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Abstract
This paper theoretically investigates the generation of isolated elliptically polarized attosecond pulses with a tunable ellipticity from the interaction of Cl2 molecule and a polarization-gating laser pulse. A three-dimensional calculation based on the time-dependent density functional theory is done. Two different methods are proposed for generating elliptically polarized single attosecond pulses. The first method is based on applying a single-color polarization gating laser and controlling the orientation angle of the Cl2 molecule with respect to the polarization direction of the laser at the gate window. An attosecond pulse with an ellipticity of 0.66 and a pulse duration of 275 as is achieved by tuning the molecule orientation angle to 40° in this method and superposing harmonics around the harmonic cutoff. The second method is based on irradiating an aligned Cl2 molecule with a two-color polarization gating laser. The ellipticity of the attosecond pulses obtained by this method can be controlled by adjusting the intensity ratio of the two colors. Employing an optimized intensity ratio and superposing harmonics around the harmonic cutoff would lead to the generation of an isolated, highly elliptically polarized attosecond pulse with an ellipticity of 0.92 and a pulse duration of 648 as.
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1. Introduction
The highly nonlinear process of high-harmonic generation (HHG) occurs when atoms or molecules interact with an intense laser pulse. HHG converts an infrared laser beam into extreme ultraviolet (XUV) radiation or soft X-rays, lasting only tens of attoseconds, through a three-step semiclassical model involving ionization, acceleration, and recombination [1–8]. Such a tabletop light source with coherent ultrashort pulses is a perfect candidate for capturing and controlling electron dynamics in atoms, molecules, and condensed matter [3,9–16]. Due to the numerous applications of circularly polarized (CP) or elliptically polarized (EP) pulses, such as the study of orbital momentum transfer [17], magnetization and spin dynamics [18–21], X-ray magnetic circular dichroism spectroscopy [22,23], and ultrafast chiral-specific dynamics of molecules [24], this topic has attracted much attention in recent years.
The electronic structure of the target and the shape of the driving laser field strongly influence the polarization state of the high-order harmonics and corresponding attosecond pulses. Indeed, manipulation of the driving laser pulse waveform significantly impacts the characteristics of the attosecond pulses generated by the HHG process. To date, various methods have been used to produce isolated attosecond pulses, such as ionization gating [25], amplitude gating [26], polarization gating (PG) [27–29], double optical gating (DOG) [30–32], attosecond lighthouse [33], few-cycle driving pulse [34], and two-color laser field [35]. In 1994, Corkum et al. [27] proposed the PG technique for the first time. An isolated attosecond pulse with a duration of 130 as was produced using this technique for the first time in an experiment in 2006 [36]. There have been many reports of single attosecond pulse generation by the PG method in atomic systems, but this method has not been widely used in molecular multi-electron systems.
In addition to limiting HHG bursts to a single event, CP or highly EP attosecond pulses have been generated using a time-delayed bicircular counter-rotating pulse. HHG resulting from the interaction of matter with bicircular counter-rotating laser pulse leads to the production of harmonics with alternating helicity. Due to the selection rules of angular momentum, the generation of 3N harmonic orders is forbidden, while 3N + 1 and 3N + 2 harmonics, are allowed. 3N + 1 harmonic orders have a similar helicity to the fundamental frequency laser pulse, whereas 3N + 2 harmonic orders have a similar helicity to the second-harmonic laser pulse. In this case, highly EP or even CP harmonics are produced, but the opposite helicity of the harmonics results in the generation of linearly polarized attosecond pulses. Several methods have been proposed to overcome this problem, such as changing the relative intensity [37–39] and relative phase [40,41] of the two laser pulses, optimizing conditions for helicity-dependent phase-matching while the HHG process occurs in hollow fibers [42], and using appropriate orbitals [43–46]. Hernández-García et al. [47] presented the production of isolated CP attosecond pulses in the interaction of a PG laser field with argon by performing single-atom simulations and solving the time-dependent Schrödinger equation (TDSE).
All of the theoretical studies regarding the generation of an isolated elliptically-polarized attosecond pulses have been based on simple models like TDSE and SAE, without considering multi-electron effects [43,47–49]. These models have been mostly successful in describing phenomena related to HHG and laser-matter interaction such as cooper minimum in molecules [50], and high harmonic interferometry of multi-electron dynamics in molecules [51]. However, sometime these simple models fail to predict and describe experimental observations. TDDFT approach that consider the multi-electron effects could be an effective method to address these limitations [52–56]. For this reason, we employ the TDDFT approach as a more comprehensive and reliable method for our calculations. To the best of our knowledge, few studies have been focused on the generation of isolated CP or highly EP attosecond pulses in multi-electron molecular systems.
