1. Introduction

Development of laser technologies is extending the research fields of laser physics. With laser intensity increasing, the laser matter interaction experiences process from linear response to perturbation in nonlinear optics successively. When the laser intensity is greater than $\sim 10^{14}$ W/cm$^2$, most of the matter will be instantaneously ionized by the laser into plasmas through multi-photon ionization and tunnel ionization of atoms [1]. Chirped pulse amplification (CPA) technique overcomes the damage threshold limitation of gain medium and optical element, resulting in several orders of magnitude improvement of laser intensity [2]. The output laser power has increased from gigawatt (GW) to petawatt (PW), and the pulse width is compressed from picosecond (ps) to femtosecond (fs). The focused laser intensity greatly exceeds the relativistic threshold of $\sim 10^{18}$ W/cm$^2$ [2,3]. The corresponding laser field is far more than the coulomb field inside the atom and thus the matter can be rapidly ionized into plasmas. The oscillating speed of plasma electrons in the laser electric fields approaches the speed of light in vacuum and the magnetic field effect is not negligible. As a result, the refractive index of plasmas exhibits strong nonlinear characteristics, and the laser matter interaction falls into relativistic nonlinear optics regime [1]. To avoid simultaneous amplification of the prepulse and the main pulse and improve the signal-to-noise ratio of the laser pulse, optical parametric CPA (OPCPA) technique combining optical parametric amplification with CPA is proposed, which is capable of further advancing the power and intensity of laser pulse [4]. Ultrashort optical pulse with intensity higher than $10^{23}$ W/cm$^2$ has been obtained in laboratories [5]. This allows us to explore strong field quantum-electrodynamics (QED) effects (e.g., radiation reaction) experimentally [6]. The classical radiation theory is not applicable any more in the energetic segment of radiation spectrum and quantum corrections are therefore necessary [7,8]. The laser interaction with matter thus enters the radiation-dominated near-QED region [9–13]. If the laser intensity continues to rise above $10^{24}$ W/cm$^2$, protons in plasmas also move relativistically in the laser fields, corresponding to the plasmas-based nuclear physics phase. Nuclear fission or fusion may occur, and particles such as neutrons and muons could be produced when nucleons with relativistic energy collide with laser or each other [14]. However, the improvement of laser intensity from $\sim 10^{23}$ W/cm$^2$ to $>10^{24}$ W/cm$^2$ remains a challenging endeavor because it is extremely difficult to fabricate large-aperture pulse compression gratings with existing manufacturing technologies.

Plasma, as an ionized production of matter, has a distinct advantage in manipulating relativistic or ultrarelativistic laser pulse because it has no thermal damage threshold. Several plasma-based laser amplifiers have been proposed, including stimulated Raman/Brillouin scattering in plasma [15–18] or magnetized plasma [19], relativistic flying parabolic mirror [20–22], relativistic self-focusing (RSF) [23–32], hollow micro-structured cone/capillary plasma channels [33–35] as well as light trapping and accumulation insides two foils [36,37]. However, the seed laser intensities in most of these cases are either low ($\sim 10^{14}$ W/cm$^2$) [15–20] and weakly relativistic ($<10^{18}$ W/cm$^2$) [23,24,36,37] or moderate ($<10^{21}$ W/cm$^2$) [21,22,25–32]. The ultimate amplified laser intensity is thus rather limited. Besides, strong-field QED effects such as radiation reaction are not taken into account in the laser-plasma interaction process in these cases. In this paper, we propose a method for efficiently producing ultrashort ultrarelativistic ($>10^{24}$ W/cm$^2$) light pulses with a sharp time rising front by the RSF and tapered-channel focusing of PW-class laser pulse in moderate density plasmas. Full three-dimensional (3D) particle-in-cell (PIC) simulations show that, with the optimized plasma parameters, almost an order of magnitude amplification in laser intensity is possible even though the invoked $\gamma$-ray radiation consumes a lot of laser energy. This method shows good robustness in a wide laser-plasma parameter range and might provide a feasible routine for achieving a laser intensity $>10^{24}$ W/cm$^2$ in present PW-class or upcoming hundreds-PW laser facilities.

