Waveform Modulation of High-Order Harmonics Generated from an Atom Irradiated by a Laser Pulse and a Weak Orthogonal Electrostatic Field

Abstract

The detailed characteristics of the harmonics emission of atoms driven via a linearly polarized laser field combined with an orthogonal, weaker electrostatic field were investigated by numerically solving the time-dependent Schrödinger equation. It was found that the direction of the laser polarization and the polarization of the attosecond light, which is synthesized from the harmonic, can be controlled by the amplitude of the electrostatic field. With the analysis of the spatial distribution of the time-dependent dipole moment and the time-dependent evolution of the electronic wave packet, the control mechanism for the harmonic characters was investigated. The generation of harmonics in the vertical direction of the laser electric field is caused by the breaking of the symmetry of the time wave packet distribution. With this mechanism, we obtained circularly polarized attosecond light.

1. Introduction

When atoms, molecules, and solids are irradiated by a strong laser pulse, one can observe the high-order harmonics generation of the driving pulse [1,2,3]. Ultrashort pulses generated by harmonics can be used to detect the dynamic evolution of electrons in matter [4,5,6,7,8,9,10,11,12,13,14]. In addition, molecular electron orbitals can be imaged through tomography using high-order harmonic radiation [10,15,16,17].

The mechanism for high-order harmonics generated from atoms and molecules driven by a linearly polarized laser can be explained by the three-step model. In the model, the atom is ionized firstly by the intense laser pulse through the tunnel progress, then the motion of the ionized electron is dominated by the laser electric field, and part of the ionized electron has an opportunity to recombine with the parent ion. The harmonics is generated in the recombining process of the ionized electron [3,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Due to the linear polarization features of the driving laser pulse, the polarization of the high-order harmonics generated from the atom is usually the same as the driving laser.

However, there are many applications that require a specific polarization for the harmonic light source [32,33,34,35,36,37,38]. For example, in the ultrafast research of magnetic materials, ultrashort pulses with a circular polarization are needed. Due to the large spectral widths of the ultrashort pulses, it is very difficult to directly change the polarization of the harmonic light source. Therefore, schemes for the generation of the required polarized harmonic radiation from the variation on the driving light field or gas target have been proposed.

Mairesse et al. observed the elliptically polarized harmonic radiation from the $N_2$ molecules irradiated by a linearly polarized laser pulse [39,40]. Using the combination of a Ti sapphire laser and its second harmonic, Becker et al. demonstrated the circular polarization characteristics of the harmonics generated from atoms [41]. Recently, the circularly polarized harmonic is generated from the atom irradiated by a counter–rotating bichromatic circularly polarized laser pulse. The circularly polarized harmonic is used as a probe for the chiral matter [1,42,43,44,45,46,47,48]. In this work, the electrostatic fields were used to control the polarization of the harmonics.

The combined pulse schemes, which include the electrostatic or the THz fields, had been proposed to control the behaviors of the harmonics [41,49,50,51,52,53,54]. Bao et al. studied the harmonics generation from the combined pulse including the electrostatic field and the laser electric field. They found that the even–order harmonic radiation appeared in the plateau part of the harmonic spectra [53,55,56]. Yuan et al. investigated the harmonics generated from ${\mathrm{H}}_2{}^{+}$ irradiated by a few–cycle elliptically polarized laser pulse combined with a terahertz pulse. They obtained the circularly polarized harmonic and the circularly polarized attosecond pulses [54,57,58,59,60,61,62].

However, the proposed schemes cannot control the spectral characteristics of the high–order harmonics due to the lack of understanding of the generation mechanism for the harmonic polarization. In this work, the modulation of the immediate behavior of the high–order harmonics can be controlled by the addition of an orthogonal electrostatic field. More importantly, the physical mechanism of the variation in the harmonic behavior can be explained by the symmetry breaking of the electronic wave packet motion. According to the understanding of the mechanism, we modulated the field amplitude of the electrostatic field to obtain circularly polarized light. The organization of this paper is as follows. In Section 2, we give the theoretical approaches to calculate high–order harmonic spectra. In Section 3, the harmonic spectra are presented and their generation mechanism is analyzed through the evolution of the wave packet. Our main conclusions are summarized in Section 4. Atomic units are used throughout unless otherwise specified.

