Reducing the Carrier-Envelope-Phase-dependence of High-Harmonic-Generation by Vectorial-Time-Polarization-Gating

Abstract: A well-known shortcoming of High Harmonic Generation (HHG) is the strong dependence of the broadband HHG spectra (HGS) on the carrier envelope phase (CEP) of the driver. Here we numerically show that compared to the current well-established scalar (linearly polarized) schemes for generating broadband HGS, namely a short driver [Amplitude gating (AG)], Polarization-Gating (PG) or Time-Gating (TG), the vectorial driver of the Vectorial-Time-Polarization-Gating (VTPG) scheme renders the cutoff HGS much less sensitive to the CEP of the driver. The polarization state (helicity) of the emitted radiation is likewise CEP-resilient. Unlike scalar schemes, where the number of recollisions heavily depends on the CEP, in VTPG the CEP keeps this number almost unchanged, and only controls the partitioning of the recollisions between two orthogonal directions. This reduces the CEP-dependence of the HGS and decreases the spectral modulations. The CEP-resilience of the VTPG scheme holds promise for a variety of applications in attosecond science benefiting from quasicontinuous, helical HHG sources liberated from the necessity to stabilize the CEP of the laser.

• The paper demonstrates a novel vectorial time-polarization gating (VTPG) scheme that suppresses CEP dependence in HHG by using orthogonal two-color driver fields.
• It employs rigorous 3D TDSE simulations under a single-active-electron model to show orders-of-magnitude reduction in CEP-induced modulation compared to traditional gating methods.
• The study indicates that VTPG produces stable, broadband XUV continua with robust polarization, enabling advanced applications in attosecond spectroscopy and ultrafast chiral detection.

Carrier-Envelope-Phase-Insensitive High-Harmonic Generation Using Vectorial Time-Polarization Gating

Introduction

High-Harmonic Generation (HHG) has developed into a core mechanism for producing coherent extreme-ultraviolet (XUV) and attosecond pulses from tabletop sources, greatly enabling experimental attoscience. However, the dependence of the HHG spectrum on the carrier-envelope phase (CEP) of the driving laser pulse fundamentally constrains the reproducibility and bandwidth of XUV generation. Particularly in gating schemes optimized for single or few recollision events—Amplitude Gating (AG), Polarization Gating (PG), and Time Gating (TG)—the cutoff region of the high-harmonic spectrum exhibits strong shot-to-shot CEP dependence due to variations in electron re-encounter trajectories. This sensitivity hampers applications such as attosecond transient absorption spectroscopy and ultrafast chiral detection, where broadband, stable, and highly-polarized XUV continua are essential.

The present work introduces and numerically investigates a Vectorial Time-Polarization Gating (VTPG) scheme utilizing a cross-linear, two-color driver field. This protocol fundamentally suppresses HHG CEP dependence through a vectorial mechanism: the total number of recollisions, and hence the resulting spectral structure, are rendered nearly CEP-invariant. Only the distribution of recollisions across temporally separated, perpendicularly-polarized gates is affected by CEP, yielding stable intensity and polarization characteristics in the spectral domain.

Mechanism of Vectorial Time-Polarization Gating

The VTPG configuration is based on the superposition of two linearly polarized fields detuned in frequency and polarized orthogonally (cross-linear configuration). Temporal "gates" of suppressed ellipticity are realized at defined intervals, confined to instants where the driver field attains near-linear polarization and supporting efficient electron recollision. Critically, the orthogonality of adjacent gates ensures that the partitioning of recollisions between the two polarization directions is CEP-dependent, but the total number—a principal determinant of spectral width and modulation—is preserved, provided an even number of gates is used.

Figure 1: The VTPG scheme with a cross-linear two-color driver demonstrates gate separation, duration tuning via detuning parameter $\delta$, and a well-defined mapping of instantaneous ellipticity.

Analysis of the instantaneous ellipticity as a function of detuning ($\delta$) confirms precise control over gate duration and separation. For example, as $\delta$ decreases, gates widen and separate, while $\delta$ increases compress both. The gating occurs with high ellipticity selectivity, restricting multicycle pulses to a minimum recollision count analogous to sub-cycle amplitude gates yet without resorting to few-cycle pulse compression or active CEP stabilization.

