1. Introduction

The strong laser field interacting with matters is usually accompanied by low energetic harmonic radiation, in which the strong terahertz (THz) emission in two-color fields is a typical exemplar [1]. Focusing the fundamental ($\omega$) and second harmonic (2$\omega$) fields in gas-phase medium produces strong ultra-broadband radiation covering from terahertz to ultraviolet band [2,3], which has potential applications in broadband and ultrafast spectroscopy. Comparing to high-harmonic generation, which has been extensively studied over the past few decades and well theoretically explained, the research on the low-harmonic generation (LHG) has been gradually carried out. The deepening of theoretical understanding benefits the applications of LHGs, including all-optical imaging for continuum electronic wavepacket [4] and all-optical broadband detection of THz waveforms [5,6].

The LHG was firstly theoretically predicted by Brunel [7], so also referred to as "Brunel harmonics", and the mechanism was found to be capable of reproducing the characteristics of third- and fifth-order harmonics in very early experiments [8]. Here, Brunel model explained the LHG in a classical and macroscopic picture: The LHG originates from the derivative of free electron current produced by strong laser fields in plasma. Then, after about 10 years, the THz radiation, i.e. zeroth-order low harmonics, created by asymmetric two-color laser field, had been successfully explained by revisiting Brunel model, often called as photocurrent (PC) model in THz community [9]. Recently, the PC model appears capable of estimating the yield ratio of LHGs from 0th to 9th harmonics, whose photon energies are far below the ionization threshold [2,3].

Besides the PC model, the strong field approximation (SFA) theory, a successful workhorse of strong-field physics, explains LHGs in a semi-classical and microscopic picture. In SFA, the LHGs mainly attribute to the continuum-continuum transition during the laser-assisted electron acceleration around ionic parent, rather than continuum-bound recollision which leads to high-harmonic generations. Within SFA framework, the continuum-continuum transition in dipole moment [10–13] and the trajectory method [14] were used to reproduce the THz generations. And recently, the continuum-continuum transition has been proved to be capable of predicting the $\omega$-2$\omega$ phase and polarization dependent yields of third harmonic generation (THG) [15].

It seems that PC and SFA explain LHGs from totally distinct aspects. The question is whether or not the connection between them exists, which will be explored in this paper. In PC model, the THz generation can be decomposed into the superposition of the contributions from individual ionization events [16,17], and the spectral shape is induced by the interference of contributions from different ionization events. Likewise, in SFA, the continuum electronic wavepacket can be represented and simplified into the ensemble of electron trajectories, and the THz generation stems from the acceleration of electron trajectories created by ionization events [14,18,19]. In this paper, the SFA-based trajectory method is used to analyse the THz characteristics, which reaches the same conclusion as the analysis of PC-based ionization events [16]. Hence, the PC and SFA can be connected with electron trajectories, which are released at series of ionization events. This work analytically builds a bridge between the PC and SFA theories by means of the trajectory method, and the trajectory method is used to analyze the THz and THG yields as a function of $\omega$-2$\omega$ phase delay.

2. Theoretical methods

Here, we start from the SFA theory and follow the analytical procedure in the previous literature [20]. In this paper, all equations are written as atomic units. The final state of electron wavefunction with asymptotic momentum ${\boldsymbol {p}}$ gives the photoelectron momentum distributions with the SFA probability amplitude,

The trajectory method can be introduced by using the saddle-point approximation to numerically solve the time integral of $M_{\boldsymbol {p}}^{(\mathrm {SFA})}$. The saddle point $t_{\boldsymbol {p}}^{(\alpha )}$ corresponds to the different ionization event. Thus, $M_{\boldsymbol {p}}^{(\mathrm {SFA})}$ can be approximated as $\tilde {M}_{\boldsymbol {p}}^{\mathrm {(SFA)}} \simeq \sum _\alpha {C_{\boldsymbol {p}}^\alpha e^{ - iS_{{\boldsymbol {p}}}^{(\alpha )}}},$ where $S_{{\boldsymbol {p}}}^{(\alpha )}=\int _{{t_{\boldsymbol {p}}^{(\alpha )}}}^\infty {\left \{ {{\frac {1}{2}{{\left [ {{\boldsymbol {p}} + {\boldsymbol {A}}(t')} \right ]}^2+I_p}}} \right \}d} t'$. ${t_{\boldsymbol {p}}^{(\alpha )}}$ is the $\alpha$th saddle point, that is determined with the stationary action equation $\frac {\partial S^{(\alpha )}_{\boldsymbol {p}}}{\partial t}|_{t_{\boldsymbol {p}}^{(\alpha )}}=0$. Here, ${t_{\boldsymbol {p}}^{(\alpha )}}$ is a complex number. The real part of ${t_{\boldsymbol {p}}^{(\alpha )}}$, $t_0 = \mathrm {Re}[{t_{\boldsymbol {p}}^{(\alpha )}}]$ is the time when an electron reaches the outer point of potential barrier, i.e. the ionization moment. The imaginary part of ${t_{\boldsymbol {p}}^{(\alpha )}}$, $t_i = \mathrm {Im}[{t_{\boldsymbol {p}}^{(\alpha )}}]$, determines the weight of the $\alpha$th trajectory.

