Third-Harmonic Generation in Photonics

Updated 2 September 2025

Third-harmonic generation (THG) is a third-order nonlinear process where three photons coherently combine to produce one photon at triple the original frequency.
Resonant enhancements, phase matching, and nanophotonic engineering (e.g., metasurfaces, plasmonics) are key to boosting THG efficiency.
THG is applied in frequency upconversion, high-resolution imaging, and material diagnostics, offering valuable insights in photonics research.

Third-harmonic generation (THG) is a third-order nonlinear optical process in which three photons at the fundamental frequency ω interact coherently within a medium to produce a single photon at the third-harmonic frequency 3ω. As a cubic effect in the electromagnetic field, THG is governed by the third-order nonlinear susceptibility tensor, χ³, or equivalently—for surfaces and 2D materials like graphene—by the third-order nonlinear conductivity. THG is an essential phenomenon for frequency upconversion, probing of symmetry, and nonlinear metrology across photonics, condensed-matter, and materials research.

1. Theoretical Foundations and Nonlinear Response

THG arises from the third-order nonlinear polarization: $P^{(3)}(3\omega) = \varepsilon_0\, \chi^{(3)}(3\omega; \omega, \omega, \omega)\, E(\omega)^3$ where ε₀ is the vacuum permittivity, χ³ is the third-order nonlinear susceptibility tensor, and E(ω) is the incident electric field. This process is symmetry-allowed in all materials, including inversion-symmetric systems, in contrast to second-harmonic generation (SHG).

In crystals, the symmetry properties of χ³ dictate the angular and polarization dependence of THG. For example, the polarization- and azimuthal-rotation-dependent THG intensity in graphene tracks the material's crystalline symmetry, producing a cos²(3θ) or sin²(3θ) variation characteristic of threefold (or sixfold) rotational symmetry (Hong et al., 2013). For tensor components, especially in anisotropic systems such as orthorhombic or monoclinic crystals, the individual elements (e.g., χ{xxxx}^{(3)}, χ{yyyy}^{(3)}) encode directional selectivity and can act as quantitative descriptors of molecular orientation, as illustrated in polarization-resolved THG imaging of starch granules (Kefalogianni et al., 27 Mar 2025).

In graphene and other 2D materials, THG can be modeled by a third-order nonlinear surface current, J^{(3)} = σ^{(3)} E(ω)^3, with σ^{(3)} the nonlinear surface conductivity. This conductivity captures both intra-band (Fermi surface) and inter-band contributions, typically calculated using diagrammatic perturbation theory and involving subtle cancellation mechanisms between power-law and logarithmic terms (Rostami et al., 2016). The cubic dependence on E(ω) leads to a highly nonlinear intensity scaling, making local field enhancements exceptionally effective for boosting THG efficiency.

2. Resonances, Enhancement Mechanisms, and Phase Matching

Enhancing THG efficacy requires careful engineering of phase matching, field enhancement, and resonance. The maximum of χ^{(3)} occurs near electronic or excitonic resonances, where the denominator of the χ^{(3)} response approaches zero: $\chi^{(3)} \propto \left[3\omega - \omega_\text{res} + i\Gamma\right]^{-1}$ with ω_res the resonance (e.g., band edge, van Hove singularity, or excitonic state) and Γ a damping parameter.

