Nonlinear propagation of an intense Laguerre–Gaussian laser pulse in a plasma channel*

Mingping Liu(刘明萍), Zhen Zhang(张震), and Suhui Deng(邓素辉)

School of Information Engineering,Nanchang University,Nanchang 330031,China

Keywords: nonlinear propagation,intense Laguerre–Gaussian laser pulse,wake-field effect,plasma channel

1. Introduction

The optical guiding of an intense laser pulse in plasma is very important to many promising applications,such as laserdriven accelerators,[1]harmonic generation,[2]laser confinement fusion,[3]and x/γ-ray radiation sources.[4]As we know,a laser pulse propagating in vacuum will diffract after a distance on the order of a Rayleigh length ZR=πr20/λ,where r0is the laser spot size at focus and λ is the laser wavelength.However,an intense laser pulse in uniform plasmas can guide itself due to the relativistic self-focusing(RSF)when the laser power exceeds the critical power Pc=17.4(ω0/ωp)2(GW),where ω0is the laser frequency and ωpis the plasma frequency.[5]If the laser power is smaller than Pc,laser diffraction will dominate over the RSF. In this case, a preformed plasma channel with a radially parabolic density profile can prevent laser diffraction and guide the latter propagating over many Rayleigh lengths,[6]which is beneficial to the above applications. In a realistic scenario,even if the laser power P ≥Pc,GeV quasimonoenergetic electron bunches can be obtained by increasing the acceleration length to a few centimeters in a laser guiding discharge capillary.[7–9]Recently,a striking result of electron bunch approaching 8 GeV has been measured in such channelguided laser accelerator with more powerful laser systems.[10]Such high energy electron bunches provide a potential approach to produce high brightness x-ray and even γ-ray radiation sources.[11–13]Laser wake-field acceleration (LWFA)is the main acceleration mechanism for the generation of such energetic electron bunch and then wake field effects play a crucial role in electron acceleration like as the plasma channel.

In these experiments, laser pulses with fundamental Gaussian(FG)mode are usually used as drivers for propagating in a preformed plasma channel. It has been shown that the propagation characteristics of an intense laser pulse in a preformed plasma channel is strongly dependent on the interactions between the laser and plasma according to the principle of the refractive guiding.[5]When electrons quiver transversely in the electric field of an intense laser pulse,their mass increase relativistically and then the radial profile of the refraction index peaks on the propagating axis, which means that the plasma acts as a positive len. Consequently,the laser pulse focuses toward the axis, i.e., the RSF occurs. A performed plasma channel with a radially parabolic density profile creates a refractive index profile with on-axis maximum,which yields the channel focusing (CF). The transverse ponderomotive force of an intense laser pulse can expel electrons away from the region where the laser intensity is the maximum, which gives rise to transverse gradient of the refractive index. This effect is well known as the ponderomotive self-channeling (PSC). All these nonlinearities discussed above present the focusing effects. Therefore, whether the laser pulse can realize guiding would mainly depend on the competition between the vacuum diffraction and the combined focusing effects of RSF, CF, and PSC. Many papers have investigated these focusing effects in the uniform plasma or preformed channel cases.[14–20]It has been found that the spot size of an laser pulse can evolve along the propagating axis with constant, periodically focusing and defocusing oscillations, and catastrophic focusing. Further, Zhang et al. has investigated the existence conditions of solitary waves in a parabolic plasma channel.[21]The solitary waves are helpful to electron acceleration in the LWFA, because any perturbation of the solitary waves would lead to the wave breaking and realize the electron self-injection in the wake-field regime.Hong et al. found that when the laser length is on the order of plasma wavelength,a wake field will be excited efficiently and focus the laser pulse during its propagation forward.[22]Most of these results are based on the laser pulses with FG modes. In recent years, there has been an increasing interest on the laser pulses with Laguerre–Gaussian(LG)modes which have hollow transverse intensity profile and carry orbital angular momentum.[23–33]LG laser pulses drive donut-shaped wake fields whose field structure is capable of accelerating positrons and significantly enhances x/γ-ray generation.[23–26]Obviously, the dynamics of LG laser pulse propagating in a preformed plasma channel would definitely affect its application. In non-relativistic limit, the condition for the matched LG laser pulse in a preformed plasma channel is the same for all modes whether it includes the finite pulse length and group velocity dispersion or not.[6,34]However,electrons can be violently accelerated in the wake field regime driven by an intense laser pulse. Thus,these nonlinearities,such as group velocity dispersion, relativistic and wake-field effects, should be considered in analyzing the LG laser dynamics.

