Fig. 3: Analytical explanation of the transition from “two-wings” state to “four-wings” state.

a IFCs at ω = 710 cm⁻¹, the red and bule curves are separated. b IFCs at ω = 718 cm⁻¹, the red and bule curves intersect. c IFCs at ω = 714 cm⁻¹, the red and bule curves are parallel to each other. d Relationship between the distribution of polaritons with the frequency ω and twist angle θ. The dashed line denotes the critical angle as a function of frequency. Black dots represent the chosen frequencies of the IFCs. The blue and orange regions represent cases in which the field distribution appears to be “two-wings” and “four-wings”, respectively. The coordinates are the normalized wavevectors q_x and q_y, resulting in unitless scale bars. Dashed dispersion curves represent higher-order modes that do not exist in real space and are not involved in coupling, while solid curves denote modes present in the field. Yellow dashed curves indicate higher-order surface modes, and blue dashed curves represent higher-order volume modes. Red solid curves denote M₀, while the blue solid curves signify M₁. The relative positioning of these curves illustrates the coupling between the two modes, greater collinearity of their wavevectors correlates with stronger interaction. The IFCs are plotted at the fixed angle of 47.5°, and are derived from the dispersion Eq. (3).