Nonlinear Silicon Metasurfaces

• Nonlinear silicon metasurfaces are planar nanophotonic structures that use resonant enhancements (Mie, Fano, and BICs) to confine light and boost weak third-order nonlinear effects.
• They exploit intrinsic nonlinearities, such as third-harmonic generation and Kerr effects, to achieve ultrafast optical switching, beam shaping, and dynamic wavefront control.
• Advanced designs incorporating anisotropy and geometric phase engineering enable functionalities like holography, polarization multiplexing, and nonreciprocity with high conversion efficiencies.

Nonlinear silicon metasurfaces are planar nanophotonic architectures composed of silicon nanoresonators that exploit resonant field confinement and the material’s intrinsic nonlinearities to enable efficient and ultrafast manipulation of light in amplitude, phase, polarization, and frequency. These metasurfaces leverage both geometric and symmetry-induced resonance effects—such as Mie-type resonances, Fano resonances, bound states in the continuum (BICs), and quasi-BICs—to dramatically enhance third-order nonlinear processes (e.g., third-harmonic generation, Kerr-induced refractive index changes), facilitate dynamic and anisotropic wavefront control, and enable functionalities including nonreciprocity, beam shaping, holography, ultrafast switching, and frequency conversion at subwavelength scales.

1. Resonant Enhancement Mechanisms: Mie, Fano, and Bound States in the Continuum

Nonlinear silicon metasurfaces rely on resonant field enhancement mechanisms that originate from the structure and symmetry of subwavelength dielectric resonators:

• Mie-Type Resonances: High-index a‑Si:H or crystalline silicon nanobricks or nanodisks (dimensions typically 100–500 nm) can support strong electric and magnetic Mie resonances, leading to strong field localization and enhancement at well-defined wavelengths. The spectral position and nature of these resonances are highly dependent on the aspect ratio, material dispersion, and symmetry of the nanostructure (Valle et al., 2017).
• Fano Resonances: In arrays with engineered asymmetry (e.g., L-shaped or double-bar elements), interference between broad and narrow resonances (bright and dark modes) creates sharp spectral features (Fano resonances) with high Q-factors and strong near-field intensities. The resonance profile is well described by:

$$f_{\text{Fano}}(\omega) = A_0 + F_0 \frac{(q+2(\omega-\omega_0)/\Gamma)^2}{1+(2(\omega-\omega_0)/\Gamma)^2}$$

where $q$ is the Fano factor, $\omega_0$ the resonance frequency, and $\Gamma$ the linewidth (Yuan et al., 2019).

• (Quasi-)Bound States in the Continuum (BICs/quasi-BICs): Geometric symmetry can protect certain optical modes from radiative losses (BICs). Symmetry breaking (via an asymmetry parameter $\alpha$) converts a true BIC to a quasi-BIC with a tunable, high but finite Q-factor ($Q \propto \alpha^{-2}$ for small asymmetry), maximizing local field enhancement for frequency conversion (Koshelev et al., 2019, Gandolfi et al., 2021, Liu et al., 29 Aug 2025).

These mechanisms enable sharp spectral confinement, field enhancement, and control that are critical for boosting weak nonlinear interactions in silicon, which is otherwise limited by material centrosymmetry to third-order (or higher) processes.

2. Dominant Nonlinear Processes and Quantitative Models

The physics of nonlinear silicon metasurfaces is governed by a combination of instantaneous and relaxation-mediated third-order processes, and in hybrid cases, by engineered second-order nonlinearities:

• Two-Photon Absorption (TPA): Instantaneous absorption of two photons modulates the imaginary part of permittivity, broadening the resonance and providing a fast (sub-picosecond) nonlinear response:

$$\Delta\epsilon_{\text{TPA}}(t) = i \frac{c n'(\nu_p)}{2\pi\nu_p} \beta_{\text{TPA}}^{\text{(eff)}} I(t)$$

where $\beta_{\text{TPA}}^{\text{(eff)}}$ is the effective coefficient, $I(t)$ pump intensity (Valle et al., 2017).

• Free-Carrier Generation and Dynamics: Pump-induced free carriers modify refractive index via the Drude response:

$$\Delta\epsilon'_{\text{FC}}(\lambda, t) = -\frac{N(t) e^2}{m^*\epsilon_0 [4\pi^2c^2\lambda^{-2}+\tau_d^{-2}]}$$

$$\Delta\epsilon''_{\text{FC}}(\lambda, t) = -\frac{\lambda \Delta\epsilon'_{\text{FC}}(\lambda, t)}{2\pi c \tau_d}$$

inducing resonance shifts and enabling sub-10 ps modulation (Valle et al., 2017).

