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Abstract
High-order harmonics generated from the interaction between a two-color circularly polarized laser and overdense plasma is proposed analytically and investigated numerically. By mixing two circularly polarized lasers rotating in the same direction with different frequencies (ω0, 2ω0), the laser envelope is modulated to oscillate at the laser fundamental frequency while the peak intensity of each cycle becomes greater than that of the monochromatic light. This feature makes the plasma oscillate more violently and frequently under the striking of the two-color laser than the monochromatic one, thereby generating stronger harmonics and attosecond pulses. In addition, the incorporation of the 2ω0 light greatly expands the spectral width of harmonics, which facilitates the production of shorter attosecond pulses. Particle-in-cell simulations prove that under the same condition, the harmonic radiation efficiency in the two-color laser case can be improved by orders of magnitude, and isolated attosecond pulses can be even generated as a bonus in some cases.
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1. Introduction
The study of electron dynamics and its correlation effect, which determine the structure and properties of matter, is of vital importance to many scientific fields, such as physics, chemistry, biology, and so on [1]. Since 2001, the generation of attosecond pulses, either in the train form or isolated form, allows us to simultaneously resolve both structure on nanometer scale and dynamics on the attosecond scale, making it possible to detect electronic dynamics in real time [2,3]. Such versatile pulses can be synthesized through the combination of broadband spectra with well-behaved phase structure. The high-order harmonics generated (HHG) in the non-linear laser-matter interaction process has been proved to be an efficient source to attosecond pulses [4,5]. So far, most attosecond pulses have been achieved by HHG in gases, and the shortest isolated pulse in the laboratory is only 43 as [6]. However, due to the limitation of the laser intensity and the difficulty of phase locking, it is almost impossible to achieve relativistically intense attosecond pulses through gas HHG [4].
Plasma, which can withstand lasers of arbitrary intensities in principle, offers a superior alternative for the generation of intense attosecond pulses [7]. When irradiated by a linearly-polarized (LP) intense laser at normal incidence, the plasma surface executes forced oscillation along the longitudinal direction under the driving of laser Lorentz force $\boldsymbol {v} \times \boldsymbol {B}_L \sim \boldsymbol {E_L}^2 (t) \propto E^2_0 f^2(t) [1+\cos (2\omega _0 t)]$. Here, $\boldsymbol {E_L}$ is the instantaneous laser electric field acting on the plasma target with an amplitude of $E_0$, the temporal profile of $f(t)$, and the oscillation frequency of $\omega _0$. Every time when plasma surface moves towards the laser, a relativistic Doppler upshift is introduced in the reflected laser frequency domain, which manifests itself as the frequency comb of HHG in the spectrum. Two main mechanisms have been identified in the relativistic regime, i.e., the relativistically oscillating mirror (ROM) [8] and the coherent synchrotron emission [9] (CSE, also called relativistic electronic spring (RES) in some Refs. [10,11]). In both mechanisms, the HHG spectrum has a universal envelope: $I(\omega ) \propto \omega ^{-p}$, where $\omega$ is the harmonic frequency, and $p$ reaches 8/3 in ROM and 4/3 in CSE respectively in the relativistic limit, implying that the HHG efficiency is several orders of magnitude higher than that in gas medium [12–15]. However, as the laser ellipticity increases, the fast oscillating term of the laser Lorentz force decreases gradually, leading to a smaller oscillating velocity of the target surface, and eventually an inefficient HHG production [16]. Especially for circularly polarized (CP) lasers, the Lorentz force $\big ( \propto E^2_0 f^2(t) \big )$ is free of the fast-oscillating component. Thus the plasma surface is slowly pushed inwards in the rising edge and pulled outwards in the falling edge, so that the HHG is almost suppressed completely [17]. Due to this intrinsic difficulty, less attention has been paid to the HHG radiation driven by a CP laser for a long period of time [18].
