Chiral Huygens Metasurfaces

• Chiral Huygens metasurfaces are engineered surfaces that combine chiral electromagnetic responses with balanced electric and magnetic dipoles to manipulate light’s phase, amplitude, chirality, and polarization.
• They utilize butterfly nanoantenna designs and twisted bilayer platforms to drive efficient nonlinear frequency conversion and perfect polarization transformations.
• Design guidelines focus on tuning material geometry, surface conductivities, and nonlinear effects to achieve high-purity vortex beams and forward-directed emission.

Chiral Huygens metasurfaces are engineered interfaces that produce highly controlled, nontrivially structured optical fields by combining chiral electromagnetic responses with balanced electric and magnetic dipole activity. These metasurfaces exploit the interplay between material geometry, surface conductivities, and nonlinear phenomena to achieve efficient manipulation of light—including phase, amplitude, chirality, and polarization transformation—at the nanoscale. Key exemplars include the butterfly nanoantenna metasurfaces for nonlinear frequency conversion and twisted atomic bilayer platforms employing magnetoelectric surface conductivities. Both regimes utilize chiral responses to enable direct conversion between input beam polarization states and output electromagnetic far-fields of arbitrary complexity, with applications demonstrated in nonlinear optics and exotic linear chiral transformations (Lesina et al., 2016, &&&1&&&).

1. Butterfly Nanoantenna-Based Chiral Huygens Metasurfaces

The butterfly nanoantenna geometry consists of two bent gold strips with uniform width $w$ and thickness $t$, joined to leave a narrow gap of size $g$. The strips' rounded ends mitigate field singularities. The rectangular footprint has a long axis $L_x$ and short axis $L_z$ ($L_x\neq L_z$), and the gap bisector in the $x$–$z$ plane makes an angle $\theta$ with respect to the $x$-axis. Key optimized parameters for a crossing mode at $\lambda_0 \approx 985.5\,\mathrm{nm}$ are $L_x=300\,\mathrm{nm}$, $L_z=200\,\mathrm{nm}$, $w=60\,\mathrm{nm}$, $t=85\,\mathrm{nm}$, $g=10\,\mathrm{nm}$; antennas are arranged on a square lattice with period $a=420\,\mathrm{nm}$.

Gold is modeled with a Drude plus two-critical-points fit and embedded in a homogeneous dielectric background (e.g., SiO$_2$ or ITO), with dielectric susceptibility $$\chi_{\mathrm{diel}}^{(3)} \gg \chi_{\mathrm{Au}}^{(3)}$$. This ensures that third harmonic generation (THG) arises from the nonlinear gap filler rather than the metal (Lesina et al., 2016).

2. Chirality, Field Enhancement, and Polarization Control

At the crossing wavelength $\lambda_0$, each butterfly nanoantenna supports two orthogonal dipolar plasmon modes that hybridize, producing constant gap-field enhancement for any incident linear polarization angle $\theta_{\mathrm{inc}}$. Full-wave FDTD analysis yields an enhancement factor $$\eta(\theta_{\mathrm{inc}}) \equiv |E_{\mathrm{gap}}(\omega_c; \theta_{\mathrm{inc}})| / |E_0| \approx \eta_0$$ for all $\theta_{\mathrm{inc}}$ (variation $< 5\%$), with $\eta_0 \approx 20$–$30$.

The phase of the gap field varies linearly with $\theta_{\mathrm{inc}}$, slope $\mp 1$, indicating left-handed (LH, $-$) or right-handed (RH, $+$) chirality. The gap field for incident linear polarization $E_0 e^{i\varphi_0}$ at angle $\theta_{\mathrm{inc}}$ takes the form $$E_{\mathrm{gap}}(\theta_{\mathrm{inc}}) = \eta_0 E_0 e^{i(\varphi_0 \mp \theta_{\mathrm{inc}})}$$, with the upper sign for LH antenna (strong LCP response) (Lesina et al., 2016).

