Ultra-intense single attosecond pulse (AP) can be obtained from circularly polarized (CP) laser interacting with overdense plasma. High harmonics are naturally generated in the reflected laser pulses due to the laser-induced one-time drastic oscillation of the plasma boundary. Using two-dimensional (2D) planar particle-in-cell (PIC) simulations and analytical model, we show that multi-dimensional effects have great influence on the generation of AP. Self-focusing and defocusing phenomena occur in front of the compressed plasma boundary, which lead to the dispersion of the generated AP in the far field. We propose to control the reflected high harmonics by employing a density-modulated foil target (DMFT). When the target density distribution fits the laser intensity profile, the intensity of the attosecond pulse generated from the center part of the plasma has a flatten profile within the center range in the transverse direction. It is shown that a single 300 attosecond (1 as = 10−18 s) pulse with the intensity of 1.4 × 1021 W cm−2 can be naturally generated. Further simulations reveal that the reflected high harmonics properties are highly related to the modulated density distribution and the phase offset between laser field and the carrier envelope. The emission direction of the AP generated from the plasma boundary can be controlled in a very wide range in front of the plasma surface by combining the DMFT and a suitable driving laser.
© 2013 Optical Society of America
New approaches to ultrafast science such as holography, nanotechnology, biological imaging and fast electron dynamics probing  have been constituted based on the extension of the pump-probe techniques to extreme ultraviolet (XUV) and soft X-ray (SXR) regime. With the generation of attosecond XUV pulse, real-time observation of a large range of phenomena involving the dynamics of electrons on the atomic timescale and energy transport in molecules  become possible. To achieve this, compact X-ray  or high harmonic sources with sufficient intensities and sub-femtosecond durations are necessary.
Several methods for phase-coherent harmonic emission using intense laser pulses have been proposed, including the nonlinear interactions with atoms in rarefied gases  and with over-dense plasmas in solid targets [5, 6]. However, the intensities of the APs obtained by laser-gas interaction are very low due to the following two restrictions. One is the power of the pump laser has to be kept low (≤ 1016 W cm−2) because of the phase mismatch effects [7, 8]. The other is the conversion efficiency which is associated with the electron density, is extremely small (10−7 ∼ 10−6) . The weak intensity of the emitted attosecond pulses has been one of the main obstacles for applications of the pump-probe techniques. Fortunately, in the scheme of laser overdense plasma interaction, the conversion efficiencies of high-order harmonic generations (HHG) in overdense plasmas were significantly increased [10, 11]. With the much higher electron density and no restriction of the phase mismatch effects (the laser intensities can be over 1018 W cm−2), the APs generated from solid targets are almost several orders of magnitude higher than those from atomic mediums [12, 13]. This feature may provide high harmonics and/or APs of sufficient intensities, fulfilling the requirements in the pump-probe experiments, and furthermore broadly expanding the scope of the applications of APs.
The linear polarization (LP) driving laser employed in harmonic generation at the over-dense plasma surface typically needs an intensity of IL > 1015 W cm−2 for the λL = 800 nm laser wavelength , which corresponds to a minimum normalized field amplitude . When an intense ultrashort laser pulse focuses on a solid target, the materials are ionized and converted into plasmas instantaneously. The incident laser pulse reflects on the dense plasma boundary which has an extremely steep interface with vacuum. The interaction process is highly nonlinear when surface electrons are compressed to a thin layer and oscillate as a plasma mirror which leads to harmonic generation. Different reflected harmonic spectra appear when the pump laser intensities vary from weak relativistic region ( ) to strong relativistic region ( ), which corresponding to two major mechanisms of harmonic generation, coherent wake emission (CWE)  and relativistic oscillating mirror (ROM) [15, 16, 11]. The CWE totally dominates the harmonic generation process for aL < 1, and contributes to a narrow frequency range of the harmonic spectrum, which typically corresponds to harmonic order less than 30. As the laser intensity raises, e.g., aL ≫ 1, the ROM process becomes the dominant mechanism. The maximum cut-off frequency of the harmonic spectrum is far beyond that from the CWE process, and extends up to , where γmax is the relativistic factor corresponding to the maximum velocity of the plasma mirror which flies towards the incoming light, and ωL is the incident laser frequency . Moreover, the harmonic intensity follows the power- law form I(ω) ∝ ω−p, where ω is the harmonic order and the exponent 5/3 ≤ p ≤ 10/3 depends on the laser pulse parameters . The harmonics are phase locked, and one can obtain a train of extremely short pulses by transforming the high harmonics spectrum into the time domain.
