Ultra-low-loss on-chip zero-index materials

Zero-index media support propagation without spatial phase advance, yielding perfect spatial coherence that can enable cloaking, arbitrary-shape waveguides, phase-mismatch-free nonlinear processes, large-area single-mode lasing, and extended superradiance. Conventional metal-based zero-index implementations suffer from severe ohmic losses, especially at optical frequencies. All-dielectric photonic crystal (PhC) slabs supporting an accidental Dirac-cone degeneracy between an electric monopole and a magnetic dipole mode provide an impedance-matched zero index while avoiding ohmic dissipation. In out-of-plane implementations, thickness and geometry constraints limit interaction length and shape. In-plane Dirac-cone materials are more fabrication-friendly and scalable but suffer from significant out-of-plane radiation loss through upward and downward channels, especially for the dipole mode, which lowers its quality factor. The research goal is to realize an on-chip, in-plane Dirac-cone zero-index material with ultra-low out-of-plane radiative loss by engineering destructive interference to create bound states in the continuum (BICs) in a practical, easily fabricated structure.

Prior approaches to zero-index materials include use of bulk metal volume plasmons near the plasma frequency, which incur large ohmic losses. Dielectric Dirac-cone PhC slabs achieve impedance-matched zero index via accidental degeneracy of monopole (ε→0) and dipole (μ→0) modes at the Γ point, demonstrated in both out-of-plane and in-plane configurations. However, in-plane implementations open two radiation channels (to air and substrate) due to momentum mismatch, leading to out-of-plane leakage for dipole modes. Symmetry-protected modes (e.g., monopole and higher-order modes) can have high Q due to forbidden coupling by symmetry. A recent low-loss Dirac-cone approach using symmetry-protected high-order modes via boundary effective medium theory was reported, but it relies on higher-order modes. The present work instead uses low-order (monopole and dipole) modes and resonance-trapped BICs, enabling a homogeneous effective-medium description and compatibility with conventional on-chip zero-index designs using simple pillar arrays.

Design: Start from a conventional in-plane Dirac-cone PhC slab consisting of a square array of silicon pillars embedded in silicon dioxide (Si pillars in SiO2) with pitch a = 733 nm, radius r = 180 nm, and thickness h initially around 1000 nm. The structure is chosen for ease of fabrication (low aspect ratio) using standard planar processes. Mode analysis: Consider first an infinite-height pillar array to identify axial supermodes near the zero-index frequency with kx ≥ 0, ky = 0. Six axial modes exist in the near-infrared: TM monopole, TE monopole, and two sets of TM dipole modes (related by 90° rotation). Monopole modes do not radiate out-of-plane due to symmetry; radiative coupling is dominated by dipole modes. Focusing on one polarization, two axial dipole modes remain: a higher-index waveguide-like dipole and a lower-index 2D-like dipole (with cutoff around 1615 nm where kz = 0, matching the 2D zero-index solution). For finite-height pillars, these two dipole modes couple to each other and to plane waves at the interfaces. Three-port and coupled-mode model: Model the top and bottom interfaces as partially reflective mirrors of a Fabry–Pérot cavity. Treat the system as a three-port network comprising the waveguide dipole (mode 1), the 2D-like dipole (mode 2), and the external plane wave (port 3), with complex scattering coefficients Sij between ports. Define half-round-trip phases φ1 = 2π h n1 / λ and φ2 = 2π h n2 / λ from effective indices n1, n2. Construct an effective 2×2 Hamiltonian H for resonance-trapped modes (those with negligible radiation) whose entries are functions of Sij and phases. Eigenvalues α of H describe round-trip decay; real α correspond to stationary resonances. The BIC condition corresponds to α = 1 (lossless round trip), leading to a phase-magnitude condition on Sij, φ1, φ2. Reciprocity implies |S12|^2 = (1 − |S11|)(1 − |S22|). The BIC occurs when φ1 = −arg(S11) and φ2 = −arg(S22), satisfied at discrete combinations of h and λ, yielding an infinite series of possible pillar heights; the shortest solution is near h ≈ 1085 nm. Numerical simulation and optimization: Use Lumerical FDTD to compute band structures, resonant wavelengths, and radiative Q-factors for finite pillar arrays across heights. Implement a material loss model with silicon’s imaginary refractive index set to 2.12 × 10^−5 to approximate experimental absorption. Compare full-wave results with coupled-mode theory (CMT) predictions. Identify the height h that maximizes the dipole-mode Q by enforcing destructive interference of upward and downward radiation (resonance-trapped BIC). Restore the Dirac-cone accidental degeneracy (monopole-dipole) by jointly tuning radius r and thickness h as two independent degrees of freedom within a narrow range, mapping contours where the monopole and dipole are degenerate while maintaining high Q for the dipole.

