Parameters of the two-mode mixing laser-plasma accelerator stage.

## Abstract

A multistage laser-plasma accelerator (LPA) driven by two mixing electromagnetic hybrid modes of a gas-filled capillary waveguide is presented. Plasma wakefields generated by a laser pulse comprising two mixing modes coupled to a metallic or dielectric capillary filled with gas provide us with an efficient accelerating structure of electrons in a substantially long distance beyond a dephasing length under the matching between a capillary radius and plasma density. For a seamless multistage structure of the capillary waveguide, the numerical model of the transverse and longitudinal beam dynamics of an electron bunch considering the radiation reaction and multiple Coulomb scattering effects reveals a converging behavior of the bunch radius and normalized emittance down to ∼1 nm level when the beam is accelerated up to 560 GeV in a 67 m length. This capability allows us to conceive a compact electron-positron linear collider providing with high luminosity of 1034 cm−2 s−1 at 1 TeV center-of-mass (CM) energy.

### Keywords

- future colliders
- lepton colliders
- laser-plasma accelerators
- multistage coupling
- CAN lasers

## 1. Introduction

In the long-standing quest for the fundamental building blocks of nature, the so-called Standard Model of particle physics, energy frontier colliders have played a central role in the forefront research for matter and interactions. For future high-energy particle colliders to explore physics beyond the Standard Model, a proton-proton circular collider at energy of 100 TeV in a 100 km circumference or electron-positron linear collider with energy of the order of 1 TeV in a 30 km length is being considered around the world, exploiting the conventional technologies such as superconducting magnets or RF systems [1]. In contrast to proton colliders that create clouds of debris, electron-positron colliders enable cleaner and more precision experiments of fundamental particle collisions. Nowadays, a diversity of electron-positron linear colliders is proposed as a potential application of advanced accelerator concepts [2], such as two beam accelerators, dielectric wakefield accelerators, beam-driven plasma wakefield accelerators, and laser-driven plasma wakefield accelerators [3], promising with much higher accelerating gradients than that of a conventional RF accelerator.

Laser-plasma accelerators (LPAs) [4, 5] can support a wide range of potential applications requiring high-energy and high-quality electron-positron beams. In particular, field gradients, energy conversion efficiency, and repetition rates are essential factors for practical applications such as compact free electron lasers [6, 7] and high-energy frontier colliders [8, 9]. Although LPAs provide enormous accelerating gradients, as high as 100 GV/m at the plasma density of 10^{18} cm^{−3}, dephasing of relativistic electrons with respect to a correct acceleration phase of the plasma wakefield with the phase velocity that is smaller than the speed of light in vacuum, and energy depletion of the laser pulse limit the electron energy gain in a single stage. A straightforward solution to overcome the dephasing and pump depletion effects is to build a multistage accelerator comprising consecutive LPA stages [3] such that a final energy gain reaches the requirement of the beam energy without loss of the beam charge and qualities through a coupling segment where a fresh laser pulse is fed to continuously accelerate the particle beam from the previous stage. The propagation of laser pulses in plasmas is described by refractive guiding, in which the refractive index can be modified from the linear free space value mainly by relativistic self-focusing, ponderomotive channeling, and a preformed plasma channel [10]. The self-guided LPA [11, 12, 13, 14] relies only on intrinsic effects of relativistic laser-plasma interactions such as relativistic self-focusing and ponderomotive channeling. On the other hand, the channel-guided LPA exploits a plasma waveguide with a preformed density channel [15, 16, 17] or a gas-filled capillary waveguide made of metallic or dielectric materials [18]. The plasma waveguide is likely to propagate a single-mode laser pulse through a radially parabolic distribution of the refractive index and generates plasma waves inside the density channel, the properties of which are largely affected by a plasma density profile and laser power [19]. In contrast with plasma waveguides, the capillary waveguide can guide the laser due to Fresnel reflection on the inner capillary wall, and plasma waves are generated in an initially homogeneous plasma, relying on neither laser power nor plasma density. The presence of the modal structure imposed by the boundary conditions at the capillary wall affects the propagation of a laser pulse through the capillary and thus the excitation of plasma waves inside the capillary. This characteristic allows us to control acceleration of electrons through the modal structure of the propagation of the laser pulse as long as the laser intensity on the capillary wall is kept below the material breakdown [20, 21].

