Open Access
Add to BibSonomy
Abstract
We investigate the above-threshold ionization (ATI) process of atoms exposed to the IR+XUV orthogonally polarized two-color laser fields by using the frequency-domain theory. It is shown that there exists a dip structure in each plateau of the angular resolved ATI spectrum. The dip structure in the first plateau is attributed to the fact that the electron cannot absorb one XUV photon when its emission direction is perpendicular to the XUV laser polarization, while the one in the second plateau is attributed to the coherent results of different channels. The emergence of dip structure is associated directly with the XUV laser field. Furthermore, by applying the saddle-point approximation, it is found that the fringes on the spectrum is caused by the interference of two trajectories for different saddle-points in the IR laser field. Finally, it is found that, in the high energy region, the probability of ATI spectrum is mainly determined by the XUV laser field, and the width of each plateau is mainly determined by the IR laser field; on the other hand, the ATI spectrum of the low energy region is only determined by the IR laser field.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Above-threshold ionization is one of the most fundamental processes in the case of atom exposed to an intense laser field, which has attracted more and more attention since its first observation [1]. The ATI can be interpreted as the process of multiphoton absorption: in addition to absorbing the minimum number of photons necessary to overcome the ionization potential, the electron also absorbs extra photons to experience the continuum-continuum transitions. In which process, the ionized electron encodes information about structures of target and laser field. Hence, until now, the ATI has been extensively studied, and a lot of research work have been done to uncover the underlying dynamics of the interaction between matter and laser field [2–4]. For example, the photoelectron angular distributions show the jet structures, which are related to the angular momentum of the atomic state [3]; it was found that the substructures of photoelectron spectrum come from ionization at different times mapping the pulse envelope [4].
Two-color laser fields have been proved to be an efficient tool to control the motion of ultrafast electron in atom and molecule [5,6]. Especially in recent years, the orthogonally polarized two-color (OTC) laser fields, usually with a fundamental laser field and its orthogonally polarized second harmonic field, have aroused great interest in the strong-field physics community, i.e., strong-field ionization, nonsequential double ionization (NSDI) [7,8], and high-order harmonic generation (HHG) [9,10]. By applying OTC pulses, it is possible to manipulate the atomic and molecular ionization [11–14]. For example, it was found that the angular-dependent asymmetries of ATI depend on the electron emission angle and the relative phase between the two-color laser fields [11], and the wave packets can be controlled with attosecond precision [13]. In the NSDI process, it was achieved that the correlated electron behavior can be controlled by tuning the phase between OTC pulses [15,16]. Also, it is demonstrated that the yield of HHG can be enhanced dramatically by OTC scheme [17,18]. Moreover, Frolov et al. analyzed the threshold phenomena in HHG, and also mentioned that the threshold phenomena in the ATI in two-color laser field are modified as compared to the case in a monochromatic laser field [19].
The availability of the two-color combined field of extreme ultraviolet (XUV) and infrared (IR) lasers offers a new way to steer and probe the motion of electron in atom and molecule [20–22]. Recently, the quantum coherent control of the photoelectron angular distribution was achieved [23]. Many studies showed that the ATI spectrum under IR+XUV two-color laser field represents a sideband structure [24–27], which can be explained by soft-photon approximation [24,28]. Especially, in ATI process, the XUV laser not only enhances the ionization probability, but also changes the ionization process, which is similar with the case of NSDI [29–31]. In the above works, the polarizations of two-color laser fields are always set to be same direction for simplicity. Recently, the dependence of ATI spectrum on the polarization direction of the two-color laser field was discussed [32]. It has been found that the ATI spectrum is independent of the angle between the two lasers’ polarization directions, and only several dips appear at certain angles, when the photon energy of one laser is larger than the atomic ionization potential. In [32], there are no clear interpretations for the emergence of dip structures.
In this paper, we investigate the ATI process of atoms exposed to IR+XUV OTC laser fields by employing the frequency-domain theory. The dip structure is shown in each plateau of the angular resolved ATI spectrum. By analyzing the interference between channels and applying the saddle-point approximation, we explain the formations of the multiplateau structure and dip structure on the spectrum. Furthermore, by changing intensities of the laser fields, we find that the IR and XUV laser fields play different roles in different regions of the ATI spectrum. Atomic units are used throughout unless otherwise stated.
