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用声学方法构造了折射系数呈余弦变化的一维光子晶体,并在介质无损耗、无电流、无磁性及各向同性假设下,把麦克斯韦方程化为一维薛定谔方程,当折射系数呈余弦变化时,薛定谔方程进一步化为了Mathieu方程。分析表明,在参数(δ,ε)平面上出现了一系列稳定和不稳定区(禁带)。当参数|ε|→0时,这些不稳定区退化为一点,给出了禁带的中心频率,并用摄动法近似地求出了禁带宽度。结果表明,一阶和二阶不稳定区宽度与介质参数和入射光子频率有关。只需适当选择这些参数,就可以有效地调节光子晶体的带结构,并按需要得到不同性能的光子晶体。
A one-dimensional photonic crystal with cosine variation of refraction coefficient was constructed by acoustical method. Maxwell’s equation was transformed into one-dimensional Schrödinger equation under the condition of no loss of medium, no current, no magnetism and isotropy. When the refractive index showed a cosine variation Schrodinger equation further to Mathieu equation. Analysis shows that a series of stable and unstable regions (forbidden band) appear in the parameter (δ, ε) plane. When the parameter | ε | → 0, these unstable regions degenerate to a point, the center frequency of the forbidden band is given, and the forbidden band width is approximately determined by the perturbation method. The results show that the first and second order instability widths are related to the medium parameters and the incident photon frequency. With proper selection of these parameters, the band structure of the photonic crystal can be effectively adjusted and photonic crystals with different properties can be obtained as needed.