Considering the broad applications of highly elliptical attosecond pulses [19–22], the interaction of CP lasers with plasma targets has attracted increasing interest. Researchers have attempted to achieve quick oscillation of the plasma target by a CP laser just as in the LP laser case. For example, in normal incidence, one-time quick oscillation of the plasma boundary can be achieved by irradiating few-cycle CP lasers onto a relativistically transparent target [23] or an ultrathin foil [24]. For oblique incidence, the component along the target normal direction of the $p$-polarization electric field of the CP laser forces the surface to oscillate and triggers the HHG process with comparable efficiency with the LP cases [25]. Based on these studies, the polarization gating technique is developed to generate isolated elliptically polarized attosecond pulses [26]. Recently, harmonic radiation driven by two-color lasers—consisting of the fundamental frequency ($\omega _0$) and its second harmonic ($2\omega _0$), has aroused great attention. For CP lasers, previous studies mainly employed two collinear or non-collinear counter-rotating CP lasers as the driving beam [27,28]. For the former case, except for the forbidden 3$n$-order harmonics, each harmonic is circularly-polarized but with alternating helicities, and the combined attosecond pulses are elliptically-polarized [27]. For the latter case, CP harmonics are dispersed and angularly separated by both helicity and frequency, which is not desirable for the synthesis of attosecond pulses [28]. In addition, Zhong et al. proposed two schemes based on LP driving lasers, providing more avenues for the generation of CP attosecond pulses [29,30].
In this paper, the HHG process driven by a pulse which consists of two co-propagating co-rotating CP lasers of $\omega _0$ and 2$\omega _0$ frequencies, as schematically displayed in Fig. 1, is proposed theoretically and investigated numerically for the first time. Unlike the threefold rosette of the electric fields in the counter-rotating case [27], the combined field here is still circularly polarized. Besides, the intensity envelope is modulated to oscillate at the laser frequency due to the participation of the 2$\omega _0$ laser. Therefore, a large oscillating velocity of the plasma surface is achieved and the HHG process can be successfully triggered at any incident angle. A series of two-dimensional (2D) particle-in-cell (PIC) simulations are carried out and the results prove that the HHG efficiency can be improved by orders of magnitude than the one-color case. More importantly, all harmonics are naturally phase-locked, which is beneficial to the generation of attosecond pulses. In addition, every attosecond pulse produced in this method is elliptically or even circularly polarized, which can be widely used to investigate ultrafast chiral-related phenomena.
Fig. 1. Schematic of the attosecond pulses generation from the interaction between a two-color ($\omega _0$, $2\omega _0$) circularly-polarized laser and a dense plasma target. The driving laser is synthesized by two circularly-polarized (CP) co-propagating co-rotating lasers, and the intensity envelope of the resultant laser oscillates at the fundamental frequency $\omega _0$ (see the red background of the incident laser). The Lissajous figures of the resultant electric fields of the two-color co-rotating (red) and counter-rotating (black) laser fields are displayed on the right side. Here, $W$ is the energy ratio of the second harmonic laser to the total incident laser.
2. Theoretical analysis
The laser pulse adopted in our scheme is achieved by the superposition of the following two CP pulses:
For comparison, a one-color laser is also employed with its normalized electric field defined as:
By squaring the laser electric field, we can get the normalized laser intensities of the two cases:
The treatment here is equivalent to normalizing the laser intensity by $I_r=1.37 \times 10^{18}$ W/cm$^2$, which is the intensity at which the quiver velocity of a single free electron approaches the speed of light $c$. Note that the Lorentz force is also proportional to the square of the laser electric field at normal incidence, so Eq. (4) and Eq. (5) also represent the longitudinal force exerted on the target by the two-color laser and the one-color laser, respectively. From Eq. (5), we can see that the longitudinal forces exerted by one-color CP laser on the target varies slowly, so the Doppler up-shift of the reflected laser is very weak [17]. However, by mixing the second harmonic, situation becomes totally different. Firstly, the participation of the second harmonic leads to quick oscillations of the Lorentz force (the second term in Eq. (4)) instead of a slowly-varying one in the one-color case, which implies that HHG can be triggered even at normal incidence. Secondly, the laser Lorentz force in the two-color case is $\left | 2a_{1} a_{2} \cos (\omega _0 \xi +\varphi ) \sin ^4(\pi \xi /\tau ) \right |$ larger than the corresponding value of the one-color case. Thus, under the same conditions, the plasma surface can oscillate more violently under the driving of the two-color CP laser so that HHG radiation can be excited more effectively.