In this work, we theoretically demonstrate the generation of an isolated nearly circularly polarized attosecond pulse by using single- and two-color PG pulses and by controlling the orientation angle of the molecule and the intensity ratio of driving lasers. The chlorine molecule has been chosen as the target for this study. The highest occupied orbital of Cl2 is a π orbital [57] and we found it to be more efficient for producing elliptically polarized attosecond pulses in comparison to atomic p orbitals of noble gasses such as Ar. Moreover, the orientation angle of the molecule adds a degree of freedom for controlling the ellipticity of the attosecond pulse and the orientation of the molecule can have an effective impact in the results. Today, the molecular orientation echoes have experimentally realized via concerted terahertz and near-IR excitations [58]. Calculations are performed in the framework of the TDDFT with considering multi-electron effects. We first study the effect of the time delay between two pulses of a single-color PG pulse and the carrier-envelope phase on the number of attosecond pulses, to find the optimized parameters for obtaining an isolated single attosecond pulse. An 800 nm laser is considered for all the single-color PG pulses. This goal is achieved only by applying a carrier-envelope phase of 90°, using a time delay in the range of 1.6 o.c. ≤ Td ≤ 2.2 o.c., and superposing harmonics in the region of harmonic cutoff. The polarization of attosecond pulses obtained by single-color-PG pulses is almost linear (e.g. Td = 1.75 o.c. leads to a 269 as pulse with an ellipticity of ε = 0.16). Then, we introduce two methods for increasing the ellipticity of output attosecond pulses: (1) adjusting the orientation angle of the Cl2 molecule with respect to the polarization direction of the single-color PG pulse (its polarization at gate window where is almost linear), and (2) employing a two-color PG pulse and controlling the intensity ratio of two colors for a parallel-aligned Cl2 molecule. The first method could lead to the generation of an elliptically polarized single attosecond pulse (with FWHM of 275 as and the ellipticity of ε = 0.66) by applying a single-color PG pulse and tuning the orientation angle to 40°. For the second method, we utilize a ω-2ω scheme for the two-color PG pulse with an 800 nm laser as the fundamental laser (ω1 = ω = 0.057 a.u. = 1.55 eV). One can obtain a 648 as pulse with a nearly circular polarization (ε = 0.92) by modifying the intensity ratio to I2ω / Iω = 2 for a parallel-aligned Cl2 molecule, which is a remarkable achievement.
This paper is organized as follows: The theoretical method and the driving PG pulses are presented in Sec. 2. In Sec. 3, we reveal our results in three subsections. In subsection 3.1, the essential conditions for generating an isolated attosecond pulse by a single-color PG are investigated. In subsection 3.2 we present the results of our first method for producing elliptically-polarized attosecond pulses based on the control of the orientation angle of the Cl2 molecule. The results regarding the generation of a nearly circularly-polarized attosecond pulse by a two-color PG pulse with the optimized intensities are described in subsection 3.3. Finally, conclusions drawn from our results are discussed in Sec. 4.
2. Theoretical methods
In the present paper, we study the interaction of a polarization-gated laser pulse with Cl2 molecule using dipole approximation. For the aim of this study, TDDFT method is applied by assuming a numerical grid in a 3D real space and real time. All equations are expressed in atomic units unless otherwise stated.
2.1 Polarization gating pulse
Polarization gating is a time-dependent ellipticity laser pulse that can be achieved from the combination of two counter-rotating circularly polarized pulses (E = E1 + E2) with a time delay between them [59], where E1 and E2 are the right- and left-circularly polarized fields, respectively. These circularly polarized fields are defined as:
We employ f(t)=sin2(πt/Tpulse) as the envelope function of the fields. Tpulse is the total pulse duration (not the FWHM) which is assumed to be five optical cycles (at 800 nm wavelength) for each pulse in all the calculations of this manuscript. Ii and ωi are the peak intensity and the central frequency of pulses, respectively. Td determines the time delay between two pulses, and φcep is the carrier-envelope phase. The polarization gating pulse defined above, has a time-dependent ellipticity which can be derived as [60]:
The ellipticity of a PG pulse is close to ±1 at the beginning and the end of the combined laser field and falls to zero at the middle part of the combined laser field where two counter-rotating pulses overlap. The time interval where the ellipticity value is less than a certain threshold value (usually ξ(t) < 0.2) is called the polarization gating width (δG).
