We study the propagation of a relativistic ultrashort laser pulse in two-dimensional inhomogeneous plasma with a density ripple based on the theoretical analysis and the particle-in-cell simulation. An analytical solution of the laser pulse propagating in the homogeneous plasma is completely obtained. It is found that the stable propagation of the relativistic ultrashort laser pulse in homogeneous plasma can be realized, and the propagation distance and intensity of the laser pulse and the spatial distribution of electron density can be effectively modulated by adjusting the pulsewidth of the incident laser and the plasma frequency. More interestingly, in the inhomogeneous plasma with a density ripple, when the wave amplitude of the density ripple is less than a critical value, the intensity of the ultrashort laser pulse is almost unchanged and the propagation characteristics of the laser in the inhomogeneous plasma are basically consistent with those in the homogeneous plasma. However, when the wave amplitude of the density ripple is larger than the critical value, the inhomogeneity of the plasma has an obvious effect on the laser characteristics, and the intensity and the spatial distribution of the laser pulse will be modulated by the plasma density ripple. The influence of the laser pulsewidth, plasma density, and plasma density ripple on the laser characteristics is discussed in detail.

## I. INTRODUCTION

The continuous development of high intensity short pulse laser technology has stimulated much interest in the laser–plasma interaction.^{1}^{,}^{2} Until now, the propagation of an intense laser pulse in underdense plasma has always been a research hotspot.^{3} One of the problems of special concern is the coupling effect of relativistic and ponderomotive force.^{4} These nonlinear effects occur when the superintense laser pulses propagate in underdense plasma [i.e., plasma with *ω*_{p} < *ω*_{0}, where $\omega p=(4\pi nee2/me)1/2$ is the plasma frequency and *ω*_{0} is the laser carrier frequency], especially when the self-focusing and self-channeling phenomena take place.^{1}^{,}^{3}^{,}^{–}^{8} In the fast ignitor concept for inertial confinement fusion (ICF), to realize the propagation of the laser pulse over considerable distances without significant energy loss, relativistic self-focusing and self-channeling offer a possibility for optical guiding of laser pulses in underdense plasma, such as the hollow channels of the ICF.^{9} Moreover, relativistic self-focusing (RSF)^{10} and self-channeling^{11} also have important applications in soft x-ray generation,^{12} laser wake field acceleration (LWFA),^{13}^{,}^{14} and new light sources.^{15} Generally, due to the expulsion of electrons by the ponderomotive force of the laser pulse, the electron density is modified and generates a low density region inside the plasma (such as the hollow channels and wake field). Meanwhile, the refractive index is also modified and becomes larger on the axis, making this region act as a positive lens.^{16}

At present, the stable transmission of the intense circularly polarized laser pulse in a homogeneous plasma is realized completely in the absence of transverse diffusion, which has great significance for the generation of the hollow channel of fusion and the acceleration of the particle in the plasma.^{3}^{,}^{17}^{,}^{–}^{23} Cattani *et al.*^{1} and Kim *et al*.^{22} obtained the exact stationary channeling solutions in 2D and 3D cylindrical geometry, respectively. In the experiment, Borghesi *et al*.^{23} observed the relativistic self-channeling phenomenon of a picosecond laser pulse in a preformed plasma with a near critical density. However, those research studies mainly focus on the analytical description of the long laser pulse $(\omega pe\u22121\u226a\tau \u226a\omega pi\u22121)$ in a homogeneous plasma.^{1,24} Comparing the longer ignition lasers, the short ignition pulse propagating in preplasma makes the fast electrons have a low divergence, which results in a high coupling efficiency of the laser and target.^{25} However, the propagation of the ultrashort laser pulse in plasma is still unclear. Clearly, the propagation of the intense short laser pulse $(2\omega pe\u22121=\tau )$ in a plasma is worthy of careful study.^{26}^{,}^{27}

