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.
Bound states in the continuum (BICs) are nonradiating eigenmodes of periodic structures that are embedded within the radiation continuum yet remain completely decoupled from it. In this context, one can distinguish between symmetry-protected BICs (S-BICs), whose symmetry is incompatible with that of the continuum modes, and accidental BICs (A-BICs), which arise from destructive interference of radiating channels [1]. In the literature, BICs are also commonly referred to as trapped modes [2][3][4], as their energy is fully confined within the structure or its near field and cannot escape due to an infinite radiative quality factor (Q factor). Moreover, in the absence of material losses or absorption, a BIC occurs at a purely real-valued eigenfrequency.
BICs have been experimentally and theoretically demonstrated in nanophotonic [5] and acoustic [6] systems. A broad and practically important class of systems supporting BICs comprises periodic arrangements of resonant scatterers, commonly referred to as metasurfaces. Upon scattering of an external wave by a metasurface, a BIC manifests itself as a dark mode [7]. In realistic finite-size arrays, translational symmetry is broken, and a genuine BIC is transformed into a quasi-BIC with a large but finite Q factor due to edge-induced diffraction losses [8,9]. However, even in infinite arrays, a genuine BIC can be converted into a quasi-BIC under certain perturbations, enabling its excitation by an external plane * nikita.ustimenko@kit.edu wave. Such a conversion can be achieved, for example, by a slight variation of the angle of incidence [10,11]. Alternatively, only structural symmetry breaking [12,13] or parameter tuning [11,14] can be employed for S-BICs and A-BICs, respectively. The resulting quasi-BIC response has been exploited in a wide range of applications [15], including field enhancement [16], nonlinear generation [17], and lasing [18].
Several theoretical approaches have been proposed to diagnose and analyze the formation of BICs. One such approach relies on expanding the electromagnetic fields into quasinormal modes [19,20] or resonant states [21][22][23], which are solutions to Maxwell’s equations with complex eigenfrequencies for an open, non-Hermitian system, satisfying the radiation boundary conditions at |r| → +∞. Recently, Laude and Wang adopted this method for describing the elastodynamics of open phononic systems [24]. Neale and Muljarov demonstrated that, in photonic-crystal slabs, BICs can arise from the hybridization of two or more resonant states [25]. This phenomenon is commonly described using non-Hermitian effective Hamiltonians [26][27][28] or temporal coupled-mode theory [29][30][31], which are equivalent to a certain extent [32,33] and, in general, do not provide an exact description of the dynamics of a resonant system [34]. Additional drawbacks of the resonant-state expansion include the divergence of resonant states in the limit |r| → +∞ and the need to compute resonant states, complex eigenfrequencies, and coupling constants for all parameters of a system that can be deterministically controlled.
One established technique for computing resonant states is the pole expansion of the scattering matrix (Smatrix), which relates incoming and outgoing scattering channels in stratified systems [35][36][37]. Blanchard, Hugonin, and Sauvan showed that a BIC corresponds to the coalescence of a pole and a zero of the S-matrix at a real frequency [38]. Moreover, Liu et al. demonstrated that S-BICs can act as chain points that connect two nodal lines of the S-matrix at the Γ-point in three-dimensional frequency-momentum space [39]. In our contribution here, we focus on another aspect of the S-matrix behavior near a BIC resonance, namely its singular value decomposition (SVD) [40]. Singular values and singular vectors of the S-matrix encode intrinsic scattering properties that govern transmission and reflection [41,42], absorption and emission [43,44], and other fundamental physical characteristics [45]. In particular, Guo et al. demonstrated how coherent perfect absorption [46] and coherent perfect extinction manifest in the topology of the singular values and vectors of the S-matrix [47].
Beyond resonant-state expansions, the decomposition of fields into spherical and cylindrical multipole waves, or simply multipoles, can also shed light on the origin of BICs in metasurfaces. Sadrieva et al. showed that the multipole content of an S-BIC with Bloch (in-plane) wave vector k BIC ∥ consists exclusively of multipoles that do not radiate along the direction specified by k BIC ∥ . In contrast, an A-BIC contains radiating multipoles whose far-field contributions destructively interfere, thereby suppressing radiation [48]. In the multipole basis, the transition matrix (T-matrix) formalism, introduced by Waterman, provides a rigorous description of the electromagnetic and acoustic response of a single resonator of arbitrary shape [49][50][51], as wel
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