Since Ahmed Zewail’s pioneering work in the field of femtosecond science, time-resolved spectroscopy with two laser beams has become a commonly used method for studying chemical bond dynamics^[1], charge transfer^[2], and carrier dynamics^[3]. The temporal overlap (i.e., the time zero between two femtosecond laser beams) of dual beams is crucial for time-resolved spectroscopy technology^[4-6]. Since the shorter pulse is sensitive to dispersion, the pulse is broadened by propagation through a dispersive medium^[7]. Especially, coherent control of terahertz (THz) emission under two-color laser excitation should not only satisfy the temporal overlap of the fundamental ($\omega$) and the second harmonic ($2\omega$) but also make good use of dispersion to control the relative phase difference between the two laser beams^[8-10]. Currently, methods for determining the temporal overlap of two light beams encompass the use of an oscilloscope in combination with a detector for observation^[11], the generation of sum frequency signals^[12-14], and the analysis of interference fringe changes^[15]. However, these methods do not provide an accurate dispersion control between two beams on temporal overlap in the time-domain. Therefore, it is necessary to find a time-domain method that determines the temporal overlap with high resolution and easily controls the dispersion between the two pulses.

Notably, THz emission spectroscopy directly characterizes time-domain information^[16,17]. Variations in the position of THz time-domain signals can occur when different optical components are introduced into the optical path of the laser beams or when the sample is reoriented^[18,19]. Additionally, the dispersion properties of optical components at various wavelengths can lead to differences in the peak positions of THz time-domain signals. Therefore, the experimental setup designed for generating THz waves with two-color laser excitation would be an effective tool to determine the temporal overlap of two femtosecond pulses.

Herein, we design an experimental system for generating THz waves in GaAs under the excitation of two-color lasers. Experimentally, the temporal overlap of the fundamental ($\omega$, 800 nm) and second harmonic ($2\omega$, 400 nm) pulses are determined using the peak positions of the THz time-domain signals. Theoretically, we calculate the dispersion caused by the components inserted in the optical paths of $\omega$ pulse and $2\omega$ pulse. We find that the experimental results are basically consistent with the theoretical calculations. Furthermore, the coherent control of THz generation by the ultrafast photocurrent in ZnSe also proves the good temporal overlap between two-color femtosecond lasers with proper dispersion manipulation. This work not only provides a method to find and detect the temporal overlap of the $\omega$ and $2\omega$ pulses in the time-domain but also realizes the dispersion control between the $\omega$ and $2\omega$ pulses.

Figure 1(a) shows the experimental setup 1 for coherent photocurrent control of THz generation in semiconductors under two-color (400 and 800 nm) excitation. A Ti:sapphire regenerative amplifier (Spitfire Ace, Spectra-Physics) generates 800 nm laser pulses with a pulse duration of 35 fs and a repetition rate of 1 kHz. After passing through the beam splitter (BS) (with a transmittance to reflectance ratio of 9:1), the laser is split into two beams: the reflected light serves as the probe beam, and the transmitted light serves as the pump beam. The time delay between the pump beam and probe beam is controlled by a high-performance mid-range travel linear stage (M-ILS200HA, Newport), which has a spatial resolution of 0.005 mm and corresponds to a temporal resolution of 0.033 ps. The pump beam is modulated by an optical chopper connected to a lock-in amplifier and excites semiconductors to generate THz waves. The 400 nm ($2\omega$) pulse is produced by the frequency-doubling process of the 800 nm ($\omega$) pulse by a $\beta \text{-}\mathrm{BBO}$ crystal. The relative phase between the $\omega$ and $2\omega$ waves is controlled by an optical wedge (OW), which is manufactured by an optical glass (N-BK7). To ensure the temporal overlap between the $\omega$ and $2\omega$ pulses, a 7.5-mm-thick $\alpha \text{-}\mathrm{BBO}$ crystal (providing negative dispersion) is employed to compensate for the positive dispersion (i.e., the spectral components with a lower frequency propagate faster than the components with a higher frequency) introduced by the OW. A Teflon plate is utilized to block the pump beam without THz wave absorption. According to the classical theory, the THz radiation propagates radially outward with an anisotropic power distribution^[20]. After passing through a pair of off-axis parabolic (OAP) mirrors, the THz waves are focused onto ZnTe ⟨110⟩ crystal, while the probe beam is aligned collinearly with THz waves onto ZnTe crystal to detect the THz emission by the electro-optic sampling method. The spatial overlap between the $\omega$ pulse (represented in red) and the $2\omega$ pulse (represented in blue) is illustrated in Fig. 1(b). The full width at half-maxima (FWHM) of 800 and 400 nm pulses are 0.976 and 0.568 mm, respectively. This is consistent with the theoretical relevance of the beam waist between $\omega$ and $2\omega$ waves as $w_{\omega }=\sqrt{2}{w}_{2\omega }$^[21]. Figure 1(c) shows the experimental setup 2 of the THz generation under only $\omega$ pulse excitation: Fig. 1(c-I) is without any additional optical components, Fig. 1(c-II) is with the $\alpha \text{-}\mathrm{BBO}$ crystal, and Fig. 1(c-III) is with both the OW and $\alpha \text{-}\mathrm{BBO}$ crystal. Figure 1(d) shows the experimental setup 3 of the THz generation under only $2\omega$ pulse excitation. As shown in Fig. 1(d-i), the $2\omega$ pulse is produced by the frequency-doubling process of the $\omega$ pulse by a $\beta \text{-}\mathrm{BBO}$ crystal, and the residual $\omega$ pulse is filtered out after passing through a short-pass filter. Figures 1(d-ii) and 1(d-iii) are the experimental setups of the THz generation under only $2\omega$ pulse excitation with the $\alpha \text{-}\mathrm{BBO}$ crystal and with both the OW and $\alpha \text{-}\mathrm{BBO}$ crystal, respectively.

