Operating limits of flying-focus two-color terahertz generation: high drive power, focal ranges beyond the dephasing length, and scaled drivers

Flying-focus pulses — laser pulses whose intensity peak travels at a programmable velocity over many Rayleigh ranges — were proposed in 2024 as a way to control the emission angle, spectrum, and focusability of terahertz (THz) radiation generated by two-color photoionization of gases, and the concept has since been demonstrated in simulation (including a 'dephasingless' phased configuration) and in a first laboratory experiment. The experimentally demonstrated results lie inside a narrow window: near-infrared 0.8-micrometer drivers, millijoule pulse energies, and focal ranges within about one two-color dephasing length, while the many-dephasing-length behaviour of the phased configuration is established in simulation only. A precise open core remains: quantitatively establishing the performance envelope and failure modes of flying-focus-driven two-color THz generation outside the experimentally demonstrated window. The open problem comprises three linked threads of the outlook of the source paper (Simpson et al., Phys. Rev. Research 6, 013041 (2024); Ref. 9): (a) the high-power regime, drive power $P \gtrsim 10\,P_c$ in units of the critical power for self-focusing $P_c$, where transverse beam breakup and multiple filamentation are expected to disrupt the ionization-driven current; (b) the achievable performance, under realistic optic tolerances and alignment imperfections, of the simulated dephasingless phased flying focus at focal ranges $L \gg L_\pi$ beyond the dephasing length; and (c) combination with mid-infrared and high-repetition-rate drivers for intensity and average-power scaling.

Background

Ultrashort, intense terahertz pulses (electromagnetic radiation at frequencies from roughly 0.1 to 30 THz) are used to probe picosecond carrier, lattice, and spin dynamics and to coherently excite matter, which has motivated a long search for bright, focusable, tabletop THz sources. In the two-color scheme, a femtosecond laser pulse at frequency $\omega$ is mixed with phase-locked radiation at its second harmonic $2\omega$; interference of the two harmonics produces an asymmetric electric field that, in a gas, ionizes atoms stepwise and leaves behind a short-lived directed current of photoionized electrons (the photocurrent mechanism; Kim, Glownia, Taylor, and Rodriguez, Opt. Express 15, 4577 (2007)). Each such current transient radiates a sub-picosecond burst of broadband THz waves. Under conventional focusing, the emission interferes constructively only on a Cherenkov-like cone: because the ionization front (the moving boundary of newly ionized gas) travels faster than the THz phase velocity, the emitted wavefronts add along a cone at an angle $\phi$ fixed by $\cos(\phi) = v_T/v_I$, where $v_I$ is the ionization-front velocity and $v_T$ the THz phase velocity in the medium (You, Oh, and Kim, Phys. Rev. Lett. 109, 183902 (2012); Johnson, Palastro, Antonsen, and Kim, Phys. Rev. A 88, 063804 (2013)). The resulting ring-shaped beam is difficult to collect and focus. Dispersion of the gas also limits the usable interaction length: the relative phase between the two harmonics slips by $\pi$ over the dephasing length $L_\pi \approx (\lambda_1/4)\,|n(\lambda_1)-n(\lambda_2)|^{-1}$, with $\lambda_1$ the fundamental wavelength and $n$ the refractive index; beyond $L_\pi$ the driven current alternates sign, the THz pulse becomes multi-cycle, and its bandwidth acquires angular dispersion.

The flying focus removes the linkage between these velocities and the focusing geometry. A flying-focus pulse is a spatiotemporally structured laser pulse whose intensity peak travels at a programmable velocity over many Rayleigh ranges (a Rayleigh range is the axial distance over which a conventionally focused beam stays near its waist), implemented through longitudinal chromatism or radial group delay (Froula et al., Nat. Photonics 12, 262 (2018)). For ultrashort pulses the standard implementation pairs an axiparabola — a reflective optic whose surface maps near-field radius to a continuum of focal lengths, producing an extended focal range — with a radial echelon, a concentrically stepped mirror imposing a prescribed radius-dependent arrival-time delay; together they let the focus travel at an arbitrarily chosen sub- or superluminal velocity while preserving femtosecond duration (Pigeon et al., Opt. Express 32, 576 (2024)). Such pulses can drive ionization fronts of arbitrary velocity (Palastro et al., Phys. Rev. A 97, 033835 (2018)).

