从几何角度理解多层膜
Understanding multilayers from a geometrical viewpoint
摘要：
我们重新阐述了无损多层膜的基本性质。我们证明了这些多层膜的传输矩阵与（2+1）维时空中的Lorentz群SO（2，1）以及几何光学中ABCD定律结构的SL（2，R）群具有基本相同的代数性质。借助于岩泽分解，我们把任何多层的作用表示为三个简单解释的矩阵的乘积。这种群论结构允许我们在复平面上引入双线性变换。多层传递函数的概念自然产生，并在单位圆盘中研究了其相应的性质。我们表明，岩泽分解反映在这个几何水平上的三个简单的行动，可以被认为是一个更深层次的理解多层行为的基本件。我们用该方法详细分析了一个简单的实际例子。
英文摘要：
We reelaborate on the basic properties of lossless multilayers. We show that the transfer matrices for these multilayers have essentially the same algebraic properties as the Lorentz group SO(2,1) in a (2+1)-dimensional spacetime, as well as the group SL(2,R) underlying the structure of the ABCD law in geometrical optics. By resorting to the Iwasawa decomposition, we represent the action of any multilayer as the product of three matrices of simple interpretation. This group-theoretical structure allows us to introduce bilinear transformations in the complex plane. The concept of multilayer transfer function naturally emerges and its corresponding properties in the unit disc are studied. We show that the Iwasawa decomposition reflects at this geometrical level in three simple actions that can be considered the basic pieces for a deeper undestanding of the multilayer behavior. We use the method to analyze in detail a simple practical example.
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