Systematic analysis of an attosecond pulse generation by a sub-cycle laser field

TL;DR

This paper analyzes high-order harmonic generation driven by sub-cycle laser fields, emphasizing intrinsic chirp and CEP effects on attosecond pulse formation. Using a time-dependent Schrödinger equation in the single-active-electron framework and analytical sub-cycle pulse models that solve Maxwell’s equations, the study reveals a center-frequency blueshift that scales as $\tau^{-5/4}$ with pulse duration $\tau$, and an attochirp that follows the same scaling. It shows CEP-dependent scaling of harmonic yield, with $Y \propto \tau^{5/4}$ for $\phi_0=0^{\circ}$ and $Y \propto \tau^{-4.1}$ for $\phi_0=-90^{\circ}$, and demonstrates that sub-cycle pulses increase the harmonic-continuum contribution, enabling intense isolated attosecond pulses when the attochirp is compensated. These findings offer practical design rules for ultrafast light sources and deepen the understanding of attosecond dynamics under sub-cycle driving fields, including strategies to mitigate attochirp via dispersive propagation.

Abstract

We investigated the influence of sub-cycle driving fields on high-order harmonic generation (HHG), with a focus on intrinsic chirp, carrier-envelope phase (CEP), and number of laser cycles. Our findings reveals that the center frequency of a laser pulse scales as $τ^{-5/4}$ with pulse duration $τ$, and that attochirp exhibits a similar dependence on pulse duration. Additionally, we identified CEP-specific trends in harmonic yield: it increases as $τ^{5/4}$ for $φ_0=0^\circ$ and decreases as $τ^{-4.1}$ for $φ_0= -90^\circ$. Although sub-cycle pulses can generate intense isolated attosecond pulses (IAPs), they also tend to produce higher attochirp and reduced cutoff energies. However, effective compensation for attochirp can mitigate these drawbacks, thereby increasing the capability of sub-cycle pulses to generate short-duration, high-intensity IAPs. These results offer valuable insights into HHG using sub-cycle pulses and have important implications for the advancement of ultrafast light sources and the understanding of ultrafast phenomena at the attosecond timescale.

Figures (9)

• Figure 1: Temporal profiles of the instantaneous frequency and electric field of the sub-cycle pulse for different FWHM durations: (a) $\tau = 1.5T_0$, (b) $\tau = 1.0T_0$, and (c) $\tau = 0.5T_0$. (d) Relative blueshift of the sub-cycle pulse frequency at the pulse center ($t_0=0$) for different FWHM durations. The frequency blueshift scales as $\propto \tau^{-5/4}$ with the pulse duration $\tau$. The points indicate computed results, while the solid line represents the scaling. Here, $T_0\ (= 2\pi/\omega_0)$ denotes one laser cycle.
• Figure 2: (a) High harmonic spectra generated for different driving pulse durations. The vertical dashed-lines indicating the cutoff energies estimated using the modified expression. For clarity, the HHG spectra for $\tau = 0.8T_0,\ 1.0T_0,\ 1.2T_0,$ and $1.5T_0$ are shifted along the y-axis by factors $10^{3},\ 10^{6},\ 10^{9},$ and $10^{12}$, respectively. (b) The time-frequency profile of the dipole acceleration for a pulse duration of $\tau = 1.0T_0$ is presented. The corresponding classical re-collision energies for ionization (green dots) and recombination (black dots) times are also shown. (c) Scaling of the attochirp $\beta$ as a function of pulse duration. The attochirp follows a scaling of $\propto \tau^{-5/4}$. The points represent computed results and solid line represents scaling.
• Figure 3: Similar to Fig. \ref{['fig2']}, but with CEP $\phi_0 = -90^\circ$. (a) HHG spectra generated for different driving pulse durations. The HHG spectra for $\tau = 0.8T_0,\ 1.0T_0,\ 1.2T_0,$ and $1.5T_0$ are shifted along the y-axis by factors $10^{3},\ 10^{6},\ 10^{9},$ and $10^{12}$, respectively. (b) Time-frequency profile of the dipole acceleration for $\tau = 1.0T_0$. (c) Scaling of the attochirp $\beta$ as a function of pulse duration, following the same $\propto \tau^{-5/4}$ scaling.
• Figure 4: Temporal profiles of isolated attosecond pulses (IAPs) generated for CEP values $\phi_0$: (a) $0^\circ$, (b) $-90^\circ$. The IAPs corresponding to different pulse durations are normalized to their respective peak intensities and shifted in time for comparison. The time on the x-axis is given in attoseconds (as). Panels (c) and (d) show the scaling of IAP duration $t_{asp}$ as a function of pulse duration for $\phi_0$: (c) $0^\circ$, (d) $-90^\circ$. The IAP duration follows a consistent scaling of $\propto \tau^{-5/4}$.
• Figure 5: Scaling of integrated harmonic yield as a function of laser pulse duration for CEP values $\phi_0$: (a) $0^\circ$, (b) $-90^\circ$. The harmonic yield is computed for an energy window of 60 eV. For $\phi_0=0^\circ$, the yield increases as $\propto \tau^{5/4}$ with pulse duration, whereas for $\phi_0=-90^\circ$, it decreases following a scaling of $\propto \tau^{-4.1}$. The points represent computed results, and the solid lines illustrate the respective scaling trends.