## Abstract

We investigate the high-order above-threshold ionization (HATI) of atoms (Ar and Xe) and molecules (N_{2} and O_{2}) subjected to strong laser fields by numerically solving time-dependent Schrödinger equation. It is demonstrated that resonance-like enhancement of groups of adjacent peaks in photoelectron spectrum of HATI is observed for Ar, Xe, and N_{2}, while this peculiar phenomenon is absent for O_{2}, which is in agreement with experimental observation [
Phys. Rev. A **88**, 021401 (2013)]. In addition, analysis indicates that resonance-like enhancement in HATI spectra of atoms and molecules is closely related to excitation of the high-lying excited states.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Above-threshold ionization (ATI) is the most fundamental process when atoms or molecules are subjected to the intense laser fields [1,2]. ATI means that more photons than necessary for ionization can be absorbed by atoms or molecules exposed to strong laser fields and the energy spectrum of the ejected photoelectron consists of a series of peaks regularly spaced by the photon energy. When the laser intensity is high, i. e., in the tunneling regime [3], the overall low-energy part of the energy spectrum drops exponentially with increasing energy, which is originated from the electron released directly without rescattering, and the aforementioned “direct electron” has a maximal classical energy of 2*U _{p}* (

*U*represents the average oscillation energy of a free electron in an oscillating electric field) [2,4]. Surprising experimental observations in this region continue to appear. For instance, a pronounced low-energy structure is detected in the low-energy part of the energy spectra for different samples, which has attracted much attention in strong-field physics recently, and its existence is unanimously attributed to the role of the ionic coulomb potential [5–11]. The high-energy part of the energy spectrum comprises of a flat plateau, which is produced by the photoelectrons recollided with the core elastically in the backward direction, and the classical “cut-off” energy is 10

_{p}*U*[2,4,12].

_{p}In addition, a peculiar structure called “resonance-like enhancement” (RLE) has been observed in the high-energy part of the energy spectrum between 6*U _{p}* and 8

*U*for rare gas atoms at certain laser intensities in experiments [13,14], and this prominent feature cannot be explained by the simpleman’s theory, which well explain the maximal kinetic energies of the direct electron and rescattering electron. It is demonstrated that RLE is very sensitive to the laser intensity, and a slight variation of the laser intensity can give rise to order-of-magnitude changes in the yield for groups of peaks. Recently, RLE of molecules has attracted considerable attention in experiments. It is found that H

_{p}_{2}exhibits the RLE behavior observed in noble gas atoms, while the distinct phenomenon is absent for N

_{2}, O

_{2}, CO

_{2}, and N

_{2}O subjected to intense laser fields [15]. Afterwards, RLE is observed for N

_{2}in experiments by Quan

*et al*. [16]. Besides the simple molecules, complex molecules such as HCOOH, C

_{2}H

_{4}, and C

_{2}H

_{6}are experimentally employed to investigate the impact of symmetry of highest-occupied molecular orbital (HOMO) and low-lying orbitals, and the different impact of different atomic centers on the RLE structure of molecules exposed to intense laser fields [17,18].

In theory, the interpretation of the underlying mechanism for the RLE structure is still under debate since two different groups of theories give rise to the similar features. Firstly, the RLE structure of atoms is ascribed to multiphoton resonance with laser-dressed excited bound states based on the numerical solution of time-dependent Schrödinger equations (TDSE) or the Floquet approach [19–25], and it is analogous to the Freeman resonances at low intensities [26], where it occurs since excited states are Stark shifted to an energy of an integer number of photons above the ground state. Secondly, the intensity-dependent hump is explained in terms of channel closing effect within the framework of strong-field approximation (SFA) [27–31], which does not take into account excited states. When the energy of *n* photons satisfies the ionization energy *nω* = *I _{p}* +

*U*(

_{p}*ω*and

*I*indicate the laser frequency and the ionization potential, respectively), the electron is liberated with near-zero kinetic energy, which can revisit its parent ion many times in the multi-cycle laser pulse. The RLE behavior shows strong dependence on the laser intensity due to the constructive interference of a multitude of revisits, which does not show up in each individual rescattering orbit. It signifies that the RLE structure becomes less evident for few-cycle pulses and eventually disappears for a single-cycle pulse, which has been verified by experiments and TDSE simulations [17, 32]. Recently, it is found that the long quantum orbits play an important role in the high-energy regime of the photoelectron spectrum of atoms subjected to elliptically polarized laser pulses [33].

