Fig. 1. (a) Schematics of the representative 1D photonic lattices for studying the fourth stop band. (b) Conceptual illustration of the photonic band structures including the first four band gaps. Guided waves are described by the complex frequency Ω=ω−iγ, where γ represents the decay rate of the mode.

Fig. 2. Comparison between the key properties of the fourth (a)–(d) and second (e)–(h) stop bands. (a), (e) Simulated dispersion relations; (b), (f) radiative Q factors; (c), (g) transmission spectra; and (d), (h) the evolution of the band edge frequencies as a function of ρ. The spatial electric field (Ey) distributions in the insets in (a) and (e) indicate that one of the band edge modes becomes the symmetry-protected BIC. As ρ varies from zero to one, the fourth stop band exhibits the closed band states three times, while the second stop band shows one band gap closure. The structural parameters ϵ0=9.00, Δϵ=2.00, ϵs=1.00, t=0.20Λ, and ρ=0.35 were used in the FEM simulations.

Fig. 3. Simulated dispersion relations near the third-order Γ point in 1D leaky-mode lattices relative to the Fourier harmonic content. The dielectric functions vary as (a) ϵ0+ϵ1 cos(Kz), (b) ϵ0+ϵ2 cos(2Kz), (c) ϵ0+ϵ3 cos(3Kz), (d) ϵ0+ϵ4 cos(4Kz), and (e) ϵ0+ϵ5 cos(5Kz). At the fourth stop band, out-of-plane radiation is induced by the interplay between the first and second Fourier harmonics, whereas the size of the gap is determined by the interplay between the first, second, and fourth Fourier harmonics. Other than the dielectric functions, the lattice parameters are the same as for Fig. 2.

Fig. 4. FEM-simulated spatial electric field distributions at the upper and lower bands of ΔΩ4 due to the fourth Fourier harmonic. The relative positions of the symmetric and asymmetric edges are reversed with the sign change of the fourth-order Fourier coefficient ϵ4=(Δϵ/2π)sin(4πρ).

Fig. 5. (a) Simulated leaky edge Q factors at the fourth stop band in the non-approximated BDG with full Fourier harmonics as a function of ρ. (b) Radiative Q factors of the leaky edge modes in the approximated lattices with dielectric functions ϵ0+ϵ1 cos(Kz) and ϵ0+ϵ2 cos(2Kz). Radiative Q factors in the lower and upper band branches in the full lattice as a function of kz when (c) ρ=0.46909 and (d) ρ=0.47.

Fig. 6. Simulated dispersion relations at the closed band states in the vicinity of the third-order Γ point in the BDG structure with (a) ρ=0.46857, (b) ρ=0.22882, and ρ=0.72479. Dirac cone dispersion can be obtained without the leaky band flattening when the out-of-plane radiation loss is suppressed.

Fig. 7. Properties of the DSG structure at the fourth stop band. (a) Simulated dispersion relations and (b) radiative Q factors near the third-order Γ point. (c) Transmission spectra exhibiting Fano resonances when θ=0° and θ=3°. (d) Evolution of the band edge frequencies as a function of ρ. (e) Leaky edge Q factors as a function of ρ. (f) Radiative Q factors in the lower and upper band branches as a function of kz when ρ=0.47307. Dispersion relations at the closed band states when (g) ρ=0.47313, (h) ρ=0.22396, and ρ=0.72591. In the FEM simulations, the average thickness t0=0.15Λ was kept constant, and we used Δt=ta−tb=0.05Λ, ϵa=12.00, and ϵs=1.00.

Fig. 8. Properties of the OSG structure at the fourth stop band. (a) Simulated dispersion relations and (b) radiative Q factors near the third-order Γ point. (c) Transmission spectra exhibiting Fano resonances when θ=0° and θ=3°. (d) Evolution of the band edge frequencies as a function of ρ. (e) Leaky edge Q factors as a function of ρ. (f) Radiative Q factors in the lower and upper band branches as a function of kz when ρ=0.50273. Structural parameters are the same as for Fig. 7.