SHG in Nonlinear Flat-Optics

• Second-harmonic generation (SHG) in nonlinear flat-optics is a frequency-doubling process using engineered quadratic nonlinearity in metasurfaces, ultra-thin films, and quantum wells.
• Innovative modal and resonance engineering, including mechanisms like Mie and Fano resonances, enhances local field strength and overcomes phase-matching limitations.
• Emerging material platforms and electrical modulation techniques enable tunable, on-chip SHG devices for applications in frequency conversion, polarimetry, and integrated photonics.

Second-harmonic generation (SHG) in nonlinear flat-optics refers to the frequency-doubling process whereby photons at a fundamental frequency ω are converted, via a material’s quadratic (χ^2) nonlinearity, into photons at the second-harmonic frequency 2ω using planar architectures such as metasurfaces, ultra-thin films, quantum wells, waveguides, and nanoantenna arrays. Flat-optical systems leverage strong electromagnetic confinement, engineered symmetry breaking, and modal resonance phenomena to achieve SHG in platforms that are integrable with photonic circuits and compatible with CMOS processes. Recent advances have circumvented bulk phase-matching limitations, extended the material base (including dielectrics, metals, 2D and quantum-confined semiconductors), and realized functionalities such as tunable conversion, electrical gating, polarization control, and on-chip parametric processes.

1. Physical Principles and Theoretical Framework

SHG arises from the electric-dipole second-order nonlinear polarization:

$$P_i^{(2)}(2\omega)=\varepsilon_0\sum_{j,k}\chi^{(2)}_{ijk}E_j(\omega)E_k(\omega)$$

where $\chi^{(2)}_{ijk}$ is the second-order susceptibility tensor and $E_j(\omega)$ are the local fields. In most metals and centrosymmetric semiconductors, bulk $\chi^{(2)}$ vanishes, restricting SHG to surface/interface contributions or engineered symmetry breaking. For non-centrosymmetric dielectrics (e.g., LiNbO₃, AlGaAs), bulk $\chi^{(2)}$ is strong and tensor elements can be harnessed via modal engineering and grating geometries (Ma et al., 2020, Menshikov et al., 1 Dec 2025).

Flat-optical systems exploit subwavelength thicknesses, tailored resonances, and field enhancements to concentrate $E(\omega)$ and maximize the product $P^{(2)}\sim\chi^{(2)}|E(\omega)|^2$, typically circumventing classical phase-matching since coherence lengths exceed structure thickness (Marino et al., 2015, Ma et al., 2020).

Modal phase matching (MPM), double-resonance enhancement, and critical/fano mode coupling serve as alternative mechanisms for achieving high SHG conversion, with output scaling as:

$$\eta_{\rm SHG} = \frac{P_{2\omega}}{P_\omega^2} \propto |\chi^{(2)}|^2 \cdot |E_{\rm loc}|^4 \cdot Q_{2\omega}$$

where $Q_{2\omega}$ is the second-harmonic quality factor, and $|E_{\rm loc}|$ is the local field enhancement factor.

2. Material Platforms and Symmetry Considerations

Centrosymmetric Media and Photo-induced SHG

Silicon, Si₃N₄ and Ge are intrinsically centrosymmetric ($\chi^{(2)}=0$ bulk); SHG can be induced via static DC fields (photogalvanic effect), interface strain, or low-dimensional symmetry breaking (Lu et al., 2020, Frigerio et al., 2021). In Si₃N₄ resonators, a photo-induced DC field $E_{DC}$ transforms intrinsic $\chi^{(3)}$ into an effective $\chi^{(2)}_F = \chi^{(3)} E_{DC}$, enabling CW SHG with efficiency $\sim 2,500$ %/W and $\sim 22\%$ absolute conversion (Lu et al., 2020).

Quantum Wells and 2D Materials

Artificial symmetry breaking in asymmetric coupled quantum wells (ACQW) of Ge/SiGe delivers giant $\chi^{(2)}$ up to $10^5$ pm/V via double-resonant intersubband transitions, with equidistant confined levels and strong quantum dipoles (Frigerio et al., 2021). In tetralayer graphene, ABCB stacking order (C₃ point group) produces pronounced SHG with six-fold azimuthal anisotropy and a sheet susceptibility $\sim2.5\times10^3$ pm/V, acting as an elemental ferroelectric flat-optic (Zhou et al., 2023).

Dielectric and Chalcogenide Metasurfaces

High-$\chi^{(2)}$ dielectrics (LiNbO₃, AlGaAs, chalcogenide glasses) allow flat-optic SHG by leveraging intrinsic tensor components (e.g., $d_{33}$ in LN, $d_{36}$ in AlGaAs, surface/interface-induced $\chi^{(2)}$ in Se), which can be enhanced via geometry and orientation (Ma et al., 2020, Gupta et al., 2021, Menshikov et al., 1 Dec 2025). Engineering Q-factors, mode volume, and resonance coupling yields strong SHG up to $10^{-6}$ W$^{-1}$ and pronounced nonlinear dichroism.

Metal Surfaces and Plasmonics

Plasmonic surfaces (e.g., monocrystalline Cu(111), Au nanorods) enable SHG via surface $\chi^{(2)}$, anisotropy, and localized field enhancements (Dayi et al., 4 Aug 2025, Xiong et al., 2016). Hyperbolic nanorod metamaterials leverage modal overlap and high-k mode density for broadband, phase-matching-free SHG exceeding planar films by $10^2$ (Marino et al., 2015).

