基于薛定谔方程的多层薄膜透射率计算方法

Calculation method of multi-layer film transmissivity based on Schrodinger equation

为减少敷膜数量、提高反射效率,提出了一种多层薄膜透射率计算方法.对一维定态薛定谔方程解的平面波函数,根据薄膜的边界条件分别求出界面含入射波、透射波振幅的关系式.通过膜层内部的透射波和反射波衔接这两个关系式,得到该薄膜层含透射波振幅的关系式,拓展得到多层薄膜含透射波振幅的关系式.根据能量因子和折射率的关系,按匹配原则选定膜层厚度,将振幅关系式变换成计算多层薄膜透射率的方程式.选用4种光学介质制成膜系,对方程进行数值模拟和对比分析.结果验证了所提方法的可行性,发现了敷膜反射效应的退变临界点.当敷膜层数为8时,膜系反射率值达到99.69%。若采用传统的"λ/4"膜系则敷膜层数为13才能达标,可见所提方法比传统方法的反射效率提高了约38%。

In order to reduce the film layer number and improve the reflection efficiency, a multi-layer film transmissivity calculation method is proposed. For the plane wave function of one-dimensional time- independent SchrSdinger equation solution, the relation formula of film surface including incident and trans- mission wave amplitudes are obtained respectively based on the boundary conditions of thin film. The two formulas are connected by transmission and reflection waves inside the film. The relation including the transmission wave amplitude of the film layer is obtained, and that of the multi-layer film is obtained. According to the relationship between energy factor and refractive index, the film thickness is selected ac- cording to the matching principle, and the amplitude relation is transformed into an equation to calculate the multi-layer film transmittance. Four kinds of optical media are used to make film system, and the equation is numerically simulated and compared. Results show that the proposed method is feasible and the degenerative critical point of coating reflection effect is found. When the film layer number is 8, the film system refiectivity reaches 99.69%. If the traditional λ/4 film system is adopted, the film layer number must be 13 for reaching standard. It shows that efficiency of the proposed method is about 38% higher than that of the traditional method.