3. Simulation results
For verification, two-dimensional (2D) particle-in-cell (PIC) simulations are carried out by using the EPOCH code [31]. The size of the simulation box is $10\lambda _0\times 16\lambda _0$, and there are 2500 $\times$ 4000 cells in the $x$ and $y$ directions, respectively. The laser is injected into the simulation box from the left boundary with the normalized electric amplitude of the one-color laser fixed at $a_0=15$, while the normalized electric amplitudes of the two beams in the two-color case are respectively: $a_1=15\sqrt {1-W}$, $a_2=15\sqrt {W}$. Here, $W$ represents the energy ratio of the second harmonic field in the total laser field. In both cases, the laser possesses a Gaussian spatial profile with a focus radius of $r=3\lambda _0$ and a temporal profile as $I{\rm _{one}}=I_{0} \sin ^2({\pi t}/{\tau })$ with a duration of $\tau =6T_{0}$. Here $\lambda _0=2\pi c / \omega _{0} =1\;\mathrm{\mu}$m is the laser wavelength, and $T_{0}=\lambda _{0}/c \approx 3.3$ fs is the laser fundamental period. A fully ionized plastic target is located at $x=8\lambda _0$ with a uniform initial electron density of $n_{e}=200n_{c}$ and a thickness of $\lambda _0/\sqrt {2}$, where $n_{c}=m_e \varepsilon _0 \omega _0^2 /q_e ^2$ is the critical density and $\varepsilon _0$ is the vacuum dielectric constant. The ions are taken to be immobile in the simulations.
Figure 2 shows the main simulation results when the laser normally irradiates on the target. The electron density along the optical axis ($y=0$) during the whole interaction process in the one-color case is shown in Fig. 2(a), while the two-color case ($W=0.7$ is chosen here) is shown in Fig. 2(b). The dotted black curves in the two figures represent the laser intensity envelopes. As analyzed before, the motion of the target surface is consistent with the laser intensity envelope in both cases. In the one-color case, since the target boundary is steadily pushed inward and outward, the Doppler blueshift of the reflected pulse is very weak, which can be confirmed from the spectrum of the reflected laser as shown in Fig. 2(c). However, through mixing the second harmonic, the target surface rapidly oscillates at the laser fundamental frequency as in the LP laser situation, with the maximum oscillating amplitude increased by $50\%$ (from 0.026 $\mathrm{\mu}$m in the one-color case to 0.038 $\mathrm{\mu}$m in the two-color case). Such quick and strong oscillations of the target surface lead to the significant Doppler blueshift of the reflected light, and the harmonic intensity is enhanced by four orders of magnitude, as shown in Fig. 2(c).
Fig. 2. The 2D simulation results at the normal incidence ($\theta =0^{\circ }$). (a) and (b) show the spatiotemporal distribution of the electron density $n_e$ on the $y=0$ axis in the one-color laser case and the two-color laser case, respectively. The dotted black curves in both figures represent the laser intensity envelope and the red lines indicate the maximum displacement of the target surface. (c) shows the spectra of the $p$-component of the reflected lasers in the one-color (black), two-color co-rotating (red) and counter-rotating (purple) CP laser cases, respectively. The energy ratio $W=0.7$ in both two-color cases. The helicities of some harmonics in the counter-rotating case are marked by purple dashed lines with rotating arrows, and the direction of rotation is defined as seen by a wave receiver. The spectra of the $s$-component is the same as the $p$-component in all three cases. (d) shows the conversion efficiency (black) and peak intensity (red) of pulses obtained by selecting 3th-10th harmonics for different energy ratio cases. Distinct enhancement is achieved in the two-color case and the optimum is reached when $W$ is around $0.6 \sim 0.7$.
According to the ROM mechanism, the HHG efficiency is positively correlated with the oscillating velocity of the target surface. Therefore, the most efficient HHG radiation can be realized through adjusting the parameters of the two beams. According to the Eq. (4), we conclude that when the energy ratio of the two beams is 1, corresponding to $W=0.5$, and the phase difference is $\varphi =0$, the peak intensity of the combined laser field reaches its maximum, and so is the target oscillation velocity. 2D PIC simulation results are shown in Fig. 2(d). Clearly, the HHG efficiency of the two-color case is significantly higher than that in one-color case (whether it is $\omega _0$ or $2\omega _0$ monochromatic field) and the maximum (about 0.76$\%$) is achieved when $W$ is around $0.6\sim 0.7$. Correspondingly, the peak intensity of the obtained pulses reaches $I{\rm _{max}}=1.1\times 10^{19}$ W/cm$^2$. The difference between the theoretical prediction and the simulation results about the optimal $W$ is due to the failure of the one-dimensional model caused by the target distortion under the driving of finite-focal-spot lasers (which is also the reason why HHG radiates under the normal incidence of the one-color CP laser), and the fact that the addition of the $2\omega _0$ beam can significantly broaden the HHG spectrum.
