0 E_{L,0} \sigma _0 }f\left (x_{0}\right )\cos \psi _{e,0}$, since $\Delta \psi _e \approx \pi$ at $x \gg x_R$. It can be seen that the maximum energy $\varepsilon (\infty )$ is proportional to $E_{L,0}\sigma _0\propto \sqrt {P_0}$ with $P_0$ the laser peak power [59]. Besides, the laser timeprofile and initial electron position $x_0$ also have a great influence on the maximum energy. For the parameters employed in the simulations, the estimated maximum energy can be up to 657.27 MeV with $\psi _{0}=\pi$ and $f(x_0)=1$, which is in excellent agreement with the achieved energy for $\sigma _0 = 3\lambda _0$ in Fig. 5(d) (628 MeV) at $t = 100T_0$.

#### 3.3 Trapping of the sub-femtosecond electron bunches

It is challenging to generate electron beams with low divergence via DLA regime, as discussed before [32,35,36]. Fortunately, the divergence angle of the generated electron bunches in our scheme is very low and gradually decreases, which may provide new opportunities for high quality electron acceleration via DLA. Figure 4(a) illustrates the angular distribution of electrons in the radially polarized and Gaussian laser pulse, respectively. The center divergence angle and the FWHM of the angular distribution of electrons generated by the radially polarized laser pulse are $0.75^\circ$ and $1.75^\circ$, respectively, which are much smaller than that generated by the Gaussian laser fields that is $7.75^\circ$ and $4.25^\circ$. It is also shown in Fig. 4(b, c) that the electrons in the radially polarized laser pulse are well confined within the range of $-1.5\lambda _{0} < y < 1.5\lambda _{0}$ while electrons in the Gaussian laser pulse are rapidly split up and spread out, which is consistent with the electron angular distributions in Fig. 4(a).

Considering $\beta _{\perp } \ll \beta _{x}$ and $B_x \ll B_{\perp }$ in our simulations, the transverse motion equation of an electron in radially polarized laser can be simplified as $\mathrm {d} \vec {p_{r}} / \mathrm {d} t=-q_0 \left (E_{r} -\beta _x B_{\varphi }\right ) \vec {e_r}$, where $B_{\varphi }$ is the azimuthal magnetic field component of the laser pulse and $\beta _x = v_x / c$ is the normalized longitudinal velocity. For simplicity, we assume $E_r = c B_{\varphi }$ [60] and the electron motion equation is

Similarly, for an electron in the linearly polarized Gaussian laser pulse with polarization along the $y$-direction, the transverse motion equation becomes

It is shown in Eq. (8) and (9) that the direction and magnitude of the net transverse force depend on the electric field force. Although the electric field force will be canceled out by the magnetic field force for both the laser pulses, it can not be canceled out completely because of $\beta _x <1$.

Figure 4(d) illustrates the evolution of the radial electric force (the red curves) and magnetic force (the blue curves) subjected by an electron in the radially polarized or Gaussian laser pulse. It is found that the two forces are comparable in magnitude but opposite in direction, which is consistent with the above theoretical analysis in Eq. (8) and Eq. (9). In addition, it is reported in the previous work that the electric field force will be canceled out completely by the magnetic field force for radially polarized laser pulse, so these electrons can be well confined [60]. However, the two forces will also cancel out for the Gaussian laser pulse, which infers that this is not the essential reason for the electron beam collimation. It is noted that the two forces in the radially polarized laser pulse are much weaker than in the Gaussian laser pulse. Since the radially polarized laser pulse has a hollow electromagnetic field distribution, the transverse laser field is much weaker when compared with that of the Gaussian laser pulse. More importantly, different from Gaussian laser pulse, the radial electric force of the radially polarized laser pulse is negative. Figure 4(e, f) shows the electron density distribution and the corresponding transverse electric field distribution in the two cases. It is shown that most electrons in the radially polarized laser field are in the phase with negative radial electric forces. The electron slices will be transversely squeezed by the transverse electric field until dephasing occurs, resulting in the small and gradually decreasing electron divergence angle, as shown in Fig. 1(c). By comparison, the electrons in the Gaussian laser field are in the phase with positive radial electric forces. Electrons gradually spread out in the Gaussian laser field, as shown in Fig. 4(c).