3. Nonlinear and Linear Huygens Source Representation

Nonlinear emitters within the butterfly gap behave as idealized Huygens sources when the crossing-point condition is met. For THG at $\omega_3=3\omega$, the third-order nonlinear polarization is $$P^{(3)}(\omega_3) = \epsilon_0 \chi^{(3)}_{\mathrm{diel}} [E_{\mathrm{gap}}(\omega)]^3$$. Equivalent surface currents:

• Electric: $$J_e(\omega_3) = -i\omega_3 P^{(3)}(\omega_3)\delta_S$$
• Magnetic: $$J_m(\omega_3) = - \hat{n} \times [ E_{\mathrm{above}}(\omega_3) - E_{\mathrm{below}}(\omega_3) ]$$

Balanced electric and magnetic dipole moments ($p_e/c = m_m$) ensure emission into the forward hemisphere with suppressed backscattering (the Huygens condition) (Lesina et al., 2016).

For twisted bilayer approaches, the interface at $z=0$ is described by linear, local surface boundary conditions coupling electric, magnetic, and magnetoelectric conductivities $\sigma_e$, $\sigma_m$, $\sigma_{\mathrm{ch}}$. The Huygens metasurface form arises in the surface current expressions:

• $$J_e = \sigma_e (E_1+E_2)/2 + \sigma_{\mathrm{ch}}(H_1+H_2)/2$$
• $$M_m = \sigma_m(H_1+H_2)/2 + \sigma_{\mathrm{ch}}(E_1+E_2)/2$$

This framework enables direct control over far-field linear polarization and amplitude transformations (Zhang et al., 2021).

4. Metasurface Phase, Amplitude Engineering, and Far-Field Structuring

Far-field structuring is attained by rotating each butterfly so its gap-normal angle $\theta$ matches the local desired polarization angle. For circular metasurfaces, $\theta(x,z) = \alpha + \gamma \varphi$ in cylindrical coordinates, with $\gamma$ the number of polarization rotations per $2\pi$. The dipole phasor distributions for third-harmonic emission ($n=3$) generate a Laguerre–Gauss beam with orbital angular momentum (OAM) $\ell = \sigma [1 - \gamma + n\gamma]$, $\sigma = \pm 1$ for LCP/RCP pump.

An example with $\ell=41$ employs $\sigma=+1$ and $\gamma=20$. The corresponding azimuthal phase step is $\Delta\Phi(x,z) = \ell \varphi$. Simulations with constant amplitude $|E_d|$ yield high-purity vortex beams, with optional radial amplitude tapers to suppress sidelobes (Lesina et al., 2016).

5. Surface Conductivities and Chiral Optics in Twisted Bilayer Systems

Twisted atomic bilayers serve as atomically thin chiral Huygens metasurfaces, characterized by three surface conductivities:

• $\sigma_e$: electric, $2\times2$ tensor (S)
• $\sigma_m$: magnetic, $2\times2$ tensor (S·m$^2$)
• $\sigma_{\mathrm{ch}}$: magnetoelectric (“chiral”), $2\times2$ tensor

For isotropic, reciprocal-chiral interfaces, $$\sigma_{\mathrm{ch}} \to [\sigma_{xx}\;\;\; \sigma_{xy}; -\sigma_{xy}\;\;\; \sigma_{xx}]$$.

Reflection $(R)$ and transmission $(T)$ matrices at normal incidence are:

$$\Lambda = \frac{1}{2}(Z_0\sigma_e + Z_0^{-1}\sigma_m) + \sigma_{\mathrm{ch}}$$

$R = -[I + \Lambda]^{-1} (2\Lambda)$

$T = [I + \Lambda]^{-1} (I - \Lambda)$

Perfect $90^\circ$ polarization conversion arises for purely chiral metasurfaces ($\sigma_e=\sigma_m=0$), and the critical condition $\sigma_{xx}\sigma_{xy} = \pm 1$ at appropriate incidence angle achieves uni­ty conversion efficiency in either transmission or reflection (Zhang et al., 2021).