2. Model and Methods

In the electric dipole approximation and the length gauge, the time–dependent Schrödinger equation for the electrons of atoms irradiated by a laser pulse is [63,64,65,66,67,68]

$$\begin{array}{c}\hfill i\frac{\partial }{\partial t}\psi (x,y,t)=H(x,y,t)\psi (x,y,t)\end{array}$$

(1)

The Hamiltonian in Equation (1) is

$$\begin{array}{ccc}\hfill H(x,y,t)& =& \frac{{\mathbf{p}}^2}{2}+V(x,y)+x{E}_x\left(t\right)+y{E}_y\left(t\right)\hfill \\ & =& \frac{{p_x}^2+{p_y}^2}{2}-\frac{Q}{\sqrt{(x^2+y^2+a^2)}}+x{E}_x\left(t\right)+y{E}_y\left(t\right)\hfill \end{array}$$

(2)

$V(x,y)$ is the soft–core Coulomb potential. Nuclear charge Q is equal to 1. The soft–core parameter a = 0.07 was selected, and the ground condition energy of this system is −24.587 eV (the ground condition energy of helium atoms). $E_x\left(t\right)$ and $E_y\left(t\right)$ are the electronic fields in the x–orientation and the driving domain in the y–orientation, respectively; $E_1$ is the linearly polarized laser pulse field, and $E^{{}^{\prime }}$ is a combination of ultrashort linearly polarized laser pulses and weak electrostatic fields, orthogonal to each other. The forms are as follows:

$$\begin{array}{c}\hfill E_1=E_y\left(t\right)={E_0}_{{}_y}f\left(t\right)sin\left(\omega t\right)\end{array}$$

$$\begin{array}{c}\hfill E^{{}^{\prime }}=E_x\left(t\right)+E_y\left(t\right)={E_0}_{{}_x}+{E_0}_{{}_y}f\left(t\right)sin\left(\omega t\right)\end{array}$$

(3)

where ${E_0}_{{}_x}$ is the electrostatic field amplitude and ${E_0}_{{}_y}$ and $\omega$ are the peak amplitude, as well as the angular frequency of the laser pulse electric domain, respectively. In this paper, we define $E_2$ for ${E_0}_{{}_x}$ = 0.003, $E_3$ for ${E_0}_{{}_x}$ = 0.005 and $E_4$ for ${E_0}_{{}_x}$ = 0.0375. The equation $f\left(t\right)$ is as follows: $f\left(t\right)=si{n}^2\left(\frac{\omega t}{2N}\right)$, where N is the cycle number of laser pulses. Two optical periods were adopted in this paper. The wave function at any time is acquired by solving Equation (1) using the numerical scheme of the split operator [69]. On this basis, the dipole acceleration $a_x\left(t\right)$ and $a_y\left(t\right)$ in the two directions can be calculated:

$$\begin{array}{c}\hfill \begin{array}{ccc}{a}_x\left(t\right)& =& \frac{d^2}{d{t}^2}〈\psi (x,y,t)\left|x\right|\psi (x,y,t)〉\hfill \\ & =& 〈\psi (x,y,t)|-\frac{\partial V(x,y)}{\partial x}-E_x\left(t\right)|\psi (x,y,t)〉\hfill \\ a_y\left(t\right)& =& \frac{d^2}{d{t}^2}〈\psi (x,y,t)\left|y\right|\psi (x,y,t)〉\hfill \\ & =& 〈\psi (x,y,t)|-\frac{\partial V(x,y)}{\partial y}-E_y\left(t\right)|\psi (x,y,t)〉\hfill \end{array}\end{array}$$

(4)

Then, the harmonic radiation power of the harmonics emission spectra ${\left|P_x\left(\omega \right)\right|}^2$ and ${\left|P_y\left(\omega \right)\right|}^2$ is counted via the Fourier transform of this time–dependent dipole acceleration:

$$\begin{array}{c}\hfill \begin{array}{cc}& P_x\left(\omega \right)=\frac{1}{{\omega }^2(t_n-t_0)}{\int }_{t_i}^{t_f}{a}_x\left(t\right)exp(-i\omega t)dt\\ & P_y\left(\omega \right)=\frac{1}{{\omega }^2(t_n-t_0)}{\int }_{t_i}^{t_f}{a}_y\left(t\right)exp(-i\omega t)dt\end{array}\end{array}$$

