A method is proposed for compressing laser pulses by fast-extending plasma gratings (FEPGs), which are created by ionizing a hypersonic wave generated by stimulated Brillouin scattering in a 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 the velocity of light, the reflected pulse is temporally overlapped and compressed. One- and two-dimensional fully kinetic particle-in-cell simulations with a laser wavelength of 1 µm show that in this regime, a pump pulse is compressed from 10–40 ps to 7–10 fs (i.e., a few optical cycles), with a two-dimensional transfer efficiency up to 60%. This method is a promising way to produce critical laser powers while avoiding several significant problems that arise in plasma-based compressors, including an unwanted linear stage, major plasma instabilities, and the need for seed preparation.
There are a variety of ways in which ultraintense lasers can be manipulated using optical techniques based on plasmas, such as plasma photonic crystals,1 plasma holograms,2,3 high-harmonic generation,4–6 frequency conversion,7–10 and laser compression.11–15 Among these techniques, plasma-based laser compression is expected to considerably extend the intensity frontier owing to the robustness of plasmas in strong laser fields. At present, backward Raman amplification (BRA)12,16,17 and strongly coupled stimulated Brillouin scattering (scSBS)13,18–20 are two promising routes to laser compression. For both BRA and scSBS, the compression is based on three-wave coupling of pump, seed, and plasma waves. The energy of a relatively long pump is scattered to a counterpropagating short seed by plasma waves—Langmuir waves or ion-acoustic waves in BRA and scSBS, respectively. Theoretical results show the possibility of producing unfocused intensities over 1017 W/cm2, which is about six orders of magnitude larger than the maximum tolerable intensity of solid-state media.
However, experimental results for both BRA and scSBS are still far from the theoretical predictions.19,21–29 For BRA, the full width at half maximum (FWHM) pump duration is compressed from ∼20 ps to ∼50 fs, but the highest transfer efficiency of 6.5% is still far from the expected value.23 For scSBS, although the transfer efficiency is close to 20%, the amplified seed intensity does not significantly exceed that of the pump, indicating that a higher laser power still cannot be obtained in this way.29
Beside technological limitations such as pulse chirp and plasma inhomogeneity encountered in experiments,30 difficulties also arise from several inherent drawbacks in the three-wave coupling model. First, it has an unwanted linear stage. In the three-wave coupling model, the plasma wave driven by the ponderomotive force of the pump and seed has to grow from near zero. The linear regime is the early stage during which the plasma wave is too weak to deplete the pump, and so the conversion efficiency is very low and the seed conversely is stretched.12 Although a strong seed enables the nonlinear stage to be reached more rapidly, this is at the cost of a reduced intensity magnification. Moreover, as the plasma wave is weak in this stage, it is vulnerable to plasma instabilities, including Landau damping, particle trapping, and thermal noise. Second, precise spatiotemporal and frequency matching of the seed and pump requires complex experimental setups. To reduce this complexity, several approaches have been proposed to avoid the need for seed preparation.31–33 However, these are still difficult to implement in practice. Third, the compression ratio (CR) is limited by plasma instabilities, in particular, modulational instability. A shorter compressed pulse can be achieved by increasing the pump intensity.12,13,18 Nevertheless, the greater plasma instability at high laser intensity imposes a limit on the extent to which the pulse duration can be reduced.
A novel approach to overcoming these problems is the use of fast-extending plasma-grating compression (FEPGC), which is shown schematically in Fig. 1. A phonon (a density wave) with spatial period half of the laser wavelength can be created by the SBS effect in a background gas. A pump pulse with an intensity below the ionization threshold first passes through the phonon region, which is possible owing due to the small Δn (where n is the refractive index) in the gas. Then, a short pulse is used to ionize the phonon, thereby creating a plasma grating expanding with the velocity of light. As Δn is much more sensitive to the plasma density than to the gas density, the resonant pump pulse is fully reflected back. It should be noted here that the reflecting surface moves with the velocity of light, and so the reflected pulse (RP) overlaps temporally with the short pulse.
