Orbital-resolved photoelectron momentum distributions of F<sup>-</sup> ions in a counter-rotating bicircular field

Jian-Hong Chen; Liang-Cai Wen; Song-Feng Zhao

doi:10.1364/OE.481153

1. Introduction

Laser-induced ionization process has laid the foundation for strong-field physics and attosecond science [1]. Both the direct ionization and the rescattering process encode sub-cycle information of ionization dynamics and provides opportunities for high temporal- and spatial-resolution imaging of atomic and molecular structure and ionization dynamics [2–4]. Much important information on the ionization process and target structure can be revealed from the photoelectron momentum distributions (PMDs) of atoms and negative ions [5–10]. For example, the PMDs induced by the elliptically polarized laser field (so-called attoclock scheme) can be employed to image the ionization process of atoms and molecules with attosecond time resolution [11–15].

Recently, the strong-field ionization induced by two-color circularly fields has also been used to manipulate the ionization process and electron dynamics [16–25]. In the two-color circularly scheme, it has been revealed that the interference pattern in the PMDs contains rich information of ionization dynamics [21–24]. For example, the time-resolved phase and the amplitude of electron wave packets can be effectively extracted from the inter-cycle interference pattern induced by the two-color co-rotating circularly field, i.e., so-called double-pointer attoclock scheme [21]. Besides that, the two-color co-rotating or counter-rotating circular fields have been exploited for imaging of the spin-resolved electron dynamics [25,26], the HHG process [27], and spin–orbit interaction [28].

On the other hand, most previous experimental results were interpreted based on the assumption of s-electron initial states even for noble-gas atoms and some negative ions carry p-electron initial states. Recently, the important role of atomic orbitals (especially for the sign of the magnetic quantum number) in strong-field ionization has been experimentally and theoretically revealed [29–42]. The most remarkable phenomenon is the dependence of ionization probability on the sign of the magnetic quantum number [30]. It has been found that strong-field ionization by circularly polarized light is sensitive to the sign of the magnetic quantum number [30]. This phenomenon was used to probe the residual-ion ring current induced by circularly polarized laser fields [36], and enabled the production of spin-polarized electron bunches [41,42]. However, there usually exists large discrepancies between experimental measurements and strong-field approximation (SFA) calculations, which is associated with the ignorance of the Coulomb potential in SFA model [23,28]. In order to eliminate the effect of the Coulomb potential, we propose the PMDs of F^- ions with p-electron initial states as an ideal platform to identify the effect of atomic orbitals on the electron dynamics during strong-field ionization. Apart from the typical advantage of the short-range potential perfectly available for strong-field approximation (SFA) calculations, the clear signature of the atomic orbital can be readily identified from the PMDs for F^- ions because of cleaner interference patterns in the PMDs at lower laser intensity.

In this paper, we theoretically study the two-dimensional PMD of F^- ions for different initial orbitals by a single-color circular field and a two-color counter-rotating circularly field, respectively. The investigation is carried out by the numerical calculations from the three-dimensional time-dependent Schrödinger equation (3D-TDSE) and the SFA method. By comparing the photoelectron energy spectra and PMDs calculated from TDSE and SFA methods, the perfect agreement indicates that the SFA model is a very good approximation as expected. We demonstrate that the PMDs of F^- ions can be modulated from an isotropic symmetric distribution into a three-lobe one by adding a weak fundamental counter-rotating field to the intense second harmonic circularly polarized one. The modulation strongly depends on the initial atomic orbital. Furthermore, based on the saddle-point (SP) method, it is found that the atomic orbital strongly affects not only the interference patterns but also electron distributions of the PMDs. We identify that the orbital-dependent electron distributions mainly originate from the orbital-dependent initial momentum at the tunnel exit. More importantly, it is demonstrated that the lobes in PMDs appear in sequential order. Atomic units are used in this paper unless stated otherwise.

