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Abstract
We theoretically study the dependence of below-threshold harmonic generation (BTHG) of atoms on the carrier-envelope phase (CEP) driven by few-cycle mid-infrared laser pulses. The BTHG spectra can be accurately and efficiently calculated by solving the three-dimensional time-dependent Schrödinger equation using the time-dependent generalized pseudospectral method. We present the BTHG spectra as a function of the laser-field CEP. CEP-dependent enhancement or suppression occurred at low laser field intensities owing to the changes in the resonant effects associated with multiple quantum trajectories. However, the BTHG of atoms driven by high laser intensities is insensitive to the CEP. The synchrosqueezing time-frequency transform of the BTHG and extended semiclassical analysis are performed to elucidate the underlying physical mechanism.
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1. Introduction
High-order harmonic generation (HHG) from atoms and molecules is a potential method for producing low- and high-frequency coherent radiation sources on an attosecond timescale. Attosecond laser pulses, which are powerful tools for the real-time observation of electronic dynamics in atomic and molecular systems [1–4], have attracted immense interest over the past decades [5]. HHG with photon energy above the ionization threshold $I_p$ is well understood by the semiclassical three-step model developed by Corkum [6] and Kulander [7].
The semiclassical model demonstrates that the maximum kinetic energy of the electron rescattering with the atomic core is 3.17$U_p$, where $U_p$ is the ponderomotive energy. Thus, the highest energy of the harmonic emission is $I_p$+3.17$U_p$. However, the conversion efficiency of the HHG above the ionization potential energy is lower than that below the ionization threshold. Recently, experimental studies [8,9] have confirmed that below-threshold harmonic generation (BTHG) provides a potential alternative method for producing a vacuum-ultraviolet frequency comb. Thus, the BTHG has attracted considerable attention in ultrafast science and technology.
Currently, considerable effort has been devoted to BTHG with regard to the exploration of dynamic processes [10,11]. Hostetter et al. [12] presented a physical picture of BTHG in a model atom using an extended semiclassical three-step model with atomic potential. Soifer et al. [13] studied near-threshold HHG from aligned molecules. Chini et al. [14] demonstrated a regime of phase-matched BTHG, in which phase matching is generated only near the resonance structures of the atomic target. Avanaki and Heslar et al. [15,16] studied the spectral characteristics of the BTHG of H$_2^{+}$ molecular ions in either linearly or nonlinearly polarized laser fields. Recently, we demonstrated the influence of nuclear symmetry on channel selection and conversion efficiency in molecular BTHG [17]. However, HHG is known to depend on the carrier-envelope phase (CEP) of few-cycle laser pulses [18]. In previous studies [19–22], CEP-dependent effects in HHG were observed, but the efforts have often been made for the above-threshold spectral regime.
Recently, Hassan et al. [23] demonstrated the CEP effects of low-energy harmonic generation from bound electrons, which provided evidence for the feasibility of tracing and controlling the nonlinear response of bound electrons on a sub-femtosecond timescale with high accuracy. Xiong et al. [24] theoretically investigated the CEP dependence of the low-order harmonic generation originating from the interference of different harmonic orders. They observed that CEP effects can only be observed when the spectrum of the driving laser is extremely wide, and the duration is very short. More recently, You et al. [25] studied the CEP dependence of the BTHG in solids driven by a mid-infrared (mid-IR) laser pulse. However, the laser-field CEP dependence in BTHG, particularly the dynamical origin associated with the selection of quantum trajectories, remains an open problem.
In this study, we present a theoretical study of BTHG from an atom in the presence of a few-cycle mid-IR laser field by solving the three-dimensional time-dependent Schrödinger equation (TDSE) accurately and efficiently using the time-dependent generalized pseudospectral (TDGPS) technique [26]. Combined with a synchrosqueezing time-frequency technique and an extended semiclassical analysis, we explore the dynamic process of the BTHG when the CEP successively varies. This allows us to identify the selection of quantum trajectories related to changes in the laser-field CEP.
The remainder of this paper is organized as follows. In Sec. 2, we briefly describe the TDSE formalism for the general treatment of the HHG of atomic systems. In Sec. 3, we analyze the BTHG spectra of atoms and identify quantum trajectories in BTHG with various laser-field CEP. We summarize our conclusions in Sec. 4.