_{p}As mentioned above, no consensus on the underlying mechanism of RLE structure of atoms has been achieved for SFA and TDSE so far. In addition, strong-field approximation is widely adopted to study the RLE behavior of atoms and molecules [16,18,27–29,31,33], and TDSE is often used to investigate the intensity-dependent enhancement of groups of HATI peaks for atoms based on single-active-electron (SAE) approximation [17, 19–21]. However, a detailed investigation of RLE structure for molecules based on TDSE is still missing. The possible reasons are that the molecules possess multicenter, which is difficult to construct a proper effective binding potential, and the computation is formidable. In the present work, we employ the model potentials of atoms and molecules to investigate the RLE behavior based on TDSE calculations, and find that the intensity-dependent enhancement of groups of HATI peaks for atoms and molecules is closely related to the highly excited states. Atom units are used throughout unless otherwise indicated.

## 2. Theoretical method

In the present work, we employ SAE approximation to investigate the ionization dynamics of atoms and molecules in intense laser fields (Ar, Xe, N_{2} and O_{2}). Within the dipole approximation and length gauge, the TDSE is given by [34,35]

*H*

_{0}(

*r*) is written as

The details of model potentials *V*(**r**) for Ar, N_{2}, O_{2}, and Xe could be found in [35,36]. The vector potential has the following form

*ω*,

*n*, and

*ẑ*denote the angular frequency of the laser field, the number of optical cycles, and the unit vector along the

*z*-axis, respectively, and the time-varying electric field is defined via

**E**(

*t*) = −

*∂*

**A**(

*t*)/

*∂t*. The polarization direction of the laser field is parallel to the molecular axis for N

_{2}and O

_{2}, and the wave function is expanded by B-splines as

*k*=7 [34, 37].

*m*indicates the magnetic quantum number, and

*ξ*= cos

*θ*. Crank-Nicolson method is employed to propagate the time-dependent wave function, and details of the TDSE method has been described in [34, 35]. In order to efficiently extract the photoelectron spectrum (PES) from the final wave function, a window operator is adopted to obtain the probability of the electron in a final-state “energy bin” of a width of 2

*γ*centered around

*E*

_{k}*μ*= 4 and

*γ*=0.0025 a.u. are employed [38]. In the present calculation, the truncated radius is

*r*=1400 a.u., and 1500 radial B-splines and 20 angular B-splines are adopted. The magnetic quantum number

_{max}*m*=0 is employed. The atoms and molecules are subjected to linearly polarized laser pulses of the frequency

*ω*=0.057 a.u. (800 nm) and 10 optical cycles, and the time step is 0.02 a.u. A cos

^{1/8}absorber function is employed near the boundary to bring down the unphysical reflections of the electron wave packet from the boundary.