According to the selection rule proposed by R. Lichters et. al., the HHG inherits the polarization of the incident laser under normal incidence [17]. Since the combined fields in our scheme are still circularly-polarized, CP harmonics are expected. For verification, several harmonics are filtered for the $W=0.7$ case and their Lissajous figures are shown in Fig. 3. Here, $a_{p}$ and $a_{s}$ represent the normalized electric fields of the $p$-polarization and $s$-polarization components, respectively. Obviously, each harmonic generated at normal incidence is purely circularly polarized and rotates in the same direction as the incident laser, as showned in Fig. 3(a)-(c). Therefore, the combined pulses are also circularly polarized as shown in Fig. 3(d), which has not been reported in previous studies. For comparison, we also present the results for the counter-rotation driving laser with the same energy ratio. In this case, the combined field oscillates three times in one laser cycle (see the Lissajous figures in Fig. 1), thus more efficient harmonic radiation is excited, as shown in Fig. 2(c). However, as demonstrated by the previous study [27], the helicities of adjacent harmonics are opposite in this case (as marked by purple dashed lines with rotating arrows in Fig. 2(c)), resulting in elliptically-polarized attosecond bursts with the major axes rotating 120 $^{\circ }$ with respect to each other [Fig. 3(e)]. These features are the same as the HHG from gas medium [32,33].
Fig. 3. 2D simulation results for the characteristics of the harmonics driving by a two-color laser ($W=0.7$) at the normal incidence ($\theta =0^{\circ }$). (a-c) show the Lissajous figures of the 4th, 5th and 7th harmonic, respectively, in the co-rotating case. The helical directions are marked by rotating arrows. (d) and (e) are the waveforms of the gated pulses by filtering 3th-10th harmonics for the co-rotating (red) and counter-rotating (purple) laser cases, respectively. Here, $a_p$ and $a_s$ are the $p$-polarization and the $s$-polarization components of the combined pulses. The rotation direction is defined as seen by a wave receiver.
For CP lasers, when obliquely incident, the normal ($p$-polarization) component of the electric field can provide extra oscillating force [17], so that the HHG efficiency can be further increased. Figure 4 shows the results when the laser irradiates the target at an incident angle of $\theta =45^{\circ }$. From the spatiotemporal evolution of the electron density along the optical axis ($y=0$), we can find that the target surface oscillates in both cases. But due to the participation of the second harmonic, both the oscillation amplitude and frequency are higher in the two-color case ($W=0.7$), which implies a larger oscillating velocity and then stronger harmonic radiation. This prediction is confirmed by the spectra of the reflected lasers as shown in Fig. 4(c) (the spectrum of the $s$-component is similar to that of the $p$-component in both cases), and more intense HHG with efficiency of two orders of magnitude higher than the one-color case is achieved in the two-color case. By filtering 10th-30th harmonics, isolated attosecond pulses are obtained in both cases, as shown in Fig. 4(d) and 4(e), but in the two-color laser case the peak intensity of the attosecond pulse is two orders of magnitude larger and the pulse width is much narrower than that in the one-color case.
Fig. 4. The 2D simulation results for the incident angle of $\theta =45^{\circ }$. Spatiotemporal distribution of the electron density $n_e$ on axis $y=0$ for (a) the one-color laser case and (b) the two-color laser ($W=0.7$) cases. (c) shows the harmonic spectra in the specular direction at $t=21T_{0}$ for the two- (red) and one-color (black) laser case, respectively. By selecting 10th-30th harmonics, the spatial distributions of the attosecond pulses are shown in (d) [the one-color laser case] and (e) [the two-color laser case], where $\tau \rm {_{FWHM}}$ is the full width at half maximum of the attosecond pulses, and the normalized intensities $I{\rm _{atto}}$ along the optical axis (which is marked by the red dashed line in (d) and (e)) are shown in (f).
The longitudinal force exerts on the target surface under oblique incidence is different from that under normal incidence, so the optimal energy ratio of the two beams should also be different. A series of 2D PIC simulations is carried out to get the optimal configuration and results are displayed in Fig. 5 and Fig. 6. Firstly, we can see that the HHG efficiency is significantly improved with the increase of the laser incident angle, as showed in Fig. 5(a). For $\theta =15^{\circ }$, the addition of the second harmonic can lead to an increase of the conversion efficiency by a maximum of 3.2 times, while for $\theta =60^{\circ }$, the conversion efficiency can reach a maximum of 61.6 times that of the one-color case. Secondly, the optimal $W$ is around 0.8 for each incident angle, deviating very much from the theoretical optimal value $W=0.5$ for which the amplitude of the synthetic electric field reaches maximum. This result indicates that the spectra broadening introduced by the second harmonic plays a more important role in improving the HHG efficiency and intensities of attosecond pulses under oblique incidence conditions.
