# Laser Compression via fast-extending plasma gratings

Zhaohui Wu  Xiaoming Zeng  Zhaoli Li  Zhimeng Zhang  Xiaodong Wang  Bilong Hu
Xiao Wang  Jie Mu  Jingqin Su  Xiaofeng Wei  Yanlei Zuo
<EMAIL_ADDRESS>Science and Technology on Plasma Physics Laboratory,
Research Center of Laser Fusion, China Academic of Engineering Physics,
Mianyang, Sichuan, China, 621900.

###### Abstract

It is proposed a new method of compressing laser pulse by fast extending
plasma gratings(FEPG), which is created by ionizing the hypersound wave
generated by stimulated Brillouin scattering(SBS) in the background gas.
Ionized by a short laser pulse, the phonon forms a light-velocity FEPG to
fully reflect a resonant pump laser. As the reflecting surface moves with a
light velocity, the reflected pulse is temporally overlapped and compressed.
This regime is supported by the simulation results of a fully kinetic
particle-in-cell(PIC) code Opic with a laser wavelength of 1$\mu$m, displaying
a pump pulse is compressed from 13ps to a few cycles(7.2fs), with an
efficiency close to 80%. It is a promising method to produce critical laser
powers due to several features: high efficiency without a linear stage,
robustness to plasma instabilities, no seed and a wide range of pump
intensity.

Plasma grating, plasma compression, Ultrashort optics, laser-plasma
interaction

††preprint: AIP/123-QED

Compression of ultrashort laser pulses by plasma has the potential to boost
the instantaneous laser power to new levels due to the robustness of plasma in
strong laser fields. Currently, stimulated raman backscattering(SRBS)
amplification Malkin, Shvets, and Fisch (1999, 2000a); Trines  _et al._
(2011); Ping, Cheng, and Suckewer (2004); Cheng  _et al._ (2005); Ren  _et
al._ (2007); Vieux  _et al._ (2017); Wu  _et al._ (2019) and strongly coupled
stimulated Brillouin scattering (SCSBS)Andreev  _et al._ (2006); Lancia  _et
al._ (2010, 2016); Peng  _et al._ (2016); Marqués  _et al._ (2019) are two
promising methods for this application. In both SRBS and SCSBS, the energy of
relative longer pump pulses are scattered to conterprorogating short seeds
pulse by the plasma waves, Langmuir wave or ion-acoustic wave, respectively.
Theoretical results show the possibility for a final amplified seed peak
intensity of over $10^{16}~{}\rm{W/cm^{2}}$, which is about 5 orders of
magnitude larger than the maximum intensity obtained by a solid-state crystal.

However, both BRA and scSBS have to experience a linear stage, in which the
plasma waves grow up from density perturbation. In this stage, the pump is
little depleted and the seed pulse is stretched, so both the transfer
efficiency and compression ratio are degraded. Moreover, due to the influence
of plasma instabilities, the linear stage in the experiment was usually much
longer than expectation. Therefore, the benefits of the theoretical prediction
are still difficult to be realized. Although a strong seed helps to achieve
the nonlinear stage quickly, it has to pay the price of reducing the
compression ratio.

The difficulties could be overcame by using a novel way, named fast-extending
plasma-grating compression(FEPGC). The scheme is shown in Fig1. The phonon—a
hypersound wave with a spatial period half wavelength of the stimulating
laser—is supposed to be created by the stimulated Brillouin scattering(SBS) of
2 counter-prorogating laser pulses in gas, which has bean widely found in
$CH_{4}$, $N_{2}$ and $H_{2}$Liu and He (2003). A pump pulse is set below the
ionization threshold of the background gas, so it can pass through the phonon
in which variation of the refraction index is still small. By using a short
pulse to ionize the phonon, a plasma grating extending with light-velocity is
created. As the refraction index is much more sensitive to density in plasma
than gas, the pump pulse satisfying the Brag condition of the grating would be
fully reflected back. To be noted is that the reflecting surface moves with a
light-velocity, so the reflected pulse(RP) is temporally overlapped to a short
one. As the plasma grating is produced by ionizing the phonon but not grows
from density perturbations, the pump can be transiently depleted and thereby
the compression process has not a linear stage.