However, when a relativistic LP pulse impinges on a solid target, the plasma boundary would oscillate twice per laser cycle as the ponderomotive force of the LP pulse includes a fast oscillating component, inducing an AP train. While for most applications single AP is preferred. Two methods have been proposed to isolate single AP from the pulse train: using a lambda-cubed (λ3) incident laser pulse [19, 20] and the polarization gating technique . Nevertheless, it is still a challenge to verify both methods for general facilities.
Motivated by the incomparably wide prospects of single APs, we have looked into the possibility of directly producing single APs from the interaction of intense few-cycle circularly pulses (CP) laser with overdense plasmas. Ji et al. found that high harmonics can be produced in the reflected light by the Doppler-effect when the CP laser pulse reflects from the laser-induced one-time oscillating plasma boundary [22, 23]. In present work, we first verify the scheme of single AP generated from CP laser interacting with overdense plasma with a flat foil by 2D PIC simulations in section 2. The generated APs are dispersed in front of the plasma surface with self-focusing and defocusing phenomena, and can hardly be used in most proposed applications. In section 3, by using density-modulated foil targets (DMFT) [24, 25] in the scheme, we find that the angular divergence of the emissions can be controlled by the plasma surface deformation. A single 300 as pulse is obtained with a designated target density distribution. The intensity of the AP is higher than 1021 W cm−2 and has a remarkable feature in the transverse direction. In section 4, an analytical model is employed to follow the spatial distributions of the plasma boundaries in detail. The numerically results agree well with the simulation runs. We also find that the emission directions of high harmonics are both sensitive to the profile of the modulated density (section 5) and to the carrier-envelope phase (CEP) of the incident CP laser pulse (section 6). The emission direction of the generated AP can be controlled in a very wide range in front of the plasma surface by combining the DMFT with an appropriate density profile and a suitable driving laser.
2. Analysis of focusing mirror by 2D PIC simulation
When a relativistic CP laser pulse impinges on a cold-plasma foil, electrons are quickly pushed forward by the steady part of the v × B force. The ions are fixed due to the short interaction period and the much larger mass. An intense space-charge field is created as the electrons are greatly compressed from the surface to inward. This compressed electron layer (CEL) intensively bounces back, and forms a one-time oscillation. A bunch of high-order harmonics will be produced in the reflected light when the CP laser pulse reflects from the laser-induced oscillating plasma boundary. The oscillation is the result of the imbalance between laser light pressure and space-charge separation field force. It is so intense that the maximum speed of the CEL can reach nearly the speed of light, which can produce an ultra-short single AP naturally . However, things make different when we investigate the AP generation by this proposed scheme in 2D geometry. In particular, the plasma boundary will be bended in the case of a laser pulse with a finite width in the transverse (y) direction . The generated radiation fields distribution at the plasma surface may differ in intensity and diffraction direction. These features lead to a process of self-focusing and then defocusing of the reflected harmonics , which might be harmful for long-distance transmission of the APs.
In order to study the self-focusing and defocusing features of the deformed plasma boundary in detail, we perform fully relativistic planar 2D PIC simulations by the code VORPAL . The simulation box is 35λ0 × 30λ0, corresponding to 10500 × 1500 cells in x and y directions, respectively, where λ0 is the incident laser wavelength. The high spatial resolution along the laser propagating direction is to ensure the calculation precision on attosecond time scales. The plane target is fully pre-ionized, with ion mass and charge of mi = 3672me and qi = −e, where me and e are mass and charge of the electron, respectively. It is initially located in the region of 28λ0 < x < 31λ0 with a density of n0 = 8nc, where nc = meω2/4πe2 is the critical density for the laser frequency ω. The normalized amplitude of the incident CP laser pulse is . where a0 = 20, τ0 = 2T0 and rL = 5λ0. This corresponds to a laser intensity of I0 = 1.1 × 1021 W cm−2 for the assumed wavelength λ0 = 1μm. The leading edge of the laser pulse enters the simulation box from the left boundary at the beginning and reaches the target at t = 28T0. Absorbing boundary conditions are used for both electromagnetic field and particles.