2. Theoretical methods
HHG can be studied by solving the following TDSE [27–29] (where atomic units are used):
The TDSE can be solved using the generalized pseudospectral (GPS) technique in spherical coordinates, which has been shown to be considerably more accurate and computationally efficient than conventional time-dependent propagation techniques that use equal-spacing grid discretization. Its validity can be verified by comparing with the experimental observations reported in our previous work [31]. For example, our numerical scheme for solving the TDSE adequately explains the Cooper minimum of the Ar atom and harmonic feature. The numerical scheme of the TDGPS method comprises two essential steps: (i) the spatial coordinates are optimally discretized in a nonuniform spatial grid using the GPS technique. This discretization, which uses only a modest number of grid points, is characterized by denser grids near the nuclear origin and sparser grids for larger distances. The semi-infinite domain $[0, \infty )$ or $[0, r_{max}]$ of the radial coordinate $r$ is transformed into a finite domain $x=[-1,1]$ by a nonlinear mapping $r$ $=r(x)=L \frac {1+x}{1-x+\alpha }$, where $L$ is a mapping parameter and $\alpha$ $=2 L / r_{\max }$ and $r_{\max }$ is the maximum radial distance used in the calculation. (ii) A second-order split-operator technique in the energy representation which allows the explicit elimination [32] of undesirable fast-oscillating high-energy components is used for the efficient temporal propagation of the wave function,
3. Results and discussions
In Fig. 1(a) we present the yields of the BTHG spectra as a function of the CEP of the 1800-nm laser field with the peak intensity $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$. In our calculation, the laser field is written as
Here $E_0$ is the electric field amplitude, $\omega$ is the laser frequency, $\phi$ is the CEP of laser field, and $T$ is the duration of one optical cycle (o.c.). The total duration is equal to 5 optical cycles. The black horizontal dash line indicates the ionization threshold marked by $I_p$, which is equal to 0.579 a.u. for the model potential of Ar [30] that coincides with the 23rd harmonic order for a laser whose wavelength is 1800-nm. Because the contribution of the multiple trajectories becomes more obvious in BTHG driven by the mid-IR laser field, we adopt a mid-IR laser pulse to explore the dynamical origin of the BTHG related to the CEP. The yields of BTHG spectra are very strong at CEP $\phi =n\pi ~(n=0,1,2,\dots )$. However, the yields of BTHG are weak at CEP $\phi =\frac {n}{2}\pi ~(n=1,3,5, \dots )$, which indicates that the emission of BTHG is suppressed and implies that the CEP of few-cycle mid-IR laser pulses plays an important role on BTHG. Figure 1(b) shows the yields of the BTHG spectra as a function of the CEP for a laser intensity $I=2.0\times 10^{13}\mathrm {~W}/\mathrm {cm}^{2}$. Figure 1(c) shows the yields of the BTHG spectra as a function of the CEP for the peak intensity $I$=$6.0 \times 10^{13} \mathrm {~W}/\mathrm {cm}^{2}$. The other laser parameters used are the same as those shown in Fig. 1(a). With the increasing of the laser intensity, the BTHG becomes less and less sensitive to the laser CEP. When the laser intensity is strong enough, the BTHG spectrum is observed to be featureless as the CEP changes.
Fig. 1. Yields of the BTHG spectra of Ar as a function of the CEP for an 1800-nm laser field whose peak intensities are (a) $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$, (b) $I$=$2.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$, and (c) $I$=$6.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$, respectively. The black horizontal dashed lines indicate the ionization threshold denoted by $I_p$.
Figure 2(a) shows the BTHG spectra of an Ar atom driven by an 1800-nm laser field with a peak intensity $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ for CEP $\phi$=0.5$\pi$ and $\phi$=$\pi$, respectively. One can see that, besides the resonance harmonic 16 (H16) and harmonic 20 (H20), only the lowest few harmonics are visible. The reason is that the ionization probability is very small for a 5-cycle 1800-nm laser pulse with the intensity I=$1.0\times 10^{13}\mathrm {~W} / \mathrm {cm}^{2}$. Thus, the harmonics in the plateau and cutoff region are absent. But this result is sufficient to be used to uncover the CEP-dependence BTHG on resonance. Because we obtain a similar result when the ionization probability is large with the increasing of the pulse duration (See the discussion in Fig. 9). Figures 2(b) and 2(c) present the wavelet time-frequency analysis of the BTHG spectra corresponding to CEP $\phi$=0.5$\pi$ and $\phi$=$\pi$, respectively, which depict the detailed spectral and temporal structures of the BTHG. The black horizontal dashed and black solid lines indicate the ionization threshold marked by $I_p$, and the laser fields, respectively. The time-frequency spectra of the BTHG are obtained using the wavelet transform of the induced dipole acceleration [34] $d_{A}(t^{\prime })$. It is expressed as
Fig. 2. (a) HHG spectra of the Ar atom driven by an 1800-nm laser field whose peak intensity is $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ for CEP $\phi$=0.5$\pi$ and $\phi$=$\pi$, respectively. (b) and (c) Corresponding to the wavelet time-frequency analysis of the BTHG spectra. The other laser parameters used are the same as those in Fig. 1(a). The black solid lines represent the time-dependent laser fields.
