1. Introduction

Many nonlinear phenomena can be generated through the irradiation of high-intensity ultrashort laser pulses with atoms, molecules, and solids, such as non-sequential double ionization, above-threshold ionization (ATI), and high-order harmonic generation (HHG) [1–19], etc. Above-threshold ionization was first observed experimentally by Agostini et al. in 1979 [20]. ATI refers to the ionization in which the bound electron of an atom has the potential to absorb more photons than the minimum number of photons required for ionization. Since the photoelectron emission spectrum resulting from ionization carries information about the quantum state of the system and the driving laser, it has been used to measure the carrier-envelope phase (CEP) of ultrashort laser pulse [21,22] and to detect the electronic structure inside atoms and molecules [23,24].

According to the conservation of energy, under the action of the laser pulse, the ground state electron of an atom absorbs energies of n photons and then ionizes, gaining kinetic energy of $E_{k} = n\omega -I_p$ ( $I_P$ is the ionization energy of the atom and $\omega$ is the central frequency of the driving laser electric field). Since the ground state electron has the opportunity to absorb more photons, several isolated peaks can appear in the photoelectron emission spectrum and the distance of every two peaks is the energy of one photon. When an atom is irradiated by intense laser electric field, the energy levels of the system will shift and the observed position of the peak energy of the photoelectron emission spectrum can commonly be determined by the equation $E_{k} = n\omega -I_p-U_p$ ($U_{P}=E_{0}^{2} / 4 \omega ^{2}$ is the ponderomotive energy acquired by the electron and $E_{0}$ is the peak amplitude in the driving laser electric field) [25]. The reason for this photoelectron emission spectrum feature is that the energy shift of the ground state of the system is smaller under the action of the laser field while the energy level shift of the continuum state is almost $U_P$.

Due to the difference in sensing the attraction of the nucleus, the energy of different bound states will move differently under the same driving light field. As the intensity of the driving laser changes, it has been found that at some specific laser intensities, electrons in the ground state have the opportunity to jump to excited states whose energy level shifts greatly, so that the electron population in the excited state enhances and then the electrons are emitted from the excited state. At this point, the photoelectron emission spectrum produces a new subpeak, which is called the Freeman resonance peak [26]. This phenomenon can be observed in numerous theoretical experiments on photoelectron emission from atoms and molecules [27–34]. For instance, Wang et al. found a subpeak structure in the photoelectron emission spectrum of a hydrogen atom and explained that the substructures due to quantum interference are entangled with those induced by the Freeman resonance [35]. Li et al. discovered Xe atoms with a double-ring structure in the photoelectron momentum spectrum under 800 nm linearly polarised laser fields and believed that this double-ring structure originated in resonant multiphoton ionization of multiple Rydberg states of atoms [36]. Yuan et al. observed a high-intensity subpeak structure in the photoelectron emission spectrum under the interaction between a hydrogen atom and an ultraviolet laser pulse and proved that this subpeak structure is generated from the interference between the ionized electrons from the ground state and the ionized electrons after a resonant jump from the ground state to the 2p state [37]. This kind of subpeak structure is also often found in molecules [38,39]. Guo et al. studied the interaction of NO molecules with 410 nm laser pulses and clarified that the peaks in the photoelectron spectrum are resulted from Freeman resonance [40]. Goto et al. examined the interaction of naphthalene (C₁₀H₈) with 775 nm laser pulses and demonstrated that the resonant structure in the photoelectron spectrum is caused by the Freeman resonance [41].

