Theoretical research on multiple rescatterings in the process of high-order harmonic generation from a helium atom with a long wavelength

High-order harmonic generation (HHG) [1, 2] is a nonlinear phenomenon coming from the interaction between intense laser field and atom [3, 4] or molecule [5, 6], which is considered as an important tool to generate a coherent attosecond laser pulse [7, 8]. Numerous schemes are presented to extend the harmonic plateau and improve the harmonic efficiency, such as few-cycle laser pulse [9–11], polarization gating technique [12–14], two-color and multicolor laser field [15–22], etc. The harmonic generation process can be well understood by the three-step model [23]: the electron is first ionized under the effect of laser field; then is accelerated in the external field; finally recombines with the nucleus after reversing the direction of laser field. While most previous harmonic researchers only pay attention to the first rescattering without considering the multiple rescatterings [24–26]. There are two main reasons for ignoring the multiple rescatterings in previous studies. On the one hand, the harmonic intensity from the multiple rescatterings is weaker as the electron should experience longer time before recombining with the nucleus. On the other hand, many features of HHG can be reasonably explained via only considering the first rescattering [27].

However, the phenomenon of multiple rescatterings has attracted more and more attention since Tate et al [28] revealed that high-order electron rescatterings can make an unexpected contribution to HHG in their study about the wavelength scaling. Ebadi [29] have investigated the influences of multiple rescatterings on the interferences induced by spatially nonhomogeneous field. He et al [27] have reported that the contribution of multiple rescatterings to HHG in long wavelength is greater than that in short wavelength. Moreover, the corresponding physical mechanism have been revealed by defining the wavelength dependence of the harmonic ratios for different rescattering orders. Miller et al [30] have controlled the emergence of multiple rescatterings in HHG process by changing the delay time between the intense infrared pulse and the isolated attosecond vacuum ultraviolet pulse. Li et al [31] have shown that the multiple rescatterings also contribute to resonance-enhanced below-threshold harmonic generation of helium (He) atom. Additionally, Le et al have confirmed that the multiple rescattering events contribute to the harmonic emission in long wavelength based on the time-dependent Schrödinger equation (TDSE) approach [32] as well as the quantum orbits theory [33]. Furthermore, they have also shown that the high-order rescattering would contribute insignificantly to the total macroscopic HHG yields for the gas jet near the laser focus; moreover, the related contribution would not survive in typical phase-matching condition especially for harmonics in higher plateau [32].

In this letter, the phenomenon of multiple rescatterings in harmonic emission from He atom with long wavelength is further investigated theoretically by solving the TDSE and the classical equation of motion. By analyzing the time-frequency distribution and the corresponding classical calculation results, the mechanism and the character of cutoff energies in multirescattering events are revealed. Furthermore, it can be proved theoretically that different initial states and laser forms will have a significant impact on the dynamic process of multiple rescatterings.

In quantum mechanical calculations, we have numerically solved the TDSE [34–36] to investigate the multiple rescatterings of He atom via the parallel quantum wave-packet computer code LZH-DICP [37, 38]. The TDSE can be expressed as follow (in atomic units (a.u.) $e=\hbar ={{m}_{\text{e}}}=1$, which is used throughout the paper unless otherwise indicated):

$$\begin{eqnarray}&&\text{i}\frac{\partial}{\partial t}\psi (x,t)=H(x,t)\psi (x,t),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&H(x,t)=-\frac{1}{2}\frac{{{\partial}^{2}}}{\partial {{x}^{2}}}+{{V}_{C}}(x)+xE(t),\end{eqnarray} \tag{ 2 }$$

where ${{V}_{C}}(x)=-1/\sqrt{a+{{x}^{2}}}$ is the 'soft-core' potential and xE(t) is the external interaction between the laser field and the He atom. The parameter a  =  0.484 is chosen to match the ionization potential (I_P) of the ground state (i.e. 24.6 eV)[39]. E(t) is the driving laser field with the form as below:

$$\begin{eqnarray}&&E(t)={{E}_{0}}\,f(t)\cos \left(\omega t\right),\end{eqnarray} \tag{ 3 }$$

where E₀ and ω represent the electric peak amplitude and the frequency of the laser field, respectively. The pulse envelope f (t) is trapezoidal pulse with seven optical cycles (o.c.) containing one ascending cycle, five constant cycles and one descending cycle.

