High-Harmonic Generation: Principles & Applications

• High-harmonic generation (HHG) is a nonlinear emission process where strong fields drive matter to emit radiation at multiples of the fundamental frequency, enabling attosecond pulse generation.
• Methodologies include semiclassical recollision, wave-based Huygens-Fresnel models, and quantum-electrodynamical approaches to resolve complex electron dynamics and optimize phase matching.
• Applications range from attosecond spectroscopy in gases and solids to quantum-light and electrical HHG, offering versatile tools for probing electronic structure and ultrafast dynamics.

High-harmonic generation (HHG) is a highly nonlinear emission process in which matter driven by strong fields emits radiation at integer multiples of the driving frequency. In atomic and molecular physics it is a principal route to attosecond pulse generation and sub-femtosecond structural and dynamical probing, while in condensed matter it has become an all-optical probe of electronic structure, crystal symmetry, and ultrafast carrier dynamics; recent work further extends the concept to electrical high-harmonic generation in transport devices (Suárez et al., 2017, Zan et al., 2024). Contemporary HHG research spans gases, crystals, amorphous solids, metals, molecular solids, topological systems, and quantum-light-driven regimes, and its theoretical description now combines semiclassical recollision models, solid-state intra-/interband currents, wave-based Huygens-Fresnel pictures, and fully quantum-electrodynamical treatments (Li et al., 2023, Gorlach et al., 2019).

1. Microscopic mechanisms and theoretical pictures

In atomic HHG, the standard microscopic framework is the three-step model: tunnel ionization, acceleration in the laser field, and recollision with recombination. Within this picture, the cutoff photon energy is

$\omega_c = 3.17\,U_p + I_p,$

with $U_p = E_0^2/(4\omega^2)$ and $I_p$ the ionization potential (Kohler et al., 2012). Analytical strong-field approximation (SFA) approaches remain central because they expose the dependence of the HHG spectrum on target structure and laser parameters, and they can resolve interference features in multicenter molecular systems through explicit bound-free and rescattering matrix elements (Suárez et al., 2017).

The conventional continuum-bound recollision mechanism is not the only coherent source of harmonics. In the over-the-barrier ionization regime, interference between two separately ionized and separately propagating free wave packets can emit coherent radiation at the kinetic-energy difference of the two returning paths. This continuum-continuum channel can dominate the coherent HHG emission after ground-state depletion and can be isolated from continuum-bound harmonics via phase matching, because the intensity-dependent phase has opposite sign for the two mechanisms (Kohler et al., 2010).

In solids, the microscopic current is commonly decomposed into intraband and interband contributions,

$J(t)=J_{\text{intra}}(t)+J_{\text{inter}}(t),$

where intraband currents arise from anharmonic motion within a band and interband currents originate from polarization associated with electron-hole creation and recombination (Li et al., 2023). The same literature emphasizes that the usual particle-like recollision language in solids is only a limiting description. The saddle-point approximation can fail for complex, non-parabolic, and anisotropic dispersions, especially near band edges and for low-order harmonics. The alternative Huygens-Fresnel picture treats HHG as the coherent superposition of electron-hole “wavelets” emitted from all birth times and crystal momenta, making interference and wavepacket deformation explicit (Li et al., 2023).

A distinct strong-nonlinearity regime arises in systems with isolated bound states well separated from the ionization threshold. There, the HHG spectrum can split into non-harmonic Mollow-type triplets rather than remaining confined to odd harmonics at $(2n+1)\omega_d$. The sidebands are tunable by the driving field and imprint the internal level structure and confining potential on the emitted spectrum (Hazanov et al., 2022). This establishes that “harmonic” spacing is not universal once strong dressing of isolated bound states is operative.

2. Phase matching, quasi-phase matching, and propagation control

Because single-emitter conversion efficiency is small, macroscopic build-up is determined by phase matching. In gases the phase mismatch is commonly written as

$\Delta k_q = qk_\omega - k_{q\omega},$

with additional contributions from neutral and plasma dispersion, Gouy phase, and atomic dipole phase (Lobachinsky et al., 2017). For high pump intensities, true phase matching is often unattainable, and quasi-phase-matching (QPM) becomes the practical route.

A software-controlled QPM scheme was demonstrated by interfering a Gaussian pump beam with a weak Bessel beam produced by a spatial light modulator. The on-axis interference creates a longitudinal periodic intensity and phase modulation with period

$\Lambda = \frac{2\pi}{\Delta k_0},$

and QPM is achieved when $\Delta k_q = 2\pi/\Lambda$ (Lobachinsky et al., 2017). This method enabled on-the-fly enhancement of harmonic orders 25 to 39, with up to 30-fold intensity enhancement, real-time selection of the enhanced order over roughly ten harmonics, and bandwidth control via the non-diffracting length of the Bessel beam (Lobachinsky et al., 2017).

