Band dynamics accompanied by bound states in the continuum at the third-order $\Gamma$ point in leaky-mode photonic lattices

Bound states in the continuum (BICs) and Fano resonances in planar photonic lattices, including metasurfaces and photonic crystal slabs, have been studied extensively in recent years. Typically, the BICs and Fano resonances are associated with the second stop bands open at the second-order Gamma ($\Gamma$) point. This paper address the fundamental properties of the fourth stop band accompanied by BICs at the third-order $\Gamma$ point in one-dimensional (1D) leaky-mode photonic lattices. At the fourth stop band, one band edge mode suffers radiation loss, thereby generating a Fano resonance, while the other band edge mode becomes a nonleaky BIC. The fourth stop band is controlled primarily by the Bragg processes associated with the first, second, and fourth Fourier harmonic components of the periodic dielectric constant modulation. The interplay between these three major processes closes the fourth band gap and induces a band flip whereby the leaky and BIC edges transit across the fourth band gap. At the fourth stop band, a new type of BIC is formed owing to the destructive interplay between the first and second Fourier harmonics. When the fourth band gap closes with strongly enhanced $Q$ factors, Dirac cone dispersions can appear at the third-order $\Gamma$ point. Our results demonstrate a method for manipulating electromagnetic waves by utilizing the high-$Q$ Bloch modes at the fourth stop band.