The wavelet time-frequency spectra are weak at the same time duration for the case of BTHG at CEP $\phi$=0.5$\pi$. Whereas those spectra for the enhanced BTHG at CEP $\phi$=$\pi$ are strong from 1.0 to 3.0 o.c. in the vicinity of the 20th harmonic order. This indicates that the yield of the BTHG is sensitive to the laser-field CEP, and the dynamical mechanism depends on the CEP.
To better understand the dynamical behavior of electrons in a CEP-dependent BTHG in a few-cycle mid-IR laser field, we perform a synchrosqueezing transform (SST) of the harmonic spectra and an extended semiclassical simulation for identifying the quantum trajectories. The results are shown in Fig. 3. The SST reveals the characteristic behavior of harmonic spectra below the ionization threshold [35,36]. Compared to other widely used time-frequency transforms, the advantage of the SST is its ability to generate sharper and clearer time-frequency distributions, which allows the identification of individual roles of multiple channels below the ionization threshold in the time-frequency domain. In addition, a semiclassical model provides a satisfactory description of the laser-driven dynamical behavior of electrons, and it has been successfully extended to extract the dynamical behaviors of laser-driven rescattering in atomic and molecular BTHG [17]. The SST is defined by
Fig. 3. (a) SST time-frequency spectra of BTHG and semiclassical returning energy of the Ar atom driven by an 1800-nm laser field with a peak intensity of $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ for CEP $\phi =0.5\pi$. (b) Probability of the returning electron at the revisit time. (c) Corresponding semiclassical trajectories as a function of the time.
The instantaneous frequency information function $\omega _f(t,\omega )$ is defined by
We adopt the semiclassical model reported by Hostetter et al. [12] to understand the process of BTHG. This semiclassical method is defined by the three-dimensional $Newton's$ equation which considers the atomic potential. It is given by
where $V(\mathbf {r})$, $E(t)$ and $\hat {z}$ represent the model potential of the $\mathrm {Ar}$ atom, the electric field of the laser pulse, and the laser polarization direction, respectively. The semiclassical simulation including the Coulomb potential is sensitive to the initial conditions. Typically, there are two types of initial conditions as defined in Ref. [12]. There are the multiphoton regime, in which electrons with nonzero kinetic energy are released along the polarized direction of the laser fields, and the tunneling regime, in which the electrons at the tunnel position are released along the polarized direction of the laser fields. In our calculation, we choose the initial conditions based on the Keldysh regime. According to the Keldysh theory [37], the nonlinear ionization regime can be quantitatively defined by introducing the Keldysh parameter $\gamma$ ($\gamma$=$\sqrt {I_p/2U_p}$). The multiphoton and tunneling ionization regimes are typically classified by $\gamma \gg 1$, and $\gamma \ll 1$, respectively. We adopt the initial condition with the multiphoton regime when the Keldysh parameter exceeds unity, whereas we use the initial condition with the tunneling regime when it is less than unity. The results obtained using these initial conditions are consistent with our quantum mechanical calculations.In Fig. 3(a), the SST time-frequency spectra and semiclassical simulation of the returning energy of the Ar atom driven by an 1800-nm laser field with the peak intensity $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ at CEP $\phi =0.5\pi$ are presented. The SST transform yields a sharper and clearer time-frequency distribution of the BTHG. The spectral and temporal characteristics of the BTHG exhibit a resonant enhancement of time-frequency spectra near H20. The Ar atom possesses a deep ionization potential $I_p$ (15.76 eV), which allows one to obtain the high-order BTHG driven by the mid-IR laser field. The yield near H20 is enhanced significantly since the absorption energy of the 20 photons at 1800 nm is equal to the single-electron transition energy of 3$p^6$-3$p^5$3$d$ (13.99 eV). Furthermore, the yield of H16 is also enhanced because the single-electron transition energy for 3$p^6$-3$p^5$4$s$ (11.68 eV) coincides with the H16 energy of the 1800-nm laser wavelength. The generation of the enhanced below-threshold harmonic is enabled only near the resonance structures of the Ar atom.