The electron trajectories ${\boldsymbol {r}}(t)$ of the saddle point ${t_{\boldsymbol {p}}^{(\alpha )}}$ can be solved according to the following boundary conditions: (I) Spatially, the bound-state electron is located at origin before the tunneling, $\mathrm {Re}[\boldsymbol {r}({t_{\boldsymbol {p}}^{(\alpha )}})]=0$. (II) All dynamical variables are real after the electron has tunneled through the time-dependent potential barrier, $\mathrm {Im} [\boldsymbol {r}(t_0)] = \mathrm {Im}[ \boldsymbol {v}(t_0)] = 0$. In order to fulfill the boundary conditions, we construct a complex trajectory as

The real part of ${t_{\boldsymbol {p}}^{(\alpha )}}$, $t_0$, is taken to obtain the position of the tunnelling exit of the electron. And the initial velocity is given as $v(t_0) = \boldsymbol {p} + \boldsymbol {A}(t_0)$. In real space, the trajectory ${\boldsymbol {r}}_{(t>t_0)}(t)$, propagating along the real-time axis from ionization moment $t_0$ to the observation time $t$, is similar to the motion of free electrons in Newton’s classical system. ${\boldsymbol {r}}_{(t>t_0)}(t)$ is written as

Analogously, the PC model can also been written as the trajectory form. In the PC model, the low-order harmonic ${\boldsymbol {E}}_{\mathrm {LHG}}(t)$ comes from the time-varying photocurrent ${\boldsymbol {J}}(t)$ induced by the electric field ${\boldsymbol {E}}(t)$, written as

The SFA-Trajectory formula in Eq. (4) and PC-Trajectory in Eq. (6) have analogous forms, and here we show the equivalence of the SFA-Trajectory and PC-Trajectory. In our calculation, the fundamental $\omega$ and second-harmonic $2\omega$ electric fields are written as the form, ${{\boldsymbol {E}}_\omega }(t) ={E}_1\sin {(\frac {{\omega t}}{{2n_1}})^2}\cos (\omega t)$ and ${{\boldsymbol {E}}_{2\omega }}(t) = {E}_2\sin {(\frac {{2\omega t}}{{2{n_2}}})^2}\cos (2 \omega t + \varphi )$, where $t \in \left [ { - \frac {{{n_1}\pi }}{\omega }, \frac {{{n_1}\pi }}{\omega }} \right ]$. The variable $\varphi$ is introduced as the phase delay of the two-color field, and $\varphi$ can also be written as the time delay $\tau$, which is calculated as $\tau = \frac {\varphi }{2 \omega }$. Hence, the $2 \omega$ field can also be expressed as the form of variable time delay, ${{\boldsymbol {E}}_{2\omega }}(t) = {E}_2\sin {(\frac {{2\omega t}}{{2{n_2}}})^2}\cos (2 \omega (t - \tau ))$. $E_1$ = 0.1 a.u. and $E_2$ = 0.05 a.u. represent the peak field strength. $n_1$ = 15 and $n_2$ = 30 are the cycle numbers of the laser pulse, corresponding to the pulse duration. And the fundamental frequency $\omega =0.05695$. Firstly, as shown in Fig. 1(a), the SFA-Trajectory and PC-Trajectory give the same predictions for THz and THG modulations versus time delay $\tau$, which well agree with the measurement [13,15]. In Fig. 1(b), we show the analogue relationship of each terms in Eq. (4) and Eq. (6). The SFA-Trajectory and PC-Trajectory predict the same ionization probability $w(t_0)$, shown as blue shadows and black line in Fig. 1(b). The electron trajectory in PC-Trajectory can be written as $\boldsymbol {r}(t) = \int ^t_{t_0}dt'' \int ^{t''}_{t_0} \boldsymbol {E}(t')H(t'-t_0) dt'$, which shows similar shape with $\boldsymbol {r}(t)$ in Eq. (4) except the initial position at the ionization moment. The exemplary electron trajectories with high weights, released at the extrema of electric field $\boldsymbol {E}(t)$, are plotted in Fig. 1(b) for comparison. The $\boldsymbol {r}(t)$ predicted by PC-Trajectory (solid lines) and SFA-Trajectory (dash lines) approximately coincides except the initial positions $\boldsymbol {r}(t_0)$. And ${{\partial ^2}{\boldsymbol {r}}(t)}/{\partial {t^2}}$ and $\boldsymbol {E}(t)H(t'-t_0)$ indicate the light waves induced by electron acceleration, shown as dot lines. Hence, if the Coulombic potential is neglected, the PC and SFA-trajectory methods give exactly the same prediction for the LHGs [22] Therefore, the equivalence of SFA-Trajectory and PC-Trajectory formulas build a bridge between the microscopic and classical theories.