As the development of research, especially in the area of ultrashort pulse technology, the ATI caused by the effect of the laser pulse envelope has gradually drawn public attention. Under higher driving laser conditions, for high-frequency photons interacting with atoms, the main ionization of the atoms occurs at the rising and falling edges of the driving laser pulse, and the ionized electrons in these two parts interfere to produce a complex multi-peak structure [42–44]. For 400 nm laser pulses, Wickenhauser et al. found the phenomenon of subpeak structures next to photoelectron peaks in the photoelectron spectra, and observed clear subpeak structures [45]. Morishita et al. demonstrated that this subpeak structural feature can also be seen in the experimental and theoretical comparison work [46]. Recently, the effect of the pulse envelope has also drawn increasing attention in studies of photoelectron emission under low-frequency lasers. By analyzing the photoelectron spectrum generated by the interaction of C₆₀ and 790 nm laser pulse, Campbell et al. found that the changes in pulse duration could affect the mechanism of electron ionization, thus resulting in fragmentation peaks in the photoelectron emission spectra [47]. Ning et al. used artificial double-pulsed electric fields and observed that interference structures appeared on the electron spectrum of varying pulse widths in the case of nonadiabatic ultrashort pulses interacting with matter. It was attributed to the composition of two ionization paths, in which some electron amplitudes of the first path are already in the continuum, while electrons in the second path are still in the ground state [48]. Through utilizing a molecular association process produced by T-field pulses interacting with thermal gas of ⁸⁵Rb, Giannakeas et al. found the corresponding conversion efficiency of molecular exhibited low-frequency interference fringes as the pulse length increased and explained that this dynamic interference phenomenon is attributed to the production of the Stückelberg phase accumulation between the low-energy continuum state and the dressed molecular state [49]. Taoutioui et al. calculated the holographic structure in the two-dimensional photoelectron momentum distribution produced by the interaction of 800 nm laser pulses with hydrogen atoms and analyzed that different numbers of optical cycles have an effect on the characteristics of the holographic structure in strong field ionization [50].

By changing the duration of the driving laser pulse to adjust the pulse envelope, the effect of the laser pulse envelope on the photoelectron emission spectrum of atoms irradiated by a 400 nm laser of high-intensity is investigated. It is proved that by varying the pulse envelope of the driving laser, additional photoelectron emission peaks can be observed in the photoelectron emission spectrum besides the photoelectron peaks and the sideband peaks mainly due to the interference of ionized electrons at different moments along the rising edge of the laser pulse envelope. The position of the peak energy of this part of the photoelectron emission does not change with the driving laser envelope, but its intensity oscillates with the pulse envelope. Analysis of the real-time population of the bound state shows that the new peaks are generated due to the reionization of the electrons after excitation to the nf state. And the reason of oscillations is the superposition between the ionized electrons generated by resonance and the photoelectron emission spectrum caused by the effect of the pulse envelope.

2. Theoretical methods

To calculate the photoelectron emission spectrum of a system, the time-dependent Schrödinger equation of an atom in the interaction of an intense laser field needs to be solved numerically. For coordinate space calculations, due to the action of multiple periods of intense laser, a larger spatial extent is required as the ionized electrons move away from the nuclear region due to the action of multiple periods of intense laser. However, the maximum kinetic energy of the ionized electrons acquired in this process is finite, thus the computational space required for calculations in momentum space is much smaller. Thus, this paper is based on momentum space to solve the corresponding the time-dependent Schrödinger equation (TDSE) by adopting the generalized time-dependent pseudospectral method [51–54]. The time-dependent Schrödinger equation for single-electron systems under a strong laser pulse (Atomic units are used throughout this paper unless explicitly stated):

Since $\nabla \cdot \vec {A}=0$ under the Coulomb gauge, Eq. (1) becomes:

Under the electric dipole approximation [55] one can get $\mathbf {A}(\mathbf {r},t)\approx \mathbf {A}(t)$.

Let $\Psi (\mathbf {r}, t)=\exp \left (-\frac {i}{2} \int \frac {1}{c^2} \mathbf {A}^2(t) d t\right ) \Psi _v(\mathbf {r}, t)$ to obtain the time-dependent Schrödinger equation under the velocity gauge as

The vector potential form of the laser pulse used in the paper is

From the Fourier transform of the wave function $\Psi _v(\mathbf {r}, t)$ in coordinate space, the wave function of momentum space $\Phi (\mathbf {p}, t)$ can be acquired, as follows:

Applying Eq. (5) into Eq. (3), the TDSE in momentum space can be obtained:

For the hydrogen atom, the potential is given by

The momentum space wave function $\Phi (\mathbf {p}, t)$ is expanded in the partial waves as [56,57]

Here $Q_l(z)$ is the Legendre function of the second kind. ${Z}$ represents the number of nuclei, and ${Z}$=1 for the hydrogen atom.