(5)

Ultrashort attosecond pulses can be generated by the coherent superposition of the selected harmonics in a certain energy range. The time–dependent intensity of the attosecond pulse can be expressed as

$$\begin{array}{c}\hfill I\left(t\right)={\left|\sum _q{a}_{q}exp\left(iq\omega t\right)\right|}^2\end{array}$$

(6)

q is the order of the harmonics; $a_q$ is

$$\begin{array}{c}\hfill a_q={\int }_{}^{}a\left(t\right)exp(-iq\omega t)dt\end{array}$$

(7)

where $a\left(t\right)$ represents the dipole acceleration.

3. Results and Discussion

We first calculated the higher–order harmonics generated by atoms driven by a combined ultrashort laser pulse and a pulse with an orthogonal weak electrostatic field. The corresponding schematic diagram is shown in Figure 1a. The x–direction harmonic spectra are presented in Figure 1a. It was found that there was no harmonics emission generated from the atom irradiated by the laser field $E_1$. For the harmonics generated from the combined pulse, the cutoff order was the 62nd harmonic. Since the driving laser is the ultrashort pulse, there were no clear odd harmonics that can be found in the harmonic spectrum. In the figure, one can observe the interference structure produced by the two emission trajectories generated in half an optical cycle. The y–direction harmonic spectra are presented in Figure 1b. It was found that there is a high–intensity harmonic emitted from the atom driven by $E_1$ and $E_3$. Compared with the harmonics generated from $E_3$, the harmonics emission cutoff energy and efficiency remained almost unchanged in the emission spectra generated from the atom driven by $E_1$. In the high–energy part of the harmonics emission plateau, more oscillatory structures with small amplitudes can be observed. In the low–energy part of the harmonics plateau, the peak position of the intra–period intervening shifts to the high energy, as shown in the inset of Figure 1b.

In addition to the change in harmonic intensity, the differences in the harmonic waveform can be also observed. In order to study the change of the harmonic waveform more visually, we present a synthetic attosecond pulse generated from the harmonic spectra whose energy was changed from the 45st to the 60th harmonic in Figure 2. In Figure 2a, the attosecond pulse was synthesized from the harmonics generated from the atom driven by $E_1$. It was found that the polarization of the attosecond pulse was linearly polarized. With the employment of the weak electric field whose amplitude was 0.003, the attosecond pulse exhibited an elliptical polarization characteristic with little ellipticity. More importantly, the ellipticity of the attosecond waveform can be controlled by the electrostatic field amplitude, as is shown in Figure 2c. From the above analysis, one knows that the addition of an orthogonal electrostatic field provides an opportunity to control the harmonic waveform.

In order to better control the harmonic waveform, one needs to analyze deeply the harmonics emission behavior of atoms irradiated by the driving field. The harmonic spectra of the atom were calculated from the time–dependent dipole moment. The dipole moment of the system is the expected value of the dipole transition operator. In the spatial region, the calculation of the time–dependent dipole moment was divided into two parts: $a_{x+}\left(t\right)$ and $a_{x-}\left(t\right)$. The dipole moments were calculated from the $x>0$ regions and the $x<0$ regions, respectively.

$$\begin{array}{c}\hfill \begin{array}{cc}& a_{x+}\left(t\right)={\int }_{-\infty }^{+\infty }{\int }_0^{+\infty }\psi (x,y,t){\psi }^{*}(x,y,t)(-\frac{\partial V(x,y)}{\partial x}-E_x\left(t\right))dxdy\\ & a_{x-}\left(t\right)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^0\psi (x,y,t){\psi }^{*}(x,y,t)(-\frac{\partial V(x,y)}{\partial x}-E_x\left(t\right))dxdy\end{array}\end{array}$$

(8)