Moreover, comparing with the homogeneous plasma, it is found that plasma targets with a good geometry design can further enhance the coupling effects of the laser and plasma.^{28}^{,}^{–}^{33} When the laser pulse with a longitudinal wave vector $hx\u0302$ (where *h* is the wave number of the laser pulse) propagates in a rippled density plasma with wave vector $qx\u0302$ (where *q* is the wave number of the density ripples of the plasma), i.e., the density modulation direction of the plasma targets is parallel to the propagation direction of the ultrashort laser pulse, the efficient acceleration of electrons and the new radiation source are realized and studied widely.^{32}^{,}^{–}^{35} Recently, it is found that the intensity of the laser pulse can be improved significantly when the density modulation direction of the plasma targets is perpendicular to the propagation direction of the ultrashort laser pulse, which can further ensure the low divergence and high energy of the electrons in the acceleration process.^{29}^{,}^{–}^{31} Snyder *et al*.^{29} successfully demonstrated the generation of high energy electrons in the interaction of the short pulse laser and the micro-channel plasma targets. Specially, people use two intersecting intense laser pulses to induce plasma density modulation in plasma to form a periodic density structure, i.e., a plasma density grating (PDG), which can make the laser maintain a relatively high intensity for a several picoseconds.^{30} Moreau *et al*.^{31} found that the energy and flux of relativistic electrons of CD_{2} nanowire arrays with near solid density can be increased significantly under the irradiation of ultra-intense laser pulses. However, fully ionized plasma can sustain higher laser intensities than solid materials and can be used repeatedly, which means that the interaction of the laser pulse and plasma with a transversely periodic density modulation is more suitable for enhancing the coupling effects of the laser and plasma to realize the efficient acceleration of electrons and the new radiation source. In the electron acceleration and the new radiation source, the stability of the spatial distribution, envelope, and intensity of the laser pulse are important for improving the energy of the electrons and the intensity of the radiation source. However, the stable propagation of the ultrashort laser pulse $(2\omega pe\u22121=\tau )$ with the longitudinal wave vector $hx\u0302$ in rippled density plasma with wave vector $qy\u0302$ is still an opening subject, which not only plays a positive role for the enhancement of the electron energy and the quality of radiation but also provides a potential modulation way of the ultrashort laser pulse without the limitation of damage threshold to replace the expensive and vulnerable optical device.^{36} The potential modulation way for the ultrashort laser pulse can further promote the development of inertial confinement fusion,^{7}^{,}^{10}^{,}^{11} electron acceleration,^{29}^{,}^{32}^{,}^{33} noninvasive terahertz imaging,^{36} and laser cladding technology.^{37}

In this paper, through the theoretical analysis and the Particle-In-Cell (PIC) simulation, we study the propagation characteristics of an intense ultrashort laser pulse with the longitudinal wave vector $hx\u0302$ in rippled density plasma with wave vector $qy\u0302$, i.e., the density modulation direction of the plasma targets is perpendicular to the propagation direction of the ultrashort laser pulse.^{38} On the one hand, we discuss the characteristics of the stable propagation of the ultrashort laser pulse in homogeneous plasma. On the other hand, we discuss the effects of inhomogeneous plasma with a density ripple on the intensity and spatial distribution of the laser pulse and the distribution of electron density. It is interesting to note that the intensity of the laser pulse and the spatial distribution of electron density can be modulated by adjusting the pulsewidth of the incident laser and the plasma frequency in homogeneous plasma. More interestingly, in inhomogeneous plasma with a density ripple, the intensity and spatial distribution of a laser pulse are almost unchanged as the wave amplitude of the density ripple is less than the critical value. As the pulsewidth of the laser decreases, the spatial variation of the laser pulse is enhanced, which means that the modulation of electron density is also enhanced. Moreover, as the wave amplitude of the density ripple increases, the intensity of the laser pulse gradually decreases, which significantly modulates the electron density of the low density region. Meanwhile, the time dependence of the related physical quantity, i.e., the envelope and the transversal spatial distribution of the ultrashort laser pulse and the spatial distribution of the plasma with the density ripple, is carefully discussed by the PIC simulation. Similar experiments have been provided to study the channel evolution, filamentation, and self-correction of the channel in the laser–plasma interaction,^{32} which are also suitable for our theoretical model.

This paper is organized as follows: In Sec. II, the Lagrange equation of the interaction between an ultrashort laser pulse and inhomogeneous plasma is derived by Maxwell’s equation and a relativistic electron motion equation, and the propagation equation of an ultrashort laser pulse in inhomogeneous plasma is obtained. In Sec. III, we study the propagation characteristics of the ultrashort laser pulse in homogeneous plasma. In Sec. IV, we numerically analyze the effect of inhomogeneous plasma with the density ripple on the laser pulse and the electron density. In Sec. V, the PIC simulations are performed and show the effects of the envelope of the laser pulse and the density ripple of the plasma on the propagation characteristics of the ultrashort laser pulse. In Sec. VI, the results are summarized.

