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Abstract
We investigate theoretically the photoelectron momentum distributions (PMDs) of the helium atom in the few-cycle nonlinear chirped laser pulse. The numerical results show that the direction of the spider-like interference structure in PMDs exhibits periodic variations with the increase of the chirp parameter. It is illustrated that the direction of the spider-like interference structure is related to the direction of the electron motion by tracking the trajectories of the electrons. We also demonstrate that the carrier-envelope phase can precisely control the opening of the ionization channel. In addition, we investigate the PMDs when a chirp-free second harmonic (SH) laser pulse is added to the chirped laser field, the numerical results show that the interference patterns can change from only spider-like interference structure to both spider-like and ring-like interference structures.
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1. Introduction
The laser-atom interaction can produce many nonlinear phenomena, such as above-threshold ionization (ATI) [1,2], high-order harmonic generation (HHG) [3–5] and nonsequential double-ionization (NSDI) [6–8]. These nonlinear phenomena play an important role in the study of the strong-field physics and attosecond science [9,10]. We can use three-step model to describe the ionization process of electrons. This model can be described in three steps [11], the electrons firstly tunnel out in the strong laser field, then are accelerated, and finally return to the parent ion. The returned electrons can collide with the parent ion elastically or inelastically [12], alternatively it can recombine with the parent ion and produce a high-energy photon [13]. Electron-ion re-collision encodes critical information about the structure and dynamics of the atomic or molecular medium used [14].
There are several common interference structures in the photoelectron momentum distributions (PMDs), such as the ATI ring that results from the inter-cycle interference [15], the temporal double-slit interference structure that results from the adjacent intra-cycle interference [16,17], and the photoelectron holography (PH) [18] that records the information about the core and electron dynamics in space and time [19]. The spider-like interference structure is a representative PH interference structure that originates from the interference of electron wave packets (EWPs) generated within the same quarter-cycle of the laser pulse [18]. Typically, all of these interferences together give rise to the final PMDs, thus the interference patterns in PMDs are too complex for people to obtain useful information from them, hence it is crucial to choose certain interference to extract information.
In order to investigate the electron dynamics behind the holographic structures, many numerical methods were developed such as the Coulomb-corrected strong-field approximation (CCSFA) [20,21], the quantum trajectory Monte Carlo (QTMC) [22] and the semiclassical two-step model (SCTS) [23], and so on. The great advantage of the semiclassical models is that they provide a visual image of the classical trajectories and allow the reproduction of the holographic structures in the tunneling state by including the phase equations associated with each classical trajectory [24].
Due to the progress of comb laser technology [25], the nonlinear chirped laser pulse has been developed, which plays an important role in the ultrafast dynamics and attosecond science [26]. Xiang et al. [27] investigated theoretically the above-threshold ionization of Hydrogen atom in the few-cycle nonlinear chirped laser pulse, they found that the cutoff of the above-threshold ionization spectrum above the threshold can be greatly extended. Carrera and Chu [28] achieved the significant cutoff extension of the high-order harmonics by using the few-cycle nonlinear chirped laser pulse. Feng and Chu [29] investigated theoretically the combined chirp effects on generations of HHG and attosecond pulse with a two-color chirped laser pulse, they found that both the harmonic cutoff energy and the supercontinuum can be remarkably extended and obtained an isolated sub-40-as pulse. Sun et al. [19] demonstrated that the interference structure of the negative direction of $x$-axis can be suppressed, the spider-like structure of the positive direction of $x$-axis gradually dominates and the region of the PMDs can be expanded with the increase of the chirp parameter by using the few-cycle chirped laser pulse. In the past work, the nonlinear chirped laser pulse plays an important role in the study of the HHG and attosecond science, while the investigation of the PMDs in the nonlinear chirped laser pulse is also very rare.
In this paper, we investigate theoretically the EWPs interference in PMDs with the few-cycle nonlinear chirped laser pulse. The numerical results show that the direction of the spider-like interference structure can periodically change with the increase of the chirp parameter. The opening of the ionization channel can be controlled by adjusting the carrier-envelope phase. In addition, the interference patterns can be from only spider-like interference structure to both spider-like and ring-like interference structure in the few-cycle nonlinear chirped laser pulse in combination with its chirp-free second harmonic (SH). Our results may be beneficial for the application of PH in probing the structure and dynamics of atoms and molecules.