The spatial distribution of electrons at t = 31T0, as seen in Fig. 1(a), is related to the combination of the ponderomotive force of the laser pulse and the restoring electrostatic force of the ions. The incident laser pulse with a transversally Gaussian profile bends the plasma surface. The red solid line in Fig. 1(b) denotes the space-charge separation field, peaking at the pulse front with the maximum intensity of 33, which plays the key role in switching the electron layer motion and forming the one-cycle oscillation. The intensity of the electrostatic field is much larger than that of the incident laser field, and has a linear profile both in the depletion region and in the compressed layer. Throughout the entire interaction time, the dented electron layer maintains as a Gaussian reflectance mirror. No obvious ripples or bubble structures which induced by multidimensional instabilities in laser-overdense plasma interaction [29, 30, 31] can be found in the plasma boundary, because the timescale for the growth of this Rayleigh-Taylor-like electron instability is in the several fs-range and thus much longer than the reflection process.
Figures 2(a)–2(c) present the spatial distributions of the reflected laser field magnitude Ey at time t = 34T0, t = 44T0 and t = 54T0, respectively. The incident laser pulse is well reflected and propagates in the −x direction. The reflected pulse is self-focused from the surface due to the curved plasma boundary. The focal spot as intense as ∼ 10I0 can be reached at t = 44T0, and the spot size is comparable to the laser wavelength. The self-focusing length is 11.6λ0 which is in good agreement with the general analytical estimate xsf = S0xRL/2.7 = 11.6 μm [32, 33], where S0 = n0/a0nc is the ultra-relativistic similarity parameter , and xRl = πrL2/λ0 is the Rayleigh length of the fundamental. The reflected pulse is defocused after the self-focusing rapidly. The whole radiation is simply diffracted away from the optical axis and transmits in diverging directions, as shown in Fig. 2(c).
The single attosecond pulse is generated due to the fact that the plasma surface is accelerated to nearly the speed of light on a short time scale (< 1T0) [35, 36]. The emission of high-frequency photons as seen at an intermediate distance from the surface occurs only during small fractions of T0. In frequency space, this leads to high harmonics spectrum which transforms into intense APs in the time domain. To liven up the picture of how the attosecond pulses propagating, the filtered pulses at different distances from the surface are investigated. After ω ≤ 3ω0 filtering of the low-frequency components of the spectrum which is observed in the position of (22λ0, 0), as seen in Fig. 2(d), we obtain a single ultra-intense AP with the duration of 356 as and its peak intensity is 1.1 × 1021W/cm2. Since the harmonics are coherent, laser field at the focal point is strongly enhanced by summing up all the field reflected from different positions of the interaction surface [27, 37]. The maximum intensity of the AP is as high as 4.0 × 1021 W cm−2 at t = 44T0 and dramatically falls to 4.6 × 1020 W cm−2 at t = 54T0, as shown in Figs. 2(e) and 2(f). Due to the extremely broad spectrum of the generated high harmonics, diffraction will exert a major influence on the spectrum propagation. The spectrum changes rapidly because of the different diffraction lengths of the harmonics and different propagation directions. The generated AP radiation will be dispersed in the far field, and can hardly be used in most proposed applications.
3. Density-modulated foil target
2D PIC simulation confirms that the reflected CP laser pulse can be self-focused. The self-focusing effect will be very useful to generate laser intensity approaching the Schwinger limit , however, it is unfavourable to produce a stable-propagating AP. As is well known, the electron dynamics in ultra-relativistic laser-plasma interaction depends on the similarity parameter, rather than on a0 and n0 separately . Taking this advantage, we propose a flat foil target with a Gaussian plasma density distribution in the transverse direction, the laser intensity profile is well matched and the self-focusing effect of the generated AP is somehow avoided. As a result, a plan-wave like AP will be generated, which will travel for a long distance without apparent distortion.