In this paper, we present a novel scheme of a gas-filled capillary accelerator driven by a laser pulse formed from two-mode mixing of the capillary eigenmodes, so-called electromagnetic hybrid modes [20]. Two coupled eigenmodes with a close longitudinal wave number can generate beating wakefields in the capillary. When the beating period is equal to the dephasing distance, the electrons experience the rectified accelerating field; thereby their energy gain can increase over many accelerating phases exceeding the linear dephasing limit and reach the saturation due to the energy depletion of a drive laser pulse in the single-stage LPA. For efficient acceleration of the electron-positron beam up to an extremely high energy such as TeV energies, the multistage accelerator comprising a series of plasma-filled capillary waveguides is a sound approach, in which the particle beam is injected into the initial stage at the right phase of the wakefield from the external injector and accelerated cumulatively in the consecutive accelerating phase of successive stages. For applications of extreme high-energy particle beams to TeV center-of-mass (CM) energy electron-positron linear colliders, minimizing the transverse normalized emittance of the beam particles is of essential importance to meet the requirement of the luminosity of the order of 10^{34} cm^{−2} s^{−1} at 1 TeV CM energy for the particle physics experiments [22]. The numerical model on the bunched beam dynamics in laser wakefields, based on the exact solution of single particle betatron motion taking into account the radiation reaction and multiple Coulomb scattering, reveals that the transverse normalized emittance and beam radius can be consecutively reduced during continuous acceleration in the presence of optimally phased recurrence of longitudinal and transverse wakefields [19]. The final properties of the particle beams reached to the objective energy meet the requirements of the luminosity without resort to an additional focusing system.

The remaining part of this paper is organized as follows. In Section 2, the complete description on the longitudinal and transverse laser wakefields generated by two electromagnetic hybrid modes with moderate intensities coupled to a gas-filled capillary waveguide is provided. In Section 3, the particle beam dynamics on energy gain, beam loading, and betatron motion in a single stage of the two-mode mixing LPA is investigated, taking into account radiation reaction and multiple Coulomb scattering with plasma ions. In Section 4, a multistage coupling with a variable curvature plasma channel is presented. For the multistage comprising two-mode mixing LPAs, the results of numerical studies on the transverse beam dynamics of a particle bunch are shown. Analytical consideration on the evolution of the normalized emittance of the particle beam in the presence of radiation reaction and the multiple Coulomb scattering is given. In Section 5, the performance of a 1-TeV CM energy electron-positron collider comprising the multistage two-mode mixing LPAs is discussed on the luminosity and beam-beam interaction. In Section 6, we conclude our investigation on the proposed laser-plasma linear collider with a summary.

## 2. Laser pulse propagation in a gas-filled capillary tube

For a large-scale accelerator complex such as the energy frontier particle beam colliders, it is axiomatically useful in assembling a long-range multistage structure for the use of long-term experimental operation at a high-precision and high-repetition rate that each electromagnetic waveguide consists of a simple monolithic structure, as referred to the design of the future electron-positron linear colliders based on radio-frequency technologies [22]. Despite the long-standing research on plasma waveguides comprising density channels generated in plasmas with laser-induced hydrodynamic expansion [23, 24] and pulsed discharges of an ablative capillary [25, 26] or a gas-filled capillary [27, 28], a length of such a plasma channel has been limited to about 10 cm. The pulsed discharge capillaries relying on collisional plasma processes have some difficulties in plasma densities less than 10^{17} cm^{−3} and the temporal and spatial stabilities of the density channel properties for the operation at a high repletion rate such as 10 kHz [5, 29]. In contrast to pulsed discharge plasma waveguides, metallic or dielectric capillary waveguides filled with gas [18, 30] will be revisited for a large-scale laser-plasma accelerator operated at a practically higher-repetition rate than 10 kHz, because of the passive optical guiding of laser pulses, the propagating electromagnetic fields of which are simply determined the boundary conditions on a static solid wall of the waveguide unless the laser intensity is high enough to cause the material breakdown on a capillary wall [20, 21]. Furthermore, the modal nature of electromagnetic fields arising from the boundary conditions on a solid wall allows us to conceive a novel scheme that can overcome a drawback of LPAs, referred to as dephasing of accelerated electron beams from a correct acceleration phase in laser wakefields.