2. Methods
To investigate the dynamics of the tunneling ionization of Helium atom in the intense laser field, we numerically solve the two-dimensional time-dependent Schrödinger equation (2D-TDSE) within the single-active-electron (SAE) approximation by using the second-order splitting-operator fast Fourier transform algorithm [30] and the two-dimensional two-step (SCTS) model [31]. Atomic units (a.u.) are used unless otherwise stated.
2.1 Time dependent Schrödinger equation (TDSE) theory
The Schrödinger equation for the electron wavefunction evolution with time under the influence of the classical electromagnetic field is as follows
In order to eliminate the singularity of the potential function at the origin, we set the soft-core parameters $a$ = 0.6 and $b$ = 1.5. With the imaginary-time propagation, we can obtain the ionization potential 0.9, which is equal to the energy of the ground state of a Helium atom [32,33].
In the dipole approximation and length gauge, the atom-field interaction is expressed as
where $E(t)$ is the electric field of the nonlinear chirped laser pulse along the $\hat {x}$ direction where $E_0$ is the amplitude of the electric field, $\omega$ is the frequency of the nonlinear chirped laser pulse, and $\varphi$ is the carrier-envelope phase. The intensity and wavelength of the nonlinear chirped laser pulse are $3 {\times } 10^{14}\;\mathrm {W} / \mathrm {cm}^2$ and 800nm. $f(t)=\sin ^2\left (\frac {\pi t}{n T}\right )$ represents the envelope of the laser pulse with the parameter $n$ = 4 and the optical period $T=\frac {2 \pi }{\omega }$. $\delta (t)=-\beta \tan \left (\frac {t-t_0}{\tau _0}\right )$ is the chirp form, the chirp parameter $\beta$ (rad) can control the frequency sweeping range, $\tau _0$ = 200 can control the steepness of the chirping function, and $t_0$ = 0 can adjust the sweep range of the electric field. Carrera and Chu [28] investigated the HHG cutoff extension when the chirp parameter increases from 1.5 to 10.25. They found a significant extension in the harmonic generation cutoff can be realized by adjusting the chirp parameter. Therefore, in order to keep the chirp parameter within a reasonable range, we choose the chirp parameter $\beta$ in the range of $0 \leq \beta \leq 10.25$.2.2 Semiclassical two-step model (SCTS) theory
We also use the two-dimensional semiclassical two-step (SCTS) model to investigate PMDs of Helium atom. We describe briefly the two-dimensional quantum trajectory Monte Carlo (QTMC) model [22,34] before introducing the SCTS model. The QTMC model combines the ADK theory with Feynman’s path-integral approach, Coulomb potential and quantum interference effect after considering the tunneling [35]. This model demonstrates that the Coulomb potential plays an important role in the trajectory and phase of tunneled electrons. The QTMC model explains the Coulomb potential in semiclassical perturbation theory, on the contrary, in the SCTS model, the phase of each trajectory is obtained by using a semiclassical expression for the quantum-mechanical propagator matrix element [36]. Therefore, the SCTS model transcends semiclassical perturbation theory to explain the binding potential [35].
The location and the momentum distribution of the tunnel exit is used as the initial conditions in the second step of classical trajectory propagation. The subsequent evolution of the tunnel-ionized electron under the above initial conditions are determined by the classical Newtonian equation of motion
where $\vec {E}(t)$ is given in Eq. (5), $V(r)=-Z / r$ is the Coulomb potential, and $Z$ = 1 is ionic charge.In the SCTS model, each trajectory is related to a matrix element of the semiclassical propagator, the SCTS phase is expressed as [31]
3. Results and discussions
Figure 1 shows the electric field of the few-cycle nonlinear chirped laser pulse (black line) and the ionization rate (red line) for different chirp parameters, where the carrier-envelope phase is set as $\varphi$ = 0. From Fig. 1 we can find that the symmetry of the electric field is broken. For the chirp parameter increased from 1 to 4 as shown in Figs. 1(a)–1(d), we observe that the maximum negative peak of the electric field appears firstly, and then the maximum positive peak of the electric field appears. As a result of the modulating influence of the chirp parameters, we can also observe that both the maximum negative and positive peaks of the electric field gradually rightward shift with the increase of the chirp parameter as indicated by the labeled blue and green lines, respectively. It can be also observed that the maximum negative peak of the electric field is gradually increased and the maximum positive peak of the electric field is gradually decreased with the increase of the chirp parameter.