2. Theoretical method
The frequency-domain theory for an atom exposed to a two-color laser field has been presented in details [25,32,33]. The theory will be briefly introduced in this paper. The ATI transition matrix from the initial state $|\Psi _i\rangle$ to the final state $|\Psi _f\rangle$ can be written as [33]
In the above, $V=-[(-i\nabla ) \cdot \textbf {A}_1(-\textbf {k}_1\cdot \textbf {r})+(-i\nabla ) \cdot \textbf {A}_2(-\textbf {k}_2\cdot \textbf {r})+\textbf {A}_1(-\textbf {k}_1\cdot \textbf {r})\cdot (-i\nabla )+\textbf {A}_2(-\textbf {k}_2\cdot \textbf {r})\cdot (-i\nabla ) ]/2+[\textbf {A}_1(-\textbf {k}_1\cdot \textbf {r}) + \textbf {A}_2(-\textbf {k}_2\cdot \textbf {r})]^2/2$ is the electron-photon interaction, where $\textbf {A}_s(-\textbf {k}_s\cdot \textbf {r})$ is the vector potential, $\textbf {k}_s$ is the wave vector of the laser field with the frequency $\omega _s$ for $s=1$ or $2$. The initial state can be expressed by $|\Psi _i\rangle =\Phi _i(\textbf {r})\otimes |n_1\rangle \otimes |n_2\rangle$, where $\Phi _i(\textbf {r})$ is the ground state wavefunction of the atom and $|n_j\rangle$ is the Fock state of the laser field with the photon number $n_j$ for $j=$1 or 2. The final state $|\Psi _f\rangle =|\Psi _{\textbf {p}_f,m_1m_2}(\textbf {r})\rangle$ is the quantized-field Volkov state in the two-color laser field [33]Using Eq. (2), Eq. (1) can be expressed by [25,32]
3. Numerical results
In this section, the ATI process of an atom in the IR+XUV OTC laser fields is considered. In this paper, the atom we choose is Xe with the corresponding atomic potential $I_p=12.1$ eV, and the ground state can be approximated by the Gaussian wavepacket, i.e., $\Phi (\textbf {p}_f)=(4\pi /\alpha )^{3/4}\exp [-(\textbf {p}^2_f/2\alpha )]$ with $\alpha =2I_p$ [34]. The frequencies of IR and XUV laser fields are $\omega _1=1.165$ eV and $\omega _2=40\omega _1$, and their initial phases are set to be zero for simplicity.
Figure 1 shows the angular resolved ATI spectra for different lasers’ intensities. Firstly, one can see that each ATI spectrum shows a multiplateau structure, where the probability of the first plateau is larger than that of the second one by about five orders of magnitude. The multiplateau structure in the spectra is similar as that for other two-color laser cases in our previous results [22,25]. Furthermore, it is noted that there exists a dip structure in each plateau, where the dips in the first plateau are located at $\theta _f\approx 0^\circ$ and $180^\circ$ (details are shown in Fig. 2(a)), and the dips in the second plateau are located at the certain angles. This result is different from the case that the dip structures only appear in the second plateau in the high-order ATI (HATI) process [22]. Then comparing with Figs. 1(a) to 1(c) or 1(d) to 1(f), it is found that the width of each plateau increases with the intensity of IR laser field. At last, comparing Figs. 1(a) with 1(d), it is found that the ionization probability of the corresponding plateau increases with the intensity of the XUV laser field. It indicates that the ionization probability is dominated by the XUV laser field, while the width of the plateau is dominated by the IR laser field, which agrees with the results shown in [25]. In the following, we will take Fig. 1(e) as an example to explain the formation of the above angular resolved ATI spectra.
Fig. 1. The angular resolved ATI spectra for different laser intensities. The top row is for the intensity of the XUV laser field $I_2=5.0\times 10^{12}$ W/cm$^2$ and different intensities of the IR laser field: $I_1=1.0\times 10^{12}$ W/cm$^2$ (a), $1.0\times 10^{13}$ W/cm$^2$ (b) and $5.0\times 10^{13}$ W/cm$^2$ (c). The bottom row is for the intensity of the XUV laser field $I_2=5.0\times 10^{13}$ W/cm$^2$ and different intensities of the IR laser field: $I_1=1.0\times 10^{12}$ W/cm$^2$ (d), $1.0\times 10^{13}$ W/cm$^2$ (e) and $5.0\times 10^{13}$ W/cm$^2$ (f). The emission angle $\theta _f$ is the angle between the IR laser’s polarization and the emission direction of the ionized electron (on logarithmic scale).
