The two-dimensional photoelectron momentum distributions (PMD) of F- ions induced by a linearly polarized few-cycle laser pulse are analyzed with the saddle-point (SP) method. The validity of the SP method is confirmed by comparing the PMD with those obtained from direct numerical integration of the transition probability amplitude in the context of strong-field approximation (SFA). We analyze the intra- and inter-cycle interference patterns in the two-dimensional PMD and show that the two-dimensional PMD can be effectively monitored by changing the carrier-envelope phase of few-cycle laser pulse. In addition, by separately calculating the two-dimensional PMD formed in the different detachment steps, we find that the rich oscillatory patterns in the two-dimensional PMD can be mainly attributed to the interference effects of electronic wave packets in the classical propagation step after the ionization, and part of intra-cycle interference fringes’ shape is affected by the sub-barrier phase.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Photoionization or detachment is the footstone for understanding numerous strong-field phenomena and attosecond physics [1,2]. Much important information about the electron dynamics can be extracted from the angle-resolved photoelectron spectra induced by strong laser field . Usually, the tunnel ionization regime is quantified by the Keldysh parameter , where is ionization potential and is the ponderomotive energy. In this regime, electron can be released by tunneling from laser-induced potential barrier near the field maxima and the dominant structures emerging in the photoelectron spectra are explained as interference between electron wave packets (WP) emitted at different times within the whole laser pulse . The earlier observed interference pattern is above-threshold ionization (ATI) peaks in the electron energy spectra , which results from the interference between wave packets emitted at time intervals separated by at least single laser cycle (so called inter-cycle interference) [6,7]. Few-cycle pulses with absolute phase stabilization have been appreciated as powerful tools to study strong-field ionization . Especially in 2005, a striking pattern called intra-cycle (or sub-cycle) interference was demonstrated with few-cycle laser pulses, which results from the temporal double-slit interference between two WPs released in consecutive laser half-cycles . Recent works show that the measurement of sub-cycle electron wave-packet interference patterns can serve as a tool to extract structure and electron dynamics in atoms and molecules on attosecond time scale [10,11].
For most strong-field ionization of neutral atoms, the ionic Coulomb field on the photoelectron gives rise to some additional structures in the two-dimensional photoelectron momentum distributions(PMD), which also distorts the interference fringes [12,13]. However, for the negative ions, the outer electron is bound by short-range potential and the Coulomb effect inside the neutral atoms is absent. Therefore strong-field approximation (SFA) model has been viewed as an accurate treatment for the photodetachment of negative ions. As a matter of fact, a series of consequent measurements about angle-resolved electron spectra for H- ions  and F- ions [15–18] implemented by Kiyan et al. have been quantitatively reproduced by using SFA model [19–24] and solving the time-dependent Schrödinger equation (TDSE) . Based on the SFA model, the saddle-point (SP) method was developed by Gribakin and Kuchiev . For the linearly polarized or two-color bicircular laser field, the SP method can provide with an intuitive description of interference effects during the strong-field photodetachment [8,14,15,24]. Most previous works on strong-field detachment of negative atomic ions focus on multicycle laser pulses. Recently, Shearer et al. generalized the SP method to describe the photodetachment of negative ions in a few-cycle linearly polarized laser pulse [25,26]. However, the possibility for monitoring the inter- and intra-cycle interference patterns in photoelectron momentum distributions (PMD) of the negative ions by few-cycle laser pulse have not yet been clarified. On the other hand, few previous works studied the effect of the phase of the tunneling wave packet on the two-dimensional PMD of neutral atoms . Very recently, Han et al. have demonstrated that the sub-barrier phase leaves definite imprint on the final photoelectron momentum distribution in tunneling ionization of Ar atoms . As far as we know, the effect of sub-barrier phase on the PMD of the negative ions with short range potential has not been analyzed in detail.
