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Abstract
Terahertz (THz) radiations from graphene are expected to provide a powerful light source for their wide applications. However, their conversion efficiencies are limited with either long-duration or few-cycle single-color laser pulses. Here, we theoretically demonstrate that THz waves can be efficiently generated from monolayer graphene by using a long-duration two-color laser pulse at normal incidence. Our simulated results show that low-frequency THz emissions are sensitive to the phase difference between two colors, the laser intensity, and the fundamental wavelength. Their dependence on these parameters can be very well reproduced by asymmetry parameters accounting for electron populations of conduction and valence bands. On the contrary, a newly defined σ parameter including the Landau-Zener tunneling probability cannot precisely predict such dependence. Furthermore, the waveform of THz electric field driven by two-color laser pulses exhibits the typical feature of a half-cycle pulse.
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1. Introduction
Over the past few decades, electromagnetic emission in the terahertz (THz) frequency band, i.e., in the far-infrared spectral region from 0.1 to 10 THz, has attracted considerable attention due to its promising applications, such as in large-scale object imaging [1,2], nonlinear THz spectroscopy [3,4], wireless communications [5], medical diagnosis [6], and military aviation [7]. With the rapid development of laser technology, intense and broadband THz waves have been generated coherently via laser-matter nonlinear interaction by using noble gases [8], air plasma [9], solids, and even liquids [10]. Different matters have shown their own features when generating THz waves. For instance, under strong driving lasers, especially a favorable scheme of synthesizing multiple-color laser pulses, air plasma and noble gases can emit the THz pulses with high amplitudes (even > 1 MV/cm) because of the absence of material damage. In 2000, Cook and Hochstrasser [11] reported that the generation of THz waves can be significantly enhanced from laser-induced air plasma by using two-color (fundamental and its second harmonic fields) laser pulses. Recently, Ma et al. [12] demonstrated that the energy of such two-color excited THz waves can be increased up to 22 times by adding a third 800-nm femtosecond laser. To interpret the generation process of THz waves, in 2007, Kim et al. [13] developed a transient current model, in which a non-vanishing electron current arise during the field ionization of air in each asymmetric optical cycle of two-color laser pulse. This model has been widely applied and further developed [14]. Besides, other models have also been established for understanding the generation mechanism of THz waves, including a four-wave mixing model [11,15], a strong-field approximation [16], and a full quantum mechanical model by solving time-dependent Schrödinger equation numerically [17–19]. In 2017, the THz wave generation by using liquids was reported by two groups [10,20], in which a thin, free-standing water film or ionized liquids were used. Meanwhile, to boost the generation efficiency of THz emission, solids have been employed due to their high atomic densities and stable physical forms. In 2008, Sagisaka et al. [21] demonstrated the first generation of THz waves caused by moving electrons on a solid target surface. Its generation mechanism can be explained generally by two laser-induced contributions in solids [22]: inter-band and intra-band currents. Another advantage of solids is to regulate THz emission by a variety of their structure features. In 2011, Hauri et al.[23] used organic electro-optic crystals to generate single-cycle THz pulses at a central frequency of 2.1 THz. In 2013, Jeong et al. [24] implemented the newly made quinolinium crystal to excite the efficient broadband THz waves.
Due to its special physical and chemical characteristics [25], graphene as a representative two-dimensional material has become one of most popularly studied solid materials under the strong laser pulses [26]. It exhibits a variety of good properties during light-field-driven processes including its broadband and ultrafast optical response, weak screening, high damage threshold [25,27], and unique electronic band structure [28–30]. Therefore, THz emission from graphene has also been investigated both in theory and in experiment. For example, Naib et al. [31] calculated the inter-band and intra-band currents from undoped-graphene at terahertz frequencies using a two-band tight-binding model and found that THz emission is sensitive to many factors, such as operating temperature, Fermi velocity, pulse duration, and so on. And they further theoretically studied the third harmonic generation from doped monolayer graphene at terahertz frequencies and optimized its strength by correcting system parameters [32]. In 2018, Hafez et al. [33] experimentally generated efficient THz harmonic generation in single-layer graphene due to the introduction of the free background Dirac electrons by applying a quasi-monochromatic linearly polarized THz laser pulse. Since graphene is a central asymmetric crystal, THz waves (below 10 THz) cannot be emitted at normal incidence with a tens or hundreds femtoseconds single-color pump beam [34]. To solve this problem, one solution is to use a scheme of oblique incidence, which has been testified by Maysonnave et al. [35] experimentally. They were able to generate a coherent THz radiation ranging from 0.1 to 4 THz via a second-order nonlinear effect by exciting the graphene with a 110-fs, 800-nm laser pulse due to the dynamical photon drag effect. Another solution is to break the symmetry of the driving laser pulse but still with normal incidence. In our previous work [36], we theoretically showed that broadband THz waves can be emitted from graphene by using a few-cycle infrared laser pulse. However, the yields of THz waves are greatly limited in these two schemes [35,36]. In the former, the laser incidence direction is different from the THz emission one. And in the latter, the input energy of a few-cycle laser is lost during the pulse compression and is thus much lower compared to a long laser pulse. Therefore, for efficient generation of THz waves from graphene, it is desirable to employ a long laser pulse at normal incidence.
