Above-threshold ionization (ATI) [1], that atoms or molecules absorb more photons than necessary for ionization, has been the hot subject of a large number of experimental [2, 3] and theoretical [4, 5] investigations in the past three decades. The intuitive semiclassical picture [6, 7], without considering the Coulomb potential after tunneling, has obtained great success in understanding ATI process, such as the cut-off energy of 10U_p (U_p = I/4ω ², the ponderomotive energy) in ATI spectrum [7] and the diffraction of electrons in angular distribution of ATI of molecules [8]. However, very recent studies [2, 9] indicated that the dynamics of ATI process was strongly influenced by Coulomb potential just like that in NSDI [10]. Due to the enhanced complexity by Coulomb potential, many puzzles about the dynamics of ATI still exist, especially for low-energy part. For example, the origin of a clear minimum at zero in the longitudinal (parallel to the laser polarization) momentum spectrum from single ionization is still under debate [9, 11, 12]. Very recently, two experiments [2, 3] show unexpected nonzero peaks at low-energy part of ATI spectra. The origin of this structure has not been unveiled soundly. The perfect interpretation of these noble structures [2, 3, 9] need further study of electron dynamic in the presence of Coulomb potential and laser field.

Compared with the longitudinal motion, the transverse (perpendicular to the laser polarization) motion of the ionized electron is more easily effected by the Coulomb potential since it is not directly influenced by the laser field. For nonsequential double ionization, previous studies have demonstrated that the Coulomb focusing on the transverse motion of the firstly ionized electron can greatly enhance the yield of double ionization [13, 14]. For single ionization in the tunneling regime, the Coulomb focusing effect in the transverse momentum distribution of the recoil-ion has been experimentally observed and reproduced by the classical simulation [15]. With the quantum calculation, Chen et al have predicted the narrowing of the transverse momentum spectra for ATI of atoms. By comparing the 2D photoelectron momentum spectra for the effective atomic potential and short-range potential modifying the “tails” of the effective atomic potential, the narrowing effect has been attributed to the attractive force of the “tails” of effective atomic potential [16]. In this paper, we have systematically investigated the Coulomb focusing effect of the electron transverse momentum with the 2D quantum calculation. By comparing the 2D photoelectron momentum spectra for the soft-core long-range potential and exponential short-range potential, the Coulomb focusing effect of the electron transverse momentum by the long-range ion-electron interaction is also demonstrated. The same results for different mode potentials obviously confirm that the Coulomb focusing effect is due to the long-range nature of Coulomb potential rather than peculiarities in the shape of the mode potentials used in calculations. More importantly, the intuitive physical picture for the focusing dynamics of the returning electron is clearly presented and the strong dependence of the focusing efficiency on the laser wavelength and intensity is systematically explored and well explained.

 

Fig. 1. The long-range Coulomb potential (blue solid curve) and the short-range Coulomb potential (red dashed curve).

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The two-dimensional time-dependent Schrödinger equation (TDSE) for atoms driven by a linearly polarized (along the x axis) intense laser pulse has the following form: $i\frac{\partial \psi (x,y,t)}{\partial t}=[-\frac{1}{2}\frac{{\partial }^2}{{\partial }^{2}x}-\frac{1}{2}\frac{{\partial }^2}{{\partial }^{2}y}+V(x,y)+xE\left(t\right)]\psi (x,y,t)$ . Where x, y are the active two-dimensional electron coordinates. E(t) is the electric field of a laser pulse. V(x,y) is the mode Coulomb potential of atoms. In our simulations, we use the long-range Coulomb potential $V_l(x,y)=-\frac{1}{\sqrt{a+(x^2+y^2)}}$ and the short-range Coulomb potential V_s(x,y) = v ₀ exp[−(x ² + y ²)/∆²]. In Fig. 1 we show the two Coulomb potential curves, where a is chosen to be 0.9, I_p=−0.4494 a.u., approaching to the ionization potential of Xe. We set v ₀=-0.903 a.u. and ∆=2.5 to make sure the equality of the binding energies of the long-range and short-range Coulomb potentials. The short-range Coulomb potential decreases rapidly to zero (about 6 a.u). In this way, we can study the effect of Coulomb potential on the ejected electrons by comparing the results of the long-range and short-range Coulomb potentials in ATI of atoms. In order to clearly show the effect of Coulomb potential on ATI process low laser intensities are used in our work.

We use the split-operator spectral method [17] to numerically solve the 2D TDSE. Following [18], the whole 2D space is partitioned into the inner (0→R_c) and outer (R_c →R_max) regions smoothly by a splitting technique [19–22]. In inner region, the wave function is propagated exactly in the presence of Coulomb potential and laser field. In outer region, which corresponds to single ionization, the Coulomb potential is neglected and the time evolution of the wave function can be performed simply by multiplications in momentum space. Our calculations use 10 optical cycles trapezoidally shaped laser pulses with different wavelengths, switched on and of linearly over 2 optical cycles. The initiate wave function is the single active electron atomic ground state obtained by imaginary-time propagation. After the end of the pulse, the wave function is allowed to propagate without laser field for an additional time of 15 fs. The final results do not change any more even though the wave function propagates without laser field for a longer additional time. At the end of the propagation, the wave function in outer region yields the electron momentum spectra from ATI.

