1. Introduction

Attosecond pulses are unique tools for detecting and controlling ultrafast microscopic phenomena due to their unprecedented spatial and temporal resolution [1], with applications including ultrafast biomedical imaging [2], probing processes in atoms and molecules [3–6], and diagnostics in inertial confinement fusion experiments [7,8]. However, clear real-time imaging and/or precise control of ultrafast phenomena necessitates single or isolated intense attosecond pulses [9–11], which greatly stimulates many efforts being devoted in this area. To date, many methods have been proposed for the generation of such short pulses, most of which have been developed to achieve linearly-polarized (LP) single-shot or isolated attosecond bursts [12]. In fact, circularly-polarized (CP) single or isolated attosecond pulses have unique characters which are required by many potential applications, e.g., interacting with matter to recognize molecular chirality [13], study the magnetic property [14–16], and probe spin dynamics [17], etc. Therefore, it is of vital importance to generate single or isolated CP attosecond pulses, which has received less attention previously.

High-order harmonics generation (HHG) from the nonlinear laser-matter interaction has been proved to be an efficient table-top source of attosecond pulses. At present, the shortest isolated CP pulse reported experimentally is around 150 attoseconds (as), which is produced by HHG from the noble gas [18]. However, due to the low conversion efficiency and the requirement of laser intensity below the ionization threshold of gases, attosecond pulses from gaseous harmonics are too weak to induce a measurable nonlinear effect in the medium, seriously restricting its applications, especially in the attosecond pump-probe technology [19]. Such a limitation on the laser intensity does not exist for the completely ionized plasma, and extensive studies by both simulations and experiments have revealed that the HHG efficiency in plasma is several orders of magnitude higher than that from gases, making it an excellent alternative for the generation of intense attosecond pulses [20]. Unfortunately, the HHG efficiency drops dramatically with the increase of the laser ellipticity for normal incident geometry, and there are barely any harmonics generation when driven by a CP laser [21]. This is because the fast oscillations of the plasma surface at nearly the speed of light, which is the main reason for triggering HHG radiation from plasma by introducing the Doppler upshift of the reflected laser frequency, do not occur when driven by the CP lasers.

A breakthrough was achieved in 2016 through obliquely impinging a laser of a suitable ellipticity onto an overdense plasma target [22,23], but the matching condition of the incident angle and the ellipticity of the driving laser is not clear. Recently, several methods based on two laser beams are proposed, including using two-color $\it {s}$-polarized laser [24], two counter-propagating lasers with orthogonal polarization [25] and two co-propagating co-rotating bichromatic CP lasers [26]. In practice, the former two approaches are very sensitive to parametric conditions and the latter one requires the normal incidence geometry at which the HHG efficiency is, however, relatively low. A single CP attosecond pulses can be isolated through all the above mentioned methods by employing few-cycle lasers and choosing appropriate filtering frequency ranges. By contrast, a single attosecond pulse can be achieved by blocking relatively arbitrary low-frequency harmonics if a one-time quick oscillation of the target surface can be realized. This phenomenon can occur in the interaction between a few-cycle CP laser and an ultrathin foil [27] or a relativistically transparent target [28]. However, both methods require a trade-off between the radiation efficiency and ellipticity of the attosecond pulses. Specifically, the target surface rebounds faster when driven by a shorter CP laser pulse, but the larger energy difference of the $\it {s-}$ and $\it {p-}$ components of the short laser falling edge due to the $\pi /2$ phase difference leads to a smaller ellipticity of the resultant attosecond pulse. While driven by a laser pulse with multiple cycles, the rebound speed of the target surface is too slow, which not only leads to the low HHG efficiency, but also results in small ellipticity of the resultant attosecond pulse due to the short-time compression. Therefore, all the above methods are only effective when few-cycle laser pulses are utilized.

