Third-Order Nonlinear Optical Susceptibility

Updated 30 July 2025

Third-order nonlinear optical susceptibility, χ^(3), is a fourth-rank tensor that characterizes a material’s cubic response to electromagnetic fields, governing processes like four-wave mixing and third-harmonic generation.
The mathematical framework employs tensor symmetry, demonstrating how Kleinman symmetry and material anisotropy reduce independent components, which is vital for modeling nonlinear optical phenomena.
Practical applications include frequency conversion, all-optical switching, and quantum photonics, with tunable, resonantly enhanced nonlinearities driving advances in modern photonic devices.

Third-order nonlinear optical susceptibility, commonly denoted as χ³, characterizes the cubic nonlinear response of a material’s polarization to applied electromagnetic fields. It is a fourth-rank tensor that governs a broad class of third-order processes, including four-wave mixing, third-harmonic generation, self-phase/mutual-phase modulation, Kerr effects, and optical switching. The formal definition, physical origins, mathematical structure, and phenomenology of χ³ underpin its central role in both fundamental nonlinear optics and emerging photonic technologies.

1. Mathematical Framework and Tensor Structure

The macroscopic polarization P in response to an electric field E(t) is expanded as a power series,

$P_i(t) = P_i^s + \epsilon_0 \sum_j \chi_{ij}^{(1)} E_j(t) + \epsilon_0 \sum_{j,k} \chi_{ijk}^{(2)} E_j(t) E_k(t) + \epsilon_0 \sum_{j,k,l} \chi_{ijkl}^{(3)} E_j(t) E_k(t) E_l(t) + \ldots$

The third-order susceptibility tensor, $\chi_{ijkl}^{(3)}$ , thus relates the induced polarization at frequency $\omega_4$ to three incident fields at frequencies $\omega_1$ , $\omega_2$ , and $\omega_3$ ,

$P_i^{(3)}(\omega_4) = \epsilon_0 \sum_{jkl} \chi_{ijkl}^{(3)}(\omega_4; \omega_1, \omega_2, \omega_3) E_j(\omega_1) E_k(\omega_2) E_l(\omega_3)$

with $\omega_4 = \omega_1+\omega_2+\omega_3$ for four-wave mixing (FWM).(Rao, 2016)

$\chi^{(3)}$ is a fourth-rank tensor with hundreds of independent elements in the general case, strongly constrained by the symmetry properties of the material and the ordering of frequencies. Under degeneracy (e.g., all $\omega_i$ equal for third harmonic generation), permutation symmetries reduce the number of independent components. In isotropic or cubic-symmetric media, the tensor can often be reduced to several key active components (e.g., $\chi_{xxxx}$ , $\chi_{xxyy}$ , etc.), as exemplified in liquid CS $_2$ or crystalline semiconductors.(Chettri et al., 2021, Ye et al., 2023)

Under Kleinman symmetry (valid far from resonance or at low frequency), further simplifications exist due to the invariance under index permutation, folding the tensor into a "D-matrix" notation.(Rao, 2016)

2. Physical Origins in Different Systems

Materials Systems

Bulk Dielectrics and Semiconductors: In centrosymmetric materials, the lowest nonvanishing term is χ³. The tensor elements are determined by interband and intraband electronic transitions, as quantified by ab initio approaches, perturbative expansions of Bloch states in the presence of fields, and DFT-based Berry phase simulations.(1012.5727, Dues et al., 2022)
Nanoscale and 2D Materials: In graphene and TMDCs, strong quantum confinement and unique band structures produce exceptionally large χ³ values, highly sensitive to chemical potential tuning, strain, and interactions near resonances (excitons, trions).(Khorasani, 2017, Jiang et al., 2017, Lafeta et al., 2021)
Metals and Nanoplasmonics: In nanostructured metals (e.g., gold nanotips), gradient field effects and longitudinal currents can drive field-forbidden χ³ contributions, giving rise to conversion efficiencies scaling inversely with system size.(Kravtsov et al., 2017)
Atomic Gases/Rydberg Ensembles: Under EIT, strong Rydberg–Rydberg interactions lead to nonlinearities scaling nonlinearly with atomic density and enable quantum nonlinearity at the single-photon level.(Bai et al., 2016)

Fundamental Processes

Four-Wave Mixing (FWM): The general paradigm for nonlinear processes mediated by χ³. In gas, vacuum, or solid, three pump/photon fields combine to generate a new field, subject to phase-matching and selection rules.(1007.0083, Rao, 2016)
Harmonic Generation: Third-harmonic generation (THG) arises as a degenerate case with all incident frequencies identical, maximizing the permutation symmetry of χ³.