In the semiclassical simulation of the returning energy shown in Fig. 3(a), the curves R$_1$, R$_2$, and R$_3$ indicate multiple quantum trajectories related to the electron initially driven by the laser fields at a specific time of the laser pulse. The laser parameters used are the same as those in Fig. 1(a); therefore, the Keldysh parameter is equal to 1.6. The initial conditions with the multiphoton regime are used, namely, the electron is released with an initial velocity along or opposite to the polarization direction of the laser field and a zero initial coordinate. Here, the initial velocity used is $|v_{0z}|$=0.5 a.u., which is closest to the energy of the excited state $3d$, and the initial position in the laser polarized direction $\hat {z}$ is zero, while the initial position in other directions used is sufficiently small. This allows the electron closest to the core and avoided the Coulomb singularity problem. Combining with the SST time-frequency spectra and an extended semiclassical return energy map, the distinct contribution of the multiple quantum trajectories can be recognized. We find that the trajectories R$_1$ and R$_2$ are responsible for the BTHG, while R$_3$ is responsible for the above-threshold HHG. The semiclassical results are consistent with the SST time-frequency analysis of the BTHG spectra of the Ar atom.
To clarify the weight of the multiple quantum trajectories predicted by the semiclassical approach, we calculate the probability of the returning electron with the corresponding return time $t$ and return energy $E$ as shown in Fig. 3(b) by using an extended semiclassical method [38]. This can be obtained from the following expression:
To understand the resonance-enhanced BTHG related to quantum trajectories, Fig. 3(c) shows the corresponding electronic trajectories calculated using the semiclassical model. In the trajectories R$_1$, the contributions of the quantum trajectories originate from two parts. The first originates from the electron released along the direction of the laser-field force and driven by the laser field at the lower laser amplitude, the contributions of these trajectories are weak from 0.7 to 0.8 o.c. The second originates from the electron released against the direction of the laser-field force and rapidly driven back to the parent core by the laser field, the contributions of these trajectories are strong from 0.8 to 1.2 o.c. Thus, the laser field plays an important role in BTHG. In the trajectories R$_2$, the electron is released along the direction of the laser-field force from the opposite side and is driven back to the parent core by the Coulomb fields at the lower laser amplitude. The contributions of these trajectories are also strong from 1.0 to 1.2 o.c. This indicates that the Coulomb field plays an important role in BTHG. In the trajectories R$_3$, the electron is driven by the laser field with a long travel time, and these trajectories are responsible for above-threshold HHG.
Figure 4(a) presents the SST time-frequency spectra of BTHG and the semiclassical returning energy of an Ar atom driven by an 1800-nm laser field with the peak intensity $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ for CEP $\phi =\pi$. The intensities of the SST time-frequency spectra in Fig. 4(a) are normalized based on Fig. 3(a). A careful comparison of the results for CEP $\phi =0.5\pi$ and CEP $\phi =\pi$ as shown in Figs. 3(a) and 4(a) present some important differences. The trajectories R$_1$ and R$_2$ in BTHG are different, and more trajectories R$_2$ can be observed. In the trajectories R$_1$, the electron is released against the direction of the laser-field force and driven back to the parent ion by the Coulomb fields. However, it is interesting to note that there are two channels for the trajectories R$_2$. One is released against the laser field force and driven back to the parent core by the laser field. Another one is released along the laser field force and driven back the core by the Coulomb fields, the contributions of the second channel of trajectory R$_2$ are strong from 1.4 to 1.7 o.c. It implies that an extra channel is opened for the BTHG for CEP $\phi =\pi$. In Fig. 4(b), the probability of the returning electron at the revisit time calculated using an extended semiclassical model is presented. It is clearly seen that the dominant contribution comes from the new channel of trajectory R$_2$, and the Coulomb field is responsible for those open channels as shown in Fig. 4(c).
Fig. 4. (a) SST time-frequency spectra of BTHG and semiclassical returning energy of Ar atom driven by an 1800-nm laser field with the peak intensity $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ on CEP=$\pi$. (b) Probability of the returning electron at the revisit time. (c) Corresponding semiclassical trajectories as a function of the time.
