1. Introduction

In the attosecond (1 as=10⁻¹⁸s) time scale, the borders between disparate scientific fields such as biology, chemistry, and physics tend to disappear, whose gaps can be bridged by the microscopic motion of electrons in atoms and molecules [1–5]. Coherent attosecond pulses in the extreme ultraviolet (XUV) and X-ray ranges can be produced from laser-matter interactions through various mechanisms [6–23], which are promising to extend human measurement and control techniques into such atomic-scale electronic dynamics. To achieve the full potential of the temporal resolution and the exposing matter that can induce nonlinear effects, attosecond light pulses with high intensity and large photon energy are required [24]. Unfortunately, the photon flux generated by a laser interacting with gas medium is too low and the energy conversion efficiencies are too small (10⁻⁸ ∼ 10⁻⁴) [24–26], which seriously limits the application prospects of these attosecond pulse generation technique.

Laser-driven high-harmonic generation (HHG) in reflection from solid targets has been investigated as a promising way for the production of intense attosecond radiation, where HHG is due to the periodic Doppler upshifted reflection of the driving laser from collective oscillations of the target plasma surface, namely, the relativistically oscillating mirror (ROM) mechanism [13–18]. However, since laser-plasma interaction occurs only in an ultrathin skin depth, where laser field drops rapidly by a factor of e⁻¹, its harmonic spectrum has a fast decay scaling of intensity on the harmonic order as I(n) ∝ n ^−8/3, which determines that the conversion efficiency from laser to the attosecond pulse in high photon energy range is rather low.

Recently, a potentially more efficient HHG mechanism, so called “coherent synchrotron emission” (CSE) [19–23], has been identified. Different from its literal meaning (like magneto-bremsstrahlung using bending magnets, undulators and/or wigglers), in this scheme, actually a compressed dense electron nanobunch is formed and dragged out from the solid target into the vacuum experiencing strong transverse acceleration caused by intense laser fields, resulting in production of high harmonics and accordingly attosecond pulses of XUV- or X-rays [19–23]. As such synchrotron emissions are produced by hot electrons from a δ-like thin and longitudinally-moving electron bunch when they interact with the transverse laser fields, they are spatially coherent with broad frequency spectrums. Previous references both theoretically [19, 20] and experimentally [21, 22] confirm that the harmonic spectrum of CSE has a slow decay scaling of I(n) ∝ n ^−4/3 to n ^−6/5, which is much flatter than ROM. That is, the conversion efficiency of CSE is much higher than ROM. However, to achieve such CSE requires a very stringent condition (such as few-cycle intense laser, high laser contrast, steep density gradient etc.) sensitively on laser and target parameters, which is still unclear so far. Characteristic CSE spectra in reflection from interaction have not been observed in experiments so far, where ROM generally dominates. Instead, only very weak CSE in the transmitted direction has been observed [21, 22], resulting in rather low intensity and small photon energy of the radiation pulse.

In this paper, we propose a new and practical route to achieve enhanced CSE by using the near-critical-density (NCD, S = n₀/a₀n_c ≈ 1) targets, so that relativistically intense attosecond X-ray pulses can be robustly obtained in both reflection and transmission. Due to the unique interaction dynamics between laser and NCD plasmas, where laser field E_y does not obviously damp and can effectively interact with a large region of plasmas, three distinct dense electron nanobunches can be formed during each half a laser cycle. Among them, two can have both strong longitudinal energy densities n_eγ_x and large transverse current density gradients ∂j_y/∂t under E_y, resulting in twice strong CSE’s in respectively the reflected and the transmitted directions. Comparing with CSE in solids, here not only the required stringent conditions above are relaxed, but also the radiation intensities are much enhanced. Moreover, because the formations of the nanobunches are spatially/temporally separated in NCD plasmas, the respective radiation are temporally rather separated, which can be easily isolated to get a single attosecond pulse. One-dimensional (1D) and two-dimensional (2D) particle-in-cell (PIC) simulations show that relativistically intense attosecond X-ray pulses with intensity ∼ 10¹⁹W/cm² and duration ~ 50as can be robustly obtained in both reflection and transmission by currently achievable drive lasers at intensities of 10²⁰W/cm².

The paper is organized as follows. In section 2, by comparing with the physical processes when a linearly polarized laser pulse irradiates solid targets, the completely different nonlinear dynamical physical processes, when NCD targets is used, is depicted. In section 3, 1D and 2D PIC simulation results are presented to confirm our theory. And results from different cases that with pre-plasmas in front of targets or different initial laser parameters are also discussed. Section 4 shows the conclusion of the practice scheme to generate intense attoseocnd pulses.