• Lattice Heating (Thermo-Optic Effect): Nonradiative recombination leads to heating and induces a slowly relaxing (10 ps–ns) refractive index change,

$$\Delta\epsilon'_L(\lambda, t) = 2 [n'(\lambda)\eta_1 - n''(\lambda)\eta_2] \Delta\Theta(t)$$

red-shifting resonances and limiting recovery time (Valle et al., 2017).

• Third Harmonic Generation (THG): Nonlinear polarization is driven by the cubic term:

$$P^{(3)}(3\omega, \vec{r}) = \varepsilon_0 \chi^{(3)} [E(\omega, \vec{r})]^3$$

with efficiency scaling as $|E_{\text{loc}}|^6$ in resonant enhancement regimes. In bound state-enhanced metasurfaces the THG signal maximizes at the critical coupling point for quasi-BICs (Koshelev et al., 2019, Koshelev et al., 2022, Sedeh et al., 8 Jul 2025).

• Kerr Nonlinearity and Nonreciprocal Transmission: The intensity-dependent Kerr effect is described by

$\epsilon = \epsilon_\ell + \chi^{(3)} |E|^2$

enabling functionalities such as self-induced nonreciprocity without external bias (Jin et al., 2020, Cotrufo et al., 2022).

• Hybrid and Cascaded Second-Order Processes: Integration with 2D materials (e.g., GaSe) introduces strong second-order effects (SHG, SFG), exploiting high-$\chi^{(2)}$ and strong local field enhancement under resonance (Yuan et al., 2019).

3. Anisotropy, Symmetry Breaking, and Chiral Nonlinear Response

• Anisotropic Nanobricks and Dynamic Polarization Control: Shape anisotropy in silicon nanobricks leads to polarization-dependent resonances. By proper spectral/polarization engineering, one can achieve independent control of ultrafast modulation for orthogonal polarization components, realizing ultrafast polarization rotators and waveplate functions (Valle et al., 2017).
• Chirality and Circular Dichroism: Nonlinear chiroptical effects—specifically third-harmonic circular dichroism—are realized by symmetry engineering (e.g., asymmetric meta-atoms, broken in-plane or out-of-plane symmetry), allowing near-unity nonlinear CD ($|CD^{(3\omega)}| > 0.9$) and independent tuning of linear and nonlinear chiral response (Gandolfi et al., 2021, Koshelev et al., 2022, Tonkaev et al., 18 Aug 2025).

Symmetry Type	Linear CD	Nonlinear CD (THG)	Example Paper
C$_4$ unperturbed	~0	Strong, cross-pol.	(Tonkaev et al., 18 Aug 2025)
Broken in-plane	Small	Large, sign-tunable	(Tonkaev et al., 18 Aug 2025)
Asymmetric (q-BIC)	Designable	Near-unity, spin-sel.	(Gandolfi et al., 2021)

• Geometric Phase and PB Theme: Rotational asymmetry in meta-atom orientation governs nonlinear geometric phase accumulation via Pancharatnam-Berry (PB) mechanisms. For THG,

$$\theta_{\text{co}} = 2\sigma\phi \quad ; \quad \theta_{\text{cross}} = 4\sigma\phi$$

allows for pixel-level phase control in wavefront engineering and holographic multiplexing (Reineke et al., 2019, Liu et al., 2020, Sedeh et al., 8 Jul 2025).

4. Device Architectures and Experimental Demonstrations

Several metasurface architectures have been realized:

• Anisotropic Arrays: Amorphous silicon nanobricks or nanofins in 2D arrays, tailored for ultrafast, anisotropy-controlled modulation (Valle et al., 2017).
• Symmetry-Engineered BIC/quasi-BIC Metasurfaces: Arrays of pairs of bars/disks with controlled geometric asymmetry ($\alpha$) to realize high-Q quasi-BICs for THG maximization and nonlinear chirality (Koshelev et al., 2019, Gandolfi et al., 2021, Liu et al., 29 Aug 2025).
• Nonlocal Metasurfaces: Topologically asymmetric (e.g., off-center-hole or perturbed) meta-atoms combine nonlocal field confinement with geometric phase manipulation—enabling simultaneous high-efficiency THG and wavefront engineering (Sedeh et al., 8 Jul 2025).
• Hybrid Si/2D-Material Platforms: Integration of monolayer GaSe or graphene for enhanced χ^2 processes, efficient SHG, SFG, and electrical/optical switching (Yuan et al., 2019, Feinstein et al., 23 Jul 2024).
• Nonreciprocal Devices: Vertical and in-plane asymmetry in bifacial/multilayer metasurfaces (e.g., Fano + Lorentz stacks) providing self-referenced, bias-free nonreciprocal transmission, with >10 dB contrast and low insertion loss (Jin et al., 2020, Cotrufo et al., 2022).
• Imaging Devices for Nonlinear Upconversion: Quasi-BIC metasurfaces for infrared-to-visible upconversion imaging with high spatial resolution ($\sim$6 μm) and conversion efficiency ($\sim$3 × 10⁻⁵ at 10 GW/cm²) (Liu et al., 29 Aug 2025).