2. Amplification of ultra-intense laser fields in moderate density plasma

The RSF occurs when the power of the incident laser pulse exceeds the critical value [23] $P_c=17(\omega _L/\omega _p)^2$ GW, where $\omega _L$ is the laser frequency, $\omega _p=(4\pi e^{2}n_{e}/\gamma m_{e})^{1/2}$ is the modified plasma frequency, $\gamma =(1+a_{0}^{2}/2)^{1/2}$ is the relativistic factor for a linearly polarized (LP) laser pulse, $n_{e}$ is the electron density of plasma, $-e$ and $m_e$ are the electron charge and rest mass, respectively. For low or moderate intensity laser pulse, RSF usually appears in underdense or near-critical density (NCD) plasmas. The normalized laser amplitude after the RSF can be expressed as [26]

Here, $a_0=eE_{0}/m_{e}\omega _{L}c$ is the maximum dimensionless amplitude of the incident laser pulse, $r_0$ and $E_{0}$ are its spot radius and peak electric field, and $n_c=m_{e}\omega _{L}^{2}/4\pi e^{2}$ is the critical plasma density. The amplification factor of laser intensity $\kappa _{s}$ can thus be written as

We find that $n_{e}$ should be increased accordingly to achieve effective laser amplification, i.e., large $\kappa _{s}$, as long as the plasma is still transparent for the incident laser pulse. From the condition of relativistic self-induced transparency $\omega _L>\omega _p$ [38], we can obtain the threshold of initial laser intensity

When $a_0>a_{0,cr}$, relativistic laser pulse can travel through the plasma and light amplification is possible. Besides, it should be mentioned that the radiation-dominated QED effects become significant if $a_0>200$. In the strong field near-QED regime, the radiation reaction force can be approximatively expressed as [9]

We perform 3D-PIC simulations by using open source code EPOCH [7] to verify the laser amplification effect in moderate density plasma. The QED modules for synchrotron radiation and electron-positron pair generation (including Breit-Wheeler process) have been implemented into the EPOCH code. Since the energy conversion efficiency of laser to electron-positron pairs in the current laser and plasma parameter range is very low (usually $\ll 1\%$), compared to that of $\gamma$-ray radiation, we only consider the radiation-dominated effect to save the computational resources. The size of simulation box is $x\times y\times z=60\lambda _0\times 20\lambda _0\times 20\lambda _0$, with a spatial resolution of 20 cells per wavelength, where $\lambda _0=1\ \mu {\rm m}$ is the laser wavelength. Each cell contains 16 macroparticles including 8 electrons and 8 protons. The open boundary conditions are used for both fields and particles. The hydrogen plasma of densities $n_e=n_p=30n_c$ is fully ionized and is located between $x=20\lambda _0$ and $x=60\lambda _0$ with a length of $L=40\lambda _0$. The initial electron temperature is 1 keV and the ions are cold. A $p$-polarized laser pulse with $a_0=280$ ($>a_{0,cr}=42.4$) and $r_0=3\lambda _0$ is normally incident into the plasma and is focused at its front surface, corresponding to a focal position of $x_{f}=20\lambda _0$. The laser field profile is given by $a=a_0\sin ^2(\pi t/2\tau )\exp (r^2/r_0^2)$, where $\tau =10T_0$ is the pulse duration and $T_0=3.3$ fs is the laser period. The corresponding laser intensity, power and total energy are $I\approx 1.1\times 10^{23}$ W/cm$^2$, $P\approx$31 PW and $\varepsilon _{L}\approx$100 J, respectively.

Figures 1(a)-(f) show the temporal evolution of the transverse electric field $E_y$ of the laser pulse when it interacts with the moderate density plasma. One can see that the spot radius becomes smaller and $E_y$ gets stronger as the laser pulse penetrates into the plasma deeply. At $t=38T_0$, the laser is focused to a highest intensity and the spot shrinks to a minimal value at $x=26.95\lambda _0$, corresponding to the RSF length of $L_f\approx 7\lambda _0$. The $yz$ cross sections of $E_y$ at $x=26.95\lambda _0$ for the initial and shaped laser pulses are presented in Figs. 1(g) and (h), respectively. We see that $a_s\approx 787$, which is about 2.8 times higher than that of the initial laser pulse. The intensity amplification ratio is thus $\kappa _{s}\approx 7.84$, which is nearly an order of magnitude improvement as compared to the initial laser pulse. At a later time, the laser pulse begins to defocus with its radius gradually enlarging and $\kappa _{s}$ is slightly weakened. $L$ should thus be comparable to $L_f$ to achieve effective light amplification and avoid the further energy loss in plasma. This result is consistent with the previous theoretical and experimental results of moderate intensity laser pulse in NCD plasma [29,32,39], although this model does not account for strong-field QED effects. Therefore, the optimal $L$ corresponds to $L\approx L_{f}$ for the possible experimental design of petawatt-class laser amplification in the near future. Besides, we note that the time rising front of laser pulse is steepened due to the laser energy absorption inside the plasma. The removal of relatively low intensity part in the time rising front is beneficial for obtaining higher laser contrast, which plays an important role in laser-driven ion acceleration in the radiation pressure acceleration [40,41] and harmonics generation [42].