### 2.1 Laser-driven wakefields generated by two capillary modes

Considering the electromagnetic hybrid modes EH_{1n} [20] to which the most efficient coupling of a linearly polarized laser pulse in vacuum occurs, the normalized vector potential for the eigenmode of the *n*-th order is written by [31].

where _{1n} mode with the vector potential *me*, and the speed of light in vacuum *n*-th zero of *n*-th mode

where _{1n} and EH_{1m} overlap to cause the beatings of the normalized vector potential, e.g., _{11} - EH_{12} mode mixing of a laser pulse with

The electrostatic potential

where

where

with the real (

The transverse focusing force is obtained from

where

The proposed scheme restricts the laser intensity such that the plasma response is within the quasi-linear regime, i.e.,

### 2.2 Coupling control for generating two capillary modes

The coupling efficiency _{1n} mode in the capillary with the radius

and for a Gaussian beam,

where _{11}-EH_{12} mode mixing with _{11}-EH_{12} mode mixing with

The radial intensity profiles for the EH_{11}, EH_{12} monomode and EH_{11}-EH_{12} mixing mode for the Airy beam case are illustrated in Figure 1c–e, respectively. As shown in Figure 1e, a high-intensity region of the mixing mode is confined within a half radius of the capillary, compared to the monomode intensity profiles, which have a widespread robe toward the capillary wall. A centrally concentrated intensity profile of the mixing mode considerably decreases the energy flux traversing on the capillary wall. The normalized flux for EH_{1n} mode at the capillary wall depends on the azimuthal angle _{11}, EH_{12} mono- and EH_{11}-EH_{12} mixing modes at ^{2} for the corresponding modes, as shown in Figure 1f. The experimental study of laser-induced breakdown in fused silica (SiO_{2}) [32] suggests that the fluence breakdown threshold is scaled to be

The coupling efficiency of an incident laser pulse to a capillary tube filled with plasma can be improved by the use of a cone-shape entrance of the capillary [34], suppressing self-focusing effects and increasing the accelerating wakefield excited in the capillary. For the propagation of a laser beam with an approximately Gaussian intensity profile _{1n} mode, i.e., _{11}-EH_{12} mixing modes in a capillary.

## 3. Beam dynamics in a single-stage two-mode mixing LPA

### 3.1 Electron acceleration

In the linear wakefields excited by two coupled modes EH_{11} and EH_{12} in the capillary waveguide, the longitudinal motion of an electron traveling along the capillary axis at a normalized velocity

where

Taking into account

where

While propagating through plasma and generating wakefields, the laser pulse loses its energy as

where

The maximum energy gain to be attainable at

Considering the mixing of two lowest order hybrid modes EH_{11} and EH_{12} with the coupling efficiencies _{11} monomode with

The average energy gain of electrons contained in the bunch with the root-mean-square (rms) bunch length

Figure 3 shows the evolution of the energy gain and the maximum attainable energy gain averaged over electrons in a Gaussian bunch with various rms lengths. It is noted that the maximum attainable energy gain at

### 3.2 Beam loading

In the linear regime, a solution of the Green’s function for the beam-driven wakefield excited by a charge bunch with bi-Gaussian density distribution

and inside the bunch (

where

where

where

and the rms energy spread due to the beam loading is estimated as

where

### 3.3 Betatron motion

In the wakefield, an electron moving along the z-axis undergoes a transverse focusing force *x* and exhibits the betatron motion. Taking into account *z*-axis

Transverse wakefield excited by the electron bunch is obtained from Eq. (20) according to the Panofsky-Wenzel theorem [39],

At the bunch center

where

and

where

The equations of motion of an electron propagating in the wakefield behind the laser pulse is written as [41].

where *x* and *t*, respectively. Here the longitudinal wakefield and focusing constant at

where

While the electron stays in the focusing region of the wakefield, i.e., *s* becomes imaginary, the amplitude of the electron trajectory increases monotonically as a result of the Bessel functions being transformed to the modified Bessel functions, leading to ejection of the electron from the wakefield [41]. Hence, the requirement of betatron oscillation in the focusing region

for a bi-Gaussian bunch with the rms radius _{11}-EH_{12} mode mixing LPA. It is noted that the minimum value of the requisite electron number occurs at the bunch length

In the bunch containing the requisite number of particles, an electron undergoes betatron motion throughout the whole accelerating phase, as shown in Figure 5, where the trajectory and momentum of the electron in the bunch with the number of electrons ^{5} segments of the laser wakefield phase excited in the plasma with density