With an FEPGC, the plasma grating (plasma wave) is no longer produced by a pump and seed, and the compression regime in FEPGC actually involves two-wave coupling (between the pump and RP). Owing to this simplification, several key problems with plasma compression can be solved. First, the plasma grating directly inherits the large amplitude of the phonon, and so the resonant pump is transiently depleted and reflected, as a consequence of which there is no linear stage in FEPGC. Second, it is not necessary to prepare a seed, as can be seen from Fig. 1. In addition, as the plasma grating is independent of the pump, the compression can still work well at a pump intensity that is sufficiently low to avoid most laser-driven plasma instabilities. As a result, the limitations imposed on pulse duration by these instabilities can be avoided.
II. ANALYTICAL SOLUTION
To describe the compression regime, we first suppose that there is a hypersonic wave in the background gas. Ionized by a short pulse, it forms an FEPG to reflect a resonant pump. Like Bragg gratings in other media, the laser–plasma interaction in the plasma grating can be described by the coupled wave equations in the variable ζ = z/c + t as follows [the full derivation of Eq. (1) is given in the supplementary material]:34
where a and b are the wave vector potentials of pump and RP normalized by mec2/e (with me and e the electron mass and charge, respectively, and c the velocity of light), at and bt are their partial derivatives with respect to t, is the coupling constant (with the plasma frequency and ωa the pump frequency), and is the normalized plasma density (with δne and the perturbed and average electron densities, respectively). Here, the plasma wave excited by the pump and RP is neglected for several reasons. First, the pump intensity below the ionization threshold of the background gas is still low. Second, because the pump is quickly reflected, the growth time for the plasma wave is around the RP duration, which is as small as a few optical cycles. Furthermore, as the RP has almost the same central wavelength as the pump, the laser-excited plasma wave is an ion-electron wave, and, owing to the participation of heavy ions, it can hardly grow significantly within such a short time.
As the plasma grating expands with the velocity of light, the boundary condition can be written as
where Z is the plasma ion charge, ρ and ρ0 are the perturbed and average gas densities, respectively, and ζ0 is the position at which the gas is fully ionized. The partly ionized layer between the gas and plasma grating is neglected because it can be made very narrow by using a sharp-fronted ionizing pulse.
where aζt is the partial derivative of a with respect to ζ and t. Equation (3) can be approximated by a self-similar function , , that satisfies the zeroth-order Bessel equation
where and are the first and second partial derivatives of with respect to ξ. With the boundary condition a(ξ = 0) = a0, the pump has a solution a = a0J0(ξ), and then
According to the properties of Bessel functions, both the pump and RP have a series of spikes. The maximum intensity of the RP is located at the first peak. Putting bt = 0 gives ξM ≈ 2.4, and then
bM grows linearly with t and a0, indicating that the amplitude magnification (AM) bM/a0 does not depend on the pump intensity. The duration of the first peak is estimated as
According to Eq. (7), the pulse duration become shorter and shorter with increasing t. However, the effective t is limited by plasma instabilities,12 particularly modulational instability with a growth rate ∝b2. For BRA and scSBS, this cannot be avoided, because a high laser intensity is required to support a sufficient growth rate.12,35 Here, this problem is mitigated by simply reducing the laser intensity without influencing the compression, and so t can be considerably prolonged to obtain a pulse duration of a few optical cycles.
III. NUMERICAL SIMULATION
Taking hydrogen with a density wave as the background gas, the ionization and compression processes are simulated with the fully kinetic particle-in-cell (PIC) codes Opic36–38 and EPOCH. The pump has wavelength λa = 1 µm and a flat-top temporal shape. A 10T 800 nm short pulse with a peak amplitude of 0.01 is used for the ionizing pulse, where T is one optical cycle of the pump wavelength. It has crossed polarization with the pump pulse to demonstrate that it does not participate in the coupling process. The ionization is represented by an Ammosov–Delone–Krainov (ADK) tunnel ionization model. The acoustic period Λ is slightly longer than λa/2 to satisfy the Bragg condition , since , where is the average refractive index in the plasma. In the simulation, an optimal value is obtained according to the maximum pump reflection.