Fig. 5. 2D Simulation results for different energy ratios $W$ and incident angles $\theta$. (a) and (b) show the dependencies of, respectively, conversion efficiency and peak intensity of attosecond pulses on the incident angle and energy ratio $W$. Here, the attosecond pulses are obtained by filtering out harmonics with orders lower than 20.
Fig. 6. 2D Simulation results for the Lissajous figures of attosecond pulses generated by the two-color laser with energy ratio of (a) $W=0.8$, (b) $W=0.6$ and (c) $W=0.4$, and the incident angles of (I) $\theta =30^{\circ }$, (II) $\theta =45^{\circ }$, (III) $\theta =60^{\circ }$ and (IV) $\theta =75^{\circ }$, respectively. Here, $a_p$ and $a_s$ are the $p$-polarization and the $s$-polarization components of the attosecond pulses synthesized by 10th-30th harmonics, and $e$ is the corresponding ellipticity.
Besides, the effects of energy ratio and incident angle on the ellipticity of the attosecond pulses are also discussed in Fig. 6. By comparing each column, we can find that under the same incident angle, there is little difference in the attosecond pulse ellipticity for different $W$. However, through adjusting the incident angle, the ellipticity of the attosecond pulses can be greatly changed. Take $W=0.8$ as an example, the ellipticity of the attosecond pulse is 0.69 at $\theta =30^{\circ }$, and decreases to only about 0.11 at $\theta =75^{\circ }$. This is due to the different responses of the $p$- and $s$- components of the CP laser to the plasma at oblique incidence. Therefore, a trade-off between the HHG efficiency and the pulse ellipticity is required before determining the incident angle, and the CP attosecond pulses can be obtained only at normal incidence or at small angles.
Finally, as pointed out by previous researches, the addition of the second harmonic can increase the difference in the amplitude of the electric field between adjacent cycles [34]. Thus the two-color CP lasers in our scheme also provide a possibility for the generation of isolated attosecond pulses. The principle is similar to the amplitude gate proposed in the gas HHG, or the few-cycle laser scheme proposed in the plasma HHG, that is, when the peak of the electric field coincides with the peak of the laser envelope, an isolated attosecond pulse can be obtained through reasonable filtering [35]. In fact, an isolated attosecond pulse has been obtained for $W=0.7$ and $\theta =45^{\circ }$, as shown in Fig. 4. Here, we display two isolated attosecond pulses generated under two different sets of parameters. For example, when $W=0.5$ and $\theta =30^{\circ }$, by synthesizing the 10th-30th harmonics, an attosecond pulse with the intensity of $1.0\times 10^{19}$ W/cm$^{2}$ and duration of about 316 as can be generated, as shown in Fig. 7. However, we point out that like other schemes, the isolated attosecond pulses can only be generated when the driving laser duration is not too large. Otherwise, the laser amplitude of adjacent cycles changes slowly, resulting in the production of an attosecond pulse train.
Fig. 7. Attosecond pulses generated for (a) $\theta =30^{\circ }$ and $W=0.5$ (by synthesizing 10th-30th harmonics) (c) $\theta =45^{\circ }$ and $W=0.3$ (by synthesizing 19th-30th harmonics). The spatial distributions of the attosecond pulses along typical axes (which are marked by the red dashed lines in (b) and (d), respectively) are shown in the purple line.
4. Conclusion
In conclusion, a novel efficient scheme for generating CP/ EP attosecond pulse has been proposed, which is based on the interaction between a two-color CP laser with a plasma target. 2D PIC simulations reveal that the participation of the co-rotating second harmonic can dramatically increase the HHG efficiency by at least two orders of magnitude, and the maximum is reached when the energy ratio of the second harmonic is about $0.7 \sim 0.8$. Through adjusting the incident angle, the HHG efficiency can be further increased but at the cost of the ellipticity decrease of the attosecond pulse. An isolated attosecond pulse can be gated by adjusting the energy ratio of the two beams and the incident angle. Such intense chiral attosecond pulses would be powerful tools to investigate materials’ structural and magnetic properties at attosecond resolution.
Funding
National Key Research and Development Program of China (2018YFA0404802); National Natural Science Foundation of China (11875319, 12004433, 12135009); Natural Science Foundation of Hunan Province (2020JJ5649, 2021JJ40657); Research Project of NUDT (ZK19-12, ZK20-36); Science and Technology Innovation Program of Hunan Province (2020RC4020).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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