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Abstract
By employing the frequency-domain theory, we investigate the influence of polarization directions on angle-resolved photoelectron energy spectrum in the above-threshold ionization (ATI) process of atoms exposed to the IR+XUV two-color laser fields, which shows the multiplateau structures. When the ionized electron is emitted along the IR laser’s polarization direction, the width of each plateau keeps a certain energy range, and the jet structures and main lobes are determined by both the emission angle relative to the polarization direction of the XUV laser field and the number of the XUV photons absorbed by the electron. While when the ionized electron is emitted along the XUV laser’s polarization direction, the width of each plateau depends on the polarization direction of the IR laser field, and the angular distribution of the ionized electron exhibits the isotropic characteristics. These results show that the ATI spectrum may be effectively controlled by changing the angle between the two laser fields’ polarization directions.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The interaction of matter and an intense laser field triggers various interesting phenomena, such as above-threshold ionization (ATI) [1–3], high-order harmonic generation (HHG) [4,5] and double ionization [6,7]. The ATI is one of the fundamental processes in the strong-field physics. So far, the ATI process has been extensively studied since its first experimental observation [8]. In this process, an electron can absorb much more photons than the minimum photon number necessary to overcome the ionization potential. The ATI spectrum encodes the information about the structure of atoms or molecules, as well as laser fields, which can uncover the underlying dynamics process [9–13]. Generally, the spectrum may consist of the low-energy and high-energy parts, which correspond to the direct ATI and high-order ATI (HATI) processes separately. The HATI process can be explained by the three-step model [14–16]. It was demonstrated that the low-energy spectrum obviously depends on the laser intensity [9,17,18], and it is also attributed to the Coulomb potential [19–21]. Recently, the carrier-envelope-phase (CEP) control of asymmetries in the ATI of xenon atoms exposed to ultrashort bichromatic fields was investigated [22]. Therefore, the ATI spectrum can be used to detect the CEP of the incident laser field [23], as well as the structures of atom and molecule [24–26].
With the development of the free-electron laser technology [27,28] and the application of HHG [29,30], the extreme ultraviolet (XUV) and the infrared (IR) two-color laser fields have become an efficient tool to investigate the strong-field physics community [31–36]. The different types of dichroism in shortpulse two-color XUV+IR multiphoton ionization were analyzed, especially for the circular dichroism [35,37–39]. The asymmetric structure of the ATI spectrum of an atom in two-color elliptically polarized laser fields was studied [40]. The first experimental investigation of linear dichroism in two-color XUV+IR ionization of He atom was performed [41], which shows that the formation of the sideband structures depends differently on the relative orientation of the IR and XUV polarization vectors. The investigations on the angular distribution of photoelectron and angle-resolved photoelectron energy spectrum are in favor of the analysis of the dichroism. The angle-resolved energy distribution of photoelectrons was studied when both lasers are linearly polarized along the same polarization direction [42]. When the energy of an XUV photon is much more than the atomic ionization potential, the angle-resolved energy distribution of photoelectrons presented a multiplateau structure [34–36,43,44]. The angular resolved ATI spectrum in IR+XUV orthogonally polarized two-color laser fields was discussed [45], which showed the dip structures of different plateaus. In recent experiments, it was shown that the amplitude of the sidebands varies strongly with the emission angle, and the angular distribution pattern is attributed to interferences between the different angular momenta for the emission electron in the multiphoton process [46]. It was further experimentally found that the photoelectron angular distribution strongly depends on the orientation of the IR polarization plane with respect to that of the XUV field [47]. These studies show that the angular distribution of the ionized electron depends on the polarization directions of two lasers and the emission directions of electron.
