High-Intensity Harmonic Generation with Energy Tunability Produced by Robust Two-Color Linearly Polarized Laser Fields

Abstract

By using the numerical solution of the time-dependent Schrödinger equation, we theoretically explored the high-order harmonic generation process under the interaction of high-intensity two-color ultrashort driving laser pulses with atoms. The symmetry of the electric field of the laser pulse will be broken. The producing electric field was controlled at the subcycle level by an IR laser and its second harmonic, which has the unique characteristic that two sequential half-cycles become distinct, rather than merely opposite in sign. Compared with the case of the atom in the fundamental laser pulse, the harmonic efficiency showed an increase of 1∼2 orders of magnitude at specific harmonic order with this combined pulse action. Through the theoretical analysis with the “three-step model”, it was demonstrated that the enhancement of the harmonic intensity is due to the fast ionization of electrons at the ionization moment and the short time from ionization to recombination of ionized electrons. In addition, effects of the peak field amplitude ratio, the full width at half maximum, the phase delay of the two-color pulses, the laser intensity and ionization probability on the harmonic efficiency enhancement were also investigated.

1. Introduction

The ultrashort intense laser interactions with atoms, molecules, and solids can produce many important nonlinear phenomena [1,2,3,4,5], such as multiphoton ionization [6], non-sequential double ionization [7,8,9], high-order above-threshold ionization [10,11,12] and high-order harmonic generation (HHG) [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. HHG has evolved into a desktop coherent light source for XUV and even soft X-ray bands because the high-order harmonic intensity with the rise in energy presents a unique plateau construction. It may also be utilized to produce attosecond pulse trains and isolated attosecond pulses [31,32,33,34,35,36,37,38,39,40]. Additionally, HHG can regulate the ultrafast motion of electrons and detect high time-resolved electron dynamics processes [41,42,43,44].

The low harmonic conversion efficiency is a key barrier to the further application of HHG. Therefore, numerous theoretical and experimental schemes have been proposed to improve the efficiency of harmonic emissions [45,46,47,48,49]. It is efficient and convenient to employ the two-color fields to increase the HHG’s intensity. Jin et al. have demonstrated that harmonic conversion efficiency can be increased by 1∼2 orders of magnitude by synthesizing two-color or three-color fields [50]. Zhang et al. found that, after adding a 0.5 fs, 62.3 nm extreme ultraviolet XUV attosecond pulse to the synthesized two-color fields at an appropriate time, the generated harmonic intensity was effectively improved by 1∼3 orders of magnitude compared with the condition of two-color fields [51]. By employing the chirped two-color fields and adjusting the laser parameters, Wang et al. selectively increased the efficiency of a single harmonic order or two harmonic orders across a wide spectrum range [52]. By using the two-color fields and the chirp parameter to control the shape of the laser field, Li et al. achieved the selective augmentation of the single-order and two-order harmonics from the He atom, which are roughly 17 times as intense as the neighborhood harmonics. This enhancement is caused by the harmonic emission peak on the short quantum trajectory of the folded structure and depends on the pulse duration of the control pulse [53].

Based on the aforementioned research, it can be proved that the pulse chirp can improve the harmonic intensity in two-color fields. However, because chirped pulses are used, the time it takes for an electron to return to the nucleus may be longer. As a result, the diffusion effect of the wave packet may be more noticeable, which could reduce the efficiency of the harmonic emission. To effectively increase the harmonic intensity, it is required to further improve the control scheme. We must provide experimentally feasible solutions from the aspects that affect the harmonic efficiency to accomplish this goal. The important factors affecting the harmonic intensity include the ionization probability of the electron that produces the harmonic and the duration of time required after ionization for the electron to return to the parent ion. High ionization rates at the moment of the electronic ionization are necessary, as is a shorter period for the ionized electrons to return to the parent ion, to achieve higher harmonic efficiency. Therefore, we applied a high-intensity two-color driven laser pulse scheme to make the electrons ionize through the over-barrier process and quickly return to the parent ion for the recombination. By adjusting the phase difference, full width at half maximum (FWHM), and peak field amplitude ratio of the driving two-color laser, it is feasible to use this technique to manage the time domain symmetry of the electrons’ quantum path and increase the conversion efficiency. The enhancement of specific harmonic efficiency can be used to obtain a high-intensity soft X-ray coherent light source. In Section 2, the theoretical methods that were employed in this paper are introduced. Section 3 contains the outcomes of our theoretical simulations and debates around them, and Section 4 is a quick overview of our study. (Atomic units are used throughout this paper, unless otherwise stated.)