 

Fig. 1. Snapshots of the normalized transverse electric-field $eE_y/m_{e}\omega _{L}c$ along the longitudinal ($z=0$) section at (a) $t=30T_0$, (b) $t=38T_0$ and (c) $t=46T_0$, respectively. (d)-(f) are the axial profiles along the $y=0$ directions of (a)-(c), and the dark red curves in (a)-(c) represent $eE_y/m_{e}\omega _{L}c$ along the black dotted lines marking the transverse sections of the maximum transverse electric fields in (d)-(f). The black curves in (d)-(f) plot the amplitude envelope of initial laser pulse. (g) shows the transverse distribution of $eE_y/m_{e}\omega _{L}c$ along the transverse section of $x=26.95\lambda _0$ at $t=38T_0$. For comparison, (h) gives the initial transverse distribution of $eE_y/m_{e}\omega _{L}c$ along the transverse section of the maximum electric field $E_y$.

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The laser RSF depends on the dynamics of relativistic electrons in the plasma. Figures 2(a) and (b) show the distributions of the normalized electron density $n_e/n_c$ along the $z=0$ and $x=26.95\lambda _0$ cross sections at $t=38T_0$. It is found that the PW-class seed laser pulse drills a deep tapered plasma channel with high density edges due to its strong transverse pondermotive force. The high-density tapered-channel edges will reflect the laser backward, resulting in the $-\pmb {x}$ direction shifting of $L_{f}$. This explains why $L_f\approx 7\lambda _0$ obtained from PIC simulation is slightly smaller than the theoretical prediction of $L_f=[(r_0^2-r^2)n_c a_0/n_e]^{1/2}\approx 8\lambda _0$ [32]. Besides, some transversely oscillating filaments also appear in the tapered channel since many attosecond electron bunches will be pulled out from the inner walls of the channel into the tapered cavity by the intense laser electric field $E_y$. All these electrons accelerated by the laser field induce the electric current, and self-generated quasi-static magnetic field $B_z$ with its maximum value $B_{z,max}=116B_{0}$ can be excited in the channel, as shown in Figs. 2(c) and (d), where $B_0=m_e\omega _{L}/e$=107.1 MG is the normalization constant. Note that $B_{z,max}=116B_{0}$ is larger than the theoretical result of $B_z=(a_{0}n_{e}/n_{c})^{1/2}m_e\omega _{L}/e=84.85B_0$ [26]. The reason is attributed to the fact that the pulled-out electron bunches, when accelerated forward by the laser, will also induce another electron current that contributes to the generation of self-generated magnetic field. $B_{z}$ tends to pinch relativistic electrons into the channel [26], leading to the redistribution of the electron density. The refractive index of plasma $\eta =(1-\omega _{p}^{2}/\gamma \omega _{L}^{2})^{1/2}$ is accordingly changed, where $\gamma =(1+a_{0}^{2}/2)^{1/2}$ is the relativistic factor. The dense plasma will act as a convex lens since the phase velocity $v_p=c/\eta$ is smaller on axis than off axis [26]. The RSF effect thus occurs inside the plasma and ultra-intense laser field can be amplified to a higher one. Besides, we observe that a large number of $\gamma$-photons are actually emitted via the synchrotron-like radiation by these pulled-out oscillating energetic electron bunches in the channel. The photons are mainly distributed along the central axis of the channel and have a very small divergence angle $\theta \propto 1/\gamma$, as shown in Figs. 2(e) and (f). It should be emphasized that the electron density in the main distribution region ($23\lambda _{0}<x<29\lambda _{0}$) of the laser electromagnetic field ranges from 0 to $71n_{c}$ (see the red arrow in Fig. 2(a)) and the tapered channel is actually not vacuumized. The RSF is dominant for ultra-intense laser amplification since Fig. 2 shows the similar features to that of moderate intensity laser pulse in NCD plasmas [25,26,29,32].

