Enhanced plasma wave excitation in a tapered plasma channel through chirped laser beatwaves

Arefnia, M.; Kim, S.; Lee, C.; Ghorbanalilu, M.; Suk, H.

doi:10.1063/5.0245587

The excitation of plasma waves by the beatwave of two-color laser beams within a tapered plasma channel has been investigated both analytically and through particle-in-cell (PIC) simulations. This study presents a promising approach for beatwave accelerators. The design aims to achieve a more stable and enhanced plasma wave amplitude, even under non-resonant conditions, compared to untapered channels. For the analytical description, we solved the plasma wave equation generated by two beating laser beams propagating through a tapered channel with a linear radial density profile. The study further validates its results by comparing the analytical predictions with PIC simulations of the beatwave phenomenon in the tapered channel. The findings confirm that this design significantly enhances plasma wave amplitude compared to untapered channels. In addition, we demonstrated that employing a chirped laser pulse with a fast rise time can significantly increase the strength of the plasma wave.

I. INTRODUCTION

Plasma-based accelerators, first proposed five decades ago, are compact devices capable of producing acceleration gradients several thousand times higher than those achieved with conventional radio frequency-based techniques.¹ In this approach, a powerful driver—such as an intense laser pulse or high-energy electron bunch—excites a trailing plasma wave, in which electrons can become trapped and accelerated over millimeter-scale distances.^2,3

The most commonly used lasers in laser-based accelerators are Ti lasers, which typically operate at a wavelength of 800 nm and deliver energies between 1 and 50 J.^4,5 Long-wavelength driver lasers were initially explored in early plasma beatwave acceleration experiments.^6–8

While advancements in the CO₂ laser system at Brookhaven National Laboratory have driven research toward self-modulated laser wakefield acceleration, plasma accelerators continue to hold their relevance.⁹ When a plasma wave is excited resonantly using a two-frequency laser beatwave, electrons can be accelerated to 50 MeV with an average gradient of 5 GeV/m.¹⁰

Although high-energy particle physics aims optimistically for the TeV range, compact plasma-based accelerators can still achieve GeV-scale energies—thousands of times more than the enormous, tens-of-kilometer-long conventional accelerators. However, the acceleration distance and energy output are limited by factors such as drive beam diffraction, dephasing, and the depletion of laser pulse energy. Diffracted lasers are unable to excite the wakefield effectively, and the acceleration distance is limited by inherent laser defocusing.¹¹ In plasma channels, however, laser pulses can be guided over much longer distances than the Rayleigh length due to the transverse density profiles.¹² Techniques such as relativistic self-guiding in the bubble regime or the use of preformed plasma channels help counteract laser diffraction, enhancing the guiding of laser pulses.^13,14

To create plasma channels, several methods are employed, such as generating a precursor plasma using laser pulse-discharged capillaries or Z-pinch discharge in gas-filled capillaries.^15,16 In addition, plasma channels have been formed by inducing time-periodic density perturbations in a uniform plasma.^17,18 The primary challenge in plasma-based accelerators is to excite large-amplitude plasma waves.^19–21

In plasma-based accelerators, as electrons are accelerated, a speed gap forms between the accelerated electrons and the plasma wave. The dephasing length, which limits the energy transfer from the wave to the electrons, can be controlled by optimizing the plasma density. To overcome this dephasing limitation, spatial tailoring of the plasma density—known as plasma tapering—has been proposed.

As the laser accelerates the particles, they tend to outrun the laser beam. This causes the plasma wavelength to shorten and the plasma density to increase. This interaction allows the particles to remain in the phase of the plasma wave.^22,23

The aim of this paper is to provide a detailed analysis of plasma wave excitation by a laser beatwave in a tapered plasma channel. We derived the plasma wave equation by coupling a fluid model with Maxwell’s equations in the plasma channel. We assume a plasma with upward density tapering along the axial direction, where the channel radius varies along its axis.

Both particle-in-cell (PIC) simulations and analytical investigations have demonstrated the generation of higher-amplitude plasma waves in the tapered plasma channel. This is attributed to the density gradient, which enhances the focusing and trapping of laser beams, resulting in more intense interaction with the plasma and the formation of higher-amplitude plasma waves.

While beatwave accelerators rely on the interaction of two laser beams to generate strong plasma waves for accelerating charged particles, dephasing from constant-frequency pulses hinders the effective generation of large-amplitude plasma waves. Chirped lasers, however, reduce dephasing by optimizing their interaction with the evolving plasma wave throughout its propagation.^24–29 This not only increases the amplitude of the generated plasma waves but also improves phase matching, ultimately leading to significantly stronger acceleration.

Although there are independent studies on plasma wave generation and electron acceleration via beatwaves, chirped laser pulses, and tapered plasma channels, a comprehensive investigation that combines the effects of these factors is currently lacking in the literature. In this paper, we explore the interaction between chirped beating laser pulses within a tapered plasma channel, the resulting plasma wave generation, and its potential for electron acceleration.

This paper is organized as follows: In Sec. II, we present a detailed analytical solution of the plasma wave equation driven by two beating laser beams in the tapered plasma channel. In Sec. III, we investigate the plasma wave behavior in a tapered plasma channel using PIC simulations and compare the results with the analytical predictions. The effects of chirped laser beams in plasma channels are discussed in Sec. IV. Finally, Sec. V provides a conclusion of the results.

II. PLASMA WAVE DYNAMICS IN A TAPERED PLASMA CHANNEL

A. Electrostatic field equation for plasma wave

In this section, we derive the equation for the excited plasma wave generated by the beatwave within the tapered plasma channel. We define the linearly polarized laser beams as

{\vec{E}}_{1, 2} = \hat{y} E_{1, 2} \sin (k_{1, 2} z - ω_{1, 2} t),

(1)

where E_1,2, ω_1,2, and k_1,2 represent the electric field amplitudes, frequencies, and wave numbers of the two laser beams, respectively. We consider a plasma with a density profile that increases linearly along the axial direction (z) and varies parabolically in the radial direction (r). This density profile is described by the following equation:

n (r, z) = n (z) + δ n \frac{r^{2}}{r_{c h}^{2} (z)},

(2)

where n(z) = n₀(1 + z/L), with n₀ representing the initial plasma density; δn is the density perturbation; and L is the plasma length. The plasma channel, illustrated in Fig. 1, has a radius r_ch(z) that varies along the z-axis. The equation governing electron momentum transfer is given by¹⁸

m n (r, z) (\frac{\partial \vec{v}}{\partial t} + \vec{v} . \nabla \vec{v}) = - \nabla p + n e (\vec{E} + \vec{v} \times \frac{\vec{B}}{c}),

(3)

where

\vec{E}

is the electrostatic field,

\vec{v}

is the electron velocity, and

\vec{v} \times \vec{B} / c

arises from the beating of two laser beams. The constants m and e refer to the electron mass and charge, respectively. The electronic pressure is given by p = Cn^γ, where C is a constant and γ = 3 for an adiabatic process.

FIG. 1.

Schematic illustrating laser beams entering a tapered plasma channel with a channel radius defined by rch=(r0/1.02)1.05+(z/L).

View large Download slide

Schematic illustrating laser beams entering a tapered plasma channel with a channel radius defined by $r_{c h} = (r_{0} / 1.02) \sqrt{1.05 + (z / L)}$ ⁠.

By applying Eq. , alongside Maxwell’s equations and second-order hydrodynamic equations, the plasma wave equation can be simplified in the linear regime as follows:

\nabla^{2} E - (α z + 0.95 λ r^{2}) E = s \exp (i Δ k z),

(4)

where

α = ω_{0}^{2} / 3 L v_{e}^{2}

⁠,

λ = 1.05 ω_{0}^{2} δ n / 3 v_{e}^{2} r_{0}^{2} n_{0}

⁠, and

s = - i e ω_{p}^{2} (r, z) Δ k E_{1} E_{2}^{*} \exp (i Δ k z) / 12 m ω_{1} ω_{2} v_{e}^{2}

⁠. The plasma frequency is given by

ω_{p} (r, z) = \sqrt{ω_{0}^{2} (1 + z / L + δ n r^{2} / n_{0} r_{c h} {(z)}^{2})}

⁠, where ω₀ = ω₁ − ω₂.