The simulation results obtained with Opic are displayed in Fig. 2 (similar results obtained with EPOCH are given in the supplementary material, Fig. S1). The pump pulse is swiftly depleted soon after it encounters the plasma grating, indicating that the nonlinear stage can be reached directly, as shown in Fig. 2(a). The initial RP at t = 150T has a FWHM duration of about 80 fs, and it is then compressed further to about 7 fs at 2000T, which is close to two cycles of λa. The RP amplitude far exceeds that of the pump. After that, there is a slight return of the duration to about 9 fs, but the amplitude still grows to 0.07 until t = 3200T, with a compression efficiency of 80%, as shown in Fig. 2(b). The overall and specific profiles of the plasma and gas are shown in Fig. 2(c), which reveals a very sharp boundary between them. After a fast Fourier transform (FFT), the logarithmic spatial frequency of the electron density is as shown in Fig. 2(e), exhibiting a first peak at fx ≈ 2/λ. This matches well with the period of Λ = 0.502 35λ for the plasma density. There is also a peak at fx ≈ 4/λ due to the double-frequency effect of the density modulation. However, this has hardly any effect on the result, because of the much smaller amplitude (∼10−4 of the first peak). Although the effect of plasma heating is weak in FEPGC owing to the very short interaction time for the pump and RP, a peak can still be found in the electron temperature, as shown in Fig. 2(f). However, the hot electrons are then swiftly dragged back and cooled down by ions, indicating that the plasma grating is little influenced by the plasma heating. As well as being due to scSBS,13,18 the ionizing plasma grating is an ion-acoustic wave, which has strong robustness against thermal effects.
The RP spectrum shows an extremely wide range from 0.5ωa to 2ωa at t = 2000T, indicating that a short pulse of a few cycles can be supported, as shown in Fig. 2(g). The spectrum could be broadened by the effect of frequency conversion in the ionization as well as by the use of a dynamic plasma grating.10 Both of these can provide a continually varying refractive index for the RP thereby leading to a frequency shift. There are two peaks at about 0.7ω and 1.8ω, respectively. However, this phenomenon is not peculiar to FEPGC. Similar spectral peaks can be found when a short laser pulse is used to ionize a uniform background gas, as shown in Fig. 2(h). Therefore, they can be attributed mainly to frequency conversion in the gas ionization.
For a higher compressed intensity, we increase the pump amplitude from 0.002 to 0.007. As shown in Fig. 3(a), the AM grows linearly with t in the initial stage of compression, with a slope exactly matching that of the analytical solution obtained from Eq. (6). As the interaction time increases, the AM departs from the analytical line and finally tends to saturation at t > 4000T. Saturation appears earlier with a higher pump intensity, which is mainly a consequence of the increase in modulational instability with increasing laser intensity. Actually, transverse modulation can also be found in a 2D simulation when the RP grows, as shown in Fig. S2 of the supplementary material. In contrast to the AM, the compression ratio (CR), defined as tpump/tRP, from the PIC simulation grows faster than the analytical solution, as shown in Fig. 3(b). This more rapid growth might be due to the effects of self-phase modulation and negative group velocity dispersion (GVD) in the plasma.38 For a longer interaction time at t > 4000T, the CR tends to grow linearly or even decrease, indicating that the RP is no longer compressed, owing to saturation. Despite this saturation, FEPGC is not sensitive to the pump intensity. The AM and CR can be greater than 25 and 1000, respectively, over a wide range of pump amplitude.