On the other hand, the generation mechanism of THz waves from solids is needed to be further uncovered [37]. Can the widely used photoelectron current model be adopted to explain it? Recently, Heide et al. [38] found that the laser-induced residual current from graphene can be related to electron populations, which may influence low-frequency THz radiation [39]. However, the direct evidences of employing the photoelectron current model to analyze the generation of THz waves are still absent.
In this work, our goal is twofold. First, we propose to efficiently generate the THz waves at normal incidence from monolayer graphene by using long-duration two-color laser pulses. The symmetry of laser pulse is expected to be destroyed by the two-color pulses consisting of the fundamental and its second harmonic fields. To demonstrate the efficient generation of THz waves, we will numerically solve the extended semiconductor Bloch equation (SBE) of a single-layer graphene, in which the divergence of dipole transition moment near Dirac points can be avoided. We will also compare of the generation of THz waves when the two-color laser pulses are normally incident or are obliquely incident to the graphene surface, see Fig. 1. Second, we suggest to extract an asymmetry parameter from electron populations at conduction and valance bands to minic the generation of THz waves. The comparison of asymmetry parameters and THz emission yields will be performed by varying the laser parameters, such as the relative phase between two colors, the peak intensity, and the wavelength component. This is another way to validate whether the photoelectron current model is applicable for analyzing the THz emission from solids. The effectiveness of the asymmetry parameter will also be examined through its comparison with the performance of the Landau-Zener tunneling transition probability. This article is organized as follows. In Sec. 2, we will give the details of extended SBE, formulation of laser-induced current and THz radiation, and the derivation of the Landau-Zener tunneling formula in graphene. In Sec. 3, we will show simulated THz spectra, analysis of laser-induced current and electron population, calculated asymmetry parameters, and summed Landau-Zener tunneling probability from the monolayer graphene. And we will conclude this work in Sec. 4.
2. Theoretical methods
2.1 Time-dependent Schrödinger equation of graphene
2.1.1 Semiconductor Bloch equation (SBE) for normal incidence
We employ the time-dependent tight-binding (T-B) approximation to study the laser-graphene interaction. With this approximation, band energies near the Dirac points can be precisely calculated unless the applied laser intensity is too higher. We first consider the normal incidence of driving laser, and time-dependent Schrödinger equation can be written as [40]
where2.1.2 Time-dependent Schrödinger equation for oblique incidence
We next consider the oblique incidence of driving laser, the vector of wave velocity in the $x-y$ plane is $\textbf {u}$ = $(u_{x},u_{y})$. It can be related to the carrier frequencies $\omega _{x}$ and $\omega _{y}$ along two orthogonal polarized directions and the wave vector $\textbf {q}$ in the $x-y$ plane as $\textbf {q}$ = $(-\omega _{x}/u_{x},-\omega _{y}/u_{y})$. Note that $u_{x}$ = $u_{y}$ = $c/\sin {\chi }$, with $c$ and $\chi$ being the vacuum velocity of light and the angle of incidence, respectively. The limit $u_{x}$, $u_{y}$ $\rightarrow$ $\infty$, i.e., $q$ $\rightarrow$ 0 corresponds to the normal incidence. The time-dependent Schrödinger equation can be written as [43]
2.2 Formulation of electron current and THz emission
The single electron current can be evaluated as
The THz spectra can be calculated via a Fourier transform of the time derivative of the electron current:
And the THz waveform in the time domain can be obtained by an inverse Fourier transform of the THz emission (including amplitude and phase) for a given spectral range.2.3 Landau-Zener tunneling transition probability near Dirac cone
To qualitatively analyze the THz radiation from graphene, one can refer to the Landau-Zener tunneling transition probability between two adiabatic states at the avoided crossing in a two-level system. The general Landau-Zener tunneling formula is
Here $E_{+}$ and $E_{-}$ are eigenenergries of two levels (or two bands), and $\Delta$ is the half of minimum energy difference between them. $P(t)$ gives the tunneling transition probability when the time goes infinite caused by the instantaneous two-level system at time $t$.We next derive the Landau-Zener tunneling formula near Dirac cone, relating to the low-frequency THz emission. Time-dependent Hamilton can be written as
According to Eq. (34) and Eq. (35),
If we only consider a linearly polarized laser (along $x$ direction for example), the Landau-Zener tunneling transition probability is written as
3. Results and discussion
A sketch of experimental generation schemes of two-color excited THz waves in a monolayer graphene sheet is shown in Fig. 1. The graphene is placed in the $x-y$ plane. $\chi$ is the incidence angle of driving laser, and $\chi$ = 0$^{\circ }$ corresponds to the normal incidence. For generally polarized two-color laser pulses, $E_{p}$ and $E_{o}$ are amplitude components of the driving electric field in two orthogonal polarization directions in the polarization plane, respectively. This plane is perpendicular to the propagation (or incidence) direction of laser pulse. The projections of $E_{p}$ and $E_{o}$ in the $x-y$ plane are $E_{x} = E_{p}\cos (\chi )$ and $E_{y} = E_{o}\cos (\chi )$. The emission direction of THz waves is always perpendicular to the graphene surface. Here we focus on two cases: normal incidence ($\chi$ = 0$^{\circ }$) and oblique incidence ($\chi$ = 45$^{\circ }$).
Fig. 1. Schematic of generating the THz waves by two-color laser pulses from monolayer graphene at normal incidence and oblique incidence.
In this work, we consider linearly polarized two-color laser pulses, thus we only keep $E_p(t)$ in the polarization plane in Fig. 1. The electric field of driving laser can be written as
3.1 THz emission from graphene by varying the relative phase between two colors
To simulate the generation of THz waves from graphene at normal incidence, we choose $I_{1} = |E_1|^2$ = 5.85 $\times$ 10$^{11}$ W/cm$^2$, $I_{2} = |E_2|^2$ = 5.85 $\times$ 10$^{11}$ W/cm$^2$, the fundamental laser wavelength $\lambda _{0}$ = 0.8 $\mu$m, and the full-width-at-half-maximum (FWHM) duration of laser pulse is 31 fs. The simulated THz spectra are presented in Fig. 2(a) by varying the relative phase $\phi _{2}$. THz emissions have a very broadband, for example, at given relative phases of $\phi _{2}$ = $\pi$/2 and 3$\pi$/2, they can be extended to about 80 THz. And the THz spectra (in terms of intensity and spectral region) change dramatically with the relative phase $\phi _{2}$. For comparison, we carry out the similar simulations with the same laser parameters but at oblique incidence, and the simulated THz spectra are shown in Fig. 2(b). They are quite similar to those at normal incidence in Fig. 2(a), however, the overall yields are much smaller.
Fig. 2. THz spectra by two-color laser pulses versus the CEP $\phi _{2}$ at normal incidence (a) and at oblique incidence (b). The comparison of THz spectra by two-color laser pulses at different incidence angles for $\phi _{2}$ = 0 (c) and $\phi _{2}$ = $\pi$/2 (d). The THz spectrum obtained by a single-color laser pulse at oblique incidence ($\chi$ = 45$^{\circ }$) is shown in (c).