2. Theoretical methods

2.1 TDSE method

The details of the three-dimensional TDSE method can be found in previous work [43–47]. In this work, we use the algorithm and the source code of Mosert and Bauer [43]. Briefly, the first step is to obtain atomic initial-state wave-function. Within the single-electron approximation, in our simulation, to match correct ground-state energy ${I_p} = 3.4eV$ of the p state for F^- ions, we use the short-range potential as follows [48],

(1)$$V(r) ={-} {a_1}\frac{{{e^{ - {a_2}r}}}}{r} - {a_3}\frac{{{e^{ - {a_4}r}}}}{r}$$

where ${a_1} = 5.137$, ${a_2} = 1.288$, ${a_3} = 3.863$ and ${a_4} = 3.545$[48]. The Hamiltonian describing the laser–atom interaction is given by

(2)$$H(t) ={-} \frac{{{\nabla ^2}}}{2} + V(r) + i{\boldsymbol A}(t) \cdot \nabla$$

where the laser field is treated in the dipole approximation, i.e., we use a time-dependent vector potential ${\boldsymbol A}(t)$ to define our field. In this work, the vector potential of the counter-rotating bi-circular laser pulse is written as

(3)$$\begin{aligned} {\boldsymbol A}(t) &= \frac{1}{{\sqrt 2 }}{\sin ^2}\left( {\frac{{\pi t}}{\tau }} \right)[{ - {A_{2\omega }}\sin (2\omega t) - {A_\omega }\sin (\omega t + \Delta \varphi )} ]\hat{x}\\ &+ \frac{1}{{\sqrt 2 }}{\sin ^2}\left( {\frac{{\pi t}}{\tau }} \right)[{{A_{2\omega }}\cos (2\omega t) - {A_\omega }\cos (\omega t + \Delta \varphi )} ]\hat{y} \end{aligned}$$

for the time interval $(0,\tau )$ and zero elsewhere. $\tau$ is the total duration of the laser pulse and $\omega$ is the angular frequency of the fundamental field. ${A_\omega }$ and ${A_{2\omega }}$ are the peak vector potentials of the fundamental laser field and second harmonic field, respectively. $\Delta \varphi$ is the phase delay between fundamental laser field and second harmonic field. $\hat{x}$ and $\hat{y}$ are the unit polarization vectors in the polarization plane, respectively. In the calculations presented below, we use the finite difference method to discretize the radial coordinate with radial range $0 < r \le 1150a.u.$ and the radial grid spacing $dr = 0.02a.u.$ The time step $dt = 0.05a.u.$, and the maximum number of partial waves ${l_{\max }} = 40$.

2.2 SFA model

Within the SFA model, the direct transition amplitude from atomic or ionic ground state to Volkov state is given by [5]

(4)$${M_{lm}}({\boldsymbol k}) ={-} i\int_{ - \infty }^\infty {\left\langle {{\boldsymbol k} + {\boldsymbol A}(t)|{\boldsymbol r} \cdot {\boldsymbol E}(t )|{\psi_n}_{lm}} \right\rangle } \exp [iS(t)]dt$$

where ${\boldsymbol E}(t )={-} {\partial _t}{\boldsymbol A}(t)$ is the instantaneous laser electric field. ${\psi _n}_{lm}$ is the hydrogen-like atomic wave function of initial ground state in coordinate representation, here n, $l$ and m are the principal, orbital and magnetic quantum numbers, respectively. $S(t)$ is the classical action which can be written as

(5)$$S(t) = \int_{ - \infty }^t {\left[ {\frac{1}{2}{{({\boldsymbol k} + {\boldsymbol A}(t^{\prime}))}^2} + {I_p}} \right]dt^{\prime}}$$

By using Fourier transform, the probability amplitude of Eq. (4) is rewritten as [41]

(6)$${M_{lm}}({\boldsymbol k}) = \int_{ - \infty }^\infty {{\boldsymbol E}(t )\cdot [{{\nabla_{\boldsymbol q}}{{\tilde{\varphi }}_{nlm}}({\boldsymbol q})} ]} \exp [iS(t)]dt$$

where ${\boldsymbol q} = {\boldsymbol k} + {\boldsymbol A}(t)$ is the canonical momentum. ${\nabla _{\boldsymbol q}}$ is the gradient calculated in canonical momentum coordinates. Here we have introduced the Fourier transform of the initial state function, ${\tilde{\varphi }_{nlm}}({\boldsymbol q}) = \int {{\varphi _{nlm}}({\boldsymbol r})\exp ( - i{\boldsymbol q} \cdot {\boldsymbol r})d{\boldsymbol r}}$. We note that both results from Eq. (4) and Eq. (6) are identical, but the derivation of Eq. (4) is much more tedious than those in Eq. (6), especially for the case of circular or bi-circular fields. Therefore, we prefer to adopt the formula of Eq. (6) in this work.