## 3. Results and discussions

Figure 1 presents the typical photoelectron energy spectra of atoms (Ar and Xe) and molecules (N_{2} and O_{2}) in the polarization direction of the laser field with various laser intensities. The low-order ATI decreases exponentially, which is followed by a plateau, and the location of the cut-off for the plateau is proportional to the laser intensity. For Ar, the enhancement of HATI is well-documented in the pioneering work of Paulus *et al*. [12, 14, 28]. In Fig. 1(a), as the laser intensity increases from 58 TW/cm^{2} to 63 TW/cm^{2}, a pronounced hump appears around 25 eV, which is in good agreement with the TDSE calculation in [20], and the location of the enhancement for Ar is close to the experiment finding [15]. This hump shrinks at a laser intensity of 68 TW/cm^{2} and subsequently is not visible at 93 TW/cm^{2}. Increasing the laser intensity to 93 TW/cm^{2}, another group of resonance-like enhancement structure becomes more evident around 32 eV. For N_{2}, as the laser intensity increases from 83 TW/cm^{2} by an interval of 5 TW/cm^{2}, it is demonstrated that a prominent hump appears near 40 eV at the laser intensity of 88 TW/cm^{2} in Fig. 1(b), which is close to the energy region of the second RLE structure of Ar. For Xe, as the laser intensity increases from 65 TW/cm^{2} to 78 TW/cm^{2}, it is shown that an evident enhancement of the group of peaks shows up at the laser intensity 73 TW/cm^{2}, which are centered around 25 eV in Fig. 1(c). It is worthwhile mentioning that the enhancement of the spectrum near 45 eV for 73 TW/cm^{2} can be attributed to large differential cross section for elastic scattering of the electron off Xe^{+} at corresponding impact energy [36]. For O_{2}, as the laser intensity increases from 43 TW/cm^{2} to 58 TW/cm^{2}, no humps in HATI appear in Fig. 1(d), which is in a good accordance with the experimental observations [15,16]. In [16], the appearance of RLE behavior for N_{2} is attributed to the constructive interference of multiple-return orbits that originate from *s* states contained in the initial HOMO, while the disappearance of the RLE structure of O_{2} is ascribed to the destructive interference of multiple-return orbits for *p* states included in the HOMO.

To further clearly show the RLE structures of HATI for Ar, Xe, N_{2} and its absence for O_{2}, we depict the two-dimensional map of the ATI evolution as a function of the laser peak intensity and kinetic energy of the emitted electrons for Ar, N_{2}, Xe, and O_{2} in Fig. 2. In general, there are evident ladder structures for all of the four samples, and the positions are in proportion to the laser intensities. In addition, a careful examination reveals that there are additional stripes in the horizontal direction for Ar, Xe, and N_{2}, which denote the above-mentioned humps formed by a series of adjacent peaks. These stripes show up at a particular laser intensity (63 TW/cm^{2} and 93 TW/cm^{2} for Ar, 88 TW/cm^{2} for N_{2}, and 73 TW/cm^{2} for Xe), which are marked by the dashed rectangles, and gradually vanish with increase of the laser intensity. However, no stripes show up for O_{2} in the horizontal direction, which is in a reasonable agreement with the finding in Fig. 1(d).

To further gain the physical insights into the RLE of atoms and molecules, we show the electron spectra below and above the ionization threshold of different samples for various fixed intensities in Fig. 3, and they are close to the particular laser intensity at which the RLE structures show up. For Ar, as the laser intensity changes from 58 TW/cm^{2} to 68 TW/cm^{2}, the ATI peaks shift toward lower energy as shown in Fig. 3(a). The above-mentioned distinct feature has been found in a recent TDSE simulation and attributed to a continuation of the ATI into the below-threshold negative energy region [39], which is well reproduced by a newly-developed quantum model that is constructed to study the excitation process of atoms and molecules exposed to intense laser fields [40]. The ATI peaks also move toward lower energy as the laser intensity is increased for N_{2}, Xe and O_{2} as shown in Figs. 3(b)–3(d). In addition, the population of excited states for the laser intensity associated with RLE is dominated over those at other laser intensities for Ar, which also occurs for N_{2} and Xe.

