All-optical reconstruction of three-band transition dipole moments by the crystal harmonic spectrum from a two-color laser pulse

Yue Qiao; Yue Qiao; Yan-Qiu Huo; Shi-Cheng Jiang; Yu-Jun Yang; Yu-Jun Yang; Yu-Jun Yang; Ji-Gen Chen; Ji-Gen Chen; Ji-Gen Chen

doi:10.1364/OE.446432

1. Introduction

The interaction between an intense laser pulse and a matter can produce a series of typical nonlinear phenomena, including high-order harmonic generation (HHG) [1–5]. The high-order harmonic can amplify the frequency of the incident laser to tens or even hundreds of times, so it can be used as a coherent light source in the extreme ultraviolet or even soft x-ray band, and it is also an important method of synthesizing ultrashort attosecond pulses [6–9]. Due to the HHG process is highly nonlinear, the harmonic intensity is usually many orders of magnitude lower than the incident laser intensity. This limits the wide application of the gas HHG in pump detection, lithography, and other aspects. Compared with atomic and molecular gases, solids have higher electronic density and periodic structure. Therefore, the solid HHG has attracted much attention in recent years in order to obtain the higher harmonic efficiency [10–17].

A lot of important works promote the development of the solid HHG. For example, Vampa et al. studied the ZnO crystal in a two-color mid-infrared laser field, and opened up a way for studying high-order harmonics of solids from the two-color light source [2]; In 2015, Vampa et al. proposed to reconstruct the momentum-dependent band gap of ZnO along the $\Gamma$-M direction by the coherent motion of electron-hole pairs under the action of mid-infrared femtosecond laser pulse [18]; In 2017, You et al. measured the carrier envelope phase (CEP) effect of the MgO crystal irradiated by a few-cycle pulse, providing a basis for attosecond pulse measurement based on solid-state HHG and multi-band electron dynamics driven by ultrafast laser [19]; You et al. studied the amorphous solid irradiated by sub-cycle pulses, the emergence of multiple platforms that the spectrum cut-off extends to the XUV region in the experiment can be explained by containing higher conduction bands, which promotes the understanding of the solid-state harmonic generation process [20]; In 2020, Uzan et al. studied the attosecond spectral singularity of MgO using a two-color field, and they demonstrated that when the electron-hole relative velocity approaches zero, the enhanced constructive interference leads to the appearance of spectral caustic in the harmonic spectra [21].

Based on the energy band theory, the physical mechanism of the crystal HHG can be explained, including the intraband current and the interband polarization. The oscillating motion of electrons (holes) in the conduction (valence) band will generate intraband currents, and the interband polarization is originated from the electronic transition between two energy bands. In 2015, Vampa et al. proposed a "three-step model" to describe the solid interband harmonic [22]: First, the laser electric field excites electrons from the valence band to the conduction band to form electron-hole pairs; then they move in conduction and valence bands respectively under the action of the laser field; when displacements of the electron and the hole match, they can recombine and emit harmonic photons. Because the informations of the band structure and transition dipole moments (TDMs) are encoded in the solid harmonic spectra, which can be applied to detect the band dispersion and TDMs of solid-state targets in principle [18,23–32].

Up to now, the all-optical reconstruction scheme was mainly focused on the energy band structure, and the research about mapping the TDM has just begun. Tancogne-Dejean et al. and our previous research found that the TDM related to the crystal momentum has a minimum value, which leads to the appearance of the Cooper minimum in the solid harmonic spectrum [3,33]. The above results show that it is possible to retrieve the TDM of solid materials by using HHG. In 2021, Uchida et al. reconstructed the structure of the two-dimensional TDM of black phosphorus at the isoenergy line in the momentum space by using the interband resonance high-order harmonic obtained from the experiment [34]. By solving the time-dependent Schr$\rm{o}^{..}$dinger equation of one-dimensional model, Hoang et al. studied the harmonic spectra of impurity-doped materials driven by strong laser pulses, and the TDM of an impurity-doped material was retrieved by the harmonic spectrum and the TDM of the reference target [35]. By using the interband harmonic spectrum of the MgO crystal along the $\Gamma$-X direction driven by a few-cycle mid-infrared laser pulse, we imaged the TDM between valence and first conduction bands [36].