Numerical Methodology

The comparative analysis involves rigorous numerical solutions of the three-dimensional time-dependent Schrödinger equation (TDSE) for argon under a single-active-electron (SAE) model. HHG spectra are computed for AG, PG, TG, and VTPG schemes, with a two-gate/three-recollision case exemplifying the vectorial platform. Standardized laser parameters ensure fair benchmarking, with the driver centered at 800 nm and a detuning $\delta = 0.1$ for two-color protocols. Temporal–spectral information is extracted via Gabor transformation of the dipole acceleration signal.

Comparative CEP Sensitivity of HHG Schemes

Direct comparison of the HHG spectra and their time–frequency evolution for multiple CEP values exposes the critical differences between scalar and vectorial gating approaches.

Figure 2: HHG spectra (and their Gabor transforms) as a function of CEP for AG, PG, TG, and VTPG schemes, highlighting the significant CEP insensitivity in the VTPG protocol.

In scalar schemes (AG, PG, TG), the number, timing, and interference of recollision trajectories, and hence the overall spectral structure, vary strongly with CEP. These variations lead to substantial modulation and instability in the XUV output. Conversely, VTPG exhibits monotonic redistribution of recollisions between orthogonal polarization directions with varying CEP, while the total recollision count and overall spectral shape remain nearly invariant. Quantitative CEP sensitivity is captured by a normalized variance metric $A(\Omega)$ evaluated over 20 equidistant CEP values. The metric for VTPG is consistently and significantly reduced compared to all scalar protocols in the cutoff region, demonstrating orders-of-magnitude suppression of shot-to-shot modulation.

An important control is provided by a dual-pulse scalar "2AG" scheme, which, despite producing two time-separated amplitude bursts, maintains polarization along a single axis. This does not suppress CEP dependence, unambiguously isolating the orthogonal polarization of gates in VTPG as causal for CEP resilience.

Figure 3: Quantitative analysis of CEP sensitivity and ellipticity as a function of CEP in the VTPG 2-gate, 3-recollision regime; panel (a) presents the $A(\Omega)$ metric for tested schemes.

Stability of Harmonic Polarization

The stability of the polarization state of emitted harmonics is essential for applications in chiral and ultrafast spectroscopy. VTPG produces highly elliptical—potentially near-circular—XUV harmonics whose polarization state is robust to CEP variations. Quantitative assessment of the harmonic ellipticity function $\varepsilon(\Omega)$ reveals limited CEP-induced fluctuation compared to spectral intensity; the residual modulation is primarily a result of the vectorial addition of perpendicularly-polarized emission bursts with varying relative phases.

Implications and Future Prospects

The introduction and validation of vectorial time-polarization gating represents a significant advance in ultrafast photonics, as it circumvents the longstanding limitation imposed by CEP sensitivity in HHG. This protocol decouples the need for strict CEP stabilization from the generation of broadband, polarization-controlled XUV continua. Experimentally, this enables attosecond and chiral measurement scenarios where active pulse compression, CEP tagging, or laser feedback control are impractical or lossy in terms of flux. In addition, the mechanism does not require sacrificing overall bandwidth and cutoff energy, as is often the case in amplitude or polarization gating. The capacity to emit highly-polarized XUV light with intrinsic CEP insensitivity will facilitate robust table-top sources for attosecond spectroscopy, circular dichroism, and XUV nonlinear optics.

Further development may involve extending VTPG concepts to more complex driver field topologies (e.g., multicolor, tailored vector beams) or integrating them with advanced phase-matching and propagation control for optimized macroscopic yields. The formalism also invites investigation of HHG dynamics in more complex atomic and molecular media, as well as probing CEP-insensitive coherent control in condensed-phase systems.

Conclusion

The vectorial time-polarization gating (VTPG) scheme fundamentally suppresses CEP dependence in high-harmonic generation by partitioning recollision events across temporally and polarization-orthogonal gates. This yields robustly stable, broadband XUV continua and polarization states, significantly outperforming existing gating methodologies in terms of CEP insensitivity and experimental versatility. The approach is broadly applicable to attosecond science and spectroscopy, presenting a viable route to precise, reliable, and polarization-engineered XUV sources without the need for stringent CEP stabilization (2604.10817).