† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11322437 and 11574010) and the National Basic Research Program of China (Grant No. 2013CB922402).
We theoretically investigate the low energy part of the photoelectron spectra in the tunneling ionization regime by numerically solving the time-dependent Schrdinger equation for different atomic potentials at various wavelengths. We find that the shift of the first above-threshold ionization (ATI) peak is closely related to the interferences between electron wave packets, which are controlled by the laser field and largely independent of the potential. By gradually changing the short-range potential to the long-range Coulomb potential, we show that the long-range potential’s effect is mainly to focus the electrons along the laser’s polarization and to generate the spider structure by enhancing the rescattering process with the parent ion. In addition, we find that the intermediate transitions and the Rydberg states have important influences on the number and the shape of the lobes near the threshold.
In the investigation of light–matter interactions in the strong fields, the photoelectron angular distributions and the photoelectron spectra provide abundant information about the ionization dynamics which reflects the information about the laser field and the target structure. For a given ionization potential Ip which is much larger than the photon energy, Keldysh[1] divided the ionization into the multi-photon regime (MPI, γ ≫ 1) and the tunnelling regime (TI, γ ≪ 1), where the Keldysh parameter
In recent years, the low energy structure produced by intense long-wavelength lasers (which is called ionization surprise) has drawn a great deal of attention from experimentalists and theorists. The appearance of the low energy structure can be attributed to the effects of the Coulomb potential during the multiple rescattering of the ionized electron which is ignored in the strong field approximation (SFA).[2–4] It shows the influence of ion structure on the photoelectron spectra in the TI regime and gives instructions to obtain holographic structures that record the spatial and temporal information for both the target ion and the electron.[5–8] The holographic structure (usually called “spider structure”) is generated by the interference between the laser-driven electron wave packet that scatters off the target ion and the electron wave packet that does not interact with the ion. It is a large scale feature that can span multiple above threshold ionization (ATI) rings. At the same time, a radial fan-like pattern has been observed in the angular distribution in the near threshold ionization both experimentally and theoretically.[9–11,14,15] The number of near-threshold radial lobes was suggested to be determined by the minimum number of photons needed to ionize the atom[15] and an empirical rule was given for the dominant orbital angular momentum. Latter, using a classical trajectory Monte Carlo method (CTMC), Arbó et al. quantitatively explained the dominant orbital angular momentum in terms of interfering classical trajectories in the presence of both the Coulomb and laser fields.[9,10] The study of photoelectron angular distributions of near zero momentum is generally related to the recapture of electrons into Rydberg states.[14] However, the CTMC method ignores the intermediate transition processes since the ionization rate is prescribed by the Ammosov–Delone–Krainov (ADK) theory, which only considers the transition rates from the bound state to the continuum states.[16,17] The comparison between TDSE calculation and the semi-classical model within a wide range of laser parameters shows the limit of its applicability.[8]
In this paper, we investigate the photoelectron angular distributions of the ionization of a hydrogen atom by solving the time-dependent Schrdinger equation (TDSE) in the TI regime with intense infrared (IR) laser pulses. We find that the shift of the first ATI ring in the momentum distribution with the changing of the wavelength reflects the interference of electron wave packets under the control of a laser field, which is largely independent of the used potential. The influences of the long-range Coulomb potential and excited states on photoelectron angular distributions are studied by changing the Yukawa potential’s parameter. By gradually changing the short potential to the long-range Coulomb potential, we can see the “spider structure” is gradually reproduced. In addition, the number of lobes and the radial pattern on the near threshold ring (first ATI ring or below the first ATI ring) is connected with the increase of the number of excited states when one changes the Yukawa potential’s parameter.
The organization of the rest of this paper is as follows. In Section 2, the numerical method is described in brief. In Section 3, we present our main results and discussion. Section 4 is a short summary.