As shown in Fig. 3, a CP laser pulse is normally incident on the density-modulated foil target (DMFT). The density profile of the DMFT is defined byFig. 3. We present the results for a DMFT with n0 = 8nc, rd = 5λ0 (matching the laser focus radius rL) and ncutoff = 0.2n0. Thus the ultra-relativistic similarity parameter can be written as
The advantage of the constant-S surface can nicely be seen in Fig. 4(a), which showing the temporal electron density distribution at t = 31T0 with gray scale. The plasma boundary is uniformly shifted along the transverse direction and keeping the shape of the compression electron front. The red dotted line is calculated from the analysis model which we will discuss later. The generation of surface plasma waves (SPW) is a vital feature of laser plasma interaction. The SPW propagates long distances along the surface, which causes several multidimensional instabilities, e.g. the Rayleigh-Taylor instability [38, 39, 40]. However, no evidence of any instability is found in our 2D simulations. In the short time-scale of a few-cycle pulse driver, the classical Rayleigh-Taylor instability is not relevant because it presumes ion motion, and the instabilities have no time to develop. It is expected that the plasma surface would remain planar during the interaction period.
The incident laser field in vacuum runs in the positive direction, while the reflected field is translated backwards in the negative direction. The spatial distribution of the laser field Ey at t = 37T0 is depicted in Fig. 4(a). The incident laser pulse is better reflected from the overdense plasma layer compared to the unoptimized case in Fig. 2(a). Evidently, no self-focusing or defocusing phenomenon occurs in front of the planar surface. These distributions look similar to the 2D simulation results reported by Ji et al. , where they employed a CP laser pulse with sufficiently flat intensity distribution across the focal spot (Super-Gaussian) to irradiate an uniform plane target. This is another possibility to keep the relativistic S-parameter constant alongside the surface. High-order harmonics are contained in the trailing edge of the reflected pulse. As the reflected fields stably propagate backwards, the profile of the plan-wave-like AP is well maintained. Figure 4(c) shows the on-axis radiation spectrum in the log-log scale. The decay in peak harmonic power levels with harmonic number ω is well represented across the range by I(ω) ∝ ω−8/3, which is in agreement with the similarity analysis and the PIC simulation spectrum . After ω ≤ 3ω0 frequency filtering, a single ultra-short light pulse is obtained, as seen in Fig. 4(b). The AP has a duration of 215 as and its peak intensity reaches 9.1 × 1020 W cm−2. The conversion efficiency is 4 orders of magnitude larger than those in laser-atom interactions . After filtering out the diploid frequency components of the spectrum which are observed at different y positions along x = 25λ0, we can see that the durations of the single APs vary slightly around 300 in the region of |y| ≤ 4λ0. The corresponding intensities are plotted in the lower panel of Fig. 4(d), where a remarkable feature is that the intensity curve shows a flatten profile within the range of |y| ≤ 2λ0 and drops linearly beyond that region. The peak intensity is ∼ 1.4 × 1021 W cm−2 and is slightly higher than the incident pulse intensity. One should be noticed once again, that the DMFT case yields a clear advantage for the attosecond pulse generation.