The wave equation becomes

\frac{\partial^{2} E}{\partial r^{2}} + \frac{1}{r} \frac{\partial E}{\partial r} + \frac{\partial^{2} E}{\partial z^{2}} - (α z + 0.95 λ r^{2}) E = s \exp (i Δ k z) .

(5)

Equation represents a second-order inhomogeneous partial differential equation (PDE) that can be solved using suitable approximation methods.

B. Fundamental wave modes in the tapered plasma channel

To solve Eq. , we first consider the homogeneous case, where there is no external source. We achieve this by separating the axial and radial variables of the electric field, expressing it as a function of R(r) and Q(z),

\frac{1}{R (r)} \frac{d R (r)}{d r} + \frac{r}{R (r)} \frac{d^{2} R (r)}{d r^{2}} + \frac{r}{Q (z)} \frac{d^{2} Q (z)}{d z^{2}} - (α r z + 0.95 λ r^{3}) = 0 .

(6)

Employing separation of variables, we derive the subsequent equations,

\frac{1}{Q (z)} \frac{d^{2} Q (z)}{d z^{2}} - α z = - K^{2},

(7a)

\frac{1}{r R (r)} \frac{d R (r)}{d r} + \frac{1}{R (r)} \frac{d^{2} R (r)}{d r^{2}} - 0.95 λ r^{2} = K^{2} .

(7b)

We solve Eq. using the Airy equation for limits of z → ±∞. Setting K² = K′²α^2/3 allows us to apply the method from Ref. 18. For non-zero K, the solution is

\begin{align} Q_{+} (+ \infty) & = {(π)}^{1 / 2} {(\frac{α}{z} (1 - \frac{K^{' 2} α^{2 / 3}}{α z}))}^{1 / 4} \\ \times \exp (- \frac{2}{3} \frac{1}{α} {(α z - K^{' 2} α^{2 / 3})}^{3 / 2}), z \to \infty, \end{align}

(8)

\begin{align} Q_{-} (- \infty) & = {(π)}^{1 / 2} {(\frac{α}{z} (1 - \frac{K^{' 2} α^{2 / 3}}{α z}))}^{1 / 4} \\ \times \exp (- \frac{2}{3} \frac{1}{α} | α z - K^{' 2} |^{3 / 2} - i π / 4), z \to - \infty . \end{align}

(9)

For K = 0, we have

Q_{+} (+ \infty) = {(π)}^{1 / 2} {(\frac{α}{z})}^{1 / 4} \exp (- \frac{2}{3} α^{1 / 2} z^{3 / 2}), z \to \infty,

(10)

Q_{-} (- \infty) = {(π)}^{1 / 2} {(\frac{α}{z})}^{1 / 4} \exp (- \frac{2}{3} α^{1 / 2} | z |^{3 / 2} - i \frac{π}{4}), z \to - \infty .

(11)

Equations and are entirely consistent with previous calculations.⁶

The radial variation is obtained by solving Eq. . We first consider the solution of Eq. in the case where K² ≠ 0. To do this, we convert Eq. into the standard form of a second-order linear differential equation,

\frac{d^{2} R_{1} (r)}{d r^{2}} + q_{1} (r) R_{1} (r) = 0,

(12)

where

R_{1} (r) = R (r) \exp (\frac{1}{2} \int \frac{1}{r^{'}} d r^{'})

and

q_{1} (r) = (- λ r^{2} + {K^{″}}^{2} λ^{1 / 2}) + \frac{1}{4 r_{0}^{2}}

with K″² = K²λ^−1/2. Equation can be rewritten as follows:

\frac{d^{2} R_{1} (r)}{d r^{2}} + (0.95 λ r^{2} + K^{'' 2} λ^{1 / 2} + \frac{1}{4 r_{0}^{2}}) R_{1} (r) = 0 .

(13)

Equation is a second-order ordinary linear differential equation that is similar to the Weber equation in connection with the parabolic cylinder. By solving Eq. , R(r) is obtained,

\begin{align} R (r) & = {(\frac{r}{r_{0}})}^{\frac{1}{2}} e^{- {K^{″}}^{2} λ^{1 / 2} r^{2}} r^{K^{2} - \frac{1}{4 r_{0}^{2}}} \{1 - \frac{(K^{2} - \frac{1}{4 r_{0}^{2}}) (K^{2} - \frac{3}{4 r_{0}^{2}})}{2 r^{2}} \\ + \frac{(K^{2} - \frac{1}{4 r_{0}^{2}}) (K^{2} - \frac{3}{4 r_{0}^{2}}) (K^{2} - \frac{5}{4 r_{0}^{2}}) (K^{2} - \frac{5}{4 r_{0}^{2}})}{2.4 r^{4}} - \dots\}, \end{align}

(14)

where K is obtained by applying the Neumann boundary condition

\frac{d R (r)}{d r} |_{r_{= 0}} = 0

⁠, as

K^{2} = \frac{1}{4 r_{0}^{2}}

⁠. On the other hand, the solution for Eq. with K² = 0 is given by the modified Bessel function K_ν(r) as follows:

K_{ν} (r) = \sqrt{\frac{π}{0.5 \sqrt{λ} r^{2}}} e^{- (0.5 \sqrt{λ} r^{2})} .

(15)

Using Eqs. –, the solution of Eq. can be derived for two specific limits,

\begin{align} E_{+} (+ \infty) & = π^{1 / 2} {(\frac{α}{z} (1 - \frac{K^{' 2} α^{2 / 3}}{α z}))}^{1 / 4} \\ \times \exp (- \frac{2}{3 α} {(α z - K^{' 2} α^{2 / 3})}^{3 / 2}) + π^{1 / 2} {(\frac{α}{z})}^{1 / 4} \\ \times \exp (- \frac{2}{3} α^{1 / 2} z^{3 / 2}) ({(\frac{r}{r_{c h} (z)})}^{1 / 2} e^{{K^{″}}^{2} λ^{1 / 2} r^{2}} \\ + \sqrt{\frac{π}{0.5 \sqrt{λ r^{2}}}} e^{- (0.5 \sqrt{λ} r^{2})}), z \to \infty, \end{align}

(16)

\begin{align} E_{-} (- \infty) & = π^{1 / 2} {(\frac{α}{| z |} (1 - \frac{K^{' 2} α^{2 / 3}}{α | z |}))}^{1 / 4} \\ \times \exp (- \frac{2 i}{3 α} {(α z - K^{' 2} α^{2 / 3})}^{3 / 2} - i \frac{π}{4}) + π^{1 / 2} {(\frac{α}{| z |})}^{1 / 4} \\ \times \exp (- \frac{2}{3} i α^{1 / 2} | z |^{3 / 2} - i \frac{π}{4}) ({(\frac{r}{r_{c h} (z)})}^{1 / 2} e^{{K^{″}}^{2} λ^{1 / 2} r^{2}} \\ + \sqrt{\frac{π}{0.5 \sqrt{λ r^{2}}}} e^{- (0.5 \sqrt{λ}) r^{2}}), z \to - \infty . \end{align}

(17)

Equations (16) and (17) show that the plasma wave propagating along the positive z-axis (z → ∞) decays, leaving only the wave traveling in the negative z-direction (z → −∞). These equations establish the foundation for the nonhomogeneous solution with a source term. By introducing the z-dependence of the channel radius [r_ch(z)], we can effectively capture how the taper profile specifically influences wave propagation driven by the source term.

