Surface topography measuring interference microscopy is a three-dimensional (3D) imaging technique that provides quantitative analysis of industrial and biomedical specimens. Many different instrument modalities and configurations exist, but they all share the same theoretical foundation. In this paper, we discuss a unified theoretical framework for 3D image (interferogram) formation in interference microscopy. We show how the scattered amplitude is linearly related to the surface topography according to the Born and the Kirchhoff approximations and highlight the main differences and similarities of each. With reference to the Ewald and McCutchen spheres, the relationship between the spatial frequencies that characterize the illuminating and scattered waves, and those that characterize the object, are defined and formulated as a 3D linear filtering process. It is shown that for the case of near planar surfaces, the 3D filtering process can be reduced to two dimensions under the small height approximation. However, the unified 3D framework provides significant additional insight into the scanning methods used in interference microscopy, effects such as interferometric defocus and ways to mitigate errors introduced by aberrations of the optical system. Furthermore, it is possible to include the nonlinear effects of multiple scattering into the generalized framework. Finally, we consider the inherent nonlinearities introduced when estimating surface topography from the recorded interferogram.

中文翻译：

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表面形貌测量干涉显微镜中的散射和三维成像

表面形貌测量干涉显微镜是一种三维（3D）成像技术，可对工业和生物医学样本进行定量分析。存在许多不同的仪器模式和配置，但是它们都具有相同的理论基础。在本文中，我们讨论了干涉显微镜中3D图像（干涉图）形成的统一理论框架。我们根据Born和Kirchhoff近似值显示了散射振幅如何与表面形貌线性相关，并重点介绍了两者的主要区别和相似之处。关于Ewald和McCutchen球体，表征照明和散射波的空间频率与表征物体的频率之间的关系，被定义为3D线性滤波过程。结果表明，对于接近平面的情况，可以在较小的高度近似下将3D滤波过程缩减为二维。但是，统一的3D框架对干涉显微镜中使用的扫描方法，干涉像散焦等影响以及减轻由光学系统像差引入的误差的方式提供了重要的补充见解。此外，有可能将多重散射的非线性效应纳入广义框架。最后，我们考虑从记录的干涉图估算表面形貌时引入的固有非线性。在较小的高度近似下，可以将3D滤波过程缩小为二维。但是，统一的3D框架对干涉显微镜中使用的扫描方法，干涉像散焦等影响以及减轻由光学系统像差引入的误差的方法提供了重要的补充见解。此外，有可能将多重散射的非线性效应纳入广义框架。最后，我们考虑从记录的干涉图估算表面形貌时引入的固有非线性。在较小的高度近似下，可以将3D滤波过程缩小为二维。但是，统一的3D框架对干涉显微镜中使用的扫描方法，干涉像散焦等影响以及减轻由光学系统像差引入的误差的方法提供了重要的补充见解。此外，有可能将多重散射的非线性效应纳入广义框架。最后，我们考虑从记录的干涉图估算表面形貌时引入的固有非线性。可以将多重散射的非线性效应纳入广义框架。最后，我们考虑从记录的干涉图估算表面形貌时引入的固有非线性。可以将多重散射的非线性效应纳入广义框架。最后，我们考虑从记录的干涉图估算表面形貌时引入的固有非线性。