Influence of chirp and carrier-envelope phase on noninteger high-harmonic generation

High-harmonic generation (HHG) is a versatile technique for probing ultrafast electron dynamics. While HHG is sensitive to the electronic properties of the target, HHG also depends on the waveform of the laser pulse. As is well known, (peak) positions, $\omega$, in the high-harmonic spectrum can shift when the carrier envelope phase (CEP), $\phi$, is varied. We derive formulæ describing the corresponding parametric dependencies of CEP shifts; in particular, we have a transparent result for the (peak) shift, $d\omega /d\phi =-2{\overline{\mathfrak{f}}}^{\prime }\omega /{\omega }_0$, where ${\omega }_0$ describes the fundamental frequency and ${\overline{\mathfrak{f}}}^{\prime }$ characterizes the chirp of the driving laser pulse. We compare the analytical formula to full-fledged numerical simulations finding only 17% average relative absolute deviation in $d\omega /d\phi$. Our analytical result is fully consistent with experimental observations.

Figure 1

High-harmonic emission spectrum $I\left(\omega \right)$ as function of the frequency $\omega$ computed from semiconductor Bloch equations (SBE), [16, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59] Eq. (5). For the SBE simulation, we employ a two-band Hamiltonian [10] to model the topological surface state of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. As driving electric field $\mathbf{E}\left(t\right)$ (inset), we use Eq. (4) with $E_0=3$ MV/cm, ${\omega }_0=2\pi ·25$ THz, $f_{\text{chirp}}=-1.25$ THz, and $\sigma =90$ fs as used in simulations in Ref. [10]. We employ two different CEPs, $\phi =0$ (blue) and $\phi =\pi /2$ (red). The driving electric field $E\left(t\right)$ is sketched in the inset.

Figure 2

(a) High-harmonics spectrum $I\left(\omega \right)$ computed from SBE [Eq. (5)] for 384 discrete CEPs $\phi \in [0,2\pi ]$ with $\mathbf{E}\left(t\right)$ and parameters as in Fig. 1. The heat map along the blue and red horizontal line (CEP $\phi =0$ and $\phi =\pi /2$) represents the emission spectra from Fig. 1. (b) Local extrema of $I\left(\omega \right)$ from (a).

Figure 3

Quantitative comparison of the peak shifts as obtained from the analytical formula (18) (background, $\mathrm{\Delta }{\omega }_{\text{peak}}$) and from SBE simulations (color-filled circles, $\mathrm{\Delta }{\omega }_{\text{peak}}^{\text{SBE}}$) in the parameter plane $({\omega }_{\text{peak}},f_{\text{chirp}})$. $\mathrm{\Delta }{\omega }_{\text{peak}}^{\text{SBE}}$ are obtained by tracing continuous maximum lines of the emission $I\left(\omega \right)$ as shown in Fig. 2 from the initial point $({\omega }_{\text{peak}}^{\text{SBE}},\phi =0)$ to the final point $({\omega }_{\text{peak}}^{\text{SBE}}+\mathrm{\Delta }{\omega }_{\text{peak}}^{\text{SBE}},\phi =2\pi )$. The dashed lines are guiding the eye; they indicate constant $\mathrm{\Delta }{\omega }_{\text{peak}}$ in the analytical formula. (For the SBE simulations, we considered 29 chirps with 384 CEPs each resulting in $11136$ SBE runs.)

Figure 4

CEP-dependency of the high-harmonics spectrum of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$; experimental data taken from Ref. [10]. We report local maxima as colored dots. We form sets of local maxima, as indicated by the various colors. For each set, we perform a quadratic fit $\omega \left(\phi \right)={\omega }_0[\alpha +\beta (\phi -\overline{\phi })+\gamma {(\phi -\overline{\phi })}^2]$ (dotted lines, $\overline{\phi }$ is fixed as the average CEP of a line segment, fit parameters $\alpha$, $\beta$, and $\gamma$ are reported in Tables 1 and Appendix pp6).

Figure 5

Maxima of $I\left(\omega \right)$ computed with parameters as in Fig. 2 for three different Monkhorst-Pack $k$-point meshes [78] $N_1\times N_2$ ($450\times 45,900\times 90,1800\times 180$), where $N_1$ is the number of $k$ points in $\mathrm{\Gamma }$-M direction and $N_2$ is the number of $k$ points orthogonal to the $\mathrm{\Gamma }$-M direction.

Figure 6

[(a) and (b)] Experimental pulse shapes from Ref. [10] that lead to HHG (a) from the topological surface state of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and (b) from the bulk of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. [(c) and (d)] Time-local chirp ${\mathfrak{f}}^{\prime }\left[t_M^{\left(0\right)}(\phi =\pi /2)\right]$ evaluated from Eq. (C2) for the pulse shapes from (a) and (b), respectively. At highest field, $M=0$, we have (c) ${\mathfrak{f}}^{\prime }\left[t_0^{\left(0\right)}(\phi =\pi /2)\right]=-1.0$ THz and (d) ${\mathfrak{f}}^{\prime }\left[t_0^{\left(0\right)}(\phi =\pi /2)\right]=-0.1$ THz.