For the F^- ions, we give the atomic wave function of initial ground state as follows [49]

(7)$${\varphi _{nlm}}({\boldsymbol r}) = B\frac{{\exp ( - \kappa r)}}{r}{Y_{lm}}(\hat{r})$$

where $\kappa = \sqrt {2{I_p}}$ and $B = 0.7$_.

Usually, the transition amplitude ${M_{lm}}({\boldsymbol k})$ is obtained by numerically integrating over time. For the F^- ions with the asymptotic wave function of Eq. (7), ${M_{lm}}$ can also be evaluated using the SP method as follows [50],

(8)$${M_{1m}} ={-} {(2\pi )^{3/2}}B\sum\limits_{{t_s}} {{Y_{1m}}({{\hat{q}}_s})\frac{{\exp (i\Phi ({t_s}))}}{{\sqrt { - i{\Phi ^{^{\prime\prime}}}({t_s})} }}}$$

where ${\hat{q}_s}$ is the unit vector in the direction of the complex canonical momentum ${\mathbf k} + {\mathbf A}({t_s})$.The saddle point ${t_s}$ follows from the root of the saddle-point equation ${({\mathbf k} + {\mathbf A}({t_s}))^2} + 2{I_p} = 0$. $\Phi ({t_s}) ={-} \int_{{t_s}}^\infty {[{{{({\mathbf k} + {\mathbf A}(t^{\prime}))}^2}/2 + {I_p}} ]} dt^{\prime}$ is the classical action and ${\Phi ^{^{\prime\prime}}}({t_s}) ={-} 2{\mathbf E}({t_s}) \cdot ({\mathbf k} + {\mathbf A}({t_s}))$ is its second derivative.

We note that, Eq. (8) not only has validity for the case of linearly polarized pulse in Ref. [50], but also for the case discussed in this paper. However, the mathematical treatment for ${Y_{lm}}({\hat{q}_s})$ are different for different schemes [46,47]. To study the effect of the orbital on the PMD, we choose the orbitals $m ={\pm} 1$ as the initial states for F^- ions, respectively. Since the bi-circular laser field of Eq. (3) rotates counterclockwise in the polarization plane, the ejected electrons for the orbital $m = 1$($m ={-} 1$) is initially co-rotating (counter-rotating) relative to the bi-circular fields. The initial electronic state for $m = 0$ won’t be discussed in this paper, since it is aligned perpendicular to the polarization plane and gives a negligible contribution to the detachment signal in the polarization plane [51].

For the initial states with the $m ={\pm} 1$, it is easy to show the transition probability amplitude Eq. (8) can be respectively rewritten as follows,

(9)$${M_{ {\pm} 1}}({\mathbf k}) ={\pm} \sqrt 3 \pi B\sum\limits_{{t_s}} {\frac{{[{{k_x} + {A_x}({t_s})} ]\pm i[{{k_y} + {A_y}({t_s})} ]}}{{i\kappa }}} \times \frac{{\exp [{i\Phi ({t_s})} ]}}{{\sqrt { - i{\Phi ^{^{\prime\prime}}}({t_s})} }}$$

Here ${k_x}$, ${k_y}$ and ${A_x}$, ${A_y}$ represent the projection of the final momenta and vector potentials on the axes $x$ and $y$ in the polarization plane, respectively. The two-dimensional PMD for the orbitals $m ={\pm} 1$ are given by ${|{{M_{ {\pm} 1}}({\boldsymbol k})} |^2}$. The photoelectron energy spectra can be obtained by integrating the PMDs over azimuthal angle as follows,

(10)$$\frac{{\partial P}}{{\partial E}} = {\int {|{{M_{ {\pm} 1}}({\boldsymbol k})} |} ^2}d{\Omega _{\boldsymbol k}} = {\int {|{{M_{ {\pm} 1}}({\mathbf k})} |} ^2}2\pi k\sin \theta d\theta$$

where $\theta$ is the emission angle of the photoelectron with respect to x axis in the polarization plane.