However, to the best of our knowledge, the all-optical reconstruction of TDMs between multi-bands by the solid HHG has not been reported. In this work, by taking the continuous harmonic spectrum from a two-color laser pulse propagated along the $\Gamma$-X direction of an MgO crystal, we propose a new all-optical method to reconstruct the three-band k-dependent TDMs of the solid. It is found that the harmonic spectrum from the two-color linearly polarized laser exists a two-plateau structure. Especially, the harmonics in the first platform are dominated by the interband polarization between the first conduction band and the valence one, and the second harmonic plateau is mainly originated from the interband current between second conduction and valence bands. Furthermore, the intensity of HHG in the two-plateau is approximately related to the square of the corresponding TDMs. Thereby, the k-dependent TDMs from three energy bands can be probed by the interband harmonic spectrum from a single quantum trajectory. Our all-optical reconstruction scheme is suitable for crystals whose polarizations between different energy bands are dominant for different harmonic plateaus [26].

2. Three-band semiconductor Bloch equations

By the solution of the three-band semiconductor Bloch equations (SBEs) in the length gauge [36], we study the interaction of the two-color linearly polarized laser field with the MgO crystal. As follows: (Atomic units are used throughout this paper unless specified otherwise)

(1)$$\begin{aligned} &\frac{\partial }{\partial t}P_{k}^{v{{c}_{1}}}={-}i\left( \left[ {{\varepsilon }_{{{c}_{1}}}}\left( k \right)-{{\varepsilon }_{v}}\left( k \right) \right]-\frac{i}{{{T}_{2}}} \right)P_{k}^{v{{c}_{1}}}\\ & -i\left( 1-f_{k}^{{{c}_{1}}}-f_{k}^{v} \right){{d}_{{{c}_{1}}v}}(k)E(t)+E\left( t \right){{\nabla }_{k}}P_{k}^{v{{c}_{1}}}\\ & +iE\left( t \right)({{d}_{{{c}_{2}}v}}\left( k \right)P_{k}^{{{c}_{2}}{{c}_{1}}}-{{d}_{{{c}_{1}}{{c}_{2}}}}\left( k \right)P_{k}^{v{{c}_{2}}}) \end{aligned}$$

(2)$$\begin{aligned} & \frac{\partial }{\partial t}P_{k}^{v{{c}_{2}}}={-}i\left( \left[ {{\varepsilon }_{{{c}_{2}}}}\left( k \right)-{{\varepsilon }_{v}}\left( k \right) \right]-\frac{i}{{{T}_{2}}} \right)P_{k}^{v{{c}_{2}}}\\ & -i\left( 1-f_{k}^{{{c}_{2}}}-f_{k}^{v} \right){{d}_{{{c}_{2}}v}}(k)E(t)+E\left( t \right){{\nabla }_{k}}P_{k}^{v{{c}_{2}}}\\ & +iE\left( t \right)({{d}_{{{c}_{1}}v}}\left( k \right)P_{k}^{{{c}_{1}}{{c}_{2}}}-{{d}_{{{c}_{2}}{{c}_{1}}}}\left( k \right)P_{k}^{v{{c}_{1}}}) \end{aligned}$$

(3)$$\begin{aligned} & \frac{\partial }{\partial t}P_{k}^{{{c}_{1}}{{c}_{2}}}={-}i\left( \left[ {{\varepsilon }_{{{c}_{2}}}}\left( k \right) -{{\varepsilon }_{{{c}_{1}}}}\left( k \right) \right]-\frac{i}{{{T}_{2}}} \right)P_{k}^{{{c}_{1}}{{c}_{2}}}\\ & +i\left( f_{k}^{{{c}_{2}}}-f_{k}^{{{c}_{1}}} \right){{d}_{{{c}_{2}}{{c}_{1}}}}(k)E(t) +E\left( t \right){{\nabla }_{k}}P_{k}^{{{c}_{1}}{{c}_{2}}}\\ & +iE\left( t \right)({{d}_{v{{c}_{1}}}}\left( k \right)P_{k}^{v{{c}_{2}}}-{{d}_{{{c}_{2}}v}}\left( k \right)P_{k}^{vc_{1}^{*}}) \end{aligned}$$