2. Materials and Methods

To simulate the harmonic emission of the atom irradiated by the two-color laser field, the time-dependent Schrödinger equation (TDSE) [54,55] satisfied by the electron needs to be solved numerically as:

$$\begin{array}{c}\mathrm{i}\frac{\partial }{\partial t}\psi (x,t)=\left[-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+E\left(t\right)·x+V\left(x\right)\right]\psi (x,t)\end{array}$$

(1)

In Equation (1), where $\psi (x,t)$ is the time-dependent spatial wave function of electrons, $\psi (x,t)$ is the soft Coulomb potential:

$$\begin{array}{c}V\left(x\right)=-\frac{q}{\sqrt{{\left|x\right|}^2+a}}\end{array}$$

(2)

where the parameters are selected as q = 1 and a = 1.4039, and the corresponding energy of the ground state of Ar is −0.579.

$E\left(t\right)$ is the driving laser electric field with a total pulse duration of five optical cycles.

$$\begin{array}{c}E\left(t\right)=F_{1}cos\left(\omega t\right)f\left(t\right)+F_{2}cos(2\omega t+\phi )f\left(t\right)\end{array}$$

(3)

where $f\left(t\right)$ is the Gauss envelope form and the peak field amplitude ratio of the two-color field is $F_1/F_2$.

In the calculation, the initial wave function of the system is obtained by using the imaginary-time propagation method. On this basis, the spectral fitting method combined with the fast Fourier transform (FFT) is used to solve Equation (1) numerically [56]. To avoid nonphysical reflection of the wave packet from the boundary, the wave function at the boundary is multiplied by a cos1/8 mask function at each time step:

$$\begin{array}{c}f\left(x\right)=\left\{\begin{array}{cc}1& \left|x\right|<x_1\\ {cos}^{\frac{1}{8}}\left(\frac{\pi \left|x-x_1\right|}{2\left(x_2-x\right)}\right)& x_1\le \left|x\right|\le x_2\end{array}\right.\end{array}$$

(4)

where $x_1$ = 550, $x_2$ = 600.

After calculating the wave function at any moment, the time-dependent dipole moment in the form of acceleration is then calculated as follows:

$$\begin{array}{c}a\left(t\right)=〈\psi (x,t)\left|-\frac{dV\left(x\right)}{dx}-E_x\left(t\right)\right|\psi (x,t)〉\end{array}$$

(5)

The corresponding harmonics can be obtained by Fourier transform of the time-dependent dipole moment:

$$\begin{array}{c}P\left(\omega \right)={\left|\frac{1}{{\omega }^2\left(t_m-t_0\right)}{\int }_{t_0}^{t_m}a\left(t\right)e^{-i\omega t}dt\right|}^2\end{array}$$

(6)

The population of bound electrons:

$$\begin{array}{c}{P}_{\mathrm{bound}}={\left|〈{\phi }_n\left(x\right)\mid \psi (x,t)〉\right|}^{.}2\end{array}$$

(7)

Time-dependent ionization probability:

$$\begin{array}{c}{P}_{\mathrm{ion}}=1-{\displaystyle \sum _{\mathrm{bound}}}{\left|〈{\phi }_n\left(x\right)\mid \psi (x,t)〉\right|}^2\end{array}$$