2. Theoretical analysis

Let’s start from analyzing the formation dynamics of dense electron nanobunches during half a laser cycle. As shown in the schematic Fig. 1, three distinct dense electron nanobunches can be formed under the dynamic balance between the laser ponderomotive force f_p and the restoring electrostatic force f_s, where f_p evolves at frequency 2ω₀ (ω₀ is laser frequency). First, initially the laser ponderomotive force f_p increases, which piles a large region of plasma electrons up to form a compressed electron layer, shown in Fig. 1(a). The maximum compressed density n_H can be estimated from the balance between the maximum f_p and f_s as 2πn_e0d =2πn_Hl_s = a₀, where d is the electron depletion length and $l_s=c/\sqrt{n_H}$ is the skin depth. Secondly, afterwards, when f_p decreases, see 1(b), the compressed electron layer splits into two parts, where electrons in the left part are pulled back by f_s due to f_s > f_p, forming a relativistic dense electron nanobunch “A” with n_H1 > n₀, and those in the right part still move forward due to f_p > f_s there, forming the bunch “B” of n_H2 > n₀. Thirdly, when the splitting of the bunches “A” and “B” further develops, electrons between them are continuously piled up into the bunch “A” by strong f_s, so that locally n_e becomes smaller than n_i, shown in 1(c). Correspondingly, the electrostatic force f_s evolves into a profile with two peaks and a negative potential f_s < 0 in the center between them [1(c) and 1(d)]. Lastly, when the laser ponderomotive force f_p increases again, it together with the double-peaked f_s drives and piles up local electrons into a new dense nanobunch “C”, where the bunch “A” decompresses, as shown in Fig. 1(d).

 

Fig. 1 Schematic profiles of ion (n_i, yellow and sky-blue), electron (n_H, pink) densities, of the ponderomotive force (f_p, blue), and of the electrostatic force (f_s, red) evolving at different times during half a cycle of intense laser interaction with NCD plasmas. They clearly show that three compressed dense electron nanobunches are formed each half a laser cycle

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Strong CSE occurs when these dense electron nanobunches move relativistically with both large longitudinal energy density n_eγ_x and strong transverse current density gradient ∂j_y/∂t ~ ∂E_y/∂t in interaction with intense laser field E_y. As shown in Fig. 1(c), the first strong CSE occurs from the bunch “A” towards the reflected direction when it moves backward and reaches a balance between f_p and f_s so that it has the highest n_H1γ_x and ∂j_y/∂t under intense E_y. Afterwards, the bunch “A” significantly decompresses [1(d)], leading to very weak emission followed. The second strong CSE occurs from the bunch “C” towards the transmitted direction when it moves forward with the highest n_H3γ_x, see Fig. 1(d). Afterwards, the bunch “C” decompresses as well. Since laser is reflected at the place of the bunch “B”, where E_y always drops to zero [1(c) and 1(d)], no CSE is generated from the bunch “B”. Therefore, we conclude that two strong CSE’s take place each half a laser cycle in respectively the reflected and the transmitted directions. Furthermore, we can see that the formations of these radiating nanobunches are spatially/temporally separated, therefore the respective sub-pulses in the radiation pulse train are temporally rather separated, which can be easily isolated to obtain a single attosecond pulse. Besides, the spectrum of these attoseocnd pulses from such nanobunches can be analytical obtained if we know the denisity profile of the bunches [27].

Note that, for solid targets, laser only penetrates into an ultrathin skin depth, where E_y drops significantly by e⁻¹, in most cases only the surface oscillation can be achieved, therefore the three dense electron nanobunches cannot be formed and no CSE occurs. Only in the tiny preplasma region, formations of the nanobunches are observed [21, 22, 28, 29]; however, under significantly dropped E_y, very weak CSE’s in transmission are observed. Furthermore, those emissions are overlapped with each other, hard for isolation [28, 29].