Fig. 2. (a) and (b) show the angular resolved ATI spectra for $s_2=1$ and $s_2=2$, respectively. The dash (black) and solid (red) lines show the beginning and cutoff positions determined by Eq. (8) for the first and second plateau, respectively. The insets in (a) and (b) presents the details for emission angle $\theta _f=0^\circ$ or $180^\circ$. (c) and (d) show the corresponding angular distributions for $s_2=1$ and $s_2=2$, respectively. The parameters of IR and XUV laser fields are the same as in Fig. 1(e). (on logarithmic scale).
Figures 2(a) and 2(b) show the contributions of atom absorbing one and two XUV photons in the ATI process, respectively. Comparing Fig. 1(e) with Figs. 2(a) and 2(b), one can see that the first and second plateaus are attributed to the atom absorbing one and two XUV photons respectively, which is the reason why the probability of the first plateau is much larger than that of the second one, while the energy value of the first plateau is lower than that of the second one. Figures 2(c) and 2(d) show angular distributions of the two plateaus. It is found that the probability of the first plateau is very low when ionized electron is emitted along the angles $\theta _f\approx 0^\circ$ and $180^\circ$. This is obviously different from the case of the second plateau, where the details are shown in the insets of Figs. 2(a) and 2(b).
In order to further investigate the spectral characteristics shown in Figs. 2 (a) and 2(b), Fig. 3 shows the ATI spectra for different emission angles. One can see that each of the ATI spectra represents a step-like structure, and the width of each step structure decreases with the increase of emission angle $\theta _f$. This is in agreement with the experimental result [35]. In Fig. 3(a), one can find that it makes greatest contributions to the ATI that the electron absorbs two XUV photons. From Figs. 3(b)–3(d), one can see that the probability of plateau decreases dramatically with the increase of the XUV photons absorbed by the atom, which is obviously different from the case shown in Fig. 3(a).
Fig. 3. The ATI spectra for different emission angles of $\theta _f=0^\circ$ (a), $30^\circ$ (b), $60^\circ$ (c), and $85^\circ$ (d). In the above, the contributions of atom absorbing all, zero, one, two and three XUV photons to ATI process are shown by the solid (black), dash (red), dot (blue), short-dash (dark cyan), short-dot (orange) lines, respectively. The parameters of IR and XUV laser fields are the same as in Fig. 1(e). (on logarithmic scale).
Through the analysis of Eq. (5), it is found that the characteristic of the angular resolved ATI spectrum is dependent to the generalized Bessel function $\aleph _{s_1 s_2}(\zeta _f)$. For the OTC laser fields, one can find that $\zeta _5=0$ and $\zeta _6=0$, hence we have that $J_{-j_5}(\zeta _5)= 1.0$ and $J_{-j_6}(\zeta _6)=1.0$ for $j_5=0$ and $j_6=0$. Therefore the generalized Bessel function $\aleph _{s_1 s_2}(\zeta _f)$ can be rewritten as
In the following, we first analyze the effect of IR laser field on the angular resolved ATI spectrum. By using the saddle-point approximation, the Bessel function ${{J_{ - {s_1}}}} ({\zeta _1},{\zeta _3})$ in Eq. (6) can be rewritten as [29]
On the other hand, one can find that the interference characteristic of the angular resolved ATI spectrum also depends on the term $J_{-s_1}(\zeta _1,\zeta _3)$. Based on Eq. (7), the interference is attributed to the term $\cos [f(t_0)-\pi /4]$. When $\cos [f(t_0)-\pi /4]=0$, the destructive interference appears, as shown by the closed squares in Fig. 4. One can see that the dark squares have a good agreement with the locations of the destructive interference fringes by the numerical calculation. Therefore, it indicates that the fringes is attributed to the interference between the two trajectories born at the different saddle-points $t_0$ and $2\pi /\omega _1-t_0$.
Fig. 4. Interference pattern of the first (a) and second (b) plateaus. The closed squares are the locations of destructive interference determined by $\cos [f(t_0)-\pi /4]=0$ in Eq. (7). The parameters of IR and XUV laser fields are the same as in Fig. 1(e). (on logarithmic scale).