 

Fig. 2. Distributions of the normalized electron number density $n_e/n_c$ [(a) and (b)], self-generated magnetic field $eB_z/m_e\omega _{L}$ [(c) and (d)] and photon number density $n_\gamma /n_c$ [(e) and (f)] along the longitudinal ($z=0$) [(a), (c) and (e)] and transverse ($x=26.95\lambda _0$) [(b), (d) and (f)] sections at $t=38T_0$, respectively. Here, $B_z$ is averaged over a laser period, and the red and black curves in (a) and (c) represent the axial (y=0) profiles of $n_e/n_c$ and $eB_z/m_e\omega _{L}$.

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Figure 3(a) shows the spectra of electrons, protons and photons at $t=38T_0$, respectively. We see that the electrons and protons with energy up to 1 GeV and 500 MeV, respectively, are generated during the laser-plasma interaction. At $t=38T_0$, the highest energy conversion efficiencies from laser to electrons and protons approach 33% and 18%, respectively [see Fig. 3(b)]. These oscillating electrons, as the intermediate, convert about 12% laser energy to $\gamma$-ray radiation with the maximum photon energy as high as 500 MeV. When $L=10\lambda _{0}$, the maximum energy conversion efficiency $\eta$ from the incident seed laser pulse to an amplified one is close to 40%.

 

Fig. 3. (a) Spectra of the electrons (red curve), protons (blue curve) and photons (purple curve) at $t=38T_0$. (b) Temporal evolution of the energy conversion efficiency from laser to electrons (red curve), protons (blue curve), and photons (purple curve), respectively. Here, the black shows the share of the electromagnetic (EM) energy left in the simulation box.

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We now discuss the dependence of $a_{s}$ and $\kappa _{s}$ on $n_{e}$, $r_{0}$, $x_{f}$ and $a_{0}$, respectively. For a LP laser pulse of $\lambda _{0}=1\mu$m and $a_{0}\geq 1$, using $P>P_{cr}$ and $\omega _{L}>\omega _{P}$, the optional density range of $n_{e}$ for RSF can be given by $0.28n_{c}/[a_{0}(r_{0}/\lambda _{0})^{2}]\leq n_{e}\leq [(a_{0}^{2}+2)/2]^{1/2}n_{c}$. For $a_{0}=280$, we have $10^{-4}n_{c}\leq n_{e}\leq 198n_{c}$. Considering that effective light amplification from the RSF occurs in the NCD plasmas for moderate intensity laser pulses, the lower limit of $n_{e}$ chosen in our PIC simulations is set to $10n_{c}$. Figure 4(a) shows that, with growing $n_{e}$, $a_{s}$ and $\kappa _{s}$ increase almost linearly for $n_{e}\leq 30n_{c}$ (i.e., Region I). This is due to that the effect of relativistic self-induced transparency occurs [39] as $n_{e}$ is not too large and the laser pulse can still propagate through the plasma. However, as $n_{e}$ increases over $30n_{c}$ (i.e., Region II), the laser-plasma interaction phenomenon is quite different. In this case, the highest compressed electron density has exceeded $200n_{c}$ [see the red curve in Fig. 2(a)]. From Eq. (3), we obtain the modified threshold amplitude $a_{0,cr}\geq 280$. This indicates that the redistributed plasma is no longer transparent for the incident laser pulse, especially for the low-intensity part on both sides of the laser propagation axis. More laser energy is thus consumed in forming the tapered channel [39] and will be reflected by the high-density electron layer edges. Consequently, $a_{s}$ and $\kappa _{s}$ are far from the theoretical values of Eq. (1). Therefore, moderate electron density plasma of a few dozen $n_{c}$ may be more suitable to obtain higher $\kappa _{s}$. In addition, we notice that $\kappa _{s}$ also weakens as $r_{0}$ becomes larger, as seen in Fig. 4(b). The reason is attributed to the fact that a lot of the lateral laser energy on both sides will be consumed in pushing the overdense plasma integrally inward, rather than expelling the plasma out to two sides since the transverse pondermotive force of the laser decreases rapidly for larger $r_{0}$. It is therefore difficult to generate the tapered-plasma channel and focus the seed laser pulse. Figure 4(c) shows that $x_{f}$ also plays a crucial role in the laser amplification. We can see that $a_{s}$ and $\kappa _{s}$ are greatest when $x_{f}$ is comparable to $L_{f}$, resulting from the synergistic effect of laser focusing and RSF. At $x_{f}=50\lambda _{0}$, $I_{s}$ is as high as $1.1\times 10^{24}$ W/cm$^2$. Figure 4(d) shows the scaling laws of both $a_{s}$ and $\kappa _{s}$ versus $a_{0}$. As expected, the laser can be amplified to a higher intensity for a larger initial $a_{0}$. However, $\kappa _{s}$ is reduced with increasing $a_{0}$ because the protons exhibit relativistic oscillations in such strong laser fields and the QED effects become more significant. It should be noted that, in principle, $\kappa _{s}$ should be decreased since about 12% laser energy is transferred to $\gamma$-photons. However, we find in Fig. 4(d) that $a_{s}$ from the PIC simulations is almost comparable to that of the theoretical model without considering strong-field QED effects. This indicates that the RSF dominates the light amplification in moderate density plasma and the focusing from the tapered channel plays a minor role. For the latter, the mechanism of laser field enhancement is similar to the case of hollow micro-structured cone targets as described in Ref. [33,34]. It should be mentioned that the scaling laws in Fig. 4(d) are obtained at a fixed plasma density of $n_e=n_p=30n_c$. Actually, the laser field enhancement can be optimized by modulating the initial plasma density.

