Complex Band Structure and Bound States in the Continuum: A Unified Theoretical Framework

Abstract

Band structure analysis is central to understanding wave propagation in periodic media; however, it becomes challenging in open systems owing to energy leakage. Photonic crystal (PhC) slabs exemplify such systems, featuring periodicity in the $x$-$y$ plane and finite extent in the $z$-direction, and supporting diverse guided-mode resonances whose interactions give rise to phenomena such as bound states in the continuum (BICs), exceptional points (EPs), and circular polarisation states. Although numerical simulations can reveal these effects, effective non-Hermitian Hamiltonians are often employed to elucidate the underlying physical mechanisms. This approach, however, relies on manually selected resonant modes and may suffer from basis incompleteness. Here, a systematic first-principles approach is presented to derive the complex band structure. The minimal channels in the scattering matrix, either open or closed, are determined by the number of propagating bulk Bloch waves. The interactions between these waves fully reveal the complex band structure. For instance, two Bloch waves predict the leading-order imaginary frequency $ω''$ and identify accidental BICs, each associated with a dual Fabry--Pérot mode, whereas three waves reveal robust Friedrich--Wintgen and symmetry-protected BICs together with the associated linewidth behaviours. Orthogonally polarised waves are further incorporated to characterise far-field polarisation and EPs. When extended to a two-dimensional periodic structure, this framework accurately predicts $ω''$, encompasses all known BICs, and tracks their evolution with system parameters. Overall, this first-principles approach provides a unified foundation for studying complex band structure and facilitates the exploration of light confinement in periodic media.

Figures (10)

• Figure 1: Scattering matrix and its poles in the complex-$\omega$ plane. The scattering matrix relates the incoming ($a^+$, $b^+$) and outgoing ($a^-$, $b^-$) waves. Inside the periodic structure, whether one-dimensional or two-dimensional, a Bloch-wave basis is adopted to describe the electromagnetic fields.
• Figure 2: EM waves in different media. Left: field profiles in (a) homogeneous medium, (b) slab waveguide, and (c) PhC slab. Right: corresponding dispersion relations. (a) The region above the light line (shaded grey) supports wave propagation. (b) FP modes (complex frequencies) lie above the light line, whereas waveguide modes (real frequencies) lie below it. (c) Waveguide modes folded into the first BZ interact with the FP modes, forming guided mode resonances with complex frequencies. Parameters: $\bar{\epsilon}=9$, $h=a=600\,$nm; perturbation strength $\delta=0.3$.
• Figure 3: Complex band structure of a 1D PhC slab. (a) Dispersion relation and imaginary part $\omega"$ (spectral width) of guided-mode resonances GR$_0^{-1}$ and GR$_1^{-1}$ at perturbation strength $\delta = 0.1$, with $\omega"$ scaled by a factor of 500 for visibility. (b) Numerical (dots) and theoretical (dashed) results for $C(q)$, defined in equation (\ref{['eq:C_q']}) using $\omega" = C(q)\,\delta^2$.
• Figure 4: Accidental BICs and their dual FP modes. (a) An accidental BIC (black dot) arises from the coupling between GR$_1^{-1}$ and FP$_3$ bands (with $\delta=0.02$, $h=1.67a$). (b) Imaginary part of the frequency for GR$_1^{-1}$ at $\delta=0.02$, 0.04, 0.06: the BIC converges to a fixed point as $\delta\to0$. (c) Accidental BICs and dual FP modes arising from impedance eigenvalue degeneracy (for $\delta=0.01$, $h=4.42a$). (d) First order diffraction amplitude on FP$_9$: dual FP modes have zero first order diffraction, whereas BICs have zero zeroth order diffraction. Right panels: electric-field profiles of zeroth and first diffraction components for BICs (top) and dual FPs (bottom).
• Figure 5: Friedrich--Wintgen BICs and their duals. (a) Dispersion of GR$_0^1$ band shows a Friedrich--Wintgen BIC arising from its coupling with GR$_2^{-1}$ at $\delta=0.01$, 0.1, 0.2 and $h=a$. The bandgap and BIC position ($\Delta q = q_* - q_c$, where $q_c$ is the band crossing point) each scale linearly with $\delta$. (b) The dual BIC emerges on GR$_2^{-1}$ near the crossing point, but is disrupted by mode hybridization.