 

Fig. 1. Time-delay dependence of THz and third-order harmonic generation. (a) THz and THG yields versus time delay $\tau$. Blue line: THz yields calculated with PC-Trajectory. Red dots: THz yields of SFA-Trajectory. Green line: THGs of PC-Trajectory. Black dots: THGs of SFA-Trajectory. (b) The electron trajectories ${\boldsymbol {r}}(t)$ and light waves induced by electron trajectories from different ionization moments $t_0$ ($1-4$). Gray line: Electric field ${\boldsymbol {E}}$. The blue shadows and black line are the ionization probability predicted by the SFA-Trajectory and PC-Trajectory. Solid lines: ${\boldsymbol {r}}(t)$ predicted by PC-Trajectory; Dash lines: ${\boldsymbol {r}}(t)$ predicted by SFA-Trajectory. Dot lines: Light waves ${{\partial ^2}{\boldsymbol {r}}(t)}/{\partial {t^2}}$ and $\boldsymbol {E}(t)H(t'-t_0)$ induced by electron trajectories.

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3. Results

The trajectory method provides a unified perspective for LHGs, and we apply the trajectory perspective to explain the THz and THG modulations versue $\omega$-2$\omega$ time delay $\tau$, which is the most robust characteristics in measurements. As shown in Fig. 1(a), the THz and THG yields show the opposite modulation manners versus $\tau$: For THz generation, the maximum and minimum values appear at $\tau = 0.33\ \mathrm {fs}$ and $\tau = 0\ \mathrm {fs}$, corresponding to the relative phase delay $\varphi = \frac {1}{2} (2n+1) \pi$ and $\varphi = n \pi$, while the $\tau$-dependent THGs are opposite.

The process of LHGs can be divided into three steps: Firstly, the electron is ionized around the extrema of the electric field, which is expressed as ionization events. In Fig. 2, the blue shadow presents the ionization probability, and the color dots marked as $1$,…, $n$ on the shadow indicate the different ionization events occurring at different ionization moments $t_1$,…, $t_n$. Secondly, the electron trajectories from individual ionization events propagate along $t$, accompanying with light wave radiation. The electron trajectory $\boldsymbol {r}(t)$ released at ionization moment $t_n$ is represented as $\boldsymbol {r}(t,t_n)$, and the exemplary trajectories are shown in Fig. 1(b). The light wave radiation induced by $\boldsymbol {r}(t,t_n)$ is represented as $\boldsymbol {E}_{n}(t,t_n)$ , which can be calculated as

 

Fig. 2. The spectral shapes induced by the superposition of ionization events at different ionization moments. (a1) and (b1) Electric fields $\boldsymbol {E}(t)$ at $\tau = 0.33\ \mathrm {fs}\ (\varphi = 0.5\pi )$ and $\tau = 0\ \mathrm {fs}\ (\varphi = 0)$. The ionization probability is shown as blue shadow. The ionization events at different ionization moments $t_1$,…, $t_n$ is marked as color dots. (a2) and (b2) The spectral shapes of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ induced by electron trajectories from individual ionization events $t_n$. (a3) and (b3) The spectral shapes induced by the superposition of different ionization events. Red line: Spectral shape of $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$, which is the sum of all $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$. Other colors: $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$ is the sum of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ and $\tilde {\boldsymbol {E}}_{m} (\nu,t_m)$ induced by two ionization events $t_n$ and $t_m$.