### 3.4 Effects of radiation reaction and multiple Coulomb scattering

A beam electron propagating in the wakefield undergoes betatron motion that induces synchrotron (betatron) radiation at high energies. The synchrotron radiation causes the radiation damping of particles and affects the energy spread and transverse emittance via the radiation reaction force. Furthermore, a notable difference of plasma-based accelerators from vacuum-based accelerators is the presence of the multiple Coulomb scattering between beam electrons and plasma ions, which counteracts the beam focusing due to the transverse wakefield and radiation damping due to betatron radiation. The comprehensive motion of an electron traveling along the *z*-axis is described as

where *x*-plane due to multiple Coulomb scattering through small deflection angles

For the classical expression of the radiation reaction force given by [42].

where

Since the scale length of the radiation reaction, i.e.,

A beam electron of the incident momentum *Ze* at impact parameter *b* in the plasma, suffers an angular deflection

where *A*. Here, the logarithm

The electron orbit and energy are obtained from the solutions of the coupled equations in Eq. (33) describing the single particle motion in the segmented phase, where

where

with

The radiated power of the electron in the classical limit is given by [42, 43].

where

### 3.5 Numerical studies of the single-particle dynamics in a single stage

Numerical calculations of the single-particle dynamics can be carried out throughout the segments in phase *i*-th particle and

where

The particle orbit and energy can be numerically tracked by using the solutions of the single particle motion (Eqs. (30) and (14)) associated with the perturbation arising from the effects of the radiation reaction and multiple Coulomb scattering, as given by Eqs. (38) and (39), respectively. The simulation of particle tracking can be carried out by using an ensemble of 10^{4} test particles, for which the initial values at the injection and the deflection angles due to the multiple Coulomb scattering at each segment in a phase step

## 4. Beam dynamics in multistage two-mode mixing LPAs

### 4.1 Seamless stage coupling with a variable curvature plasma channel

A gas-filled capillary waveguide made of metallic or dielectric materials can make it possible to comprise a seamless staging without the coupling section, where a fresh laser pulse and accelerated particle beam from the previous stage are injected so as to minimize coupling loss in both laser and particle beams and the emittance growth of particle beams due to the mismatch between the injected beam and plasma wakefield. For dephasing limited laser wakefield accelerators, the total linac length will be minimized by choosing the coupling distance to be equal to a half of the dephasing length [9]. A side coupling of laser pulse through a curved capillary waveguide [46, 47, 48] diminishes the beam-matching section consisting of a vacuum drift space and focusing magnet beamline [9]. Furthermore, the proposed scheme comprising seamless capillary waveguides can provide us with suppression of synchrotron radiation from high-energy electron (positron) beams generated by betatron oscillation in plasma-focusing channels and delivery of remarkably small normalized emittance from the linac to the beam collision section in electron-positron linear colliders.

Since the electron beam size with a finite beam emittance causes a rapid growth in a vacuum drift space outside plasma [41], the coupling segment must be used for spatial matching of the electron beam with the transverse wakefield as well as temporal phase matching with the accelerating wakefield in a subsequent stage. A proof-of-principle experiment on two LPA stages powered by two synchronized laser pulses through the plasma lens and mirror coupling has been reported, showing that an 120 MeV electron beam from a gas jet (the first stage) driven by a 28 TW, 45 fs laser pulse was focused by a first discharge capillary as an active plasma lens to a second capillary plasma channel (the second stage), where the wakefield excited by a 12 TW, 45 fs separate laser pulse reflected by a tape-based plasma mirror with a laser-energy throughput of 80% further increased an energy gain of 100 MeV [49]. In this experiment, a trapping fraction of the electron charge coupled to the second stage was as low as 3.5% [3]. Such a poor coupling efficiency could be attributed to the plasma mirror inserted at a vacuum drift space. To avoid a rapid growth in the vacuum drift space and improve coupling efficiency, a multistage coupling using a variable curvature plasma channel [48] enables off-axial injection of a fresh laser pulse into the LPA stage without a vacuum gap in the coupling segment; thereby an electron bunch is continuously accelerated through the plasma-focusing channel over the consecutive stages only if the temporal phase-matching between the laser and electron beams can be optimized [3].