Figure 1: Illustration of the laser compression by fast-extending plasma
grating produced by a short pulse(brown pulse, cross-polarized) and the
phonon(transparent green fringes) in background gas. As boundary of the plasma
grating(green fringes) moves with the ionizing pulse, the reflected pulse(blue
pulse) of the pump(red pulse) is compressed.

Plasma gratings created by laser-plasma interaction have been widely studied
for manipulating high-power lasersForslund, Kindel, and Lindman (1975); Wu,
Sheng, and Zhang (2005, 2006); Lehmann and Spatschek (2016); Wu, Sheng, and
Zhang (2017); Lehmann and Spatschek (2018); H, Marquès, and L. Lancia (2019);
H.Peng  _et al._ (2019). However, they are usually difficult to be controlled
due to plasma instabilities. Take a different approach, we consider generating
plasma gratings from more-stable acoustic waves. The hypersound wave excited
and maintained by the SBS effect in gas can be described by a material
equationDamzen  _et al._ (2003)

$\displaystyle\begin{array}[]{rcl}\frac{\partial\rho}{\partial
t}+(-i\delta\omega+\frac{\Gamma_{B}}{2})\rho=-\frac{i\gamma_{e}\varepsilon_{0}k_{B}}{4\upsilon}E_{0}E_{1}^{*}\\\
\end{array}$ (2)

where $E_{0,1}$ and $\omega_{0,1}$ are the E-field and frequencies of the
laser pulses; $n$ is the refraction index of the gas; $c$ is the light
velocity in vacuum; $\varepsilon_{0}$ is the permittivity of free space;
$\gamma_{e}=\rho_{0}(\partial\varepsilon/\rho)$ is the electrostriction
coefficient, and it can be written as $(n^{2}-1)(n^{2}+2)/3$ in
centrosymmetric materials; $k_{B}$ is the Brillouin wavevector of the acoustic
wave; $\rho$ is the amplitude of the density perturbation; $\rho_{0}$ is the
medium density; $\Gamma_{B}$ is the inverse of the acoustic lifetime
$\tau_{B}$ given by

$\displaystyle\begin{array}[]{rcl}\Gamma_{B}=\frac{1}{\tau_{B}}=\frac{k_{B}^{2}\eta}{\rho_{0}}\end{array}$
(4)

where $\eta$ is the is the viscosity. Assuming the $E_{0}$, $E_{1}$ and
density wave satisfy the resonant condition, $\rho$ has a steady-state value
for a sufficient interaction time

$\displaystyle\begin{array}[]{rcl}\frac{\rho}{\rho_{0}}=\frac{I\gamma_{e}}{\upsilon
cnk_{B}\eta}\end{array}$ (6)

where, $I=\varepsilon_{0}cnE_{0}E^{*}_{1}/2$ is the interfered intensity of
the laser pulses, and $\upsilon$ is the acoustic velocity. By employing the
parameters of several gases, the steady-state $\rho/\rho_{0}$ are shown in
Tab.1. A particular value of $\rho/\rho_{0}$ can be obtained by adjusting the
laser intensity which is mainly limited by the threshold of multiphoton
ionization. For hydrogen, the measurable ionization was observed at an
intensity of some $10^{13}~{}\rm{W/cm^{2}}$ with laser wavelength of 1064
nmWolff  _et al._ (1988).