• The in-plane Dirac-cone PhC slab can host resonance-trapped dipole modes (BICs) by exploiting interference between two coupled axial dipole modes (waveguide-like and 2D-like) and the two interfaces acting as a Fabry–Pérot cavity.
• An optimal pillar height near h ≈ 1085 nm at λ ≈ 1550 nm yields a lossless round-trip condition (α = 1) for the hybrid dipole, strongly suppressing out-of-plane radiation. The corresponding dipole-mode radiative quality factor exceeds 10^5, as extracted from simulations and CMT.
• At h ≈ 1085 nm, the effective wavelength of the waveguide resonance matches the monopole resonance (λ ≈ 1550 nm), achieving accidental degeneracy and an impedance-matched zero index.
• The BIC condition requires at least two coupled axial modes; BICs cannot form if only one axial mode propagates or below the 2D-like dipole cutoff, where its out-coupling vanishes.
• Phase conditions for BICs reduce to φ1 = −arg(S11) and φ2 = −arg(S22); solutions occur at discrete (h, λ) pairs, forming an infinite series of allowed pillar heights, with the first near 1085 nm.
• There is a broad parameter region with high dipole Q for heights around ~1080 nm and pillar radii r ≈ 160–184 nm. The monopole–dipole degeneracy line is nearly horizontal in r–h space, indicating strong sensitivity to thickness and weaker dependence on radius.
• Coupled-mode theory accurately predicts resonant wavelengths and Q-factors, corroborated by full-wave FDTD simulations. Residual losses at the BIC condition arise primarily from weak silicon absorption (imaginary refractive index 2.12 × 10^−5).
• The structure uses a square array of low-aspect-ratio Si pillars in SiO2, compatible with standard planar fabrication, and behaves as an effective medium with ε ≈ 0 and μ ≈ 0.

The work directly addresses the central challenge of out-of-plane radiative loss that limits the scalability of in-plane, all-dielectric zero-index materials. By engineering interference between two axial dipole modes and the two interfaces, the design creates bound states in the continuum that suppress radiation from the dipole mode while maintaining the accidental monopole–dipole degeneracy necessary for an impedance-matched zero index. Achieving Q > 10^5 for the dipole mode at telecom wavelengths drastically reduces propagation loss and enables the large-area, arbitrarily shaped zero-index regions envisioned for applications in linear, nonlinear, and quantum optics. The low-order mode basis supports a homogeneous effective-medium description (ε = μ = 0), improving robustness and simplifying integration compared with approaches relying on higher-order symmetry-protected modes. The agreement between CMT and FDTD strengthens confidence in the physical mechanism and provides clear design rules (phase and thickness conditions) for realizing ultra-low-loss zero-index PhC slabs in practical pillar-array geometries.

The authors demonstrate a practical route to ultra-low-loss, on-chip, in-plane zero-index materials by introducing resonance-trapped bound states in the continuum within a square array of silicon pillars embedded in SiO2. Through coupled-mode analysis and full-wave simulations, they identify discrete pillar heights (notably ~1085 nm at λ ≈ 1550 nm) where destructive interference at both interfaces eliminates out-of-plane radiation from the hybrid dipole mode, yielding radiative Q-factors exceeding 10^5. By simultaneously tuning pillar radius and thickness, they restore the accidental degeneracy of monopole and dipole modes, realizing an impedance-matched zero index in a structure compatible with standard planar fabrication. This approach paves the way to exploiting perfect spatial coherence over large areas for applications in linear, nonlinear, and quantum photonics. Future work could explore experimental realizations across different wavelengths, tolerance to fabrication variations, integration with active/nonlinear components, and extension to other array and unit-cell geometries supporting monopole and dipole modes.

• The BIC mechanism requires at least two coupled axial dipole modes; it cannot be realized when only one axial mode is present or below the cutoff of the 2D-like dipole mode.
• The high-Q condition and monopole–dipole degeneracy are highly sensitive to slab thickness; small deviations from the optimal height significantly reduce Q.
• Optimizing thickness alone can break the accidental degeneracy and open a photonic bandgap; concurrent tuning of radius and thickness is needed to recover the Dirac cone.
• Residual loss remains due to material absorption in silicon, even when radiative losses are minimized.
• The provided design rules target specific wavelength regions; dispersion implies discrete height–wavelength solutions, potentially constraining bandwidth.