Coupled-wave formalism for bound states in the continuum in guided-mode resonant gratings

We present simple yet extremely accurate coupled-wave models describing the formation of bound states in the continuum (BICs) in one-dimensional periodic guided-mode resonant gratings consisting of a slab waveguide layer with a binary grating attached to one or both of its interfaces. Using these models, we obtain simple closed-form expressions predicting the locations of the BICs and quasi-BICs in the $\omega \text{−}{k}_x$ parameter space. We study two mechanisms of the BIC formation: coupling between two counterpropagating guided modes and coupling between a guided mode and a Fabry-Pérot mode. The BIC conditions corresponding to the considered mechanisms are formulated in terms of the scattering coefficients of the binary grating. The predictions of the presented models are in excellent agreement with the results of full-wave simulations obtained using the Fourier modal method.

Figure 1

Guided-mode resonant gratings consisting of a slab waveguide layer with a binary grating attached to (a) one or (b) both of its interfaces. (c) Light lines of $\pm 1\mathrm{st}$ and $0\mathrm{th}$ diffraction orders in the substrate and superstrate (blue solid lines) and in the waveguide layer (red dashed lines).

Figure 2

Propagating plane waves considered in the coupled-wave model.

Figure 3

(a) Rigorously calculated transmission spectrum of the considered GMRG and (b) two magnified fragments demonstrating the formation of a BIC (top panel) and of a quasi-BIC (bottom panel). (c) Quality factors of the modes corresponding to the magnified fragments.

Figure 4

Propagating plane waves considered in the coupled-wave model of a GMRG with a horizontal symmetry plane.

Figure 5

(a) Rigorously calculated transmission spectrum of the considered GMRG possessing horizontal symmetry. (b) Model spectrum calculated using Eq. (16). (c)–(e) Quality factors of the modes corresponding to the fragments bounded by black rectangles. The red dashed line in (e) shows the quality factor of the modes of the GMRG having a small absorption.