As discussed above, some trajectories associated with the resonant effects are suppressed during BTHG of the Ar atom at CEP $\phi =0.5\pi$ at the lower laser intensity. But for CEP $\phi =\pi$, BTHG is enhanced owing to multiple opening trajectories. However, the mechanism underlying the influence of the laser intensity on trajectory selection and efficiency on BTHG related to the CEP must be thoroughly explained. Figure 5(a) shows the SST time-frequency spectra of BTHG and semiclassical simulation of the returning energy of the Ar atom for CEP $\phi =0.5\pi$ for the laser intensity $I$=$6.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$. The Keldysh parameter is lower than unity for this laser intensity and we use the initial condition with the tunneling regime for the semiclassical simulation. The curves marked by R$_1$, R$_2$, and R$_3$ depict the semiclassical return energy as a function of the emission time. The results are consistent with the SST time-frequency analysis of the BTHG spectra of the Ar atom. In Fig. 5(b), we present the probability of the returning electron at revisit time calculated using an extended semiclassical model. We can find that the contribution of trajectory R$_2$ dominates the BTHG. In Fig. 5(c), the corresponding electronic trajectories calculated by the semiclassical model are presented. The dynamical processes of trajectories R$_1$, R$_2$, and R$_3$ are similar to those shown in Fig. 3. Figure 6(a) shows the SST time-frequency spectra of BTHG and the semiclassical simulation of the returning energy of the Ar atom for CEP $\phi =\pi$ for the laser intensity $I$=$6.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$. The probability of the returning electron at revisit time calculated using an extended semiclassical model is shown in Fig. 6(b). Similar to Fig. 5(b), the dominant contribution to BTHG comes from trajectory R$_2$. Figure 6(c) shows the corresponding electronic trajectories. The dynamics of the trajectories R$_1$, R$_2$, and R$_3$ resemble the results shown in Figs. 5(a) and 5(c). It implies that the BTHG is insensitive to CEP for high laser fields because the dominant contribution to BTHG comes from the Coulomb field rather than the laser field.
Fig. 5. (a) SST time-frequency spectra of BTHG and semiclassical returning energy of the Ar atom driven by an 1800-nm laser field with a peak intensity $I$=$6.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ on CEP $\phi =0.5\pi$. (b) Probability of the returning electron at the revisit time. (c) Corresponding semiclassical trajectories as a function of the time.
Fig. 6. (a) SST time-frequency spectra of BTHG and semiclassical returning energy of the Ar atom driven by an 1800-nm laser field with a peak intensity $I$=$6.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ on CEP $\phi =\pi$. (b) Probability of the returning electron at the revisit time. (c) Corresponding semiclassical trajectories as a function of the time.
Figures 7(a) and 7(b) show the yields of the BTHG spectra as a function of CEP for the wavelengths of $2400\mathrm {~nm}$ and $3600 \mathrm {~nm}$, respectively, at a laser peak intensity of $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$. The other laser parameters used are the same as those in Fig. 1(a). The dependence of the CEP is insensitive to the laser wavelength at low laser intensities. The yields of the BTHG spectra are also very high for CEP $\phi =n\pi ~(n=0,1,2,\dots )$, but they are weak near CEP $\phi =\frac {n}{2}\pi ~(n=1,3,5,\dots )$. This result is similar to those shown in Figs. 1(a) and 1(b).
Fig. 7. BTHG spectra as a function of the CEP at a laser peak intensity of $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ for the wavelength (a) $2400\mathrm {~nm}$ and (b) $3600 \mathrm {~nm}$, respectively. The black horizontal dashed lines indicate the ionization threshold marked by $I_{p}$.
To check the sensitivity of the CEP dependence to the atomic species, we present the yields of the BTHG spectra as a function of the CEP for hydrogen and helium atoms driven by the 1800-nm laser pulse as shown in Figs. 8(a) and 8(b), respectively. In Fig. 8(a), the laser intensity is $I$=$8.6 \times 10^{12} \mathrm {~W} / \mathrm {cm}^{2}$, and the Keldysh parameter is the same as that shown in Fig. 1(a). We obtain results similar to those shown in Fig. 1. Namely, the emission of BTHG spectra is strong when CEP $\phi =n\pi ~(n=0,1,2,\dots )$, but weak when CEP $\phi =\frac {n}{2}\pi ~(n=1,3,5,\dots )$. Figure 8(b) shows the yields of the BTHG spectra as a function of the CEP for the helium atom. For the comparison between the different atoms, we use the laser intensity with the same Keldysh parameter as the case of the hydrogen atom shown in Fig. 8(a). The Keldysh parameter $\gamma$ is 1.6 for the hydrogen atom, which corresponds to the multiphoton ionization. For the helium atom, the laser peak intensity $I=1.6 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ is chosen to maintain the same ionization regime as in the case of the hydrogen atom. The CEP dependence exhibits the same structure as the hydrogen atom in BTHG, which indicate that the dependence of the CEP in BTHG is insensitive to the atomic species in the multiphoton ionization regime.
