Enhancing high-order-harmonic generation by time delays between two-color, few-cycle pulses

Use of time delays in high-order-harmonic generation (HHG) driven by intense two-color, few-cycle pulses is investigated in order to determine means of optimizing HHG intensities and plateau cutoff energies. Based upon numerical solutions of the time-dependent Schrödinger equation for the H atom as well as analytical analyses, we show that introducing a time delay between the two-color, few-cycle pulses can result in an enhancement of the intensity of the HHG spectrum by an order of magnitude (or more) at the cost of a reduction in the HHG plateau cutoff energy. Results for both positive and negative time delays as well as various pulse carrier-envelope phases are investigated and discussed.

Figure 1

Comparison of calculated two-color ($\omega -2\omega$) HHG spectra for the H atom produced by the linear combinations $A_{\mathrm{NTD}}\left(t\right)$ and $A_{\mathrm{TD}}\left(t\right)$ of 1600-nm ($6\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$, 8 fs) and 800-nm ($4\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$, 5.6 fs) pulses [see Eqs. (1) and (2)] for four different time delays $\tau$ and corresponding CEPs ${\varphi }_2$, with $\tau$ and ${\varphi }_2$ related according to Eq. (8). In each panel, the solid (red) line is for the TD case, and the dashed (blue) line is for the NTD case. In all panels, ${\varphi }_1={\varphi }_2^{\prime }=0$. The arrows indicate the locations of the cutoff energies, ${\mathrm{\Omega }}_c$, the values of which are given in Table 1. The inset in each panel shows the electric fields $F_{\mathrm{NTD}}\left(t\right)$ and $F_{\mathrm{TD}}\left(t\right)$ corresponding to the vector potentials in Eqs. (1) and (2), respectively.

Figure 2

Calculated HHG spectra for the H atom produced by the same two-color ($\omega -2\omega$) pulse as in Fig. 1 but with different time delays $\tau$ and CEPs ${\varphi }_1$ and ${\varphi }_2$. In each panel, the solid (red) line is for the TD case, and the dashed (blue) line is for the NTD case. For all TD pulses, ${\varphi }_2^{\prime }=0$. The arrows indicate the locations of the cutoff energies, ${\mathrm{\Omega }}_c$, which are given in Table 2. The inset in each plot shows the electric fields $F_{\mathrm{NTD}}\left(t\right)$ and $F_{\mathrm{TD}}\left(t\right)$ corresponding to the vector potentials in Eqs. (1) and (2), respectively.

Figure 3

Calculated two-color ($\omega -3\omega$) HHG spectra for the H atom produced by a linear combination of 1500-nm ($6\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$, 7.5 fs) and 500-nm ($4\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$, 3.5 fs) pulses with different phases and time delays. ${\varphi }_1={\varphi }_2^{\prime }=0$ for all panels. In each panel, the dashed (blue) line is for the NTD case, and the solid (red) line is for the TD case: (a) $\tau =2.3$ fs, ${\varphi }_2=0.8\pi$; (b) $\tau =1.2$ fs, ${\varphi }_2=1.4\pi$. The inset in each plot shows the electric fields $F_{\mathrm{NTD}}\left(t\right)$ and $F_{\mathrm{TD}}\left(t\right)$ corresponding to the vector potentials in Eqs. (1) and (2), respectively. The arrows indicate the locations of the cutoff energies, ${\mathrm{\Omega }}_c$, which are given in Table 3.

Figure 4

Time-frequency analysis of the TDSE spectra in Fig. 1. (a) Time-frequency results for the laser electric field $F_{\mathrm{NTD}}\left(t\right)$ with $\varphi =1.6\pi$. (b) Time-frequency results for the laser electric field $F_{\mathrm{TD}}\left(t\right)$ with $\tau =2.1$ fs. In both (a) and (b) the intensities of the spectra are plotted on a color-coded log scale shown at the right of the figure. (c) The electric fields $F_{\mathrm{NTD}}\left(t\right)$ and $F_{\mathrm{TD}}\left(t\right)$ [which correspond to the vector potentials in Eqs. (1) and (2), respectively, and are the same fields as in the inset in Fig. 1] are plotted vs time on the same scale as in (a) and (b), and the dots are the ionization and recombination times of the second trajectory in Table 4.

Figure 5

Comparison of analytic and TDSE HHG spectra for the two laser fields shown in Fig. 4. (a) Analytic HHG spectra calculated from Eq. (10) are multiplied by a constant factor of 27.6 so that the low-energy plateau has the same intensity as the TDSE results. (b) TDSE HHG spectra calculated from Eq. (5) [which are the same results as in Fig. 1].

Figure 6

Comparison of analytic and TDSE two-color HHG spectra for laser pulses of 1600 nm ($1.2\times {10}^{14}{\mathrm{W}/\mathrm{cm}}^2$, 5.3 fs) and 800 nm ($8\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$, 3.8 fs) having no time delay with ${\varphi }_1=0$ and ${\varphi }_2=1.2\pi$. The solid (red) line is the analytic result, $\rho \left(\mathrm{\Omega }\right)$, and the dashed (blue) line is the TDSE result, $S\left(\mathrm{\Omega }\right)$. Note that $\rho \left(\mathrm{\Omega }\right)$ is multiplied by an overall constant factor of 32.2.

Figure 7

Two-color HHG spectra predicted by TDSE calculations for laser pulses having positive (solid red line) and negative (dashed blue line) time delays. Results in panels (a)–(d) are for the same $\omega -2\omega$ pulses as in Fig. 1; results in panels (e) and (f) are for the same $\omega -3\omega$ pulses as in Fig. 3. Positive and negative time delays are calculated using, respectively, Eqs. (8) and (28). The arrows indicate the HHG plateau cutoff energies, the values of which are given in Table 5.

Figure 8

Same as Fig. 4, but for the HHG spectra in Fig. 7.