C. Non-homogeneous solution of the plasma wave equation

At this stage, we analyze Eq. using the Green’s function method. The general solution is expressed as E = E_h + E_p, where E_h and E_p represent the homogeneous and nonhomogeneous solutions, respectively. The nonhomogeneous solution is given by the following equation:

E_{p} (r, z) = \frac{1}{β} \frac{\int_{0}^{r} d ξ \int_{0}^{z} s (ξ, χ) e^{- i Δ k (χ) χ} E (ξ, χ) d χ}{⟨ E, E ⟩} E (r, z),

(18)

where β is a constant that will be determined from boundary conditions and χ and ξ represent the axial and radial variables from the beginning and center of the channel, respectively.

First, we need to solve the denominator (⟨E, E⟩) of Eq. , where E is a homogeneous solution obtained in Subsection II B. Since the plasma wave exists in the limit z → −∞, we will solve the general solution as follows:

⟨ E, E ⟩ = \int_{0}^{r} d ξ \int_{0}^{z} {(E (χ, ξ))}^{2} d χ .

(19)

We solve this equation for outgoing waves in z → −∞ as follows:

\begin{align} ⟨ E_{-}, E_{-} ⟩ & = \int_{0}^{r} \int_{0}^{z} ((π α^{1 / 2} \frac{1}{| χ |} {(| χ | - K^{' 2} α^{- 1 / 3})}^{1 / 2} + π α^{1 / 2} \frac{1}{| χ |^{3 / 4}} {(| χ | - K^{' 2} α^{- 1 / 3})}^{1 / 4} \exp (- \frac{2 i α^{1 / 2}}{3} {(| χ | - K^{' 2} α^{- 1 / 3})}^{3 / 2} - | χ |^{3 / 2}) \\ + π α^{1 / 2} \frac{1}{| χ |^{1 / 2}} + π α^{1 / 2} \frac{1}{| χ |^{3 / 4}} {(| χ | - K^{' 2} α^{- 1 / 3})}^{1 / 4} \exp (\frac{2 i α^{1 / 2}}{3} {(| χ | - K^{' 2} α^{- 1 / 3})}^{3 / 2} - | χ |^{3 / 2})) \\ \times (\frac{ξ}{r_{c h} (z)} e^{{2 K^{″}}^{2} λ^{1 / 2} ξ^{2}} + \frac{π}{0.5 \sqrt{λ} ξ^{2}} e^{- (0.5 \sqrt{λ} ξ^{2})} + 2 {(\frac{2 π \sqrt{L}}{ξ r_{c h} (z) \sqrt{λ}})}^{1 / 2} e^{({K^{″}}^{2} λ^{1 / 2} ξ^{2} - 0.5 \sqrt{λ} ξ^{2})})) d χ d ξ . \end{align}

(20)

By solving the integral for the axial component first, we can derive the equation for the radial component, leading to the general solution for the denominator (see Appendix A for Φ₁–Φ₁₂ definitions),

⟨ E_{-}, E_{-} ⟩ = \sum_{j = 1}^{12} Φ_{j} .

(21)

Now, we solve the numerator of Eq. in the form of

\begin{align} \int_{0}^{r} d ξ \int_{0}^{z} s (ξ, χ) e^{- i Δ k (χ) χ} E (ξ, χ) d χ = - \frac{i}{12} \frac{e}{m} \frac{E_{1} E_{2}^{*} Δ k ω_{0}^{2}}{ω_{1} ω_{2} v_{e}^{2}} \int_{0}^{r} d ξ \int_{0}^{z} (\frac{π^{1 / 2} α^{1 / 4} {(χ - K^{' 2} α^{- 1 / 3})}^{1 / 4}}{χ^{1 / 2}} \\ \times \exp (- \frac{2 i α^{1 / 2}}{3} {(χ - K^{' 2} α^{- 1 / 3})}^{3 / 2} - \frac{i π}{4}) + \frac{π^{1 / 2} α^{1 / 4}}{χ^{1 / 4}} \exp (- \frac{2 i α^{1 / 2}}{3} χ^{3 / 2} - \frac{i π}{4})) ({(\frac{ξ}{r_{c h} (χ)})}^{1 / 2} e^{{K^{″}}^{2} λ^{1 / 2} ξ^{2}} \\ + \sqrt{\frac{π}{0.5 \sqrt{λ ξ^{2}}}} e^{- (0.5 \sqrt{λ} ξ^{2})}) (1 + \frac{χ}{L} + \frac{δ n}{n_{0}} \frac{1.04 ξ^{2}}{r_{0}^{2} ((1.05 + \frac{χ}{L}))}) \exp (- i Δ k (ξ, χ) χ) d χ = (\sum_{j = 1}^{4} Ψ_{j}) . \end{align}

(22)

Therefore, the general solution of Eq. using Eqs. , , and is obtained as follows (see Appendix B for Ψ₁–Ψ₄ definitions):

E_{p} (r, z) = \frac{1}{β} \frac{\sum_{j = 1}^{4} Ψ_{j}}{\sum_{j = 1}^{12} Φ_{j}} E_{-} (- \infty) .

(23)

Equation is the non-homogeneous solution for Eq. . Combining this with the homogeneous solution results in the complete solution, expressed as follows:

E (r, z) = E_{p} (r, z) + E_{-} (r, - \infty) .

(24)

D. Characterizing beatwave-excited plasma waves in the tapered plasma channel

Figure 2 presents a comparison of plasma waves generated by beatwave lasers in both tapered and untapered plasma channels. The plots reveal distinctly different wave profiles, illustrating the impact of channel geometry on wave behavior. These results are based on specific parameters: CO₂ laser beam wavelengths of λ = 10.6 μm and λ = 10.3 μm, amplitudes of E_1,2 = 2.7 × 10⁸ V/cm, a laser intensity of 10¹⁴ W/cm², and an electron temperature in the kilo-electron volt range. The unperturbed plasma density is approximately n₀ ≈ 10¹⁶ cm⁻³, with a channel radius of r₀ = 10 × 10⁻³ cm. Within the non-relativistic regime (a₀ ≪ 1), the normalized amplitudes are calculated as a₀ = 0.09 for the first laser and a₀ = 0.088 for the second laser. However, within the plasma channel, the plasma wave field can reach significant values. The maximum field occurs when all plasma electrons oscillate, resulting in E_max = E_WB. The non-relativistic wave-breaking field, E_WB (V/cm), is ∼0.96n^1/2 (cm⁻³), where n is the plasma density in cm⁻³. Therefore, the plasma wave amplitudes in the figures are normalized by the maximum wave-breaking field.

FIG. 2.

View large Download slide

Normalized plasma wave amplitude in tapered and untapered plasma channels as a function of distance along the longitudinal z-axis, near the radial edge of the channel (r = 0.01 cm).

A tapered plasma channel features a varying density profile along its length, resulting in a gradient in the plasma frequency, which is the natural frequency at which electrons in the plasma oscillate. In such a tapered channel, the varying plasma frequency facilitates a broader range of wave coupling.

Even if the combined frequency of the beating waves does not perfectly match the plasma frequency at any given point (in the non-resonant regime), there will always be regions within the channel where the local plasma frequency is sufficiently close to enable efficient excitation.

This localized coupling within the tapered channel concentrates energy into a smaller region, leading to stronger plasma waves compared to those in a uniform plasma. The excitation mechanism is not restricted to a specific resonance frequency, allowing a wider range of beating wave frequencies to effectively excite plasma waves.

Overall, the tapered plasma channel functions like a focusing lens for wave coupling, enhancing plasma wave generation even in the non-resonant regime.