To further compare of THz emissions at different incidence angles, we choose two relative phases at $\phi _{2}$ = 0 and $\pi$/2 to replot the THz spectra in Figs. 2(c) and 2(d). It is known that a single-color long-duration laser pulse can only excite THz waves at oblique incidence. For comparison, we set its peak intensity at $1.17 \times 10^{12}$ W/cm$^2$, wavelength at 800 nm, and pulse duration at 31 fs with the same incident angle of $\chi$ = 45$^{\circ }$. The calculated THz spectrum is also shown in Fig. 2(c). One can see that at $\phi _{2}$ = 0, there is almost no low-frequency THz radiation (below 5 THz, red line) generated by two-color laser pulses at normal incidence, and THz radiations in the same spectral region are greatly enhanced at oblique incidence by either two- (blue line) or single-color (black line) laser pulses. This can be understood qualitatively that the photon drag effect at oblique incidence can lead to the breaking of symmetry of solid system [34]. At $\phi _{2}$ = $\pi$/2, THz radiations up to 50 THz are greatly enhanced no matter at normal incidence or oblique incidence. In the spectral region of 0.1 to 10 THz, the enhancement factor can be as high as more than 4-5 orders of magnitude compared to that in Fig. 2(c). On the other hand, in Fig. 2(d), intensities of THz waves in the whole extended spectral region at normal incidence are about 5-6 times stronger than that at oblique incidence. This implies that breaking symmetry of laser pulse plays a much more important role in the emission process of THz waves than the photon drag effect manifested at oblique incidence.
3.2 Analysis of THz emission by using residual current, asymmetry parameter, and Landau-Zener tunneling probability
3.2.1 Laser induced intra-band current of graphene
To understand the physical mechanism of THz emissions in Fig. 2, especially to identify the role of two-color laser waveform, we first analyze the influence of laser-driven electron trajectories on the electron current. We choose two cases of $\phi _{2}$ = 0 and $\pi$/2 at normal incidence for illustration. In Figs. 3(a) and 3(b), two types of electron trajectories near Dirac points are shown: one for an electron starting at initial negative wave number, $p_{x^-} = -0.2 A_{0}$, and the other with a positive initial wave number, $p_{x^+} = 0.2 A_{0}$, where $A_{0}$ is the peak value of vector potential. The black filled curves denote the Landau-Zener tunneling transition probability of graphene, calculated by Eq. (41). We consider the movements of electrons near the main peak of laser field (the regions marked by vertical dashed lines), where the considerable tunneling ionization occurs. In Fig. 3(a), at $\phi _{2}$ = 0, the two-color laser waveform (black line) is symmetric, the integration of momentum $k(t)$ over time for $p_{x^+}$ (blue shaded region) has the same value as that for $p_{x^-}$ (red shaded region), but with the opposite sign. So the total summation of contributions from both $p_{x^+}$ and $p_{x^-}$ is equal to zero. If the two-color laser waveform (black line) is not symmetric in Fig. 3(b), the total summation of momentum integrations over time from $p_{x^+}$ and $p_{x^-}$ is a non-zero value. We then calculate the induced current from different electron trajectories. The intra-band current can be calculated as
Fig. 3. Electron momentum $k(t)$ [(a, b)] and intra-band current $j_{\rm {Intra}}^{x}(t)$ [(c, d)] induced by electrons with initial positive and negative wave numbers ($p_{x^+}$ and $p_{x^-}$) under a two-color laser field. The black filled areas are calculated Landau-Zener tunneling transition probabilities in (a) and (b). The first column is for $\phi _{2}$ = 0 and the second one is for $\phi _{2}$ = $\pi$/2. Note that the time is expressed in terms of the optical period of 800-nm laser and $j_{\rm {Intra}}^{x}(t)$ is in the unit of the Fermi-velocity $v_{F}$.
3.2.2 Asymmetry parameter of conduction and valence bands
Next we refer to the electron population to interpret the generation of low-frequency THz emission. The electron populations of conduction band $\rho _{cc}$ at time ($t = +\infty$) long enough after the laser pulse are shown in Figs. 4(a) and 4(b). The results are for $\phi _{2}$ = 0 and $\pi /2$ at normal incidence. Figures 4(c) and 4(d) are enlarged views of the areas marked by the black rectangles in Figs. 4(a) and 4(b). One can see that the electron population is symmetric about $k_{x}$ = 0 when $\phi _{2}$ = 0 while it is not when $\phi _{2}$ = $\pi /2$. To further show the difference between two relative phases, we calculate $S_{c} = \int _{-\infty }^{+\infty } \rho _{cc} dk_{y}$ and $S_{v} = \int _{-\infty }^{+\infty } \rho _{vv} dk_{y}$ as well with the electron population of valence band $\rho _{vv}$ (not shown). The calculated results are shown in Figs. 4(e) and 4(f). It is more clearly displayed that both $S_{c}$ and $S_{v}$ are symmetric about $k_{x}=0$ at $\phi _{2}$ = 0 and they are obviously not symmetric at $\phi _{2}$ = $\pi /2$. Similar asymmetrical electron populations from graphene have been discussed in Ref. [48] by modifying the laser waveform.