Figure 2 displays 2D photoelectron momentum spectra for ATI by a 10 cycles trapezoidally shaped laser pulse with wavelength of 1200 nm at the peak intensity of 8×10¹³ W/cm ². Figures 2(a) and 2(b) represent the long-range and short-range Coulomb potentials respectively. At first glance, the 2D momentum spectra appear to be fairly similar for the two different Coulomb potentials, in terms of the ring structure that corresponds to the separated peaks of ATI spectrum. Each ring consists of some discrete maximal portions, which has been shown in Ref. [16]. A closer look reveals that there are some differences. Within the first ring, these maximal portions in Fig. 2(a) are close to horizontal axis and their intensities decrease with the transverse momentum increasing; whereas those maximal portions in Fig. 2(b) have larger momenta and their intensities increase with the transverse momentum increasing. Similarly, within the second ring, major electrons have transverse momenta below 0.04 a.u. in Fig. 2(a), but considerable electrons have larger transverse momenta in Fig. 2(b). These distinctions between Fig. 2(a) and 2(b) indicate that the transverse distribution in Fig. 2(a) is much narrower than that in Fig. 2(b) within the first and second rings. The narrowing effect of the transverse momentum distribution in Fig. 2(a) still exists for higher order rings, however is weaker than that for lower order rings. That is to say, the transverse momentum distribution for the long-range Coulomb potential is much narrower than that for the short-range Coulomb potential.

 

Fig. 2. 2D photoelectron momentum spectra for ATI by a 10 cycles trapezoidally shaped laser pulse with wavelength of 1200 nm at the peak intensity of 8×10¹³ W/cm ². (a) corresponds to the long-range Coulomb potential; (b) corresponds to the short-range Coulomb potential. The maximum differential probabilities in the photoelectron momentum spectra are normalized to 1 respectively.

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Fig. 3. Transverse momentum spectra from the 2D momentum spectra of Fig. 2. Blue solid curve: the long-range Coulomb potential; Red dashed curve: the short-range Coulomb potential. The curves are normalized so that the area under each curve is unit.

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In order to further elucidate this phenomena, in Fig. 3 we present the transverse momentum spectra integrated over longitudinal momentum for Fig. 2, where blue solid curve and red dashed curve correspond to the long-range and short-range Coulomb potentials respectively. The curves are normalized so that the area under each curve is unit. Clearly the transverse momentum spectrum for the long-range Coulomb potential exhibits a sharp cusp-like distribution near zero momentum, which is consistent with previous study [23]. Moreover, the transverse momentum distribution for the long-range Coulomb potential is much narrower than that for the short-range Coulomb potential. In other words, the ejected electrons are focused to zero momentum by the long-range interaction between the parent cores and electrons.

We consider the spatial distribution of the laser intensity. The calculations are performed at 10 different intensities in the range 0.0153–0.08 PW/cm ² for the Gaussian spatial distribution in a laser focal volume at the wavelength of 1200 nm, as shown in Fig. 4. One can find that these rings corresponding to ATI peaks tend to be washed out due to the spatial average. However, the focusing of the electron transverse momentum is still apparent by comparing Fig. 4(a) with 4(b).

 

Fig. 4. 2D photoelectron momentum spectra for ATI with spatial averaging over the Gaussian intensity distribution in a laser focal volume with the wavelength of 1200 nm at the intensity of 8×10¹³ W/cm ² in the focal center. (a) corresponds to the long-range Coulomb potential; (b) corresponds to the short-range Coulomb potential. The maximum differential probabilities in the photoelectron momentum spectra are normalized to 1 respectively.

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According to the classical model [7], the electron ejected within 12.5° phase after the peak of the electric field will return at least three times. Because most of electrons are ejected near the peak of the electric field, the electrons with multiple returns are considerable and thus Coulomb focusing involving multiple returns is not ignored. Due to initial transverse momentum v _⊥ most of these returning electrons will miss the parent cores during the returns. However, these missing electrons will still interact with the parent cores when they return the vicinity of the parent cores. Due to the Coulomb attraction of the parent core, the electron is focused gradually into an area close to the parent core in the transverse direction and its transverse momentum also decreases gradually whenever they approach the parent core. In this way, the focusing of the electron transverse momentum in ATI is well interpreted. According to the analysis above, the electron transverse momentum can be manipulated by controlling the dynamics of the returning electron. The longitudinal momentum of the returning electron is $\stackrel{.}{x}=-\frac{E\lambda }{2\pi c}(\cos \omega t_1-\cos \omega t_0)$ by integrating the Newton motion equation. Here t ₀ is the ejected time of the electron, t ₁ is the return time of the electron. The longitudinal momentum of the returning electron is directly proportional to the intensity of the laser electric field and wavelength. With the increase of the laser wavelength and intensity the longitudinal momentum of the returning electron become large and thus the returning electron overpasses the parent core more quickly. Consequently, the time of the interaction between the parent core and the returning electron become short for the long wavelength and high intensity. Thus we can predict that the focusing of the electron transverse momentum becomes gradually weak with the laser wavelength and intensity increasing by the intuitive physical picture.

