Omnidirectional Polarization-Independent Nonreciprocity

• Omnidirectional polarization-independent nonreciprocity is achieved by engineering multilayer and metasurface structures that enable nonreciprocal emission across broad angles and polarizations.
• Design strategies leverage magnetic Weyl semimetals, InAs layers, and self-biased vortex states to eliminate the need for external bias or complex patterning.
• Performance metrics such as the nonreciprocity index and directional dichroism validate enhanced thermal emission and transmission contrast in advanced photonic systems.

Omnidirectional polarization-independent nonreciprocity refers to optical or thermal emission processes in which the transmission, absorption, or emission of energy is nonreciprocal (direction-dependent), robust to the polarization state of the incident field (i.e., both $p$ and $s$ waves), and maintained over a broad angular range. Such functionality surpasses conventional magneto-optical (MO) and metamaterial-based nonreciprocal devices, which generally exhibit pronounced polarization or angular dependence and rely on external biasing or complex patterning. Recent advances have established technical platforms enabling broadband, bias-free, and pattern-free realization of this effect using multilayer heterostructures or metasurfaces with intrinsic or synthetic time-reversal symmetry breaking (Do et al., 30 Dec 2025, Máñez-Espina et al., 15 Oct 2025).

1. Theoretical Formalism for Nonreciprocal Emission

A rigorous description of nonreciprocity in optical and thermal systems invokes Maxwell’s equations incorporating anisotropic and bianisotropic response tensors. In static MO media with magnetization $\mathbf{M}$, the permittivity tensor $\boldsymbol{\varepsilon}(\omega,\mathbf{M})$ encodes off-diagonal terms responsible for nonreciprocity. For a Voigt configuration, the tensorized form includes: $$\boldsymbol{\varepsilon} = \begin{pmatrix} \varepsilon_{xx} & 0 & \varepsilon_{xz} \ 0 & \varepsilon_{yy} & 0 \ -\varepsilon_{xz} & 0 & \varepsilon_{zz} \end{pmatrix}$$ with Drude–Lorentz expressions for $\varepsilon_{xx}$ and $\varepsilon_{xz}$ incorporating the cyclotron frequency $\omega_c$. In Weyl semimetals, similar tensor forms emerge, with $\omega_c$ replaced by a topologically induced parameter. The corresponding boundary-value problem solved via a $4 \times 4$ scattering matrix yields nonreciprocal reflection and transmission coefficients for both $p$ and $s$ polarizations.

In bianisotropic metasurfaces, the Maxwell-Ampère and Maxwell-Faraday laws are further generalized by magnetoelectric tensors $\xi$ and $\zeta$, as in: $$\mathbf{D} = \hat{\varepsilon} \mathbf{E} + \hat{\xi} \mathbf{H},\quad \mathbf{B} = \hat{\mu} \mathbf{H} + \hat{\zeta} \mathbf{E}$$ where $\hat{\xi}$ and $\hat{\zeta}$ are antisymmetric and critically responsible for nonreciprocal, synthetic-motion effects that mimic moving media (Máñez-Espina et al., 15 Oct 2025).

2. Pattern-Free Multilayer and Synthetic-Motion Metasurface Designs

Polarization-independent and angularly robust nonreciprocity is implemented via two principal mechanisms:

A. Multilayer Heterostructures:

Stacks of InAs (magneto-optical semiconductor) and magnetic Weyl semimetals on Ag are configured with each InAs layer ($d_n \approx 1.2\,\mu$m, $n_{e,n} = 3.5$–$5.5 \times 10^{17}\,\mathrm{cm}^{-3}$) and thinner Weyl layers ($d_n = 200$ nm, $E_{F,n}=0.05$ eV). Magnetization directions $\{\varphi_n\}$ of consecutive layers are controlled to disrupt simple vector summation and enable constructive nonreciprocal phase accumulation over the stack. Cross-polarization conversion and multi-layer interference ensure the nonreciprocity persists for both $p$ and $s$ polarizations across all incidence angles (Do et al., 30 Dec 2025).

B. Metasurfaces via Synthetic Motion:

Arrays of ferrite nanodisks (diameter $D$, height $h$) in self-magnetized vortex states (no external bias) serve as meta-atoms. Their symmetry-protected quasi-bound states in the continuum (quasi-BICs) are engineered for strong electric and magnetic dipolar resonances, hybridized through antisymmetric magnetoelectric coupling $$\alpha_m \propto \varepsilon_a \langle M(\mathbf{r}) \rangle$$. Stamp-assisted vortex writing offers deterministic, uniform control of vortex configuration over large areas (Máñez-Espina et al., 15 Oct 2025).