3. Symmetry, Anisotropy, and Tensorial Properties

Crystallographic symmetry, orientation, and physical geometry exert a profound influence on the anisotropy and magnitude of χ³:

Structural Anisotropy: Systems such as TeO $_2$ exhibit dramatic anisotropy in χ³, with helical chain motifs and cooperative nonlocal polarization effects amplifying specific tensor components (e.g., $\chi_{zzzz}$ or $\chi_{yyyy}$ depending on phase).(1012.5727)
Orientation-Dependent Nonlinearity: In cubic semiconductors such as ZnSe and GaP, the crystal face and beam polarization define the symmetry of observed nonlinearities (e.g., uniform 4-fold for (100), asymmetric for (110)), and precise rotation-dependent periodic modulation in two-photon absorption and two-beam coupling experiments directly tracks tensorial anisotropy.(Ye et al., 2023)
2D Material-Specific Effects: In graphene, time-resolved OHD-OKE measurements confirm that the in-plane tensor components ( $\chi_{xyxy}+\chi_{xyyx}$ , $\chi_{xxyy}$ ) dominate the nonlinear response, while out-of-plane components (e.g., $\chi_{xzxz}$ ) are experimentally suppressed to below 10% of in-plane values.(Dremetsika et al., 2017)
Role of Magnetic and Geometric Factors in Vacuum: In a QED-corrected vacuum, geometric factors dependent on polarization and beam arrangement are essential, differentiating it from the symmetry-defined susceptibilities in matter.(1007.0083)

4. Resonant Enhancement, Divergences, and Density Dependence

The amplitude and spectral characteristics of χ³ can be greatly affected by resonances and intraband/excitonic phenomena:

Resonant Enhancement: When photon energies approach interband gaps, exciton, or trion energies, the nonlinear response is dramatically enhanced, as observed in resonant four-wave mixing in TMD monolayers, where χ³ sheet susceptibilities reach $10^{-29}$ m $^3$ /V $^2$ near absorptive resonances.(Lafeta et al., 2021)
Intraband Divergences: In doped or excited 2D systems, as the sum of incident frequencies or combinations of frequencies approaches zero, divergences in χ³ emerge due to non-perturbative intraband conductivity, controlled (regularized) by carrier relaxation rates. This mechanism yields large, robust nonlinearities that can be tuned via chemical potential in graphene or via carrier injection in 2D semiconductors.(Cheng et al., 2018)
Quantum Coherent Effects: In graphene, coherent summation of resonant contributions from one-, two-, and three-photon processes, modulated via chemical potential or gating, produces tunable and, in some difference-frequency mixing scenarios, potentially divergent third-order effects.(Jiang et al., 2017)
Density Dependence in Rydberg Systems: In cold-atom EIT media, the Rydberg–Rydberg interaction-induced nonlinearity becomes quadratically dependent on atomic density, producing “giant” nonlinearities far surpassing photon–atom interaction contributions. Fifth-order processes even scale cubically with density.(Bai et al., 2016)

5. Measurement Techniques, Computational Approaches, and Benchmark Values

Experimental Measurement

Z-scan Technique: Standard for quantifying nonlinear absorption (β) and the nonlinear refractive index (n₂), both directly related to χ³. Critical corrections include phase-mismatch, pulse duration effects, and two-photon absorption contributions (especially in wide-gap materials like diamond).(Almeida et al., 2017, Sharma et al., 2021)
Degenerate Four-Wave Mixing (DFWM): Provides access to the full tensorial components, their time dependence, and the interplay between instantaneous electronic and slower molecular reorientation contributions, as systematically resolved in CS₂.(Chettri et al., 2021)
THG/FWM Microscopy: Used for direct sheet and effective bulk χ³ extraction in 2D materials; calibration against references such as fused quartz is routine for absolute quantification.(Woodward et al., 2016, Khorasani, 2017)
Enhanced OHD-OKE: Enables isolation of temporal and tensorial components with sub-100 fs resolution and discrimination of in-plane/out-of-plane responses, especially in 2D materials.(Dremetsika et al., 2017)

Computation

First-Principles (DFT, Berry-Phase): Permits ab initio prediction of χ³ components, accounts for band structure and anisotropy (as in TeO $_2$ and ferroelectric oxides), and captures multi-photon resonant features subject to scissors-shift corrections for bandgaps.(1012.5727, Dues et al., 2022)
Semi-classical/TB/Perturbation Theory: Enables large-scale computation of tensor components in low-symmetry 2D crystals; tight-binding models accurately capture valley-dependent, spin–orbit, and multi-band effects in TMDCs.(Khorasani, 2017)