 

Fig. 4. Dependence of the amplitude $a_{s}$ of the amplified laser pulse and the intensity amplification factor $\kappa _{s}$ on (a) the normalized electron density of plasma $n_{e}/n_{c}$, (b) initial spot radius $r_{0}/\lambda _{0}$, (c) focal position $x_{f}/\lambda _{0}$ and (d) amplitude $a_0$ of the laser pulse, respectively (symbols and solid curves). Here, the dashed curves in (a) and (d) are from Eqs. (1)–(2), and the solid lines are from the PIC simulations. $n_{e}/n_{c}$, $r_{0}/\lambda _{0}$, $x_{f}/\lambda _{0}$ and $a_0$ are changed in turn and other parameters remain the same as the above simulation.

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3. Role of radiation reaction, geometric dimensionality and pulse duration in laser field amplification

Figures 5(a)-(b) show the electron, ion energy spectrum and the corresponding laser energy conversion efficiencies for the case without considering the QED effect (here, radiation reaction). Compared with Fig. 3, we can see that both the maximum particle energy and energy conversion efficiencies are somewhat increased. This is due to that the laser matter interaction enters into the radiation-dominated near-QED region as the laser intensity ranges from $10^{22}$ W/cm$^2$ to $10^{24}$ W/cm$^2$. Electrons become extremely relativistic and energetic electrons radiate bright $\gamma$-rays when they oscillate in the laser fields [as shown in Figs. 2(e) and (f)], leading to the share reduction of charged particle energy. Meanwhile, by comparing Fig. 5(c) and 2(a), we find that fewer electrons are trapped inside the tapered-plasma channel due to the disappearance of radiation trapping effect in the case without considering the QED effect [11,13]. The energy loss of relativistic electrons is therefore relatively small and the strength of quasi-static self-generated magnetic field $B_z$ is thus slightly enhanced, resulting in a small improvement of $a_s$, as shown in Figs. 5(d)-(f). It is exciting to see that although the $\gamma$-ray radiation leads to more energy loss, the amplified laser intensity $a_{s}$ is still comparable to the results without considering the QED effect.

 

Fig. 5. Simulation results without considering the QED effect (here, radiation reaction): (a) Spectra of the electrons (red curve) and protons (blue curve) at $t=38T_0$. (b) Temporal evolution of the energy conversion efficiencies from laser to electrons (red curve), protons (blue curve) and the share of EM fields left in the box (black curve). (c) Distribution of the electron number density at $t=38T_0$. (d) Self-generated magnetic field $eB_z/m_e\omega _{L}c$ averaged over a laser period and its axial profile along the $y=0$ direction at $t=38T_0$ (black curve). (e) Axial and (f) transverse profiles of the normalized transverse electric-field $eE_y/m_{e}\omega _{L}c$ along the $y=0$ and $x=26.95\lambda _{0}$ directions (w/o, blue curves) at $t=38T_0$. For comparison, the simulation results with considering the QED effect (w, red curves) are given in (e) and (f).