6. Microscopic Origins: Twist Angle and Interlayer Coupling

Nonzero magnetoelectric surface conductivity $\sigma_{\mathrm{ch}}$ emerges due to twisting and quantum interlayer tunneling, which break in-plane mirror symmetry. Within a continuum model:

• $$\sigma_e(\omega; \theta) = (e^2/4\hbar) F_e(\omega; \theta, t_\perp)$$
• $$\sigma_m(\omega; \theta) = (\mu_0 e^2 v^2/4\hbar\omega^2) F_m(\omega; \theta, t_\perp)$$
• $$\sigma_{\mathrm{ch}}(\omega; \theta) = (e^2/\hbar) F_{\mathrm{ch}}(\omega; \theta, t_\perp)$$

$$F_{\mathrm{ch}}(\omega, \theta) \propto \sin\,3\theta$$ maximizes near the “first magic” angle $\theta \approx 1.1^\circ$ (twisted graphene), allowing the tuning of $\sigma_{xx}/\sigma_{xy}$ to reach the conversion condition at desired $\omega$. $F_e$, $F_m$ are even in $\theta$, and $\sigma_e$ is suppressed when the Fermi level lies in flat bands; $\sigma_{\mathrm{ch}}$ remains finite (Zhang et al., 2021).

7. Conversion Efficiency and Simulation Parameters

Nonlinear conversion efficiency under undepleted plane-wave pumping for THG is:

$$P^{(3)}(3\omega) = \epsilon_0 \chi^{(3)}_{\mathrm{diel}} [E(\omega)]^3$$

$$\eta_{3\omega} = P_{3\omega}/P_\omega \propto |\chi^{(3)}_{\mathrm{diel}}|^2 I(\omega)^2 L_\mathrm{eff}^2 / [n(\omega)^2 n(3\omega)]$$

Large metasurface simulations used full 3D-FDTD (mesh 2 nm), with gold modeled through Drude+2CP, dielectric via Lorentz+instantaneous Kerr, radius $\sim 30\,\mu$m, $\sim 3600$ antennas, and $\gamma=20$. Evaluations were run on IBM BlueGene/Q (~32K cores, $\sim 3 \times 10^{10}$ Yee cells) with near-to-far-field transforms at $3\omega$ 10 nm above gaps to $R=30\,\mu$m.

Resultant third harmonic far-field presents a single doughnut ring (OAM $\ell=41$) and >98 % mode purity. The forward/backward emission ratio exceeds 10 dB, corroborating Huygens-source operation. Chirality-enforced selectivity leads to %%%%94$\lambda_0 \approx 985.5\,\mathrm{nm}$95%%%% weaker THG when pumped with opposite handedness (Lesina et al., 2016).

8. Design Guidelines for Chiral Huygens Metasurfaces

• Select bilayer materials supporting strong interlayer tunneling ($t_\perp$) and absent in-plane mirror symmetry (graphene, MoS$_2$, $\alpha$-MoO$_3$).
• Tune twist angle $\theta$ for maximal $|F_{\mathrm{ch}}|$ and $\left|\sigma_{xx} \sigma_{xy}\right| \approx 1$ at the operating frequency.
• Suppress $\sigma_e$, $\sigma_m$ via chemical potential or patterning to maximize chiral effects.
• Set incidence angle $\theta_i$ as $\cos \theta_i = \sqrt{|\sigma_{xx}/\sigma_{xy}|}$ to select conversion mode.
• Embed in symmetric dielectrics for simplified boundary conditions.
• Exploit weak $F_{\mathrm{ch}}$ dispersion for broadband operation or stack with varied twist angles.
• Stabilize chiral condition via electrostatic or dielectric environment engineering (Zhang et al., 2021).

Chiral Huygens metasurfaces, whether realized via nanoantenna arrays or twisted bilayer platforms, offer complete amplitude and polarization control of scattered light. The exploitation of magnetoelectric coupling underlies both nonlinear and linear regimes, facilitating vortex beam generation and perfect polarization transformation. These results are supported by simulation and theoretical frameworks, confirming precision structuring of light at the nanoscale (Lesina et al., 2016, Zhang et al., 2021).