## II. THEORETICAL MODEL

In order to understand the effects caused by the action of ponderomotive force and relativistic nonlinearity, we consider the interaction of the relativistic ultrashort laser pulse and inhomogeneous plasma. The intensity of the laser must be high enough to make sure that the electrons propagate with the relativistic quiver motion.^{1}^{,}^{21}^{,}^{39}^{,}^{–}^{41} In this case, we describe the laser–plasma system by Maxwell’s equations and relativistic electron equations of motion. Compared with the ponderomotive force, the pressure gradient of the electron is neglected and the electrons are treated as cold. The basic equations are composed of Maxwell’s equations in the Coulomb gauge (** ∇** ·

**A**= 0), and the equations of motion of cold electron plasma are given as

Through equality

The plasma is considered to be in an irrotational state, and we get ** ∇** × (

**p**−

*e*

**A**/

*c*) = 0 with

**p**=

*γm*

**v**. Thus, Eq. (5) can be reduced to

Through Eq. (6), we obtain

where ∇^{2} = *∂*^{2}/*∂x*^{2} + *∂*^{2}/*∂y*^{2}, *γ* is the relativistic factor, *e* is the electron charge, *m* is the rest mass of an electron, and *n*_{e} is the electron density. The plasma is taken to be space-periodically modulated along the *y*-direction with a density ripple of *n*_{0} = *N*_{0}[1 − *β* cos(*qy*)], which means that the density modulation direction of the plasma targets (with wave vector $qy\u0302$) is perpendicular to the propagation direction of the ultrashort laser pulse (with wave vector $hx\u0302$). Here, *N*_{0} is the density of the background plasma and *β* and *q* are the amplitude and wave number of the density ripple, respectively. In the experiment, plasma with the density ripple can be achieved by using many techniques, such as wave-mixing mechanism of the cross beams,^{40}^{,}^{41} plasma density grating (PDG),^{30} transmissive ring grating, and pattern mask.^{42}^{,}^{43}^{,}**A** is the electromagnetic vector potential, *φ* is the electrostatic scalar potential, and *ψ* is a scalar function. As the mass of ions is much larger than the electron mass, the ion motion can be ignored. We will look for a quasisteady state, i.e., the time dependence of Eq. (8) can be ignored. Since the plasma is cold, the velocity of electrons is zero before the laser pulse arrives, i.e., *ψ* = 0.^{44}^{,}^{45} Based on the slowly varying envelope approximation, the normalized vector potential of the ultrashort laser pulse is assumed to be

where *x* is the propagation direction of the laser, *y* is the transverse direction, $vg=c(1\u2212\omega p2/\omega 02)1/2$ is the group velocity, *ω*_{p} is the plasma frequency, *τ* is the laser pulsewidth, and *ω*_{0} is the laser frequency. Using the paraxial approximation, *k*_{⊥} ≪ *h*, where $k\u22a5(=k1\u2212h2/k2)$ is the transverse component of the laser wave vector, *h* is the longitudinal propagation constant, and *k* is the magnitude of the vacuum wave vector of the laser pulse. Substituting Eq. (9) into Eq. (1), we get

and Eq. (2) becomes

Since *ψ* = 0, through Eq. (8), we have

Here,

where *a*″ = *d*^{2}*a*/*dy*^{2}, *ϕ*″ = *d*^{2}*ϕ*/*dy*^{2}, *k* = *ω*_{0}/*c*, *n* = *n*_{e}/*N*_{0}, $\gamma =(1+|a|2)1/2$ is the relativistic factor, and $Ncr=m\omega 02/4\pi e2$ is the critical density, respectively. Since the amplitude of the high frequency oscillation is not ignored and the effects of the longitudinal component of the ultrashort laser pulse are considered effectively, the Poisson equation [Eq. (11)] is modified by the factor of *η*, which exactly explains the formation of the multifilament structures of the laser and avoids the appearance of the negative electron densities effectively.^{24} Here, the transversal length *y* is dimensionlessed by the transversal wave vector of the laser pulse $y\u223cky1\u2212h2/k2$.^{18} The other physical quantities are dimensionlessed as follows: *ϕ* ∼ *φ*/(*mc*^{2}), *ω* ∼ *ω*/*ω*_{0}, *ω*_{p} ∼ *ω*_{p}/*ω*_{0}, *t* ∼ *ω*_{0}*t*, *τ* ∼ *ω*_{0}*τ*, and *x* ∼ *x*/*λ*, where *λ* = 2*π*/*k* is the laser wavelength.