(4)$$\begin{aligned} \frac{\partial }{\partial t}f_{k}^{v}=&2\operatorname{Im}\left[ {{d}_{{{c}_{1}}v}}\left( k \right)E\left( t \right)P_{k}^{vc_{1}^{\text{*}}}+{{d}_{{{c}_{2}}v}}\left( k \right)E\left( t \right)P_{k}^{vc_{2}^{\text{*}}} \right]\\ &+E(t){{\nabla }_{k}}f_{k}^{v}- \frac{{\rm{1}}}{{{\rm{2}}{{\rm{T}}_1}}}(f_k^v - f_{ - k}^v) \end{aligned}$$

(5)$$\begin{aligned} \frac{\partial }{\partial t}f_{k}^{{{c}_{1}}}=&2\operatorname{Im}\left[ {{d}_{{{c}_{1}}{{c}_{2}}}}\left( k \right)E\left( t \right)P_{k}^{{{c}_{2}}c_{1}^{\text{*}}}+{{d}_{{{c}_{1}}v}}\left( k \right)E\left( t \right)P_{k}^{vc_{1}^{\text{*}}} \right]\\ &+E(t){{\nabla }_{k}}f_{k}^{{{c}_{1}}}- \frac{{\rm{1}}}{{{\rm{2}}{{\rm{T}}_1}}}(f_k^{{c}_{1}} - f_{ - k}^{{c}_{1}}) \end{aligned}$$

(6)$$\begin{aligned} \frac{\partial }{\partial t}f_{k}^{{{c}_{2}}}=&2\operatorname{Im}\left[ {{d}_{{{c}_{2}}{{c}_{1}}}}\left( k \right)E\left( t \right)P_{k}^{{{c}_{1}}c_{2}^{\text{*}}}+{{d}_{{{c}_{2}}v}}\left( k \right)E\left( t \right)P_{k}^{vc_{2}^{\text{*}}} \right]\\ & +E(t){{\nabla }_{k}}f_{k}^{{{c}_{2}}}- \frac{{\rm{1}}}{{{\rm{2}}{{\rm{T}}_1}}}(f_k^{{c}_{2}} - f_{ - k}^{{c}_{2}}) \end{aligned}$$

where ${{\varepsilon }_{\lambda }}\left ( k \right )\left ( \lambda =v,{{c}_{1}},{{c}_{2}} \right )$ represents the energy of the valence (conduction) band, $f_k^\lambda (\lambda = v,{c_1},{c_2})$ is the hole (electron) density in the valence (conduction) band, $P_k^{{\lambda _1}{\lambda _2}}({\lambda _1}{\lambda _2} = v{c_1},v{c_2},{c_1}{c_1})$ and ${{d}_{{{\lambda }_{1}}{{\lambda }_{2}}}}\left ( k \right )$ are the micropolarization and TDMs between three bands. The energy band structure and TDM of MgO are calculated by the density functional theory (DFT) package in VASP using the Perdew-Burke-Ernzeroff generalize gradient approximation functional. The cutoff energy of the plane wave is 500 eV, and the k-point Monkhorst pack mesh in the Brillouin zone equals to 20$\times$20$\times$20, and the HSE06 hybrid function with the parameter AEXX=0.43 is adopted. The band structure of the MgO crystal along the $\Gamma$-X direction and TDMs between three bands (one valence band and two conduction bands) are consistent with those given by Jiang et al. in 2020 [37]. T$_2$ is the dephasing time, which describes the decoherence between two bands. In our work, T$_2$ is set to about a quarter-cycle of the incident fundamental laser pulse. T$_1$ is the thermalization time of the electron nonstationary energy distribution. We found that T$_1$ doesn’t change the interband harmonic shape, since the reconstruction of TDMs mainly utilizes the harmonics with the energy above the minimum bandgap between the first conduction band and the valence one, we take T$_1$ to infinity in our simulations. In this paper, the form of the two-color field is $E\left ( t \right ) = \left [ {{E_1}\cos \left ( {{\omega _1}t} \right ) + {E_2}\cos \left ( {{\omega _2}t + \varphi } \right )} \right ]f\left ( t \right )$ [38], and $f(t) = \exp \left [ {{{ - 2\ln 2{{(t - {t_0})}^2}} \mathord {\left / {\vphantom {{ - 2\ln 2{{(t - {t_0})}^2}} {\tau _p^2}}} \right. } {\tau _p^2}}} \right ]$ is the Gaussian envelope. By solving the SBEs, the intraband current ${{J}_{\operatorname {int}ra}}\left ( t \right )$ and the interband current ${{J}_{\operatorname {int}er}}\left ( t \right )$ can be calculated:

(7)$${{J}_{\operatorname{int}ra}}\left( t \right)\text{=}\sum_{\lambda \text{=}{{c}_{1}},{{c}_{2}},v}{\int_{BZ}{{{v}_{\lambda }}}\left( k \right)f_{k}^{\lambda }}dk$$

(8)$${{J}_{\operatorname{int}er}}\left( t \right)=\sum_{{{\lambda }_{1}}{{\lambda }_{2}}}{\frac{\partial }{\partial t}}\int_{BZ}{{{d}_{{{\lambda }_{1}}{{\lambda }_{2}}}}\left( k \right)}P_{k}^{{{\lambda }_{1}}{{\lambda }_{2}}}dk+c.c.$$

where $\lambda$ is the energy band index and $BZ$ represents the Brillouin zone. Here ${{v}_{\lambda }}(k)={{\nabla }_{k}}E(k)$ is the group velocity. The harmonic spectrum can be obtained by the Fourier transform of total current:

(9)$${{S}_{HHG}}\propto {{\left| \int_{-\infty }^{\infty }{\left[ {{J}_{\operatorname{int}ra}}(t)+{{J}_{\operatorname{int}er}}(t) \right]{{e}^{i\omega t}}dt} \right|}^{2}}$$

3. Continuous harmonic spectrum from the two-color laser pulse

Figure 1 shows the harmonic spectrum of MgO along the $\Gamma$-X direction driven by the two-color laser pulse. The peak intensities of the two-color field are ${{E}_{1}}=2.1\times {{10}^{13}}W/c{{m}^{2}}, {{E}_{2}}=0.25{{E}_{1}}$. The center wavelengths of the two-color laser field are 1700 nm and 5100 nm, respectively. The relative phase between the two-color field is $\varphi \text {=}0$. The Full width at half maximum (FWHM) of the laser field is 11.34 fs. Figure 1(a) presents the total (black solid line), intraband (red dotted line), and interband (blue dashed line) harmonic spectra. It can be observed that, the total harmonic spectrum with energy higher than the minimum bandgap between the first conduction band and the valence one is largely attributed to the interband polarization. Moreover, the harmonic spectrum from the minimum bandgap to the cutoff frequency exists a double platform structure, and is almost continuous and smooth. The energy range of the first platform is located in the energy difference 7.8-18 eV between the first conduction band and the valence band, and the energy range of the second plateau corresponds to the bandgap 21-24.7 eV between the second conduction band and the valence band. The harmonic intensity of the first platform is about one order of magnitude higher than that of the second plateau. In order to analyze the generation mechanism of the interband harmonic more clearly, the total interband harmonic is separated, and harmonics between the first conduction band and the valence band (red dashed line), harmonics between the second conduction band and the valence band (pink dotted line), and harmonics between the first conduction band and the second conduction band (green dash-dotted line) are given in Fig. 1 (b), respectively. It can be seen that for the first platform, the polarization between the first conduction band and the valence band plays a dominant role, and the second platform is mainly controlled by the interband current between the second conduction band and the valence band, which indicates that the interband polarization between other bands have little effect on them.

Fig. 1. (a) The total harmonic spectrum (black solid line) calculated by the three-band model and the harmonic spectra generated by interband (blue dashed line) and intraband currents (red dotted line). (b) The total interband harmonic spectrum (blue solid line), the harmonic spectrum from the interband current between the first conduction band and the valence band (red dashed line), the harmonic spectrum from the polarization between the second conduction band and the valence band (pink dotted line), and the harmonic spectrum from the interband current between the first conduction band and the second conduction band (green dash-dotted line).