4. Analytical model
In order to have a better understanding of the different electron dynamics between uniform density target case and DMFT case, we propose an analytical model as shown in Fig. 5 . When a CP pulse impinges on an overdense plasma layer, the electrons at the plasma vacuum interface are greatly compressed and accelerated forward by the ponderomotive force of the incident laser pulse, creating an intense space-charge field with the immobile step-like ion background. The momentum of the CEL is determined by the imbalance between the pondero-motive and space-charge forces, which implies an uniform electron density np0 and a thickness of ls = x2 − x1 equalling the skin depth . In view of the simulation result shown in Fig. 1(b), we assume that the electrostatic field peaks at the CEL left surface (x = x1) with the intensity of Ex0 = 4πn0e(x1 − x0), and has a linear profile both in the depletion region (x0 < x < x1) and in the compressed layer (x1 < x < x2). The ponderomotive force experienced by the electrons can be written as41, 42]. The areal density σ can be obtained by invoking the global charge conservation np0λs = n0(x + λs) and I(x, y, t) = cE2(x, y, t)/4π is the CP laser intensity with Eq. (3) and the charge conservation relationship into Eq. (5), one can get the normalized displacement of the electrons
Equation (6) can be solved numerically with the corresponding initial conditions x(0) = 0 and u(0) = 0. The plasma boundary profiles of the uniform density target case and the DMFT case calculated from the analytical model are plotted in Fig. 6(a). The incident laser pulses and the targets density profiles are the same as the corresponding simulation parameters employed in section 2 and section 3, respectively. The spatial distribution of the CEL along the transverse in the DMFT case (red solid line) is uniformly shifted in the region of |y| ≤ 6.4λ0, but presents a Gaussian shape in the uniform density target case (blue dotted line). The numerically results fit nicely with the simulation results, as seen in Fig. 1(a) and Fig. 4(a). Furthermore, we trace the momentum of the on-axis electron according to the exact calculating of Eq. (6), as seen in Fig. 6(b). It is clear that, the electron is first accelerated forward by the laser ponderomotive force, and then intensively bounced back at t = 3T0 due to the intense space-charge separation field force. The maximum shifting distance of the electron is less than 0.7λ0, it is tiny comparing to the spot size of incident laser (rL = 5λ0). The speed of the plasma boundary can reach nearly the speed of light in 1T0, and high harmonics are generated relaying on the Doppler-effect when the CP laser pulse reflects from the one-time oscillation plasma boundary.
5. Control of the attosecond pulse emit direction
The incident laser pulse is well reflected from the plasma boundary with a constant ultra-relativistic S-parameter in the transverse direction. Attosecond peaks can be obtained and the intensity is comparable to the incident laser pulse. In order to interpret the changes of the reflected pulses with the relativistic S-parameter alongside the layer surface, we varied the transverse density profile parameter rd from 1λ0 to 7λ0 with the other parameters unchanged, so that
In Fig. 7(a), the distribution of the electron-layer surface driven by the laser pulse looks like a convex lens in case of rd = 3λ0, thus the reflected radiation spread as a relevant curved diffusion. Surprisingly, the reflected radiation has been split into two separate impulses. Each of them is moving in its own direction. The less intense impulse (1) propagates along the center axis, and the more intense impulse (2) is directed towards the lower left-hand direction. The maximum intensity value is as high as 0.6I0 at t = 42T0, and falls slowly by the influence of diffraction. Tracing this impulse, it follows near 10° direction with respect to normal incidence direction, and also has a divergence of ∼ 10°. When rd decreases to 2λ0, the reflected impulse direction increases to ∼ 15°. The impulse direction increases while the density profile parameter rd decreases, which can be explained by the direct relationship between the reflected radiation and the electron-layer surface deformation.
Being exposed to the laser, the plasma surface is curved and shaped appropriately. The reflected high harmonics which consist of several pulses propagating in different directions can be focused to high intensities if rd > rl. Two snapshots of the impulse intensities at the time of t = 42T0 and t = 47T0 for the target profile rd = 7λ0 are shown in Fig. 7(b). The reflected harmonics focus to the center axis and enhance the intensities of the APs. The focal distance is much further comparing to the radiation focusing from the planar surface in section 2. This is due to the smaller curvature of the plasma surface in this case. The intensity gain of this work is about proportional to a0, which is not as strong as that in the coherent harmonic focusing (CHF) . But, comparing to the CHF, our simulation conditions are more realistic and achievable. To obtain higher intensity gain, it is needed to maximize the solid angle and to minimize the focal distance which will make the maximum use of the temporal focusing caused by the relativistic surface effects . These two features can be optimized by modulating the DMFT profile parameters of this work.
6. Effect of the incident pulse phase offset
The laser-plasma interaction is phase sensitive due to the short incident pulse duration. Thus, as the driver pulses we employed in present work come to a few-cycle laser pulse regime, the effects of phase offset between laser field and carrier envelope should be taken into account [19, 43]. Simulations are performed to study this effect, where the phase of the incident pulse ϕ is changed from 0 to 2π by every π/2 in the DMFT scheme with rd = 3λ0 and rd = 7λ0 separately. Then we measured the filtered radiation pulse intensities with different pulse phases..