In this paper, by employing the frequency-domain theory based on the nonperturbative quantum electrodynamics [48,49], we will discuss the angle-resolved PES in the ATI process of atoms exposed to the IR+XUV two-color laser fields. Comparing with the time-dependent theory (e.g. TDSE), in this approach there are three advantages to deal with the strong-field physics [50]. Firstly, the key feature of this theory is time-independent, which can greatly save the computation time. Secondly, this approach provides a different concept for the physical processes of the strong-field, especially for recollision process (e.g. HATI, NSDI, HHG), which can generally be treated as a direct ATI followed by a laser-assisted collision or laser-assisted recombination so as to investigate the role of these subprocesses separately. Thirdly, all of the recollision processes can be investigated under a unified theoretical frame, which is in favor of analyzing the relationship among all these recollision processes. However, the investigation addressing the influence of the polarization directions on the ATI process is rare in detail. Here we will investigate the influence of polarization directions on angle-resolved PES in the ATI process of atoms exposed to the IR+XUV two-color laser fields. It is found that the formation of the spectrum depends on the angle between the emission direction of the ionized electron and polarization direction of the XUV laser field.
Atomic units are used throughout unless otherwise stated.
2. Theoretical method
In the frequency-domain theory based on the nonperturbative quantum electrodynamics, the transition matrix from the initial state $|\Psi _i\rangle$ to the final state $|\Psi _f\rangle$ in the ATI process can be expressed as [49]
where $V$ represents the potential of interaction between the electron and two laser fields, which can be expressed by $V=-[(-i\nabla ) \cdot \textbf {A}_1(-\textbf {k}_1\cdot \textbf {r})+(-i\nabla ) \cdot \textbf {A}_2(-\textbf {k}_2\cdot \textbf {r})+\textbf {A}_1(-\textbf {k}_1\cdot \textbf {r})\cdot (-i\nabla )+\textbf {A}_2(-\textbf {k}_2\cdot \textbf {r})\cdot (-i\nabla ) ]/2+[\textbf {A}_1(-\textbf {k}_1\cdot \textbf {r}) + \textbf {A}_2(-\textbf {k}_2\cdot \textbf {r})]^2/2$. Here $\textbf {A}_\gamma (-\textbf {k}_\gamma \cdot \textbf {r})=(2\omega _\gamma V_\gamma )^{-1/2}[\hat {\epsilon }_\gamma \exp ({i\textbf {k}_\gamma \cdot \textbf {r}})a_\gamma +c.c.]$, $\textbf {k}_\gamma$ is the wave vector, $a_\gamma (a^+_\gamma )$ is the annihilation (creation) operator, $V_\gamma$ is the normalization volume of the laser field, and $\hat {\epsilon }_\gamma =[\varepsilon _x\cos (\xi _\gamma /2)+i\varepsilon _y\sin (\xi _\gamma /2)]\exp [i\phi _\gamma /2]$ is the polarization vector, $\phi _\gamma$ is the initial phase of the two laser fields of the laser frequency $\omega _\gamma$, the angle $\xi _\gamma$ is the degree of polarization, $\xi _\gamma =0$ corresponds to the linear polarization for $\gamma$ = 1 or 2. The initial state $|\Psi _i\rangle$ can be written as with the initial energy being $E_i = - I_{p} + ({n_1} + 1/2){\omega _1} + ({n_2} + 1/2){\omega _2}$. Here $\Phi _i(\textbf {r})$ is the ground state of the atom, $|n_1,n_2\rangle$ is the Fock state of the laser field with the photon number $n_\gamma$ for $\gamma =$1 or 2, $I_{p}$ is the ionization potential of the atom. Based on the strong-field approximation, the influence of the Coulomb potential of the ion on the ionized electron and the effect of the laser field on the bound electron in the ground state are neglected. The final state of system $|\Psi _f\rangle =|\Psi _{\textbf {p}_f,q_1q_2}(\textbf {r})\rangle$ is the quantized-field Volkov state in the two-color laser fields, which can be expressed as [49]In the above, $q_3$, $q_4$, $q_5$ and $q_6$ can be considered dummy variables and be summed from $-\infty$ and $\infty$ [49]. By using Eqs. (2) and (3), the transition matrix for the ATI can be expressed as [43]
3. Numerical results
In this section, we will consider the ATI process of an atom exposed to the IR+XUV two-color linearly polarized laser fields, where the atomic ionization potential is $I_p=12.1$ eV. The frequencies of the IR and XUV laser fields are $\omega _1=1.165$ eV and $\omega _2=20\omega _1$, and their initial phases are set to be zero for simplicity. Under the present lasers’ conditions, we only consider the direct ATI process, because its contribution is much larger than the rescattering contributions.