(8)

In order to analyze the HHG mechanism, the wavelet transform can be used to analyze the time-frequency behavior, which is given by [57,58]:

$$\begin{array}{c}{A}_{\omega }\left(t_0,\omega \right)={\int }_{t_i}^{t_t}a\left(t\right)w_{t_0,\omega }\left(t\right)dt\end{array}$$

(9)

where $t_i$ = 1, $t_t$ = 1100, $t_0$ = 550, the kernel of wavelet transform is:

$$\begin{array}{c}{w}_{t_0,\omega }\left(t\right)=\sqrt{\omega }W\left[\omega \left(t-t_0\right)\right]\end{array}$$

(10)

where $W\left(x\right)=\frac{1}{\sqrt{\tau }}{e}^{ix}{e}^{-\frac{x^2}{2{\tau }^2}}$, $\tau =10$.

In addition, to explain the results of the quantum simulations, it is also necessary to numerically solve the motion equation of classical electrons under the action of a strong laser. The velocity and space position of the ionized electron at t moment are:

$$\begin{array}{c}V\left(t\right)={\int }_{t_i}^t-E\left(t^{\prime }\right)d{t}^{\prime }\end{array}$$

(11)

$$\begin{array}{c}X\left(t\right)={\int }_{t_i}^{t}V\left(t^{\prime }\right)d{t}^{\prime }\end{array}$$

(12)

3. Results and Discussion

Figure 1a shows the laser pulses used for this paper. The black dashed line represents the fundamental frequency laser field, which has a peak field amplitude of $6.9\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, a wavelength of 1600 nm, and an initial phase of 0. The high frequency laser field, which has an 800 nm wavelength and a peak field amplitude of $4.2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, is shown by the green dotted line. The orange solid line represents the change of the electric field amplitude of combined pulses with the time. Figure 1b displays the harmonic spectra (in fundamental light single photon energy units) generated by the three respective laser fields. The cutoff energy of the high frequency laser field’s harmonic emission spectrum (green dotted line) is small, being in the 62nd order. In contrast, the cutoff energy of the harmonic spectrum under the fundamental frequency laser field (black dashed line) is in the 688th order. The harmonic efficiency also decreases more quickly as the harmonic energy increases. The cutoff position of the harmonic spectrum (orange solid line) in the combined pulse action increases to the 706th order, and the conversion efficiency is significantly enhanced in the energy range of the plateau region compared to the harmonic generated by the fundamental frequency laser field. It is interesting to observe the harmonic emission peak in the two-color fields near the 192nd harmonic, and its strength is significantly larger than in the case of the fundamental laser pulse.

In the following, we examined the time-frequency behavior of the harmonic emission to determine the cause of the sharp rise in the harmonic efficiency under monochrome field and combined laser beams, as shown in Figure 2. The wavelet transform is used to display the harmonic emission’s time-frequency characteristics. Figure 2a,b show the time-frequency distributions of the high-order harmonic emission generated in monochromatic laser fields with driving laser wavelengths of 1600 nm and 800 nm. Figure 2c shows the time-frequency distribution of the high-order harmonic emission generated by the interaction of the combined two-color laser field with atoms at wavelengths of 1600 nm and 800 nm. There are five emission paths in the time-frequency diagram. The emission times are around 420, 500, 600, 650, and 750, respectively. The cut-off energy of the first emission trajectory is the 50th order, the second emission path and the fourth emission trajectory intersect near the 760th order, the cut-off energy of the third emission trajectory is about the 150th order, and the fourth and fifth emission trajectories intersect at the order of 250th. The main photon emission time is from 500 to 550. It can be seen from the black solid line in the figure that the primary emission path in the quantum simulation can be accurately reproduced using the classical three-step model calculation. When the emission time is around 551, as indicated by the red box in the figure, it is evident from the time-frequency behavior of harmonics that there is a large emission near the 192nd order. Instead of following the typical harmonic trajectory, the harmonic emission at this energy has a folded structure. From the red box of Figure 2a,b, we label the main long and short trajectories related to the folding structure of the enhanced orders. The electron that gets the most kinetic energy when the electrons are ionized departs the atom after each peak of the field and impacts the nucleus again. Less kinetic energy is returned by electrons with later (short trajectories A and C) or earlier (long trajectories B and D) nucleus hits. As a result, the recolliding electron emits two contributions to the HHG radiation, one from the long trajectory and the other from the short trajectory, both of which are up- or down-chirped. This folded form can also be accurately reproduced using the conventional three-step model calculation. Next, we will analyze the electronic ionization and recombination instants through the classical simulation to understand the reason for this part of the harmonic enhancement.