Using the Hamilton principle, we can write the Lagrangian of systems (10)–(12) in the form^{24}^{,}^{42}^{,}^{46}^{,}^{–}^{48}

where *g*(*a*) is a metric and *V*(*a*) is the potential of the system, both yet to be determined, and the prime indicates the derivative with respect to *y*. Using Lagrange’s equation,

we can obtain

and

According the conservation law *H* = *H* (*y* → ∞) and using the boundary condition *y* → ∞, *a*(*y*) → 0, and *a*′(*y*) → 0, we have *H* = −*η*[1 − *β* cos(*qy*)], which results in

Equation (23) governs the propagation of the relativistic ultrashort laser pulse in inhomogeneous plasma with a density ripple. The analytical solution of Eq. (23) can be obtained when *β* = 0, i.e., in the homogeneous plasma, which is discussed in Sec. III. When *β* ≠ 0, i.e., in plasma with a density ripple, we can obtain the numerical solution of Eq. (23) by the fourth order Runge–Kutta method, and the propagation characters of the laser pulse in inhomogeneous plasma are discussed in Secs. IV and V. Correspondingly, the electron density is calculated using Eqs. (11) and (12). We can write

In Eq. (25), we can see that the electron density is determined by the laser and the plasma density ripple, while the change of laser is controlled by *η* and *μ* and is related to the propagation constant *h* and wave number *k* of the laser.

## III. PROPAGATION CHARACTERISTICS OF ULTRASHORT LASER PULSE IN HOMOGENEOUS PLASMA

In homogeneous plasma (*β* = 0), an exact analytical solution of the laser is obtained by integrating Eq. (23),

The electron density is also given in Eq. (25),

where *α*^{2} = *η* − *μ*, *η* > *μ* > 0. For the long pulse laser (*σ* = 1, *μ* = 1), the propagation characters of the laser were carefully studied by Naseri *et al.*^{3} For an ultrashort laser pulse, as *μ* ≠ 1 and *α*^{2} ≠ *η* − 1, the effects of the ultrashort laser pulse will generate obvious effects on the transverse profile and intensity of the laser.

In order to investigate the propagation characteristics of a relativistic ultrashort laser pulse in homogeneous plasma, the appropriate parameters of the laser and plasma are selected to analyze the physical mechanism. The laser and plasma parameters are defined as follows: the ultrashort laser pulse with an initial intensity of *I*_{0} = 1.109 04 × 10^{20} W/cm^{2} and the laser frequency and pulsewidth of *ω*_{0} = 12 × 10^{15} Hz and *τ* = 1 fs, respectively, which can be generated by a Free Electron Laser (FEL).^{47}^{,}^{49}^{,}^{50} The initial electron density is *N*_{0} = 1.255 12 × 10^{21} cm^{−3}, and the corresponding plasma eigenfrequency is *ω*_{p} = 20 × 10^{14} Hz. The effects of laser pulsewidth *τ* on the propagation of laser and the modulation of electron density are shown in Fig. 1. For the ultrashort laser pulse with *τ* = 18 and *τ* = 30 propagating in homogeneous plasma, the intensity of the laser is the same [Figs. 1(b)–1(d)]. Correspondingly, the minimum value of electron’s density located at *x* = 125.61 generated by the laser with *τ* = 18 and *τ* = 30 is also the same [Figs. 1(f)–1(h)]. However, the longitudinal distribution of the electrons in the *τ* = 18 and *τ* = 30 cases is different significantly [Figs. 1(f) and 1(g)]. It is interesting to note that when the pulsewidth of the laser is near *τ* = 6 [Figs. 1(a) and 1(d)], the high intensity of the laser is needed to have stable propagation of the laser and the corresponding minimum value of electron’s density will be more smaller [Figs. 1(e) and 1(h)]. The quantitative comparison of the intensity of laser pulse and beam width at different pulsewidths is clearly shown in Fig. 1(d). When *τ* = 18 and *τ* = 30, the intensity of a laser pulse is 1.19. When *τ* = 6, the intensity (beam width) of laser is enhanced (reduced). As the pulsewidth of the laser changes from 30 to 6, the intensity of the laser is enhanced from 1.19 to 1.41, which results in the electron’s density of the low density region reaching 0.47 [Fig. 1(h)]. That is, when the ultrashort laser pulse propagates in homogeneous plasma, the intensity of the laser pulse is enhanced obviously as the pulsewidth of the laser decreases. Correspondingly, the modulation of electron density is more obvious. Therefore, the adjustment of the pulsewidth of the incident laser is the key factor for the modulation of the distribution of electron density.