Fig. 1. Electric field of the few-cycle nonlinear chirped laser pulse (black line) and the ionization rate (red line) for different chirp parameters (a) $\beta$ = 1, (b) $\beta$ = 2, (c) $\beta$ = 3, (d) $\beta$ = 4, (e) $\beta$ = 5, (f) $\beta$ = 6, (g) $\beta$ = 7 and (h) $\beta$ = 8, where the carrier-envelope phase is set as $\varphi$ = 0.
Compared to Figs. 1(a)–1(d), for the chirp parameter increased from 5 to 8 as shown in Figs. 1(e)–1(h), we can observe that the maximum positive peak of the electric field appears firstly, and then the maximum negative peak of the electric field appears. There is also a rightward shift of the maximum positive and negative peaks of the electric field with the increase of the chirp parameter as indicated by the labeled blue and green lines, respectively. It can be also observed that the maximum positive peak of the electric field gradually is increased and the maximum negative peak of the electric field is gradually decreased with the increase of the chirp parameter.
For the chirp parameter increased from 1 to 4 as shown in Figs. 1(a)–1(d), we define the ionization time channels corresponding to the maximum negative and positive peaks as $t_1$ and $t_2$ channels, which also correspond to the ionization rate. Figures 1(a)–1(c) show that the EWP is dominantly generated in the ionization channel $t_1$ and $t_2$, we can observe that the ionization channel $t_1$ is enhanced, while the ionization channel $t_2$ is weakened when the chirp parameter increases from 1 to 3. For $\beta$ = 1, the ionization rate of $t_1$ is significantly smaller than that of $t_2$ as shown in Fig. 1(a). For $\beta$ = 2, the ionization rate of $t_1$ is approximately equal to that of $t_2$ as shown in Fig. 1(b). For $\beta$ = 3, the ionization rate of $t_1$ is greater than that of $t_2$ as shown in Fig. 1(c). By adjusting the chirp parameter to 4, we can observe that the EWP is almost generated in the ionization channel $t_1$. Additionally, the ionization channel $t_1$ is enhanced and the ionization channel $t_2$ is suppressed as shown in Fig. 1(d).
For the chirp parameter increased from 5 to 8 as shown in Figs. 1(e)–1(h), we define the ionization time channels corresponding to the maximum positive and negative peaks as $t_1$ and $t_2$ channels, which also correspond to the ionization rate. In Figs. 1(e)–1(g), we observe that the EWP is predominantly released from channels $t_1$ and $t_2$ for $\beta$ = 5, $\beta$ = 6 and $\beta$ = 7. As the chirp parameter increases from 5 to 7, there is the corresponding enhancement in the ionization channel $t_1$, while the weakening in the ionization channel $t_2$. For $\beta$ = 7, the ionization channel $t_2$ is almost suppressed. By adjusting the chirp parameter to 8, the EWP is only generated in the ionization channel $t_1$, the ionization channel $t_1$ is enhanced and the ionization channel $t_2$ is suppressed as shown in Fig. 1(h).
Figure 2 shows the PMDs simulated by the SCTS model with the few-cycle nonlinear chirped laser pulse for different chirp parameters. Figures 2(a)–2(d) show the PMDs correspond to Figs. 1(a)–1(d). From Fig. 2(a) we can see the emergence of the temporal double-slit interference structure on the left side and the distinct spider-like interference structure on the right side for $\beta$ = 1. For $\beta$ = 2 and $\beta$ = 3, the PMDs show the similar patterns that the intensity of the temporal double-slit interference structure of the left side gradually becomes weak and the spider-like interference structure of the right side gradually disappears as shown in Figs. 2(b)–2(c). As the chirp parameter increases to 4, the spider-like interference structure appears on the left side, the temporal double-slit interference structure appears on the right side as shown in Fig. 2(d).