Figure 7

Local extrema of $I\left(\omega \right)$ computed from SBEs for the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$-surface-state Hamiltonian as in Fig. 2 but with varying pulse duration and with varying chirp in the driving electric field (4): (a) $f_{\text{chirp}}=-2.5$ THz, $\sigma =90$ fs, (b) $f_{\text{chirp}}=0$, $\sigma =90$ fs, (c) $f_{\text{chirp}}=1.25$ THz, $\sigma =90$ fs, and (d) $f_{\text{chirp}}=0$, $\sigma =50$ fs,

Figure 8

Local extrema of $I\left(\omega \right)$ computed from SBEs for the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$-surface-state Hamiltonian as in Fig. 2 but with varying pulse duration [(a) and (b)] $\sigma =50\mathrm{fs}$ and [(c) and (d)] $\sigma =200\mathrm{fs}$ [in Fig. 2: $\sigma =90\mathrm{fs}]$. For (a) and (c), we choose the shape of $\mathbf{E}\left(t\right)$ [Eq. (4)] as in Fig. 2, in particular $f_{\text{chirp}}=-1.25$ THz. For (b) and (d), we choose $f_{\text{chirp}}=1.25$ THz, which leads to an inversion of the tilt angle, fully in line with our analytical formula Eq. (17).

Figure 9

Local extrema of $I\left(\omega \right)$ computed from SBEs for the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$-surface-state Hamiltonian as in Fig. 2, but with a varying dephasing time of [(a) and (b)] $T_2=1$ fs and [(c) and (d)] $T_2=100$ fs (dephasing time in Fig. 2: $T_2=10$ fs). For (a) and (c), we choose the driving field $\mathbf{E}\left(t\right)$ [Eq. (4)] as in Fig. 2, in particular $f_{\text{chirp}}=-1.25$ THz. For (b) and (d), we choose $f_{\text{chirp}}=1.25$ THz which leads to an inversion of the tilt angle, fully in line with our analytical formula Eq. (17).

Figure 10

Local extrema of $I\left(\omega \right)$ computed from SBEs for the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$-surface-state Hamiltonian as in Fig. 2 but with varying driving field amplitude [(a) and (b)] $E_0=0.1\mathrm{MV}/\mathrm{cm}$ and [(c) and (d)] $E_0=30\mathrm{MV}/\mathrm{cm}$ [in Fig. 2: $E_0=3$ MV/cm]. For (a) and (c), we choose the shape of $\mathbf{E}\left(t\right)$ [Eq. (4)] as in Fig. 2, in particular $f_{\text{chirp}}=-1.25$ THz. For (b) and (d), we choose $f_{\text{chirp}}=1.25$ THz which leads to an inversion of the tilt angle, fully in line with our analytical formula Eq. (17). We show extremal lines in (a) and (b) only up to harmonic order $\omega /{\omega }_0=26$ due to numerical noise present in higher harmonic orders.

Figure 11

Local extrema of $I\left(\omega \right)$ computed from SBE's for the topological surface state [(a)–(c)] as in Fig. 2, the Bulk Hamiltonian of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ [(d)–(f)] from Ref. [10] and a one-dimensional model with constant dipole and symmetric cosinelike bands [(g)–(i)]. For the latter, we use a gap of 0.15 eV and a band width of 8.7 eV. The Brillouin zone has a size $2\pi /3\AA$ and dipole is $d=3e\AA$ (e: electric charge). We use three different chirp-frequency-ratios, [(a), (d), and (g)] $2\pi f_{\text{chirp}}/{\omega }_0=-0.05$, [(b), (e), and (h)] $f_{\text{chirp}}/{\omega }_0=0$, and [(c), (f), and (i)] $2\pi f_{\text{chirp}}/{\omega }_0=+0.05$. For (a)–(c), the driving pulse is parametrized from Eq. (4) with $E_0=3\mathrm{MV}/\mathrm{cm}$, ${\omega }_0=2\pi ·25\mathrm{THz}$ and $\sigma =90\mathrm{fs}$, as already used in Fig. 2. For (d)–(i), the parameters are $E_0=3\mathrm{MV}/\mathrm{cm}$, ${\omega }_0=2\pi ·40\mathrm{THz},$ and $\sigma =50\mathrm{fs}$ as in Ref. [10] for simulating HHG from the semiconducting bulk of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. The respective electric fields for CEP $\phi =0$ are shown in the insets. We employ a dephasing time $T_2=10$ fs for the surface state model [(a)–(c)] and $T_2=1$ fs in the bulk case [(d)–(i)] as in Ref. [10]. Additionally, the band occupations are damped toward the ground state with a damping time $T_1=10$ fs [only for (d)–(i)] as in Ref. [10].