In order to explain the dip structure in the first plateau of the angular resolved ATI spectrum, we analyze the Bessel function ${J_{ - {s_2} }}({\zeta _2},{\zeta _4})$. It is noticed that $\zeta _2=0$ for a fixed value $\zeta _4$, since $J_n(0)=\delta _{n0}$, and thus ${J_{ - {s_2} }}(0,{\zeta _4})={J_{ - {s_2/2}}}({\zeta _4})$ for even $s_2$ and $j_4=s_2/2$. Therefore, the value of ${J_{ - {s_2} }}(0,{\zeta _4})$ does not vanish only for even $s_2$ and $j_4=s_2/2$. For the first plateau of the angular resolved ATI spectrum, i.e., $s_2=1$: since $\zeta _2=0$ if $\theta _f$=0$^\circ$ or 180$^\circ$, and hence ${J_{ - {s_2} }}(0,{\zeta _4})={J_{ - {j_4}}}({\zeta _4})=0$ with $j_4\neq s_2/2$, which leads to ${\aleph _{{s_1} {s_2}}}(\zeta _f)=0$, resulting in a zero point of the generalized Bessel function. Thus the probability of the first plateau is zero when the ionized electron is emitted along the polarization of IR laser field, leading to the emergency of dip structure in the first plateau. Moreover, we can also understand this phenomena from the viewpoint of system parity: based on the previous works [36], the electron can only absorbs even photons from the laser field when it is emitted along the direction perpendicular to the laser polarization, hence the probability for the electron absorbing one XUV photon is zero since the ionized electron is emitted along the direction perpendicular to the XUV laser polarization.
Next in order to explain the formation of dip structure in the second plateau, where $s_2=2$, we define the channel by the parameter $j_4$. Figures 5(a) and 5(b) show the channel contributions for $j_4=0$ and $j_4=1$, while the contributions of other channels are negligible. It is noticed that the probability of $j_4=0$ is very low when the ionized electron is emitted along $\theta _f$=0$^\circ$ and 180$^\circ$, while the channel of $j_4=1$ makes an important contribution to the ATI process because of $j_4=s_2/2$. Figures 5(c) and 5(d) show the incoherent and coherent summation of channels of $j_4=0$ and $j_4=1$. It is found that the dip structure in the second plateau is attributed to the coherent summation of both channels of $j_4=0$ and $j_4=1$, which is similar with the case of atom absorbing two photons in monochromatic IR laser field [37].
Fig. 5. (a) and (b) show the angular resolved ATI spectra for channels of $j_4=0$ and $j_4=1$. (c) and (d) show the incoherent and coherent results of (a) and (b). The parameters of IR and XUV laser fields are the same as in Fig. 1(e). (on logarithmic scale).
From the above results, it is noticed that the order $-s_2+2j_4$ of the Bessel function $J_{ - {s_2} + 2{j_4} }({\zeta _2})$ in the generalized Bessel function ${J_{ - {s_2} }}({\zeta _2},{\zeta _4})$ represents the total number of the photons that the atom absorbs during the interaction between the electron and the XUV laser field [33]. The electron does not get extra energy when $\zeta _2=0$, thus the number of XUV photon absorbed by the electron satisfies the relationship $s_2/2=j_4$. For the first plateau, the relationship will be violated if $\theta _f$=0$^\circ$ or 180$^\circ$, therefore the ionization probability is very low. For the second plateau, there exits a channel satisfying the relationship, so the probability of electron ionized along the IR laser field is not zero. we also notice that the forming reasons of the dip structures in the ATI process are entirely different from that in the HATI process with the parallel polarized laser pulses [22].
Finally, Fig. 6 shows the ATI spectra of the ionized electron emitted along the angles $\theta _f=0^\circ$ and $30^\circ$ for different laser intensities. One can see that, no matter how the laser intensities change, the ATI spectra for $\theta _f=0^\circ$ are attributed to the process that electron absorbs even XUV photons, which is quite different from the situation for $\theta _f=30^\circ$. Furthermore, it is found that, in the higher energy region, the probability of ATI spectrum is mainly determined by the XUV laser field, and the width of each plateau in the spectrum is mainly determined by the IR laser field; however, in the lower energy region, the ATI spectrum is only determined by the IR laser field, especially for the lower ATI region for $\theta _f=0^\circ$.