III. A PIC SIMULATION APPROACH

The complicated method that was studied in Sec. II D made us confirm our findings through simulation. We utilized the 2D particle-in-cell (PIC) code EPOCH to model the interaction between the laser beatwave and the plasma. In this section, we explain the details of the simulation method based on the introduced plasma density. We use a computational dimension region 4000 μm × 220 with the number of grid points 10 000 × 80. We use electron cells and a neutralizing immobile ion background. All simulation parameters, including laser beam profiles (plane wave), were chosen to match the analytical model. Importantly, the simulation duration (15 ps) should exceed the beatwave interaction time within the plasma length. The PIC simulation results were then compared with the analytical solution, demonstrating good agreement between the two approaches.

Figure 3 presents a PIC simulation comparing the plasma wave amplitude in a tapered plasma channel vs an untapered channel. The x-axis represents the distance along the plasma channel. It is evident that the tapered plasma channel demonstrates an increased plasma wave amplitude in the tapered region. This enhanced excitation can be attributed to the non-resonant coupling between the beating waves and the plasma in the tapered channel. The waveguide analogy helps explain the focusing effect, but the main benefit comes from the changing plasma density. This variation creates a gradient in the plasma frequency, allowing for efficient wave coupling even if the combined frequency of the beating waves does not perfectly match the plasma frequency anywhere (non-resonant regime). This non-resonant coupling leads to stronger localized energy deposition and, consequently, a more intense plasma wave.

FIG. 3.

View large Download slide

Contour plot of the normalized plasma wave amplitude based plasma channel radius in terms of propagation distance for (a) tapered and (b) untapered plasma channels. Variation of the normalized plasma field amplitude against propagation distance for (c) tapered and (d) untapered plasma channels, near the channel’s wall.

IV. PLASMA WAVES WITH CHIRPED LASER BEAMS

We investigate the impact of chirped laser beams on plasma wave generation in tapered and untapered plasma channels. The chirp function, typically linear, modifies the pulse spectrum and beat frequency, potentially leading to broader bandwidths. We utilize a time-dependent frequency evolution description given by ω_1,2 → ω_1,2 + (ω_1,2β_1,2t/2), where β_1,2 = ω_1,2α_1,2 with α_1,2 as the chirp parameters for exploring interaction within the tapered plasma channel.

A comparison of Figs. 4 and 5 illustrates the behavior of chirped laser beams in untapered and tapered plasma channels. This analysis covers situations where one or both lasers are chirped, allowing for the examination of the impact of chirp and channel geometry on plasma wave excitation. These findings suggest that the chirp of laser beams affects their ability to create waves. This effect is even stronger when both beams change color. Our simulations reveal a further enhancement in plasma wave amplitude achieved by increasing chirp parameters, such as chirp rate or frequency range. This indicates a stronger interaction between the chirped laser and the plasma.

FIG. 4.

View large Download slide

Contour plot of the normalized plasma wave amplitude based plasma channel radius in terms of propagation distance for (a) chirped laser beam by α₁ = 0.000 55, (b) chirped laser beams by α₁ = 0.000 55 and α₂ = 0.000 55, and (c) chirped laser beams by α₁ = 0.000 56 and α₂ = 0.000 56 for untapered plasma channel near the channel’s wall.

FIG. 5.

View large Download slide

Contour plot of the normalized plasma wave amplitude based plasma channel radius in terms of propagation distance for (a) chirped laser beam by α₁ = 0.000 55, (b) chirped laser beams by α₁ = 0.000 55 and α₂ = 0.000 55, and (c) chirped laser beams by α₁ = 0.000 56 and α₂ = 0.000 56 for tapered plasma channel near the channel’s wall.

Figure 6 presents the electron density distributions for both untapered and tapered plasma channels. A comparison of the two reveals that the tapered plasma channel exhibits a less localized electron density profile. This means that the electron density is less concentrated in a specific region and is more spread out across the channel. This variation can be attributed to the modified geometry of the plasma channel. The tapering of the channel alters the plasma density gradient, which in turn improves the phase-matching conditions between the plasma wave and the beatwave. Improved phase matching allows for more efficient energy transfer from the laser beams to the plasma wave, resulting in a stronger plasma wave generation.

FIG. 6.

View large Download slide

Contour plot of electron density as a function of plasma channel radius vs propagation distance for (a) untapered and (b) tapered plasma channels, both at t = 15 ps and utilizing the same parameters.

As illustrated in Fig. 7(a), a beatwave interaction manifests as a periodic evolution of laser intensity. This phenomenon arises from the combined effects of two lasers with differing frequencies (ω₁ and ω₂). The periodicity observed in the figure is directly linked to the difference in frequencies (Δω) between the lasers. A larger Δω translates to a shorter period, resulting in faster intensity oscillations in the figure.

FIG. 7.

View large Download slide

Electric field representing the laser (laser evolution) indicated by the propagation distance at t = 15 ps for (a) unchirped laser beams and (b) two chirped laser beams with α₁ = 0.000 55 and α₂ = 0.000 55.

However, when chirped laser pulses are employed [Fig. 7(b)], the physics of the interaction becomes more intricate. In chirped laser pulses, the frequency of the light itself varies continuously within the pulse duration. This time-dependent frequency variation changes the constant phase relationship that existed with unchirped lasers. Physically, the changing frequency in a chirped pulse translates to a continuously changing wave vector (k) within the pulse. As the frequency changes in a chirped pulse, the wave vector also changes, leading to a mismatch in the phase velocities of different frequency components within the pulse. This mismatch disrupts the formation of a stable, constant phase difference between the two chirped laser pulses during the beatwave interaction. Consequently, the intensity pattern observed in the beatwave with chirped lasers deviates from the simple periodicity seen with unchirped lasers.

Figure 8 presents a snapshot of momentum of electrons with propagation distance at time t = 15 ps for three situations: (a) unchirped, (b) one chirped, and (c) two chirped laser beams. The observed periodic oscillation in Fig. 8 matches the period of the plasma beatwave generated within the 4 mm plasma channel.

FIG. 8.

Normalized momentum against distance for (a) unchirped laser beams, (b) single chirped laser beam with α1 = 0.000 55, and (c) two chirped laser beams with α1 = 0.000 55 and α2 = 0.000 55.

View large Download slide

Normalized momentum against distance for (a) unchirped laser beams, (b) single chirped laser beam with α₁ = 0.000 55, and (c) two chirped laser beams with α₁ = 0.000 55 and α₂ = 0.000 55.

As the electron travels through this modulated plasma, it experiences periodic forces that cause its momentum to oscillate in a similar pattern. Figure 8(c) shows that using two chirped lasers, the electrons’ momentum swings much higher than with a single-chirped laser beam [Fig. 8(b)] or no chirping at all [Fig. 8(a)]. This enhancement can be attributed to the synergy between the two chirped laser pulses. Their specific chirp characteristics enable them to effectively couple their energy to the plasma, leading to a more intense modulation of the plasma density and, consequently, stronger forces acting on the electrons. Furthermore, the momentum oscillation in the two-chirped laser configuration exhibits a remarkable property, and it remains undamped even at the end of the plasma channel. This lack of damping signifies that the energy transfer mechanism between the laser pulses and the plasma remains efficient throughout the interaction, allowing the electrons to maintain their high momentum oscillation. This sustained momentum gain holds substantial implications for various applications, such as laser-driven particle acceleration and high-energy electron beam generation.