Selected Benchmark Values (as reported):

Material/System	χ³ (m²/V² or esu)	Notable Features/Context
SWCNT film (fully dense)	$6.32 \times 10^{-11}$ esu	Three orders of magnitude > fused silica
MoS₂ monolayer (sheet)	$1.7 \times 10^{-28}$ m³/V²	Sheet value, telecom C-band(Woodward et al., 2016)
Graphene monolayer (sheet)	(varies, see THG/FWM)	Gate-tunable, stepwise enhancement(Jiang et al., 2017)
TeO₄ crystalline	$\sim10^{-12}$ esu	Two orders of magnitude > SiO₂(1012.5727)
[111] Si nanowire	$\sim 1.29 \times 10^{-17}$ m²/V²	~100x > bulk Si (Shiri, 2017)
Diamond (n₂/β-based)	up to $1.7 \times 10^{-19}$ m²/W	2PA-driven, ultrafast(Almeida et al., 2017)
QED vacuum (effective, optimal)	$4 \times 10^{-40}$ m²/V²	geometry- and polarization-dependent(1007.0083)

6. Practical Implications and Device Prospects

Large or tunable χ³ is foundational for:

Frequency Conversion: Efficient THG, FWM, and parametric amplification in materials such as TMDCs, aligned SWCNTs, and nanoplasmonic metals has direct relevance for wavelength-agnostic photonic circuit elements.(Morris et al., 2013, Kravtsov et al., 2017)
All-Optical Switching and Bistability: Nonlinear refractive index changes governed by χ³ in Kerr media (and in layered TMDC-on-Kerr-dielectric stacks) enable ultrafast switching, bistability, and on-chip logic, with additional tuning by strain and chemical gating.(Sengupta et al., 2015)
Quantum Photonics: Rydberg EIT systems, squeezed-light generation in high nonlinearity microresonators (TMDC-covered), and optomechanical coupling all exploit enhanced, density-dependent or resonant χ³.(Bai et al., 2016, Khorasani, 2017)
Optical Limiting and Protection: Highly nonlinear materials such as NiCo $_2$ O $_4$ nanoflowers and classical fluids (e.g., CS $_2$ ) serve as optical limiters by exploiting strong nonlinear absorption and refraction over a broad ultrafast intensity range.(Sharma et al., 2021, Chettri et al., 2021)
Damage Control and Anisotropic Processing: In semiconductors, the relationship between χ³(θ), orientation, and damage thresholds now guides laser machining and materials engineering at the micron/nanometer scale, avoiding saturation even as threshold intensities are approached.(Ye et al., 2023)

7. Limitations, Open Problems, and Future Directions

Interaction-Length-Driven Saturation: By reducing the optical interaction length (using thin films or tight focusing), the non-saturation regime for nonlinear processes can be maintained up to the damage threshold, ensuring that intrinsic anisotropies and crystalline signatures of χ³ are fully expressed.(Ye et al., 2023)
Resonant Losses and Damping: Enhancement via resonance is constrained by increased absorption and finite-lifetime broadening; accurate modeling must incorporate relaxation rates and excitonic damping, regulating divergences in both strong and weak field regimes.(Almeida et al., 2017, Cheng et al., 2018)
Material Engineering: Opportunities exist in deliberate symmetry breaking (surface termination, strain), nanostructuring, and heterostructure assembly to maximize and tune χ³ for targeted device function.(1012.5727, Shiri, 2017)
Quantum Electrodynamics and Vacuum Effects: Although the effective χ³ for the QED vacuum is exceedingly small (∼10⁻⁴⁰ m²/V²), formal semiclassical treatments demonstrate the existence and measurable consequences of such nonlinearity for high-intensity, phase-matched four-wave mixing, connecting nonlinear optics directly to quantum field theory corrections.(1007.0083)
Tensor Metrology and Ultrafast Dynamics: Advanced all-optical, polarization- and time-resolved probes now enable direct measurement of tensor-resolved and temporally separated χ³ components, facilitating rigorously calibrated databases and improved theoretical-experimental correspondence in both 2D and 3D media.(Dremetsika et al., 2017, Chettri et al., 2021)

The third-order nonlinear optical susceptibility, χ³, thus constitutes a quantitatively precise, physically rich, and technologically vital material property, encoding the nonlinear, anisotropic, and dynamic response of matter (and, under QED, even the vacuum) to intense electromagnetic fields. Its intricate tensor structure, resonance behavior, and tunability underpin diverse applications, from scalable quantum photonics to ultrafast nonlinear optics, while continuing to pose challenges and opportunities at the interface of condensed matter, field theory, and device engineering.