In this paper, we investigate theoretically the two-dimensional PMD of F- ions induced by a linearly polarized few-cycle laser pulse with the generalized SP method. There are two main goals in this paper. First, based on the SP method, we analyze the intra- and inter-cycle interference patterns in two-dimensional PMD of F- ions induced by few-cycle laser pulse in detail. Second, based on the imaginary time method (ITM) and two-step semiclassical model for strong-field phenomena [27–29], we study the PMD formed in each step of photodetachment process and the effect of the sub-barrier phase on the interference patterns. The effect of the carrier-envelope phase (CEP) on the PMD is also discussed. Atomic units are used in this paper unless stated otherwise.
2. Theoretical methods
In the SFA model without rescattering mechanism, the direct transition probability amplitude from the atomic or ionic ground state to the continuum state with momentum is given by [8,18]26]30].
The final state is approximately by the Volkov state
The vector potential of the laser pulse is defined by . The electric field of a linearly polarized laser pulse (along the axis) can be written as
After performing the integration over space coordinates in Eq. (1) analytically, for the magnetic quantum number , we obtain the transition probability amplitude for F- ions with,
In Eq. (5), is a canonical momentum of the electron in a laser field and is the projection of in the polarization direction of laser pulse.
The transition amplitude in Eq. (5) is usually calculated by numerically integrating over time (hereafter to be called the numerical integration method). Alternatively, can also be evaluated using the SP method of Shearer et al.  as follows
The first part , where is calculated along the path parallel to the imaginary axis down to the real axis. The second part is evaluated along the real axis. In terms of the imaginary time theory  and semiclassical two-step model , the first part in Eq. (10) denotes the step that an electron tunnels through the barrier of negative ions, while the second part represents the step that the electron classically moves toward a detector in a laser pulse, i.e. classical propagation.
The differential detachment probability of an electron with energy in the direction of is given by
3. Results and discussion
3.1 Saddle-point distributions and the validity of the SP method
In this paper, we investigate the direct photodetachment from F- ions in a few-cycle linearly polarized laser pulse by the SP method. In our calculations, we use a three-cycle laser pulse with wavelength of 1400 nm and peak intensity of W/cm2.
Figure 1 shows the electric field of three-cycle laser pulse with and the corresponding SP distributions in the upper half-plane of complex for the fixed photoelectron energy of 3.4eV. In Fig. 1(b), each group of points depict the saddle points for the angles from to with a step size of . It is found that there are eight SPs as the solutions of Eq. (7) for the three-cycle laser pulse with . In terms of the ITM method , the real parts of saddle points SP3, SP4, and SP5 in Fig. 1(b) represent the moments when the electrons are released more probably.
Next we establish the validity of the SP method by comparing with numerical integration method. In Figs. 2(a) and 2(b), we show the two-dimensional PMD for, by numerical integration method and SP method, respectively. and represent the momentum components perpendicular and parallel to the polarization direction of the laser’s electric field, respectively. By comparing Fig. 2(a) with Fig. 2(b), one can see that the structures of two-dimensional PMD using the numerical integration method and the SP method agree very well with each other. In Fig. 2(c), we show the PMD by considering coherently superposition of three dominant SPs, i.e., SPs 3-5. The main structures of Fig. 2(a) can be well reproduced by only considering the contribution of the SPs 3-5. Therefore three central SPs, i.e. SP3, SP4 and SP5, dominate the interference patterns of PMD in the three-few cycle pulse with .