The laser field synthesized by trapezoidal laser field and the static electric field can be written as:

$$\begin{eqnarray}&&E(t)={{E}_{0}}\,f(t)\cos \left(\omega t\right)+\beta {{E}_{0}},\end{eqnarray} \tag{ 4 }$$

the parameter β is used to adjust the intensity of the static electric field.

To research the influence of the initial state on the multiple rescattering, the initial state is prepared in the ground state |1s〉 and a coherent superposition of the ground state |1s〉 and the first excited state |2s〉, respectively. The coherent superposition state is expressed as:

$$\begin{eqnarray}&&\psi \left(x,{{t}_{0}}\right)=\frac{1}{\sqrt{2}}\left(|1s\rangle \,+|2s\rangle \right).\end{eqnarray} \tag{ 5 }$$

According to the Ehrenfest theorem [40, 41], the dipole acceleration can be written as:

$$\begin{eqnarray}&&{{d}_{A}}(t)=\langle \psi (x,t)|\,-\,\frac{\partial {{V}_{C}}}{\partial x}+E(t)|\psi (x,t)\rangle.\end{eqnarray} \tag{ 6 }$$

The time-frequency distribution is given by means of the wavelet transform [42, 43]:

$$\begin{eqnarray}&&{{d}_{\omega}}(t)={\int}^{}{{{d}}_{A}}\left({{t}^{{{\prime}^{{}}}}}\right)\sqrt{{{\omega}_{0}}}W\left({{\omega}_{0}}\left({{t}^{{{\prime}^{{}}}}}-t\right)\right)\text{d}{{t}^{{{\prime}^{{}}}}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} W(x)=\frac{1}{\sqrt{\tau}}{{\text{e}}^{\text{i}x}}{{\text{e}}^{-{{x}^{2}}/2{{\tau}^{2}}}}. \end{array}\end{eqnarray} \tag{ 8 }$$

In the classical calculations, the electron velocity can be calculated by integrating the classical equation of electron motion. With the ionization moment (t_i) and recombination moment (t_r), the electron velocity in external field is $v=-{\int}_{{{t}_{\text{i}}}}^{{{t}_{\text{r}}}}E(t)\text{d}t$. The related kinetic energy of the returning electron is ${{E}_{\text{k}}}=\frac{1}{2}{{v}^{2}}$ and the harmonic energy is given by $E={{I}_{\text{P}}}+{{E}_{\text{k}}}$.

The time-frequency distribution of He atom in ground state |1s〉 under the effect of laser field (2000 nm, $3\times {{10}^{14}}$ W cm⁻²) is presented in figure 1(a) and the laser form is expressed as equation (3). In figure 1(a), the horizontal and vertical axes represent the emission time and the harmonic energy, respectively. Moreover, the intensity of the color represents the harmonic intensity. It is generally known that the electron recombines only once every 0.5 o.c. for the atom, however, two or more additional harmonic emissions with different emission times and cutoff energies are presented obviously every 0.5 o.c. from 1.5 o.c. to 5.0 o.c. in figure 1(a). According to references [27, 28], these additional harmonic emissions are attributed to the multiple rescatterings, which is not presented so clearly under the effect of short wavelength laser pulse for the decreased energy separation between coincident rescattering events [30]. In order to better understand the multiple rescatterings, the classical returning-kinetic-energy map is shown in figure 1(b), which conforms well with the quantum results presented in figure 1(a) about the cutoff energies and the emission times. Obviously, the electronic recombination around 1.2 o.c. (red circles), 1.7 o.c. (magenta circles) and 2.2 o.c. (purple circles) are the first rescattering, second rescattering and third rescattering induced by the electron ionized around 0.5 o.c., respectively. In addition, the fourth, the fifth and the sixth rescatterings can also be distinguished clearly in figure 1(b). Nevertheless, it is difficult to pick out each rescattering from the time-frequency distribution in figure 1(a) after 2.5 o.c., because more and more multiple rescatterings are superimposed on together and interfere each other after that time. Furthermore, it can be observed clearly from figure 1(b) that the cutoff energy of the 2nth rescattering is lower than that of the (2n-1)th and the (2n  +  1)th rescatterings as shown in figure 1(b) ($n=1,2,3\cdots$). For example, the cutoff energy of the second rescattering is lower than that of the first and the third rescatterings.