In this paper, we study the propagation characteristics of an intense LG laser pulse in a preformed plasma channel used by variational technique,including the combined effects of RSF, CF, and PSC. In order to effectively excite the wake field, the pulse length is defined as L=λpin our model,[1,5]where L and λpare the pulse length and plasma wavelength,respectively. The evolution equation of the LG laser spot size is obtained and then the conditions of laser and plasma channel parameters for propagating with constant spot size,periodically focusing, and defocusing oscillations, catastrophic focusing and solitary waves are identified in detail. All these nonlinear effects mentioned above are also illustrated numerically.

2. Evolution equation of laser spot size

The propagation along the z-direction of a linearly polarized laser pulse in a preformed plasma channel will be considered in our model. The density of the plasma channel is assumed with a profile of the form nch(r) = n0+Δnr2/r2ch,where n0is the initial axial electron density,Δn and rchare the channel depth and radius,respectively. The normalized vector potential of the laser field is given by where a=eA/m0c2is the normalized amplitude of the vector potential of the laser field,e and m0are the charge and the rest mass of electron,c is the light velocity in vacuum,ω =ck/βgis the laser center frequency,βg=vg/c is the normalized group velocity, ˆx is the unit vector along the x axis,and c.c. denotes the complex conjugate.

where Llnis the generalized Laguerre polynomial,φ is the polar angular coordinate, and as(z), θ(z), rs(z), and α(z) are the real functions of the propagation distance z and represent the amplitude, phase shift, spot size, and curvature, respectively. For the sake of simplicity, the first order LG mode(l=1,n=0)is considered in our model,which has been used by most papers to study electron acceleration and radiation sources.[23–33]By inserting Eq. (4) into Eq. (3), the solution for the perturbed plasma density inside the pulse(0 ≤ξ ≤L)is given by

where a0and r0are the initial(z=0)amplitude and spot size of the LG laser pulse,respectively. Considering the effects of relativistic,channel-coupling nonlinearity and wake field,one can obtain the evolution characteristics of the laser spot from Eqs. (6)–(8). Equation (6) indicates the power conservation relation=. The first term on the right-hand side of Eq. (8) shows the vacuum diffraction, the second term represents the RSF,the third term yields the CF,the fourth and fifth terms present the wake-field effects induced by the transverse and longitudinal ponderomotive forces of the laser pulse, respectively. It obviously shows that the wake field also leads to the focusing of the LG laser pulse. Compared with the results of a FG laser pulse, such as Eq. (8) in Ref. [22], coefficient of the term of vacuum diffraction decreases by half and coefficients of the terms of the combined focusing also decrease by a quarter except the third term of the CF.Therefore,the hollow transverse intensity profile of the LG laser pulse significantly influences on its nonlinear propagation characteristics.

3. Propagation characteristics of the LG laser pulse

Equation (10) has the same form of an energy equation for a particle in a potential well V(rs). Here, rshas the role of the space coordinate and z is the“time”. Thus,it is more instructive to discuss the solution of Eq.(10)according to the value of the potential V(rs), which in turn depends on the initial laser and plasma parameters.