2.2 Time-dependent Kohn-Sham equation
The time-dependent response of the Cl2 molecule under the influence of the PG laser is calculated by solving the time-dependent Kohn-Sham (TDKS) equation:
Vxc is the exchange-correlation (XC) potential, VH is the Hartree potential, and Vne is the potential caused by the Coulomb interaction between the electron and the nucleus. The external potential due to the laser field is given as Vext = −E.r. ψ(r,t) represents the TDKS wave function and the time-dependent electron density of a spin compensated N-electron system can be determined by
The dipole acceleration of the system is determined by Ehrenfest theorem [61]
Once the dipole acceleration is calculated, the HHG spectrum could be obtain from the Fourier transform of dipole acceleration (a(ω)) as follow:
The intensity of the output attosecond pulse produced by high-order harmonics between ω1 to ω2 could be calculated by an inverse Fourier transform of the spectrum as follows:
For a more detailed analysis of the HHG process, the time-frequency profile is calculated according to the following equation, which is known as the Gabor transform [62]:
The ellipticity of each harmonic can be calculated as follows:
Finally, the ellipticity of an attosecond pulse produced from superposing harmonics in a specific range could be calculated by averaging over ε(ω) in that frequency range.
3. Results and discussions
In all simulations, a spatial grid with a spacing of Δr = 0.35 a.u. is considered in a parallel-piped box with a size of 180 × 180 × 70 a.u. Since the laser polarization direction lies always in the xy-plane, larger dimensions are considered along the x and y direction. To prevent the reflection of liberated electrons from the boundaries of the simulation box, a mask function (with a width of 30 a.u. in the x and y directions, and a width of 15 a.u. in the z direction) is applied in the boundaries. The time step for propagating KS orbitals in time is assumed to be Δt = 0.1 a.u. = 9.1 × 10−4 optical cycle (o.c.). All units of o.c. used throughout this manuscript are for 800 nm laser wavelength (1 o.c. = 2.67 fs). The Cl2 molecule is aligned in the x direction. A self-interaction corrected local density approximation [63] is used for the XC potential. This is a proper potential which gives a reasonable value of ionization energy (11.42 eV) for the Cl2 molecule in comparison to the experimental value (11.48 eV [57]). The calculations have been converged with respect to all the effective parameters such as the size of the simulation box, grid spacing, and time step. We employ the OCTOPUS package to carry out the calculations [64].
3.1 Generation of isolated attosecond pulse using single-color polarization gating
As the starting point, we investigate the generation of an isolated attosecond pulse from the Cl2 molecule using a single-color polarization gating pulse. This pulse (Fig. 1) is produced by combining two counter-rotating circularly polarized with equal intensities of I1 = I2 = 1 × 1014 Wcm-2, equal central wavelengths of λ1 = λ2 = 800 nm, and equal pulse duration of Tpulse = 5 o.c. As can be seen from Fig. 1, the single-color PG pulse is circularly polarized at both the tailing and leading edges of the pulse, and linearly polarized in the central part. In the tailing and leading edges, the freed electron deviates from its path towards the parent ion due to the transverse component, which prevents the electron from recombining. However, in the central part of the laser pulse where the polarization is almost linear, electrons can recombine and generate attosecond pulses. The time interval where the ellipticity ξ(t) < 0.2, is defined as the polarization gate width (δG) in this manuscript.
Fig. 1. The electric field of the single-color polarization gating pulse in (a) three dimensions and (b) two dimensions with I1 = I2 = 1 × 1014 Wcm-2, wavelengths of λ1 = λ2 = 800 nm, Td = 1.75 o.c., and φcep = 0. Black dashed curve in (b) denotes the time-dependent ellipticity of the pulse (ξ(t)). The green shaded area shows the time interval where the ellipticity is less than 0.2 and the pulse is almost linear. The polarization gate width for this pulse is δG = 0.52 o.c.