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According to the three-step model [23], the harmonic cutoff energy is mainly influenced by the acceleration process. So the electron motion needs to be discussed in order to explain the role cutoff energies followed in multirescattering events above mentioned. To make the multiple rescatterings process easily be understood, only the first, the second and the third rescatterings induced by the electron ionized around 1.0 o.c. are chosen as example in the following investigation. The sketch of laser field is presented in figure 2(a) and the parameters are same as those in figure 1. By solving the classical equation of motion, the electron dynamics can be obtained. In figure 2(b), the red dashed line (${{t}_{\text{i}}}=1.09$ o.c., ${{t}_{\text{r}}}=1.58$ o.c.), red solid line (${{t}_{\text{i}}}=1.05$ o.c., ${{t}_{\text{r}}}=1.70$ o.c.) and red dotted line (${{t}_{\text{i}}}=1.00$ o.c., ${{t}_{\text{r}}}=1.91$ o.c.) are selected to analyze the electron dynamics process in the first rescattering. As described by the red dashed line and red solid line, the electron recombines with the nucleus only once under the interaction of the laser field. Nevertheless, the electron is ionized again and recombines with the nucleus after 1.91 o.c. as presented by the red dotted line. In this case, the electron may also contribute to the high-order rescatterings. Based on the three-step model [23], the electron depicted by red solid line can gain maximum kinetic energy in external field which determines the cutoff energy of the first rescattering. As to the second rescattering and the third rescattering, we only show the dynamics of electron obtaining the maximum kinetic energy in external field in figure 2(c) (${{t}_{\text{i}}}=1.02$ o.c., ${{t}_{\text{r}}}=2.23$ o.c.) and figure 2(d) (${{t}_{\text{i}}}=1.01$ o.c., ${{t}_{\text{r}}}=2.73$ o.c., respectively.

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In order to make the electron motion more intuitive and clearer, the physical model for the first, the second and the third rescatterings are established in figure 2(e). The red and blue circles represent the nucleus and the electron, respectively. Combining with figures 2(a) and (e), firstly, the electron ionized around A is accelerated under the effect of laser field; then it is decelerated when the laser field reverses its direction and the distance between the electron and the nucleus reaches the maximum around B; finally the electron recombines with the nucleus around C releasing the harmonics of the first rescattering. For the second rescattering, the electron is supposed to miss the nucleus around C and is continually accelerated in the external field. When the direction of laser field reverses once again, the electron is decelerated and begins to remove to the nucleus around D. Eventually, the harmonics of the second rescattering are emitted when the electron recombines with the nucleus around E. So for the second rescattering, the distance between the electron and the nucleus reaches the maximum twice around B and D as shown in figure 2(e). As to the electron motion in the third rescattering, the electron misses the nucleus twice (i.e. around C and E). Moreover, the electron experiences acceleration process and deceleration process after E, and then it removes back to the nucleus around F. In the end, the harmonics of the third rescattering are generated around G. Thus the distance between the electron and the nucleus reaches the maximum three times (i.e. around B, D and F) in the third rescattering.

Combining with the above analysis, the maximum kinetic energies contributing to the cutoff energies correspond to the processes from B to C, from D to E and from F to G for the first, the second and the third rescatterings, respectively. According to the classical theory [34], the electron velocity is $v=-{\int}_{{{t}_{\text{i}}}}^{{{t}_{\text{r}}}}E(t)\text{d}t$, so the areas of the shaded parts in figure 2(a) represent the electron velocities. Moreover, the areas of S₁, S₂ and S₃ represent the maximum velocities when the electron recombines with the nucleus in the first, the second and the third rescatterings, respectively. In figure 2(a), the area of S₂ is smaller than that of S₁ and S₃, which means that the maximum velocity electron obtaining in the second rescattering is less than that in the first and the third rescatterings. Considering the formula ${{E}_{\text{k}}}=\frac{1}{2}{{v}^{2}}$, the maximum kinetic energy released in the second rescattering is less than that of the first and the third rescatterings. As a result, the cutoff energy of the second rescattering is less than that of the first and the third rescatterings. With the same method, the electron motion in other multiple rescatterings can also be analyzed, so do the cutoff energies.