Figure 1 shows the angle-resolved PESs for different laser intensities, where the ionized electron is emitted along the IR laser’s polarization direction, and $\theta$ is the angle between the polarization directions of IR and XUV lasers. It can be seen that each of the PESs shows a multiplateau structure, where the probability of the first plateau is much larger than that of other plateaus, and there exist several minimum values (dip structures) in each plateau when the angle between the direction of emitted electron and the polarization of the XUV laser field changes. If the angle $\theta =$ 0$^o$ or 180$^o$, the ATI spectrum will also show the terracelike structure similar with the previous results results [43]. Furthermore, for the electron emitted along the polarization of the IR laser field, it is found that the width of each plateau keeps a certain energy range. From Figs. 1(a)-(c) or (d)-(f), one can find that the width of each plateau increases with the intensity of the IR laser field, and even the overlap between the adjacent plateaus occurs. By comparing Figs. 1(a) and (d), it is shown that the ionization probability of the ATI process increases with the intensity of the XUV laser field, and the number of the plateaus also increases. It indicates that the IR and XUV laser fields play different roles in the ATI process, where the XUV laser field plays a crucial role to the ionization probability of the ATI process, and the IR laser field makes a dominant contribution to the width of each plateau, which are consistent with the cases in the IR+XUV orthogonally polarized two-color laser fields [45] and the experimental results [36].
Fig. 1. The angle-resolved PESs as a function of the angle $\theta$ between polarization directions of the XUV and IR two-color laser fields, where the electron is emitted along the IR laser polarization direction. The XUV intensity of (a)-(c) is $I_2=3.6\times 10^{13}$ W/cm$^2$, and its IR intensities are (a) $I_1=1.0\times 10^{11}$ W/cm$^2$, (b) $I_1=1.0\times 10^{12}$ W/cm$^2$ and (c) $I_1=1.0\times 10^{13}$ W/cm$^2$; the XUV intensity of (d)-(f) is $I_2=3.6\times 10^{14}$ W/cm$^2$, and its IR intensities are (d) $I_1=1.0\times 10^{11}$ W/cm$^2$, (e) $I_1=1.0\times 10^{12}$ W/cm$^2$, and (f) $I_1=1.0\times 10^{13}$ W/cm$^2$. (on logarithmic scale).
Next we show the angle-resolved PESs for the ionized electron being emitted along the polarization of the XUV laser field, as shown in Fig. 2. It can be seen that each of these spectra also shows a muliplateau structure, the ionization probability of the ATI process increases with the intensity of the XUV laser field, while the width of each plateau increases with the intensity of the IR laser field. These results are consistent with the above results in Fig. 1. Furthermore, it is noticed that the width of the ionized electron energy strongly depends on the angle between the emission direction of the ionized electron and the polarization direction of the IR laser field. The width of each plateau decreases with the change from $\theta =$ 0$^o$ (180$^o$) to 90$^o$, which is in well agreement with the experimental results [46]. Especially, the energy range is the largest for $\theta =$ 0$^o$ or 180$^o$, the energy range is the narrowest for $\theta =$ 90$^o$. This result can be also observed in the experiment for the first plateau [47]. By comparison with Fig. 2 and Fig. 1, the width of each plateau is determined by the angle between the emission direction of the ionized electron and the polarization direction of the IR laser; while the minimum value is determined by the angle between the emission direction of the ionized electron and the polarization direction of the XUV laser.