3.2 Analysis of interference patterns in the two-dimensional PMD
To further explore the formation of the interference structure in PMD by SP method, it is necessary to distinguish the contributions of the dominant SPs in the formation of main shape of PMD. In Figs. 3(a), 3(b) and 3(c), we show the PMD by coherently adding the contributions of SP3 + SP4, SP4 + SP5, SP3 + SP5, respectively. As known, the inter-cycle interference arises from the coherent superposition of electron wave packets released at complex times during different optical cycles, whereas intra-cycle interference comes from the coherent superposition of electron packets released in the same optical cycle. The interference patterns of Fig. 3(a) and Fig. 3(b) are actually originated from the intra-cycle interference and the interference patterns of Fig. 3(c) are originated from the inter-cycle interference. By comparing Fig. 2(a) with Fig. 3(a) and Fig. 3(b), it is found that the main interference patterns in the lower-half plane of Fig. 3(a) are similar to the lower-half part in the Fig. 2(a), while the main interference patterns in the upper-half plane of Fig. 3(b) are similar to upper-half part in the Fig. 2(a). It indicates that the main interference patterns in the lower half plane of Fig. 2(a) originate from the intra-cycle interference from wave packets for SP3 and SP4, while the main oscillatory patterns in the upper half plane of Fig. 3(a) originate from the intra-cycle interference from wave packets for SP4 and SP5. Figure 3(c) has the form of concentric rings centered at zero momentum in the momentum plane. The rings in Fig. 3(c) is associated with above-threshold detachment (ATD) rings of the photoelectron spectrum. The radius of each ATD ring can be calculated from an energy conservation equation ,Fig. 2(a) with Fig. 3(c), we find that the interference fringes with the concentric rings centered at zero momentum in Fig. 2(a) originates from the inter-cycle interference from wave packets for SP3 and SP5. Therefore the main structures of the two-dimensional PMD are attributed to the interplay of the intra- and inter-cycle interferences of those SPs.
Based on the ITM method described in Sec.2, the action in Eq. (6) can be split into two parts [see Eq. (10)], i.e., and . The sub-barrier action represents the process of quantum tunneling detachment. The action describes the classical propagation process of photoelectron in the few-cycle laser field. The PMD that are contributed by and can be calculated by using Eq. (12) and Eq. (13). Although it is impossible to separately detect these detachment probability distributions in the experiment, it is useful to understand the origin of interference structure of PMD with the theoretical analysis.
In Figs. 3(d)-3(f), we show the PMD formed in the classical propagation step by coherently adding the contributions of SP3 + SP4, SP4 + SP5, SP3 + SP5, respectively. We can clearly see that the main shape and number of the destructive interference fringes in Figs. 3(d)-3(f) agree with those in Figs. 3(a)-3(c). It indicates that the interference patterns in PMD of Fig. 2(a) can be attributed to the interference effects of electronic wave packets during the classical propagation process. Further careful inspection shows that the interference fringe at the bottom of Fig. 3(d) is more curved than those in Fig. 3(a), and the interference fringe at the top of Fig. 3(e) is more curved than those in Fig. 3(b). The reason is that the sub-barrier phase is absent in the calculations for Figs. 3(d)-3(f).
The PMD formed in the tunneling step are given in Figs. 3(g)-3(i). In practice, Figs. 3(g)-3(i) show the typical feature of tunneling probability distributions for the negative ions (or atoms) with the ground states of p state, in which two distinct maxima of probability deviating from zero momentum coexist along the laser polarization axis. As seen in Figs. 3(g)-3(i), the tunneling probability distributions are asymmetric about the horizontal axis for SP3 + SP4 and SP4 + SP5, but symmetric about the horizontal axis for SP3 + SP5. In the complex-time plane at the CEP of , the positions of SP3 and SP5 are symmetric about the horizontal axis and both higher than the position of SP4, which suggests that the contribution for the tunneling probability from SP4 are greater than those from other two SPs, but the contributions from SP3 and SP5 are equivalent.
3.3 The effect of the sub-barrier phase on the interference patterns
In terms of ITM method, the real part of the sub-barrier action is the accumulated sub-barrier phase when electrons tunnel through the potential, and the imaginary part is related to the ionization probability [12,27–29].