Simpson, Pigeon, Ambat, Miller, Ramsey, Weichman, Froula, and Palastro (Phys. Rev. Research 6, 013041 (2024)) simulated two-color THz generation driven by such an ultrashort flying focus. In their simulations (a unidirectional pulse propagation equation in cylindrical geometry, argon, 0.8-micrometer fundamental, about 0.3 mJ total pulse energy, focal range $L = 1.5$ cm, roughly half a dephasing length), tuning the focal velocity — and hence the ionization-front velocity $v_I$ relative to the THz phase velocity $v_T$ — controlled the emission angle, spectrum, and spatiotemporal profile: a subluminal front produced single-cycle, on-axis, highly focusable THz radiation, while a superluminal front produced the shortest pulses at the Cherenkov angle. The conversion efficiency grew approximately linearly with focal range — the THz energy growing quadratically while the laser pulse energy grew linearly at fixed far-field intensity — for drive powers below roughly ten times the critical power for self-focusing, $P_c$ (the power at which Kerr self-focusing overcomes diffraction), and for $L \lesssim L_\pi$. The same group then showed in simulation that a phased flying focus — adding a weak diffractive lens (a phaser) that precompensates the harmonic relative phase along the trajectory — removes the dephasing limit, generating a half-cycle THz pulse with almost no angular dispersion over multiple dephasing lengths at a controlled emission angle and with one quarter of the angular divergence of a conventional configuration (Simpson et al., Sci. Rep. 14, 26587 (2024)). A first experimental demonstration followed: a two-color ultrashort flying focus (3 mJ, 1 kHz, 50 fs, 800 nm, axiparabola-echelon pair, air) generated THz radiation at about $10^{-4}$ power conversion efficiency with a velocity-programmable beam profile (on-axis versus ring), a tunable spectrum, an improved focal spot (about 70 micrometers versus about 120 micrometers for a conventional filament source of comparable length), and a THz focal velocity near $1.01c$ independent of the programmed velocity (Pigeon et al., Nat. Commun. (2026)). An independent experiment steered the THz radiation from a single-color laser filament to large and even backward angles using a superluminal flying-focus-driven ionization front, albeit at very low efficiency (Fu et al., Phys. Rev. Lett. 134, 045001 (2025)).

The experimentally demonstrated results establish the technique inside a specific window: near-infrared 0.8-micrometer drivers, millijoule energies, and focal range comparable to one dephasing length. Whether the flying-focus advantage survives, degrades, or grows outside this window — at higher power, over longer interaction lengths, and with scaled drivers — is the open problem.

Problem Statement

Quantitatively establish the performance envelope and failure modes of flying-focus-driven two-color terahertz generation outside the parameter window in which it has been established by experiment and unphased simulation (near-infrared 0.8-micrometer drivers, millijoule pulse energies, and focal ranges within roughly one two-color dephasing length $L_\pi$, the propagation distance over which gas dispersion rotates the relative phase between the fundamental and its second harmonic by $\pi$); the many-dephasing-length behavior of the phased 'dephasingless' configuration is established in simulation only, and its robustness to realistic optic tolerances and alignment imperfections is open. The problem has three linked facets, corresponding to the open threads identified in the source outlook:

(a) High-power regime. For drive powers $P \gtrsim 10\,P_c$, where $P_c$ is the critical power for self-focusing in the target gas, transverse beam breakup and multiple filamentation (the splitting of a self-focused beam into several parallel filaments) are expected to disrupt the spatial and temporal coherence of the ionization-driven current and to degrade THz conversion efficiency and beam quality. Determine quantitatively whether and where this degradation sets in (thresholds and scaling with $P/P_c$ and focal geometry).

(b) Dephasing-dominated regime. Establish, by modeling and tolerance analysis, the achievable performance of the dephasingless phased-flying-focus configuration at focal ranges $L \gg L_\pi$ — in which the relative phase between the fundamental and second harmonic is precompensated along the trajectory — under realistic optic tolerances and alignment imperfections, confirming or bounding the predicted formation of half-cycle, low-angular-dispersion THz radiation at controlled emission angles, quantifying the achievable THz energy and focusability relative to the unphased configuration, and identifying the tolerances and design requirements that an experimental realization would have to satisfy.

(c) Driver scaling. Quantify the THz intensities and average powers achievable when the two-color flying focus is combined with mid-infrared ultrashort drivers — a route the source outlook suggests could provide focused THz pulses approaching relativistic intensities ($\sim 2\times 10^{13}\,\mathrm{W/cm^2}$) — and with ultra-high-repetition-rate laser systems, for which the source outlook targets directed and controlled THz pulses at 500 kHz approaching watt-level average powers.

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