In order to overcome the limitation on the pulse duration while maintaining the high ellipticity of single attosecond pulses, a novel method is proposed and investigated in this paper, as schematically shown in Fig. 1(a), which employs a multi-cycle CP laser pulse with a sharp falling edge followed by a weak tail to interact with a near-critical density plasma target. Such unique laser pulses can be synthesized by two CP laser pulses that have the same helicities but different frequencies and pulse widths. Under the matching conditions, which are theoretically derived and numerically verified, the resultant driving laser is modulated into two short pulses through the beat effect: the first one can excite a strong one-time oscillation of the target surface, and the second weak tail can provide enough cycles so that the energy to be compressed by the rebounding target surface in the two polarization directions is as equal as possible. Therefore, a single CP attosecond pulse can be robustly produced via our method. Furthermore, the loose restriction on the pulse duration makes our method a new approach for the generation of single CP attosecond pulse in the current laboratories worldwide, which will greatly facilitate related studies, such as ultrafast dynamics in chiral systems and magnetic materials.

 

Fig. 1. (a) Schematic of a single CP attosecond pulse generation from the interaction between a two-color circularly-polarized laser and near-critical density plasma target. (b) The dependencies of, respectively, conversion efficiency (red) and ellipticity (purple) of attosecond pulses (synthesized by 10th-30th harmonics) on the pulse width for the one-color lasers. (c) Two situations when the matching condition is not satisfied and the details are described in the Theoretical analysis part.

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2. Theoretical analysis

For clarity, all the physical quantities associated with longer and shorter components of the driving laser are distinguished by subscripts 1 and 2. The driving pulse adopted in our method propagates along the $x$ direction and is achieved by the superposition of the following two CP components:

Then the Lorentz force $\boldsymbol {v} \times \boldsymbol {B} \sim \boldsymbol {E}^2$ exerted on the target when the laser is normally incident can be obtained by squaring the laser electric field, and its normalized value equals the laser intensity normalized by $I_r\lambda _1^2=1.37\times 10^{18}$ W/cm$^2\cdot\;\mathrm{\mu}$m$^2$, thus it can be expressed by:

For the sake of simplicity, $x=0$ is taken here. We can find that the beat term $I {\rm _{beat}}$ introduced by the time delay, initial phase difference, and frequency difference of the two beams plays a crucial role in determining the resultant laser waveform. Assuming $\omega _1=\omega _2$, then the cosine term is a constant. For positive cosine term, the resultant Lorentz force increases, but the effect is equivalent to employing an intenser driving laser. On the contrary, a negative cosine term leads to a reduction of the Lorentz force, resulting in a waste of energy. While for the case with, $\omega _1\neq \omega _2$, the sign of the beat term may change over time. This provides a chance to modulate the resultant pulse into two short pulses, where the main pulse with a higher intensity and steeper falling edge is expected to excite a violent oscillation of the targe surface, and the weak tail pulse is expected to provide sufficient cycles to guarentee the high ellipticity of the attosecond pulse without leading to addtional significant oscillation of the surface. As mentioned above, the latter is barely realized for one-color laser cases as shown in Fig. 1(b).

In order to achieve the desired waveform, some conditions need to be met. First of all, the main pulse, whose width is determined by the beat frequency, should be shorter than the initial sub-pulses, that is $|\pi /(\omega _2-\omega _1)| < \tau _2$. Otherwise, as shown in Fig. 1(c), the superposed field (solid brown line shows the intensity envelope) can be longer than the shorter driving component (represented by dashed brown line), and the effect is also equivalent to utilizing an intenser multi-cycle driving laser. However, if the beat frequency is too high, multiple sub-pulses appear in the resultant field, as shown by the red line in Fig. 1(c), leading to multiple attosecond bursts (see the blue line and our previous research [26]). To realize single attosecond pulses production, only two sub-pulses are allowed in the overlapping region of the two beams. The number of sub-pulse pairs is determined by the pulse width of the shorter component and beat period. Specifically, when the pulse width of the short pulse is equal to twice the beat period, $\tau _2=2|\pi /(\omega _2-\omega _1)|$, one sub-pulse pair is generated. In principle, if two pairs appear in the resultant field, i.e., $\tau _2=4|\pi /(\omega _2-\omega _1)|$, more than one attosecond pulses would be produced. However, we find from extensive simulation studies, single or isolated attosecond pulses can still be generated by carefully adjusting the parameters of the laser beams when $\tau _2=4|\pi /(\omega _2-\omega _1)|$. Still when $\tau _2 \ge 6|\pi /(\omega _2-\omega _1)|$, multiple attosecond pulses are generated at any parameter conditions. Then the beat period should be larger than $\tau _2/6$. To sum up, the first condition that needs to be met to get a single CP attosecond pulse is:

In addition to the waveform, laser intensity also affects the radiation efficiency of the harmonics. To take full advantage of the two beams to excite the strongest HHG radiation, the peaks of the two beams should be added when the combined field reaches its peak, which means that the phase difference between the two beams should be an integer multiple of $2\pi$ at the corresponding time:

Here, $K$ is an integer, $t_{peak}$ is the time when the laser peak irradiates the target surface, and its magnitude depends on the energy ratio of the two beams. In particular, if we use W to denote the energy ratio of the shorter component to the total beam, then $t_{peak}\approx \tau _2/2{+\Delta d}$ when $W\rightarrow 1$, while $t_{peak}\approx \tau _1/2$ when $W\rightarrow 0$.

3. Simulation results

In order to demonstrate the feasibility of the method mentioned, one-dimensional (1D) particle-in-cell (PIC) simulations are first carried out by using the open source EPOCH code [29]. The length of the simulation box is $14\lambda _1$ and is divided into 14000 grids in total. The number of quasiparticles placed in each grid for calculating electrons and ions is 200 and 64, respectively. A fully ionized near-critical density plasma target is located between $x=12\lambda _1$ and $x=13\lambda _1$, and the initial electron number density is $n_e=8n_c$, where $n_c=m_e\varepsilon _0\omega _1^2/q_e^2$ is the critical density and $\varepsilon _0$ is the vacuum dielectric constant. The ions are mobile in all simulations. As an example, two laser beams with sinusoidal temporal profile and wavelengths $\lambda _1=1~\mathrm{\mu}$m and $\lambda _2=0.8~\mathrm{\mu}$m are used, and their pulse widths are $\tau _1=10T_1$ and $\tau _2=6T_2=4.8T_1$ with $T=\lambda /c$. The energy of the shorter sub-pulse accounts for $W=0.84$ of the overall energy and is incident into the simulation box with a delay time of $\Delta d=1.5T_1$ relative to the longer one. For both sub-pulses, the initial phase $\varphi =0$. The resultant laser waveform is shown in the schematic diagram Fig. 1(a). For comparison, two one-color lasers with the same total energy but respective wavelengths $\lambda _1$ and $\lambda _2$ and pulse widths of $\tau _1$ and $\tau _2$ are utilized in case A and case B, respectively, while the two-color laser is utilized in case C. The amplitude of the normalized electric field of the one-color pulse is $a_1=15\sqrt {2}$ for case A, $a_2=a_1\times \sqrt {\tau _1/\tau _2}$ for case B, and $a=a_1\times \sqrt {1-W}+a_2\times \sqrt {W}$ for case C.

The electron density evolutions in the whole interaction process of the three cases are respectively shown in Fig. 2(I). In all cases, it can be seen that the target surface is continuously and steadily pushed inward by the laser Lorentz force and then bounces back, performing only a one-time oscillation. In each case, a typical electron whose trajectory exactly matches the target surface can be found, as shown in the yellow circles of Fig. 2(I). The electrostatic forces $F_s=q_eE_x$ and the Lorentz forces $F_l=q_e{(\textbf {v} \times \textbf {B})}_x$ felt by typical electrons throughout the whole interaction process are represented by red lines and blue lines in Fig. 2(II), respectively. Both forces are normalized by $(m_e\omega _1 c)^{-1}$. In the push-forward stage when the laser rising edge acts on the target, a strong electrostatic field is developed due to the charge separation, which constitutes the restoring force against the laser Lorentz force. Under the combined action of the two forces, the longitudinal velocity $\beta _x=v_x/c$ of the typical electron (shown by purple lines in the Fig. 2(II)) first increases, and then decreases to 0 when the target surface is compressed to the limit. From the electron density distributions at the corresponding time which are shown respectively in the insets in Fig. 2(I), it’s clearly to see that the maximum compression lengths in the three cases are respectively $0.58~\mathrm{\mu}$m (for case A), $0.69~\mathrm{\mu}$m (for case B), and $0.78~\mathrm{\mu}$m (for case C). The deeper compression of our method over the other two one-color laser cases is due to the amplification of the laser peak intensity after the superposition of the two-color circularly polarized lasers [26], which in turn produces a maximum electrostatic field $E_{x,max} \approx 36.87E_r$, an increase of at least $10\%$ compared to the one-color laser cases.