Table 1:  Calculation of steady-state $\rho/\rho_{0}$ for several gases. Both the laser pulses have the wavelength of 1 $\mu$m. $I$ is used the unit of $\rm{10^{12}~{}W/cm^{2}}$. The gas pressure is $P=1~{}atm$ temperature is $T=20^{\circ}C$. Gas | $\eta(10^{-6}\rm{Pa.s})$ | $\upsilon(\rm{m/s})$ | n | $\gamma_{e}(10^{-4})$ | $\rho/\rho_{0}$
---|---|---|---|---|---
$H_{2}$ | 8.8 | 1295 | 1.00013 | 2.6 | 0.06$I$
$CH_{4}$ | 10.8 | 440 | 1.00044 | 8.8 | 0.47$I$
$N_{2}$ | $16.6$ | 334 | 1.00030 | 6.0 | 0.28$I$

After the phonon is ionized, the laser-plasma interaction can be depicted by
three-waves equations as followsKruer (1988):

$\displaystyle\begin{array}[]{rcl}(\frac{\partial^{2}}{\partial
t^{2}}-c^{2}\Delta^{2}+\omega^{2}_{pe})\textbf{A}_{a}=-\frac{4\pi
e^{2}}{m_{e}}\delta n_{e}\textbf{A}_{b}\\\ (\frac{\partial^{2}}{\partial
t^{2}}-c^{2}\Delta^{2}+\omega^{2}_{pe})\textbf{A}_{b}=-\frac{4\pi
e^{2}}{m_{e}}\delta n_{e}\textbf{A}_{a}\\\ (\frac{\partial^{2}}{\partial
t^{2}}-c^{2}_{s}\Delta^{2})\delta
n_{e}=\frac{Ze^{2}{n_{e}}_{0}}{m_{e}m_{i}c^{2}}\Delta^{2}(\textbf{A}_{a}\textbf{A}_{b})\end{array}$
(10)

where $\textbf{A}_{a,b}$ are laser vector potentials;
$c_{s}=\sqrt{ZT_{e}/m_{i}}$ is the ion acoustic velocity; $Z$ is the plasma
ion charge; $T_{e}$ is the plasma temperature; $e$ is the electron charge;
$m_{e(i)}$ is the electron(ion) mass; $\delta n_{e}$ is the amplitude of
electron density perturbation, and ${n_{e}}_{0}$ is the average electron
density.

By using the slowly-varying envelope approximation, the 1-dimensional Eq.10
can be simplified to

$\displaystyle\begin{array}[]{rcl}&&a_{t}+2a_{\zeta}=-Vbf\\\
&&b_{t}=Vaf^{*}\\\
&&(\partial_{t}+{\partial_{\zeta}}-i\frac{k^{2}_{B}c^{2}_{s}-\omega_{B}^{2}}{2\omega_{B}})f=\frac{Ze^{2}}{m_{e}m_{i}c^{2}}ab^{*}\end{array}$
(14)

in variable $\zeta=z/c+t$, where $a$ and $b$ are the normalized amplitude of
the pump and RP, $V\equiv i\frac{\omega^{2}_{pe}}{2\omega_{a}}$ is coupling
constant; $\omega_{pe}=\sqrt{n_{e}e^{2}/m_{e}\varepsilon_{0}}$ is plasma
frequency; $f\equiv\frac{\delta n_{e}}{{n_{e}}_{0}}$ is the normalized plasma
density. As the plasma grating is directly obtained by ionizing the phonon,
the acoustic frequency $\omega_{B}=v/\lambda$ and wave number $k_{B}$ are
remained. The RP would still satisfy the resonant condition
$\omega_{b}=\omega_{a}-\omega_{B}\approx\omega_{a}$ and
$k_{B}=k_{a}+k_{b}\approx 2k_{a}$, where $\omega_{a,b}$ and $k_{a,b}$ are the
laser frequencies and wave numbers. Commonly, it has $v\ll c_{s}$, so the
growth of $f$ has a frequency detuning of
$(k^{2}_{B}c^{2}_{s}-\omega_{B}^{2})/{2\omega_{B}}$ , as shown in Eq.14. As
the boundary of the plasma grating moves with a light velocity, the boundary
condition can be written as $f(\zeta=\zeta_{0})=f_{0}=Z\rho/\rho_{0}$, where
$\zeta_{0}$ is the position where the gas starts to be fully ionized. The
laser-plasma interaction in the partly-ionized section is neglected because it
can be very short by using a sharp-front ionizing pulse. Moreover, for the
plasma grating, the laser reflection becomes significant only when the average
density is close to the Brag condition.