A different gas-phase route exploits propagation rather than externally imposed periodicity. In a strongly overdriven regime with near-infrared peak intensities up to $1.6\times10^{16}\,\mathrm{W/cm^2}$ in high-density rare-gas jets, ionization-induced self-phase modulation and plasma defocusing self-regulate the driver intensity to the phase-matchable range. Experiments reported ten-fold broadening of the near-infrared spectrum and continuous XUV and soft-X-ray spectra from 18 to 140 eV despite 50-fs driving pulses (Major et al., 2022). This directly contradicts the usual assumption that such overdriving is necessarily incompatible with phase-matched HHG.

Phase-matching control has also been reformulated as spectral selectivity. Using a Bessel-Gauss driving beam and a periodically modulated gaseous medium, one can impose the condition

$\int_0^{z_p}\Delta k_q(z)\,dz = 2\pi n$

for only one desired harmonic order, while neighboring orders accumulate nonintegral phase and are suppressed (Finke et al., 2024). Analytical and numerical results show direct monochromatic high-order harmonic generation from the source, without external monochromatizing optics (Finke et al., 2024).

In solids, the phase-matching problem is unusually severe because the wavevector mismatch is large and traditional gas-phase methods are ineffective. A recently proposed strategy uses two crystalline flakes with an interlayer twist angle $U_p = E_0^2/(4\omega^2)$0, which imparts a nonlinear Pancharatnam-Berry phase

$U_p = E_0^2/(4\omega^2)$1

to the $U_p = E_0^2/(4\omega^2)$2-th harmonic polarization (Ma et al., 11 Mar 2025). For hexagonal boron nitride, this twist-induced nonlinear optical phase was designed to compensate the intrinsic mismatch and yielded a record-high fifth-harmonic conversion efficiency of $U_p = E_0^2/(4\omega^2)$3 in a total thickness of only $U_p = E_0^2/(4\omega^2)$4, with quadratic intensity scaling versus the number of flakes and a broad operational bandwidth of $U_p = E_0^2/(4\omega^2)$5 nm (Ma et al., 11 Mar 2025). The same work states that existing solid-state methods either fail for higher-order harmonics or are controversial in their effectiveness for true single-step HHG (Ma et al., 11 Mar 2025).

3. Solids, surfaces, and nanostructured emitters

Solid-state HHG does not reduce to a single microscopic origin across all materials, but several experimentally distinct regimes have been identified. One reported view, supported by z-scan and interface-sensitive measurements, is that HHG in some solids dominantly originates from strongly localized surface and interface states rather than the bulk. In that work, bulk contributions were suppressed by at least 1–2 orders of magnitude in perturbative conditions and by at least 3–4 orders of magnitude in non-perturbative conditions; for one third-harmonic measurement the bulk contribution was at least 4400 times smaller than the surface contribution (Seres et al., 2018). The interpretation is that localized surface states possess a sizable transition dipole, whereas delocalized bulk states are penalized both by small dipole matrix elements and poor phase matching (Seres et al., 2018).

Nanostructuring can strongly amplify the driving field and thereby reshape the HHG source term itself. An all-dielectric Si metasurface with a sharp Fano resonance, described as a classical analogue of electromagnetically induced transparency, enhanced non-perturbative harmonic emission by more than two orders of magnitude relative to unpatterned Si (Liu et al., 2017). The enhancement was highly anisotropic with excitation polarization and selective to excitation wavelength, with the seventh harmonic exceeding the unpatterned film by more than 100× at $U_p = E_0^2/(4\omega^2)$6 and the fifth harmonic enhanced by about one order of magnitude (Liu et al., 2017). Because the resonance exists only for polarization along the bar antenna, the enhanced HHG is an explicitly mode-engineered process rather than a bulk material response.

These surface- and nanostructure-based results define an important design axis in solid HHG: emission can be governed by localization, interface geometry, and resonant near fields as much as by bulk crystal periodicity. A plausible implication is that “solid HHG” is better understood as a family of strongly driven condensed-matter emitters whose macroscopic efficiency depends on where coherence is accumulated and how the nonlinear polarization is phased.

4. Material diversity beyond conventional transparent crystals

The current literature shows that HHG is not confined to transparent crystalline dielectrics or simple gases. The following platforms illustrate the breadth of the phenomenon.