 

Fig. 2. Simulation results of (a) case A, (b) case B, and (c) case C. (I) shows the spatiotemporal distributions of the electron density $n_e$, where positions of the maximum compression of the target surface are indicated by red arrows and the electron density distributions at the corresponding time are shown in the insets. Here, the yellow circles represent the trajectories of the typical electrons. (II) shows the time-varying images of the longitudinal force $F_x$ including the normalized electrostatic force $F_s$ (red) and the Lorentz force $F_l$ (blue), and the longitudinal velocity $\beta _x=v_x/c$ (purple) of the typical electrons, respectively. (III) shows the relativistic factor $\gamma _x=1/\sqrt {1-\beta _x}$ at the HHG radiation moment.

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In the pull-back stage when the Lorentz force decreases on the laser falling edge, the accumulated electrostatic energy is converted into the electron kinetic energy, as the electron longitudinal velocities $v_x$ in the three cases increases negatively to near the speed of light $c$. The release time of electrostatic energy is inversely proportional to the length of the laser falling edge, that is, in the $\lambda _1$ case, the release time is longest, about $2T_1$, while in the other two situations, the release time is less than $T_1$. Therefore, a rapid compression of the laser by the target surface is expected in the last two cases. The $\gamma _x$ distributions of electrons at the HHG radiation moment are shown in the Fig. 2(III). Here, $\gamma _x$ is defined by $\gamma _x=(\sqrt {1-\beta _x^2})^{-1}$, which determines the harmonic cutoff according to the BGP theory [30]. Obviously, the maximum $\gamma _x$ in the one-color laser cases are 2.19 and 5.25 respectively, while in the two-color laser case, the value of the maximum $\gamma _x$ can reach 20.77, which promises a more efficient and violent HHG radiation.

Figure 3 displays the characteristics of both reflected lasers and resultant attosecond pulses in detail. Notice that the spatial distributions of the fields here are diagnosed at $t=24T_1$ when the lasers travel in vacuum after being reflected from the plasma target, which are indeed consistent with the temporal distributions of those that are diagnosed at $x=\lambda _1$. As expected, in all cases, the Doppler effect causes the rising edge of the laser to be stretched due to the spectrum redshift and the falling part to be compressed as a result of spectrum blueshift. The laser distortion after the quick compression by the target surface is particularly pronounced for the last two cases as shown in Fig. 3(I). The spectra of reflected lasers are depicted in Fig. 3(II). In all cases, a supercontinuous rather than discrete spectral structure is formed, implying the occurrence of a single XUV burst. Figure 3(III) shows the single attosecond pulses generation after a filter that transmits radiation with frequency $10\omega _1\sim 30\omega _1$ and the Lissajous figures are correspondingly shown in Fig. 3(IV). It is evident that our method does indeed result in a single attosecond pulse with larger ellipticity (about 0.9) and higher amplitude. In contrast, the ellipticities of the attosecond pulses are very low in one-color laser cases due to insufficient compression as shown in Fig. 3(a-IV) and Fig. 3(b-IV). Here, the reasons for inadequate compression in the one-color laser cases are different. As mentioned above and verified in the simulation, the peak amplitude of the long pulse is smaller under the condition of the same energy, resulting in a small rebound velocity of the target surface. Furthermore, the target surface is rapidly decelerated by the slowly decreasing falling edge, thus leading to a very short effective compression. While for the short pulse case, the target surface bounces off quickly, but the short falling edge leads to an early termination of the compression process. Moreover, the slight energy difference between the two polarization directions at the steep short falling edge leads to a reduction of the ellipticity of the final attosecond pulse. Therefore, for the one-color laser cases, high-intensity multi-cycle lasers are required to get a single attosecond pulse with high ellipticity, which means a large pulse energy is required. Through the superposition of two-color pulses, not only can we construct a main pulse with a higher intensity and a shorter pulse width than the initial sub-pulses to excite a one-time abrupt oscillation of the target surface, but also a weak tail pulse to provide as much as possible the same energy in both polarization directions to guarantee the high ellipticity of the attosecond pulse. In order to clearly show the influence of the tail pulse on the ellipticity of the attosecond pulse, the Lissajous Figure of the attosecond pulse producecd under the driving of the main pulse is displayed in Fig. 3(c-IV). Clearly, without the tail pulse, the ellipticity of the attosecond pulse is reduced from 0.9 (black line) to 0.6 (purple dotted line). Thus, our method effectively overcomes the inherent problem existing in one-color laser cases and provides an efficient way to trigger the radiation of single highly elliptical attosecond pulses by a multi-cycle pulse with relatively low energy.