3. Results and discussion

In this paper, we theoretically study the photoelectron momentum spectra of F^- ions in the two-color counter-rotating bi-circular field. The two-color counter-rotating bi-circular field is synthesized by a three-cycle circularly polarized laser pulse at 1000 nm and the second harmonic (500 nm) circularly polarized laser pulse. The intensity of the second harmonic field is fixed at $1.7 \times {10^{13}}$W/cm² while the intensity of the fundamental field is $2.7 \times {10^{12}}$W/cm², such that the second harmonic field and the fundamental field can be regarded as the strong field and the weak field, respectively. To reveal the modulation effect of the weak field, we firstly display the energy and momentum spectra of photoelectrons in a single-color strong field for contrast. In Fig. 1, we show the photoelectron energy spectra in a single 500 nm strong field rotating counterclockwise in the polarization plane. For comparison, the SFA results are normalized to first above threshold detachment (ATD) peak of TDSE ones. We note that the SFA results are calculated by numerically integrating over time based on Eq. (6) throughout the paper. One can clearly see that the SFA results show good agreement with those obtained by TDSE calculations, suggesting that the SFA method is a good approximation for the current condition. In addition, there are three striking features in Fig. 1: (i) The electron yield for $m ={-} 1$ is about five times of magnitude greater than that for $m = 1$; (ii) the main shape of the photoelectron energy spectra dramatically depends on the atomic orbital; (iii) atomic orbital slightly affects the location of the main ATD peaks. For different orbitals, the energies corresponding to the first ATD peaks are 1.4 eV and 1.27 eV, and the energies corresponding to the second ATD peaks are 3.74 eV and 3.66 eV. It is noted that the locations for the ATD peaks are obtained from TDSE results. The above findings qualitatively agree with previous work [51]. In Fig. 2, we display the two-dimensional photoelectron momentum spectra. For simplicity, the momentum distributions are normalized to 1.0 at the maximum of detachment probability. The calculations by the SFA model show similar distributions with the results by our TDSE calculations and the results by R-matrix calculations [51]. A series of ATD rings and the isotropic symmetric feature are well reproduced.

Fig. 1. Angle-integrated photoelectron energy spectra of F^- ions by the six-cycle circularly polarized laser pulse with a peak intensity of $1.7 \times {10^{13}}$W/cm² and wavelength of 500 nm. The energy spectra from the SFA method are normalized to the first peak of the TDSE results.

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Fig. 2. Two-dimensional PMDs of F^- ions by the same laser pulse as Fig. 1. The values in the first column are calculated by the TDSE method. The values in the second column are calculated by the SFA method. The values in the first row are the results for the $m = 1$ orbital, and the values in the second row are those for the $m ={-} 1$ orbital. The data are normalized to the maximum probability.

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In the following, we study the case of two-color counter-rotating bi-circular field by adding a weak 1000 nm field rotating clockwise in the polarization plane. In Fig. 3, we show the photoelectron energy spectra in two-color (500nm + 1000 nm) counter-rotating bi-circular field with $\Delta \varphi = 0$. Again, the SFA results show good agreement with those obtained by TDSE calculations. In addition to the three common features mentioned in Fig. 1, there are two obvious uniqueness in Fig. 3: (i) the energy of the main ATD peaks is slightly less than those for the single-color case. For different orbitals, the energies corresponding to the first ATD peaks are 1.21 eV and 1.01 eV, and the energies corresponding to the second ATD peaks are 3.41 eV and 3.28 eV. (ii) In the energy regions between the adjacent ATD peaks or below the first ATD peak, there are some new peaks, which are called “sidebands” in previous work [21–24]. Physically, the sidebands relate to the absorption or emission of a single 1000 nm photon from adjacent ATD peaks, as shown in the electron energy spectra at 0.22 eV and 2.17 eV for $m = 1$, and 0.21 eV and 2.14 eV for $m ={-} 1$, respectively. We observe the first sideband are significantly enhanced for $m = 1$ in contrast to those for $m ={-} 1$. And the situation of the second sideband is reversed.

Fig. 3. Angle-integrated photoelectron energy spectra of F^- ions by the two-color (500nm + 1000 nm) counterrotating bicircular field with $\Delta \varphi = 0$. The energy spectra from the SFA method are normalized to the first peak of the TDSE results.