We first analyze the potential function V(rs). By solving V(rs)=0,one can obtain three solutions as

First of all, we discuss the condition for a matched(nonevolving) spot size of the LG laser pulse in the plasma channel. Only case (vi) mentioned above can ensure the test particle rest in the potential well V(rs) as shown in Fig. 1(f).Thus,if the initial laser and plasma parameters satisfy P=Pth2and Nc＞Nth,the spot size rs=1 will retain during the propagation of the LG laser pulse in the plasma channel.Second,the test particle will be unstable and eventually move boundlessly to the position rs→0, which can occur in the cases (i), (ii),(iv),and(ix),respectively.In this case,the spot size will be focused catastrophically and contracted to a geometric point. It should be pointed that the singularity will never occur since the laser intensity will be increased infinitely,which is beyond the weakly relativistic limit of our model.According to the parameter conditions of cases(i), (ii), (iv), and(ix), one can obtain the comprehensive parameter regions for catastrophic focusing,i.e.,P ＞Pth1for any Ncand Pth2＜P ≤Pth1for Nc≤Nth.Third,one can find from Figs.1(e)and 1(h)that the test particle will oscillate back and forth between rs1and rs2. So the structure of the spot size rsis a train of finite amplitude waves with finite wavelength. The test particle will perform periodically defocusing oscillation in the case (v) whereas perform periodically focusing oscillation in the case(viii). Last but not least,in cases(vii)and(iii),the test particle initially lies in or moves to a critical point. It is very interesting to discuss the two critical cases for the propagation of LG laser pulse in a preformed plasma channel.

Fig.1. Potential well V(rs)as a function of rs. Panels(a)–(i)correspond to the enumerated cases(i)–(ix)in the paper discussed above rs1,rs2,and rs3 are the three roots of V(rs)=0. The red point in each figures represents the initial position of a test particle with rs=1 and ∂rs/∂z=0. The normalized laser amplitude is a0=0.3 for all cases.

According to the nonlinear dynamics theory,equation(9)can be reduced to the following autonomous system’s equations:

Fig. 2. Propagation characteristics of a LG laser pulse (black curves) and a FG laser pulse (red curves) for a0 =0.3. The circles correspond to the critical parameters Nth. The solid curves 4 represent the parameters of the existence of the solitary waves,while the dashed curves 2 denote the propagation of laser pulse with constant spot size. There are three regions,which correspond periodically defocusing oscillations (region 1), periodically focusing oscillations (region 3), and catastrophic focusing (region 5). The results of the FG laser pulse have calculated according to the formulas of Ref.[22].

In order to get the full view of the propagation characteristics of the intense LG laser pulse, the initial parameters of laser and plasma channel for each cases discussed above are plotted in Fig.2(black curves). For comparison,the propagation characteristics of a FG laser pulse in a preformed channel are also shown in Fig. 2 (red curves). The solid curves 4 are the critical power of the laser pulse for the existence of the solitary waves under different the channel parameter Nc, and the dashed curves 2 denote the laser power for its propagation with constant spot size. The two types of curves classify the Nc–P plane with three regions corresponding to the periodically defocusing oscillation(region 1),periodically focusing oscillation(region 3)and catastrophic focusing(region 5),respectively. As expected, the effects of relativistic, channelcoupling nonlinearity and wake field greatly release the total critical power P for the laser focusing, which is far smaller than 1. It is obviously seen from Fig.2 that the critical power of the LG laser pulse for the existence of the solitary waves almost coincides with that of the FG laser pulse,while the laser power for its propagation with constant spot size in the case of the LG laser pulse is decreased remarkably compared with the case of the FG laser pulse. When Nc=0.5,the powers for the existence of the solitary waves are 0.525 for the LG laser pulse and 0.515 for the FG laser pulse,respectively. However,the power for the propagation with constant spot size is very weak, i.e., P0, in the case of the LG laser pulse, which is more less than that (P=0.305) in the case of the FG laser pulse. Therefore,one can note that there is a overlapping area between the periodic focusing region of the LG laser pulse and the periodic defocusing region of the FG laser pulse.

4. Numerical results and discussion

In this section, the propagation characteristics of the LG laser pulse in a performed plasma channel are numerically investigated according to Eq. (9), which is solved using the fourth-order Runge–Kutta method with the parameters presented in Fig. 1 and the initial conditions rs|z=0= 1 and∂rs/∂z|z=0=0. The evolution of laser spot size rsand amplitude aswith the propagation distance z will be shown in the following Figs.3–5 for the cases of constant spot size, catastrophic focusing, periodically focusing (or defocusing) oscillation,and solitary waves,respectively.