## 4. Discussion

The influence of the target and laser parameters on the achieved bunch energy $\varepsilon (x)$ and charge are investigated respectively, such as the fiber radius $R$, the fiber length $L$, the dimensionless parameter $a_0$ and beam waist radius $\sigma _0$, as summarized in Fig. 5. Here, only one parameter is changed in each case and all other parameters keep fixed. According to Eq. (6), the achieved bunch energy $\varepsilon (x)$ is proportional to $E_x$. As the target radius $R$ increases, the acceleration electric field $E_x$ in the stage I gradually decreases, while the increasement of the length of the target $L$ will extend the duration of stage I. We also conclude that the maximum energy under ideal conditions $\varepsilon \left (\infty \right )$ is proportional to $\sqrt {P_0}$ which increases with $a_0$ and $\sigma _0$. As a result, the achieved bunch energy will decrease with $R$ and $L$, but rise with $a_0$ and $\sigma _0$, which agrees with the results in Fig. 5. It is noted that we get the values of the cut-off energy in the PIC simulations at $t = 50T_0$ in (a-c) and at $t = 100T_0$ in (d), since the acceleration length of the electrons can be extended by the increased $\sigma _0$.

According to Fig. 2, most electrons generated by the radially polarized electric field are ejected from the left tip, while a small fraction of electrons is from the lateral surface. Since the increase of $R$ will enlarge the area of the left tip, more electrons will join in the bunches as $R$ increases. The increase of $L$ will also enlarge the number of electrons ejecting from the lateral surface, but it contributes little to the total beam charge. Therefore, the total beam charge will increase with $R$ and $L$ but weakly depends on $L$. As is demonstrated by Fig. 5(a,b), the beam charge is 3.36 nC with $R=1\lambda _0$ and 2.26 nC with $L = 10\lambda _0$, respectively. The effect of laser parameters on the total beam charge is also considered, which shows that the beam energy and charge are mainly determined by the laser power experienced by the left tip of the target. It is easy to figure out that the laser power experienced by the left surface increases with $a_0$ and varies little with $\sigma _0$, so that the total beam charge increases with $a_0$ but varies little with $\sigma _0$, as can be seen from Fig. 5(c,d). We also consider the influence of the laser pre-pulse and the misalignment of the laser pulse off the nanofiber in experiments. It is shown that the generated electron bunches still keep intact in these cases, which demonstrates the robustness of our scheme (see Supplement 1).

## 5. Conclusion

To sum up, we propose a novel DLA scheme to generate high-quality ultrashort electron beams by irradiating a radially polarized laser pulse onto a nanofiber. It is demonstrated that collimated monoenergetic electron bunches can be generated via 3D PIC simulations. The energy spread, the center divergence angle and the FWHM of the divergence angle of the selected electron slice at $t=50T_0$ are 14%, $0.75^{\circ }$ and $1.75^{\circ }$, respectively. Since electrons are continuously squeezed by inward radial electric field force, the beam divergence angle gradually decreases as electrons transport stably along the laser pulse. The well collimated electron bunches are effectively accelerated by the circularly-symmetric longitudinal electric field component, so the electrons in the same bunch will experience almost the same longitudinal laser electric field. The relative energy spread can also gradually decrease, since the peak energy gradually increases with the FWHM of the energy spectrum almost unchanged. A 1D model predicts the maximum energy achieved, which is given and verified by the simulation results. Besides, the location of the electrons ejected from the target is also significantly different from that of the Gaussian laser scheme, since the hollow electromagnetic distribution of the radially polarized laser pulse leads to a weak laser intensity near the axis on the left tip of the nanofiber. The narrow energy spread and low divergence angle of generated the electron bunches make it possible for the stable electron acceleration in the DLA regime. The well collimated monoenergetic electron bunches may have diverse applications such as such as 4D electron diffraction and microscopy [14], relativistic electron mirrors [15] and ultrashort radiation sources [16].

## Funding

National Key Research and Development Program of China (2018YFA0404802); National Natural Science Foundation of China (11775305, 11875319, 11991074); Science Challenge Project (TZ2018005); Fok Ying Tung Education Foundation (161007); Science and Technology Program of Hunan Province (2020RC4020); Research Project of NUDT (ZK18-02-02, ZK19-22); NUDT Young Innovator Awards (20190102); Hunan Provincial Research and Innovation Foundation for Graduate Students (CX20190017, CX20190018); Teaching Reform Program for Advanced Electrodynamics; The Open Fund of the State Key Laboratory of High Field Laser Physics (SIOM).

## Disclosures

The authors declare no conflicts of interest.

## 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.

## Supplemental document

See Supplement 1 for supporting content.

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