This work explores a novel approach for exciting plasma waves using a beatwave within a tapered plasma channel. This method offers advantages for excitation with long laser wavelengths. We demonstrate that a chirped laser in this setup allows for achieving a larger plasma wave amplitude, leading to enhanced electron acceleration in the linear regime. A major limitation in using untapered channels for laser wakefield accelerators (LWFAs) is the dephasing length, which restricts the effective acceleration distance for electrons.^23,30 Tapered channels significantly increase this dephasing length, allowing for larger electron energy gains due to the extended interaction with the accelerating plasma wave. This highlights the importance of studying laser propagation in tapered channels for optimizing LWFAs. Previous studies have analyzed laser beam propagation in plasma channels.^22,31 However, our work goes beyond this by investigating the combined effects of a beatwave, a chirped laser, and a tapered channel on plasma wave excitation and electron acceleration. This combined approach holds promise for achieving more efficient and powerful electron acceleration. While recent studies explored autoresonant PBWA with chirped lasers,³² their acceleration gradients remained limited, typically achieving less than double the value reported by Rosenbluth and Liu.⁶ In contrast, our work utilizing a beatwave in a tapered channel with a chirped laser achieves a more than threefold increase in the acceleration gradient. This improvement arises from the tapered channel addressing the dephasing limitation, allowing for a longer electron–plasma wave interaction. Another investigation proposed a modified plasma beatwave accelerator using a chirped laser for high electric fields.³³ However, our work surpasses this by employing a beatwave in a tapered channel, achieving a significantly higher acceleration gradient.

V. CONCLUSIONS

This study investigated the influence of a tapered plasma channel and chirp on the characteristics of a plasma wave excited by the interaction of two laser beams. We derived analytical solutions for the plasma wave behavior within the tapered channel and compared these results with those from untapered channels.

A tapered plasma channel amplifies the plasma wave amplitude, effectively acting as a waveguide that focuses the laser and minimizes energy loss. In contrast to untapered channels, tapered channels exhibit significantly more stable plasma waves that persist until the end of the channel. The tapering optimizes phase matching between the laser and the plasma wave, leading to more efficient energy transfer and a stronger, more stable wave. This optimization reduces non-linear effects that can cause wave breaking and instability.

The enhanced stability signifies the improved energy confinement, which opens exciting possibilities for applications such as laser-driven particle acceleration, where achieving high-energy beams relies on efficient energy transfer within the plasma wave.

Employing chirped laser beams, as opposed to unchirped ones, presents a remarkable strategy for manipulating and enhancing plasma waves. Our findings suggest that the tailored frequency spectrum within a chirped pulse plays a pivotal role in more effectively exciting the plasma wave. This effect becomes even more pronounced when both lasers driving the interaction are chirped, resulting in a synergistic amplification effect. Interestingly, further enhancement of the chirp parameters, such as the chirp rate or frequency range, empowers the plasma wave by enabling even more efficient energy transfer from the lasers.

Moreover, utilizing a tapered plasma channel adds another layer of advantage. As the wave propagates through the channel, the gradually increasing plasma density naturally minimizes damping effects, leading to a significantly higher plasma wave amplitude at the end of the channel compared to other regions. The energy preservation mechanism offered by the tapered channel further underscores the benefits of combining chirped lasers with this specific geometry for optimal plasma wave generation.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

M. Arefnia: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal). S. Kim: Data curation (equal); Software (equal); Writing – original draft (equal). C. Lee: Data curation (equal); Software (equal); Writing – original draft (equal). M. Ghorbanalilu: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). H. Suk: Software (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available by a reasonable request from the corresponding author.

APPENDIX A: DEFINITION OF PARAMETERS OF EQ. (21)

ϕ_{1} = \frac{1.02 π α^{1 / 2}}{r_{0}} (\frac{e^{{2 K^{″}}^{2} λ^{1 / 2} r^{2}}}{4 K^{″} λ^{1 / 2}} - \frac{1}{4 K^{″} λ^{1 / 2}}),

(A1)

ϕ_{2} = {(L - K^{' 2} α^{- 1 / 3})}^{1 / 2}, ϕ_{3} = {(K^{' 2} α^{- 1 / 3})}^{1 / 2},

(A2)

ϕ_{4} = ϕ_{2} - ϕ_{3} \tan^{- 1} (\frac{ϕ_{2}}{ϕ_{3}}) - i ϕ_{3} + ϕ_{3} \tan^{- 1} (1),

(A3)

ϕ_{5} = \frac{ϕ_{2}^{3}}{3} + ϕ_{3}^{2} ϕ_{2} - i (\frac{ϕ_{3}^{3}}{3}) - i ϕ_{3}^{3} + ϕ_{3}^{3} (\tan^{- 1} \frac{ϕ_{2}}{ϕ_{3}} - \tan^{- 1} (1)),

(A4)

Φ_{1} = ϕ_{1} (2 ϕ_{4} + \frac{1}{L} (ϕ_{5} + ϕ_{3}^{2} ϕ_{4})) .

(A5)

ϕ_{6} = \frac{2 π^{2} \sqrt{α}}{0.5 \sqrt{λ}} {(0.5 \sqrt{λ})}^{1 / 2} Γ (- \frac{1}{2}),

(A6)

Φ_{2} = ϕ_{6} ϕ_{4} .

(A7)

ϕ_{7} = \frac{1.02 π \sqrt{α}}{\sqrt{r_{0}}} \sqrt{π \sqrt{\frac{1}{λ}}} ({({K^{″}}^{2} λ^{1 / 2} - 0.5 \sqrt{λ})}^{- 1 / 4} Γ (- \frac{1}{4})),

(A8)

ϕ_{8} = \frac{ϕ^{3}}{3} + ϕ_{2} ϕ_{3}^{2} - i \frac{ϕ_{3}^{3}}{3} + i ϕ_{3}^{3} + {ϕ_{3}}^{3 / 2} (\tan^{- 1} \frac{ϕ_{2}}{ϕ_{3}} - \tan^{- 1} (1)),

(A9)

Φ_{3} = ϕ_{7} (2 ϕ_{4} - \frac{1}{L} ϕ_{8} - \frac{1}{2 L} ϕ_{3}^{2}) .

(A10)

ϕ_{9} = \frac{2}{3} {(\frac{2 i α^{1 / 2}}{3})}^{- 5 / 6} Γ (\frac{5}{6}),

(A11)

ϕ_{10} = \frac{1}{2} {(- \frac{2 i α^{1 / 2}}{3})}^{1 / 2} Γ (- \frac{1}{2}),

(A12)

ϕ_{11} = \frac{2 i α^{1 / 2}}{3} {(- \frac{2 i α^{1 / 2}}{3})}^{- 1 / 2} Γ (\frac{1}{2}),

(A13)

ϕ_{12} = {(- \frac{2 i α^{1 / 2}}{3})}^{- 1 / 6} Γ (\frac{1}{6}),

(A14)

ϕ_{13} = \frac{2 i α^{1 / 2}}{3} {(- \frac{2 i α^{1 / 2}}{3})}^{- 7 / 6} Γ (\frac{7}{6}),

(A15)

\begin{align} ϕ_{14} & = (z^{- 3 / 4} - \frac{z^{1 / 4}}{2 L}) \exp (\frac{2 i α^{1 / 2}}{3} z^{3 / 2}) \\ + ϕ_{10} + \frac{1}{8 L} ϕ_{12} - ϕ_{11} + \frac{1}{L} ϕ_{13}, \end{align}

(A16)

Φ_{4} = ϕ_{1} ϕ_{9} ϕ_{14} .

(A17)

Φ_{5} = ϕ_{9} \frac{π^{2} \sqrt{α} z^{- 3 / 4}}{0.5 \sqrt{λ}} {(0.5 \sqrt{λ})}^{1 / 2} Γ (- \frac{1}{2}) \exp (\frac{2 i α^{1 / 2}}{3} z^{3 / 2}) .

(A18)

Φ_{6} = ϕ_{7} ϕ_{9} ϕ_{14} .

(A19)

Φ_{7} = \frac{20 π \sqrt{α} L^{1 / 2}}{3 r_{0}} (\frac{e^{{2 K^{″}}^{2} λ^{1 / 2} r^{2}}}{4 K^{″} λ^{1 / 2}} - \frac{1}{4 K^{″} λ^{1 / 2}}) .