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In order to explain the continuous characteristic of the harmonic spectrum generated by the MgO crystal, we use the time-frequency analysis method to investigate the emission time of the continuous harmonic. Figures 2(a) and (b) present the time-frequency distribution of the total harmonic spectrum (corresponding to two harmonic plateaus of the black solid line in Fig. 1(a)) and the color represents the harmonic intensity. It can be noticed that the harmonic emission time of the second platform within the energy range of 21-24 eV is later than the harmonic emission instant of the first platform within the energy range of 7.8-18 eV, and the harmonic intensity of the second plateau is about one order of magnitude lower than that of the first platform. It is important that the harmonics of the two platforms are mainly generated by a single quantum path, so the harmonic spectrum is continuous and smooth. Through the semiclassical recollision model [22,39], the relationship between the harmonic photon energy and the emission time is exhibited by the purple circle in the figure. It can be seen that the classical and quantum calculation results are in good agreement. Therefore, it is possible to use the harmonic spectra from the single quantum orbit in the two plateaus to probe the k-dependent TDMs between the first, second conduction bands, and the valence one.

Fig. 2. Time-frequency distribution of the total harmonic spectrum corresponding to the black solid line in Fig. 1(a). The purple circle is the relationship between the photon energy and the emission time calculated by the semiclassical recollision model. (a) The energy range is the bandgap between the first conduction band and the valence band; (b) the energy range is the energy difference between the second conduction band and the valence band.

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In Fig. 3(a), we give the electric field of the two-color linearly polarized laser pulse (blue solid line) and time-dependent electronic populations of the first conduction band (black dotted line) and the second conduction band (red dashed line). It can be found that the ionization mainly occurs in the central half optical period of the laser field, which explains why the harmonic spectrum from a single quantum path can be generated in this two-color laser field. Meanwhile, in order to understand time profiles of the harmonics more clearly, we exhibit the time evolution of the electron density distribution related to k on first and second conduction bands in Figs. 3(b) and (c). It can be observed that at the peak of the laser electric field, the electrons are excited from the valence band to the first conduction band with the greatest probability of transition. Under the action of the driving laser field, the electrons oscillate accordingly. When the electric field becomes to zero, a lot of electrons can recombine holes in the valence band. If some of excited electrons on the first conduction band move to the boundary of the Brillouin zone, and have the opportunity to transiting to the second conduction band. Electrons on the first conduction band start to distribute around k=0, and electrons on the second conduction band begin to appear around k=-1. The electron density distribution of the second conduction band is weaker than that of the first conduction band, and the recombination probability of electrons in the second conduction band to holes in the valence band is smaller than that of the first conduction band. This also explains why the the harmonic intensity of the second platform is lower than that of the first platform.

Fig. 3. (a) Electric field of the two-color linearly polarized laser pulse (blue solid line), time-dependent electronic populations of the first conduction band (black dotted line) and the second conduction band (red dashed line). Time evolution of the electronic density distribution at different crystal momentums in (b) the first conduction band (c) and the second conduction band.

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By using the semiclassical recollision model, Figs. 4 (a) and (b) give two typical trajectories of electrons and holes in the real coordinate space. The black solid line in Fig. 4(a) represents the trajectory of the hole with the emission energy of 9.11 eV, and the red solid line is the trajectory of the electron with the emission energy of 9.11 eV. The black solid line in Fig. 4(b) shows the trajectory of the hole with the emission energy of 22.8 eV, and the blue solid line represents the trajectory of the electron with the emission energy of 22.8 eV. The position of the pink dashed line is the time when the electron transits from the first conduction band to the second conduction band. The photon is emitted when trajectories of the electron and the hole intersect in real space. It can be seen that under the driving of the two-color linear polarization field, electrons in the two conduction bands have only one chance to collide with holes in the valence band, which results in the continuum HHG.

Fig. 4. Real space trajectories of electrons and holes corresponding to the emissions on (a) the first platform (b) and the the second platform.