In the propagation of a laser pulse through a curved plasma channel, the radial equilibrium position of the laser pulse is shifted away from the channel axis due to the balance between the refractive index gradient bending the light rays inward and the centrifugal force pulling them outward. As a result, the minimum of the effective plasma density, which is proportional to a guiding potential, is located outward from the channel axis [47]. Thus, a direct guiding of a laser pulse from the curved channel with a constant curvature to the straight channel causes large centroid oscillations in the straight channel even though the laser pulse is injected to the equilibrium position, leading to loss of the laser energy and electron beam transported from the previous stage as a result of off-axis interaction with plasma wakefields [48]. To diminish the mismatching at the transition from a curved channel to a straight one [3], a variable curvature plasma channel has been devised such that the equilibrium position guides the laser centroid gradually along the channel axis from the initial equilibrium position to the channel center, where the straight channel axis merges together, as shown in Figure 7a. A seamless acceleration in two-stage LPA coupled with a variable curvature plasma channel has been successfully demonstrated for the guided laser intensity of 8.55 × 10^{18} W/cm^{2} (normalized vector potential *a0* = 2) by the three-dimensional particle-in-cell simulations, as shown in Figure 7b–f, indicating that the injection trapping efficiency increases with the initial beam energy and approaches 100% at energies higher than 2 GeV.

### 4.2 Betatron motion of the particle beam in the seamless multistage

For

where *k*-th stage is given by

where *k*-th stage, respectively, assuming an approximately constant focusing strength *k*-th and *l*-th stages, the ratio of the accelerating field amplitude is given by

where *Ns* stages, the final betatron amplitude yields

where *Ns*-th stage, respectively, and

Here we consider the evolution of transverse normalized emittance for the particle beam acceleration in the multistage capillary accelerator. The definition of transverse normalized emittance given by Eq. (42) is expressed as

where

This indicates that in the absence of radiation and multiple Coulomb scattering, the transverse normalized emittance of an initially matched beam is conserved in the laser wakefield acceleration when the amplitude of accelerating field has no variation, i.e.,

the initial amplitude of betatron oscillation at the next stage is

Accordingly, the emittance at the *k*-th stage is calculated as

Assuming *k*-th stage yields

where

For an application of laser-plasma accelerators to electron-positron colliders, it is of most importance to achieve the smallest possible normalized emittance at the final stage of the accelerator system, overwhelming the emittance growth due to the multiple Coulomb scattering off plasma ions, being increased in proportion to the square root of the beam energy. We consider the effect of multiple Coulomb scattering on the emittance growth and evaluate an achievable normalized emittance at the end of the accelerator system comprising *Ns* stages. Using the growth rate of the mean square deflection angle in Eq. (36) due to the multiple Coulomb scattering, the growth rate of the transverse normalized emittance is estimated as [8, 44].

where

At the *Ns*-th stage, the normalized emittance can be obtained from

Assuming that the beam energy at the *k*-th stage is approximately given by

for

where *Ns*-th stage is simply calculated as

As expected, the normalized emittance in the multistage accelerator operated with the constant accelerating field is conserved to the initial normalized emittance and then limited by an increasing growth due to multiple Coulomb scattering. For

### 4.3 Numerical studies of the single-particle dynamics in multistages

Numerical studies on transverse beam dynamics of an electron bunch accelerated in the multistage mode mixing LPA have been carried out by calculating the ensemble of trajectories of test particles throughout consecutive stages, using the single-particle dynamics code based on the analytical solutions of the equations of motion of an electron in laser wakefields with the presence of effects of the radiation reaction and multiple Coulomb scattering, as described in Section 3. Figure 8a shows examples of the phase space distribution of 10^{4} test particles on the

The detailed study on the evolution of the transverse normalized emittance in the multistage two-mode mixing LPA has been investigated for three cases with the different stage phases, i.e.,

## 5. Considerations on electron-positron collider performance

Electron and positron beams being reached to the final energies in the multistage two-mode mixing LPA are extracted at a phase corresponding to the minimum transverse normalized emittance, followed by propagating a drift space in vacuum and a final focusing system to the beam-beam collisions at the interaction point. In a vacuum drift space outside plasma, the particle beam changes the spatial and temporal dimensions of the bunch proportional to the propagation distance due to the finite emittance and energy spread of the accelerated bunch. The evolution of the rms bunch envelope ^{+}e^{−} linear colliders in the TeV center-of-mass (CM) energies, for which two major effects must be taken into account, namely, the disruption effect bending particle trajectories by the oncoming beam-generated electromagnetic fields and the beamstrahlung effect yielding radiation loss of the particle energies induced by bending their trajectories due to the disruption [52]. The radiative energy loss due to beamstrahlung for a Gaussian beam can be estimated in terms of the beamstrahlung parameter _{11} and EH_{12} is shown in Table 1.