Fig. 6. The ATI spectra with lasers intensities: [(a),(b)] $I_1=1.0\times 10^{13}$ W/cm$^2$, $I_2=5.0\times 10^{14}$ W/cm$^2$; [(c),(d)] $I_1=1.0\times 10^{13}$ W/cm$^2$, $I_2=5.0\times 10^{13}$ W/cm$^2$; and [(e),(f)] $I_1=1.0\times 10^{14}$ W/cm$^2$, $I_2=5.0\times 10^{13}$ W/cm$^2$. In the left column, the angle $\theta _f$ is chosen to 0$^\circ$; in the right column, the angle $\theta _f$ is chosen to 30$^\circ$. (on logarithmic scale).
4. Conclusion
In this paper, we have investigated the ATI process of atoms exposed to the IR+XUV OTC laser fields. It has been found that there exists a dip structure of each plateau in the angular resolved ATI spectrum. By the analysis, the dip structure in the first plateau is caused by the fact that the electron cannot absorb one XUV photon when its emission direction is perpendicular to the XUV laser polarization, while the one in the second plateau is from the coherent results of different channels. Moreover, it has been found that the fringes of the angular resolved ATI spectrum is attributed to the interference between two trajectories born at the different saddle-points $t_0$ and $2\pi /\omega _1-t_0$ in IR laser field. Finally, by changing intensities of the laser field, it has been shown that, in the high energy region, the probability of ATI spectrum is mainly determined by the XUV laser field, and the width of each step is mainly determined by the IR laser field; on the other hand, the ATI spectrum of the low energy region is only determined by the IR laser field.
Funding
National Natural Science Foundation of China (NSFC) (11474348, 11774411, 11804264, 11847146); the Research Fund for the Doctoral Program of Xi’an Aeronautical University (2018KY0205, 2018KY0206, 2018KY0207).
References
1. P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, “Free-free transitions following six-photon ionization of xenon atoms,” Phys. Rev. Lett. 42(17), 1127–1130 (1979). [CrossRef]
2. W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. Paulus, and H. Walther, “Above-threshold ionization: From classical features to quantum effects,” Adv. At., Mol., Opt. Phys. 48, 35–98 (2002). [CrossRef]
3. M. Li, Y. Liu, H. Liu, Y. Yang, J. Yuan, X. Liu, Y. Deng, C. Wu, and Q. Gong, “Photoelectron angular distributions of low-order above-threshold ionization of xe in the multiphoton regime,” Phys. Rev. A 85(1), 013414 (2012). [CrossRef]
4. R. Della Picca, A. A. Gramajo, C. R. Garibotti, S. D. López, and D. G. Arbó, “Nonconstant ponderomotive energy in above-threshold ionization by intense short laser pulses,” Phys. Rev. A 93(2), 023419 (2016). [CrossRef]
5. M. Richter, M. Kunitski, M. Schöffler, T. Jahnke, L. P. H. Schmidt, M. Li, Y. Liu, and R. Dörner, “Streaking temporal double-slit interference by an orthogonal two-color laser field,” Phys. Rev. Lett. 114(14), 143001 (2015). [CrossRef]
6. M. Li, W.-C. Jiang, H. Xie, S. Luo, Y. Zhou, and P. Lu, “Strong-field photoelectron holography of atoms by bicircular two-color laser pulses,” Phys. Rev. A 97(2), 023415 (2018). [CrossRef]
7. Z. Yuan, D. Ye, Q. Xia, J. Liu, and L. Fu, “Intensity-dependent two-electron emission dynamics with orthogonally polarized two-color laser fields,” Phys. Rev. A 91(6), 063417 (2015). [CrossRef]
8. A. Chen, M. F. Kling, and A. Emmanouilidou, “Controlling electron-electron correlation in frustrated double ionization of triatomic molecules with orthogonally polarized two-color laser fields,” Phys. Rev. A 96(3), 033404 (2017). [CrossRef]