The calculated harmonic emission energy in three-step model fluctuation with the ionization time is depicted in Figure 3a. As can be observed, there are basically two portions of the ionization period after the ionization of the electrons: 420–470 and 575–625, in which the ionization time of the 192nd order harmonic is close to 451. The previous study has shown that, in order to produce harmonics with high intensities, the ionization rate at the ionization instant must be sufficiently high, the associated electron ionization chance must be high, and the electron must be able to quickly return to the parent ion. As a result, we further examined the relationship between the ionization time and the fluctuation of the recombination time, as shown in Figure 3b. The graphic shows that the electron ionizes at 451 and returns to the parent ion at 551. This duration is less than half the optical period of the fundamental frequency driving laser field, and the wave packet dispersion effect produced by the ionization electron is small. The time-dependent amplitude of the driving laser and the ionization probability change with time, as shown in Figure 3c,d. The electrons have the chance to ionize quickly at this time, creating more ionized electrons with the potential to return to the nucleus. This is because the peak field amplitude of the driving pulse is higher at the instant 451, which is $4.7\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The ground state is already over the potential barrier created by the atom and the driving laser field at this point, as exhibited in the inset of Figure 3d, which means that the electrons have the chance to ionize quickly by over-barrier ionization. This fast ionized electron returns to the parent ion in a short time to produce the harmonic radiation, which is significantly more efficient than the other harmonics.

The phase delay between the two-color laser pulse will be crucial for the harmonic emission as well as the attosecond pulse modulation. Therefore, we carefully examined how the phase delay between the two-color field affects the harmonic efficiency. Figure 4a shows the harmonic emission spectra when the phase difference changes from −0.4–0.4 $\pi$. It can be seen from the figure that, with the change of the phase difference, the harmonic behavior is much different. Only when the phase difference is −0.2–0.2 $\pi$, can the enhancement of a specific order harmonic be observed in the harmonic emission spectrum. An IR laser and its second harmonic, which has the special property that two subsequent half-cycles become distinct instead of just opposite in sign, are used to regulate the generating electric field at the subcycle level. With the phase delay, the symmetry of the laser pulse’s electric field will vary, the symmetry of the pulse in the time domain will fluctuate, thus regulating the time domain symmetry of the quantum trajectory of the electrons generating the high-order harmonic. In this range, the order of the enhanced harmonic also increases rapidly from 50th to about 300th with the increase of the phase difference. Based on the analysis of the ionization rate and the duration from the ionization to the recombination, we also checked the effect of different phase delays on HHG. Figure 4b shows the change of the time-dependent ionization rate with the phase delay. As can be found in the figure, there is a noticeable ionization rate augmentation near 450 and 475. The three-step model calculation shows that a higher ionization rate occurs at the harmonic ionization moment with greater efficiency (black triangular shape). The harmonic with improved efficiency’s recombination moment (red triangle shape) is also shown in the illustration. As can be observed, the dispersion effect is minimal and the produced harmonics are extremely effective since the time between the ionization and the recombination in the harmonic enhancement region is very short (less than half the optical cycle of the fundamental laser field).

