Singular value decomposition to describe bound states in the continuum in periodic metasurfaces

TL;DR

This work tackles the challenge of understanding and engineering bound states in the continuum (BICs) in periodic metasurfaces by developing a unified singular-value-decomposition (SVD) framework for two key representations: the effective T-matrix in a multipole basis and the S-matrix in a plane-wave basis. The authors derive general BIC criteria in terms of the largest singular value and its inverse, $\sigma_1$ and $\xi_1$, plus orthogonality conditions $\langle \Psi_{\rm out} | \mathbf{u}_1 \rangle = 0$ and $h^1 = f^1 = 0$, capturing both symmetry-protected and accidental BICs and their quasi-BIC transitions under perturbations. They validate the approach across diverse electromagnetic and acoustic metasurfaces, including infinite and finite arrays, substrates, degenerate BICs, and Fabry-Perot configurations, demonstrating how the BIC position and radiative losses are tracked by the behavior of the leading singular values and vectors. The framework provides a practical, scalable route to design high-Q resonances by controlling incidence and structural parameters, enabling targeted control of radiative properties in complex metastructures. Overall, the SVD-based diagnostic bridges multipole and plane-wave descriptions, offering a versatile toolkit for diagnosing and engineering BICs and quasi-BICs in advanced wave systems.

Abstract

Understanding how bound states in the continuum (BICs) emerge in periodic metasurfaces is essential for the controlled design of high-Q resonances and their systematic manipulation. Here, we investigate the singular value decomposition (SVD) of the effective transition matrix and the scattering matrix of periodic metasurfaces within a parameter range where the metasurface sustains a BIC. Our analysis yields general and practically applicable conditions on the singular values and singular vectors that enable BIC formation. At the BIC eigenfrequency, the inverse of the largest singular value of both matrices vanishes, and the corresponding left (right) singular vector is orthogonal to outgoing (incoming) plane waves that propagate in the directions of open diffraction orders. Our SVD-based approach predicts the spectral position of the BIC and provides detailed information about its properties, including the expansion coefficients in the multipole and plane-wave bases, as well as its behavior under perturbations that transform the BIC into a quasi-BIC. The approach is numerically validated by considering both symmetry-protected and accidental BICs in arrays of scatterers supporting electromagnetic or acoustic multipole resonances. The presented SVD framework offers a broadly applicable foundation for engineering BICs and quasi-BICs in complex metasurfaces, potentially enabling new routes for wave-based devices with tailored radiative properties.

Figures (7)