9. Y. Zheng, H. Diao, Z. Zeng, X. Ge, R. Li, and Z. Xu, “Manipulating electron-ion recollision in a midinfrared laser field,” Phys. Rev. A 92(3), 033417 (2015). [CrossRef]
10. S. Stremoukhov, A. Andreev, B. Vodungbo, P. Salières, B. Mahieu, and G. Lambert, “Origin of ellipticity of high-order harmonics generated by a two-color laser field in the cross-polarized configuration,” Phys. Rev. A 94(1), 013855 (2016). [CrossRef]
11. S. Luo, M. Li, H. Xie, P. Zhang, S. Xu, Y. Li, Y. Zhou, P. Lan, and P. Lu, “Angular-dependent asymmetries of above-threshold ionization in a two-color laser field,” Phys. Rev. A 96(2), 023417 (2017). [CrossRef]
12. M. Han, P. Ge, Y. Shao, M.-M. Liu, Y. Deng, C. Wu, Q. Gong, and Y. Liu, “Revealing the sub-barrier phase using a spatiotemporal interferometer with orthogonal two-color laser fields of comparable intensity,” Phys. Rev. Lett. 119(7), 073201 (2017). [CrossRef]
13. L. Zhang, X. Xie, S. Roither, D. Kartashov, Y. Wang, C. Wang, M. Schöffler, D. Shafir, P. B. Corkum, A. Baltuška, I. Ivanov, A. Kheifets, X. Liu, A. Staudte, and M. Kitzler, “Laser-sub-cycle two-dimensional electron-momentum mapping using orthogonal two-color fields,” Phys. Rev. A 90(6), 061401 (2014). [CrossRef]
14. Q. Ji, J. Wang, P. Lu, H. Li, X. Gong, Q. Song, K. Lin, W. Zhang, J. Ma, M. F. Kling, H. Zeng, J. Wu, and F. He, “Subcycle characterization of photoelectron emission with multicycle laser pulses,” Phys. Rev. A 96(5), 053423 (2017). [CrossRef]
15. L. Chen, Y. Zhou, C. Huang, Q. Zhang, and P. Lu, “Attosecond-resolved electron emission in nonsequential double ionization,” Phys. Rev. A 88(4), 043425 (2013). [CrossRef]
16. L. Zhang, X. Xie, S. Roither, Y. Zhou, P. Lu, D. Kartashov, M. Schöffler, D. Shafir, P. B. Corkum, A. Baltuška, A. Staudte, and M. Kitzler, “Subcycle control of electron-electron correlation in double ionization,” Phys. Rev. Lett. 112(19), 193002 (2014). [CrossRef]
17. I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]
18. L. Brugnera, F. Frank, D. J. Hoffmann, R. Torres, T. Siegel, J. G. Underwood, E. Springate, C. Froud, E. I. C. Turcu, J. W. G. Tisch, and J. P. Marangos, “Enhancement of high harmonics generated by field steering of electrons in a two-color orthogonally polarized laser field,” Opt. Lett. 35(23), 3994–3996 (2010). [CrossRef]
19. M. V. Frolov, N. L. Manakov, T. S. Sarantseva, A. A. Silaev, N. V. Vvedenskii, and A. F. Starace, “Control of threshold enhancements in harmonic generation by atoms in a two-color laser field with orthogonal polarizations,” Phys. Rev. A 93(2), 023430 (2016). [CrossRef]
20. L.-Y. Peng, W.-C. Jiang, J.-W. Geng, W.-H. Xiong, and Q. Gong, “Tracing and controlling electronic dynamics in atoms and molecules by attosecond pulses,” Phys. Rep. 575, 1–71 (2015). [CrossRef]
21. F. Jin, J. Chen, Y. Yang, X. Liu, Z.-C. Yan, and B. Wang, “Nonsequential double ionization of helium in IR+XUV two-color laser fields II: collision-excitation ionization process,” J. Phys. B: At., Mol. Opt. Phys. 51(3), 035601 (2018). [CrossRef]
22. F. Jin, F. Li, Y. Yang, J. Chen, X. Liu, and B. Wang, “Angle-resolved photoelectron energy spectrum from the high-order above-threshold ionization process in IR+XUV two-color laser fields,” J. Phys. B: At., Mol. Opt. Phys. 51(24), 245601 (2018). [CrossRef]
23. E. V. Gryzlova, A. N. Grum-Grzhimailo, E. I. Staroselskaya, N. Douguet, and K. Bartschat, “Quantum coherent control of the photoelectron angular distribution in bichromatic-field ionization of atomic neon,” Phys. Rev. A 97(1), 013420 (2018). [CrossRef]