Fig. 2. The angle-resolved PESs as a function of the angle between polarization directions of the XUV and IR two-color laser fields, where the electron is emitted along the XUV laser polarization direction. The parameters of two laser fields are same as the case in Fig. 1. (on logarithmic scale).
In order to explain the reason of the formation of multiplateau structures shown in Figs. 1 and 2, we define the channel by the number $s_2$ of the XUV photons absorbed by the electron in the ATI process. Taking Fig. 1(b) as an example, Fig. 3 shows the channel contributions of atom absorbing one (a), two (b), and three (c) XUV photons, where other channels are not shown because of their little contributions. Comparing Fig. 3 and Fig. 1(b), the first, second and third plateaus in Fig. 1(b) are separately corresponding to the processes of atom absorbing one, two and three XUV photons. It indicates that the ionization probability rapidly decreases with an increase number of atom absorbing XUV photons. To explain the formation of Fig. 2(b), as an example, the corresponding channel contributions are shown in Fig. 4. One can see that the angle-resolved PES of Fig. 2(b) is also from contributions of atoms absorbing one, two and three XUV photons.
Fig. 3. Channel contributions of electron absorbing one (a), two (b) and three (c) XUV photons for the ionized electron being ionized along the polarization of the IR laser field $\varphi =0$. The upper triangular and lower triangular curves are plotted by the expressions $E_{max}=[\sqrt {2(s_2 \omega _2-I_p)-4U_{p_1}sin\varphi ^2}+2\sqrt {4U_{p_1}}cos\varphi ]^2/2$ and $E_{min}=[\sqrt {2(s_2 \omega _2-I_p)-4U_{p_1}sin\varphi ^2}-2\sqrt {4U_{p_1}}cos\varphi ]^2/2$, respectively, indicating the maximal and minimal energies obtained by electron in the corresponding channel. The intensities of IR and XUV laser fields are $I_1=1.0\times 10^{12}$W/cm$^2$ and $I_2=3.6\times 10^{13}$W/cm$^2$. (on logarithmic scale).
Fig. 4. Channel contributions of electron absorbing one (a), two (b) and three (c) XUV photons for the ionized electron being ionized along the polarization of the XUV laser field. The upper triangular and lower triangular curves indicate the maximal and minimal energies obtained by electron in the corresponding channel. The intensities of IR and XUV laser fields are $I_1=1.0\times 10^{12}$ W/cm$^2$ and $I_2=3.6\times 10^{13}$ W/cm$^2$. (on logarithmic scale).
So as to understand the formation of each plateau in the angle-resolved PES, we will analyze the generalized Bessel function $\aleph _{s_1 s_2}(\zeta _f)$ in the transition matrix Eq. (6). In present laser conditions, the arguments of the generalized Bessel function $\zeta _i\approx 0.0$ for $i=5$ and $6$, hence we set $J_{s_i}(\zeta _i)\approx 1.0$ and $s_i=0.0$ for $i=5$ and $6$. Therefore, the generalized Bessel function $\aleph _{s_1 s_2}(\zeta _f)$ can be reduced to [43]
In the following, we will discuss the characters of each plateau caused by the IR laser field, which is determined by the Bessel function $J_{ - {s_1}} ({\zeta _1},{\zeta _3})$. The ionized electron can absorb or emit many photons in the ATI process for an intense IR laser field. Hence the IR laser field can be treated as a classical field, $\textbf {A}_{cl}(t)=\hat {\varepsilon }_1 E_1/\omega _1 \cos (\omega _1 t)$ with $E_1$ being the amplitude of the laser field. With the help of the saddle-point approximation, the ionization time $t_0$ satisfies the energy conservation relationship in the ATI process [45]