Applying Eqs. (9) and (10) to (6), we obtain an integro-differential equation for the momentum space radial wave function

Since the Legendre function of the second kind has a logarithmic singularity at $p=p^{\prime }$, the momentum-space Coulomb potential $V_l\left (p, p^{\prime }\right )$ has a logarithmic singularity at $p=p^{\prime }$ in partial wave form. This poses a difficulty in solving the momentum-space time-dependent Schrödinger equation. To remove this singularity, the Landé subtraction technique [58] is used.

Here, we map the semi-infinite domain $[0, \infty ]$ to the finite one $[-1, 1]$, then use the Gaussian quadrature to discretize the grids in the momentum space. The mapping function is taken as

Solving Eq. (6) by a generalized time-dependent pseudospectral scheme gives the time-dependent wave function of the system. Since a linearly polarized laser pulse with the initial state of 1s is used, the equation of the scattering state is given by

The corresponding single differential scattering cross section is calculated through the projection of the scattering continuous state onto the wave function of the system at the end of the laser:

To achieve ionization information, the bound state wave functions of the system are calculated and projected onto the total wave function to obtain the population of the bound state at any moment. And when $t=t_{fin}$ the bound state population at the end time of the laser pulse is produced as follows:

Under the condition that the selected number of grid points for the calculation was 2000 and the partial waves of 25, the total number of bound states was found to be 3833. To obtain the ionization probability at the end of the laser pulse, we employed the approach of subtracting the population of all bound states from 1 :

In addition, we also calculated the photoelectron emission spectrum for the strong field approximation (SFA) method [59,60]. The Schrodinger equation for the amplitude of the corresponding continuum states $b\left (\mathbf {p}, t_{f i n}\right )$ reads

Here ${d}_{{z}}(\mathbf {v})=\left \langle \phi _{\mathrm {v}}|\mathbf {z}| 0\right \rangle$ denotes the atomic dipole matrix element for the bound-free transition and $\phi _{\mathrm {v}}=(2 \pi )^{-3 / 2} \exp (i\mathbf {v} \cdot \mathbf {r})$ is a plane-wave.

3. Results and discussions

We systematically investigated the photoelectron emission spectra of the hydrogen atom under the effect of different durations of the laser pulse at a light intensity of $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$, as shown in Fig. 1(a). It can be seen that among multiple peak structures in the photoelectron emission spectrum, photoelectron peak is the one with relatively high intensity under the action of the laser pulse with relatively short duration (e.g., near energy 0.028 in Fig. 1(a)). This photoelectron peak is generated by the interference between the ionized electrons produced by the laser pulse at different optical cycles, and the width of its peak decreases with the increase of the pulse duration. The energy position of the photoelectron peak can be determined by the energy conservation equation $E_{k} = n\omega -I_p-U_p$. In addition to the generation of photoelectron peaks, subpeaks can be observed near the photoelectron peaks. The number of subpeaks gradually increases with the increasing of the pulse duration. The energy position of the subpeak moves towards lower energies with the increase of the driving laser pulse duration. It should be noted that there are subpeaks whose energy positions do not vary with the duration of the laser pulse (e.g., the subpeak at energy 0.08 in the pink solid box in Fig. 1(a)) and the peak intensities of these subpeaks show an oscillatory behavior as the change of the pulse duration. These subpeaks still have high intensities when the pulse duration is relatively longer, even higher than the intensity of photoelectron peak. At the same time, we calculated the variation of the population of the atom’s total ionization state with the duration of the incident laser pulse at the light intensity of $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$, as presented in Fig. 1(b). From Fig. 1(b), with the increasing of the incident laser pulse duration, the population of the atom’s total ionization state first enhances rapidly and then increases slowly with periodic oscillations. And it is consistent with the oscillation behavior of the subpeak at energy 0.08 on the photoelectron emission spectra in Fig. 1(a). We found that these subpeaks can be roughly divided into two structures. The first one is the sideband peak with changing energy position and no oscillatory behavior at different pulse durations (e.g., the sideband peak with such behavior at the black dashed line in Fig. 1(a)). The second one is the subpeak with constant energy position and oscillatory behavior at different pulse durations (e.g., the subpeak at energy 0.08 in the pink solid box in Fig. 1(a)).

 

Fig. 1. (a) Dependence of the photoelectron emission spectrum with the duration of the incident laser pulse with a light intensity of $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$. (b) Variation of the population of the total ionization of the hydrogen atom with the duration of the incident laser pulse.