The compression process can be described by numerically solving the Eq.14.
However, in the limit of a low pump intensity below the ionization threshold,
the material equation in Eq.14 can be neglected for several reasons. Firstly,
the resonant laser exponentially decaying in the plasma grating can only enter
into a very thin plasma layer. Secondly, as the ionization pulse moves
forward, the plasma layer is continually replaced by new ones, resulting in a
very short interaction time for laser and plasma interaction, which is down to
a few optical cycles. Due to the participation of heavy ion, the density
variation driven by laser is quite small compared with the plasma grating.

As the interaction directly enters into the nonlinear stage, it has
$\frac{\partial a}{\partial t}\ll\frac{\partial a}{\partial\zeta}$, so Eq.14
is given as

$\displaystyle\begin{array}[]{rcl}&&2a_{\zeta}=-Vb\delta f\\\ &&b_{t}=Va\delta
f^{*}\\\ &&f=f_{0}\end{array}$ (18)

Eq.18 can be written as

$\displaystyle\begin{array}[]{rcl}a_{\zeta t}=V^{2}f_{0}^{2}a/2\end{array}$
(20)

It can be approximated by a self-similar function
$a(\zeta,t)=\tilde{a}(\sqrt{\xi})$, $\xi\equiv Vf_{0}\sqrt{2\zeta t}$, that
satisfies the zero-class Bessel function

$\displaystyle\begin{array}[]{rcl}\tilde{a}_{\xi\xi}+\tilde{a}_{\xi}/\xi=\tilde{a}\end{array}$
(22)

With the boundary condition of $a(\xi=0)=a_{0}$, the pump has a solution
$a=a_{0}J_{0}(\xi)$, then

$\displaystyle\begin{array}[]{rcl}b=2a_{0}Vf_{0}J_{1}(\xi)t/\xi.\end{array}$
(24)

Let $b_{t}=Va\delta f^{*}=0$, the RP has the maximum value at the first peak
for $\xi_{M}\approx 2.4$, gives

$\displaystyle\begin{array}[]{rcl}b_{M}\approx 0.43a_{0}Vf_{0}t.\end{array}$
(26)

In the ideal case, $b_{M}$ grows linearly with time. However, just as BRA and
sc-SBS, the output intensities are limited by several plasma instabilities.
The modulation instability corresponds the variations in plasma density driven
via the ponderomotive force by intensity modulations in the light beam, with a
growth rate of $\sim b{{}^{2}}\omega{{}^{2}_{pe}}/2\omega_{a}$Malkin, Shvets,
and Fisch (1999). The longitudinal modulation instability can make the RP peak
depart from the maximum gain and gradually tend to saturation, while the
transversal modulation instability suppresses the compression mainly by
causing filamentation. Combining with Eq.26, it gives the limited time of
$t_{md}=\frac{2.53\Lambda_{md}}{V}(a_{0}f)^{-2/3}$, where $\Lambda_{md}$ is
the number of exponentiations which can be tolerated for modulation.
Furthermore, as the laser pulse is compressed to a few optical cycles, the
group velocity dispersion(GVD) given as
$\beta=\frac{\partial^{2}\omega\bar{n}}{c\partial\omega}|_{\omega=\omega_{0}}\approx\frac{\omega^{2}_{pe}}{\omega^{3}_{0}c}$
can not be neglected. For a transform limit pulse with $T_{0}$, the pulse is
stretched to $T=\sqrt{T^{2}_{0}+\beta ct/T^{2}_{0}}$. It limits the shortest
pulse can be obtained by the compression.