Fig. 4. Electron populations ($t=+\infty$) of conduction band driven by two-color laser pulse with $\phi _{2}$ = 0 (a) and $\phi _{2}$ = $\pi$/2 (b). (c, d) plot enlarged views of the rectangle areas in (a) and (b), respectively. (e, f) show $S_{c}$ and $S_{v}$ versus $k_{x}$, i.e., the integrated electron populations of conduction and valence bands over $k_{y}$, respectively.
Fig. 5. The integrated THz yields (black lines) in the spectral regions of 0.1 - 10 THz and 40 - 50 THz as a function of $\phi _{2}$. The square of asymmetry parameter $\xi ^2$ (red line) normalized with the integrated yield at $\phi _{2}$ = $\pi$/2 is also shown in each figure.
From Eq. (43), the band velocities have the following relation: $\textbf {v}_{c}$ = -$\textbf {v}_{v}$ in the two-band T-B model, and electron populations from both conduction and valence bands contribute to the intra-band current. We thus introduce an asymmetry parameter $\xi$ as
We then use the asymmetry parameter to minic the generation of THz emission, i.e., the dependence of THz yields on the relative phase. At normal incidence, the low-frequency THz yields (black line) integrated from 0.1 to 10 THz as a function of the relative phase $\phi _{2}$ are plotted in Fig. 5(a). The THz yields display a periodic variation with the relative phase of $\pi$. And the square of asymmetry parameter $\xi ^2$ (red line) is presented in the same figure, and is normalized with the integrated THz yields at $\phi _{2}$ = $\pi /2$. It shows that the asymmetry parameter can perfectly reproduce the dependence of THz emission yields on the relative phase. We also plot the integrated high-frequency THz yields from 40 to 50 THz (black line) in Fig. 5(b), however, they are quite deviated from the asymmetry parameters (red line) as shown in the same figure. At oblique incidence, similar comparisons of THz yields and asymmetry parameters are shown in Figs. 5(c) and 5(d). The low-frequency THz yields mostly agree well with the asymmetry parameters except for some small deviations around the relative phase $\phi _{2}$ = 3$\pi /2$ in Fig. 5(c). This can be understood by the existence of photon drag effect in $\textbf {k}(t)$ at oblique incidence, in which $\textbf {k}(t)=\textbf {k}+\textbf {A}(t+\textbf {q}\cdot \textbf {r}/\omega )$ and the factor of $\textbf {q}\cdot \textbf {r}/\omega$ is equivalent to an additional phase of laser pulse in the time domain. This effect leads to some changes in the velocity of conduction and valence bands in Eq. (43), however, it cannot be fully included in the asymmetry parameter in Eq. (44). For high-frequency THz yields in Fig. 5(d), their differences from the asymmetry parameters are enlarged compared to those in Fig. 5(b) at normal incidence. In all, Fig. 5 evidently tells that the low-frequency THz emission can be well described by the asymmetry parameter, which is mainly determined by the symmetry of laser waveform, while the high-frequency THz emission has considerable deviations with the asymmetry parameter because it not only depends on the symmetry of laser pulse, but also on the nonlinear properties of materials [35].
3.2.3 Summed Landau-Zener tunneling transition probability
One can use Landau-Zener tunneling probability to analyze the generation of THz emission. At a given wave vector $k$, electron populations from different energy bands are related to the tunneling transition probability. Near Dirac cone, this probability can be calculated by using the Landau-Zener tunneling formula given in Eq. (39). According to Dirac-Fermi distributions, the probabilities are added up near the Dirac cone, in which $|\textbf {k}-\textbf {k}_{D}|<0.2$ a.u. with $\textbf {k}_{D}$ as the position of Dirac cone. We can obtain
We then define a parameter $\sigma$ to show the difference of summed Landau-Zener tunneling probability between the leading and falling edges of driving laser pulse as The comparison of the square of the parameter $\sigma$ and the yield integration from 0.1 to 10 THz as a function of the relative phase $\phi _{2}$ is shown in Fig. 6. One can see that for both normal and oblique incidence cases, the $\sigma ^2$ mostly follows the change of yield integration with the relative phase, however, some small deviations can also be observed visually. This indicates that the parameter $\sigma$ can in general reflect the change of laser waveform by considering the effect of electron tunneling transition between conduction and valence bands.Fig. 6. The comparison of the integrated THz yields (0.1 - 10 THz) and the square of summed Landau-Zener tunneling probability $\sigma ^2$ versus $\phi _{2}$. Note that the two sets of data are normalized at $\phi _{2}$ = $\pi$/2.