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To understand the reasons for the generation of sideband peaks in photoelectron emission spectra. Under the laser pulse with a peak intensity of $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$, the ionized electron was calculated separately for each optical cycle under the action of a laser pulse with a full width at half maximum of 20 optical cycles (o.c.). Since there are a total of 40 optical cycles of the laser pulse from the beginning to the end, we have analyzed the photoelectron emission of these 40 optical cycles. As shown in Fig. 2(a), we calculated the ionized electrons for 40 optical cycles, and the photoelectron emission spectrum (red dash dot line) caused by the coherent superposition of the wave packets of ionized electrons for each optical cycle is consistent with the photoelectron emission spectrum (black solid line) calculated from the wave function of total continuum state at the end of the laser pulse. The photoelectron emission spectrum corresponding to the continuum state wave packet ionized from the bound state for each optical cycle is shown in Fig. 2(b). It can be seen from the figure that the intensity of the photoelectron emission spectrum corresponding to the ionized electrons from the 8th to the 18th cycle is stronger than that corresponding to the ionized electrons from the other cycles. Thus, the photoelectron emission produced from the 8th to the 18th cycle plays a major role. Moreover, the photoelectron emission spectra produced by each optical cycle do not show the interference structure characteristic of the alternating peaks and valleys in the region of the sideband peaks, which is prominent in Fig. 2(a). We further calculated the photoelectron emission spectra produced by the coherent superposition of the ionized electron wave packets at each cycle from 0 to N cycles $(\mathrm {N}=1, 2, 3, \ldots, 40)$, as depicted in Fig. 2(c). The horizontal axis represents the time and the colors represent the variation of the intensity of the photoelectron emission spectrum obtained by the coherent superposition of the ionized electron wave packets from each cycle from 0 to N cycles $(\mathrm {N}=1, 2, 3, \ldots, 40)$. From Fig. 2(c), the photoelectron emission spectrum obtained by coherent superposition of the ionized electron packets in periods of 8 to 18 cycles is generally consistent with the photoelectron emission spectrum produced at the end of the laser. The interference characteristics of the alternating peaks and valleys in the sideband peak region are found to be obvious with the increasing number of cycles of superposition. To see more clearly the reasons for the generation of the sideband peaks in the photoelectron emission spectrum, we give the variation of the peak intensity with the superimposed cycles N at the sideband peak energy of 0.272 in Fig. 2(c) (indicated by the black arrow in Fig. 2(a)), as shown in Fig. 2(d). From the figure, it can be found that the intensity of the photoelectron emission peak produced in periods of 8 to 18 cycles shows the fastest increase. Thus, the sideband peaks in the photoelectron emission spectrum are generated mainly due to the interference of ionized electrons at different moments along the rising edge of the laser pulse envelope.

 

Fig. 2. Photoelectron emission spectra caused by the coherent superposition of the wave packets of ionized electrons for each optical cycle (red dash dot line) and the wave function of the total continuum state at the end of the laser pulse (black solid line). (b) Photoelectron emission spectrum from the continuum state wave packet ionized from the bound state at each optical cycle. (c) Photoelectron emission spectra are produced by the coherent superposition of the ionized electron wave packets at each cycle from 0 to N cycles $(\mathrm {N}=1, 2, 3, \ldots, 40)$. (d) Variation of the photoelectron emission peak intensity with the superimposed cycles N at the sideband peak energy of 0.272 in Fig. 2(c).