Fig. 8. BTHG spectra as a function of the CEP for (a) hydrogen and (b) helium atoms driven by laser peak intensities of $I=8.6 \times 10^{12} \mathrm {~W} / \mathrm {cm}^{2}$ and $I=1.6 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$, respectively. The laser wavelength used is 1800 nm. The other laser parameters used are the same as those in Fig. 1(a). The black horizontal dashed lines indicate the ionization threshold marked by $I_{p}$.
As discussed above, we present the dependence of BTHG of atoms on the CEP driven by the few-cycle laser field. To confirm our findings also apply to the multi-cycle mid-IR case, we calculate the harmonic spectra of Ar atoms driven by a multi-cycle 1800-nm laser pulse. Figure 9(a) shows a comparison of the harmonic spectra from Ar driven by 1800-nm laser pulse with pulse duration of 5, 20, and 40 optical cycles, respectively. With the increasing of the pulse duration, the absent harmonics appear in the plateau and cutoff region, but the structure of the BTHG is the same as those in the few-cycle case. In Fig. 9(b) and 9(c), we present the yields of BTHG spectra as a function of laser CEP of an 1800-nm laser pulse with intensity $1.0\times 10^{13}\mathrm {~W} / \mathrm {cm}^{2}$ and $6.0\times 10^{13}\mathrm {~W} / \mathrm {cm}^{2}$, respectively. The total pulse duration is 40 optical cycles. We can find that the CEP-dependence of BTHG keeps similar results as shown in Fig. 1.
Fig. 9. (a) Comparison of HHG spectra of the Ar atom driven by 1800-nm laser fields with different pulse duration for peak intensity $I$=$1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ and CEP $\phi$=0. Yields of BTHG spectra of Ar as a function of the CEP for 40-cycle 1800-nm laser field with peak intensity (b) $I=1.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$ and (c) $I=6.0 \times 10^{13} \mathrm {~W} / \mathrm {cm}^{2}$, respectively. The other laser parameters used are the same as those in Fig. 1(a).
4. Conclusion
In conclusion, we have studied the CEP-dependent BTHG in a few-cycle mid-IR laser field by solving the TDSE using the TDGPS method accurately in space and time. The yields of BTHG spectra are observed to be strong when CEP $\phi =n\pi ~(n=0,1,2,\dots )$, while they are weak when CEP $\phi =\frac {n}{2}\pi ~(n=1,3,5,\dots )$ for the low laser field intensities. This result indicates the important role of CEP of few-cycle mid-IR laser pulses in BTHG. To understand the dynamical origin of BTHG associated with the laser CEP, we analyze the quantum trajectories by combining with semiclassical energy maps and time-frequency spectra. The contribution of multiple quantum trajectories depend on the laser CEP at low laser field intensities. For the enhanced BTHG associated with the laser CEP, several quantum trajectories are responsible for the resonant BTHG at CEP $\phi =n\pi$. For the BTHG at CEP $\phi =\frac {n}{2}\pi$, the absence of some quantum trajectories suppress resonant effects in BTHG. However, the BTHG is insensitive to the CEP for high laser fields because the dominant contribution to BTHG comes from the Coulomb fields. In addition, we verify the sensitivity of the CEP dependence to the laser intensity, wavelength, and atomic species. We find that the CEP-dependent BTHG in the few-cycle intense laser field is sensitive to the laser intensity rather than the laser wavelengths, and atomic species in the multiphoton ionization regime. Our findings also apply to the multi-cycle mid-IR case.
Funding
National Natural Science Foundation of China (No. 12074239, No. 91850209); Natural Science Foundation of Guangdong Province (No. 2020A1515010927, No. 2020ST084); Department of Education of Guangdong Province (No. 2019KTSCX038, No. 2020KCXTD012); Shantou University (No. NTF18030).
Acknowledgments
We would also like to acknowledge the partial support of the Shantou University.
Disclosures
The authors declare no conflicts of interest.
Data availability
The 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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