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We found that the spectral shapes induced by individual ionization events (Fig. 2(a2) and (b2)) and the superposition of all ionization events (red lines in Fig. 2(a3) and (b3)) are essentially different. In Fig. 2, we perform the back-analysis of electron trajectories to investigate how the superposition of trajectories released at different ionization events influences the spectral shapes and yields of LHGs. Here, three typical combinations of ionization events are considered, the superposition of trajectories released within (i) the same cycles, (ii) the adjacent cycle and (iii) the interval cycles. Several conclusions have been drawn from the trajectory back-analysis in frequency domain.

Firstly, the comb-like structure in spectrum stems from the intercycle superposition of trajectories released at adjacent cycles. $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$ induced by intercycle superposition of trajectories is shown as the blue lines in Fig. 2(a3) and (b3). $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$ presents the harmonic spectrum, whose energetic peaks are evenly spaced by the photon energy, which is similar to the sum of all trajectories (red curve). Here, the comb-like structure in $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$ results from the interference of the electron trajectories from adjacent cycles, which shares analogous explanation as high-harmonic generation.

Secondly, the superposition of ionization events from one-cycle interval leads to the minutia structure of harmonic spectra. As the green lines shown in Fig. 2(a3) and (b3), the superposition of the trajectories released from interval cycles results in rapid modulation along $\nu$ in $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$, leading to the sideband peaks between the main peaks.

Thirdly, the THG yield modulation versus $\tau$ stems from the intracycle superposition of electron trajectories. When two-color time delay $\tau = 0.33\ \mathrm {fs}$, there are two electric-field extrema within one cycle, shown as blue shadow in Fig. 2(a1). The intracycle interference of trajectories from the two electric-field extrema produces the dip in the 3rd harmonic frequency (black line in Fig. 2(a3)). When $\tau = 0\ \mathrm {fs}$, only one electric-field extremum exists within one cycle, so the intracycle destructive interference in 3rd harmonics disappears, shown as black line in Fig. 2(b3).

Fourthly, the THz yield modulation versus $\tau$ stems from the $\tau$-dependent THz yield of the individual trajectory. Compared to $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ at $\tau = 0.33\ \mathrm {fs}$ (Fig. 2(a2)), $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ at $\tau = 0\ \mathrm {fs}$ (Fig. 2(b2)) has a clear dip at 0rd harmonics (THz region). The trajectory interference does not only influence the THz yield, but changes the spectral shape in THz region.

In the interference of electron trajectory, the phase relationship of individual ionization events is a critical factor to determine the yield and spectral shape of LHGs. The same angular diagram as the previous literature [16] is used to illustrate the interference effect of electron trajectories. Figure 3 shows the phase analysis of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ in THz region, where $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ are induced by the propagation of electron trajectories released at different ionization events. At $\tau = 0.33\ \mathrm {fs}$, Fig. 3(a1) shows that $\tilde {\boldsymbol {E}}_{n} (\nu = 3\ \mathrm {THz},t_n)$ of different ionization events are almost aligned in polar diagram. $\tilde {\boldsymbol {E}}_{n} (\nu = 3\ \mathrm {THz},t_n)$ have the same phases, which means that the trajectories from both inter-cycle and intra-cycle undergo the constructive interference at $\nu = 3\ \mathrm {THz}$. At $\nu = 15\ \mathrm {THz}$ and $\nu = 30\ \mathrm {THz}$, $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ is dispersively distributed in polar diagram, as shown in Fig. 3(a2) and (a3). The dephasing suppresses the constructive interference of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ in high-frequency THz region, thus leading to the cutoff shape in high-frequency THz spectrum. The phase slopes of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ in Fig. 3(a4) show that the phases of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ coincide at low frequency and diverge at high frequency, creating the peak structure at the THz region (red line in Fig. 2(a3)). The similar analysis has been implemented by PC-based ionization events [16,17], which reaches the same conclusion as the SFA-based trajectory method.

 

Fig. 3. Trajectory analysis in THz region. (a1) - (a4) Phase analysis at $\tau = 0.33\ \mathrm {fs}\ (\varphi = 0.5 \pi )$. (a1) - (a3) Polar diagrams of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ at 3 THz, 15 THz and 30 THz. The circle dots with different colors correspond to different ionization events in Fig. 2(a1). The size of circle dots represents the weight $w(t_n)$ of the ionization event. The radius of the polar coordinate, i.e. the distance between the dot and origin, represents the yield of terahertz or third harmonics induced by individual ionization event at ${t_n}$. (a4) Phase of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ in THz region at $\tau = 0.33\ \mathrm {fs}\ (\varphi = 0.5 \pi )$. The different color lines correspond to $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ released at different ionization events. (b1) - (b4) Phase analysis at $\tau = 0\ \mathrm {fs}\ (\varphi = 0)$. (b1) - (b3) The circle dots with different colors correspond to ionization events in Fig. 2(b1).

