Discussion on the analytical solutions of graphene surface modes and the relationship with quantum systems

The TM surface mode supported by the air-graphene-air structure is analytically calculated under the ultra-thin approximation. A third order closed form of the effective refractive index is derived, which can be used to improve accuracy of the calculation. This is verified by numerical calculations. It is also pointed out that such an optical field can be interpreted as the wave function of a quantum particle in a double-delta potential with some interesting features.