The stimulated Raman backscattering(SRBS) from the pump and forward Raman
scattering(FRS) from the RP have the same interaction time as RP. Both of them
are based on Langmuir wave, with growth rates of
$\gamma_{b}=a\sqrt{\omega_{pe}\omega_{a}}/2$ and
$\gamma_{fw}=b\sqrt{\omega^{3}_{pe}/8\omega_{a}}$Forslund, Kindel, and Lindman
(1975); Kruer (1988), respectively. The growth of Langmuir waves are strongly
damped by the frequency detuning due to the strong density inhomogeneity of
the plasma grating. The effective growth rates of these 2 instabilities can be
roughly estimated as $\gamma^{2}/\Delta\omega_{pe}$Malkin and Fisch (2014),
where $\Delta\omega_{pe}$ is the frequency variation of Langmuir wave due to
plasma inhomogeneity.

Several major plasma instabilities suppressing the plasma gratings can be
avoided here. Plasma kinetic effected is significantly migrated due to a weak
plasma-heating effect. In the scheme of FEPGC, the pump is set below the
ionization of background gas. The compression is just behind the ionizing
process, leaving a very short distance before the RP for plasma heating and
precursors generation by the pump. The regime is quite similar with seed-
ionizingMalkin, Shvets, and Fisch (2000b); Clark and Fisch (2002); Zhang  _et
al._ (2017) or flying-focusTurnbull  _et al._ (2018); Froula  _et al._ (2018);
SAINTE-MARIE, GOBERT, and QUÉRÉ (2017) BRA. Moreover, the collision and Landau
damping have little threat to the compression because they can hardly impact
the plasma grating within the interaction time of the laser and plasma layer(a
few optical cycles). In addition, the wavebreaking is far from approach. The
wavebreaking has a time of $t_{wb}=\sqrt{2m_{i}/m_{e}}/\omega_{a}b\approx
60/\omega_{a}b$ for hydrogenForslund, Kindel, and Lindman (1975); Hüller,
Mulser, and Rubenchik (1991), showing the toleration of laser amplitude can be
up to $b>1$ for a duration of a few optical cycles.

To be noted is that the Brag condition of the grating is varied after the
ionization. In a typical Brag grating, the Brag condition is written as
$\Lambda=m\lambda_{0}/2\bar{n}$, where $\Lambda$ is the spatial period of the
grating, $m$ is the diffractive number, $\lambda_{0}$ is the resonant laser
wavelength in gas, $\bar{n}$ the average refraction index. $\bar{n}\approx 1$
in the background gas, so $\Lambda\approx\lambda_{0}/2$ . After the gas is
ionized, the refraction index in plasma becomes
$\bar{n}=\sqrt{1-\frac{{n^{2}_{e}}_{0}e^{2}}{m_{e}\varepsilon_{0}\omega^{2}}}<1$,
indicating the resonant laser wavelength is slightly down shifted. That’s why
the reflectivity decreases after the gas is ionized in the SBS
experimentGorbunov  _et al._ (1983). The pump with a small incidence angle or
a shorter wavelength length can satisfy the new Brag condition. For instance,
in hydrogen with $\rho_{0}=0.01n_{c}$, where
$n_{c}=\omega^{2}_{a}m_{e}\varepsilon_{0}/e^{2}$, it requires
$\theta=8.1^{\circ}$ or $\lambda_{a}=0.99\lambda_{0}$, where $\theta$ is the
incidence angle and $\lambda_{a}$ is the resonant wavelengths in plasma,
respectively.

By employing hydrogen with a hypersound wave as the background gas, the
ionization and compression process are simulated with a fully kinetic
particle-in-cell code OpicZhang  _et al._ (2012, 2014, 2017). The simulation
was carried out in a moving window of $800\lambda_{a}$ with cells of
$\Delta_{z}=\lambda_{a}/20$, and $32$ particles for each cell. A 10$T$ 800-nm
short pulse with a peak amplitude of 0.03 is used for ionization pulse, where
$T$ is one optical cycle of $\lambda_{a}$. It has a crossed polarization with
the pump pulse in order not to affect the compression. The ionization process
is represented by an Ammosov-Delone-Krainov tunnel ionization model(ADK). The
pump pulse has a wavelength of $\lambda_{a}=1~{}\mu m$ and an amplitude below
the ionization threshold. To satisfy the Brag condition for the pump, an
optimal $\Lambda>0.5\lambda_{a}$ of the density wave is obtained according to
the maximum pump depletion in the simulation.