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In Fig. 4, we present the two-dimensional PMDs in two-color counter-rotating bi-circular field. We note that the momentum distributions are normalized to 1.0 at the maximum of detachment probability. The SFA results show good agreement with those obtained by TDSE calculations. As shown in Fig. 4, the PMD exhibits a typical three-lobe distribution on the first ATD peak, and weak sideband peaks emerge between adjacent ATD peaks. In contrast to single-color case, the most striking characteristic is that the main structures of the two-dimensional PMD strongly depend on the atomic orbital. Especially, two striking differences for different orbital can be observed.

Fig. 4. Two-dimensional PMDs of F^- ions by the same laser pulse as Fig. 3. The values in the first column are calculated by the TDSE method. The values in the second column are calculated by the SFA method. The values in the first row are the results for the $m = 1$ orbital, and the values in the second row are those for the $m ={-} 1$ orbital. The data are normalized to the maximum probability. The data are normalized to 1.0 at the maximum probability.

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First, along the electron emission angle $\theta$, the lobes on the first ATD peak locate at different angle position for different orbital. For $m = 1$, ATD electrons prefer to emit along the emission angles $90^\circ$, $225^\circ$ and $315^\circ$. The situation is strongly different for $m ={-} 1$, i.e., the majority of electrons are distributed along the emission angles $45^\circ$, $135^\circ$ and $270^\circ$.

Second, the PMDs on the sidebands strongly depend on the atomic orbital. For $m = 1$, there are two obvious maxima with symmetry about the ${k_y}$ axis on the first sideband, while there are two less obvious maxima along the ${k_y}$ axis on the first sideband for $m ={-} 1$. Besides that, the shape of the PMDs on the second sideband are different for different orbitals.

In order to shed more light on the dependence of the PMDs on the atomic orbital, we analyze the PMDs using the SP method. All the saddle points are obtained by numerically solving the saddle-point equation. Figure 5 shows the electric field of the two-color counter-rotating bi-circular laser pulse and the corresponding SP distributions. In Fig. 5, each group of points depict the saddle points for the energy from 0 to 0.18 a.u. with a step size of 0.027 a.u. and the emission angle from 1 degree to 360 degree with a step size of 1 degree. It is found that there are three SPs (SPs 4-6 or SPs 5-7) per laser cycle as the solutions of saddle-point equation, which is consistent with the analytical result for a long bicircular field [52]. The six SPs during two laser cycles, i.e., SP3-SP8 are the lowest SPs in the upper half plane of complex time. Actually, the imaginary part of the complex time is related to the ionization probability, the lower SPs give the larger contributions to the ionization probability. Therefore SPs 3-8 give the significant contributions to the PMDs. In Figs. 6(a) and 6(b), we show the two-dimensional PMDs by considering coherently superposition of SPs 3-8. The main structures of Fig. 4 from the TDSE calculations can be well reproduced by only considering the contribution of the SPs 3-8. Therefore SPs 3-8 dominate the interference patterns of PMD. In Figs. 6(c) and 6(d), we give the PMDs without the interference patterns, i.e., the electron distributions of the PMDs. By comparing Fig. 6(c) with 6(d), we find that the electron probability distributions strongly depend on the atomic orbitals. For $m = 1$, near the first ATD peak, one can see that there is a typical three-lobe distribution with threefold symmetry along the emission angle, each maximum locates at the same angular position with the corresponding interference fringes in Fig. 6(a). However, for $m ={-} 1$, only one maximum locates at the center of the Fig. 6(d).

Fig. 5. The temporal sketch of the two-color counterrotating bicircular field with $\Delta \varphi = 0$ and the corresponding saddle-point distributions. Each group of points depict the saddle points for the energy from 0 to 0.18 a.u. with a step size of 0.027 a.u. and the emission angle from 1 degree to 360 degree with a step size of 1 degree. The numbers of several dominant SPs are also marked in the complex-time plane.

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Fig. 6. Two-dimensional PMDs obtained from the SP method by considering six dominant SPs (SPs 3–8). The panels in first row are the PMDs by considering coherently superposition of SPs 3-8. The panels in second row are the PMDs without the interference patterns. The values in the first column are the results for the $m = 1$ orbital, and the values in the second column are those for the $m ={-} 1$ orbital. The data are normalized to the maximum probability. The two white circles marked in each panel are plotted according to the photoelectron ATD energy gained from Fig. 3. The arrow in each panel represents the direction of the emission angle $\theta = 90^\circ$.