3. Simulation analysis

The dynamics analyzed above are verified by 1D and 2D PIC simulations using the code, EPOCH [30], which has been widely used for modeling intense laser-plasma interactions [31–35]. In 1D simulations, a P-polarized laser pulse with peak intensity I₀ = 8.6 × 10²⁰W/cm² (a₀ = 20) and λ = 800nm propagates from the left boundary at x₀ = −20λ and irradiates a NCD carbon plasma target at x = −λ. A Gaussian-temporal profile of the laser nomalized electric field amplitude $a_y=a_0\exp [-\sqrt{2\mathit{ln}2}{(t-\sigma )}^2/{(\tau )}^2]$ with full-width-at-half-maximum (FWHM) duration τ = 9.5fs and σ = 5T₀= 13.fs is chosen, where the “starting” intensity of the leading edge of the pulse is at about 10¹⁷W/cm², which is below the relativistic value, so that the laser ponderomotive force and the acceleration/expansion of hot electrons is very weak and negligible within such short duration. The target has electron density n₀ = 20n_c satisfying S = n₀/a₀n_c = 1 and thickness d = 2λ. The simulation box is 40λ long with a resolution of λ/4000. 1000 quasiparticles per cell are taken. For comparison, the simulation with solid targets at n₀ = 100n_c (S = 5) is also carried out.

Figures 2(a) and 2(b) plot the spatial-temporal evolutions of electron densities n_e(x, t) for respectively the NCD and solid target cases in the half cycle around the peak driving laser intensity from time t = 22.5 to 23.0T₀. In NCD plasmas (S = 1) [2(a)], it can be clearly seen that three dense electron nanobunches “A”, “B” and “C” are formed successively, which is consistent with our theoretical expectation, while only the surface oscillation is shown for the solid case [see 2(b)]. Figures. 2(c)–2(e) plot the profiles of n_e overlaid with the ponderomotive f_p = v_yb_z/mcω and the electrostatic f_s = eE_x/mcω forces at t = 22.61, 22.72, and 22.78T₀ when respectively the bunches “A”, “B”, and “C” are formed. It clearly shows three nanobunches are formed under the offset effect between f_p and f_s at different times, which agrees well with our theory shown in Fig. 1. The corresponding phase space distributions of respectively the electron longitudinal (x, p_x) and transverse (x, p_y) momentums at these times are plotted in Figs. 2(f)–(h). In 2(f), we see that two bunches “A” and “B”, moving in two opposite directions, are formed. Fig. 2(g) shows that the bunch “A” is accelerated to a relativistic speed (p_x > 1) at time t = 27.22T₀ when strong emission occurs and some electrons from bunch “A” are pushed forward again in the region f_s < 0. At t = 22.78T₀, three distinct bunches, that bunch “A" moves backward but “B” and “C” move forward, are formed [2(h)].

 

Fig. 2 Formation of three dense electron nanobunches in 1D PIC simulations of our scheme. (a) spatial temporal distribution of the electron density n_e(x, t) (normalized by n₀) from time t = 22.5 to 23.0T₀ corresponding to the half cycle at the peak intensity of the driving laser. The signs “A”, “B” and “C” mark where and when the bunches are formed, “RE” (emissiosns propagating along the reflected direction) and “TE” (emissions in the transmitted direction) present where and when the intense emissions take place in the two directions, respectively corresponding to the most intense attosecond pulses in Figure 3(e) and (f). “WRE” and “WTE” are those for the weak emissions, corresponding to the inset figures. (b) plots those for the case of solid targets (S = 5). (c)–(e) are the profiles of n_e (left), overlaid with the ponderomotive f_p and electrostatic f_s forces (right) at respectively t = 22.61, 22.72, and 22.78T₀, which have verified our theory, agreeing well with those shown in Schematic Fig. 1. (f) – (h) show the phase space distributions of normalized longitudinal (p_x, red) and transverse (p_y, blue) momentums along x-axis at respectively t = 22.61, 22.72, and 22.78T₀.

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The harmonic spectra of the obtained radiations in the reflected and the transmitted directions are plotted in Figs. 3(a) and 3(b) respectively. We see that in both directions, the radiations in NCD plasmas (S = 1) show a much slower decay scaling of the intensities on the harmonic orders than those in solid targets (S = 5), in particular, for the high photon energy range (> 100eV). This indicates much intenser attosecond pulses can be obtained in NCD plasma targets as proposed, see Figs. 3(c) and 3(d), whose intensities are above relativistic and more than two orders higher than those in solid targets [insets of 3(c) and 3(d)] in both directions. Also because the formations of the bunches “A” and “C” (and their corresponding strong CSE’s) are separated by a long distance (time duration), we see from Figs. 3(c) and 3(d) that the sub-pulses in the radiation pulse train are separated by about 1300as, where a single attosecond pulse can be easily isolated. The most intense attosecond pulses emitted at the half cycle around the peak laser intensity are plotted in Figs. 3(e) and 3(f) for respectively the reflected and the transmitted directions. In the reflected direction, a relativistically intense attosecond X-ray pulse with peak intensity I = 6.73 × 10¹⁹W/cm² and FWHM duration τ = 57.7as is produced, while in the latter an intense pulse with I = 1.5 × 10¹⁹W/cm² and τ = 53.3as is also obtained, which is the most intense attosecond X-ray pulse obtained in transmission so far.