Figure 2: PIC Simulation results of pulse compression by FEPG at
$\rho_{0}=0.01n_{c}$, $\rho/\rho_{0}=0.5$, $a_{0}=0.002$,
$\Lambda=0.50238\lambda_{a}$, and $\lambda_{a}=1~{}\mu m$, corresponding to
$T=3.3~{}fs$. (a) The pump(red) and RP(blue) at t=100 $T$. (b) Evolution of
the RP shape at different interaction time. Overall(c) and specific(d)
distribution of plasma grating and background gas at t=2000 $T$. (e) The
spectrum of SBS at $t=100~{}T$ and $t=2000~{}T$.

The simulation results are displayed in Fig2. The pump pulse is swiftly
depleted soon after it encounters the plasma grating, implying the nonlinear
stage can be directly reached, as shown in Fig2(a). The initial RP at $t=200T$
has a half-maximum full width(HMFW) duration around 80 fs, and then is further
compressed, with an amplitude far exceeding that of the pump. The shortest
duration of 7.2 fs is obtained at the interaction time of $2000T$, which is
close to 2 cycles of the pump wavelength. After that, the duration is slightly
returned back to around 9 fs, but the amplitude still grows up to 0.07 until
$t=3200T$, with a compression efficiency close to 80%. The overall and
specific profiles of plasma grating are shown in Fig2(c)-(d). It has a very
sharp boundary between the background gas and plasma zone. The density
fluctuation gradually decreases as it is far from the boundary, because of the
thermodynamic motion of the plasma. However, the compression is not impacted
because only a very thin plasma layer in Fig.2(d) participates the
interaction. The RP spectrum displays an extremely wide range from
$0.5\omega_{a}-2\omega_{a}$ at $t=2000~{}T$, implying a short pulse of a few
cycles can be supported, as shown in Fig2(d).

For a higher compressed intensity, the pump amplitude is increased from
0.002-0.007. To compare the compression results , we define amplitude
magnification(AM) as the maximum $|b|/a_{0}$ and compression ratio(CR) as FWHM
$t_{Pump}/t_{RP}$. In the beginning stage of compression, AM grows linearly
with $t$, which satisfies well to Eq.26. As the interaction time becomes
longer, it gradually departs from Eq.26 and then tends to saturation, as shown
in Fig.3(a). The saturation appears earlier with a high pump intensity,
probably due to the modulation instability and GVD. As well as AM, the CR also
slightly degraded as the pump intensity increases, as shown in Fig.3(b).
However, excellent results are still obtained in a wide range of pump
intensity, with optimal AM and CR are above 25 and 1000, respectively. The
highest amplitude of 0.17 is obtained by the pump amplitude of 0.007 at
$t=3200~{}T$, corresponding to an intensity of $4\times
10^{16}~{}\rm{W/cm^{2}}$, which is close to the result obtained by BRAMourou
_et al._ (2012); Clark and Fisch (2002) at pump wavelength $\sim$1 $\mu$m, but
with a much lower pump intensity or interaction length.

Figure 3: Simulation results of pulse compression by fast-extending plasma
gratings with various pump amplitudes($a_{0}$), phonon amplitudes($\rho$) and
gas densities($\rho_{0}$) (a)Amplitude magnifications with different pump
intensities. (b) Compression radios with different pump intensities. (c)
Amplitude magnifications with different phonon amplitudes $\rho$ for
$\rho_{0}=0.01~{}n_{c}$, $a_{0}=0.006$, $\Lambda$ is adjusted according to the
maximum AM. (d) Amplitude magnifications with different gas densities
$\rho_{0}$ for $\rho/\rho_{0}=0.5$, $a_{0}=0.006$.