As GaAs can generate THz waves with a large amplitude under both 800 nm ($\omega$)^[22] and 400 nm ($2\omega$)^[23,24] excitation, we utilize it as a model sample to align the temporal overlap of $\omega$ and $2\omega$. The time delay between the 400 and 800 nm pulses is 0.026 ps after passing through the $\beta \text{-}\mathrm{BBO}$ crystal (0.2 mm). Hence, the impact of this result on the accuracy of our experimental method is negligible since the temporal resolution is 0.033 ps. Figure 2(a) depicts the normalized THz time-domain signals of GaAs excited by either $\omega$ (red) or $2\omega$ (blue) pulses under the setup 2-I [Fig. 1(c-I)] and setup 3-i [Fig. 1(d-i)] conditions. The peak positions of the time-domain THz signals excited by only $\omega$ (red) and only $2\omega$ (blue) excitations are located at 3.210 ps ($\pm 0.033\mathrm{ps}$) and 10.410 ps ($\pm 0.033\mathrm{ps}$), respectively. This time-domain peak position shift between the $\omega$ and $2\omega$ pulses is introduced by the filter as shown in Figs. 1(c-I) and 1(d-i). This result indicates that the filter causes a 7.200 ps delay, which can be determined directly from the THz time-domain signals under the excitation by only $\omega$ and $2\omega$ pulses. This positive dispersion between the $2\omega$ and $\omega$ lasers can be compensated by a negative dispersion crystal ($\alpha \text{-}\mathrm{BBO}$ crystal) in the optical path. As illustrated in Fig. 2(b), when the $\alpha \text{-}\mathrm{BBO}$ crystal is introduced [as shown in Figs. 1(c-II) and 1(d-ii)], the peak positions of the normalized THz time-domain signals excited by only the $\omega$ (red) and only the $2\omega$ (blue) move to 20.190 and 25.720 ps, respectively. Compared to Fig. 2(a), the $\alpha \text{-}\mathrm{BBO}$ crystal introduces a delay of 16.980 ps for $\omega$ pulse, while it introduces a delay of 15.310 ps for $2\omega$ pulse. The result indicates that the $2\omega$ pulse propagates 1.670 ps faster than the $\omega$ pulse after passing through the $\alpha \text{-}\mathrm{BBO}$ crystal, which means the 7.5-mm-thick $\alpha \text{-}\mathrm{BBO}$ crystal can compensate for the positive dispersion with a value of 1.670 ps. To ensure temporal overlap between the $\omega$ and $2\omega$ pulses, an OW with suitable thickness (6.9 mm in our experiments) is selected. Figure 2(c) depicts the normalized THz time-domain signals of GaAs excited by either $\omega$ (red) or $2\omega$ (blue) pulses after passing through both the $\alpha \text{-}\mathrm{BBO}$ crystal and OW as shown in Figs. 1(c-III) and 1(d-iii). The peak positions of the normalized THz time-domain signals excited by only $\omega$ (red) and only $2\omega$ (blue) excitations move to 19.351 and 26.557 ps in the time-domain, respectively. Compared to Fig. 2(a), the $\alpha \text{-}\mathrm{BBO}$ and OW introduce a delay of 16.141 ps for $\omega$ pulse and a delay of 16.147 ps for $2\omega$ pulse. This result suggests that both $\omega$ and $2\omega$ pulses can focus on GaAs with the temporal overlap simultaneously with the proper dispersion compensation.