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In addition, in order to understand the behavior of the subpeak at 0.08 energy in the photoelectron emission spectrum, the partial wave photoelectron emission spectra of the hydrogen atom have been calculated respectively with laser pulse durations of 20 optical cycles and 50 optical cycles by using Eq. (6), as shown in Figs. 3(a) and 3(b). The insets in Figs. 3(a) and 3(b) show partial waves for $l$=0, $l$=1, $l$=3, and $l$=5 with relatively small weights in the vertical coordinate range 0-4. The six partial waves with the largest contribution are given here and compared with the total photoelectron emission spectrum. From these figures, it can be seen that the first peak in the first group of peaks in the photoelectron emission spectrum is the photoelectron one (e.g., energy 0.024 a.u. in Fig. 3(a) or energy 0.021 a.u. Figure 3(b)). The $l$=0, $l$=2, and $l$=4 partial waves are the main contributions, when the pulse duration is 20 optical cycles and the photoelectron peak is at an energy of 0.024. However, the $l$=2 and $l$=4 partial waves are the main contributions when the pulse duration is 50 optical cycles and the photoelectron peak is at an energy of 0.021. By calculating the weight of each partial wave, it is observed that for the enhanced subpeak at energy 0.08 a.u., when the pulse duration is 20 optical cycles, the partial wave with $l$=4 has the largest part in the weight 0.71475 (normalized) of the total partial wave. And the second largest part is the partial wave with $l$=2 at 0.19813 (normalized). However, at the pulse duration of 50 o.c., the $l$=4 partial wave has the greatest weight of 0.82354 (normalized), which plays a leading part, followed by the $l$=2 partial wave with a relatively small weight of 0.13839 (normalized). As can be seen from the figure, the main contribution of the photoelectron peak usually originates from more than two partial waves, whereas for the peak at energy around 0.08 a.u., the photoelectron yield mainly originates from one partial wave. In order to see more clearly the sources of the sideband peaks and the generation of subpeaks in the energy of 0.08, the photoelectron momentum distributions of the hydrogen atom were calculated for laser pulse durations of 20 o.c. and 50 o.c., respectively, as shown in Figs. 3(c) and 3(d). For the case of the linearly polarized laser pulse, the scattering state is projected to the wave function of the system at the end of the pulse, and the photoelectron momentum distribution can be obtained. The ring marked by the pink dash dot line circle in the figure is the photoelectron peak, and the ring pointed by the white arrow is the location of subpeaks in the energy of 0.08 (at the momentum of about 0.4 a.u.). From the figure, it can be seen that the photoelectron momentum distribution of the photoelectron peaks and the sideband peaks basically show similar structures, which are clearly different from the photoelectron momentum distribution at energy 0.08. At the pulse duration of 20 o.c., the peak intensity of the photoelectron momentum distribution at energy 0.08 is higher than that of the other sideband peaks, while the peak width of the photoelectron momentum distribution at energy 0.08 is narrower than that of the other sideband peaks. When the pulse duration is 50 o.c., the peak intensity of the photoelectron momentum distribution at energy 0.08 is higher than that of the other sidebands while the peak width of the photoelectron momentum distribution at energy 0.08 is narrower than that of the other sidebands. Meanwhile, the photoelectron momentum distribution at energy 0.08 is clearer and narrower than that at pulse duration of 20 o.c. The photoelectron momentum distribution at an energy of 0.08 is significantly different from that of the other sideband peaks. There is a difference, which may result from the different ionization channels of the ionized electrons in these two parts. Therefore, the population of the bound state of the atoms at the end of the laser has been further analyzed.

 

Fig. 3. Partial waves photoelectron spectra of the hydrogen atom at an incident laser intensity of $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$ and laser pulse durations of (a) 20 o.c. and (b) 50 o.c., respectively. (The inset shows partial waves for $l$=0, $l$=1, $l$=3, and $l$=5 with relatively small weights in the vertical coordinate range 0-4.) (c) Photoelectron momentum distribution with a pulse duration of 20 optical cycles. (d) Photoelectron momentum distribution with a pulse duration of 50 optical cycles.

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Figures 4(a) and (b) give the variation of the ground state and total excited state population of the atom with the duration of the laser electric field after the end of the laser. From the figures, with the increasing pulse duration, the ground state population decreases rapidly and shows a characteristic of small oscillation, while the population of the system’s total excited state shows a clear oscillatory behavior. In order to analyze the reason for the oscillation of the total excited state population, the variation of the population of the 2p, 3p, 4f, 5f, 3d, 4p, 5p, 2s, 4d, 5d, 5g, 3s, 4s and 5s states of the atom with the duration of the laser pulse after the end of the laser was calculated, as shown in Figs. 4(c)-(f). From the figure, it can be examined that the population of the 4f state is the largest compared to the population of the other states. That is, during the pulse duration, the excitation process occurs mainly between the ground state and the 4f state. The oscillation characteristics of the periodic peaks and valleys with laser pulse duration are synchronous with the total excited state population of the atom. And also, they reach the maximum or minimum at the same time. It follows that the excited 4f state, which contributes to the ionization, is likely to arise mainly from the absorption of four photons with the same spin and a photon with the opposite spin by the electron at a laser pulse with a frequency of 0.114. At the same time, according to the energy conservation, the ground state electron has the opportunity to jump to the 4f state due to the larger energy level shift in the laser field at higher light intensities, resulting in the increase of the population of the 4f state, followed by the ionization from this energy state. The change in the population of more bound states with the laser pulse duration was further investigated, as shown in Fig. 4(g). It can be observed from the figure that the population of the excited state also shows a clear oscillation with the laser pulse duration in the energy range of −0.055 a.u. to −0.005 a.u.. In order to analyze the connection between the bound state and the peak of photoelectron oscillation, the time-containing population of the bound state and the photoelectron emission spectrum at different times were analyzed.