In case of rd = 3λ0, the generated impulses in three phase offsets π/2, π, 3π/2 are shown in Figs. 8(a)–8(c), respectively. The direction of the peak intensity presents a clockwise rotation while increasing the phase offset. When the phase is changed by π, the deflected light shows an inversion symmetry about the incident laser direction, e.g., the distributions in Fig. 8(a) and 8(c). All the reflected radiations split into two impulses. The transmission directions of the less intense impulses in three phase offsets change slightly alongside the normal direction, which is because the motion of the reflecting surface around the center position is almost irrelevant to the phase offset of the incident laser. The divergence and the FWHM width of the reflected radiation with ϕ = π/2 are larger than those in ϕ = π, which leads to a little bit smaller peak intensity. The symmetrical structures also can be found in the self-focus process with rd = 7λ0 as shown in Figs. 8(d)–8(f). With these finer snapshots, we can find that the focused radiations are also composed of two impulses. The main reflected pulse (1) is transmission on the center axis, and the direction of the less intense pulse (2) is phase sensitive. Two impulses overlap with each other in ϕ = 0 or ϕ = π, which enhance the peak intensity.
It is well known, the direction of the resulting AP is difficult to be controlled, with the features we discussed above. The reflected AP from the plasma surface has a strong dependence on the plasma density profile in the transverse direction, and is also sensitive to the phase offset of incident laser pulse. Thus, the direction of the reflected radiation can be controlled by combining the DMFT with an appropriate density profile and a suitable driving laser.
7. Conclusion and Discussion
In the case of atomic harmonics, ionization gives rise to the saturation intensity, which limits the intensities of the APs. Fortunately, the theoretical treatment of the generation process of harmonics from laser-overdense plasma interaction reveals that there is no fundamental limitation in using the highest possible laser intensities available. The attosecond pulses are directly generated by reflecting from the plasma surface, therefore, the higher the laser intensity the stronger the nonlinear response of the medium. When a CP laser pulse impinges on the plasma surface, high harmonics can be produced in the reflected light relying on the Doppler-effect. An ultra-intense single attosecond light pulse can be naturally generated when the few-cycle relativistic CP laser pulse reflects from the laser-induced one-time oscillating plasma boundary. We investigated this scheme by 2D PIC simulations and analysis model. The spatial distributions of the compressed electron layer calculated from the analytical model agree well with simulation runs.
The incident laser pulse with a transversally Gaussian profile bends the plasma boundary. The reflected pulses are focused down to a small spot by the naturally curved surface and rapidly defocused after the self-focusing. The whole radiation is simply diffracted away from the optical axis and transmits in diverging directions, which can hardly be used in most proposed applications. We presented that the reflected high harmonics can be controlled by employing a density-modulated foil target. When the target density distribution fits that of the laser intensity profile, the incident laser pulse is better reflected from the overdense plasma boundary with no self-focusing or defocusing phenomena in front of the planar surface. This feature is extremely useful since the optical filters have to be placed inside the far-field in a real experiment. The peak intensity of the AP generated from the center part of the plasma boundary is as high as 1.4×1021W cm−2. The intensity curve shows a flatten profile within the range of |y| ≤ 2λ0. and drops linearly beyond that region in the transverse direction. The FWHMs vary slightly around 300 as in the region of |y| ≤ 4λ0. This could be a valid solution to get an intense single AP with good contrast ratio by combining DMFT with the use of an optical filter.
The prepulse plays an important role in ultra-intense laser plasma interaction, as reported by Zheng et al. , one can control the emission attosecond pulses by changing the target surface profile with proper preplasma. In our results, the emission directions of high harmonics are also sensitive to the profile of the modulated density and to the phase offset between the laser field and the carrier envelope. By modifying transverse density profiles and phase offsets, the reflected impulses are defocused in given directions or focused down to a spot with different focal lengths. With these features, the emission direction can be controlled in a very wide range in front of the plasma surface by combining the DMFT with an appropriate density profile and a suitable driving laser. These results show very encouraging scientific potentials in a lot of fields.
This work is supported by National Natural Science Foundation of China (Grant Nos. 11075069, 11205243, 91026021, 11075068, 11175076, 91026005 and 11135002), the Research Project of NUDT, and “Strategic Priority Research Program” of the Chinese Academy of Sciences, Grant No. XDA01020304.
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