Except the modulation of the pulsewidth of the incident laser, the modulation of plasma frequency is important for the control of the spatial distribution of the laser pulse and electron density. The spatial variation of the laser pulse and the electron density under the different plasma density is shown in Fig. 2. As the plasma frequency increases, the spatial distribution of the laser pulse changes, which includes the decrease of the beam width and the increase of the laser’s intensity [Figs. 2(a)–2(d)]. Meanwhile, the electron density is also modulated with the appearance of a more lower density (*n* = 0.66) of the electron [Figs. 2(e)–2(h)]. The quantitative comparison of the laser’s transverse profile and the corresponding distribution of the electron’s density are shown in Figs. 2(d) and 2(h), respectively. The intensity of the laser pulse can be enhanced from 1.09 to 1.23, and the electron’s density of the low density region can be modulated from 0.77 to 0.66 by increasing the plasma frequency. Clearly, the intensity and the spatial distribution of the laser with stable propagation character and the corresponding electron density can be controlled by adjusting the plasma frequency, which means that the effective modulation for the intensity of laser pulse and electron density is realized.

## IV. PROPAGATION CHARACTERISTICS OF ULTRASHORT LASER PULSE IN INHOMOGENEOUS PLASMA

In the previous part, we carefully discussed the effects of the plasma frequency and the pulsewidth of laser on the spatial distribution of the laser pulse and electron density in homogeneous plasma. For the inhomogeneous plasma (*β* > 0), the transverse spatial distribution of the laser given by Eq. (23) can be numerically obtained by the fourth order Runge–Kutta method. Correspondingly, the electron density is also obtained by using Eq. (25). Figure 3 shows the effect of the wave amplitude *β* of the density ripple on the intensity of the laser pulse in different plasma densities. The numerical solution is in good agreement with the analytical solution when *β* = 0 (i.e., the homogeneous plasma). When 0 < *β* < 0.15, the inhomogeneity of the plasma has a very weak effect on the intensity of the laser pulse. With the increase of *β*, the intensity of the laser pulse gradually decreases, which is different from the case in homogeneous plasma. Clearly, when 0 < *β* < 0.15, the propagation characteristics of the laser in inhomogeneous plasma are basically consistent with those in homogeneous plasma. However, when *β* > 0.15, the inhomogeneity of the plasma will obviously affect the propagation of the laser pulse.

The effects of wave number *q* of the density ripple on the spatial distribution of the laser pulse and electrons density are shown in Fig. 4. When the ultrashort laser pulse propagates in inhomogeneous plasma with the density ripple, the wave number of the density ripple has a weak effect on the spatial distribution of the laser pulse [Fig. 4(a)]. Moreover, since the ponderomotive force of the laser at the axis of the laser is larger than that at the edge of the laser, an obvious low density region appears at the wave trough of the plasma density ripple [Fig. 4(e)]. As the wave number of the density ripple increases, there is a weak variation of the intensity of the laser pulse [Figs. 4(b) and 4(c)]. However, the width of the transverse distribution of the low density region gradually decreases as the wave number of the density ripple increases [Figs. 4(f) and 4(g)]. The quantitative comparison of the intensity of the laser pulse and the corresponding distribution of electron density under the different density ripple wave number are shown in Figs. 4(d) and 4(h). Clearly, when *β* = 0.1, due to the influence of the plasma density ripple, the transverse profile of the laser pulse is changed with the formation of the platform located at the axis of the laser. However, the intensity of the laser pulse is almost unchanged. As the wave number of the density ripple increases from *q* = 3 to *q* = 9, the intensity of the laser pulse is enhanced from 1.21 to 1.24, which results in the reduction of the electron density in the low density region at *x* = 125.86 to *n* = 0.197 and narrowing of the width of the transverse distribution of the low density region [Fig. 4(h)]. Therefore, the wave number of the density ripple has a weak effect on the spatial distribution of the laser pulse. However, it plays a key role in the transverse distribution of electron density.