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Above we discuss the amplification of PW-class laser pulse in the 3D geometry. Now we focus on the effect of geometric dimensionality on the laser focusing. 2D-PIC simulation with the same laser-plasma parameters is carried out for reference. Figures 6(a)-(b) show the axial and transverse profiles of $eE_y/m_{e}\omega _{L}c$ at $t=38T_0$ in both cases. One can see that the amplified laser field $E_y$ declines in the 2D case, in good accordance with the previous literatures such as Ref. [26,29], showing the different effect of the 2D and 3D RSF in moderate density plasma. The reason is attributed to that the laser pulse penetrates a shorter distance in moderate density plasma and the refractive index gradient of the plasma lens becomes smaller [see Fig. 6(c)] for the 2D case. The charged-particle energy and their energy conversion efficiencies are thus somewhat decreased, as shown in Figs. 6(d)-(e). As a result, the quasi-static self-generated magnetic field $B_z$ and the RSF effect are obviously weakened [see Fig. 6(f)]. In addition, we observe that, due to the more transversely oscillating electrons in the 2D geometry, the energy conversion efficiency from laser to photons is slightly improved, as compared by Fig. 6(e) with Fig. 3(b).

 

Fig. 6. 2D-PIC simulation results: (a) Axial and (b) transverse profiles of the normalized transverse electric-field $eE_y/m_{e}\omega _{L}c$ along the $y=0$ and $x=26.95\lambda _{0}$ directions at $t=38T_0$ (blue curves). (c) Distribution of the electron number density at $t=38T_0$. (d) Spectra of the electrons (red curve), protons (blue curve) and photons (purple curve) at $t=38T_0$. (e) Temporal evolution of the energy conversion efficiencies from the seed laser to electrons (red curve), protons (blue curve), photons (purple curve) and the share of EM fields left in the box (black curve). (f) Self-generated magnetic field $eB_z/m_e\omega _{L}c$ averaged over a laser period and its axial profile along the $y=0$ direction at $t=38T_0$ (black curve). For comparison, 3D-PIC simulation results (3D, red curves) are also given in (a) and (b).

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It should be emphasized that although almost one order of magnitude amplification of the laser intensity is achieved, the energy conversion efficiency $\eta$ from initial seed laser to the resultant pulse is below 40%. We simulate the cases of longer $\tau$, as shown in Fig. 7. We find that $\eta$ rises accordingly with increasing $\tau$. However, $\eta$ will not be 100% efficient since some laser energy is lost in forming the tapered-plasma channel and generating $\gamma$-photon radiation. It is worth noting that the laser pulse is most effectively amplified at $\tau =25T_0$, where $\eta$ is as high as 60%. $a_{s}$ actually reduces as $\tau$ extends further, which is due to that the front of the tapered channel has a flat lateral distribution with much higher density, as seen in Fig. 7(i). The tapered channel focusing for laser pulse will change into the rectangular channel focusing because the plasma will be continuously expelled outwards by the laser ponderomotive force in the case of long pulse duration. The wave interference between the incoming and the reflected lights will be dominant. As a result, Fig. 7(f) shows that the peak amplitude of the electric field ($a_{s}\approx 560$) is twice of that of the incident light ($a_{0}=280$). In this case, $n_{e}$ should be optimized for efficient light amplification.

 

Fig. 7. PIC simulation results at (a)-(c) $\tau =25T_0$, (d)-(f) $\tau =40T_0$ and (g)-(i) $\tau =60T_0$: Temporal evolution of the energy conversion efficiencies from the seed laser to electrons (red curve), protons (blue curve), photons (purple curve) and EM fields (black curve) in three cases; Axial profile of the normalized transverse electric-field $eE_y/m_{e}\omega _{L}c$ along the $y=z=0$ direction, and distributions of the electron number density at $t=57T_0$, $72T_0$ and $85T_0$, respectively. Here, $t=57T_0$, $72T_0$ and $85T_0$ correspond to the moments when the peak amplitude is highest in each case.

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4. Conclusion

In summary, we investigate the dynamics of ultrashort ultra-intense ($>10^{23}$ W/cm$^2$) laser pulse propagating through the moderate density plasma. 3D-PIC simulation results show that a tapered plasma channel is generated by the powerful ponderomotive force of the seed laser pulse. Strong self-generated quasi-static magnetic field is excited inside the channel by the electric current, which tends to pinch extremely energetic electrons into the channel. The electron density is redistributed, resulting in a change in refractive index of plasma. Relativistic self-focusing and focusing from the tapered plasma channel occur in moderate density plasma simultaneously and almost an order of magnitude amplification in laser intensity is achieved with appropriate laser plasma parameters in the radiation-dominated near-QED regime. An ultrarelatistic laser pulse of $>10^{24}$ W/cm$^2$ with a steep time rising edge is obtained and the energy conversion efficiency from initial seed laser to an amplified one can exceed 60% as the laser duration properly lengthens. This method is useful for the continued ascension of laser intensity for present PW-class or upcoming hundreds-PW laser facilities in the near future.