 

Fig. 3. Characteristics of the reflected lasers and the produced attosecond pulses in (a) case A, (b) case B, and (c) case C. (I) Spatial distributions of the reflected lasers, and the corresponding spectra are shown respectively in (II). (III) is the normalized intensities of the attosecond pulses (filtered from $10-30\omega _1$), and the corresponding Lissajous figures are shown in (IV). The purple dotted line in (c-IV) is the result when case C lacks of tail pulse. Here, $e$ is the corresponding ellipticity.

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In order to verify the effectiveness of our mechanism, a series of 1D simulations are carried out and the results are displayed in Fig. 4 and Fig. 5. First and foremost, we can check the optimal matching conditions of the delay time $\Delta d$ and energy ratio $W$, which is given in the Theoretical analysis part, as shown in Fig. 4. Compared with the laser as discussed above, the laser used in Figs. 4(a)-(c) only changes $\Delta d$ and $W$, while for Figs. 4(d)-(f), longer initial pulses are used with $\tau _1=18T_1$ and $\tau _2=12T_2\approx 11T_1$, and the ellipticity of attosecond pulses generated by one-color pulses of such durations are less than 0.3. To satisfy Eq. (5), the frequency of the short pulse is changed to $\omega _2=1.1\omega _1$. According to Eq. (6), to get the optimal $\Delta d$, $t_{peak}$ needs to be determined first. Since $t_{peak}$ represents the moment when the laser electric field reaches its peak, which is determined by the linear superposition of the two sub-pulses, the theoretical optimal $\Delta d$ for any $W$ can be obtained according to the two limit cases: $W=0$ and $W=1$, which are drawn in the figures with black dotted lines. It’s clear that the theoretical values are in good agreement with the simulation results in both cases. From Fig. 4(c) we can find that, in a wide parametric range, the ellipticities of single attosecond pulses produced by two-color fields is higher than 0.8. For longer pulse case, although the optimal parametric range is narrower than the shorter pulse case, still single attosecond pulses with ellipticity over 0.8 can be efficiently produced via our method, which is almost impossible for one-color laser cases.

 

Fig. 4. The dependencies of conversion efficiency, peak intensity and ellipticity of the attosecond pulses (filtered from $10-30\omega _1$) on the time delay $\Delta d$ and energy ratio $W$ in the cases of (a)-(c) $\omega _2/\omega _1=1.25$, $\tau _1=10T_1$, $\tau _2=6T_2$ and (d)-(f) $\omega _2/\omega _1=1.1$, $\tau _1=18T_1$, $\tau _2=12T_2$. Other parameters are the same as above. Here, the black dotted lines represent the match conditions calculated theoretically.