The polarization gate width is critical for confining the generation of attosecond pulses to an isolated attosecond pulse. Each three-step process takes place in about 0.75 o.c. Therefore, to have only one major HHG process and to create an isolated attosecond pulse, the gate width should be less than 0.75 o.c. Since the gate width could be controlled by tuning the time delay between counter-rotating pulses (Td), the effect of time delay on HHG is studied in the following. In Fig. 2, the time-frequency profiles of HHG obtained by different time delays are presented. The upper and lower panels correspond to φcep = 0 and φcep = 90°, respectively. The Cl2 molecule is assumed to be aligned parallel to the polarization of the PG pulse (the direction of polarization at the gate window). This orientation where the angle between the molecular axis and the laser polarization (the orientation angle) is zero, is called as the “parallel-aligned” case in this manuscript. One can see that by increasing the time delay, the gate width decreases and the number of HHG processes is reduced. However, only by employing φcep = 90° and choosing a time delay in the range of Td ≥ 1.6 o.c. (δG ≤ 0.58 o.c.), HHG occurs one time and a single attosecond pulse will be expected to be observed. It should be noted that increasing the time delay in the range of Td = 1.6-2 o.c. leads to weaker recombination and emission and no HHG signal will be observed for time delays larger than 2.2 o.c. Therefore, we choose the time delay of Td = 1.75 o.c. (where we have almost a single clean HHG burst with high enough intensity) to examine the possibility of producing a single attosecond pulse in the following.
Fig. 2. The time-frequency profiles of HHG from a parallel-aligned Cl2 molecule driven by a single-color polarization gating pulses with different delay times, φcep = 0 (uper panel) and φcep = 90° (lower panel). Right- and left-circularly polarized pulses are assumed to have equal intensities I1 = I2 = 1 × 1014 Wcm-2 and equal wavelengths of λ1 = λ2 = 800 nm for all cases. The width of the Gaussian window function in the Gabor transform is σ = 0.03 o.c.
Figure 3 shows the HHG spectrum and the output attosecond pulse related to the time-frequency profile presented in Fig. 2(i). As expected, by employing Td = 1.75 o.c. and φcep = 90°, and superposing harmonics around the cutoff harmonic order (H36-H50) a single attosecond pulse with a FWHM of 269 as and an ellipticity of ε = 0.16 is generated.
Fig. 3. (a) The HHG spectrum from interaction of a parallel-aligned Cl2 molecule and a single-color PG laser pulse with I1 = I2 = 1 × 1014 Wcm-2, λ1 = λ2 = 800 nm, Td = 1.75 o.c., and φcep = 90°. (b) The attosecond pulse is generated from the superposition of harmonics from H36 to H50 (dashed vertical lines in (a)).
The attosecond pulse generated in Fig. 3 has an ellipticity value of 0.16. One can conclude that that irradiating a parallel-aligned molecule with a single-color PG pulse could only lead to production of attosecond pulses with a linear polarization. To overcome this limitation, we will present two alternative methods for generating isolated attosecond pulses with an elliptical polarization. The first method involves utilizing a single-color PG pulse while controlling the angle between the laser polarization and the molecular axis. In the second method, a two-color PG pulse is applied while the intensity of the second color is tuned to optimize the ellipticity of the attosecond pulse.
3.2 Generation of elliptically polarized pulses using polarization gating and control of laser-molecule alignment angle
This section investigates how the orientation of the Cl2 molecule affects the ellipticity of the output attosecond pulses. The orientation angle is defined as the angle between the molecular axis and the polarization direction of PG pulse at the gate window (green shaded area in Fig. 1(b) where the PG pulse is almost linear). The impact of different orientation angles on the resulting HHG spectrum and attosecond pulse is illustrated in Fig. 4.
Fig. 4. (a) The HHG spectra from a Cl2 molecule with different orientation angles. A single-color PG pulse with I1 = I2 = 1 × 1014 Wcm-2, λ1 = λ2 = 800 nm, Td = 1.75 o.c., φcep = 90° is assumed as the driving laser. (b) The attosecond pulses due to different orientation angles of the molecule which all are generated from the superposition of harmonics from H34 to H50. (c) The variation of the attosecond pulse ellipticity with respect to the orientation angle. (d) The variation of the peak intensity and FWHM of the attosecond pulses with respect to the orientation angle.
Figure 4 demonstrates that adjusting the orientation of the molecule leads to changes in both the cut-off harmonics and the intensity of harmonics within the cut-off region. A fixed range of harmonics (H34-H50) are superposed to obtain isolated attosecond pulses and compare for different orientation angles as shown in Fig. 4(b).
Setting the orientation angle to 40° results in a dramatic increase in the ellipticity of the attosecond output pulse (Fig. 4(c)), reaching values as high as ε = 0.66, which is a significant result. Also, Fig. 4(d) reveals that the attosecond pulse obtained by the orientation angle of 40° has a pulse duration of 275 as which is about 150 as shorter than parallel-aligned case (zero orientation angle). Moreover, the orientation angle of 20° produces the most intense output attosecond pulse.