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4. Reconstruction of k-dependent TDMs between three bands

According to our previous work [36], the yield of HHG from the interband current due to a single quantum trajectory is approximately proportional to:

(10)$${P({{k}_{r}})\propto {{\left| {{d}_{cv}}({{k}_{r}}){{e}^{-({{t}_{r}}-t_{i}^{'})/{{T}_{2}}}} \right|}^{2}}}$$

The difference between the recombination time and the ionization time as a function of the photon energy is shown in Fig. 5. Where $t_r$ is the instant of the electron recombining with the hole, $t_i$ is the time of the electron transition from the valence band to the first conduction band and $t_j$ is the time of the electron transition from the first conduction band to the second conduction band. It can be seen from the Fig. 5(a) that there is roughly a smooth upward trend from 9 eV to 16 eV. However, from 16 eV to 18 eV, the time difference shows a sharp increase with the change of photon energy. In the range of 21 to 24 eV, there is approximately a linear trend, as shown in Fig. 5(b). After reasonably ignoring exponential factors about $t_r$-$t_i$ and $t_r$-$t_j$, then the harmonic yield is directly related to the square of the absolute value of TDM at the emission instant. Therefore, we have the opportunity to directly extract TDMs at k points where photons emitted.

Fig. 5. (a) The dependence of the difference between the recombination instant and the ionizatin instant (the time of the electron transition from the valence band to the first conduction band) with the photon energy; (b) the dependence of the difference between the recombination instant and the ionizatin instant (the time of the electron transition from the first conduction band to the second conduction band) with the photon energy.

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In Figs. 6 (a) and (b), we explore the dependence of the TDM amplitude between the first conduction band and the valence band and the TDM amplitude between the second conduction band and the valence band on the bandgap, which are calculated from the first-principles calculations. The blue dashed line in the figure represents TDM between different bands, and the red solid line is the total harmonic spectrum in the two platforms. It can be found that the shape of TDM is almost the same as that of harmonic spectrum. When the electron recombines with the hole, the energy of the harmonic photon is equal to the bandgap determined by the corresponding crystal momentum. Therefore, there is a one-to-one correspondence between the harmonic photon energy and the crystal momentum at the moment of the emission. According to the above analysis, it is possible to reconstruct the TDM between the first conduction band and the valence band and the TDM between the second conduction band and the valence band by using the harmonic spectra in the two energy ranges derived from a single quantum trajectory.

Fig. 6. (a) The TDM amplitude between the first conduction band and the valence band as a function of the bandgap; (b) the TDM amplitude between the second conduction band and the valence band as a function of the bandgap. The solid red line in the figure is the total harmonic spectrum.

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Next, we reconstruct the the k-dependent TDM through the harmonic spectrum under the two-color field. Figure 7(a) shows the reconstruction of the k-dependent TDM between the first conduction band and the valence band by directly using the harmonic spectrum from the black solid line in Fig. 1(a). Here, the harmonic intensity is normalized to the highest intensity harmonic in the plateau. We select the harmonic with the engery n eV in the platform region and mark the energy in the bandgap, as exhibited in Fig. 7 (blue circle). After considering the exponential factor about $t_r$-$t_i$, the indirect reconstruction of TDM between the first conduction band and the valence band through the harmonic spectrum is presented in Fig. 7(b). Figure 7(c) presents the reconstruction of the k-dependent TDM between the second conduction band and the valence band by directly adopting the harmonic spectrum from the black solid line in Fig. 1(a). By calculating the exponential factor about $t_r$-$t_j$, the indirect reconstruction of TDM between the second conduction band and the valence band is exhibited in Fig. 7(d). In order to compare results, the TDM (black solid line) and bandgap (red solid line) obtained by the first-principles calculations are also given in Fig. 7. In these conditions, the mapped k-dependent TDM in most k points (diamonds in Fig. 7) agrees well with the real TDM. In particular, there is not much difference between direct and indirect reconstructions by the harmonic spectrum. The above results confirm that the three-band k-dependent TDMs can be detected by the harmonic spectrum of the two-color laser pulse.

Fig. 7. Reconstruction of the TDM (diamonds) between the first conduction band and the valence band by directly (a) or indirectly (b) using the HHG spectrum. Reconstruction of the TDM (diamonds) between the second conduction band and the valence band by directly (c) or indirectly (d) using the HHG spectrum. Black solid line and red solid line are TDM and bandgap calculated by the first-principles, respectively. The harmonic photon energy for mapping TDM is marked by blue circles.