(A20)

Φ_{8} = 2 \frac{π^{2} \sqrt{L α}}{0.5 \sqrt{λ}} {(0.5 \sqrt{λ})}^{1 / 2} Γ (- \frac{1}{2}) .

(A21)

Φ_{9} = \frac{22}{6} L^{1 / 2} ϕ_{7} .

(A22)

ϕ_{15} = \frac{1}{2} {(\frac{2 i α^{1 / 2}}{3})}^{1 / 2} Γ (- \frac{1}{2}),

(A23)

ϕ_{16} = \frac{1}{8 L} {(\frac{2 i α^{1 / 2}}{3})}^{- 1 / 6} Γ (\frac{1}{6}),

(A24)

ϕ_{17} = \frac{2 i α^{1 / 2}}{3} {(\frac{2 i α^{1 / 2}}{3})}^{- 1 / 2} Γ (\frac{1}{2}),

(A25)

ϕ_{18} = \frac{2 i α^{1 / 2}}{3 L} {(\frac{2 i α^{1 / 2}}{3})}^{- 7 / 6} Γ (\frac{7}{6}),

(A26)

\begin{align} Φ_{10} & = ϕ_{1} ϕ_{9} ((z^{- 3 / 4} - \frac{z^{1 / 4}}{2 L}) \exp (- \frac{2 i α^{1 / 2}}{3} z^{3 / 2}) \\ + ϕ_{15} + ϕ_{16} - ϕ_{17} + ϕ_{18}) . \end{align}

(A27)

ϕ_{19} = \frac{2}{3} {(- \frac{2 i α^{1 / 2}}{3})}^{- 5 / 6} Γ (\frac{5}{6}),

(A28)

Φ_{11} = \frac{1}{2} ϕ_{19} Φ_{8} z^{- 5 / 4} \exp (- \frac{2 i α^{1 / 2}}{3} z^{3 / 2}) .

(A29)

ϕ_{20} = (z^{- 3 / 4} - \frac{z^{1 / 4}}{4 L}) \exp (- \frac{2 i α^{1 / 2}}{3} z^{3 / 2}) + ϕ_{10} + \frac{1}{16} ϕ_{17} - ϕ_{18},

(A30)

Φ_{12} = ϕ_{7} ϕ_{19} ϕ_{20} .

(A31)

APPENDIX B: DEFINITION OF PARAMETERS OF EQ. (23)

ψ_{1} = - \frac{i Δ k π^{1 / 2} α^{1 / 4}}{12} \frac{e}{m} \frac{E_{1} E_{2}^{*} ω_{0}^{2}}{ω_{1} ω_{2} v_{e}^{2}} e^{- i π / 4},

(B1)

ψ_{2} = \frac{{({K^{″}}^{2} λ^{1 / 2})}^{- 3 / 4}}{3} \frac{1.02}{\sqrt{r_{0}}} {(\frac{2 i α^{1 / 2}}{3})}^{- 5 / 6} Γ (\frac{5}{6}) Γ (\frac{3}{4}),

(B2)

ψ_{3} = \frac{{({K^{″}}^{2} λ^{1 / 2})}^{- 7 / 4}}{3} \frac{{1.02}^{5 / 2}}{r_{0}^{5 / 2}} \frac{δ n}{n_{0}} {(\frac{2 i α^{1 / 2}}{3})}^{- 5 / 6} Γ (\frac{5}{6}) Γ (\frac{7}{4}),

(B3)

ψ_{4} = ({(z^{2} + \frac{z^{3}}{L})}^{- 1 / 4} - L^{- 1 / 2}) - i Δ k ({(z^{- 2} + \frac{z^{- 1}}{L})}^{- 1 / 4} - L^{1 / 2}),

(B4)

ψ_{5} = ({(z^{- 2} + \frac{z^{- 1}}{L})}^{- 1 / 4} - L^{1 / 2}) - i Δ k ({(z^{- 6} + \frac{z^{- 5}}{L})}^{- 1 / 4} - L^{3 / 2}),

(B5)

\begin{align} ψ_{6} & = ({(z^{2 / 5} + \frac{z^{7 / 5}}{L})}^{- 5 / 4} - L^{- 1 / 2}) \\ - i Δ k ({(z^{- 2 / 5} + \frac{z^{3 / 5}}{L})}^{- 5 / 4} - L^{1 / 2}), \end{align}

(B6)

Ψ_{1} = ψ_{1} (ψ_{2} ψ_{4} + ψ_{2} \frac{ψ_{5}}{L} + ψ_{3} ψ_{6}) .

(B7)

ψ_{7} = \frac{1}{3} \sqrt{\frac{π}{0.5 \sqrt{λ}}} {(\frac{2 i α^{1 / 2}}{3})}^{- 5 / 6} Γ (\frac{5}{6}),

(B8)

ψ_{8} = \frac{4}{3 r_{0}^{2}} \frac{δ n}{n_{0}} {(0.5 \sqrt{λ})}^{- 1} {(\frac{2 i α^{1 / 2}}{3})}^{- 5 / 6} Γ (\frac{5}{6}),

(B9)

ψ_{9} = (z^{- 1 / 2} - i L^{- 1 / 2}) - i Δ k (z^{1 / 2} - i L^{1 / 2}),

(B10)

ψ_{10} = (z^{1 / 2} + \frac{1}{2} L^{1 / 2}) - i Δ k (z^{3 / 2} + L^{3 / 2}),

(B11)

\begin{align} ψ_{11} & = {(z^{1 / 2} + \frac{z^{3 / 2}}{L})}^{- 1} - i Δ k {(z^{- 1 / 2} + \frac{z^{1 / 2}}{L})}^{- 1} \\ + (2 L^{- 1 / 2} - i Δ k (2 L^{1 / 2})), \end{align}

(B12)

Ψ_{2} = ψ_{1} (ψ_{7} ψ_{9} + ψ_{7} \frac{ψ_{10}}{L} + L ψ_{8} ψ_{11}) .

(B13)

ψ_{12} = \frac{{({K^{″}}^{2} λ^{1 / 2})}^{- 3 / 4}}{3} \frac{1.02}{r_{0}^{1 / 2}} {(\frac{2 i α^{1 / 2}}{3})}^{- 2 / 3} Γ (\frac{2}{3}) Γ (\frac{3}{4}),

(B14)

ψ_{13} = \frac{{({K^{″}}^{2} λ^{1 / 2})}^{- 7 / 4}}{3} \frac{{1.02}^{5 / 2}}{r_{0}^{5 / 2}} \frac{δ n}{n_{0}} {(\frac{2 i α^{1 / 2}}{3})}^{- 2 / 3} Γ (\frac{2}{3}) Γ (\frac{7}{4}),

(B15)

ψ_{14} = ({(z + \frac{z^{2}}{L})}^{- 1 / 4} - L^{- 1 / 4}) - i Δ k ({(z^{- 3} + \frac{z^{- 2}}{L})}^{- 1 / 4} - L^{3 / 4}),

(B16)

ψ_{15} = ({(z^{- 3} + \frac{z^{- 2}}{L})}^{3 / 4} - L^{- 1 / 4}) - i Δ k ({(z^{- 7} + \frac{z^{- 6}}{L})}^{- 1 / 4} - L^{7 / 4}),

(B17)

\begin{align} ψ_{16} = & ({(z^{1 / 5} + \frac{z^{6 / 5}}{L})}^{- 5 / 4} - 5 {(2 L)}^{- 1 / 4}) \\ - i Δ k ({(z^{- 3 / 5} + \frac{z^{2 / 5}}{L})}^{- 5 / 4} - 5 {(2 L)}^{3 / 4}), \end{align}

(B18)

Ψ_{3} = ψ_{1} (ψ_{12} ψ_{14} + ψ_{12} \frac{ψ_{15}}{L} + ψ_{13} ψ_{16}) .