3.3 Generation of isolated elliptically polarized attosecond pulse using two-color polarization gating
In this section, we have applied a two-color polarization gating pulse as the driving laser to investigate the possibility of generating EP attosecond pulses. For this purpose, a right-circularly polarized laser with the wavelength of 800 nm (the fundamental pulse) is combined with a left-circularly polarized laser with the wavelength of 400 nm (ω2 = 2ω1). Without considering any delay time between the pulses, the combined pulse will be a bicircular laser pulse with a frequency ratio of 1:2 and a three-fold symmetry through for all optical cycles of the pulse. With increasing the time delay, this three-fold symmetry will be limited to the gate window (the middle part of the combined pulse where two pulses overlap). For the few-cycle lasers which we have used in this paper, one cannot see a perfect threefold symmetry even in the gate window (see Fig. 5(a)). The red petal highlighted in Fig. 5(a) shows the part of driving laser ellipse which fits in the gate window (where ξ(t) < 0.2) and is responsible for the generation of harmonics around the cutoff frequency. The ellipticity of this petal can be controlled by changing the ratio of intensities between two pulses γ = I2 / I1 = I2ω / Iω. Before studying the effect of γ, we first present the results obtained by a two-color PG pulse with γ = 1. Figure 5(a) shows the driving two-color PG pulse with φcep = 0, γ = 1, and Td = 1.25 o.c. (with optical cycle period of the fundamental pulse with the wavelength of 800 nm). The delay time of 1.25 o.c. is selected to observe the HHG process only one time at the gate window and to achieve a single attosecond pulse at the output. Figure 5(b) demonstrates the resulting HHG spectrum, and the vertical dashed lines denote the range of harmonics superposed for producing the attosecond pulse (H22-H30). An isolated EP attosecond pulse with an ellipticity of 0.7 is generated. The temporal profile of attosecond pulse and the shape of electric field in the xy-plane are shown in Figs. 5(c, d). This attosecond pulse has an ellipticity of ε = 0.73 and a FWHM = 733 as.
Fig. 5. (a) The ellipse of the driving two-color PG pulse with λ1 = 2λ2 = 800 nm, Td = 1.25 o.c., φcep = 0, and γ = 1 (I1 = I2 = 1 × 1014 Wcm-2). The red highlighted part fits in the gate window and is responsible for the generation of harmonics around the cutoff. (b)The HHG spectrum from a parallel-aligned Cl2 molecule. (c) The time profile of the output attosecond pulse produced by superposing harmonics from H22 to H30 (dashed lines in (b)). (d) The electric field ellipse of the output attosecond pulse. Units of harmonic order and o.c. are based on the fundamental pulse with the wavelength of 800 nm.
In the following, the effect of molecule orientation angle on the ellipticity of attosecond pulses is studied. Figure 6 displays the variation of the ellipticity, peak intensity, and pulse duration of the produced attosecond pulses with respect to the orientation angle of the Cl2 molecule. As can be seen, a parallel-aligned molecule could emit an attosecond pulse with the maximum ellipticity and maximum peak intensity. Moreover, the pulse duration of the attosecond pulses for the parallel-aligned case is a bit longer (∼150 as) than the minimum value which is observed for the orientation angles of 10° and 30°. Consequently, we can conclude that the parallel alignment is the optimized case for producing elliptically polarized attosecond pulses by two-color PG pulses.
Fig. 6. (a) The variation of the attosecond pulse ellipticity with respect to the orientation angle of the Cl2 molecule. (b) The variation of the peak intensity and pulse duration of the attosecond pulse with respect to the orientation angle. A two-color PG pulse with λ1 = 2λ2 = 800 nm, I1 = I2 = 1 × 1014 Wcm-2, φcep = 0, and Td = 1.25 o.c. is considered as the driving laser. Harmonics from H22 to H30 (based on the fundamental frequency ω1) are superposed for calculating the time profile of attosecond pulses.
The ellipticity of the high-order harmonics and corresponding attosecond pulses could be increased by increasing the driving laser ellipticity. On the other hand, the ellipticity of a driving two-color PG pulse can be controlled by tuning the intensity ratio γ. Therefore, we can manipulate and enhance the ellipticity of output attosecond pulses with controlling γ. The effect of γ on the ellipticity of output attosecond pulses generated from the interaction of a parallel-aligned Cl2 molecule with two-color PG pulses is illustrated in Figs. 7(a, b). The intensity ratio γ is altered by changing I2 and keeping I1 fixed at 1 × 1014 Wcm-2. To keep the ionization probability of the Cl2 molecule below 1%, the upper limit of the ratio of intensities is set to γ = 3. One can see that using the optimized value of γ = 2 leads to a highly elliptical attosecond pulse with an ellipticity of ε = 0.92 (almost a circularly polarized pulse). Figure 7(b) shows the variation of peak intensity and pulse duration of attosecond pulse with respect to the intensity ratio γ. Increasing the intensity ratio from γ = 0.5 to γ = 3 leads to an overall increase of the attosecond pulse intensity and oscillation of the pulse duration in the range of 500-900 as.