 

Fig. 3 Coherent synchrotron emissions from the 1D PIC simulation. (a) and (b) The harmonic spectra of the obtained radiations in respectively the reflected and the transmitted directions for both NCD (S = 1) and solid (S = 5) target cases. (c) and (d) The produced attosecond pulse trains in both directions by selecting harmonics above 20ω₀ through spectral filter, where insets show those for the solid target case. (e) and (f) The normalized intensities |E_y|² $(m_e^2{c}^2{\omega }_0^2/e^2)$ of the most intense attosecond pulses emitted at the half cycle around the peak laser intensity respectively progating along the reflected and transmited directions, where insets show the following weak emissions when the bunches decompress in the same half cycle. (g)–(l) The spatial profiles of the bunch longitudinal energy density n_eγ_x (in units of n₀) and the transverse E_y (m_ecω₀/e) at the times when the emissions in (e) and (f) occur.

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Besides, from the insets of Figs. 3(e) and 3(f), we see that the relativistically intense attosecond pulses in both directions are followed with a weak unrelativistic emission at later separate times of respectively 900 and 415as, which are within our expectation above, also marked in Fig. 2(a). To explain this, the spatial profiles of the bunch electron longitudinal energy density n_eγ_x, overlaid with the laser field E_y at the times when these emissions occur are plotted in Figs. 3(g) and 3(l) respectively. The strong CSE’s are obtained when the bunches have both large nγ_x and strong transverse acceleration under E_y. When the bunch “A” moves back at t = 22.72T₀ [3(g)], it is compressed and accelerated with nγ_x > 800n₀ and E_y ≈ −18, therefore strong CSE takes pace, resulting in an intense radiation pulse shown in Figs. 3(e). However, while the bunch “A” moves forward [3(h)], it undergoes heavy decompression with nγ_x significantly decreased and broadened, leading to weak emission in the transmitted direction [inset of 3(f)]. In contrary, the bunch “C” generates strong emission [see 3(l) and correspondingly 3(f)] in transmission, but the weak emission in reflection [3(k) and inset of 3(e)]. For the bunch “B”, because E_y drops to zero and is reflected there [3(i) and 3(j)], which can hardly have emissions. Therefore, electrons from the bunches “A” and “C” can both emit an intense attosecond pulse and later a weak one in the reflected and transmitted directions. That corresponds to “RE” at near t = 22.72T₀ and “WTE” at near t = 22.85T₀ from the bunch “A”, and “TE” at near t = 22.78T₀ and “WRE” at near t = 22.78T₀ from the bunch ’‘C” [See Fig. 2(a) and/or Fig. 3].

To check the robustness of our scheme, simulations for NCD targets with the preplasma scale length up to L = 2λ are carried out, where the pre-plasma profile is n_e = n₀ exp (−x/L). As shown in Fig. 4(a), an intense transmitted attosecond pulse is also obtained. Of course, with the existence of a large scaling preplasma, its intensity drops a little and, in some way, its duration is also enlarged a little. However, as mentioned in previous works [19,20], for CSE in solid targets, the preplasma scale length should satisfy L ≤ 0.1λ. According to the classical hydrodynamic theory, after some straightforward calculations, we can estimate that at least a laser intensity contrast of of 10⁻⁹ is required for NCD targets (L = 2λ) and 10⁻¹³ is required for solid targets (L = 0.1λ). Therefore, comparing with CSE in solid targets, our proposed scheme has much less stringent requirements on the laser contrast. Simulations for the driving laser at different intensities of I₀ = 2 × 10²⁰ (a₀ = 10) and 2 × 10²¹W/cm² (a₀ = 30) are also carried out, which proves that CSE can be robustly achieved by using the NCD plasma targets [see 4(b) and 4(c)].

 

Fig. 4 The square of the normalized amplitude |E_y|² of the transmitted attosecond pulse [corresponding to Fig. 3(f)] for the different cases. (a) is the attosecond pulse when a preplasma of scale length L = 2λ exists (a₀ = 20). (b) and (c) are corresponding to results when the driving laser amplitude decreases to a₀ = 10 (n₀ = 10n_c), and when it increases to a₀ = 30 (n₀ = 30n_c), respectively. (d) shows the normalized peak intensity |E_2as,0| (m_e²ω₀²c²/e²) of the attosecond pulses in the reflected direction by selecting high order harmonics above 20ω₀, which are obtained by changing the similar parameter S when the initial laser parameters respectively satisfy a = 10 (red), a = 20 (black) and a = 3 (blue).

