Strong resonances on periodic arrays of cylinders and optical bistability with weak incident waves

A one-dimensional periodic array of circular dielectric cylinders surrounded by air is a simple structure on which guided modes above the light line, also called bound states in the continuum (BICs), may exist. Recent studies reveal that such an array supports not only antisymmetric standing waves which are symmetry-protected BICs, but also propagating Bloch BICs and symmetric standing waves. Near a BIC, there is a family of resonant modes (depending on the Bloch wave number $\beta$) with arbitrarily large quality factors. Using a perturbation method, we show that the quality factor of the resonant mode typically depends on $\beta$ like $1/{(\beta -{\beta }_{*})}^2$, where ${\beta }_{*}$ is the Bloch wave number of the BIC, but near a symmetric standing wave $({\beta }_{*}=0)$, the quality factor blows up like $1/{\beta }^4$. This indicates that strong resonances can be more easily induced near a symmetric standing wave. As an application, we numerically study optical bistability for the periodic array assuming the cylinders have a Kerr nonlinearity. With the nonlinear effects enhanced by the resonances, it is possible to have optical bistability for weak incident waves. The numerical results confirm that the weakest incident wave for optical bistability is realized through the resonances near the symmetric standing waves.

Figure 1

A 1D array of circular cylinders with radius $a$, dielectric constant ${\epsilon }_1$, and nonlinear coefficient ${\gamma }_1$. The array is periodic in $y$ with period $L$.

Figure 2

Wave field patterns (i.e., the real parts of $u$) of three BICs on a periodic array of circular cylinders with radius $a=0.382L$. (a) Antisymmetric ($y$-odd) standing wave for ${\epsilon }_1=5$ and $\omega L/\left(2\pi c\right)=0.532688$. (b) Propagating BIC for ${\epsilon }_1=5$, $\omega L/\left(2\pi c\right)=0.647949$, and $\beta L/\left(2\pi \right)=0.07228$. (c) Symmetric ($y$-even) standing wave for ${\epsilon }_1=6.44974$ and $\omega L/\left(2\pi c\right)=0.573935$. The $x$ and $y$ axis are given in unit $L$.

Figure 3

Logarithmic relations between $-\text{Im}\left(\omega \right)L/\left(2\pi c\right)$ and $(\beta -{\beta }_{*})L/\left(2\pi \right)$ for the three BICs shown in Fig. 2.

Figure 4

Reflection coefficient of a nonlinear diffraction problem related to the BIC shown in Fig. 2, for different incident amplitude $A$ (with unit V/m), wave number $\beta L/\left(2\pi \right)=0.01$, and frequency $\omega L/\left(2\pi c\right)=0.53246$.

Figure 5

Reflection coefficient of a nonlinear diffraction problem related to the BIC shown in Fig. 2, for different incident amplitude $A$ (with unit V/m), wave number $\beta L/\left(2\pi \right)=0.08228$, and frequency $\omega L/\left(2\pi c\right)=0.65040$.

Figure 6

Reflection coefficient of a nonlinear diffraction problem related to the BIC shown in Fig. 2, for different incident amplitude $A$ (with unit V/m), wave number $\beta L/\left(2\pi \right)=0.01$, and frequency $\omega L/\left(2\pi c\right)\approx 0.574086$.