 

Fig. 5. 2D photoelectron momentum spectra for ATI by 10 cycles trapezoidally shaped laser pulses with intensities and wavelengths: panels (a)–(d) I = 8×10¹³ W/cm ²; (e), (f) I =4×10¹³ W/cm ²; (a), (b) λ =1600 nm, (c)–(f) λ =2000 nm. (a), (c) and (e) correspond to the long-range Coulomb potential; (b), (d) and (f) correspond to the short-range Coulomb potential. The maximum differential probabilities in the photoelectron momentum spectra are normalized to 1 respectively.

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We further calculate 2D photoelectron momentum spectra for ATI by 10 cycles trapezoidally shaped laser pulses with wavelengths of 1600 nm [Figs. 5(a), 5(b)], 2000 nm [Figs. 5(c)–5(f)], at the peak intensities of 8×10¹³ W/cm ² [Figs. 5(a)–5(d)], 4×10¹³ W/cm ² [Figs. 5(e), 5(f)]. Figures 5(a), 5(c) and 5(e) represent the long-range Coulomb potential. Figures 5(b), 5(d) and 5(f) represent the short-range Coulomb potential. Comparing the two momentum spectra in a row in Fig. 5, the focusing of the electron transverse momentum by the long-range interaction is clear for all the laser wavelengths and intensities we investigated. From the momentum spectra of Figs. 2(a), 5(a) and 5(c), Obviously, the transverse momentum distribution for the long wavelength is much broader than that for the short wavelength, i.e., the focusing of the electron transverse momentum is suppressed for the long wavelength with respect to the short wavelength. From Figs. 5(c) and 5(e), we can find that the focusing of the electron transverse momentum at low intensity is much stronger than that at high intensity. The intensity and wavelength dependence of the focusing effect can be more clearly seen if we integrate over longitudinal momentum of 2D photoelectron momentum spectra, and the results are shown in Fig. 6. In Fig. 6(a), the wavelength is 2000 nm and the intensities are 0.08 PW/cm ² (blue solid curve) and 0.04 PW/cm ² (red dashed curve), respectively. In Fig. 6(b), the intensity is 0.08 PW/cm ² and the wavelengths are 1200 nm (blue solid curve), 1600 nm (green solid curve), and 2000 nm (red dashed curve), respectively. The strong dependence of the focusing of the electron transverse momentum on the laser wavelength and intensity by quantum mechanical calculations accords well with the prediction from the intuitive physical picture.

 

Fig. 6. Transverse momentum spectra for ATI by intense laser fields with different intensities and wavelengths. (a) the laser intensities are I = 8×10¹³ W/cm ² (Blue solid curve), I = 4×10¹³ W/cm ² (Red dashed curve), and the wavelength is 2000 nm for both curves; (b) the laser intensity is I = 8×10¹³ W/cm ² for all of the cures, and the wavelengths are 1200 nm (Blue solid curve), 1600 nm (Green solid curve), and 2000 nm (Red dashed curve), respectively.

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The analysis above shows that Coulomb focusing effect on the electron transverse momentum is significantly influenced by the longitudinal momentum of the returning electron. The longitudinal momentum of the returning electron can be easily controlled by changing the wavelength or intensity of a laser pulse. Consequently, the electron transverse momentum distribution can be manipulated by controlling the wavelength or intensity of a laser pulse in ATI. Thus the diffraction pattern of ATI of molecules [8] can be manipulated by controlling the wavelength or intensity of a laser pulse.

In conclusion, we have investigated the 2D photoelectron momentum spectra from ATI of atoms for the long-range and short-range Coulomb potentials at varied laser wavelengths and intensities by numerically solving the 2D TDSE. By comparing the 2D momentum spectra for the long-range and short-range Coulomb potentials, the focusing of the electron transverse momentum by the long-range interaction is clearly revealed. Analysis based on the classical model indicates that the Coulomb attraction of the parent core to the returning electron is responsible for the focusing of the electron transverse momentum. Moreover, the focusing of the electron transverse momentum becomes gradually weak with the laser wavelength and intensity increasing. Our results provide a powerful tool to manipulate the diffraction pattern of ATI of molecules.