(B19)

ψ_{17} = \frac{1}{3} \sqrt{\frac{π}{0.5 \sqrt{λ}}} {(\frac{2 i α^{1 / 2}}{3})}^{- 2 / 3} Γ (\frac{2}{3}),

(B20)

ψ_{18} = \frac{1.04}{3 r_{0}^{2}} \frac{δ n}{n_{0}} {(0.5 \sqrt{λ})}^{- 1} {(\frac{2 i α^{1 / 2}}{3})}^{- 2 / 3} Γ (\frac{2}{3}),

(B21)

ψ_{19} = (z^{- 1 / 4} - L^{- 1 / 4}) - i Δ k (z^{3 / 4} - L^{3 / 4}),

(B22)

ψ_{20} = (z^{3 / 4} + L^{3 / 4}) - i Δ k (z^{7 / 4} + L^{7 / 4}),

(B23)

\begin{align} ψ_{21} & = {(z^{1 / 4} + \frac{z^{5 / 4}}{L})}^{- 1} - i Δ k {(z^{- 3 / 4} + \frac{z^{1 / 4}}{L})}^{- 1} \\ + (2 L^{- 1 / 4} - i Δ k (2 L^{- 3 / 4})), \end{align}

(B24)

Ψ_{4} = ψ_{1} (ψ_{17} ψ_{19} + ψ_{17} \frac{ψ_{20}}{L} + ψ_{18} ψ_{21}) .

(B25)

REFERENCES

1

T.

Tajima

and

J. M.

Dawson

, “

Laser electron accelerator

,”

Phys. Rev. Lett.

43

,

267

(

1979

).

https://doi.org/10.1103/physrevlett.43.267

Google Scholar

Crossref

2

J. B.

Rosenzweig

,

B.

Breizman

,

T.

Katsouleas

, and

J. J.

Su

, “

Acceleration and focusing of electrons in two-dimensional nonlinear plasma wake fields

,”

Phys. Rev. A

44

,

R6189

(

1991

).

https://doi.org/10.1103/physreva.44.r6189

Google Scholar

Crossref

PubMed

3

D. N.

Gupta

,

K.

Gopal

,

I. H.

Nam

,

V. V.

Kulagin

, and

H.

Suk

, “

Laser wakefield acceleration of electrons from a density-modulated plasma

,”

Laser Part. Beams

32

,

449

(

2014

).

https://doi.org/10.1017/s0263034614000378

Google Scholar

Crossref

4

A. R.

Maier

,

N. M.

Delbos

,

T.

Eichner

,

L.

Hübner

,

S.

Jalas

,

L.

Jeppe

,

S. W.

Jolly

,

M.

Kirchen

,

V.

Leroux

,

P.

Messner

,

M.

Schnepp

,

M.

Trunk

,

P. A.

Walker

,

C.

Werle

, and

P.

Winkler

, “

Decoding sources of energy variability in a laser-plasma accelerator

,”

Phys. Rev. X

10

,

031039

(

2020

).

https://doi.org/10.1103/physrevx.10.031039

Google Scholar

Crossref

5

M.

Rezaei-Pandari

,

M.

Mirzaie

,

C. I.

Hojbota

,

T. G.

Pak

,

C. H.

Nam

,

S. B.

Kim

,

G. W.

Lee

,

R.

Massudi

,

A. R.

Niknam

,

S. K.

Lee

, and

K. Y.

Kim

, “

Laser wakefield electron acceleration with polarization-dependent ionization injection

,”

Phys. Rev. Appl.

20

,

034026

(

2023

).

https://doi.org/10.1103/physrevapplied.20.034026

Google Scholar

Crossref

6

M. N.

Rosenbluth

and

C. S.

Liu

, “

Excitation of plasma waves by two laser beams

,”

Phys. Rev. Lett.

29

,

701

(

1972

).

https://doi.org/10.1103/physrevlett.29.701

Google Scholar

Crossref

7

R. J.

Noble

, “

Plasma-wave generation in the beat-wave accelerator

,”

Phys. Rev. A

32

,

460

(

1985

).

https://doi.org/10.1103/physreva.32.460

Google Scholar

Crossref

8

S. Y.

Tochitsky

,

R.

Narang

,

C. V.

Filip

,

P.

Musumeci

,

C. E.

Clayton

,

R. B.

Yoder

,

K. A.

Marsh

,

J. B.

Rosenzweig

,

C.

Pellegrini

, and

C.

Joshi

, “

Enhanced acceleration of injected electrons in a laser-beat-wave-induced plasma channel

,”

Phys. Rev. Lett.

92

,

095004

(

2004

).

https://doi.org/10.1103/physrevlett.92.095004

Google Scholar

Crossref

PubMed

9

N.

Hafz

,

M. S.

Hur

,

G. H.

Kim

,

C.

Kim

,

I. S.

Ko

, and

H.

Suk

, “

Quasimonoenergetic electron beam generation by using a pinholelike collimator in a self-modulated laser wakefield acceleration

,”

Phys. Rev. E

73

,

016405

(

2006

).

https://doi.org/10.1103/physreve.73.016405

Google Scholar

Crossref

10

Y.

Kitagawa

,

T.

Matsumoto

,

T.

Minamihata

,

K.

Sawai

,

K.

Matsuo

,

K.

Mima

,

K.

Nishihara

,

H.

Azechi

,

K. A.

Tanaka

,

H.

Takabe

, and

S.

Nakai

, “

Beat-wave excitation of plasma wave and observation of accelerated electrons

,”

Phys. Rev. Lett.

68

,

48

(

1992

).

https://doi.org/10.1103/physrevlett.68.48

Google Scholar

Crossref

PubMed

11

T.

Katsouleas

, “

Physical mechanisms in the plasma wake-field accelerator

,”

Phys. Rev. A

33

,

2056

(

1986

).

https://doi.org/10.1103/physreva.33.2056

Google Scholar

Crossref

12

W. P.

Leemans

,

C. W.

Siders

,

E.

Esarey

,

N. E.

Andreev

,

G.

Shvets

, and

W. B.

Mori

, “

Plasma guiding and wakefield generation for second-generation experiments

,”

IEEE Trans. Plasma Sci.

24

,

331

(

1996

).

https://doi.org/10.1109/27.509997

Google Scholar

Crossref

13

S. Y.

Kalmykov

,

A.

Beck

,

S. A.

Yi

,

V. N.

Khudik

,

M. C.

Downer

,

E.

Lefebvre

,

B. A.

Shadwick

, and

D. P.

Umstadter

, “

Electron self-injection into an evolving plasma bubble: Quasi-monoenergetic laser-plasma acceleration in the blowout regime

,”

Phys. Plasmas

18

,

5

(

2011

).

https://doi.org/10.1063/1.3566062

Google Scholar

Crossref

14

N. E.

Andreev

,

L. M.

Gorbunov

,

V. I.

Kirsanov

,

K.

Nakajima

, and

A.

Ogata

, “

Structure of the wake field in plasma channels

,”

Phys. Plasmas

4

,

1145

(

1997

).

https://doi.org/10.1063/1.872186

Google Scholar

Crossref

15

A.

Butler

,

D. J.

Spence

, and

S. M.

Hooker

, “

Guiding of high-intensity laser pulses with a hydrogen-filled capillary discharge waveguide

,”

Phys. Rev. Lett.

89

,

185003

(

2002

).

https://doi.org/10.1103/physrevlett.89.185003

Google Scholar

Crossref

PubMed

16

T.

Hosokai

,

M.

Kando

,

H.

Dewa

,

H.

Kotaki

,

S.

Kondo

,

N.

Hasegawa

,

K.

Nakajima

, and

K.

Horioka

, “

Optical guidance of terrawatt laser pulses by the implosion phase of a fast Z-pinch discharge in a gas-filled capillary

,”

Opt. Lett.