We also investigated the impact of FWHM as well as the peak field amplitude ratio of the two-color driving laser pulses on the harmonic intensity after clarifying the harmonic enhancement mechanism. With a peak field amplitude of $6.9\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ for the electric field’s fundamental frequency and $4.2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ for the high frequency electric field, Figure 5a shows the change in the harmonic emission with the FWHM for $\phi$ = 0. As can be seen from the figure, the enhancement of specific order harmonics can still be observed as the driving laser pulse width increases, and the energy of the enhanced harmonics increases slightly. The harmonic enhancement peak at high energy vanishes when the driving laser pulse width is greater than 325. Correspondingly, a peak of the harmonic enhancement can be noticed near the low energy region. Figure 5b displays the modification of the high harmonic emission spectrum with the change of the driving laser electric field amplitude ratio for $\phi$ = 0, FWHM = 310. When the peak field amplitude ratio is less than 1, the harmonic enhancement of the specific order is not obvious and mainly reflects the behavior of the monochromatic field. When the peak field amplitude ratio is more than 1, the phenomenon of the harmonic intensity enhancement of a certain order starts to show itself and, as the amplitude ratio rises, this harmonic order moves to the high energy region. The above analysis indicates that the strong ultrashort two-color pulse scheme can be used to produce a particular order harmonic emission with high intensity through over-barrier ionization to boost the ionization rate and control the return of electrons to the parent ion in a shorter period. The particular order harmonic with high intensity may be efficiently altered by adjusting the peak field amplitude ratio, the delay of the two-color field, or other factors.

It is worth noting that ionization has a great influence on harmonics. When the laser intensity is high during the interaction with the gas target, numerous ionized electrons are generated for pulses with long laser pulse widths. Usually, the ionization probability of atoms under the influence of a long pulse width cannot be too high, and the ionization probability is typically less than 0.1. This is because the ionized electrons will cause the driving laser to de-focus and diminish the strength of the driving laser field. However, under the condition of low gas density, the defocusing effect of the light field will be weakened and this limit can be broken for short-pulse laser action by regulating the density of the gas target and laser focusing conditions, and high intensity harmonic emission and attosecond pulses can still be observed at an ionization probability of 1 [59]. In this work, the driving laser is an ultrashort pulse, so the effect of ionization can still be controlled by modulating the driving laser focusing conditions to enhance the intensity of specific harmonics. The effect of ionization probability on harmonic enhancement was systematically investigated by varying the laser intensity and is shown in Figure 6. As can be seen from the figure, the harmonic intensity is enhanced to different degrees at specific orders for both weaker and stronger fields. The blue curve in the figure shows the corresponding ionization probability changes with the laser field strength. It is evident that, as the field strength grows, the probability of ionization similarly rises, at first slowly, then quickly at the $F_2$ strength of 0.06–0.08, and finally progressively approaching 1. The blue line is the reference line with ionization of 0.1. We can clearly see the enhancement of the high-order harmonic intensity even under the ionization probability of less than 0.1. Figure 6 shows that, as the field strength becomes larger, the order in which the enhancement appears changes and the enhancement becomes more pronounced.

4. Conclusions

In summary, by utilizing high-intensity two-color driven laser beams, we theoretically suggested a method to improve the efficiency of a particular order harmonic. This scheme is based on the control of the atomic ionization by the instantaneous intensity of the high-intensity driving laser, allowing the harmonic-producing ionized wave packet to swiftly recombine with the parent ion by over-barrier ionization. Further analysis of the relationship between the phase delay, the peak field amplitude ratio, FWHM, the laser intensity and ionization probability of the two-color laser pulse and the enhancement of a specific order harmonic reveals that the scheme allows for the precise control of the required harmonic with high efficiency, opening up new possibilities for producing affordable and useful extreme ultraviolet and even X-ray coherent light sources.