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At $\tau = 0\ \mathrm {fs}$, the THz radiation has the minimum yield, not only because the low amplitude of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ of the individual trajectories, but also because the phase relationship of trajectories. As shown in Fig. 3(b1), $\tilde {\boldsymbol {E}}_{n} (\nu = 3\ \mathrm {THz},t_n)$ of trajectories from inter-cycles have opposite direction in polar diagram, leading to the destructive interference. The inter-cycle superposition of $\tilde {\boldsymbol {E}}_{n} (\nu = 3\ \mathrm {THz},t_n)$ attributes to the deep dip in $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$ at low THz region (red line in Fig. 2(b3)). At $\nu = 15\ \mathrm {THz}$ and $\nu = 30\ \mathrm {THz}$ (Fig. 3(b2) and (b3)), $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ are dispersively distributed in polar diagram, where the complete destructive interference does not occur any more. Hence, $\tilde {\boldsymbol {E}}_{\mathrm {LHG}} (\nu )$ increases in THz region as $\nu$ increasing. The phase slopes at $\tau = 0\ \mathrm {fs}$ (Fig. 3(b4)) indicate that the phase difference of inter-cycle trajectories is close to $\pi$, so the contributions of inter-cycle trajectories coherently cancel each other.

The phase analysis can be also applied to explain the THG modulation versus $\tau$. When $\tau = 0.33\ \mathrm {fs}$, there are two separated ionization windows in one cycle, as shown in Fig. 2(a1). The electron trajectories from the two temporal windows radiate $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ with opposite phase at the THG peak ($\nu = 1125\ \mathrm {THz}$), as shown in Fig. 3(a2), and the phase differences between the two types of trajectories are presented in Fig. 3(a4). The superposition of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ from the two ionization windows gives rise to the intra-cycle destructive interference, which leads to the minimum THG yield at $\tau = 0.33\ \mathrm {fs}$.

In asymmetric two-color electric field at $\tau = 0\ \mathrm {fs}$, only one ionization window appears within one cycle, so the intra-cycle destructive interference is suppressed. Thus, only inter-cycle interference can be observed in polar diagram, as shown in Fig. 4(b1) - (b3). The trajectories from inter-cycle emit $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$, whose phases coincides at $\nu = 1125\ \mathrm {THz}$, leading to constructive interference, and the phases of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ dephase around 1125 THz, which forms the peak structure at 3rd harmonic frequency.

 

Fig. 4. Trajectory analysis in third-order harmonic (1125 THz) region. (a1) - (a4) Phase analysis at $\tau = 0.33\ \mathrm {fs}\ (\varphi = 0.5 \pi )$. (a1) - (a3) Polar diagrams of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ at 1100 THz, 1125 THz and 1150 THz. The circle dots with different colors correspond to ionization events in Fig. 2(a1). The size of circle dots represents the weight $w(t_n)$ of the ionization event. The radius of the polar coordinate represents the yield of terahertz or third harmonics induced by individual ionization event at ${t_n}$. (a4) Phase of $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ in 3rd order harmonic region at $\tau = 0.33\ \mathrm {fs}\ (\varphi = 0.5\pi$). The different color lines correspond to $\tilde {\boldsymbol {E}}_{n} (\nu,t_n)$ released at different ionization events. (b1) - (b4) Phase analysis at $\tau = 0\ \mathrm {fs}\ (\varphi = 0)$. (b1) - (b3) The circle dots with different colors correspond to ionization events in Fig. 2(b1).

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4. Conclusion

In summary, the analytic form of trajectory method is used to explain the low-order harmonic radiation, which connects photocurrent model and strong-field approximation. The trajectory method bridges the gap between the macroscopic and microscopic explanation for the low-order harmonics. The feasibility of trajectory method is verified by explaining the THz and 3rd harmonic yields versus two-color time delay $\tau$. The $\tau$-dependent THz yield can be explained by both the $\tau$-modulated THz amplitudes and destructive interference of different trajectories, whereas the $\tau$-dependent 3rd harmonics stems from intra-cycle and inter-cycle interference of multiple trajectories released at different tunnelling windows.

The trajectory method provides a new analysis method for low-order harmonic generation, which clearly decouples the processes of electron ionization, propagation of electron trajectory and coherence superposition of trajectories. In the future, the non-adiabatic dynamics and structural information, such as tunnelling time delay and Coulombic potential, can be incorporated into trajectory method, which is used to investigate the low-order harmonics signature of the reshaping of tunneling electron wavepacket.