THz Assistance Achieves Soft X-Ray Region Harmonic Generation Cutoff Extension Near

Researchers are pushing the boundaries of high-order harmonic generation (HHG) to create compact sources of soft X-rays, vital for diverse applications from materials science to medical imaging. Doan-An Trieu (Duy Tan University), Duong D. Hoang-Trong (Van Lang University), and Cam-Tu Le, with colleagues including Sang Ha, Ngoc-Hung Phan (Ho Chi Minh City University of Education) and F. V. Potemkin, demonstrate a surprisingly effective method for extending the energy range of HHG using readily available terahertz (THz) fields. Their simulations, detailed in a new report, reveal that even weak THz assistance can significantly broaden the harmonic plateau , a crucial step towards generating softer X-rays , and they’ve identified a robust mechanism for this extension, confirmed through both classical and quantum calculations. This work provides practical guidance for designing tabletop, high-energy HHG sources and offers a novel approach to observing ultrafast electron dynamics in real time, representing a significant advance in the field.

THz Fields Extend EUV Harmonic Cutoff significantly

Scientists have demonstrated a pathway to extend the high-harmonic cutoff using readily accessible terahertz (THz) fields, paving the way for more compact and efficient tabletop sources of extreme ultraviolet (EUV) and soft X-ray radiation. Researchers from multiple institutions in Vietnam and Russia comprehensively investigated THz-assisted high-order harmonic generation (HHG) through rigorous time-dependent Schrödinger equation simulations, bolstered by classical trajectory analysis and Bohmian-based quantum dynamics. By meticulously mapping the evolution of the harmonic plateau as a function of THz field strength, the study reveals that even weak THz fields can significantly extend the cutoff, creating a distinctive “fish-fin” structure in the harmonic spectrum. This structure exhibits prominent rays that saturate near Ip+8Up, where Ip represents the ionization potential and Up the ponderomotive energy, a crucial benchmark for HHG performance.
The team traced this extension to remarkably long electron excursions, spanning several optical cycles before recombination, and provided a fully consistent explanation using both classical analysis and Bohmian trajectories. These calculations demonstrate that the extended cutoff isn’t merely a theoretical possibility, but a robust phenomenon persisting across diverse atomic species, including hydrogen, helium, argon, and neon. Importantly, the cutoff extension remains largely insensitive to variations in the driving laser parameters, suggesting a high degree of control is achievable even with laboratory-scale THz fields. This robustness is a key advancement, simplifying the engineering of coherent high-energy HHG sources and offering a practical route towards tracking ultrafast electron motion in real time.

Experiments show that the conventional gas-phase cutoff, typically described as Ip + 3.17Up, can be surpassed through strategic field engineering. While previous attempts to broaden the spectrum often required strong static electric fields, around 39% of the fundamental laser field, or approximately 100 MV/cm, the current research establishes that THz fields offer a dynamic and more accessible alternative. Recent advances in THz technology have enabled the generation of these fields on a tabletop scale, but achieving significant cutoff extension previously demanded high-amplitude THz fields, limiting widespread adoption. This work, however, demonstrates that cutoff control is achievable with weaker, laboratory-accessible THz fields, offering practical guidelines for engineering coherent high-energy HHG.

The research establishes a clear connection between weak THz fields and the emergence of multiple HHG plateaus, indicating an extended cutoff, and provides a systematic investigation of cutoff behavior as a function of THz field strength. By solving the time-dependent Schrödinger equation for atoms exposed to combined IR and THz fields, and complementing these calculations with classical and Bohmian trajectory analyses, the scientists uncovered the underlying physics driving this phenomenon. The findings not only advance the development of compact EUV and soft X-ray sources, but also provide a robust pathway for probing ultrafast electron dynamics with unprecedented precision.

THz Field Extension of Harmonic Cutoff Spectroscopy

Scientists engineered a novel methodology to investigate terahertz (THz)-assisted high-order harmonic generation (HHG), crucial for advancing tabletop coherent extreme ultraviolet and soft X-ray sources. The research team comprehensively mapped plateau evolution versus THz field strength to demonstrate that even weak THz fields can extend the harmonic cutoff, producing a distinctive “fish-fin” structure with prominent rays saturating near a specific intensity. This extension arises from long electron excursions spanning several optical cycles before recombination, a phenomenon meticulously traced using both classical trajectory analysis and Bohmian-based dynamics. The study pioneers a fully consistent explanation of this cutoff extension, revealing its remarkable robustness across diverse atomic species and insensitivity to variations in driving parameters.

To achieve these insights, the work directly solved the time-dependent Schrödinger equation (TDSE) for atoms exposed to combined infrared (IR) and THz fields, expressed in atomic units as i ∂ ∂tψ(r, t) = −1/2∇2 + Vc(r) + r · E(t) ψ(r, t). Here, ψ(r, t) represents the time-dependent wavefunction, and Vc(r) denotes the atomic potential, with detailed specifications provided in Appendix A. The combined electric field, E(t) = E0f(t) cos(ω0t) + ET cos(ωT t), incorporates the IR driving laser pulse (amplitude E0, frequency ω0) and the THz field (amplitude ET, frequency ωT), both polarized along the x-axis. The TDSE was solved using a split-operator algorithm, ensuring convergence through careful parameter selection and initial wavefunction preparation via imaginary-time propagation.

Experiments employed Ehrenfest’s theorem to calculate the laser-induced acceleration dipole, subsequently obtaining the HHG intensity via Fourier transformation of this dipole. To analyse the temporal emission of HHG, the team computed harmonic time-frequency profiles using the Gabor transform. Complementing the TDSE method, scientists analysed electron motion using both classical and quantum trajectory methods. Classical trajectories were obtained by solving the equation of motion x(t) = −E(t), with initial conditions x(ti) = x(ti) = 0, for varying ionization times, identifying returning electrons that satisfy the condition x(tr) = 0 at the return time tr.

Furthermore, the study harnessed Bohmian mechanics, applying the pilot-wave theory to describe electron dynamics guided by the wavefunction ψ(r, t), using the guiding equation dr(t) dt = Im ∇ψ(r, t) / ψ(r, t). Each trajectory was assigned a weight proportional to the initial probability within a small vicinity of 0.02 atomic units around the initial electron position, ensuring statistical consistency with standard quantum mechanics. These innovative. This structure arises from extended electron excursions spanning several optical cycles before recombination, a phenomenon consistently explained by both classical and Bohmian dynamics.

The findings establish a robust mechanism for cutoff control, proving it remains effective across different atomic species and is largely unaffected by changes in driving parameters. Specifically, the study identified a direct relationship between THz amplitude, electron excursion time, and the resulting cutoff position, offering a practical rule for predicting and enhancing HHG efficiency. The authors acknowledge that the simulations require a large numerical box due to the extended electron motion induced by the THz field, potentially increasing computational cost. Future work could focus on experimental validation of these findings and exploring the application of this control mechanism in diverse ultrafast spectroscopic techniques, ultimately paving the way for more accessible and efficient tabletop EUV and soft X-ray sources.