Bound states in the continuum on periodic structures surrounded by strong resonances

Bound states in the continuum (BICs) are trapped or guided modes with their frequencies in the frequency intervals of the radiation modes. On periodic structures, a BIC is surrounded by a family of resonant modes with their quality factors approaching infinity. Typically the quality factors are proportional to $1/|\mathbf{\beta }-{\mathbf{\beta }}_{*}{|}^2$, where $\mathbf{\beta }$ and ${\mathbf{\beta }}_{*}$ are the Bloch wave vectors of the resonant modes and the BIC, respectively. But for some special BICs, the quality factors are proportional to $1/|\mathbf{\beta }-{\mathbf{\beta }}_{*}{|}^4$. In this paper, a general condition is derived for such special BICs on two-dimensional periodic structures. As a numerical example, we use the general condition to calculate special BICs, which are antisymmetric standing waves, on a periodic array of circular cylinders, and show their dependence on parameters. The special BICs are important for practical applications, because they produce resonances with large quality factors for a very large range of $\mathbf{\beta }$.

Figure 1

(a) Periodic array of circular cylinders surrounded by air. (b) $F_e$ (with unit $L$) of the first ASW as a function of radius $a$ for ${\epsilon }_1=8.2$.

Figure 2

Wave field patterns of two special $x$-even ASWs on periodic arrays of circular cylinders with ${\epsilon }_1=8.2$. (a) The first ASW for $a=0.4404L$. (b) The fifth ASW for $a=0.4323L$.

Figure 3

(a) Parameters of the periodic array for two special ASWs: the first ASW (red solid curve) and the fifth ASW (blue dashed curve). (b) The corresponding frequencies of the two special ASWs. Points $\mathsf{A}$ and $\mathsf{B}$ correspond to Figs. 2 and 2, respectively.

Figure 4

(a),(b) Diffraction solutions ${\phi }_e$ corresponding to the special ASWs shown in Figs. 2 and 2, respectively.

Figure 5

Imaginary parts of the complex frequencies vs $\beta$ for resonant modes near ASWs shown in Fig. 2 ($\circ$) and Fig. 2 ($*$), respectively.