Platform	Reported observation	Representative paper
Amorphous fused silica / crystal quartz	Harmonics up to $U_p = E_0^2/(4\omega^2)$7 eV in both; quartz up to 4× more efficient at highest field	(You et al., 2017)
Silver thin films	Nonperturbative harmonics up to $U_p = E_0^2/(4\omega^2)$8 eV; multi-shot damage threshold $U_p = E_0^2/(4\omega^2)$9	(Gholam-Mirzaei et al., 7 Mar 2025)
Fullerenes $I_p$0	Multiple plateaus bounded by excitonic lines; $I_p$1 stronger by one to two orders of magnitude	(Avetissian et al., 2021)
Pentacene single crystals	Odd harmonics from 3rd to 17th order; strong polarization dependence on intermolecular coupling	(Wiechmann et al., 5 May 2025)

The observation of HHG in amorphous fused silica established that long-range periodicity is not necessary for nonperturbative coherent XUV generation in solids. Both amorphous fused silica and crystal quartz generated harmonics up to about 25 eV, but the spectral signatures differed: fused silica exhibited only odd harmonics and strong carrier-envelope-phase dependence with large atto-chirp, whereas quartz exhibited both odd and even harmonics, a minimum around 17–18 eV separating two plateau regions, and up to fourfold larger efficiency at the highest field (You et al., 2017). The reported comparison decouples long-range periodicity from atomic density and indicates that disorder on the scale studied does not preclude coherent HHG (You et al., 2017).

Metallic HHG was long considered unfavorable because of high infrared reflectivity, but silver thin films were shown to emit nonperturbative harmonics extending to the extreme ultraviolet, up to 20 eV, under few-cycle near-infrared excitation at near-normal incidence (Gholam-Mirzaei et al., 7 Mar 2025). Despite roughly 99.5% reflectivity at 800 nm, the residual field within the penetration depth was sufficient to drive sub-cycle electron dynamics. Time-dependent density-matrix simulations based on maximally localized Wannier functions attributed low-order harmonics to conduction electrons near the Fermi surface, mainly $I_p$2- and $I_p$3-type orbitals, and higher harmonics to bound $I_p$4-electron excitations (Gholam-Mirzaei et al., 7 Mar 2025). The strong orientation dependence of the emitted harmonics was used to argue against a plasma-driven origin.

Molecular and molecular-solid HHG broaden the landscape further. Dynamical Hartree-Fock calculations for fullerene molecules predicted multiple plateaus whose borders are defined by excitonic lines, with plateau cutoffs scaling linearly with field amplitude and increasing in photon energy as the pump frequency increases, in contrast to the atomic case (Avetissian et al., 2021). The same study reported that increasing electron-electron interaction suppresses the overall HHG rate, and that $I_p$5 exhibits richer spectra and one to two orders of magnitude higher intensity than $I_p$6 (Avetissian et al., 2021).

In thin organic molecular crystals, HHG has now been demonstrated experimentally in perfectly aligned pentacene single crystals. Harmonics from the 3rd through the 17th order were observed under mid-infrared excitation, and the polarization dependence showed that even weak van der Waals intermolecular coupling governs the HHG response. Higher harmonic orders were found to be especially sensitive to nearest- and next-nearest-neighbor couplings, while model calculations indicated that weaker intermolecular interactions require higher harmonic orders to resolve the crystal structure (Wiechmann et al., 5 May 2025). This places organic molecular crystals in an intermediate regime between isolated-molecule and band-structure HHG.

5. Symmetry, Berry curvature, and topological information

HHG has become a probe not only of energetic band structure but also of geometric and topological information. A particularly clear demonstration was obtained in a spin scalar chiral state on a triangular lattice, where reversing the scalar chirality flips the sign of the Berry curvature and Chern numbers without changing the energy band structure (Ono et al., 2024). In that setting, the transverse HHG phase shifts by $I_p$7 when the chirality is reversed, but the more striking effect at low optical frequency relative to the gap is a pronounced chirality dependence of the longitudinal HHG intensity and its polarization dependence (Ono et al., 2024).

The proposed mechanism combines anomalous velocity and interband recombination. The anomalous velocity term,

$I_p$8

modulates the real-space trajectories of electron-hole pairs, and the resulting change in recombination conditions alters the amplitude and phase of interband currents along the field direction (Ono et al., 2024). This is notable because linear-response intuition would suggest that Berry-curvature sensitivity should be predominantly transverse; the reported HHG calculations show that in the nonperturbative, low-frequency regime the longitudinal channel can carry the stronger geometric fingerprint (Ono et al., 2024).

Symmetry sensitivity also appears in less overtly topological settings. In quartz, the lack of inversion symmetry allows even harmonics and modifies the carrier-envelope-phase periodicity relative to amorphous fused silica (You et al., 2017). In silver, suppression of specific harmonics under rotation reflects the crystal’s six-fold lattice symmetry (Gholam-Mirzaei et al., 7 Mar 2025). In pentacene, the directions of maximal harmonic yield track intermolecular directions rather than isolated-molecule transition dipoles (Wiechmann et al., 5 May 2025). Taken together, these results show that HHG observables can encode Berry curvature, inversion breaking, rotational symmetry, and intermolecular electronic connectivity in a single strong-field measurement.