The regime is further demonstrated by 2D simulations with Opic, in which both the pump and ionizing pulses have Gaussian distributions with the same waist of 50 µm. In contrast to the 1D simulation, the ionizing pulse with a peak amplitude of 0.01 has the same polarization and wavelength as the pump. Actually, the ionizing pulse can be artificially polarized, because it does not participate in the compression process. Here, it has the same polarization as the pump to collect its energy to the RP. As shown in Fig. 3(c), the RP intensity grows to 9 × 1016 W/cm2 after t = 5500T, with a compressed FWHM duration of about 10 fs, corresponding to an AM of ∼40 and a CR of 3667. The total transfer efficiency is 35%. However, the efficiency is not uniformly distributed, with ∼90% at the beam center but only ∼10% at the edge. This is mainly due to beam divergence, which makes the edge region depart from the Bragg condition. This problem can be avoided by using a large-scale pump beam with a much smaller divergence angle. The efficiency reaches more than 60% in our simulation when 200 µm (waist) sixth-order super-Gaussian beams are employed for the pump and ionizing pulses (see the supplementary material, Fig. S2).
To optimize FEPGC, a series of 1D PIC simulations are carried out with different modulation depths ρ and average plasma densities ρ0. ρ and ρ0 influence FEPGC mainly by changing the penetration depth of the pump, which is defined as the width from the pump peak to the first valley, as shown in Fig. 4(a). It is an important parameter representing the reflective feature of the plasma grating. Usually, a smaller penetration depth is expected for a shorter RP duration and higher RP intensity, indicating that higher ρ and ρ0 are preferred for FEPGC, as shown in Fig. 4(b). However, plasma instabilities also increase with increasing ρ and ρ0, leading to an earlier saturation time, as shown in Figs. 4(c) and 4(d). Above all, an FEPG with larger ρ and ρ0 has advantages for the compression of short-duration pumps. Conversely, a low plasma density and small modulation depth are more suitable for longer-time compression. For instance, the AM can still grow well at t > 8000T for ρ0 = 0.005nc, as shown in Fig. 4(d), illustrating the potential of FEPGC for laser compression from hundreds of picoseconds to a few optical cycles.
The robustness of compression is tested with inhomogeneous plasmas. Figure 5(a) shows the density profiles of two plasma gratings: one with a random density fluctuation and the other with a 500λa periodic density fluctuation, both with amplitude up to 50%ρ0. For FEPGC, the pump wavelength and plasma density have to satisfy the Bragg condition , and since , the compression will be mainly affected by a nonuniform ρ0. However, the simulation results in Fig. 5(c) for gratings with random and periodic density fluctuations show that the AM can still grow well in the presence of high- and low-frequency density fluctuations, respectively. Although some high-frequency peaks appear in the spatial frequencies of these plasma gratings, the dominant frequency is still fx ≈ 2/λa, as can be seen in Fig. 5(b), indicating that regions are always present in which the Bragg condition is satisfied. Hence, RP growth will be slowed down, but not fully halted, and thus it is maintained in an inhomogeneous plasma. The spectra of the RP become slightly narrower in the case of an inhomogeneous plasma, because of the weak RP intensity. However, a wide range of spectra are still available to support a laser duration of a few optical cycles, as can be seen in Fig. 5(d).
Several major plasma instabilities that would otherwise suppress the plasma grating can be avoided here. Kinetic effects in the plasma pose little threat, owing to the low plasma temperature. In FEPGC, the compression is just behind the ionization process, leaving little distance before the RP for plasma heating by the pump. Hence, the plasma temperature can be controlled at a very low level. This regime is quite similar to seed-ionizing31,38,39 or flying-focus40–42 BRA. Moreover, the wavebreaking is far from onset. The growth time of wavebreaking is given by for hydrogen,43,44 and thus a laser amplitude b > 1 can be tolerated for a twb of a few optical cycles.