It can be seen from Eq. (9) that the propagation characteristics of the LG laser pulse strongly depend on the competition between the vacuum diffraction and the combined focusing effects of relativistic, channel-coupling nonlinearity, and wake field. When the effect of the vacuum diffraction is exactly canceled by the combined focusing effects, the equilibrium solution with constant spot size rsand constant amplitude asare shown in the solid curves of Fig.3 corresponding to the case(vi)of Fig.1. If the combined focusing effects significantly overwhelm the vacuum diffraction,the catastrophic focusing will happen. One can see the cases of the catastrophic focusing plotted in Fig.3 with the dashed curves,dotted curves and dash–dotted curves, which correspond to the cases of Figs.1(a),1(d),and 1(i),respectively.With the nearly same laser power P, the spot size rsof the case of Fig. 1(a)is focused catastrophically more faster than those cases of Figs. 1(d) and 1(i) because of the higher channel parameter Nc. As mentioned early, the catastrophic focusing will never occur since the laser intensity will increase infinitely when the spot size is focused to zero. Then,the effects of higher-order relativistic nonlinearities and other terms will prevent the spot size collapse in the real laser-plasma interactions. Actually,the high-intense laser pulse (a0≥1) will expel local plasma electrons and create an electron-free blowout. This effect will lead to the diffracting propagation for the laser pulse as in the vacuum.[36,37]Even so,it is still out of the weakly relativistic limit used in our model.

Fig. 3. The evolution of laser spot size rs (a) and amplitude as (b) with the propagation distance z. The propagation characteristics of the LG laser pulse with constant spot size are plotted with the parameters of Fig. 1(f)(solid curve), while the cases of catastrophic focusing are plotted with the parameters of Fig. 1(a) (dashed curve), (d) (dotted curve), and (i) (dash–dotted curve).

If the initial parameters of laser and plasma are chosen as in region 1 of Fig. 2, the periodically defocusing oscillations will occur as shown in Figs. 4(a) and 4(c) responding to the case of Fig.1(e). In this region,the combined focusing effects are weaker than the vacuum diffraction in the beginning, and the laser spot size in turn increases at first,which can be verified by rswith greater than 1 in Fig.4(a). With the increase of the spot size,the CF will be enhanced but the vacuum diffraction becomes weaker. Then, the defocusing behavior of the spot size will be prevented and it will decrease after reaching a peak. Thus, the periodically defocusing oscillations of the laser spot size will form as shown in Fig. 4(a). In the region 3 of Fig. 2, with the parameters of laser and plasma in this region,e.g. the the case(viii)of Fig.1,the laser spot size decreases at first and will be less than 1, which leads to enhancing the vacuum diffraction but decreasing the CF. Thus,the periodically focusing oscillations can build as shown in Figs.4(b)and 4(d).

The propagation characteristics of solitary waves are presented in Fig.5. One can see that the homoclinic orbits in rs–∂rs/∂z plane are depicted in Figs.5(a)and 5(b)corresponding to the parameters of Figs.1(c)and 1(g),respectively. It means that the evolution of the spot size rsand amplitude aswith the propagation distance z will be solitary waves. However,in general it is hard to get the exact solutions of the solitary wave for Eq.(9)since the numerical error is unavoidable. Hence,a train of solitary-like waves of the spot size rs,of course amplitude as,are shown in Figs.5(c)–5(f). It should note that in the case of Fig.1(c),the initial spot size is not at a saddle point and a solitary-like structure is formed at first,and then the spot size keeps constant for a long distance about 7ZR. However,in the case of Fig. 1(g), the spot size initially lies on a saddle point and in turn keeps constant at first for a longer distance about 150ZR. It is worth pointing out that the solitary-like waves have the focusing effect in Figs. 5(a), 5(c), and 5(e) because of the higher channel focusing, while the solitary-like waves have the defocusing effect in Figs.5(b),5(d),and 5(f).

Fig. 4. The evolution of laser spot size rs (a) and amplitude as (c) shows the periodically defocusing oscillation with the parameters of Fig.1(e). The evolution of laser spot size rs(b)and amplitude as(d)shows the periodically focusing oscillation with the parameters of Fig.1(h).