 

Fig. 4. Variation of the total ionization of the hydrogen atom with the duration of the laser pulse, (b) variation of the excited state population with the duration of the laser pulse, population of (c) 2p, 3p, 4f, 5f, (d) 3d, 4p, 5p, (e) 2s, 4d, 5d, 5g, (f) 3s, 4s and 5s states as a function of laser pulse duration, and (g) variation of the bound state populations of the hydrogen atom with the duration of the laser pulse. Here, the intensity of the laser pulse is $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$.

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The change of 1s, 4f, 5f, and 6f states of the hydrogen atom over time were calculated when the duration of the laser pulse was 5511 a.u., as shown in Figs. 5(a), 5(b), 5(c), and 5(d). From these figures, it can be found that the 1s state population decreases between t=0 and t=3306 a.u. In contrast, the 4f, 5f, and 6f state populations first increase and then decrease with time. It is noteworthy that the population of the 1s state is changeless over time when t > 3306 a.u., with the population of 0.0011 at the end of the laser (t=5511 a.u.). However, the populations of the 4f, 5f, and 6f states are still obvious, and are respectively 0.0046, 0.0053, and 0.0019 at the end of the laser (t=5511 a.u.). To deeply comprehend the causes of resonance peaks in the photoelectron emission spectrum, the photoelectron emission process was analyzed in detail. The photoelectron emission spectra at t=3306 a.u. and t=5511 a.u. were further calculated, as shown in Fig. 5(e). It can be seen from the figure that the photoelectron emission spectrum at t=3306 a.u. produces almost identical photoelectron peaks at t=5511 a.u. except that the intensity of the subpeaks at energies of 0.081 a.u., 0.093 a.u., and 0.099 a.u. is weaker. Figure 5(f) shows the photoelectron emission spectra calculated by the time-dependent wave function at t=4409 a.u. and t=5511 a.u. From Fig. 5(f), the photoelectron emission spectrum calculated when t=4409 a.u. is consistent with the one calculated when t=5511 a.u. at the end of the laser. The Fig. 5 shows that at energies of 0.081 a.u., 0.093 a.u., and 0.099 a.u, the produced intensity of the subpeaks in the photoelectron emission spectrum at the moment t=5511 a.u. is stronger than t=3306 a.u.. This is related to the population of the bound states. In other words, the appearance of the resonance peaks at energies of 0.081 a.u., 0.093 a.u., and 0.099 a.u in the photoelectron emission spectrum can be attributed to the ionization that occurs after resonance from the 1s state to the 4f, 5f, and 6f states respectively.

 

Fig. 5. With the laser intensity of $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$ and the duration of the laser pulse of 50 o.c. (a) is the population of the ground state of the hydrogen atom over time, (b) is the population of the 4f state over time, (c) is the population of the 5f state over time, (d) the population of the 6f state over time, and (e) and (f) are the photoelectron emission spectra of the hydrogen atom at different moments, respectively.