Fig. 2. PMDs simulated by the SCTS model in the few-cycle nonlinear chirped laser pulse for different chirp parameters (a) $\beta$ = 1, (b) $\beta$ = 2, (c) $\beta$ = 3, (d) $\beta$ = 4, (e) $\beta$ = 5, (f) $\beta$ = 6, (g) $\beta$ = 7 and (h) $\beta$ = 8, where the carrier-envelope phase is set as $\varphi$ = 0.
Figures 2(e)–2(h) show the PMDs correspond to Figs. 1(e)–1(h). For $\beta$ = 5, a clear spider-like interference structure appears on the left side and the temporal double-slit interference structure appears on the right side as shown in Fig. 2(e). For $\beta$ = 6 as shown in Fig. 2(f), the pattern shows that the spider-like interference structure of the left side gradually vanishes and the intensity of the temporal double-slit interference structure of the right side is gradually weakened. For $\beta$ = 7 as shown in Fig. 2(g), the interference structure of the left side disappears and the temporal double-slit interference structure appears on the right side. By adjusting the chirp parameter to 8, a clear spider-like interference structure becomes visible on the right side, while the interference structure of the left side almost disappears as shown in Fig. 2(h). Through the above analysis, it is indicated that the interference structure of the negative direction of the $x$-axis can be suppressed, the spider-like interference structure of the positive of the direction of the $x$-axis gradually dominates when the chirp parameter increases from 5 to 8. The results are similar with that demonstrated in Ref. [19].
By comparing Figs. 2(a)–2(d) with Figs. 2(e)–2(h), we can find that the direction of the spider-like interference structure has an opposite change when the chirp parameter increases from 1 to 4 and increases from 5 to 8. The above phenomenon is associated with the change in the sequence of appearance of the maximum negative and positive peaks of the electric field in the ionization channels $t_1$ and $t_2$ as shown in Figs. 1(a)–1(h). Additionally, from Figs. 2(a)–2(h) we can observe that the spider-like interference structure need four optical cycles to change its direction for the chirp parameter increased from 1 to 4 and increased from 5 to 8. It is indicated that the direction of the spider-like interference structure in PMDs exhibits periodic variations with the increase of the chirp parameter.
In order to make sense of the dynamical mechanism of the electron, we can separate the PMDs of the EWP for different ionization channels according to the ionization rate for different chirp parameters. Figures 3(a)–3(g) show the PMDs of the EWP which released from different ionization channels, where the carrier-envelope phase is set as $\varphi$ = 0. Figures 3(a)–3(b) show the photoelectron momentum distribution for the case where the electron emitted from $t_1$ and $t_2$ channels for $\beta$ =1. From Fig. 3(a) we can find the obvious spider-like interference structure which points to the left side, while in Fig. 3(b), the spider-like interference structure points to the right side. We do the same analysis for $\beta$ = 2 and $\beta$ = 3 as shown in Figs. 3(c)–3(d) and Figs. 3(e)–3(f), the numerical results are similar with that for $\beta$ =1. From Fig. 1(d) we can see the EWP is almost generated from channel $t_1$, thus our exclusive focus is on presenting the spider-like interference structure formed in the ionization channel $t_1$ for $\beta$ =4. In Fig. 3(g), we can find the formation of the spider-like interference structure pointed to the left side in the ionization channel $t_1$. Through above analysis, we can observe that the ionization channel $t_1$ contributes to the formation of the spider-like interference structure pointed to the left side, and the ionization channel $t_2$ contributes to the formation of the spider-like interference structure pointed to the right side when the chirp parameter increases from 1 to 4.
Fig. 3. PMDs of the EWP which released from different ionization channels and different chirp parameters in the few-cycle nonlinear chirped laser pulse, where the carrier-envelope phase is set as $\varphi$ = 0. (a) $\beta$ = 1, ionization channel $t_1$; (b) $\beta$ = 1, ionization channel $t_2$; (c) $\beta$ = 2, ionization channel $t_1$; (d) $\beta$ = 2, ionization channel $t_2$; (e) $\beta$ = 3, ionization channel $t_1$; (f) $\beta$ = 3, ionization channel $t_2$; (g) $\beta$ = 4, ionization channel $t_1$ and (h) PMDs simulated by the TDSE method for $\beta$ = 4.