In Fig. 4, we give the PMD of F- ions for without sub-barrier phase, by considering the contributions from different saddle points. The PMD in Fig. 4(a) are obtained from coherently adding three dominant saddle points SP3-SP5 . It can be seen that the main interference fringes at larger momenta are more curved than those in Fig. 2(a). Figs. 4(b),(c),(d) show the PMD by coherently adding the contributions of SP3 + SP4, SP4 + SP5, SP3 + SP5, respectively. By comparing Figs. 4(b), 4(c), and 4(d) with Figs. 3(a), 3(b) and 3(c), we find that part of the intra-cycle interference fringes’ shape is changed, if the sub-barrier phase is absent. However, the inter-cycle interference fringes are almost independent of the sub-barrier phase.
3.4 The effect of carrier-envelope phase on the interference patterns
In this section, we study the effect of the CEP on the interference patterns in the two-dimensional PMD. In Fig. 5, we show the electric field of three-cycle laser pulse with and the corresponding SP distributions for the fixed photoelectron energy of 3.4eV. It is found that there are eight SPs as the solutions of Eq. (7) for the three-cycle laser pulse with . In Fig. 5(b), SP4 and SP5 are the lowest two SPs in the upper half-plane of complex . It indicates that the saddle points SP4 and SP5 make more significant contributions to the ionization probability and interference structure. Fig. 6(a) illustrates the two-dimensional PMD for by numerical integration method. The momentum distribution of Fig. 6(a) is asymmetric about the horizontal axis. Comparing with Fig. 2(a), one can find that the distinctive feature of the two-dimensional PMD of Fig. 6(a) is absence of ATD rings. It indicates that the inter-cycle interference between electronic wave packets is suppressed for the case of . In Fig. 6(b), we show the PMD formed by coherently superposition two dominant SPs, i.e., SP4 and SP5. We find that the most striking feature of the two-dimensional PMD of Fig. 6(a) can be reproduced by only considering the SP4 and SP5. Therefore the main structures of the two-dimensional PMD for are dominated by of the intra-cycle interferences. Figure 6(c) and Fig. 6(d) show the PMD by coherently adding the contributions of SP4 + SP5, formed in the classical propagation step and quantum tunneling step, respectively. In Fig. 6(c), we can clearly see that the similar shape and number of the destructive interference fringes appeared in Figs. 6(b). Therefore the interference patterns in PMD of Fig. 6(a) is dominated by the intra-cycle interference effects between electronic wave packets during the classical propagation process. In Fig. 6(d), it is found that the two discrete maxima of probability are not symmetric about the horizontal axis, even though the contribution for the tunneling probability from SP4 and SP5 are equivalent and the positions of SP3 and SP5 are symmetric about the horizontal axis in the complex-time plane. It indicates that the CEP has an important influence on the tunneling probability.
In summary, we have analyzed the two-dimensional PMD of F- ions in a three-cycle laser pulse by the saddle-point method. By distinguishing the contributions of the dominant SPs, we find that the main structures of PMD for the CEP are attributed to the interplay of the intra- and inter-cycle interferences between electronic wave packets. For the PMD with , the intra-cycle interference between electronic wave packets comes into the dominant role, the inter-cycle interference is suppressed. Moreover, based on the two-step semiclassical model and ITM method, we separately calculate the PMD formed in quantum tunneling and classical propagation steps. It is found that the rich interference patterns in the PMD can be mainly attributed to the interference effects of coherent quantum wave packets in the classical propagation step of detachment processes, and part of intra-cycle interference fringes’ shape is affected by the sub-barrier phase. Since the sub-barrier phase originates from the tunneling, it definitely imprints its effect on the coherent processes in tunneling-related phenomena . The theoretical analysis clearly indicates that the tunneling process transfers the initial phase onto the two-dimensional PMD of F- ions with p-state valence electrons and short-range potential. The present work is meaningful for further controlling and probing electron dynamics in the detachment (or ionization) process of negative ions (atoms) with p-state electrons with the few-cycle laser pulse.
National Natural Science Foundation of China (NSFC) (11434002, 11664035, 11647150); the Young Talents Program of Gansu Province (2016); the Scientific Research Program of the Higher Education Institutions of Gansu Province (2016A-068).
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