Fig. 5. Propagation characteristics of solitary-like waves corresponding to the cases of Fig.1(c)(left column)and 1(g)(right column). (a)and(b)Orbits on the phase plane rs–∂rs/∂z, the initial position is marked in red; (c)and(d)the evolution of the laser spot size rs with the propagation distance z; (e) and (f) the evolution of the laser amplitude as with the propagation distance z.

In order to further understand the influence of the hollow transverse intensity profile of the LG laser pulse on its propagation characteristics, it is necessary to investigate the behaviors of the spot size of the LG and FG laser pulses with the same initial parameters. Figure 6 shows the evolution of the spot size rsof the LG (solid curves) and FG (dashed curves) laser pulses with the same parameters of laser power and plasma channel in the different regions of Fig. 2, i.e.,P=0.2 and Nc=0.2 in region 1,P=0.2 and Nc=0.5 in the overlapping region,P=0.4 and Nc=0.5 in the region 3,and P=0.6 and Nc=0.8 in the region 5.In region 1 of Fig.2,both of the LG and FG laser pulses perform periodically defocusing oscillations as shown in Fig.6(a). However, the effect of vacuum diffraction of the LG laser pulse is weaker than that of the FG laser pulse and then the spot size of the latter increases to the peak about 2.05,close to 1.6 times the peak value of the former. When the channel parameter Ncincreases to 0.5,i.e.,in the overlapping region of Fig.2,the combined focusing effects overwhelm the vacuum diffraction effect for the LG laser pulse and it in turn performs periodically focusing oscillations as shown in Fig. 6(b). However, due to the stronger vacuum diffraction,the FG laser pulse still remains the periodically defocusing oscillations but with the smaller amplitude of oscillation. In region 3 of Fig.2,the combined focusing continually overwhelms the vacuum diffraction for the LG laser pulse and then its oscillation amplitude is further enhanced as shown in Fig.6(c). Due to the enhanced laser power,the FG laser pulse converts to present periodically focusing oscillations with the smaller amplitude of oscillation.When the parameters of laser power and plasma channel locate in the region 5 of Fig.2,the focusing characteristics of the LG and FG laser pulses coincide with each other except the spot size rsclose to 0 as shown in Fig.6(d),which will never occur.

Fig. 6. The evolution of spot size rs with propagation distance z for the LG (black solid curves) and FG (red dashed curves) laser pulses. The parameters of laser pulse and plasma channel are(a)P=0.2 and Nc=0.2 in region 1 of Fig. 2, (b) P=0.2 and Nc =0.5 in the overlapping region of Fig. 2, (c) P=0.4 and Nc =0.5 in the region 3 of Fig. 2, and (d) P=0.6 and Nc=0.8 in the region 5 of Fig.2,respectively.

5. Conclusion

In a summary, the propagation characteristics of an intense LG laser pulse in a performed plasma channel are investigated, which takes into account the combined effects of relativistic,channel-coupling nonlinearity,and wake field. An evolution equation of the laser spot size is derived by means of the variational method. It is proved that the LG laser pulse can propagate with constant, periodically defocusing and focusing oscillations,catastrophic focusing,and solitary waves,respectively. The behaviors of the laser spot size and amplitude strongly depend on the initial parameters of the LG laser pulse and plasma channel,i.e.,laser amplitude a0,laser power P,and channel parameter Nc.Comparing with the propagation characteristics of a FG laser pulse, the effect of the vacuum diffraction is reduced by half and the effects of relativistic and wake-field focusing are decreased by one quarter because of the hollow transverse intensity profile of the LG laser pulse.However, the CF effect is the same order of magnitude with that of the FG laser pulse. Correspondingly,the matched condition for the LG laser pulse with constant spot size is reduced obviously,while the parameters of the laser and plasma for the existence of solitary waves nearly coincides with those of the FG laser pulse.Therefore,the characteristic region of the periodically defocusing oscillation is enlarged compared with the case of the FG laser pulse and the characteristic region of the periodically focusing oscillation in turn is reduced.