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Unified analysis and mathematical representation of film-thickness behavior of film-substrate systems

机译：薄膜-基材系统的薄膜厚度行为的统一分析和数学表示

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The ellipsometric function rho of a film-substrate system is studied as the film thickness d is kept constant and the angle of incidence psi is changed. The generated constant-thickness contours (CTCs) are characterized by an introduced mathematical behavior indicator that represents a group of CTCs. The behavior of each group is developed and studied in the four planes psi-d, X, Z, and rho, where X is the film-thickness exponential function and Z is a previously introduced intermediate plane. In the phi-d plane the film-thickness domain is identified and divided into a sequence of disconnected thickness subdomains (DTSs), depending on only N_(0) and N_(1), and their number depending on the range in which N_(0)/N_(1) lies. The behavior of the CTCs in the successive planes X, Z, and rho is then studied in each DTS, and the CTC's space is divided into disconnected subfamilies according to the behavior indicator. Equivalence classes that reduce the infinite number of subfamilies into a finite number are then introduced. The transformation from each plane to the next is studied with the origin of the Z plane mapped onto the point at infinity of the rho plane, forming a singularity. A multiple-film-thickness inequality is derived to determine the unique solution of the film thickness. The type of reflection being internal or external at both ambient-film and film-substrate interfaces affects the analysis and is also considered. To conclude we introduce the design of polarization-preserving devices and a novel oscillating single-element ellipsometer to fully characterize zero film-substrate systems as examples of applying the knowledge developed here.
机译：研究了膜-基底系统的椭圆函数rho，因为膜厚度d保持恒定并且入射角psi发生了变化。生成的等厚轮廓线（CTC）由引入的代表一组CTC的数学行为指标表征。在四个平面psi-d，X，Z和rho上开发和研究了每个组的行为，其中X是膜厚指数函数，Z是先前引入的中间平面。在phi-d平面中，确定薄膜厚度域并将其分为一系列不连续的厚度子域（DTS），仅取决于N_（0）和N_（1），其数量取决于N_（ 0）/ N_（1）说谎。然后在每个DTS中研究CTC在连续平面X，Z和rho中的行为，并根据行为指标将CTC的空间划分为不连续的子族。然后引入等价类，这些等价类将无限数量的子族减少为有限数量。研究从每个平面到下一个平面的转换，将Z平面的原点映射到rho平面无穷大的点上，从而形成奇点。得出多个薄膜厚度不等式，以确定薄膜厚度的唯一解。在环境膜和膜-基材界面的内部或外部反射类型会影响分析，因此也应予以考虑。总而言之，我们将介绍保偏装置的设计和一种新颖的振荡单元素椭圆偏振仪，以充分表征零薄膜-衬底系统，作为应用此处开发的知识的示例。

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