When the chirp parameter increases from 1 to 4, Figs. 1(a)–1(d) show that the ionization channel $t_1$ is gradually enhanced, the spider-like interference structure formed in the ionization channel $t_1$ (as shown in Figs. 3(a), 3(c), 3(e) and 3(g)) dominates in PMDs, thus we can observe that the spider-like interference structure pointed to the left side gradually appears as shown in Figs. 2(a)–2(d). While the ionization channel $t_2$ is gradually suppressed when the chirp parameter increases from 1 to 3 as shown in Figs. 1(a)–1(c), the spider-like interference structure formed in the ionization channel $t_2$ (as shown in Figs. 3(b), 3(d) and 3(f)) is suppressed in PMDs, hence we can find the spider-like interference structure pointed to the right side gradually disappears as shown in Figs. 2(a)–2(c).
In Fig. 3(h), we show the PMDs by using the TDSE method [30] for $\beta$ = 4. It can be explicitly seen the spider-like interference structure appears on the left side and the temporal double-slit interference structure appears on the right side. The result is in good agreement with that presented in Fig. 2(d) by using the SCTS model as that demonstrated in [31,39].
Figures 4(a)–4(g) show the PMDs of the EWP which released from different ionization channels, where the carrier-envelope phase is set as $\varphi$ = 0. In Fig. 4(a), it exhibits a spider-like interference structure pointed to the right side in the ionization channel $t_1$ for $\beta$ = 5, while in Fig. 4(b) , it exhibits a spider-like interference structure pointed to the left side in the ionization channel $t_2$ for $\beta$ = 5. We do the same analysis for $\beta$ = 6 as shown in Figs. 4(c)–4(d), the results are similar with that for $\beta$ = 5. Figure 4(e) shows the PMD for the case that the electron emitted from the ionization channel $t_1$ for $\beta$ = 7. Figure 4(f) shows the PMD for the case that the electron emitted from the ionization channel $t_2$ for $\beta$ = 7. From Fig. 4(e) we can see the obvious spider-like interference structure which points to the right side. While in Fig. 4(f), the interference structure disappears completely since the ionization channel $t_2$ is closed (as shown in Fig. 1(g)). From Fig. 1(h) we can observe that the EWP is only generated in the ionization channel $t_1$ and the ionization channel $t_2$ is closed for $\beta$ = 8, thus we need only to consider the ionization channel $t_1$. In Fig. 4(g), it is indicated that the spider-like interference structure points to the right side. Through the above analysis, it becomes apparent that the electrons emitted from $t_1$ can generate a spider-like interference structure pointed to the right side, whereas the electrons emitted from $t_2$ can generate a spider-like interference structure pointed to the left side when the chirp parameter increases from 5 to 8.
Fig. 4. PMDs of the EWP which released from different ionization channels and different chirp parameters in the few-cycle nonlinear chirped laser pulse, where the carrier-envelope phase is set as $\varphi$ = 0. (a) $\beta$ = 5, ionization channel $t_1$; (b) $\beta$ = 5, ionization channel $t_2$; (c) $\beta$ = 6, ionization channel $t_1$; (d) $\beta$ = 6, ionization channel $t_2$; (e) $\beta$ = 7, ionization channel $t_1$; (f) $\beta$ = 7, ionization channel $t_2$; (g) $\beta$ = 8, ionization channel $t_1$ and (h) PMDs simulated by the TDSE method for $\beta$ = 8.
For the chirp parameter increased from 5 to 8 as shown in Figs. 1(e)–1(h), the ionization channel $t_1$ is gradually enhanced, the spider-like interference structure formed in the ionization channel $t_1$ (as shown in Figs. 4(a), 4(c), 4(e) and 4(g)) dominates in PMDs, thus we can observe that the spider-like interference structure pointed to the right side gradually appears as shown in Figs. 2(e)–2(h). While the ionization channel $t_2$ is gradually suppressed when the chirp parameter increases from 5 to 6 as shown in Figs. 1(e)–1(f), the spider-like interference structure formed in the ionization channel $t_2$ (as shown in Figs. 4(b), 4(d)) is suppressed in PMDs, hence we can find the spider-like interference structure pointed to the right side gradually disappears as shown in Figs. 2(e)–2(f).