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To further confirm the origin of this peak, we have chosen the initial state of the atom to be the 1s state and excluding the scattering term, then the strong field approximation theory is used to calculate the photoelectron emission spectrum as a function of the laser pulse duration under the same laser conditions, as shown in Fig. 6(a). It can be found from the figure that, compared to Fig. 1(a) obtained by numerically solving the TDSE, only one main photoelectron peak and multiple subpeaks are shown. The peak position can be given by, $E_{k} = n\omega -I_p-U_p$ and the resonance peak at an energy of 0.08 a.u. disappears. In the SFA calculations, only the ionization of the atom from the ground state to the continuum state is considered, neglecting the effect of the excited state. For the TDSE calculations, however, the contribution of the excited state is included. Through this comparison, it appears that the peak production near the energy of 0.08 a.u. should be in connection with the contribution of the excited state. For purpose of quantitatively analyzing the contribution of the excited state, the initial state of the atom was chosen as the 4f state, and then the photoelectron emission spectra were calculated with different durations of the laser pulse under the same driving laser conditions, which can be seen in Fig. 6(b). It can be observed that the energy position of the resonance peak through the variation of the pulse duration when the peak energy of the photoelectron emission is 0.08 a.u. is consistent with the situation when the energy is 0.08 a.u. in the photoelectron emission spectrum in Fig. 1(a). Thus, the peak with changing oscillation in the photoelectron emission spectrum can be attributed to the jumping of the ground state to a higher excited state 4f due to the resonance and then the direct ionization of the 4f state. It is important to be noted that the peak intensity of the photoelectron emission directly ionized from the 4f state does not oscillate as the pulse duration of the driving laser changes. However, the intensity of this peak generated from the ionization of the ground state exhibits oscillatory behavior. The reason for this feature is that there is an energetic superposition between the ionization yield after this resonance transition and the subpeak produced by the effect of the laser envelope, resulting in a greater intensity of the emission in the part of the energy superposition between the two cases, which then becomes smaller and greater again, showing an oscillatory feature.

 

Fig. 6. (a) Variation of the photoelectron emission spectrum under the duration of the incident laser from SFA. (b) Change of the photoelectron emission spectrum of the atom whose initial state is the 4f state under the duration of the incident laser.

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To see more clearly the effect on the photoelectron emission spectrum between light intensity and pulse duration, we calculated the variation of the photoelectron emission spectrum with the light intensity of the driving laser with pulse durations of 5 o.c., 20 o.c., and 50 o.c., as shown in Fig. 7. It can be noticed from this figure that the peak position of the photoelectron emission spectrum shifts towards lower energies as the light intensity increases. In addition to the main peak value of the photoelectron emission spectrum, a sideband with less intensity can be observed. As the pulse duration increases, the number of sideband peaks increases and the peak spacing becomes progressively smaller. At the pulse duration of 5 o.c., the resonance is not visible as the light intensity increases. When the pulse duration increases to 20 o.c., we can see that when the light intensity is stronger than $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$, there is a clear resonance peak with the energy position located at about 0.08 a.u., and its peak intensity shows oscillatory behaviour as the change of laser pulse duration. When the pulse duration continues to increase to 50 o.c., the energy position where the resonance peak is located is still about 0.08 a.u., and the peak intensity of the resonance peak has exceeded the intensity of the photoelectron emission peak. This is due to the fact that when the pulse duration is relatively long, it is possible to make the electron resonance jump to the excited state and then emit from the excited state. Thus the distinct resonance peak can be observed in the photoelectron emission spectrum for a specific range of light intensities, and it shows oscillatory behaviour with the light intensity, which is generated by the superposition of the resonance peak with the subpeak next to the photoelectron peak. At other light intensities, the resonance peak in the photoelectron emission spectrum cannot overlap energetically with the sideband peak next to the photoelectron peak. Therefore, resonance peaks with energies of about 0.08 are not observed.

 

Fig. 7. Variation of the photoelectron emission spectrum with the light intensity of the driving laser under the duration at (a) 5 o.c., (b) 20 o.c., and (c) 50 o.c., respectively.

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We further investigated the variation of the photoelectron emission spectrum with CEP for laser pulses with durations of 2 o.c., 4 o.c., and 20 o.c., as shown in Figs. 8(a)-(c). From the figure, it can be found that when the pulse duration is 2 o.c., the photoelectron emission spectrum varies very significantly with CEP, and when the pulse duration is 4 o.c., the photoelectron emission spectrum no longer varies significantly with CEP. When the pulse duration is 20 o.c., the photoelectron emission spectrum does not change with CEP. We also calculated the ground state population for CEP=0 and CEP=$\pi /4$ with time under laser pulses with durations of 2 o.c., 4 o.c. and 20 o.c., as shown in Figs. 8(d)-(f). Thus, for relatively long pulse durations, the peak positions in the photoelectron emission spectrum do not differ significantly under different CEP.