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In the following, we explore the reason leading to the main differences of PMDs in Fig. 6. In Fig. 7(a), we display the photoelectron energy distribution from Figs. 6(a) and 6(b) along the emission angle $\theta = 90^\circ$. It can be found that the first peak for $m ={-} 1$ locates between the first peak and the second peak for $m = 1$. To understand this feature, we show the photoelectron energy distribution obtained by coherently adding SPs 5 and 6 in Fig. 7(b). We observe that the above feature of Fig. 7(a) can be reproduced, although the peak location is slightly shifted. This shows that the interference pattern is very different comparing electronic initial stated with $m = 1$ and $m ={-} 1$. Besides that, we display the photoelectron energy distribution from Figs. 6(c) and 6(d) along the emission angle $\theta = 90^\circ$ in Fig. 7(c). As shown in Fig. 7(c), the momentum difference of the peaks for different orbital is larger than 0.2 a.u. Eckart et al. have demonstrated that the radial shift of the momentum can be viewed as a fingerprint of the orbital-dependent initial velocity at the tunnel exit [36]. At the tunnel exit ${r_t}$, the velocity difference for the ${\pm} |m |$ orbitals should be $\Delta v = \Delta {l_z}/{r_t}$, where $\Delta {l_z}$ is the angular momentum difference, which is typically in the range of zero to $2|m |$. By letting ${r_t} \approx {I_p}/{E_{2\omega }}$, thus the maximum of the velocity difference at the tunnel exit should be ${(\Delta v)_{\max }}$=0.249 a.u. The estimated maximum of the velocity difference is also marked in Fig. 7(c). In practice, the electron’s initial velocity distribution at the tunnel exit can be extracted by using the relationship between the initial velocity ${{\boldsymbol v}_i}$ and the final momentum k, that is ${\boldsymbol k} = {{\boldsymbol v}_i} - {\boldsymbol A}({t_0})$. ${t_0}$ is the real part of the complex time ${t_s}$. In this work, the peak vector potentials ${A_\omega }$ and ${A_{2\omega }}$ are fixed as 0.194 a.u. and 0.242 a.u, respectively. In Fig. 7(d), we show the electron’s initial velocity distribution at the tunnel exit. We note that the electron’s initial velocity distribution is obtained based on the electronic wave packet for SP5. In practice, the results from the electronic wave packets for SP6 is same as Fig. 7(d). One can clearly see that the detachment probability for $m ={-} 1$ is maximal when the initial velocity is -0.02 a.u., while the detachment probability for $m = 1$ is maximal when the initial velocity is 0.14 a.u. Moreover, at the peak of the detachment probability for $m ={-} 1$, the magnitude of the combined vector potential at SP5 is 0.04 a.u. Thus, the magnitude of the initial velocity is smaller than the magnitude of the negative vector potential. Further, the final momentum equals to the sum of the initial velocity and the negative vector potential. Hence, the final momentum for $m ={-} 1$ approximately points to the direction of the negative vector potential. The situation is reversed for $m = 1$. The magnitude of the combined vector potential at SP5 is 0.087 a.u. The magnitude of the initial velocity is larger than the magnitude of the negative vector potential, the initial velocity and the negative vector potential are antiparallel, thus the final momentum points to the opposite direction of the negative vector potential. Therefore, it can be concluded that the initial velocity difference at the exit has been definitely mapped onto the final PMDs and leads to the main difference of final PMDs for different orbitals.

Fig. 7. (a) The photoelectron energy distribution from Figs. 6(a) and (b) along the emission angle $\theta = 90^\circ$. (b) The photoelectron energy distribution along the emission angle $\theta = 90^\circ$ obtained by coherently adding SPs 5 and 6. (c) The photoelectron energy distribution from Figs. 6(c) and (d) along the emission angle $\theta = 90^\circ$, where ${(\Delta v)_{\max }} = 0.249a.u.$ represents the estimated maximum of the initial velocity difference at the tunnel exit (see text). (d) The electron’s initial velocity distribution at the tunnel exits. The data in each panel are normalized to 1.0 at the maximum of the probability.