where $s_2$ represents the number of XUV photons absorbed by the electron. Based on Eq. (8), if the angle $\varphi$ between the emission direction of the ionized electron and polarization direction of the IR laser field is lower than 90$^o$, the maximum energy of ionized electron can be expressed as $E_{max}=[\sqrt {2(s_2\omega _2-I_p)-4U_{p_1}sin^2\varphi }+2\sqrt {U_{p_1}}cos\varphi ]^2/2$, and the minimum energy can be expressed as $E_{min}=[\sqrt {2(s_2\omega _2-I_p)-4U_{p_1}sin^2\varphi }-2\sqrt {U_{p_1}}cos\varphi ]^2/2$. If $\varphi$ is larger than 90$^o$, the maximum and minimum positions exchange. When the ionized electron is emitted along the IR laser field, here the corresponding maximum and minimum positions are shown by the upper and lower triangular curves in Figs. 3(a)-(c). Also, as the ionized electron is emitted along the XUV laser field, the corresponding maximum and minimum positions are shown by the upper and lower triangular curves in Figs. 4(a)-(c). One can see that the classical results agree well with the quantum calculations. These results indicate that the ATI process of atoms in the IR+XUV two-color laser fields can be considered as following: the multiphoton ionization occurs by electron absorbing $s_2$ photons from the XUV laser field, which leads to the emergence of multiplateau structures; and then the ionized electron can be accelerated or decelerated by the IR laser field, leading to the formation for width of each plateau, where the energy range is equal to $\Delta E=4\sqrt {[2(s_2\omega _2-I_p)-4U_{p_1}sin^2\varphi ]U_{p_1}}cos\varphi$, which depends on the angle $\varphi$ between the emssion direction of the ionized electron and polarization direction of the IR laser field. It indicates that the energy width of each plateau is determined by the projection of the interaction between electron and the IR laser field on the polarization direction of the IR laser field. Especially, as the electron is emitted along the IR laser field i.e. $\varphi =0$, the range of energy remains unchanged $\Delta E=4\sqrt {2(s_2\omega _2-I_p)U_{p_1}}$. In Fig. 3, the ranges of energy $\Delta E$ for the first, second and third plateaus are 6.16 eV, 10.80 eV, and 13.98 eV, respectively.Figure 5 shows the ATI spectra as a function of the angle between the polarization directions of the two laser fields when the electron is emitted along the direction of the IR [(a)-(c)] or XUV [(d)-(f)] laser field. In Fig. 5(a), one can see that the ionized electron emitted along the polarization direction of the XUV laser field forms the main lobes, which is similar to the experimental result of the monochromatic XUV laser field [47]. It is noticed that the ionization probability of the ionized electron along the IR polarization is very low when the XUV polarization is perpendicular. This phenomenon had been explained by a zero point of the generalized Bessel function $\aleph _{s_1 s_2}(\zeta _f)$ [45] and the selection rule [51]. In Figs. 5(b) and (c), the jet structures appear between the main lobes. For electron absorbing even XUV photons, there exists a central jet; while for electron absorbing odd XUV photons, there is no central jet, and there are side jets in the case of electron absorbing three XUV photons. This result agrees with the experimental results in the monochromatic IR laser field [52,53]. On the other hand, when the ionized electron is emitted along the XUV laser field as shown in Figs. 5(d)-(f), the ATI spectra are independent of the the angle and exhibits the isotropic characteristics, no matter the number of XUV photons absorbed by the ionized electron is odd or even. As can be seen from Fig. 5, the jet structures in the ATI spectra are determined by both the angle between emission direction of the ionized electron and polarization direction of the XUV laser field and the number of photons absorbed by the emitted electron.
Fig. 5. The ATI spectra of the ionized electron as a function of the angle between the polarization directions of two lasers for the first [(a), (d)], second [(b), (e)] and third [(c), (f)] plateaus, when the ionized electron is emitted along the polarizations of the IR [(a)-(c)] and XUV [(d)-(f)] laser fields. The laser intensities for [(a)-(c)] and [(d)-(f)] are same as the parameters in Figs. 3 and 4, respectively.
