  Recent studies have demonstrated that ultrahigh-$Q$ resonances, which are
robust to fabrication imperfections, can be realized by merging multiple bound
states in the continuum (BICs) in momentum space. The merging of multiple BICs
holds significant promise for practical applications, providing a robust means
to attain ultrahigh-$Q$ resonances that greatly enhance light-matter
interactions. In this study, we introduce a novel approach to achieve the
merging of BICs at the edges of the fourth stop band, which opens at the
third-order $\Gamma$ point, in one-dimensional leaky-mode photonic lattices.
Photonic band gaps and BICs arise from periodic modulations in lattice
parameters. However, near the third-order $\Gamma$ point, out-of-plane
radiation arises by the first and second Fourier harmonic components in the
lattice parameters. Accidental BICs can emerge at specific $k$ points where an
optimal balance exists between these two Fourier harmonic components. We
demonstrate that these accidental BICs are topologically stable, and their
positions in momentum space can be precisely controlled by adjusting a specific
lattice parameter that influences the strength of the first and second Fourier
harmonic components. Furthermore, we show that accidental BICs can be merged at
the third-order $\Gamma$ point, with or without a symmetry-protected BIC, by
finely adjusting this specific lattice parameter while keeping other parameters
constant.