25

,

10

–

12

(

2000

).

https://doi.org/10.1364/ol.25.000010

Google Scholar

Crossref

PubMed

17

B.

Hafizi

,

D. F.

Gordon

,

A.

Zigler

, and

A.

Ting

, “

Electron trajectories in the magnetic field of capillary discharge: Application to laser wakefield accelerators in plasma channel

,”

Phys. Plasmas

10

,

2545

(

2003

).

https://doi.org/10.1063/1.1571824

Google Scholar

Crossref

18

M.

Arefnia

,

M.

Ghorbanalilu

, and

A. R.

Niknam

, “

Excitation and enhancement of wakefield by beating of two laser beams in a preformed plasma channel: An analytical study

,”

Phys. Plasmas

29

,

072305

(

2022

).

https://doi.org/10.1063/5.0096499

Google Scholar

Crossref

19

G.

Shvets

,

J. S.

Wurtele

,

T. C.

Chiou

, and

T. C.

Katsouleas

, “

Excitation of accelerating wakefields in inhomogeneous plasmas

,”

IEEE Trans. Plasma Sci.

24

,

351

(

1996

).

https://doi.org/10.1109/27.509999

Google Scholar

Crossref

20

S. N.

Razavinia

and

M.

Ghorbanalilu

, “

Effect of external magnetic field on wakefield generation in underdense plasma slab using backward semi-Lagrangian Vlasov code

,”

Phys. Rev. Accel. Beams

22

,

111305

(

2019

).

https://doi.org/10.1103/physrevaccelbeams.22.111305

Google Scholar

Crossref

21

M.

Arefnia

,

M.

Ghorbanalilu

, and

A. R.

Niknam

, “

Large-amplitude plasma wave generation by laser beating in inhomogeneous magnetized plasmas

,”

Heliyon

10

,

e32813

(

2024

).

https://doi.org/10.1016/j.heliyon.2024.e32813

Google Scholar

Crossref

PubMed

22

W.

Rittershofer

,

C. B.

Schroeder

,

E.

Esarey

,

F. J.

Grüner

, and

W. P.

Leemans

, “

Tapered plasma channels to phase-lock accelerating and focusing forces in laser-plasma accelerators

,”

Phys. Plasmas

17

,

063104

(

2010

).

https://doi.org/10.1063/1.3430638

Google Scholar

Crossref

23

A.

Pukhov

and

I.

Kostyukov

, “

Control of laser-wakefield acceleration by the plasma-density profile

,”

Phys. Rev. E

77

,

025401

(

2008

).

https://doi.org/10.1103/physreve.77.025401

Google Scholar

Crossref

24

X.

Zhang

,

B.

Shen

,

L.

Ji

,

W.

Wang

,

J.

Xu

,

Y.

Yu

,

L.

Yi

,

X.

Wang

,

N. A. M.

Hafz

, and

V.

Kulagin

, “

Effect of pulse profile and chirp on a laser wakefield generation

,”

Phys. Plasmas

19

,

5

(

2012

).

https://doi.org/10.1063/1.4714610

Google Scholar

Crossref

25

M.

Akhyani

,

F.

Jahangiri

,

A. R.

Niknam

, and

R.

Massudi

, “

Optimizing chirped laser pulse parameters for electron acceleration in vacuum

,”

J. Appl. Phys.

118

,

183106

(

2015

).

https://doi.org/10.1063/1.4935283

Google Scholar

Crossref

26

S. N.

Razavinia

and

M.

Ghorbanalilu

, “

Semi-Lagrangian Vlasov simulation of the interaction between femtosecond chirped and double laser pulses with a thin plasma slab

,”

Plasma Phys. Controlled Fusion

62

,

045007

(

2020

).

https://doi.org/10.1088/1361-6587/ab73dc

Google Scholar

Crossref

27

M.

Arefnia

,

M.

Sharifian

, and

M.

Ghorbanalilu

, “

Terahertz radiation generation by beating of two chirped laser pulses in a warm collisional magnetized plasma

,”

Chin. Phys. B

30

,

094101

(

2021

).

https://doi.org/10.1088/1674-1056/abfb5d

Google Scholar

Crossref

28

Y.

Cui

,

G.

Zhang

,

Y. Y.

Ma

,

D. B.

Zou

,

X. H.

Yang

et al, “

Beam quality improvement of ionization injected electrons by using chirped pulse in wakefield acceleration

,”

Plasma Phys. Controlled Fusion

61

,

085023

(

2019

).

https://doi.org/10.1088/1361-6587/ab249e

Google Scholar

Crossref

29

B.

Nikrah

and

S.

Jafari

, “

Laser wakefield acceleration of electrons in a magnetically controlled plasma

,”

Plasma Phys. Controlled Fusion

65

,

115008

(

2023

).

https://doi.org/10.1088/1361-6587/ad0178

Google Scholar

Crossref

30

K. C.

Tzeng

,

W. B.

Mori

, and

T.

Katsouleas

, “

Electron beam characteristics from laser-driven wave breaking

,”

Phys. Rev. Lett.

79

,

5258

(

1997

).

https://doi.org/10.1103/physrevlett.79.5258

Google Scholar

Crossref

31

P.

Sprangle

,

B.

Hafizi

,

J. R.

Penano

,

R. F.

Hubbard

,

A.

Ting

,

C. I.

Moore

,

D. F.

Gordon

,

A.

Zigler

,

D.

Kaganovich

, and

T. M.

Antonsen

, Jr., “

Wakefield generation and GeV acceleration in tapered plasma channels

,”

Phys. Rev. E

63

,

056405

(

2001

).

https://doi.org/10.1103/physreve.63.056405

Google Scholar

Crossref

32

M.

Luo

,

C.

Riconda

,

I.

Pusztai

,

A.

Grassi

,

J. S.

Wurtele

, and

T.

Fülöp

, “

Control of autoresonant plasma beat-wave wakefield excitation

,”

Phys. Rev. Res.

6

,

013338

(

2024

).

https://doi.org/10.1103/physrevresearch.6.013338

Google Scholar

Crossref

33

R. R.

Lindberg

,

A. E.

Charman

,

J. S.

Wurtele

, and

L.

Friedland

, “

Robust autoresonant excitation in the plasma beat-wave accelerator

,”

Phys. Rev. Lett.

93

,

055001

(

2004

).

https://doi.org/10.1103/physrevlett.93.055001

Google Scholar

Crossref

PubMed

This work is licensed under a Creative Commons Attribution 4.0 International License .

Enhanced plasma wave excitation in a tapered plasma channel through chirped laser beatwaves

I. INTRODUCTION

II. PLASMA WAVE DYNAMICS IN A TAPERED PLASMA CHANNEL

A. Electrostatic field equation for plasma wave

B. Fundamental wave modes in the tapered plasma channel

C. Non-homogeneous solution of the plasma wave equation

D. Characterizing beatwave-excited plasma waves in the tapered plasma channel

III. A PIC SIMULATION APPROACH

IV. PLASMA WAVES WITH CHIRPED LASER BEAMS

V. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: DEFINITION OF PARAMETERS OF EQ. (21)

APPENDIX B: DEFINITION OF PARAMETERS OF EQ. (23)

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Enhanced plasma wave excitation in a tapered plasma channel through chirped laser beatwaves

I. INTRODUCTION

II. PLASMA WAVE DYNAMICS IN A TAPERED PLASMA CHANNEL

A. Electrostatic field equation for plasma wave

B. Fundamental wave modes in the tapered plasma channel

C. Non-homogeneous solution of the plasma wave equation

D. Characterizing beatwave-excited plasma waves in the tapered plasma channel

III. A PIC SIMULATION APPROACH

IV. PLASMA WAVES WITH CHIRPED LASER BEAMS

V. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: DEFINITION OF PARAMETERS OF EQ. (21)

APPENDIX B: DEFINITION OF PARAMETERS OF EQ. (23)

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only