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As well known, for the case of the linearly polarized fields, the inter-cycle interference arises from the coherent superposition of electron wave packets released at complex times during different optical cycles, whereas sub-cycle interference comes from the coherent superposition of electron packets released during the same optical cycle [53]. As mentioned earlier, SPs 4–6 or SPs 5–7 not only appear during one cycle of fundamental field but also correspond to the peaks of the synthesized fields. In the following, we explore the formation of the interference patterns in PMDs for different orbitals by distinguishing the contributions of the SPs 4–6 and SPs 5–7 on PMDs.

First, we analyze the interference patterns in PMDs for $m = 1$. Figures 8(a) and 8(c) show the PMDs for $m = 1$ by considering coherent superpositions of SPs 4–6 and SPs 5-7, respectively. It is found that the main shape of the PMD on the left-half plane of Fig. 4(a) can be reproduced using only three SPs, i.e., SP4-SP6, while the main shape on the right-half plane of Fig. 4(a) can be reproduced using SP5-SP7. Besides, the three-lobe structure on the second ATD peak are reproduced both in Figs. 8(a) and 8(c). Therefore, we can comprehend the main interference patterns of Fig. 4(a) from two seemingly opposite but unified aspects. On the one hand, the main shape of the PMDs on the left-half plane of Fig. 4(a) originates from the inter-cycle interference from wave packets for SPs 4-6. On the other hand, since SPs 4–6 appear during one cycle of fundamental fields, the interference pattern from wave packets for SPs 4-6 can be also regarded as the sub-cycle interference during one cycle of fundamental fields. We can also understand the interference patterns from the shape of interference fringes. The inter-cycle interference pattern is usually associated with the ATD rings, thus the main interference pattern in Fig. 4(a) appears near the ATD rings and distributes along the emission angle. However, the interference fringes are partitioned into three-lobe structures, and the lobes have obvious radial momentum distributions which is similar to the sub-cycle interference pattern.

Fig. 8. The PMDs obtained from the SP method by coherently adding three saddle points during one cycle of fundamental field. The values in the first row are the results obtained by considering SPs 4–6, and the values in the second row are obtained by considering SPs 5–7. The values in the first column are the results for the $m = 1$ orbital, and the values in the second column are those for the $m ={-} 1$ orbital. The data are normalized to the maximum probability. The two white circles marked in each panel are plotted according to the photoelectron ATD energy gained from Fig. 3.

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Second, we study the interference patterns in PMDs for $m ={-} 1$. Figures 8(b) and 8(d) show the PMDs for $m ={-} 1$. In Figs. 8(b) and 8(d), an obvious three-lobe momentum distribution appears on the first ATD peak, and its shape and the angular position are dramatically different from the case for $m = 1$. By comparing with Figs. 8(a) and 8(c), we observe that the three-lobe momentum distribution is significantly enhanced while the momentum distribution on the first sideband is weakened. It suggests that the interference patterns along the radial momentum are changed dramatically by atomic orbital.

In Fig. 9, we display the PMDs by considering coherent superpositions of two adjacent SPs. Interestingly, the PMDs of Fig. 9 clearly exhibit how each lobe (i.e., the maximum on the first ATD peak) of the three-lobe momentum distribution dynamically shapes along with the detachment time. As well known, the real part of the saddle-point time ${t_s}$, i.e., $Re [{t_s}]$ represents the detachment instant of electrons. Figures 9(a)–9(c) show the time-dependent formation of each lobe of the PMDs for $m = 1$. One can see that, at start time, the interference patterns from wave packets for SP4 and SP5 forms the lobe located near emission angle $\theta = 225^\circ$. Subsequently, the lobe located near emission angle $\theta = 90^\circ$ results from the interference patterns from wave packets for SP5 and SP6. And finally, the lobe located near emission angle $\theta = 135^\circ$ originates from the interference patterns from wave packets for SP6 and SP7.

Fig. 9. The PMDs obtained from the SP method by coherently adding two adjacent saddle points during one cycle of fundamental field. The values in the first row are the results for the $m = 1$ orbital, and the values in the second row are those for the $m ={-} 1$ orbital. The values in the first column are the results obtained by considering SPs 4 and 5, the values in the second column are obtained by considering SPs 5 and 6, and the values in the third column are obtained by considering SPs 6 and 7. The data are normalized to the maximum probability. The two white circles marked in each panel are plotted according to the photoelectron ATD energy gained from Fig. 3.