3. Resonance Engineering and Device Architectures

Mie and Fano Resonances

Nanoridge arrays in LN offer tunable Mie resonances; s-polarized ($E_\perp$) input excites magnetic-dipole modes, elevating SHG at controlled wavelengths (e.g., $400$ nm at $2.0\times10^{-6}$ efficiency for $600$ nm period) (Ma et al., 2020). Fano resonances in symmetry-broken metasurfaces (e.g., L-shaped Si meta-atoms with GaSe) generate strong local fields, facilitating nonlinear coupling with 2D materials; CW-pumped SHG enhances by $10^2$ over bare Si (Yuan et al., 2019).

Mode Coupling in Chalcogenide Systems

Amorphous Se metasurfaces use coupled particle-lattice resonances. The Rayleigh anomaly ($\lambda_{RA}=P n_{sub}$) and Mie magnetic modes hybridize, producing high-Q Fano resonances with maximal $|E_{loc}|^2$ scaling as $|P - P_c|^{-2}$, delivering $\sim10^4$ performance increase over films (Gupta et al., 2021).

Plasmonic Nanoantennas and Hyperbolic Slabs

Surface nonlinearity in MR-NA and PIC-NA designs yields double-resonant field overlap and directional SH emission tunable via geometry and gap asymmetry (Xiong et al., 2016). Hyperbolic slabs amplify SHG by two orders due to enhanced modal overlap, subwavelength thickness, and the absence of bulk phase-matching (Marino et al., 2015).

4. Electrical, Temporal, and Polarization Control

Parametric Modulation

Time-varying metasurfaces (a-Si nanobar arrays) achieve electrical gating of SHG via heterodyne interference between pump and DC fields, producing a dynamic phase modulation governed by the Mathieu equation:

$$\frac{d^2a}{dt^2}+[\omega_0^2+\delta\cos(\Omega t)]a=F e^{i\omega t}$$

Resulting SHG scales super-quadratically ($I_{2\omega}\propto E_{dc}^{2.1-2.5}$) with an on/off ratio $>15,000\times$ and modulation depth of $4.05\times10^4$ %/V (Guo et al., 2020).

Polarimetric Imaging and Dichroism

AlGaAs grating metasurfaces (d$_{36}\approx$ 188 pm/V) form superpixels with four orientation-twisted units, enabling full Stokes vector retrieval via polarization-selective SHG with nonlinear circular dichroism up to $0.98$ at $1.55\,\mu$m (Menshikov et al., 1 Dec 2025). Crystal symmetry controls tensor coupling, with C$_{3v}$, C$_3$, or C$_{2v}$ configurations dictating the angular SH emission pattern (Zhou et al., 2023, Dayi et al., 4 Aug 2025).

5. Conversion Efficiency, Performance, and Applications

Representative SHG Conversion Efficiencies

Platform/Material	Peak $\eta_{\rm SHG}$	Wavelength Regime
Si$_3$N$_4$ microring	$2,500$ %/W (norm.), $22\%$ abs.	1560/780 nm (TE modes)
Asymmetric Ge/SiGe QW	$10^5$ pm/V (χ$^{(2)}$)	9-12$\,\mu$m (mid-IR)
LN metasurface (grating)	$2.0\times10^{-6}$	400–500 nm
Chalcogenide metasurface	$1\times10^{-6}$	680–780 nm
Monocryst. Cu(111) flake	Not reported (modulation depth)	515 nm (cross-pol.)
Time-varying a-Si metasurface	$>15,000\times$ enhancement	770 nm
Hyperbolic Au nanorod slab	$10^2\times$ over film	400–1,400 nm
Si metasurface + GaSe	$2.5\times10^{-5}$ W$^{-1}$	771.9 nm

Efficiencies and nonlinearities vary widely according to material platform, resonance engineering, symmetry control, and excitation conditions. Dominant optimization levers are mode overlap, Q-factor, local field enhancement, symmetry breaking, and active modulation.

Application Domains

• On-chip frequency doublers for telecom bands, mid-IR, or visible (Frigerio et al., 2021, Lu et al., 2020).
• Polarimetric imaging in NIR via polarization-sensitive SHG (Menshikov et al., 1 Dec 2025).
• Electrically tunable, ultrafast modulators, all-optical switches, dynamic wavefront engineering (Guo et al., 2020).
• Nonlinear holography and compact nonlinear light sources (Gupta et al., 2021, Ma et al., 2020).
• Surface-specific or domain-engineered diagnostics (crystalline orientation mapping, catalysis) (Zhou et al., 2023, Dayi et al., 4 Aug 2025).

6. Integration, Scalability, and Future Directions

Platforms explored span wafer-scale integration (CVD-grown 2D/quantum well materials, thin-film dielectrics), electrically and optically switchable metasurfaces, and flexible substrates for wearables. Modal phase matching, double resonance, and time/electrical modulation are emerging paradigms for device scaling and tunability.

Future avenues include:

• Extension to higher-order nonlinearities (THG, FWM)
• Active materials integration: phase-change, ferroelectric, 2D Janus compounds
• Dynamic and reconfigurable metasurfaces with electrical or optical phase encoding
• Integration into CMOS-compatible photonic ICs for signal processing, clocking, and frequency comb generation
• Flat-optical quantum light sources and entanglement platforms

Flat-optics SHG architectures provide a unifying framework combining material science, symmetry engineering, modal optics, and integrated photonics, delivering functionalities beyond classical bulk nonlinear optics.