Fig. 7. (a) The variation of the attosecond pulse ellipticity with respect to the ratio of driving pulse intensities γ. (b) The variation of the peak intensity and pulse duration of the attosecond pulse with respect to γ. A parallel-aligned Cl2 molecule and a two-color PG pulse with λ1 = 2λ2 = 800 nm, fixed I1 = 1 × 1014 Wcm-2, φcep = 0, and Td = 1.25 o.c. is considered. For producing attosecond pulses, a proper range of harmonics around the cutoff frequency are superposed in which a maximum ellipticity is achieved for each case.
Finally, results of the optimized intensity ratio (γ = 2) are presented in Fig. 8. This figure demonstrates the possibility of producing a circularly polarized attosecond pulse from a parallel-aligned Cl2 molecule shined by an optimized two-color PG pulse. To produce such an attosecond pulse, proper harmonics around the cutoff frequency (H36-H50 in Fig. 8(b)) should be superposed. This attosecond pulse has an ellipticity of ε = 0.92 and a pulse duration of 648 as shown in Figs. 8(c, d) which is a significant achievement.
Fig. 8. (a) The ellipse of the driving two-color PG pulse with λ1 = 2λ2 = 800 nm, fixed I1 = 1 × 1014 Wcm-2, Td = 1.25 o.c., φcep = 0, and γ = 2. The red highlighted part fits in the gate window and is responsible for the generation of harmonics around the cutoff. (b)The HHG spectrum from a parallel-aligned Cl2 molecule. (c) The time profile of the output attosecond pulse produced by superposing harmonics from H36 to H50 (dashed lines in (b)). (d) The electric field ellipse of the output attosecond pulse. Units of harmonic order and o.c. are based on the fundamental pulse with the wavelength of 800 nm.
4. Conclusion
In conclusion, we have theoretically investigated the generation of a single attosecond pulse with a tunable ellipticity from the interaction of the Cl2 molecule and a polarization gating laser. Calculations were performed based on TDDFT in a three-dimensional system and considering multi-electron interactions. At first, we demonstrated that by adjusting the time delay and the carrier-envelope phase of a single-color PG pulse an isolated attosecond pulse is achieved from a parallel-aligned Cl2 molecule. Applying a PG laser (with λ1 = λ2 = 800 nm, I1 = I2 = 1 × 1014 Wcm-2, Td = 1.75 o.c., and φcep = 90°) and superposing harmonics around the harmonic cutoff lead to the generation of an isolated attosecond pulse with a FWHM of 269 as an ellipticity of ε = 0.16. Then, we have introduced two parameters for controlling the ellipticity of the attosecond pulses, the orientation angle of the molecule and the intensity ratio of the combined pulses. Regarding the first one, we have shown that using a single-color PG pulse while adjusting the orientation angle of the Cl2 molecule on 40°, could increase the ellipticity of the attosecond pulse to ε = 0.66. The second control parameter (the intensity ratio of the combined pulses) is examined by simulating the interaction of a parallel-aligned Cl2 molecule with a two-color PG pulse (λ1 = 2λ2 = 800 nm, Td = 1.25 o.c., and φcep = 0°). We have fixed the intensity of the fundamental pulse to I1 = 1 × 1014 Wcm-2 and altered the intensity of the second pulse (I2). The results show that by using an optimized value for the intensity ratio γ = I2 / I1 = 2 and adding proper harmonics around the harmonic cutoff, a single attosecond pulse with an ellipticity of ε = 0.92 and a pulse duration of 648 as is produced. Generation of such an isolated circularly polarized attosecond pulse is a great achievement that helps for studying the chiral sensitive light-matter interactions in attosecond scale and measuring the ultrafast phenomena.
Acknowledgments
The authors are grateful to Tarbiat Modares University for supporting this research. M. M. gratefully acknowledges financial support by the SFB 1375 NOA and the Thuringian State Government within initiative Quantum Hub Thuringia (FGI 0043 and IZN 0026).
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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