As shown in Fig. 3, the populations of excited states within a specific range of energy are closely related to the emitted electron, so we depict the time-dependent probabilities of excited states for different species exposed to various laser pulses in Fig. 4. For Ar, 4*f* state is chosen for analysis as an example (other states with close energies behave similarly). It is found that the population of 4*f* state firstly raises as a function of time and subsequently decreases with increasing time at the laser intensity of 58 TW/cm^{2} in Fig. 4(a). However, as the laser intensity is further enhanced to 63 TW/cm^{2}, where enhancement of a series of peaks in the plateau shows up, the probability of 4*f* state increases as a function of time, which is remarkably different from that of 58 TW/cm^{2}, and the population of 4*f* state overtakes that of the laser intensity 58 TW/cm^{2} after 5*T*. At the laser intensity of 68 TW/cm^{2}, the time-varying population of 4*f* state exhibits an analogous behavior of that of 58 TW/cm^{2}. For N_{2}, it is also demonstrated that the probability of 14*σ _{g}* state increases as a function of time at a well-defined laser intensity of 88 TW/cm

^{2}, which surpasses that of 93 TW/cm

^{2}after 6

*T*in Fig. 4(b), and the population for 83 TW/cm

^{2}and 93 TW/cm

^{2}exhibits a moderate decrease after 3

*T*and 4

*T*, respectively. For Xe, the population of 8

*p*state increases with time at the intensities of 65 TW/cm

^{2}and 73 TW/cm

^{2}, while the time-varying population of 8

*p*state shows a slight increase before a quick decrease at 78 TW/cm

^{2}as shown in Fig. 4(c), which is different from that of the intensity 73 TW/cm

^{2}where the RLE structure appears. It implies that the time-dependent probabilities of excited states show strong dependence on laser intensities, which is similar to the RLE structures, and the hump formed by a series of peaks appears at the laser intensities of 63 TW/cm

^{2}, 88 TW/cm

^{2}, and 73 TW/cm

^{2}for Ar, N

_{2}, and Xe, respectively. This observation is apparently in agreement with the result in [24]. However, similar behavior can be found in the Rydberg population of O

_{2}that the probability of, e. g. 5

*π*state, raises monotonously with time at the laser intensities of 43 TW/cm

_{u}^{2}but almost keeps unchanged or even drops with time at 48 TW/cm

^{2}and 53 TW/cm

^{2}in Fig. 4(d). It is noteworthy that the Rydberg state excitation (RSE) of O

_{2}is strongly suppressed comparing with that of Xe and Ar and N

_{2}behave similarly in RSE [35]. Since all the species possess similar Rydberg levels due to the large orbits of the decoupled electron, the tight relationship between RLE and RSE cannot be explained by the multiphoton Freeman-resonance picture, implying that both of them originate from electron that at first tunnels out and then evolves in the continuum state. The RSE corresponds to capture of the electron by the Rydberg state when it moves away from the core [40] and the RLE occurs when the intensity satisfies the condition of constructive interference of multiply revisiting trajectories.

## 4. Conclusions

In summary, resonance-like enhancement of atoms and diatomic molecules are numerically investigated by TDSE in combination with model potentials, B-splines and the Crank-Nicolson method. It is demonstrated that the enhancement of a series of peaks shows up for Ar, N_{2} and Xe in the plateau of ATI, which shows extreme dependence on the laser intensity, but no RLE can be found in HATI for O_{2}. In addition, a careful examination reveals that some high-lying excited states are significantly populated at particular laser intensities for which the RLE structure appears for Ar, N_{2} and Xe, and the above-mentioned populations change remarkably by a slight variation of the laser intensity, implying that the RLE of atoms and molecules is closely related to the highly excited states. Interestingly, such sensitive dependence of the RSE on the intensity can also be found for O_{2}. Since the RSE and RLE are both strongly suppressed for O_{2} comparing with Xe, the RSE and RLE can be attributed to be captured of and multiple rescattering of the electron after it is pumped into the continuum state by the laser field, respectively.

## Funding

National Key Research and Development program of China (No. 2016YFA0401100); National Natural Science Foundation of China (NSFC) (11334009 and 11425414).

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