24. P. Radcliffe, M. Arbeiter, W. B. Li, S. Düsterer, H. Redlin, P. Hayden, P. Hough, V. Richardson, J. T. Costello, T. Fennel, and M. Meyer, “Atomic photoionization in combined intense XUV free-electron and infrared laser fields,” New J. Phys. 14(4), 043008 (2012). [CrossRef]
25. K. Zhang, J. Chen, X.-L. Hao, P. Fu, Z.-C. Yan, and B. Wang, “Terracelike structure in the above-threshold-ionization spectrum of an atom in an ir+xuv two-color laser field,” Phys. Rev. A 88(4), 043435 (2013). [CrossRef]
26. K. Zhang, M. Liu, B.-B. Wang, Y.-C. Guo, Z.-C. Yan, J. Chen, and X.-J. Liu, “Role of XUV photons in atomic high-order above-threshold ionization processes in IR+XUV two-color laser fields,” Chin. Phys. Lett. 34(11), 113201 (2017). [CrossRef]
27. S. Minemoto, H. Shimada, K. Komatsu, W. Komatsubara, T. Majima, T. Mizuno, S. Owada, H. Sakai, T. Togashi, S. Yoshida, M. Yabashi, and A. Yagishita, “Ar 3p photoelectron sideband spectra in two-color XUV + NIR laser fields,” J. Phys. B: At., Mol. Opt. Phys. 51(7), 075601 (2018). [CrossRef]
28. A. J. Galán, L. Argenti, and F. Martin, “The soft-photon approximation in infrared-laser-assisted atomic ionization by extreme-ultraviolet attosecond-pulse trains,” New J. Phys. 15(11), 113009 (2013). [CrossRef]
29. F. Jin, Y. Tian, J. Chen, Y. Yang, X. Liu, Z.-C. Yan, and B. Wang, “Nonsequential double ionization of helium in IR+XUV two-color laser fields: Collision-ionization process,” Phys. Rev. A 93(4), 043417 (2016). [CrossRef]
30. F. Jin, J. Chen, Y. Yang, Z.-C. Yan, and B. Wang, “Intensity dependence of nonsequential double ionization of helium in IR+XUV two-color laser fields,” J. Phys. B: At., Mol. Opt. Phys. 49(19), 195602 (2016). [CrossRef]
31. S. X. Hu, “Boosting photoabsorption by attosecond control of electron correlation,” Phys. Rev. Lett. 111(12), 123003 (2013). [CrossRef]
32. M. Liu, Y.-C. Guo, and B.-B. Wang, “Dependence of above-threshold ionization spectrum on polarization directions of two-color laser fields,” Chin. Phys. B 24(7), 073201 (2015). [CrossRef]
33. D.-S. Guo and G. W. F. Drake, “Stationary solutions for an electron in an intense laser field. II. multimode case,” J. Phys. A: Math. Gen. 25(20), 5377–5394 (1992). [CrossRef]
34. P. Fu, B. Wang, X. Li, and L. Gao, “Interrelation between high-order harmonic generation and above-threshold ionization,” Phys. Rev. A 64(6), 063401 (2001). [CrossRef]
35. S. Düsterer, G. Hartmann, F. Babies, A. Beckmann, G. Brenner, J. Buck, J. Costello, L. Dammann, A. D. Fanis, P. Geßler, L. Glaser, M. Ilchen, P. Johnsson, A. K. Kazansky, T. J. Kelly, T. Mazza, M. Meyer, V. L. Nosik, I. P. Sazhina, F. Scholz, J. Seltmann, H. Sotoudi, J. Viefhaus, and N. M. Kabachnik, “Angle resolved photoelectron spectroscopy of two-color XUV–NIR ionization with polarization control,” J. Phys. B: At., Mol. Opt. Phys. 49(16), 165003 (2016). [CrossRef]
36. P. A. Korneev, S. V. Popruzhenko, S. P. Goreslavski, T.-M. Yan, D. Bauer, W. Becker, M. Kübel, M. F. Kling, C. Rödel, M. Wünsche, and G. G. Paulus, “Interference carpets in above-threshold ionization: From the coulomb-free to the coulomb-dominated regime,” Phys. Rev. Lett. 108(22), 223601 (2012). [CrossRef]
37. R. Reichle, H. Helm, and I. Y. Kiyan, “Photodetachment of ${H}^{{-}}$ in a strong infrared laser field,” Phys. Rev. Lett. 87(24), 243001 (2001). [CrossRef]