Resonant Enhancement: In monolayer graphene, strong THG is observed when the pump is tuned such that three photons sum to an exciton-shifted van Hove singularity at the M-point, resulting in a resonant enhancement (Hong et al., 2013). Optical enhancement in semiconductors is similarly seen near exciton-polariton resonances (e.g., 1s-excitons in bulk GaAs), with magnetic field–induced increases in exciton oscillator strength delivering up to 50× enhancement in THG (Warkentin et al., 2018).
Metasurfaces and Plasmonics: Patterned graphene metasurfaces supporting localized surface plasmon polaritons (SPPs) can confine the pump field, achieving electric field enhancements >100×, especially when backed by metallic reflectors and tuned to photonic or Fabry–Pérot resonances (Jin et al., 2017). Metal-based dielectric nanostructures, such as silicon disks on a metal substrate, combine the perfect electric conductor (PEC) surface effect and coupling between electric dipole resonances (EDRs) of the dielectric and propagating surface plasmon resonances (PSPRs) of the metal, yielding more than eight orders of magnitude increase in THG relative to all-dielectric configurations (Yao et al., 2019).
Strong Coupling and Quasi-Bound States in the Continuum (quasi-BICs): Structures exploiting hybridization between TE and TM-polarized quasi-guided modes (QGMs) can enter the strong-coupling regime, indicated by avoided crossings (Rabi splitting), and reach Q-factors up to 10¹², with THG conversion efficiencies ≥10⁻² demonstrated (Huang et al., 20 Aug 2025). Quasi-BICs in metasurfaces with C₄ᵥ symmetry yield polarization-independent THG and allow conversion efficiencies up to 1.03×10⁻⁵ at pump intensities of 5.85 GW/cm² (Liu et al., 22 Dec 2024), regulated by precise engineering of nanodisk asymmetry and lattice parameters.
Temperature and Hot-Carrier Effects: In graphene, the nonlinear conductivity σ^{(3)} is highly sensitive to carrier temperature Tₑ. Under resonant THz excitation, hot-carrier effects (elevated Tₑ and reduced μ_SLG) can significantly modulate nonlinear response, enhancing conversion efficiency by a factor of three compared to the isothermal case (Doukas et al., 15 Jul 2025). Modelling must incorporate these thermal effects self-consistently for accurate device performance predictions.
Phase Matching: In bulk nonlinear crystals, efficient THG requires phase-matching so that generated 3ω waves constructively interfere along the propagation direction. In diamond, near-ideal phase-matching is realized via a Cherenkov-type process, producing a measurable angular deflection matching theoretical predictions (e.g., a Cherenkov angle of ~12.7°) (Abulikemu et al., 2023). Multilayer grating techniques achieve quasi-phase matching by introducing stratified sub-gratings with engineered phase jumps, resulting in coherent summation of TH fields and enhancement factors up to N in N-layer devices (Yizhou et al., 2013).

3. Quantum and Many-Body Effects

Electronic correlations and many-body effects introduce complexity and functionality to THG:

Excitonic Effects: Inverse third-order nonlinearity can be enhanced near excitonic resonances both in semiconductors and in correlated systems such as excitonic insulators. In the latter, both bare (three-photon) and vertex-corrected (order-parameter–driven) contributions appear, resulting in multiple resonant features in THG susceptibility at ℏΩ = Δ_g/3, Δ_g/2, and Δ_g, where Δ_g is the band gap. The amplitude of vertex-corrected peaks provides an optical signature of the excitonic collective mode – especially visible in the BCS regime (Tanabe et al., 2021).
Superconductors: In multiband superconductors, THG is strongly enhanced below T_c by resonant excitation of lattice-modulated charge fluctuations (the "LCF channel") when the pump frequency matches the superconducting gap. The resulting nonlinear response is tensorial, displaying polarization dependence, though this dependence may be suppressed by non-resonant instantaneous contributions and finite pulse widths (Cea et al., 2017). In s-wave dirty superconductors with paramagnetic impurities, the amplitude mode of the order parameter appears as a vertex correction that dominates the THG intensity; breaking time-reversal symmetry destroys this mode and reduces the THG peak (Jujo, 2017).

4. Nanophotonic and Metasurface Engineering

Metasurface design is central to modern THG enhancement, uniting field localization, symmetry control, and modal hybridization:

Structure	Enhancement Mechanism	Resultant THG Properties
Graphene metasurface	Localized SPPs, Fabry–Pérot cavity	Electric field enhancement, dynamic tunability
Silicon nanodisk	PEC effect, PSPR–EDR coupling	Dual resonance, 8+ orders of THG increase
SSH photonic chain	Topological edge-state localization	3 orders of magnitude THG boost, robust modes
Quasi-BIC metasurface	High-Q doubly degenerate BICs	Polarization-independent, 10⁻⁵ efficiency
Bilayer with cubes	Strong TE/TM mode hybridization	Dynamical Q, ≥10⁻² efficiency, tunable by pol.

These approaches leverage modal selectivity, high Q-factors, resonant field enhancement, and topology (e.g., SSH chain edge states (Yuan et al., 2021)) to maximize the EM field at the nonlinear active region, directly impacting THG performance.