IV. PHONON GENERATION
A key precondition for compression is the presence of a hypersonic wave generated by SBS in the background gas. This has been experimentally realized and is widely used for phase conjunction45,46 and laser compression.47,48 The density modulation due to the SBS effect is usually small. However, it can be boosted by using a strong counterpropagating laser pulse to replace the weak SBS, which can be described by combining Maxwell’s equations and the Navier–Stokes equations. With slowly varying envelope approximation, the 1D equations are as follows:49
where E0,1 and ω0,1 are the electric fields and frequencies of the laser pulses, α is the absorption coefficient, γe = ρ0∂ɛ/∂ρ is the electrostriction coefficient, which can be written as (n2 − 1)(n2 + 2)/3 in a centrosymmetric material, ɛ0 is the permittivity, δω is the frequency detuning, kB is the wavenumber of the acoustic wave, η is the viscosity, and υ is the acoustic velocity in the background gas.
Within a short region of low-density gas, the influence of the phonon on the laser pulses can be neglected, since δn is small. Under the assumption that E0, E1, and the density wave satisfy the resonant condition, ρ/ρ0 has an analytic solution
indicating that ρ/ρ0 tends to a steady-state value of Iγe/(υcnkBη) for an interaction time t > 1/τ, where is the laser intensity and τ = η/(2k2) is the phonon lifetime. The steady-state value of ρ/ρ0 is mainly limited by multiphoton ionization. For hydrogen, this has been observed at an intensity of order 1013 W/cm2 with a 1064 nm laser,50 corresponding to a maximum ρ/ρ0 ∼ 0.6. The steady-state values of ρ/ρ0 for several gases are given in Table I.
|Gas .||η (10−6 Pa s) .||υ (m/s) .||n .||γe (10−4) .||ρ/ρ0 .|
|Gas .||η (10−6 Pa s) .||υ (m/s) .||n .||γe (10−4) .||ρ/ρ0 .|
Simulation results for phonon generation by two counterpropagating laser pulses are shown in Fig. 6. As the two beams symmetrically enter the simulation window, their intensity profiles overlap, as shown in Fig. 6(a). After 3 ns of interaction, the laser intensities decrease slightly as they propagate along the gas, owing to the reflection of the phonon. However, the transmissions are still more than 90%, implying that they are not strongly influenced. This is mainly due to the low gas density (1 atm pressure) and short gas length (6000λ). As well as the laser intensity, the density modulation is slightly lower at the center of the gas. However, this inhomogeneity does not affect the FEPGC, because it does not change the Bragg condition of the plasma grating. The phonon amplitude rises above 0.5 after 3 ns, which is consistent with the analytical solution, as shown in Figs. 6(b) and 6(c).
As well as the SBS effect, a high-frequency gas grating has been reported to be created by the interfering ionization of two intersecting ultrashort laser pulses.51 In this regime, both plasma and gas gratings are initially created by the interfering ionization. The plasma grating disappears rapidly, owing to the thermal motion of the particles, while the gas grating remains. Such a gas grating has been demonstrated by both simulation and experiment, with a ρ/ρ0 up to 0.5. It can probably also satisfy the requirements for FEPGC.
We have proposed a novel plasma-based laser compressor. The use of density modulation in a gas combined with a short ionizing pulse enables the creation of a plasma grating whose boundary moves with the velocity of light, and so the reflection of the resonant pump is compressed. In this regime, the plasma grating is generated from a gas grating rather than from density perturbations, and so the traditional three-wave coupling model of laser compression [involving stimulated Raman scattering (SRS) and SBS] is simplified to a two-wave coupling model. This method of compression has several excellent features: a high compression efficiency without a linear stage, robustness to plasma instabilities, no need for a seed, and an ultrashort compressed duration. Finally, the required density modulation can be generated by either the SBS effect or interfering ionization in the background gas.
The compression regime is similar to, although not exactly the same as the concept of a “flying mirror.” A flying mirror formed by high-velocity particles has been proposed for laser compression and x-ray generation.52 To effectively reflect the pump, the plasma density has to be overcritical for the laser frequency. Moreover, as the particles have a high velocity, there is strong frequency up-conversion due to the Doppler effect. For FEPGC, the pump is reflected by the plasma grating, which requires a much lower plasma density (∼0.01nc), provided the Bragg condition is satisfied. In addition, although the boundary of plasma grating expands at nearly the velocity of light, the particles remain almost static, indicating that the Doppler effect is negligible. Therefore, the RP laser is compressed, but with the same central wavelength as the pump.