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To achieve our scheme, on the one hand, the laser field cannot be seriously damped during the interaction with the target, and on the other hand, the target should not be transparent to the laser (otherwise no nanobunches can be formed). Both of these lead to that the electron density of the target should be close to the relativistic critical density, n₀ ≥ γn_c i.e., the parameter S = n₀/a₀n_c ≥ 1 should be required. Furthermore, to further discuss the valid parameter region, a series of 1D PIC simulation results from the cases of different initial laser and target parameters are plotted in Figure. 4(d). It clearly shows that only when the similar parameter S satisfies 1 ≤ S < 4, the radiation intensities of attosecond pulses can be larger than the relativistic limit (|E_as,0|² > 1). Especially, when S ≈ 1, the peak intensity reaches to the maximum. In view of the above, we conclude that the proposed strong CSE regime works well in the range of 1 ≤ S < 4. In particular, S ≈ 1 is the optimal value.

To consider the multi-dimensional effects, 2D PIC simulations are carried out, where the simulation box is composed of 23λ × 10λ. Due to the limitation of computational resources, we reduce the spatial resolution as λ/1000 × λ/1000 and the number of particles per cell as 50 only. The laser pulse has a gaussian transverse profile with FWHM radius r0 = 3.5λ. All other parameters are the same as the above 1D simulation for NCD plasmas. Figures 5(a) and 5(b) plot the electron density maps at t = 14.93 and 15.0T₀ in half a laser cycle, which clearly shows that three dense electron nanobunches “A”, “B” and “C” are also consecutively formed in the multi-dimensional case. These nanobunches result in strong CSE of two relativistically intense attosecond pulses in respectively the reflected and the transmitted directions, shown in Figs. 5(c) and 5(d). Both pulses have relativistic intensities around 10¹⁹W/cm² and durations < 100as. Multi-dimensional effects can also be seen in Fig. 5. On the one hand, the emitted attosecond pulses [5(c) and 5(d)] are spatially curved due to the transversely Gaussian-distributed drive laser; on the other hand, due to transverse electron loss in the extra dimension, the formed nanobunches have lower densities and larger thicknesses [5(a) and 5(b)], leading to lower intensities and longer durations of the emitted attosecond pulses [5(c) and 5(d)]. Note that, at the later stage after the strong CSE occurs, with laser intensity drops later, the hot electrons disperse over the interaction region, smearing out the electrostatic force, and instabilities grow up, making the electron phase space incoherent, which together leads to heavy decompression/deformation of the formed electron nanobunches. Eventually, the coherent high harmonic emission terminates. Even some high harmonics can be generated by incoherent electron scattering, they are extremely weak. And these incoherent emissions also have a large angular divergence, propagating along different directions. Thus, the high harmonics generated at the early stage by CSE can be easily isolated from the latter incoherent emissions.

 

Fig. 5 2D PIC simulation results of CSE’s in NCD targets. (a) and (b) The electron density maps n e (in units of n 0) at t = 14.93 and 15.0T₀, which shows that three dense electron nanobunches “A”, “B” and “C” are also consecutively formed here. (c) and (d) The obtained intense attosecond pulses E_y in respectively the reflected and the transmitted directions after selecting harmonics > 20ω₀. Insets show the corresponding longitudinal profiles |E_y|² at y = 0.

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4. Conclusion

In summary, we have proposed a practical and efficient scheme for laser-driven attosecond pulse generation by using NCD plasma targets, where the required stringent conditions on laser and targets are much relaxed. Our 1D and 2D simulation results confirm that, due to the unique nonlinear dynamics between the interaction of laser and NCD plasmas that three distinctly dense electron nanobunches can be formed during each half a laser cycle, two electron bunches can have both strong longitudinal energy densities n_eγ_x and large transverse current density gradients ∂j_y/∂t under large E_y, resulting in twice generations of attosecoond pulses in respectively the reflected and the transmitted directions. In both directions, the two attosecond pulses, one at relativistic intensity and the other at unrelativistic, are separated by about 1000 as, which can be easily isolated from each other to obtain single intense attosecond pulse. More importantly, in the scheme, relativistically intense attosecond pulses with more than two orders of magnitude enhancement in intensities can be obtained in both reflection and transmission, comparing with attosecond light pulses from solid targets. Such intense pulses can be applied for pump-probing a wide range of bound electron exception and relaxation dynamics, which are inaccessible today.