3.3 Dependence of low-frequency THz radiation on the intensity of pump laser
Intensity is a key factor in the generation of THz waves. Here we investigate the dependence of low-frequency THz yields on the intensity of two-color laser pulses. In our simulations, we set the fundamental wavelength at $\lambda _{0}$ = 0.8 $\mu$m, the FWHM pulse duration as 31 fs, $\phi _{1}$ = 0, and $\phi _{2}$ = $\pi$/2. The intensity $|E_1|^2$ = $|E_2|^2$ is varied from 8.775 $\times$ 10$^{9}$ W/cm$^2$ (0.0005 a.u.) to 8.775 $\times$ 10$^{11}$ W/cm$^2$ (0.005 a.u.). The integrated THz yields (0.1 - 10 THz) as a function of laser amplitude $E_1$ (black lines) are plotted in Figs. 7(a) and 7(b) for normal incidence and oblique incidence, respectively. For comparison, in these figures, we also present the square of asymmetry parameter as a function of laser strength (blue lines), which is normalized with THz yields at $E_1$ = 5 $\times$ 10$^3$ a.u.. One can see that the asymmetry parameter can perfectly reproduce the exponential increase of THz yields with the laser strength no matter normal incidence or oblique incidence. We also plot the square of parameter $\sigma$ defined in Eq. (46) in terms of Landau-Zener tunneling formula in Figs. 7(c) and 7(d). The parameter $\sigma ^2$ agrees with the integrated THz yields when the laser strength is large, and with the decrease of laser strength, the difference between them becomes bigger and bigger. This trend is valid for both normal and oblique incidence. Figure 7 clearly shows that using the asymmetry parameter is a more accurate way to describe the generation of low-frequency THz emissions than using the $\sigma$ parameter in terms of Landau-Zener tunneling transition probability.
Fig. 7. (a, b) The comparison of the integrated THz yields (0.1 - 10 THz) (black lines) and the square of asymmetry parameter $\xi ^2$ (blue lines) versus the strength of driving laser. (c, d) also show the square of summed Landau-Zener tunneling probability $\sigma ^2$ (red lines). Note that THz yield is normalized with the value of $\xi ^2$ (or $\sigma ^2$) at the laser strength of 0.005 a.u.. See text for detailed laser parameters.
3.4 Dependence of low-frequency THz radiation on the wavelength of pump laser
The wavelength of pump laser is another key factor in the generation of THz waves. We also investigate the dependence of THz yields on the fundamental wavelength of two-color laser pulse. In our calculations, we set the fundamental wavelength varying from 0.2 to 2.0 $\mu$m, fix the intensity of $|E_{1}|^2$ (= $|E_{2}|^2$) at 5.61 $\times$ 10$^{11}$ W/cm$^{2}$, the CEP phases $\phi _{1}$ = 0 and $\phi _{2}$ = $\pi /2$, and the FWHM-duration of laser pulse at 31 fs. We show the integrated THz yields (0.1 - 10 THz) as a function of the fundamental wavelength (black lines) in Fig. 8. For both cases (normal incidence and oblique incidence), the THz yields increase with the fundamental wavelength $\lambda$, scaling as $\lambda ^{1.6}$. To understand these results, we adopt two previously mentioned methods. One is to use the asymmetry parameter. We show the square of asymmetry parameter varying with the fundamental wavelength (blue lines) in Figs. 8(a) and 8(b). The other is to use the $\sigma$-parameter obtained by the Landau-Zener tunneling probability. We plot the $\sigma ^2$ as a function of the fundamental wavelength (red lines) in Figs. 8(c) and 8(d). One can see that the asymmetry parameter can perfectly reproduce the change of THz yields with the fundamental wavelength for both normal incidence and oblique incidence, while the $\sigma$-parameter can simulate the general trend of THz yields, but with some small deviations.
Fig. 8. (a, b) show the integrated THz yields (0.1 - 10 THz) (black lines) and the asymmetry parameter (blue lines) varying with the fundamental wavelength in the two-color laser pulses. The square of newly defined $\sigma$ parameter (red lines) accounting for the Landau-Zener tunneling probability are plotted in (c) and (d) as well. See text for other related laser parameters.
