Terahertz-field-induced near-cutoff even-order harmonics in a femtosecond laser

High-order harmonic generation by a femtosecond laser pulse in the presence of a moderately strong terahertz (THz) field is studied under the strong field approximation. We demonstrate a simple proportionality of near-cutoff even-order harmonic (NCEH) amplitude to the THz electric field. The formation of the THz-induced NCEHs is analytically shown for both continuous-wave and Gaussian pulses. The perturbation analysis with regard to the frequency ratio of the THz field to the femtosecond pulse shows the THz-induced NCEHs originate from its first-order correction of the action in the presence of the THz vector potential, and the available parametric conditions for the phenomenon are also clarified. As the complete characterization of the time-domain waveform of the broadband THz field is essential for a wide variety of applications, the paper provides an alternative time-resolved field-detection technique, allowing for a robust broadband characterization of pulses in THz spectral range.

Figure 1

Schematics of the reconstruction of the THz field with near-cutoff even-order harmonics, which uses a femtosecond laser pulse to scan over a THz field and to record the generated harmonics as a function of the time delay between the fields. Panel (a) shows an exemplary THz field with ${\omega }_1=2\times {10}^{-4}$ (1 THz), and $E_1=2\times {10}^{-5}$ (100 kV/cm). Without the THz field, panel (b) shows the harmonics generated by a femtosecond pulse with ${\omega }_0=0.0353\left(1.3\mu \mathrm{m}\right),E_0=0.06(I=1.3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$) and $\sigma =2106$ (FWHM of 120 fs). With the THz field, panel (c) shows the all-order harmonics versus the time delay. The difference $\mathrm{\Delta }|\tilde{d}\left(\omega \right)|$ between the harmonics with THz field as given in (c), and the one without a THz field as shown in (b) is presented in panel (d) in the linear scale. All near-cutoff even-order harmonics as indicated by the box around $E_{\mathrm{cutoff}}=I_p+3.17U_p$ exhibits synchronous change with the intensity of the THz field.

Figure 2

The function $2|C_0\left({\phi }_{\tau }\right)|$ (solid line) and $|C_1\left({\phi }_{\tau }\right)|$ (dashed line) versus ${\phi }_{\tau }$ of the return time. The maximum of the $2|C_0\left({\phi }_{\tau }\right)|$ is associated with the maximum kinetic-energy gain that corresponds to the cutoff energy of the HHG as indicated by the black line for $3.17U_p$. The shaded area (light blue) highlights ${\phi }_{\tau }$ contributing to the near-cutoff harmonic spectrum of the the energy between $2.4U_p$ (red line) and $3.17U_p$ (black line).

Figure 3

Emission of all-order harmonics under the laser field of a continuous wave accompanied by a THz field. The laser parameters are given by ${\omega }_0=0.0353(1.3\mu \mathrm{m}$) and $E_0=0.06$ (intensity $1.3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$) for the infrared laser, and ${\omega }_1=3.53\times {10}^{-4}$ (2.3 THz) and $E_1=2\times {10}^{-5}$ (100 kV/cm) for the THz field. The $|\tilde{d}\left(\omega \right)|$ (blue line), evaluated from the Fourier transform of $d\left(t\right)$ as the direct numerical integration of Eq. (1), is compared with analytical formulas Eqs. (6) and (11) for odd- (green) and even-order (orange) harmonics, respectively. The even-order harmonics are maximized with initial phase $\varphi =\pi /2$.

Figure 4

Comparison of contributing phases in actions as a function of ${\phi }_t$ between fields of the continuous wave and of the Gaussian envelope. Parameters ${\phi }_{\sigma }=50$ and ${\phi }_{\tau }=4$ are used which are typical for the generation of NCEHs.

Figure 5

Effect of the pulse envelope on harmonic generation. The panels show the near-cutoff harmonics under (a) a continuous monochromatic laser of the same parameters used in Fig. 3, and (b) a Gaussian-enveloped pulse of $\sigma =1755$ (FWHM of 100 fs). The red marks label all harmonics of odd orders, whose distribution is highlighted by orange curves. Black dashed lines indicate $E_{\mathrm{cutoff}}$.

Figure 6

Comparison of $|\tilde{d}\left(\omega \right)|$ evaluated by direct numerical integration of Eq. (1) (blue) and by using the derived action, $S=S_0+\mathrm{\Delta }S$ with $S_0$ and $\mathrm{\Delta }S$ given by Eqs. (12) and (15), respectively. The same parameters of femtosecond pulse as in Fig. 5 are used except for $\sigma =2106$ (FWHM of 120 fs). The THz field is parametrized by ${\omega }_1=1\times {10}^{-4},E_1=2\times {10}^{-5}$, and $\varphi =\pi /2$.

Figure 7

Near cut-off harmonics without a THz field (blue) and with a THz field (orange) when (a) $\varphi =\pi /2$ and (b) $\varphi =\pi$. The inset of each panel shows the time center of the femtosecond pulse (blue) relative to that of the THz field (orange) for different $\varphi$'s. Marks “$\circ$” (“$\times$”) label the even-order harmonics with (without) the THz field. The position of $E_{\mathrm{cutoff}}$ is indicated by the black dashed line. The same parameters as in Fig. 6 are used.

Figure 8

Reconstruction of time-domain spectrum of THz waves. (a) The parameters are the same as used in Fig. 1. ${\omega }_0=0.0353\left(1.3\mu \mathrm{m}\right),E_0=0.06(I=1.3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2),\sigma =2106$ (FWHM of 120 fs), ${\omega }_1=2\times {10}^{-4}$ (1 THz), and $E_1=2\times {10}^{-5}$ (100 kV/cm). (b) The reconstruction for THz wave of higher frequency with ${\omega }_0=0.1139$ (400 nm), $E_0=0.06(I=1.3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2),\sigma =702$ (FWHM of 40 fs), ${\omega }_1=0.003$ (20 THz), and $E_1=2\times {10}^{-5}$ (100 kV/cm). The insets show the temporal profiles $E_0\left(t\right)$ and $E_1\left(t\right)$ of the femtosecond pulse and the THz field, respectively.

Figure 9

Available parametric range for THz reconstruction with NCEHs with regard to THz electric field amplitude $E_1$, the intensity, and the wavelength of incident femtosecond laser pulse. The detailed conditions are specified in the main text.