To grasp the rule of cutoff energies for the multiple rescatterings, the intensity-dependent maximum kinetic energies for different rescatterings are presented in figure 3(a). The horizontal axis represents the intensities of laser pulse with 2000 nm and the vertical axis represents the maximum kinetic energy. From this figure, it can be seen that the maximum kinetic energy (the ponderomotive potential U_P as a unit) in each rescattering is nearly independent of the laser intensity. With $E={{I}_{\text{P}}}+{{E}_{\text{k}}}$, the cutoff energies of the first, second, third, fourth, fifth and sixth rescatterings can be obtained, i.e. ${{I}_{\text{P}}}+3.17{{U}_{\text{P}}}$, ${{I}_{\text{P}}}+1.54{{U}_{\text{P}}}$, ${{I}_{\text{P}}}+2.42{{U}_{\text{P}}}$, ${{I}_{\text{P}}}+1.73{{U}_{\text{P}}}$, ${{I}_{\text{P}}}+2.27{{U}_{\text{P}}}$, ${{I}_{\text{P}}}+1.81{{U}_{\text{P}}}$, respectively. The maximum kinetic energies of multiple rescatterings are given in figure 3(b). The horizontal axis represents the Nth rescattering and the vertical axis represents the maximum kinetic energies. Through figure 3(b) the character of cutoff energies mentioned above is verified again. These cutoff energies are obtained in fixed wavelength, however, U_P is also influenced by the laser frequency, then the wavelength is changed to 1600 nm and 1800 nm to testify the generality of the cutoff energies mentioned above. Calculations show that the rule of the cutoff energies given above is also applicable even if the wavelength is changed. For the time-frequency distributions with the wavelength of 1600 nm and 1800 nm are similar to that in figure 1(a),which are not presented here.

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As shown in figure 1(a), both the intra-cycle interference and the inter-cycle interference of multiple rescattering events exist in harmonic emission for the case of long wavelength. Here, a scheme is proposed to simplify the process of the multirescattering and weaken the interference by preparing the coherent superposition state expressed equation (5) as the initial state. The time-frequency distribution and the corresponding classical calculation (black line) for the case of the superposition state are presented in figure 4 and the other parameters are same as those in figure 1. Notably, the result based on the quantum theory accords well with that based on the classical method. Compared with the time-frequency distribution of the ground state shown in figure 1(a), there is only a dominating harmonic emission around every recombination moment. It means that the intra-cycle interference caused by multiple rescatterings is suppressed in the superposition state. Furthermore, the cutoff energies of multiple rescatterings in the superposition state are identical with that in the ground state, i.e. the character of cutoff energies mentioned above is also applicable for the case of the superposition state.

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The harmonic intensity is mainly affected by ionization probability, then the time-dependent ionization probabilities corresponding to the ground state and the superposition state are given in figures 5(a) and (b) to explain the great difference in the time-frequency distributions. In figure 5 the horizontal axis and the vertical axis represent ionization time and ionization probability, respectively. The trend of ionization probability is totally different in two cases. The ionization probability gradually increases in figure 5(a), which means the number of electrons ionized at every laser peak is almost same. However, the ionization probability quickly increases to 0.5 in figure 5(b) around 0.5 o.c.. It indicates that the ionization takes place very rapidly in a short time interval and there are few ionized electrons after 0.5 o.c.. The initial state in figure 5(b) is the superposition state composed by the ground state and the first excited state, moreover, the electron in the first excited state is more easily ionized under the effect of laser pulse than that in the ground state. So the electron in the first excited state is almost totally ionized around 0.5 o.c.. According to the analysis about the multiple rescatterings shown in figure 1, the electron ionized around 0.5 o.c. contributes to the first rescattering around 1.2 o.c., the second rescattering around 1.7 o.c. and the third rescattering around 2.2 o.c. etc. So these multiple rescatterings should be more intense, which are presented explicitly in figure 4. With few electrons ionized from the ground state after 0.5 o.c., the related contribution to the harmonic generation is little. As a result, by changing the initial ground state to the superposition state the harmonic generation of multiple rescatterings is mainly ascribed to the electron ionized around 0.5 o.c.. It decreases the interference caused by the multiple rescatterings of the electrons ionized around different moments, i.e. the intra-cycle interference is suppressed with the employment of the superposition state in monochromatic trapezoidal field. So the harmonic generation process of multiple rescatterings becomes much clearer and easier to be understood. Furthermore, it may also provide an effective way to decrease the intra-cycle interference of multiple rescatterings in the experiment using long wavelength laser pulse.