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Fig. 5. Simulation results in (a) $\omega _2/\omega _1=1.25$, $\tau _1=10T_1$, $\tau _2=6T_2$, $\Delta d=1.5T_1$, $W=0.84$ and (b) $\omega _2/\omega _1=1.25$, $\tau _1=18T_1$, $\tau _2=12T_2$, $\Delta d=4.1T_1$, $W=0.47$ (filtered from $10-30\omega _1$). (I) Spatial distributions of the incident lasers, and the generated single CP attosecond pulses are shown respectively in (II). Here, $a_p$ and $a_s$ are the $p$- and the $s$-polarization components of the normalized electric field, respectively. (c) Spatial distribution of the electron density. In both cases, the maximum electron density is $n_e=8n_c$ with a scale length of $L=0.1\lambda _1$.

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From a practical point of view, preplasma is unavoidable due to the laser pre-pulse. Therefore, the effect of the preplasma is also considered in our simulations. Figure 5 depicts the correspondingly simulation results for the same two lasers adopted in Fig. 4, where the energy ratio is $0.84$ in the shorter case and $0.47$ for the longer case, and an exponential preplasma with a density scale length of $L=0.1\lambda _1$ exists in front of the target. Apparently, in both cases, single elliptical attosecond pulses are still produced robustly with ellipticity of 0.9 and 0.7, respectively, proving that our mechanism is very practical and reliable.

Last but not least, two-dimensional (2D) particle-in-cell (PIC) simulations are also carried out to examine multi-dimensional effects of our method. The size of the simulation box is $20\lambda _1 \times 40\lambda _1$, and there are $20000\times 40000$ cells in the $x$ and $y$ directions, respectively. A target with a density of $n_0=8n_c$ is located in the region $16-18\lambda _1$ in the $x$ direction. A two-color driving laser with a Gaussian spatial profile is used here, and other parameters the same as in Fig. 2. Figure 6(a) shows the intensity distribution of the reflected laser, with the corresponding electric fields along the optical axis overlaid. Obviously, similar to the 1D results, the falling edge of the reflected pulse is also highly compressed into a sharp peak, thus a single CP attosecond pulse can be still obtained (see Fig. 6(b)). But the transverse distortion of the target surface during the interaction process leads to the deformation of the wavefront of the attosecond pulses, as clearly seen in Fig. 6(b). Besides, the peak amplitude of the attosecond pulse is slightly larger in 2D simulation than in 1D simulation due to the focusing effect of the target surface. Since the propagation direction near the central axis ($y=20\lambda _1$) is almost along the $x$ axis, the pulse spatial distribution and Lissajous Figure are displayed in Figs. 6(b) and (c), which show basically the same characteristics as in the 1D simulation, e.g., a duration of 300 as, a peak intensity of $1.1 \times 10^{20}$ W/cm$^2$, and an ellipticity of nearly 0.9, confirming the robustness of our method once again.

 

Fig. 6. 2D PIC simulation results. (a) Spatial distribution of the intensity of the reflected laser where the two orthogonal components $E_y$ (red) and $E_z$ (blue) along the axis are displayed. (b) Spatial distribution of the electric field $E_y$ of the obtained attosecond pulse (filtered from $10-30\omega _1$) with a duration of about 300 as, and the corresponding Lissajous curve is shown in (c).

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4. Conclusion

In conclusion, we propose a novel method for producing single intense CP attosecond pulses, where a near-critical density plasma target is irradiated by a superposition field of two-color co-rotating CP lasers with different pulse widths. When the matching condition is satisfied, the two sub-pulses form a shorter and intenser main pulse together with a weak tail pulse through the beat phenomenon effect. Therefore, one-time quick oscillation of the target surface is realized due to steepening of the laser falling edge, leading to a violent compression of the reflected laser and efficient radiation of the supercontinuum spectrum. Besides, the weak tail pulse provides enough cycles to be compressed by the rebounding target surface, preserving the high ellipticity of the synthesized attosecond pulses. Through a series of simulations across a wide parametric range, including multi-cycle lasers, targets with preplasma, and multi-dimensional simulations, the theoretical matching conditions are verified and the robustness of our method is confirmed. We expect our method could be validated experimentally, and the robust production of single intense CP attosecond pulses would undoubtedly facilitate research in the attosecond pump-probe experiments and the study of chirality-sensitive light-matter interactions.