In Fig. 4(h), we show the PMDs by using the TDSE method [30] for $\beta$ = 8. The pattern shows that a spider-like interference structure appears on the right side, while the temporal double-slit interference structure appears on the left side. By comparing Fig. 2(h) with Fig. 4(h), we can find that the structures on the right side of the two are almost similar, but there is a noticeable interference structure on the left side in Fig. 4(h). The reason of the difference between the TDSE calculations and the SCTS results may be that the ionization rate predicted by the ADK theory is not accurate and the nonadiabatic effect is significant when the chirp parameter is 8.
To better understand the physical mechanism of the spider-like interference structure, we take $\beta$ = 1 and $\beta$ = 4 as examples to explore the motion trajectories of the re-scattered electrons for different ionization channels as shown in Figs. 5(a)–5(c). For the ionization channel $t_1$ (as shown in Fig. 5(a) for $\beta$ = 1 and 5(c) for $\beta$ = 4), we can find that the electron first moves to the positive direction along $x$-axis, then returns to the parent ion for a forward re-scattering in the interaction of Coulomb force and laser field, the trajectory is located at the last position of $x$ < 0 plane, the interference between the re-scattering trajectory and the direct trajectory can form a spider-like interference structure in the negative direction [19], which explains the phenomena as shown in Figs. 3(a) and 3(g), respectively. For the ionization channel $t_2$ (as shown in Fig. 5(b) for $\beta$ = 1), the final position of the trajectory distributes in $x$ > 0 plane, interference between this re-scattering trajectory and the direct trajectory can generate a spider-like interference structure in the positive direction [40], which explains the phenomenon as shown in Fig. 3(b). By the above analysis, we observe the direction of the spider-like interference structure is associated with the orientation of electron motion.
Fig. 5. Typical trajectories of the re-scattered electrons which released from different ionization channels with different chirp parameters (a) $\beta$ = 1, ionization channel $t_1$ (blue line); (b) $\beta$ = 1, ionization channel $t_2$ (blue line); (c) $\beta$ = 4, ionization channel $t_1$ (red line).
We investigate the influence of the carrier-envelope phases for PMDs. Figure 6 shows the PMDs for the fixed chirp parameter $\beta$ = 1 with different carrier-envelope phases. Figures 6(a)–6(b) show the electric field of the few-cycle nonlinear chirped laser pulse (black line) and the ionization rates (red line) for carrier-envelope phases $\varphi =0.5 \pi$ and $\varphi =1.0 \pi$. We define the ionization time channel corresponding to the first and second maximum negative peaks as $t_1$ and $t_3$ channel and the first maximum positive peak as $t_2$ , which also corresponds to the ionization rate. For $\varphi$ = 0 and $\beta$ = 1, the EWP is released from the ionization channels $t_1$ and $t_2$ as shown in Fig. 1(a), the ionization channel $t_3$ is closed. For $\varphi =0.5 \pi$ and $\varphi =1.0 \pi$, the EWP is released from the ionization channels $t_1$, $t_2$ and $t_3$ as shown in Figs. 6(a)–6(b), the ionization channel $t_3$ is opened. It is indicated that the carrier-envelope phase can precisely control the opening of the ionization channel in the few-cycle nonlinear chirped field. Additionally, when the carrier-envelope phase changes from $0.5\pi$ to $1.0\pi$, the ionization rate of the ionization channels $t_1$, $t_2$ and $t_3$ slowly enhanced as shown in Figs. 6(a)–6(b).
Fig. 6. Electric field of the few-cycle nonlinear chirped laser pulse (black line) and the ionization rate (red line) for the fixed chirp parameter $\beta$ = 1 with different carrier-envelope phases (a) $\varphi =0.5 \pi$, (b) $\varphi =1.0 \pi$. PMDs simulated by SCTS model for the chirp parameter $\beta$ = 1 with different carrier-envelope phases (c) $\varphi =0.5 \pi$, (d) $\varphi =1.0 \pi$.