 

Fig. 8. Photoelectron emission spectra with CEP driven by laser pulses with durations of (a) 2 o.c., (b) 4 o.c., and (c) 20 o.c.. Ground population for CEP=0 and CEP=$\pi /4$ with time driven by laser pulses with durations of (d) 2 o.c., (e) 4 o.c., and (f) 20 o.c.. Here, the intensity of the laser pulse is $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$.

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The above analysis shows that the sideband photoelectron emission peak generated due to the laser pulse envelope can be enhanced by superimposing with the photoelectrons produced by resonance. In addition, the hydrogen atom under the action of an intense laser will be strongly ionized at the rising edge of the pulse, and the rapid ionization due to this envelope can also be detected using the generated sideband photoelectron emission peak. In Fig. 9, the photoelectron emission spectra driven by laser pulse durations of (a) 20 o.c., (c) 25 o.c., and (e) 30 o.c. are given. The red dashed line is the envelope of the photoelectron emission spectrum that we obtained using the high frequency filtering method. As can be seen, the photoelectron emission spectra shows a multipeak structure, and the energy positions of the peak with the highest intensity are different. Since the source of the subpeaks are the interference produced by electrons ionized at different intensities of the laser envelope, the variations of intensity should be related to the time-dependent ionization. The variation of the population of the ground state, excitation population of the atom and the laser pulse envelope over time under the action of laser pulses with pulse durations of 20 o.c., 25 o.c., and 30 o.c. are respectively given in Figs. 9(b), 9(d), 9(f). As can be seen, the atoms ionizes rapidly under the action of a high-intensity laser pulse. And because of the ground state loss effect, the electric field corresponding to the rapid ionization gradually move towards the rising edge of the laser pulse. At the falling edge of the laser pulse envelope, there is also a significant population in the excited state and the atom can be ionized. By using the immediate light intensity (green dot) of the laser electric field envelope at the time corresponding to the fastest ionization of the ground state (red rectangle), the immediate $U_p$ can be obtained. According to the energy position formula $E_{k} = n\omega -I_p-U_p$, the corresponding energy position of the emission peak of the photoelectron can be calculated. The calculation reveals that the immediate intensity of the laser pulse corresponding to the rapid decline of the ground state population produces a photoelectron emission peak. And this peak is consistent with the one with the largest intensity among the multiple subpeaks observed (e.g., the red arrow in the figure). The position of the peak indicated by the red arrow roughly coincides with the position of the envelope peak of the photoelectron emission spectrum. The energy position of the strongest subpeak is therefore determined by the time of the fastest ionization of the ground state, corresponding to the immediate $U_p$ of the electric field envelope. Thus, the real-time ionization can be detected by the sidebands of the photoelectron peaks.

 

Fig. 9. With the laser intensity of $I=2.84 \times 10^{14} \,\mathrm {W} / \mathrm {cm}^{2}$, (a), (c), and (e) are respectively the photoelectron emission spectra and spectral envelope driven by laser pulses with duration of 20 o.c., 25 o.c., and 30 o.c.. In the same condition, (b), (d), and (f) are respectively the variation of the ground state population, excitation population of atoms and the envelope of the laser field under the action of laser pulses over time when the duration are 20 o.c., 25 o.c., and 30 o.c..

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4. Conclusion

Using numerically solving the time-dependent Schrödinger equation scheme based on momentum space, the hydrogen atom’s photoelectron emission spectrum under the action of a 400 nm high-intensity laser with different durations is theoretically investigated in this paper. In the photoelectron emission spectra, photoelectron peaks from ionization in different periods,sideband peaks from the ionization at different moments due to the effect of the pulse envelope, and resonance peaks from the resonances of the ground state and nf states are observed. It is discovered that the intensity of the resonance peaks in the photoelectron emission spectrum shows an oscillatory structure as the duration of the driving laser pulse increases, and this oscillation is analyzed to be generated by the ionization superposition of the resonance peaks with the photoelectron sideband subpeaks. And the sideband peaks in the photoelectron emission spectrum are mainly due to the interference of ionized electrons at different moments along the rising of the laser pulse envelope.