5. Nonlinear Wavefront Control, Holography, and Multiplexing

Nonlinear silicon metasurfaces provide versatile wavefront control of frequency-converted light:

• Wavefront Engineering: Nanofin/nanodisk rotation enables geometric phase control of THG, achieving full $2\pi$ phase manipulation and enabling beam deflection, vortex beam generation, and arbitrary phase mask encoding (Reineke et al., 2019, Hähnel et al., 2022, Sedeh et al., 8 Jul 2025).
• Holography and Polarization Multiplexing: Polarization-selective channels (co-polarized, cross-polarized, or chiral states) allow for robust holographic image reconstruction in the THG band, as well as multiplexing by polarization switching (Reineke et al., 2019, Liu et al., 2020).
• Uniform Amplitude and Phase Sampling: Monte Carlo and sampling-based design approaches optimize meta-atom dimensions for uniform THG amplitude and full phase coverage, maximizing nonlinear hologram quality and intensity in the principal diffraction order (amplification up to $\sim$500× relative to previous work) (Hähnel et al., 2022).

6. Functional Implications, Applications, and Limitations

Nonlinear silicon metasurfaces provide a platform for a range of advanced nanophotonic applications:

• Ultrafast Optical Switching: Sub-20 ps all-optical modulation by exploiting the balance of TPA, free-carrier, and lattice heating processes (Valle et al., 2017).
• Nonreciprocal and Logic Devices: Self-induced nonreciprocity, all-optical diodes, optically switchable isolators for integrated photonics, enabled via Kerr nonlinearity and asymmetric resonances (Jin et al., 2020, Cotrufo et al., 2022).
• Nonlinear Sensing and Imaging: High-resolution IR-to-visible upconversion imaging, highly sensitive to pump intensity and spatial patterning (Liu et al., 29 Aug 2025). Applications extend to night vision, biomedical sensing, and IR security imaging.
• Chiral and Quantum Photonics: The ability to engineer nonlinear chiral responses while maintaining linear achirality opens avenues for advanced chiral sensors and secure communications (Tonkaev et al., 18 Aug 2025). Metasurface-enhanced harmonic generation is also relevant for on-chip quantum light sources (Gennaro et al., 2022).
• Design Limitations and Modeling Considerations: For highly resonant and structured metasurfaces (e.g., PhC nanocavity arrays), homogenization approaches are only qualitatively accurate, especially near sharp resonances where field inhomogeneity dominates. Full-wave numerical simulation is generally required for quantitative prediction and device optimization (Ren et al., 2019).

7. Mathematical Tools and Critical Coupling Principles

Key theoretical frameworks and formulas employed include:

• TCMT for Quasi-BICs: The resonance Q-factor, field enhancement, and optimal nonlinear conversion are governed by critical coupling. Maximum THG efficiency is achieved when radiative and nonradiative loss rates are equal:

$$Q_r = Q_{\text{nr}} \implies \alpha_{\text{cr}} = \sqrt{Q_0/Q_{\text{nr}}}$$

and

$$P_{3\omega} \propto [\alpha Q(\alpha)]^6 P_\omega^3$$

for THG (Koshelev et al., 2019, Koshelev et al., 2022).

• Wavefront Control via PB Phase:

$\phi_{\text{PB},n} = \sigma (n \pm 1) \phi$

for n-th harmonic and incident helicity $\sigma$ (Reineke et al., 2019).

• Nonlinear Circular Dichroism:

$$\text{CD}^{(3\omega)} = \frac{I_{\text{RCP}} - I_{\text{LCP}}}{I_{\text{RCP}} + I_{\text{LCP}}}$$

(measured for the third-harmonic signal) (Tonkaev et al., 18 Aug 2025, Koshelev et al., 2022).

• THG Conversion Efficiency:

$$\eta = \frac{P_{\text{THG}}^{\text{avg}}}{(P_{\text{Pump}}^{\text{avg}})^3}$$

with a normalized version independent of pump power for device benchmarking (Liu et al., 29 Aug 2025).

These expressions form the basis for quantitative design, modeling, and benchmarking of nonlinear silicon metasurface performance.

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Nonlinear silicon metasurfaces, by virtue of their resonance-driven field enhancement and versatility in symmetry and geometric phase engineering, deliver ultrafast, low-loss, and efficient control over nonlinear light-matter interactions. This capability, together with the maturity of silicon nanofabrication, positions them as leading candidates for next-generation ultrafast photonic, optoelectronic, sensing, imaging, and quantum devices.