Super-bound states in the continuum through merging in grating

We consider bound states in the continuum (BICs) in grating composed of infinitely long silicon rods of rectangular cross-section. We reveal merging off-$\mathrm{\ensuremath{\Gamma}}$ Friedrich-Wintgen BIC with symmetry protected BIC. We present CMT and multipole decomposition theory, complementing each other, to analyze the merging phenomenon. The theories show a crossover of the behavior of $Q$ factor from standard inverse square law ${k}_{x,z}^{\ensuremath{-}2}$ towards extremely fast boosting law ${k}_{x,z}^{\ensuremath{-}6}$ in momentum space. In turn that crossover gives rise to another crossover from $Q\ensuremath{\sim}{N}^{2}$ to $Q\ensuremath{\sim}{N}^{3}$ 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 can also be suppressed considerably if grating is stretched from the ends.