Additionally, a series of theoretical calculations are conducted to confirm the dispersion compensation between the $\omega$ and $2\omega$ pulses. As shown in Fig. 2(d) (purple line), the refractive index ($n$) of OW with the wavelengths ($\lambda$) is described as^[25]$$\begin{gathered} n^2=\frac{1.03961212{\lambda }^2}{{\lambda }^2-0.00600069867}+\frac{0.231792344{\lambda }^2}{{\lambda }^2-0.0200179144} \\ +\frac{1.01046945{\lambda }^2}{{\lambda }^2-103.560653}+1. \end{gathered}$$(1)

Therefore, the refractive indices of the $\omega$ and $2\omega$ pulses are calculated as $n(800)=1.5108$ and $n(400)=1.5308$, respectively. The group refractive index ($n_g$) can be written as^[26]$$n_g=n-\lambda \frac{\mathrm{d}n}{\mathrm{d}\lambda }.$$(2)

Hence, the difference in the group refractive index between the $\omega$ and $2\omega$ pulses is calculated as $\mathrm{\Delta }{n}_{g1}=0.0575$ by Eqs. (1) and (2). As depicted in Fig. 2(d), since the $\alpha \text{-}\mathrm{BBO}$ crystal is a birefringent crystal, the refractive indices for the ordinary (o) wave (the red dashed line as $n_o$) and the extraordinary (e) wave (the blue dashed line as $n_e$) are described as^[27,28]$$\begin{gathered} n_o^2=2.7359+\frac{0.01878}{{\lambda }^2-0.01822}-0.01354{\lambda }^2 \\ {n}_e^2=2.3753+\frac{0.01224}{{\lambda }^2-0.01667}-0.01516{\lambda }^2. \end{gathered}$$(3)

In our experiments, the $\omega$ pulse is horizontally polarized, and the $2\omega$ pulse generated by frequency doubling in the $\beta \text{-}\mathrm{BBO}$ crystal is vertically polarized. When propagating through the $\alpha \text{-}\mathrm{BBO}$ crystal, the $\omega$ pulse propagates in the same direction as the o-light, while the $2\omega$ pulse travels in the same direction as the e-light. Hence, the refractive indices of the $\omega$ and $2\omega$ pulses are calculated as $n(o/800)=1.6591$ and $n(e/400)=1.5678$. The difference in the group refractive index between o-light and e-light is calculated by Eq. (2) as $\mathrm{\Delta }{n}_{g2}=0.0529$. Based on the above discussion, the time delay ($\mathrm{\Delta }\tau$) of the THz time-domain signal under the $\omega$ and $2\omega$ pulse excitation is calculated as $$\mathrm{\Delta }\tau =\frac{l\mathrm{\Delta }{n}_g}{c},$$(4)where $l$, $\mathrm{\Delta }{n}_g$, and $c$ are the thickness of the optical component, the difference of group refractive index between the $\omega$ and $2\omega$ pulses, and the velocity of light ($3\times {10}^8\mathrm{m}/\mathrm{s}$), respectively. Figures 2(e) and 2(f) illustrate the dependence of the time delay between the $\omega$ and $2\omega$ pulses on the thickness of the $\alpha \text{-}\mathrm{BBO}$ and the OW, respectively. The negative dispersion introduced by the 7.5-mm-thick $\alpha \text{-}\mathrm{BBO}$ and the positive dispersion introduced by the 6.9-mm-thick OW are calculated to be 1.322 and 1.322 ps, respectively. These theoretical values also prove the temporal overlap with proper dispersion compensation between $\omega$ and $2\omega$ pulses. Besides, we have calculated the dispersion compensation of a 3.5-mm-thick and 4-mm-thick $\alpha \text{-}\mathrm{BBO}$ for $\omega$ and $2\omega$ pulses, and the theoretical results (0.617 and 0.783 ps) are in good agreement with the experimental results (0.597 and 0.799 ps) as indicated in Fig. 2(e).