The effects of laser pulsewidth *τ* on the laser propagation and the modulation of electron density are shown in Fig. 5. For the ultrashort laser pulse with *τ* = 24 and 36 propagating in inhomogeneous plasma, the intensity of the laser pulse with different pulsewidths is the same [Figs. 5(b) and 5(c)]. Correspondingly, the variation of electron density is not obvious in the low density region [Figs. 5(f) and 5(g)]. However, there is a significant difference in the spatial distribution of electrons in the longitudinal direction [Figs. 5(f) and 5(g)]. As the pulsewidth of the laser decreases, the intensity of the laser pulse is enhanced significantly [Fig. 5(a)], which results in the formation of the low density region in the longitudinal direction with more low electron density [Fig. 5(e)]. The quantitative comparison of the intensity of the laser pulse and the corresponding distribution of electron density at different pulsewidth *τ* are clearly shown in Figs. 5(d) and 5(h). As the pulsewidth of the laser decreases from 36 to 12, the intensity of the laser pulse can be enhanced from 0.93 to 1.08. Correspondingly, the electron density of the low density region can be modulated from 0.43 to 0.19. That is, when the ultrashort laser pulse propagates in inhomogeneous plasma with the density ripple, the intensity of the laser pulse is enhanced significantly and the modulation of the distribution of electron density is more obvious as the pulsewidth of laser decreases. Hence, the pulsewidth of the incident laser plays an important role in the modulation of electron density in inhomogeneous plasma with the density ripple.

Here, the effects of the wave amplitude *β* of the density ripple on the intensity of laser pulse and the distribution of electron density are shown in Fig. 6. Clearly, comparing with the results of *β* = 0.1 [Fig. 4(a)], when *β* increases from 0.1 to 0.2, the intensity of the laser pulse gradually decreases and the intensity of the laser pulse decreases from 1.21 to 1.08, but the inhomogeneity of the spatial distribution of the laser pulse is enhanced obviously. When the ultrashort laser pulse propagates in inhomogeneous plasma with *β* = 0.2, the spatial distribution of the laser pulse is obviously affected by the wave amplitude of the density ripple and the inhomogeneity of the spatial distribution of the laser pulse is more obvious [Fig. 6(a)]. When *β* increases from 0.2 to 0.3, the intensity of the laser pulse gradually decreases [Figs. 6(a) and 6(b)], and the corresponding electron density slightly changes in the longitudinal direction [Figs. 6(d) and 6(e)]. The quantitative comparison of the intensity of the laser pulse and the corresponding distribution of electron density under the wave amplitude of different density ripples are clearly shown in Figs. 6(c) and 6(f). When *β* increases from 0.2 to 0.3, the wave amplitude of the density ripple has a significant effect on the transverse profile of the laser pulse and the intensity of the laser pulse decreases from 1.08 to 0.87; correspondingly, the electron density in the low density region at *x* = 125.86 is also modulated from 0.19 to 0.28 [Fig. 6(f)]. Clearly, the intensity of the laser pulse is obviously modulated by the wave amplitude of the density ripple, which is consistent with the result obtained in Fig. 3. Therefore, the propagation characteristics of the laser in inhomogeneous plasma are basically consistent with those in homogeneous plasma when 0 < *β* < 0.15. However, when *β* > 0.15, the intensity of the laser pulse gradually decreases, and the inhomogeneity of the spatial distribution of the laser pulse is more obvious, which means that the inhomogeneity of the plasma will affect the stable propagation of the laser pulse obviously.