3.5 Temporal THz waveform from graphene driven by two-color laser pulse
Finally, we check the temporal electric field of the THz emission. Figure 9(a) shows the THz spectra generated by the two-color laser pulse with the FWHM durations of 31 fs and 62 fs. The fundamental wavelength is fixed at 0.8 $\mu$m. The electric waveforms by synthesizing the THz radiations from 0.1 to 10 THz are plotted in Fig. 9(b). The waveforms exhibit the structure of half-cycle pulse. With a longer driving laser, the duration of temporal pulse becomes longer, and the peak field becomes higher.
Fig. 9. (a) THz spectra by 800 + 400-nm laser pulses with durations of 31 and 62 fs. $|E_{1}|^2$ = $|E_{2}|^2$ = 5.85 $\times$ 10$^{11}$ W/cm$^{2}$, and CEP phases $\phi _{1}$ = 0 and $\phi _{2}$ = $\pi$/2. (b) Temporal electric waveforms by synthesizing the THz radiations (including amplitude and phase) from 0.1 to 10 THz in (a).
We then maintain the FWHM duration of laser pulse at 62 fs, and keep the same laser intensity. The THz spectra by the two-color laser pulse with the fundamental wavelengths of 0.8 $\mu$m and 1.6 $\mu$m are plotted in Fig. 10(a). By spectrally filtering the THz emission from 0.1 to 10 THz, the resulted waveforms (after normalization) are presented in Fig. 10(b). One can see that with the same duration of driving laser pulse, the THz waveform does not change with the fundamental wavelength. The peak intensity of waveform is higher at a longer fundamental wavelength.
Fig. 10. (a) The comparison of THz spectra generated by 800 + 400-nm and 1600 + 800-nm laser pulses, respectively. $|E_{1}|^2$ = $|E_{2}|^2$ = 5.85 $\times$ 10$^{11}$ W/cm$^{2}$, CEP phases $\phi _{1}$ = 0 and $\phi _{2}$ = $\pi$/2, and pulse duration of 62 fs. (b) Waveforms of THz electric field by spectrally filtering 0.1 - 10 THz radiations in (a). Note that the waveform of 800 + 400-nm laser pulse is multiplied by a factor of 7.8 for easy comparison.
4. Conclusions
In summary, we demonstrated that the efficient generation of low-frequency terahertz emission (0.1–10 THz) can be achieved by the long-duration two-color laser pulses from monolayer graphene at normal incidence. We found that the yields of THz emission are very sensitive to the relative phase between two colors, i.e., the symmetry of two-color laser pulse. The extended semiconductor Bloch equation of graphene was numerically solved under the tight-binding approximation. To understand the generation mechanism of THz radiation through the photoelectron current model, we first employed the laser-induced intra-band currents and electron populations of conduction and valence bands to distinguish the symmetry of two-color laser pulses. And then we introduced an asymmetry parameter to reflect the effect of electron populations of two bands, and for comparison, we also defined a $\sigma$-parameter by considering the commonly used Landau-Zener tunneling transition probability, which can be related to the laser-induced current. We showed that the low-frequency THz yields as a function of the relative phase between two colors, the peak intensity, and the fundamental wavelength in the two-color pulse can be perfectly reproduced by the asymmetry parameter at normal incidence, which behaved much better than the $\sigma$-parameter. At oblique incidence, the asymmetry parameter still performed better than the $\sigma$-parameter although the asymmetry parameter cannot well reproduce the THz emission for a few cases. The typical waveform structure of half-cycle pulse was always presented by synthesizing the low-frequency THz emissions no matter changing the duration or the fundamental wavelength of driving two-color laser pulses.
Our work provides with an effective way to enhance the THz radiation from solids at normal incidence through modifying the sub-cycle waveform of driving laser pulse. Except for combining the fundamental laser and its second harmonic field, this can also be realized easily by the current laser technology. Due to the advance in optical parametric amplification (OPA) and optical parametric chirped pulse amplification (OPCPA), it is possible to generate any optical waveform in principle by accurately controlling the synthesis of multi-color laser pulses [49–51]. We expect that it will be a promising method to generate the low-frequency THz radiation by controlling and optimizing the multi-color laser waveform.
Funding
National Natural Science Foundation of China (91950102, 11774175, 11834004, 11864037, 91850209).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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