The robustness of compression is demonstrated with various $\rho$ and
$\rho_{0}$. With a larger $\rho$ , the AM grows more quickly, implying the
compression radio can be enhanced. However, affected by plasma instabilities,
the compression tends to saturation earlier, so the maximum AM with a longer
interaction time does not significantly impacted, as shown in Fig3(c).
Besides, a higher $\rho_{0}$ may be necessary for a sufficient $\rho$. As
$\rho_{0}$ goes up, the compression can be further accelerated and the
reflecting bandwidth for the pump is broadened. However, the modulation
instabilities and GVD becomes more serious. The maximum AM and saturated time
descends as $\rho_{0}$ increases, as shown in Fig.3(d). In the experiment,
optimal $\rho$ and $\rho_{0}$ can be chosen according to the pump duration and
bandwidth.

Figure 4: PIC Simulation results of FEPGC with inhomogeneous plasmas. (a)
Profiles of plasma grating with random and periodic density fluctuations.(b)AM
with inhomogeneous plasmas.

As well as the Brag gratings in other mediums, the reflectivity is very
sensitive to $\rho_{0}$, so AM drops sharply as $\rho_{0}$ departs from the
Brag condition. The maximum AM decreases from 25 to 5 as $\rho_{0}$ is reduced
by 10%. However, despite of a fluid medium, it is not difficult to obtain a
precise average density $\rho_{0}$ in plasma. Instead, an unavoidable issue is
the density fluctuations. The compression shows a strong robustness against to
density fluctuations. For an inhomogeneous density along the laser
propagation, the plasma grating is still effective as long as it contains
sections satisfying the Brag condition. Therefore, the compression is little
influenced by high-frequency density fluctuations, as shown Fig.4(b) with a
random density flu of $50\%$ $\rho_{0}$. For a low-frequency fluctuation, the
compression divided into ’fast’ and ’slow’ seasons due to the detuning period
is prolonged but broken down. The AM keeps well with a density fluctuations of
$50\%\rho_{0}$ and period of 500$\mu m$, as shown in Fig.4(b).

The scheme is easily realized in the experiment. For example, a hypersound
wave can be firstly generated by the 1064 nm nanosecond laser pulses in
Hydrogen with density of $0.01~{}n_{c}$, corresponding to the plasma grating
satisfying the Brag condition of an 1053-nm pump. Then, an 800-nm, 30-fs
Ti:Sapphire CPA laser can be used for the ionization pulse, in order to
created the FEPG. Finally, laser compression is obtained by using the FEPG to
dynamically reflect the Pump. Besides, the generation of FEPG is not limited
by using hypersound waves, other paths such a gas jet with periodic structures
may also available. Also, the pump wavelength is not limited to around 1
$\mu$m but can extend from extreme ultraviolet(EUV) to microwave by adjusting
the acoustic period of the phonon.

In summary, a novel way of laser compression via FEPGC is proposed, showing
several excellent features as follows: First, the linear stage is absent as
the plasma grating created by ionizing the phonon owns a large amplitude from
birth. Consequently, a very high efficiency and short pulse can be obtained.
Second, without a growth process, the plasma grating has a strong robustness
against plasma instabilities. Third, it is not necessary to prepare a seed as
long as the grating and the pump satisfy the Brag condition. In addition, the
generation of plasma grating does not depends on the pump, indicating any pump
intensity below the ionization threshold works well.

The simulation result of a fully kinetic PIC code Opic displays that a 10-30
ps pump with wavelength of $1~{}\mu m$ is compressed to a few optical
cycles(7-9 fs), having a transfer efficiency near to $80\%$. Supposing a
20-kJ, 20-ps, 1053-nm pump laser is produced by the Nd:glass CPA
techniqueMourou  _et al._ (2012), it would obtain a peak power around 2
EW($1~{}\rm{EW}=10^{18}$ W) after the compression. Moreover, as the pulse is
compressed to a few optical cycles, it can provide an extremely high-bright
source for the generation of attosecond laser pulses.

This work was partly supported by National Key Program for S$\&$T Research and
Development (Grant No. 2018YFA0404804), the National Natural Science
Foundation of China(Grant No. 11875240) and the Science and Technology on
Plasma Physics Laboratory Fund (Grant No. 6142A0403010417).

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