3. Performance Metrics and Analytical Constraints

Two central metrics quantify omnidirectional, polarization-independent nonreciprocity:

Nonreciprocity Index (NRI):

$$\mathrm{NRI} = \min_{\theta \in [0, \pi/2],\,\phi \in [0,2\pi]} \left[ \eta_{\rm unpol}^+(\theta,\phi) - \eta_{\rm unpol}^-(\theta,\phi) \right],\quad \eta_{\rm unpol} = \frac{\eta_p + \eta_s}{2}$$

where a positive NRI indicates the directional emission of thermal radiation (emissivity exceeds absorptivity in all directions).

Directional Dichroism for Unpolarized Light:

$\Delta T = T_{+} - T_{-}$

with transmission functions $T_{\pm}$ for forward/backward incidence. At resonance and under critical coupling and strong inter-modal coupling ($g \gg \gamma$), $\Delta T \to 1$ is possible, corresponding to near-unity nonreciprocal transmittance contrast for unpolarized light.

Emissivity and absorptivity are nontrivially related: $$\varepsilon_{+} - \alpha_{+} = \varepsilon_{-} - \alpha_{-} = \Delta T$$ showing maximal nonreciprocal heat flow coincides with maximal transmittance contrast (Do et al., 30 Dec 2025, Máñez-Espina et al., 15 Oct 2025).

4. Optimization Strategies

Pareto-Optimal Multilayer Design:

Magnetization directions $\boldsymbol{\varphi}$ in multilayer stacks are treated as multi-objective optimization variables, targeting simultaneous maximization of spectrally integrated $\eta_p$ and $\eta_s$. The non-dominated sorting genetic algorithm II (NSGA-II) is used:

• Initialize populations of $\{\boldsymbol{\varphi}\}$.
• Evaluate objective functions through fast scattering-matrix calculations.
• Evolve populations via selection, crossover, and mutation until reaching a well-distributed Pareto front. This approach enables selection of designs with desired tradeoffs between $p$- and $s$-wave nonreciprocity, optimizing the unpolarized NRI (Do et al., 30 Dec 2025).

Metasurface Resonance Engineering:

Maximal directional dichroism is achieved by co-locating electric and magnetic dipole resonances (Huygens condition), ensuring critical coupling ($\gamma_e = \gamma_i$), and maximizing nonreciprocal dipole hybridization $g \gg \gamma$. Quality factor scaling, lattice constant, and nanodisk geometry are tuned for the target wavelength, with critical scaling set by material parameters (e.g., $\varepsilon_a$ of ferrite, magnetization $M_s$, etc.) (Máñez-Espina et al., 15 Oct 2025).

5. Representative Numerical Results

• In optimized InAs+Weyl multilayer stacks, dual-polarization nonreciprocity $\Delta r_p(\theta) \gtrsim 20\%$ for $\theta \in [0^\circ,85^\circ]$, and $\Delta r_s(\theta) \gtrsim 0.5\%$.
• Broad, multiband enhancement is attained near the ENZ points of layers (5–40 μm).
• Angular robustness: For metasurfaces, full-wave calculations verify $\Delta T(\theta) \geq 0.8$ for $|\theta| \lesssim 15^\circ$, with an effective angular bandwidth of $\Delta \theta \approx \pm 25^\circ$.
• Material specifics: For Bi$_3$Fe$_5$O$_{12}$, $\varepsilon_r=8.077-0.016i$, $\varepsilon_a=0.0733-0.007i$ at $\lambda=650$ nm. For YIG, $\varepsilon_r=4.84-6.6\times10^{-7}i$, $\varepsilon_a=4\times10^{-4}$ at $\lambda=1.27\,\mu$m. Disk diameters $D=0.62a$, with lattice period $a=435$ nm (BIG) or $850$ nm (YIG).

6. Design Guidelines and Practical Implementation

Multilayer Stacks:

• Gradient-doped InAs layers ensure a spread of ENZ points over the operational band.
• Weyl semimetals are interleaved to induce intrinsic, magnet-free nonreciprocity.
• Layer-by-layer rotation of magnetization (difference of tens of degrees between adjacent layers) enhances omnidirectionality and polarization independence via phase-matched accumulation.
• Target material parameters: $\varepsilon_{xx} \to 0$ (for ENZ effect), maximal $\kappa = \varepsilon_{xz}/\varepsilon_{xx}$.

Metasurfaces:

• Self-biased vortex magnetization in nanodisks eliminates the need for external bias.
• Symmetry-protected quasi-BICs facilitate high-$Q$ resonances for dual-polarization operation.
• Deterministic, large-area patterning of the vortex state is implemented by a stamp-assisted nucleation protocol using cobalt nanobar arrays in external fields, yielding persistent and uniform vortex configurations.

A direct implication is that omnidirectional, polarization-independent nonreciprocity is no longer constrained by the need for complex magnetic circuitry, lithographically patterned metamaterials, or restricted to a single linear polarization. This paves the way for scalable, tailorable nonreciprocal thermal emitters, compact nonreciprocal photonic devices, and robust radiative control in advanced energy harvesting and photonic engineering contexts (Do et al., 30 Dec 2025, Máñez-Espina et al., 15 Oct 2025).