The features of the ATI spectra in Fig. 5 are caused by the XUV laser field, we analyze the Bessel function $J_{ - {s_2}} ({\zeta _2},{\zeta _4})$ in Eq. (7). The parity property of the Bessel function is related to its order $s_2$ because of $J_{ - {s_2}} ({\zeta _2},{\zeta _4})=(-1)^{s_2}J_{ {s_2}} ({\zeta _2},{\zeta _4})$. It indicates that the angular distribution for electron absorbing even XUV photons has a central jet, while the angular distribution for electron absorbing odd photons has no central jet.
Figure 6 presents the Bessel function $|{J_{ - {s_2} }}({|\zeta }|,{\zeta _4})|^{2}$ as a function of the variable $|{\zeta }|$ for $s_2=$1(a), 2(b) and 3(c), respectively. In Fig. 6(a), when $|\zeta |=0.0$, $J_{ - {s_2}} ({|\zeta }|,{\zeta _4})\approx 0.0$, which leads to the disappearance of the central jet; then the Bessel function increases with $|\zeta |$, which leads to the formation of the main lobes. In Fig. 6(b), there exists a maximum value at $|\zeta |=0.0$, while there is a minimum value at $|\zeta |\approx 0.03$. The change from $|\zeta |=0.0$ to 0.03 leads to the formation of the central jet, and the formation of the main lobes comes from $|\zeta |>0.03$. In Fig. 6(c), the interval $[0, 0.048]$ of $|\zeta |$ leads to the side jets, and the interval $[0.048, 0.09]$ leads to the main lobes. As the order $s_2$ increases, the number of the minimum values of the Bessel function also increases, and these minimum values are sources of the dip structures in Fig. 1. Hence the jet structures and main lobes are determined by both the emission angle relative to the polarization direction of the XUV laser field and the number of the XUV photons that the electron absorbs.
Fig. 6. The Bessel function $|{J_{ - {s_2} }}({|\zeta }|,{\zeta _4})|^{2}$ as a function of the variable $|{\zeta }|$ for $s_2=$1(a), 2(b) and 3(c), respectively. The insets are the calculated angular distributions of $|{J_{ - {s_2} }}({|\zeta }|,{\zeta _4})|^{2}$. $|{\zeta }| = 2\sqrt {2u_{p_2}(s_2\omega _2-U_{p_1}-U_{p_2}-I_p)/{\omega _2}} |\cos \theta |$. The laser intensities are same as the parameters in Fig. 3.
4. Conclusion
In this paper, we have investigated the influence of polarization directions of two laser fields on the angle-resolved PES in the ATI process of atoms exposed to the IR+XUV two-color laser fields. The IR and XUV laser fields play different roles in the ATI process, where the IR laser field determines the width of each plateau, and the XUV laser field plays a crucial role for ionization probability. The angle-resolved PES shows the multiplateau structure. When the ionized electron is emitted along the IR laser’s polarization direction, the energy width of each plateau remains unchanged, and the jet structures in the ATI spectrum are determined by both the angle between direction of the emitted electron and polarization direction of the XUV laser field and the number of the photons that the electron absorbs. When the ionized electron is emitted along the XUV laser’s polarization direction, the width of each plateau is mainly determined by the angle between the emission direction of the ionized electron and the polarization direction of the IR laser, and the ATI spectrum shows the isotropic characteristics. Furthermore, by analyzing of the generalized Bessel function, it is found that the energy width of each plateau can be treated as the projection of the interaction between electron and the IR laser field on the polarization direction of the IR laser field. The angular distribution for electron absorbing even XUV photons has a central jet, while the angular distribution for electron absorbing odd photons has no central jet.
Funding
National Natural Science Foundation of China (11947080, 11804264, 11774215, 11774411, 12074418, 91950101).
Disclosures
The authors declare no conflicts of interest.
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