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It is well known that the long and short quantum paths contribute to the harmonic generation, which is very important to understand the HHG process. So it is a significant point to reveal the physical mechanism of the multiple rescatterings that which path contributes more to the multiple rescatterings. Then a scheme is proposed to find the underlying mechanism via changing the form of laser field and preparing the initial state at the superposition state mentioned above. The form of laser field is expressed as equation (4), which is synthesized by trapezoidal laser field (2000 nm, $3\times {{10}^{14}}$ W cm⁻²) and static electric field. The sketches of laser fields are presented in figure 6 with $\beta =0.03$ (black line), $\beta =0.06$ (red line) and $\beta =0.09$ (blue line), respectively. Meanwhile, the time-frequency distributions in the cases of $\beta =0.03,0.06$ and 0.09 are shown in figure 7. As described from figures 7(a)–(c), the long quantum path of the first rescattering around 1.4 o.c. gradually weakens with the increase of β. Furthermore, it is very interesting that the multiple rescatterings (such as the second rescattering around 1.7 o.c. and the third rescattering around 2.2 o.c.) are always accompanied by the long quantum path around 1.4 o.c.. According to the three-step model [23], the harmonic emissions around 1.4 o.c. come from the contribution of the electron ionized around 0.5 o.c., and the acceleration process is influenced by the peak intensity around 1.0 o.c.. The figure 6 shows that the laser intensity decreases around 0.5 o.c. while increases around 1.0 o.c. with β increasing, which will lead to the extension of the acceleration process. So the ionized electron of the long quantum path will experiences longer time before recombining with the nucleus, which maybe responsible to the suppression of the long quantum path. In a word, the weaker intensity of the multiple rescatterings shown in figure 7(c) strongly recommends that the ionized electron of long quantum path is perhaps more likely to miss the nucleus to generate multiple rescatterings than that of the short one. Furthermore, the related inter-cycle interference is also much weaker than that shown in figure 4.

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In summary, the phenomenon of multiple rescatterings of a He atom in a long wavelength is investigated theoretically by solving the time-dependent Schrödinger equation and the classical equation of motion. The results show that the cutoff energy of the 2nth rescattering is lower than that of the (2n-1)th and the (2n  +  1)th rescatterings due to the different dynamical process of the ionized electron. Based on the time-frequency distribution and corresponding classical returning-kinetic-energy map, the physical picture of multiple rescatterings is built and the physical mechanism is revealed in detail. Additionally, when the initial state is changed into the superposition state, the harmonic generation is mainly contributed by the ionized electron located at the excited state around 0.5 o.c., which decreases the intra-cycle interference of the multiple rescatterings obviously. Furthermore, the phenomenon of multiple rescatterings almost disappears simultaneously with the suppression of the long quantum in the synthesized field of the trapezoidal laser field and the static electric field, which indicates that the long quantum path of the first rescattering controls the multiple rescattering events.

The authors sincerely thank Prof. Keli Han and Dr Ruifeng Lu for providing us the LZH-DICP code. This work is supported by National Natural Science Foundation of China (Grant No. 11404204, 11447208, 11504221), Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant No. 2015021023), and Program for the Top Young Academic Leaders of Higher learning Institutions of Shanxi Province, China.