## V. PARTICLE-IN-CELL SIMULATIONS

A two-dimensional (2D) particle-in-cell (PIC) simulation is performed to verify the theoretical analysis. The physical dimension of the simulation is 60 × 10 *µ*m^{2} with 6000 × 1000 cells, resolving each 0.01 *µ*m and Δ*t* = 9.2*as*. Under the Courant–Friedrichs–Lewy (CFL) condition, the same results are also obtained when Δ*t* = 18.4*as*, which means that the numerical dispersion will not happen in this PIC simulation. Each cell contains 2 × 4 macroparticles, representing the ensemble of real charged particles. The absorbing boundary is used in PIC simulations. The other parameters of laser and plasma remain constant with Sec. III. The key results of the spatial distribution and envelope of the laser pulse and the plasma density at different times (*t* = 223, 554, 885, 1216) are shown in Fig. 7. In the initial process (*t* ≤ 554), a stable propagation of the intense ultrashort laser pulse in plasma is realized effectively [Figs. 7(a1) and 7(a2)]. Clearly, the envelope of the incident laser pulse is not affected effectively [Figs. 7(c1) and 7(c2)]. The transversal spatial distribution of the laser pulse at *x* = 20 holds the stable soliton structure [Fig. 7(d1)], which is predicted by our theoretical analysis (see Fig. 1). Meanwhile, a low density region with a circular structure is generated in plasma behind the ultrashort laser pulse [Figs. 7(b1) and 7(b2)] due to the ponderomotive force of the laser pulse that will make the electrons to be expelled from the laser axis. When *t* = 885, there is a weak variation of the laser pulse, including the slightly deformation of the envelope and decrease of the intensity of the pulse [Fig. 7(a3)]. The falling edge of the envelope of the laser pulse broadens and the intensity of the laser pulse decreases weakly [Fig. 7(c3)]. The transversal spatial distribution of the laser pulse is also deformed slightly with the generation of the negative electric field of the laser pulse at *x* = 130 [Fig. 7(d3)]. The circular structure of the low density region of the plasma is also deformed [Fig. 7(b3)]. When *t* ≥ 1000, the envelope and the transversal spatial distribution of the laser pulse will be gradually broken. Figure 7(a4) clearly shows the deformation of the envelope and transversal spatial distribution of the laser pulse at *t* = 1216. It has to be noted that the falling edge of the envelope of the laser pulse not only broadens effectively but also forms a significant structure of the wiggling tail [Fig. 7(c4)]. Correspondingly, the ultrashort laser pulse no longer propagates steadily with the disappearance of the soliton structure of the transversal spatial distribution of the laser pulse [Fig. 7(d4)]. Although there is still a low density region of the plasma, the spatial distribution of the low density region of the plasma becomes a triangle structure due to the deformation of the laser pulse [Fig. 7(b4)]. Clearly, the deformation of the envelope of the laser pulse from the **J** × **B** heating effect and the self-generated electric field of the plasma affect the stable propagation of the laser pulse in the plasma, which is not carefully considered in theoretical analysis.

To further increase the stable propagation distance of the laser pulse, the modulation for the plasma density can be a key way to realize the stable long distance propagation of the ultrashort laser pulse. In the low density plasma (*N*_{0} = 0.42 × 10^{21} cm^{−3}), the variation of the envelope and the transversal spatial distribution of the laser pulse with different times are shown in Fig. 8. As the time increases, the intensity of the laser pulse gradually decreases. However, the envelope of the laser pulse does not show any deformation [Fig. 8(a)]. Meanwhile, the transversal spatial distribution of the laser pulse at the falling edge of the envelope of the laser pulse in the different times still holds the soliton structure without any significant deformation [Fig. 8(b)]. Thus, the ultrashort laser pulse with the soliton structure in the transversal direction can realize a stable propagation in low density plasma.