The simulation results of the fully kinetic Opic PIC code reveal that a 10–40 ps pump with a wavelength of 1 µm is compressed to a few optical cycles (7–10 fs), with a 2D transfer efficiency up to 60%. As the pulse is compressed to a few optical cycles, it can provide an extremely high-brightness source for the generation of isolated attosecond pulses. Moreover, assuming that comparable efficiency can be obtained in the 3D case, and that a 20 kJ, 20 ps, 1053 nm pump laser can be produced by Nd:glass chirped pulse amplification (CPA),53 the potential exists to produce a peak power above 1 EW (1018 W) after compression. However, as 2D simulations do not fully reveal the physical processes involved in FEPGC, 3D simulations will be required for future development of this technique.
As well as laser compression, the acoustic plasma grating proposed here provides new ways for laser manipulation. Compared with dynamic gratings in a plasma,1,54,55 the acoustic grating generated by the SBS effect in a gas is much more stable, with a reflectivity up to 95%.49 As the acoustic plasma grating is obtained directly from a gas grating, it may retain those benefits and thereby significantly enhance the quality of plasma optics.
VI. MATERIALS AND METHODS
The PIC codes Opic and EPOCH were used for the simulation of FEPGC. As the PIC code cannot describe phonon generation, we took hydrogen with a modulation period around λa/2 as the background gas. The physical processes, including optical ionization, particle motion, and the evolution of electromagnetic field, were then simulated by the PIC code based on Maxwell’s equations.
The 1D EPOCH simulation was carried out in a 3000λa static window, with 20 cells per λa and 32 particles per cell. The 1D Opic simulation used an 800-wavelength moving window, with 20 cells per λa and 32 particles per cell. The simulation results showed good consistency with each other. The 2D Opic simulation with 50 μm-waist Gaussian beams was carried out in a moving window of 300λa × 100λa, with five cells per λa longitudinally and ten cells per λa transversely, and 32 particles per cell. For the 2D Opic simulation with 200 μm-waist super-Gaussian beams, we used a moving window of 300λa × 500λa, with ten cells per λa longitudinally and 1 cell per λa transversely, and 32 particles per cell.
The simulation of phonon generation by the SBS effect in gas was based on Maxwell’s equations and the Navier–Stokes equations. The numerical solution used the split-step fast Fourier transform (SFFT) and a Runge–Kutta method. For the simulation, a static window with ten steps per λa and a total length of 6000λa was used. Hydrogen at a pressure of 1 atm and a temperature of T = 20 °C was taken as the background gas. Two counterpropagating 3 ns flat-top pulses with the same intensity of 1013 W/cm2 were taken as the interfering lasers.
The authors gratefully acknowledge helpful discussions with Professor Nathaniel J. Fisch and Syzmen Suckewer. This work was partly supported by the 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).
Conflict of Interest
The authors have no conflicts to disclose.
Zhaohui Wu: Conceptualization (equal); Methodology (equal); Writing – original draft (equal). Yanlei Zuo: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal). Xiaoming Zeng: Funding acquisition (equal). Zhaoli Li: Formal analysis (equal); Methodology (equal). Zhimeng Zhang: Investigation (equal); Software (equal). Xiaodong Wang: Data curation (equal); Resources (equal). Bilong Hu: Validation (equal); Visualization (equal). Xiao Wang: Visualization (equal); Writing – review & editing (equal). Jie Mu: Investigation (equal); Validation (equal); Visualization (equal). Jingqin Su: Project administration (equal); Supervision (equal). Qihua Zhu: Funding acquisition (equal); Project administration (equal); Supervision (equal). Yaping Dai: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal).
All data included in this study are available upon request by contact with the corresponding author.