Our method is based on the numerical solution to the TDSE, whose details can be found in previous work.[21–24] In this work, the hydrogen in its ground 1s state is exposed to a linearly polarized laser pulse, with a vector potential given by
In this section, we first calculate the photoelectron spectra in the TI regime with different laser wavelengths. Then, we fix the laser parameter and gradually change the Yukawa potential to approach the shape of the Coulomb potential. By comparing with the calculations using the Coulomb potential, we can evaluate the roles of the long-range Coulomb potential and intermediate states in the photoelectron angular distributions. Atomic units are used throughout.
In our previous work,[24] we have extended the electron wave packet interference from two single-cycle pulses to one multicycle pulse, and verified that the ionization yield can be modulated by varying the wavelength. In Figs.
Comparing Fig.
![](cpb_25_9_093201/cpb160785ieqn3.gif)
According to the above analysis, one can also explain the main interference structures in Fig.
![]() | Fig. 1. Photoelectron momentum distributions on the log scale for different wavelengths, calculated for the Coulomb potential. The laser has a total duration of 8 cycles at a fixed peak intensity of I0 = 2 × 1014 W/cm2. The wavelengths are marked on the top of each panel. Panels (a)–(c) are for the case with the maximum ionization yield and panels (d)–(f) are for the minimum ionization yield, as discussed in Ref. [24]. The “spider” structures are marked with black dashed lines for guiding the eye. |
![]() | Fig. 2. The same as those Fig. |
In this way, we can see the main structure of the photoelectron spectra is greatly influenced by the laser field (the strength of electric field and the frequency). It is a character for the TI regime since the tunneling electrons are mainly emitted at the peak of the field and the generated electron wave packets at different subcycles interfere with each other. The position of the ATI ring is the reflection of this interference controlled by a laser field, while the lobes of the ring which are different for the Yukawa potential and the Coulomb potential show the inner structure of the atoms. Hereafter, we will change the potential shape and the number of excited states to see the effects of the long-range potential and intermediate transitions.
The importance of the long-range Coulomb potential in the low energy momentum spectra has been studied in previous works.[15,27] Here, we would like to study to what extent the long-range Coulomb potential effects the shape of the first ring and the number of the lobes. For this purpose, we gradually change the Yukawa potential to a long-range potential by adjusting the parameters Z and κ to reproduce the ground state energy of the H atom. The potentials with different parameters as a function of r are shown in Fig.
![]() | Table 1. The energy levels (in a.u.) of excited states with different values of Yukawa potential. . |
We select six Yukawa potentials as shown in Fig.
The number of excited states used in Figs.
As we change the Yukawa potential to the long-range potential, the number of excited states does not change in a linear fashion with the shape of potential since the excited states become more and more denser near the continuum states. It shows that the combined effect of a series of Rydberg states plays an important role in the formation of a different number of the radial lobes near the zero momentum. One can obtain the momentum distribution by projecting the final wave function onto the scattering states. The scattering states are a description of the electron’s scattering by the Coulomb potential, and the projection can be expanded into a series of spherical harmonics. The l means different scattering trajectories that the electrons are emitted from different initial states or experienced the scattering process, but these trajectories do not accurately describe the quantum states when the tunneling electrons recombine with the ion after the scattering.
![]() | Fig. 5. The same as those in Fig. |
![]() | Table 2. The dominant angular momentumls for the first ATI ring. . |
![]() | Fig. 6. The normalized expansion coefficients of the spherical harmonics for the momentum |
According to Eqs. (
We theoretically investigate the low energy structure in the TI regime for different potentials at different laser parameters. By changing the wavelength, we find that the shift of the first ATI ring on the momentum distribution is consistent with the ionization modulation, which can be reproduced using both the Yukawa and the Coulomb potential. It is a result of the interference of wave packets emitted at different times, which reflects the nature of the laser field. The radial fan structure on the momentum distribution along with the number of lobes on the first ATI ring is greatly influenced by the atomic intermediate states. The long-range potential’s influence is the Coulomb focusing effects and the generation of the “spider” structure along the laser’s polarization direction. By changing the number of bound states in the Yukawa potential, we find that the radial shape of the lobes near the threshold is a result of resonant ionization while the number of the lobes is related to the combined effect of a series of Rydberg states.
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