Based on the experimental setup [Fig. 1(a)], we further demonstrate the coherent control of ultrafast photocurrent in ZnSe with proper temporal overlap between $\omega$ and $2\omega$ pulses^[29]. Figure 3(a) shows the contour map of the THz time-domain waveforms of ZnSe with the two-color laser excitation under 40° incident angle. The THz time-domain signals are controlled by changing the relative phase difference ($\mathrm{\Delta }\phi$) between the $\omega$ and $2\omega$ pulses, which is attributed to the dispersion manipulation in the OW for both $\omega$ and $2\omega$ pulses. In addition, the relative phase difference between the $\omega$ and $2\omega$ pulses can also be controlled by rotating the $\alpha \text{-}\mathrm{BBO}$ crystal^[30,31]. The black lines represent the THz time-domain waveforms when $\mathrm{\Delta }\phi =-\pi /2$ (bottom line) and $\mathrm{\Delta }\phi =\pi /2$ (upper line). To clearly observe the coherent control of the photocurrent in ZnSe, the peak-to-valley values of THz time-domain dependence on the relative phase are extracted from Fig. 3(a) after taking into account the background effect as shown in Fig. 3(b). Three black points represent the peak-to-valley values of THz time-domain signals when $\mathrm{\Delta }\phi$ is $-\pi /2$, 0, and $\pi /2$, respectively. The peak-to-valley values of THz amplitude exhibit a sine function symmetry distribution around the zero axis, corresponding to the relative phase difference between the $\omega$ and $2\omega$ pulses. This result is a signature of quantum interference. The coherent control of THz generation in ZnSe further confirms the temporal overlap between the $\omega$ and $2\omega$ pulses. In principle, the THz emission field can be described as^[31]$$E_{\mathrm{THz}}\propto \mathrm{d}{J}_i/\mathrm{d}t=2T_{ijkl}{E}_{\omega }{E}_{\omega }^{*}{E}_{2\omega }\sin (\mathrm{\Delta }\phi )-J_i/{\tau }_R,$$(5)where $T_{ijkl}$, $E_{\omega }$, $E_{2\omega }$, and $\mathrm{\Delta }\phi$ represent the third-order nonlinear susceptibilities, the electric fields of the $\omega$ and $2\omega$ pulses, and the relative phase difference between the $\omega$ and $2\omega$, respectively; ${\tau }_R$ represents the characteristic carrier relaxation/recombination time. All-optical two-color THz emission can occur in three situations: $E_g<\hslash \omega <2\hslash \omega$, $\hslash \omega <E_g<2\hslash \omega$, and $\hslash \omega <2\hslash \omega <E_g$. For graphene ($E_g<\hslash \omega <2\hslash \omega$)^[32] and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ ($\hslash \omega <E_g<2\hslash \omega$)^[33], the THz emission originates from the quantum interference between single-photon absorption and two-photon absorption. However, when the photon energy follows $\hslash \omega <2\hslash \omega <E_g$, the fast nonlinearity induced by intra-band excitation can be regarded as the origin of coherent THz wave generation^[34]. Besides, surface optical rectification can also participate in all-optical two-color THz emission as observed in InAs^[30], ZnTe^[31], and ITO^[35]. As for the plasma/gas under two-color excitation, the THz emission is related to the photoelectrons produced in a symmetry-broken laser field composed of $\omega$ and $2\omega$ pulses^[36], as well as the radiation pressure force and the ponderomotive force^[37]. These physical mechanisms are different from that in solids under two-color excitation^[30-35].

In summary, we proposed a time-domain experimental method for high-resolution temporal overlap measurement with dispersion compensation manipulation under two-color femtosecond laser excitation. We experimentally determined the temporal overlap of two femtosecond lasers by comparing the peak positions of the THz time-domain signals from GaAs under the $\omega$ and $2\omega$ pulse excitation. We also theoretically calculated the dispersion of the optical elements for the $\omega$ and $2\omega$ pulses, which are consistent with the experimental results. Additionally, THz time-domain signals of ZnSe exhibit a sine function dependence on the relative phase difference between the $\omega$ and $2\omega$ pulses, which also proves the good temporal overlap between the two femtosecond laser beams. This work not only provides a method for determining temporal overlap but also lays the foundation for the coherent control of ultrafast photocurrents.