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Then we investigate the influence of the relative phase on the harmonic spectrum in the two-color field, as shown in Fig. 8, and the other laser parameters are same with Fig. 1. It can be observed from the total harmonic spectra in Fig. 8(a) that there exists two plateaus. As the relative phase increases, the harmonic peak gradually move from the low-energy region to the high-energy region. This is similar to the experimentally observed results for the monochromatic field by You et al. in 2017 [19]. And, it can be found from this figure that, when the relative phase is 0-0.15$\pi$, the harmonic spectra in the two plateaus are continuous. Figures 8(b) and 8(c) display the harmonic spectra from the interband current between the first conduction band and the valence band, and the spectra from the polarization between the second conduction band and the valence one, respectively. One can see that the first platform of the harmonic spectra is mainly leaded by the interband current between the first conduction band and the valence band, while the second platform of the harmonic spectra is dominated by the interband polarization between the second conduction band and the valence one. The above results indicate that the interband polarization between other bands have little effect on them. Therefore, the interband harmonic spectra in the relative phase from 0 to 0.15$\pi$ can be chosen to retrieve three-band TDMs. Finally, we investigate the influence of the relative phase of the two-color laser on the k-dependent TDMs reconstruction. It can be seen from Fig. 9 that when the relative phase changes from 0.05$\pi$ to 0.15$\pi$, the direct reconstruction results are still good. Thus, the reconstruction scheme can be realized in a wide range of the relative phase in the practical experimental implementation.

Fig. 8. The dependence of the harmonic spectra on the relative phase of the two-color field. (a)The total harmonic spectra. (b)The harmonic spectra from the interband current between the first conduction band and the valence band. (c) The harmonic spectra from the polarization between the second conduction band and the valence band.

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Fig. 9. Reconstruction of TDMs through the harmonic spectrum generated by two-color laser pulses with different relative phases.(a) (b) $\varphi$=0.05$\pi$ (c) (d) $\varphi$=0.1$\pi$ (e) (f) $\varphi$=0.15$\pi$

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5. Conclusion

In summary, by solving the SBEs under the three-band model including one valence band and two conduction bands. The harmonic spectrum of the MgO crystal along the $\Gamma$-X direction under the irradiation of the two-color linearly polarized laser pulse is numerically studied. We demonstrated that the supercontinuum harmonic emission with the two plateaus from a single quantum orbit can be produced by such laser pulse. According to the one-to-one correspondence between the photon energy and the crystal momentum at the emission instant, the k-dependent TDM between first conduction and valence bands can be retrieved by the harmonics in the first plateau, and the TDM from second conduction and valence bands is directly probed by the harmonics in the second plateau. Furthermore, we examine the effect of the relative phase on the harmonic spectra. It is demonstrated that the three-band TDMs can be mapped well by directly using the harmonic spectra, when the relative phase of the two-color laser pulse is changed from 0 to 0.15$\pi$. Compared with the all-optical reconstruction method based on the ultrashort one-color pulse or the two-color field with the frequency ratio 1:2, our scheme can map three-band transition dipole moments in a wide range of the relative phase, which become feasible in practical experimental implementation. In addition, when the peak intensity ratio of the two-color laser pulse is altered from ${{E}_{2}}=0.2{{E}_{1}}$ to ${{E}_{2}}=0.3{{E}_{1}}$, the harmonic spectra in the two plateaus have more continuous, and we can still well retrieve three-band TDMs. In the near future, based on the quantitative rescattering theory [40–42], we hope to propose a method to retrieve the multi-band TDMs , which is independent of the incident laser parameters.

Funding

Outstanding Youth Project of Taizhou University (Grant No. 2019JQ002); National Natural Science Foundation of China (No. 11627807, No. 11774129, No. 11975012); National Key Research and Development Program of China (No. 2017YFA0403300, No. 2019YFA0307700).

Acknowledgments

The authors acknowledge the High Performance Computing Center of Jilin University for supercomputer time.

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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All-optical reconstruction of three-band transition dipole moments by the crystal harmonic spectrum from a two-color laser pulse

Abstract

1. Introduction

2. Three-band semiconductor Bloch equations

3. Continuous harmonic spectrum from the two-color laser pulse

4. Reconstruction of k-dependent TDMs between three bands

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (10)

Optics Express