Figures 6(c)–6(d) show the PMDs correspond to Figs. 6(a)–6(b) by using the SCTS model. For $\varphi$ = 0, the temporal double-slit interference structure appears on the left side and a clear spider-like interference structure appears on the right side as shown in Fig. 2(a). As the carrier-envelope phase increases from $\varphi$ = 0 to $\varphi =1.0 \pi$, we can observe the temporal double-slit interference structure of the left side as shown in Fig. 2(a) transforms into a spider-like interference structure that is shown in Fig. 6(d), while the opposite change occurs on the right side, it shows that the spider-like interference structure of the right side as shown in Fig. 2(a) transforms into the temporal double-slit interference structure that is shown in Fig. 6(d). The above analysis demonstrates that the interference structures are sensitive to the carrier-envelope phase.
We also investigate the PMDs when a chirp-free second harmonic (SH) laser pulse is added to the chirped laser field. The chirp-free SH is linearly polarized along the $x$-axis. The wavelength and intensity of the chirp-free SH laser pulse is 400nm and 4 ${\% }$ of the chirped laser field, respectively. Figures 7(a)–7(c) show the electric field of the laser pulse and the ionization rate for different chirp parameters, where the gray line marked the few-cycle chirped laser pulse, the blue line marked the chirp-free SH laser pulse and the red line marked the ionization rate. For $\beta$ = 0, Fig. 7(a) shows that the laser pulse is symmetrical, while the symmetry of the electric field is broken for $\beta$ = 2 and $\beta$ = 4 as shown in Figs. 7(b)–7(c). The electric field transforms linearly polarized field into two-color parallel linearly polarized field when a chirp-free second harmonic (SH) laser pulse is added to the chirped laser field, the ionization channel correspondingly changes. From Fig. 7(a) we can see that the EWP can be released from one ionization channel when the chirp parameter is 0, which is recorded as $w_1$. In Fig. 7(b), it shows that the EWP is dominantly generated in the ionization channel $w_2$ for $\beta$ = 2. When the chirp parameter increases to 4, the EWP can emit from two ionization channels, which are recorded as $w_3$ and $w_4$, respectively. The ionization rates of $w_3$ and $w_4$ are approximately identical as shown in Fig. 7(c).
Fig. 7. Electric field of the few-cycle nonlinear chirped laser pulse and the chirp-free second harmonic (SH) laser pulse and the ionization rate for different chirp parameters (a) $\beta$ = 0, (b) $\beta$ = 2, (c) $\beta$ = 4, where the gray line marked the few-cycle nonlinear chirped laser pulse, the blue line marked the chirp-free SH laser pulse, the red line marked the ionization rate, and the relative phase is set as 0. PMDs simulated by SCTS model for different chirp parameters (d) $\beta$ = 0, (e) $\beta$ = 2, (f) $\beta$ = 4.
Figures 7(d)–7(f) show the PMDs corresponding to Figs. 7(a)–7(c) by using the SCTS model. For $\beta$ = 0, the spider-like interference structure points to the right side as shown in Fig. 7(d). For $\beta$ = 2, the spider-like interference structure points to the left side as shown in Fig. 7(e). As the chirp parameter increases to 4, the spider-like interference structure points to the right side again and a ring-like interference structure appears as indicated in the red border as shown in Fig. 7(f). In Fig. 7(c), we can see that the electrons in the ionization channels $w_3$ and $w_4$ are emitted around 1.5o.c. and 2.5o.c. respectively, the time interval of two channels is one optical cycle, the interference between the trajectories from the ionization channels $w_3$ and $w_4$ can generate the ring-like interference structure [15], which agrees with the red border shown in Fig. 7(f). From Figs. 7(d)–7(f) we can observe that the interference pattern from only spider-like interference structure to spider-like and ring-like interference structure, which are similar with that illustrated in Ref. [40].
4. Conclusion
In summary, we investigate theoretically the PMDs of Helium atom in the few-cycle nonlinear chirped laser pulse. We demonstrate that the spider-like interference structure can exhibits periodic variations with the increase of the chirp parameter. We also demonstrate that the carrier-envelope phase can precisely control the opening of the ionization channel in the few-cycle nonlinear chirped laser field. In addition, the interference patterns can change from only spider-like interference structure to both spider-like and ring-like interference structure when a chirp-free second harmonic (SH) laser pulse is added to the chirped laser field.
Funding
National Natural Science Foundation of China (12074142).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data that support the findings of this study are available from the corresponding author upon reasonable request.
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