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However, the situation is some different for $m ={-} 1$. In Figs. 9(d)–9(f), one can see that there are two maxima on the first ATD peak, one of them is enhanced and the other is weakened because of the interference effect. We identify that the lobe on the first ATD peak relates to the enhanced maximum by comparing with Fig. 6. It can be found that the lobe caused by the interference patterns from wave packets for SP5 and SP6 locates near the emission angle $\theta = 270^\circ$ for $m ={-} 1$, while the lobe for $m = 1$ locate near emission angle $\theta = 90^\circ$. The situation is similar for other lobes. It can be concluded that the angle difference of the lobes for different orbital is nearly $180^\circ$. Therefore, main difference of the PMDs for different orbital in Fig. 4 results from the dependence of the interference patterns on the atomic orbital, especially the dependence of the interference patterns from the adjacent wave packets on the atomic orbital.

4. Conclusions

In summary, we have carried out a systematical theoretical study on the PMDs of F^- ions in a counter-rotating bi-circular laser field, for the ground-state orbital of $m = 1$ and $m ={-} 1$, respectively. For both cases of the single-color field and the counter-rotating bi-circular laser field, the calculation by the SFA method show good agreement with those obtained by the benchmark TDSE calculations, demonstrating that the SFA method is a good approximation for negative ion systems. For the case of single-color field, we well reproduce the main features of previous calculations by R-Matrix method [51], especially in three aspects: (i) The electron yield for $m ={-} 1$ is greater than that for $m = 1$; (ii) the main shape of the photoelectron energy spectra and the PMDs dramatically depends on the atomic orbital; (iii) atomic orbital slightly affects the location of the main ATD peaks. For the case of the counter-rotating bi-circular laser field, in addition to the above main features, the PMD exhibits a typical three-lobe distribution on the first ATD peak, and weak sideband peaks emerge between adjacent ATD peaks or below the first ATD peak. The three-lobe distribution and the entire PMDs strongly depend on the sign of the magnetic quantum number of the bound electron prior to ionization.

Based on the SP method, further investigations have confirmed that, the atomic orbital not only strongly affects the PMDs without the interference patterns but also the interference patterns of electron wave packets. It has been revealed that the radial momentum shift of PMDs for different orbitals is definite imprint of orbital-dependent initial momentum at the tunnel exit. Moreover, it is found that the interference patterns from the three saddle points during one cycle of fundamental fields play an important role in forming the three-lobe distributions of PMDs, despite the absence of threefold symmetry. More interestingly, we intuitively reveal how each lobe of the three-lobe momentum distribution dynamically shapes along with the detachment time, by considering coherent superpositions of two adjacent SPs. It is demonstrated that the lobes sequentially appear in the momentum plane along the clockwise direction. The weak fundamental field rotates clockwise in the polarization plane, which dominates the appearance of the lobes. In practice, the shape of synthesized fields strongly depends on the phase delay of the weak fundamental field. The forming time and the angular positions of the lobes in the momentum plane can be efficiently controlled by manipulating the phase delay of the weak fundamental field. Therefore, the scheme discussed in this work can be regarded as a controllable rotating temporal Young’s two-silt interferometer. With this temporal Young’s two-silt interferometer, in principle one can extract some important information of ionization dynamics. The present work not only has potential implications for imaging the orbital-resolved electron dynamics of negative ions with valence p orbitals, but also provides a valuable reference for the strong-field ionization associated with noble-gas atoms and oriented molecules.

Funding

National Natural Science Foundation of China (12064023, 12164044); State Key Laboratory of Artificial Microstructure and Mesoscopic Physics; Natural Science Foundation of Gansu Province (20JR5RA209); the Scientific Research Program of the Higher Education Institutions of Gansu Province (2020A-125).

Acknowledgments

The present work is also supported by the State Key Laboratory of Artificial Microstructure and Mesoscopic Physics in Peking University. The anonymous reviewers have also contributed considerably to the publication of this paper.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Orbital-resolved photoelectron momentum distributions of F^- ions in a counter-rotating bicircular field

Abstract

1. Introduction

2. Theoretical methods

2.1 TDSE method

2.2 SFA model

3. Results and discussion

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (10)

Optics Express