The corresponding harmonic spectra are $|P_{x+}\left(\omega \right){|}^2,|P_{x-}\left(\omega \right){|}^2$, respectively. The total harmonic power can be expressed as $|\mathrm{P}\left(\omega \right){|}^2=|{\mathrm{P}}_{x+}\left(\omega \right)+{\mathrm{P}}_{x-}\left(\omega \right){|}^2$. Here,

$$\begin{array}{c}\hfill \begin{array}{cc}& P_{x+}\left(\omega \right)=\frac{1}{{\omega }^2(t_n-t_0)}{\int }_{t_i}^{t_f}{a}_{x+}\left(t\right)exp(-i\omega t)dt\\ & P_{x-}\left(\omega \right)=\frac{1}{{\omega }^2(t_n-t_0)}{\int }_{t_i}^{t_f}{a}_{x-}\left(t\right)exp(-i\omega t)dt\end{array}\end{array}$$

(9)

The corresponding harmonic phase ${\varphi }_{x-}$ and ${\varphi }_{x+}$ can be written as:

$${\varphi }_{x-}=arctan(P_{{x-}_{im}}\left(\omega \right)/P_{{x-}_{re}}\left(\omega \right)),{\varphi }_{x+}=arctan(P_{{x+}_{im}}\left(\omega \right)/P_{{x+}_{re}}\left(\omega \right)),$$

where $P_{{x-}_{im}}\left(\omega \right)(P_{{x+}_{im}}\left(\omega \right)$ and $P_{{x-}_{re}}\left(\omega \right)(P_{{x+}_{re}}\left(\omega \right)$ represent the imaginary part and the real part of the harmonics emission in the $x<0$ regions ($x>0$ regions), respectively. The phase difference between the two spatial regions is

$$\begin{array}{c}\hfill \Delta {\varphi }_x={\varphi }_{x-}-{\varphi }_{x+}\end{array}$$

(10)

One can quantitatively analyze the mechanism of the harmonic waveform control using harmonic analysis in different spatial regions. Figure 3a presents the x–directional harmonic spectra generated from the atom in field $E_1$ in different spatial regions. It was found that the intensities of the harmonic spectra in different spatial regions are the same. However, the intensity of the harmonic spectra in the whole spatial range is zero. The disappearance of the harmonics emission in the x–direction is caused by the interference of the harmonics generated in the two parts of space. It can be proven by the phase difference between these two parts of the harmonics emission, as shown in Figure 3b. Furthermore, the harmonics emission analysis of the atom in the $E_2$ field was performed, as shown in Figure 3c. It was found that the harmonic intensity of the two spatial regions was also close to being the same. The intensity difference of the harmonic spectra is presented in Figure 3b, which is marked as a pink solid line. The intensity difference is larger than the intensity calculated from the whole spatial range. This intensity weakening can also be explained by the interference between the harmonics generated in the two spatial regions. The corresponding phase difference is present in Figure 3c,d. Next, we quantitatively studied the interference effects of the harmonics in different spatial regions.

For the quantitative analysis, we defined ${\Delta }_P=|{\mathrm{P}}_{x-}\left(\omega \right)|-|{\mathrm{P}}_{x+}\left(\omega \right)|$, $\mathrm{P}=|\mathrm{P}\left(\omega \right){|}^2$ and $\Delta {\varphi }_x={\varphi }_{x-}-{\varphi }_{x+}$. The total harmonic power of the system can be written as

$$\begin{array}{ccc}\hfill |\mathrm{P}\left(\omega \right){|}^2& =& |{\mathrm{P}}_{x+}\left(\omega \right)+{\mathrm{P}}_{x-}\left(\omega \right){|}^2\hfill \\ & =& |{\mathrm{P}}_{x+}\left(\omega \right){|}^2+|P_{x-}\left(\omega \right){|}^2+2|{\mathrm{P}}_{x+}\left(\omega \right)\left|\right|P_{x-}\left(\omega \right)|cos\Delta {\varphi }_x\hfill \\ & =& 2|{\mathrm{P}}_{x+}\left(\omega \right){|}^2(1+cos\Delta {\varphi }_x)+2{\Delta }_P\left|P_{x+}\left(\omega \right)\right|(1+cos\Delta {\varphi }_x)+{{\Delta }_P}^2\hfill \\ & =& {\mathrm{P}}_1+{\mathrm{P}}_2+{\mathrm{P}}_3\hfill \end{array}$$

(11)

Here, $P_1=2|{\mathrm{P}}_{x+}\left(\omega \right){|}^2(1+cos\Delta {\varphi }_x)$, $P_2=2{\Delta }_P\left|P_{x+}\left(\omega \right)\right|(1+cos\Delta {\varphi }_x)$, $P_3={{\Delta }_P}^2$.