5. Advanced Imaging and Material Characterization

THG extends beyond frequency conversion to quantitative imaging and diagnostics:

Microscopy: Synthetic aperture holographic THG microscopy enables wide-field, label-free imaging with both amplitude and phase resolution of the THG field. Phase recovery is achieved via off-axis holography and computational aberration-correction techniques (e.g., modified DASH and SVD algorithms), significantly improving field-of-view, resolution, and contrast over conventional point-scanned intensity imaging (Farah et al., 6 Feb 2024).
Polarization-Resolved Structural Imaging: Polarization-dependent THG (P-THG) resolves sub-micron anisotropy in biological structures by mapping the angular dependence of the nonlinear susceptibility tensor. In starch granules, spatial variation of the anisotropy ratio AR = χ{xxxx}^{(3)}/χ{yyyy}^{(3)} reveals sharp transitions in molecular organization between core and shell, a feature inaccessible to standard intensity-only THG (Kefalogianni et al., 27 Mar 2025).
Material and Device Characterization: THG microscopy is sensitive enough to distinguish graphene from glass with high background-free contrast (>2 orders of magnitude), permitting detection of lattice symmetry, strain, or defect-induced deviations from the expected polarization dependence (Hong et al., 2013).

6. Design Guidelines, Practical Implementations, and Challenges

Implementing efficient THG in devices or experiments involves optimizing several interrelated parameters:

Field Localization and Q-Factor: Design for maximal field enhancement and mode confinement—using high-Q resonant modes, quasi-BICs, plasmonic resonators, or topological localization—to increase the local intensity experienced by the χ³ medium.
Tuning and Phase Matching: Align pump wavelength with electronic/excitonic or photonic resonances; in frequency-conversion devices, engineer phase-matching by structural tuning (thickness, periodicity) or using quasi-phase-matching schemes (layered or grating-assisted).
Material Purity and Dispersion: High-quality, low-loss materials (e.g., electronic-grade diamond with sub-ppb impurity concentration) are crucial for minimizing linear absorption losses and achieving high conversion efficiency (Abulikemu et al., 2023). For 2D materials, model dispersion and losses with explicit inclusion of nonlinear and dissipative terms.
Fabrication Tolerances: In multilayer grating schemes, random error in layer thicknesses directly limits the effective coherence and efficiency (e.g., σ_T must be <12% of the coherence length in a 9 mm, N=9 layer structure to avoid >50% loss) (Yizhou et al., 2013).
Thermal and Hot-Carrier Effects: At high pump intensities, photoinduced hot-carrier effects and Kerr nonlinearities (index shifting) can perturb resonance conditions, requiring multiphysics modeling for accurate design of tunable or high-intensity THG platforms (Doukas et al., 15 Jul 2025, Yao et al., 2019).
Polarization Control: Device geometry and symmetry (C₄ᵥ, orthorhombic, SSH-type) dictate the angular dependence and polarization selectivity of the THG process. Polarization-independent geometry is desirable for broadband or unpolarized sources, while polarization-resolved schemes enable structural diagnostics.

7. Applications and Future Directions

THG finds applications in:

Nonlinear frequency conversion (IR to visible/UV, on-chip frequency combs)
Quantum light sources (as room-temperature upconvertors and photon pair generators)
Quantum sensing and communications (ultrapure diamond-based systems (Abulikemu et al., 2023))
Nonlinear THz spectroscopy and imaging (using tunable, efficient graphene metasurfaces (Jin et al., 2017, Doukas et al., 15 Jul 2025))
Label-free biological imaging (wide-field and phase-resolved THG tomography (Farah et al., 6 Feb 2024))
Metamaterial-enabled wave mixing and ultrafast photonics (polarization-agnostic, robust metasurface upconverters (Liu et al., 22 Dec 2024), tunable through incident polarization (Huang et al., 20 Aug 2025))
Structural and orientation-sensitive diagnostics (molecular heterogeneity in biological and soft matter (Kefalogianni et al., 27 Mar 2025))

Future research will likely push conversion efficiencies toward unity in nanoscale platforms, harness collective many-body or topological effects for new dynamical regimes, and couple THG with quantum-well, heterostructure, or excitonic resonance engineering for expanded spectral flexibility and device functionality. The integration of hot-carrier dynamics, high-Q resonance control, and phase-resolved detection schemes positions THG as a versatile probe and tool across modern photonics and condensed matter research.