Super bound states in the continuum through merging in grating

Bound states in the continuum (BICs) in gratings composed of infinitely long silicon rods of rectangular cross-section are considered. We reveal merging off-$\Gamma$ Friedrich-Wintgen BIC with symmetry protected BIC. We present CMT and multipole decomposition theory complementary each other to analyze the merging phenomenon. The theories show a crossover of the behavior of $Q$-factor from standard inverse square law $k_{x,z}^{-2}$ towards to extremely fast boosting law $k_{x,z}^{-6}$ in momentum space. In turn that crossover gives rise to another crossover from $Q\sim N^2$ to $Q\sim N^3$ for symmetry protected quasi BIC in finite grating of $N$ rods owing to suppression of radiation leakage of quasi BIC mode from surface of grating. As a result the $Q$-factor of quasi BIC is determined by residual leakage from ends of grating. We show numerically that this leakage also can be suppressed considerably if to stretch grating from the ends.