Plasma density n_{e} | 1 × 10^{8} cm^{−3} |

Plasma wavelength λ_{p} | 33.4 μm |

Capillary radius R_{c} | 152.6 μm |

Capillary stage length | 16.75 cm |

Laser wavelength λ | 1 μm |

Laser spot radius r_{0} | 91 μm (51 μm) |

Laser pulse duration τ | 25 fs |

Normalized vector potential a_{0} | 1 |

Electromagnetic hybrid mode | EH_{11} and EH_{12} |

Coupling efficiency for an Airy beam (a Gaussian beam) | C_{1} = 0.4022 (0.5980)C _{2} = 0.4986 (0.3531) |

Bunch initial and final phase | Ψi = 0, Ψf = 4.5π |

Average accelerating gradient | 8.3 GeV/m |

Laser peak power PL | 95 TW (18 TW) |

Laser pulse energy UL | 2.4 J (0.44 J) |

Repetition frequency fc | 50 kHz |

Laser average power per stage | 120 kW (22 kW) |

Laser depletion η_{pd} | 77% |

An embodiment of the LPA stage may be envisioned by exploiting a tens kW-level high-average power laser such as a coherent amplification network of fiber lasers [53]. Table 2 summarizes key parameters on the performance of 1 TeV CM energy electron-positron linear collider.

CM energy | 1.12 TeV |

Beam energy | 559 GeV |

Injection beam energy | 1 GeV |

Particle per bunch N_{b} | 1 × 10^{8} |

Collision frequency fc | 50 kHz |

Total beam power | 0.9 MW |

Geometric luminosity 0 | 3.6 × 10^{32} cm^{−2} s^{−1} |

Effective luminosity | 1.76 × 10^{34} cm^{−2} s^{−1} |

Effective CM energy | 1.09 TeV |

rms CM energy spread | 8.4% |

rms bunch length σz | 16 μm |

Beam radius at IP σb* | 3.3 nm |

Beam aspect ratio R | 1 |

Normalized emittance at IP εnf | 3.7 pm-rad |

Distance between LPA and IP L_{col} | 0.2 m |

Beamstrahlung parameter ϒ_{*} | 0.94 |

Beamstrahlung photons nγ | 0.52 |

Disruption parameter D | 12 |

Luminosity enhancement HD | 49 |

Number of stages per beam Ns | 400 |

Linac length per beam | 67 m |

Power requirement for lasers | 95 MW (18 MW) |

## 6. Conclusions

The electron acceleration and beam dynamics of the two-mode mixing LPA comprising a gas-filled metallic or dielectric capillary have been presented for the performance of the single-stage and multistage configurations. As shown in Table 1, when a laser pulse with an Airy beam (or a Gaussian beam) profile of the spot radius _{11} and EH_{12} are generated with the coupling efficiency _{11} and EH_{12} with a Gaussian temporal profile can efficiently excite a rectified accelerating wakefield, where relativistic electrons dominantly propagate in the accelerating phase and continuously gain the energy until depletion of the laser pulse energy, whereby a nearly 100% of the laser energy can be transferred to wakefields in the single stage.

In the two-mode mixing LPA multistage coupled with a variable curvature plasma channel, the transverse dynamics of the electron bunch is dominated by seamless recurrence of the accelerating wakefield in the stages, where the cumulative nature of the particle trajectories is determined by the amplitude ratio of the accelerating field at the final phase ^{34} cm^{−2} s^{−1} at 1 TeV center-of-mass energy in a very compact size.

In conclusion, a novel scheme of 1 TeV electron-positron linear collider comprising properly phased multistage two-mode mixing LPAs using gas-filled capillary waveguides can provide a unique approach in collider applications. This scheme presented resorts two major mechanisms pertaining to laser wakefield acceleration, that is, dephasing and strong focusing force as well as very high-gradient accelerating field. The multistage scheme using two-mode mixing capillary waveguides filled with plasma may provide a robust approach leading to the supreme goal for LPAs. The numerical model developed for study on beam dynamics in large-scale LPAs will be useful for assessing effects of underlying physics and the optimum design for future laser-plasma-based colliders. Although the present model has been developed to study the simplest two-dimensional phase-space model of electron beam dynamics in laser wakefield acceleration, the analysis of higher multi-dimensional phase-space model as well as the quantum plasma effect will be extensively pursued in the future work.

## Acknowledgments

The work was supported by the NSFC (No. 11721091, 11774227), the Science Challenge Project (No.TZ2018005), and the National Basic Research Program of China (No. 2013CBA01504).