• Figure 1: Schematic illustration of the notation and approaches. (a) The effective T-matrix describes the acoustic or optical response of a unit cell $\lvert\Psi_{\rm sca}\rangle$ to an incident plane wave $\lvert\Psi_{\rm inc}^d\rangle$ (depicted for $d = \uparrow$), accounting for interactions among unit cells in a spherical or cylindrical multipolar basis. (b) The S-matrix links outgoing and incoming plane waves (diffraction orders), while the description is valid for $|z| > z_c/2$, where $z_c$ is the thickness of the metasurface.
• Figure 2: Electromagnetic S-BIC in a single metasurface of spherical particles in free space. (a) Reflectance of the metasurface [for $\ell_{\rm max} = 4$] as a function of the angle of incidence of a TE polarized plane wave and the wavelength detuning $\Delta \lambda = \lambda - \lambda_{\rm R}$, where $\lambda_{\rm R} = 1459.16$ nm is the BIC wavelength. The lattice constant is $L = 1037.78$ nm. The inset in panel (a) depicts the metasurface and the illumination conditions. The inverse of the largest singular values of the (b) effective T-matrix [$\ell_{\rm max} = 1$] and (c) S-matrix [$\left| \mathbf{G}_{n_x, n_y}\right|_{\rm max} = 2\pi/L$]. (d) Absolute values of scalar products of the singular vectors of $\mathbf{T}_{\rm eff}(\mathbf{k}_{\parallel}, \omega)$ and plane waves: $|\braket{\Psi^\downarrow_{\rm out}}{\mathbf{u}_1}|$ (blue solid) and $|\braket{\mathbf{v}_1}{\Psi^\downarrow_{\rm inc}}|$ (orange dashed), corresponding to minima in panel (b). (e) Absolute values of components of vector $\lvert\mathbf{u}_1\rangle$ in the spherical-wave basis labeled by three indices $(\ell,m,p)$. (f) Absolute values of zeroth-order components of the left and right singular vectors of $\mathbf{S}$, $|h^1_{\mathbf{k}_{\parallel},\downarrow,\mathrm{TE}}|$ and $|f^1_{\mathbf{k}_{\parallel},\downarrow,\mathrm{TE}}|$, corresponding to minima in panel (c).
• Figure 3: Development of the electromagnetic S-BIC in finite-size arrays of $N \times N$ spheres. (a) Inverse of the largest singular value of the effective T-matrix [$\ell_{\rm max} = 1$]. The red line indicates the BIC wavelength in the infinite array ($\lambda_{\rm R}$). (b) Wavelength minimum value of $\sigma_1^{-1}$ in panel (a) as a function of $N$. (c) Normalized absolute values of components $|u^1_{i,1,0,{\rm TE}}|/\max\limits_i|u^1_{i,1,0,{\rm TE}}|$ of the left singular vector $\lvert\mathbf{u}_1\rangle$ for $N = 23$ and $\lambda_{\rm R} = 1459.465$ nm. $u^1_{i,1,0,{\rm TE}}$ is proportional to the $z$ component of the magnetic dipole moment ($m_z$) of the $i$th particle.
• Figure 4: Electromagnetic S-BIC in a single metasurface on a substrate. (a) Inverse of the largest singular value of the total S-matrix [$\left| \mathbf{G}_{n_x, n_y}\right|_{\rm max} = 8\pi/L$]. Inset: absolute values of zeroth-order components of the singular vectors $|h^1_{\mathbf{k}_{\parallel},\downarrow,\mathrm{TE}}|$ (blue solid), $|f^1_{\mathbf{k}_{\parallel},\downarrow,\mathrm{TE}}|$ (orange dashed), $|h^1_{\mathbf{k}_{\parallel},\uparrow,\mathrm{TE}}|$ (green dashed-dotted), and $|f^1_{\mathbf{k}_{\parallel},\uparrow,\mathrm{TE}}|$ (red dotted). (b) Reflectance for a TE polarized plane wave. Inset: a Si-particle metasurface on a SiO$_2$ semi-infinite substrate and the geometry of incidence. The black and white dashed lines in panels (a) and (b), respectively, indicate the diffraction threshold in the substrate $\lambda(\theta) = L \left( n_{\rm sub} + \left| \sin \theta\right|\right)$, where $n_{\rm sub} = 1.4$.
• Figure 5: Doubly-degenerate BICs in a single metasurface of particles with $\varepsilon = \mu$. The resonance wavelength is 1280.72 nm, and the lattice constant is 910.90 nm. Panels (a) and (b) show the inverse of the largest singular values of the effective T-matrix [$\ell_{\rm max} = 1$] and the S-matrix [$\left| \mathbf{G}_{n_x,n_y}\right|_{\rm max} = 2\pi/L$], respectively. The insets in panels (a) and (b) show nonzero components of the first and second right singular vectors as degenerate BICs with pure TE and TM polarizations, obtained from the SVD of the effective T-matrix and the S-matrix, respectively. Please note for the inset in panel (b) that $h^i_{\mathbf{0}, \rm{TM/TE}} = \left( \left| h^i_{\mathbf{0}, \uparrow,\rm{TM/TE}} \right|^2 + \left| h^i_{\mathbf{0}, \downarrow,\rm{TM/TE}} \right|^2 \right)^{1/2}$.