6. Quantum-light-driven and electrical HHG

A major conceptual expansion of HHG comes from treating the driving field quantum mechanically. A fully quantum-electrodynamical theory predicts shifted frequency combs arising from transitions between perturbed states of the driven atom, new spectral features associated with breakdown of the dipole approximation for emission, and the result that each emitted HHG photon is a superposition of all frequencies in the spectrum, i.e. a single-photon comb (Gorlach et al., 2019). The same framework distinguishes many-atom from single-atom limits: semiclassical terms scale coherently, whereas quantum corrections become comparatively important in few-emitter regimes (Gorlach et al., 2019).

A complementary quantum-statistical theory shows that the plateau and cutoff depend on the photon statistics of the driving light. Coherent and Fock states preserve the established cutoff law, while thermal and squeezed states substantially exceed it at the same average intensity (Gorlach et al., 2022). In that formulation, the emitted spectrum is averaged over the Husimi $I_p$9-function of the driving field,

$J(t)=J_{\text{intra}}(t)+J_{\text{inter}}(t),$0

so rare high-amplitude excursions in broad photon-number distributions disproportionately enhance the high-energy tail (Gorlach et al., 2022).

This quantum-control perspective has already been pushed toward low-energy driving. In monolayer transition-metal dichalcogenides, a weak classical field with energy about 1000 times less than that used in conventional HHG experiments, perturbed by a bright squeezed vacuum pulse with less than 0.5% of the driving laser energy, was proposed as a “quantum dial” for tuning nonlinear emission, electron dynamics, and ionization (Wang et al., 16 Sep 2025). In that work, the bright squeezed vacuum perturbation enables even harmonics and strongly modifies the distribution of harmonic emission in momentum space (Wang et al., 16 Sep 2025).

The HHG concept has also migrated from optics to transport. Electrical HHG was observed up to the 300th order in dual-gated, h-BN-encapsulated bilayer graphene, with clear non-perturbative behavior, both even and odd harmonics, and periodic oscillations as a function of reciprocal drive current (Zan et al., 2024). The measured $J(t)=J_{\text{intra}}(t)+J_{\text{inter}}(t),$1-th harmonic Hall response is written as

$J(t)=J_{\text{intra}}(t)+J_{\text{inter}}(t),$2

but for high orders the scaling exponent is much smaller than $J(t)=J_{\text{intra}}(t)+J_{\text{inter}}(t),$3, demonstrating a non-perturbative transport regime (Zan et al., 2024). Theoretical simulations connect these oscillations to a singular distribution of Berry curvature near the band edges (Zan et al., 2024).

7. Applications, barriers, and unresolved questions

HHG remains the workhorse for attosecond pulses and a central source of coherent XUV radiation, but the application space has diversified. The literature explicitly identifies transient absorption and reflection spectroscopy, vacuum-ultraviolet spectroscopy, high-precision frequency-comb metrology, compact ultraviolet and attosecond tabletop sources, and on-chip or chip-integrable photonic devices as active targets (Kohler et al., 2012, Major et al., 2022, Seres et al., 2018, Ma et al., 11 Mar 2025). In solids and molecular crystals, HHG is also framed as an all-optical probe of crystal structure, intermolecular coupling, Berry curvature, and ultrafast electron dynamics (Wiechmann et al., 5 May 2025, Ono et al., 2024).

The principal barriers remain regime-dependent. For multi-keV atomic HHG, phase matching and relativistic drift are identified as the main limitations, with long-wavelength drivers, quasi-phase-matching schemes, tailored fields, and x-ray assistance proposed as routes forward (Kohler et al., 2012). In solids, phase mismatch is exceptionally large, and the recent twist-phase-matching literature explicitly notes that existing approaches can fail for higher-order harmonics or remain controversial for true single-step HHG (Ma et al., 11 Mar 2025). Another unresolved issue is interpretive rather than purely technical: the particle recollision picture in solids can misestimate emission timing when saddle-point assumptions break down, so extraction of attosecond dynamics may require wave-based descriptions (Li et al., 2023).

A further point of contention concerns the spatial origin of solid-state emission. Surface-state-dominated models, bulk interband-intraband pictures, and penetration-depth-limited metallic dynamics are all supported in different systems (Seres et al., 2018, Li et al., 2023, Gholam-Mirzaei et al., 7 Mar 2025). This suggests that a single universal microscopic narrative for all solid HHG is unlikely. What is firmly established is that HHG is now a multi-platform, multi-regime phenomenon: it can be phase matched by structured propagation or by crystal twist, enhanced by dielectric resonances, sustained in amorphous matter and noble metals, tuned by photon statistics, and extended into electrical transport.