The terms $P_1$ and $P_2$ are associated with the phase difference of the harmonics generated from different spatial regions, and $P_3$ is only associated with the intensity difference of the amplitude of harmonics emission. Figure 4a presents P, $P_1$, and the harmonic spectra calculated from the TDSE. It was found that the $P_1$ term had little effect on the harmonics emission. $P_3$ played a dominant role in the harmonic spectra, as is shown in Figure 4b. Through the above analysis, it was found that the symmetry of the dipole moment had an important impact on the harmonic yield. If this symmetry is broken, clear harmonics emission will be observed in the vertical direction of the driven laser field. Next, we analyzed the reason for the symmetry breaking by using the time–dependent evolution of the wave packet.

The electron density distributions at different instants are present in Figure 5. The instants are (a,e) $t_1$ = 1.29 o.c., (b,f) $t_2$ = 1.36 o.c., (c,g) $t_3$ = 1.43 o.c., and (d,g) $t_4$ = 1.50 o.c. The red solid line is the laser electric field, and the corresponding moments are marked with yellow dots. The distributions are symmetrical in the x–direction when the driven field is $E_1$, as is shown in Figure 5a–d. It is clearly observed that the wave packet re–collides with the parent ion. Interference structures generated by ionized and re–collision wave packets can be observed. For the driven field $E_2$, the electronic density distributions are not symmetrical in the x–direction. The motion of electrons is affected by the electrostatic field. Therefore, the magnitudes of the harmonic spectra in the two spatial regions are no longer symmetrical. This leads to the generation of the harmonics in the vertical direction of the laser electric field.

By the above analysis, when the electrostatic field was not added in the x–direction, the electron wave packet density was symmetric about the y–axis, and there were no harmonics in the x–direction, so the synthetic line polarization laser was in the y–direction. However, when an electrostatic field was added in the x–direction, the symmetry of the original wave function was destroyed, so the x–polarization direction produced the harmonics. Furthermore, by adjusting the amplitude of the electrostatic field, the adjustment of the harmonic intensity in the x–direction can be realized. Therefore, under a certain electrostatic intensity amplitude condition, harmonics emissions with close intensity in two directions can be produced. After the electrostatic field and the driving light field act together at the moment of the harmonics emission, the direction of electron movement returning to the parent ion was deflected, and the corresponding phase difference was 0.5$\pi$, and a circularly polarized ultrashort pulse emission could be observed. In this process, by adjusting the electrostatic field, it was possible to achieve a phase difference of 0.5$\pi$ between the phases of certain order harmonics, which led to the production of circularly polarized ultrashort pulses. Other orders of the harmonics emission spectrum would generate ultrashort pulses close to linear polarization. When the amplitude of electrostatic field was 0.0375, the harmonic spectrum was as presented in Figure 6a. By using the harmonics emission ranging from the 40th to the 47th harmonic, one can obtain an ultrashort pulse whose polarization is close to circular polarization. The amplitude evolution of the ultrashort pulse with time is presented in Figure 6b.

4. Conclusions

In summary, we theoretically studied the high–order harmonics generated from an atom irradiated by an electric field, which included a laser field and an orthogonal weak electrostatic field. It was found that the electrostatic field broke the symmetry of the electron wave packet density distribution and generated harmonics perpendicular to the laser polarization. Further, modulating the amplitude of the electrostatic field can well control the waveform of the harmonics. The variety of the harmonic waveform was determined by the motion of the ionized electrons. Applying this mechanism, a circularly polarized attosecond light pulse was generated. The finding of this work provides a simple and reliable method to achieve the optimization of the harmonic waveform.