Despite the effects of the plasma density on the propagation of the ultrashort laser pulse, inhomogeneous plasma (*β* > 0) can also modulate the spatial distribution of the laser pulse effectively. Based on the PIC simulation, inhomogeneous plasma with the density ripple (*β* = 0.2) modulates the transversal spatial distribution of the laser pulse effectively [Figs. 9(a1) and 9(a4)]. When *t* = 800, the envelope of the laser pulse is not affected by the density ripple of the inhomogeneous plasma [Figs. 9(a1) and 9(a3)], which is similar to the propagation of the ultrashort laser pulse in the homogeneous plasma [Fig. 7(c3)]. However, there are four platforms in the transversal distribution of the laser pulse [Fig. 9(a4)], which results in the effects of the inhomogeneous plasma with the density ripple. That is, the density ripple of inhomogeneous plasma can effectively change the transverse distribution of the laser pulse to form the novel structure of the laser pulse. Correspondingly, the distribution of the plasma is also modulated and generates three low density regions inside the plasma [Fig. 9(a2)]. As a potential modulation way for the ultrashort laser pulse, the generation of the multi-platforms of the laser pulse can replace the expensive digital micromirror device to effectively modulate the transversal structure of the laser pulse without the limitation of damage threshold.^{36} Correspondingly, the modulation for the density ripple of the plasma can also be an effective way to obtain the ultrashort laser pulse with the multi-platform structure. This is in good agreement with theoretical predictions (Figs. 5 and 6). However, as the time further increases (*t* > 1000), the spatial distribution of the laser pulse is also broken [Fig. 9(b1)], which is the same with the case of the homogeneous plasma [Fig. 7(a4)]. The envelope of the laser pulse generates the weak deformation [Fig. 9(b3)]. The intensity of the laser pulse with the multi-platform structure gradually decreases [Fig. 9(b4)]. However, the distribution of the plasma still holds three low density regions inside plasma when *t* > 1000 [Fig. 9(b2)].

Then, as predicted by theoretical analysis (Fig. 3), the amplitude of the ripple density of the plasma will affect the transversal spatial distribution of the laser pulse effectively. When *β* < 0.15, the ultrashort laser pulse can propagate in inhomogeneous plasma steadily, which is similar to that of homogeneous plasma. When *β* > 0.15, the ultrashort laser pulse will be effectively modulated in the intensity and spatial distribution of the laser pulse.

Through PIC simulation, Fig. 10 shows the dependency of the transversal spatial distribution of the laser pulse on the time with the different amplitude of the ripple density of the plasma. Here, we focus on the transversal spatial distribution of the laser pulse at the location of the maximum electric field intensity of the laser pulse. For *β* = 0.1 < 0.15, the transversal spatial distribution of the initial laser pulse has the soliton structure. As the time increases, PIC simulation shows that the soliton structure of the laser pulse presents weak diffusion and slightly distortion in the transversal direction [Fig. 10(a)], which is in good agreement with the theoretical predictions (Fig. 3). For *β* = 0.2 > 0.15, although the transversal spatial distribution of the initial laser pulse still holds the soliton structure, PIC simulation shows that the soliton structure of the initial laser pulse will be modified and forms the structure with the multi-platforms in plasma with the ripple density when *t* > 20. This agrees with our theoretical prediction shown in Figs. 3–6. In the time region of 20 < *t* ≤ 1600, the multi-platforms of the laser pulse are held steadily in plasma with the ripple density. There is only a weak oscillation of the intensity of the laser pulse. As the time further increases, the multi-platforms of the laser pulse will be gradually affected by the diffusion effect of the laser pulse [Fig. 10(b)].

## VI. CONCLUSION

In conclusion, we have investigated the interaction of the ultrashort laser pulse and inhomogeneous plasma with the density ripple. In homogeneous plasma, the propagation distance and intensity of the laser pulse and the spatial distribution of electron density can be modulated by adjusting the pulsewidth of the incident laser and the plasma frequency. In inhomogeneous plasma with the density ripple, when the wave amplitude of the density ripple is less than the critical value, the intensity of the laser pulse is almost unchanged and the propagation characteristics of the laser in inhomogeneous plasma are basically consistent with those in homogeneous plasma. Moreover, when the wave amplitude of the density ripple is larger than the critical value, the spatial distribution of the laser pulse changes significantly, the intensity of the laser pulse gradually decreases, and the inhomogeneity of the plasma has an obvious effect on the propagation of the laser pulse. Therefore, the wave number and amplitude of the density ripple play an important role in modulating the intensity and the spatial distribution of the laser pulse.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11865014, 11765017, and 11764039, the Natural Science Foundation of Gansu Province under Grant No. 17JR5RA076, and the Scientific Research Project of Gansu Higher Education under Grant No. 2016A-005.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

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