To take advantage of the singular properties of matter, as well as to characterize it, we need to interact with it. The role of optical spectroscopies is to enable us to demonstrate the existence of physical objects by observing their response to light excitation. The ability of spectroscopy to reveal the structure and properties of matter then relies on mathematical functions called optical (or dielectric) response functions. Technically, these are tensor Green’s functions, and not scalar functions. The complexity of this tensor formalism sometimes leads to confusion within some articles and books. Here, we do clarify this formalism by introducing the physical foundations of linear and non-linear spectroscopies as simple and rigorous as possible. We dwell on both the mathematical and experimental aspects, examining extinction, infrared, Raman and sum-frequency generation spectroscopies. In this review, we thus give a personal presentation with the aim of offering the reader a coherent vision of linear and non-linear optics, and to remove the ambiguities that we have encountered in reference books and articles.

中文翻译：

---

一阶至三阶介电响应的统一数学形式主义：在特定于表面的两色振动光谱学中的应用

为了利用物质的奇异特性并对其进行表征，我们需要与之相互作用。光学光谱学的作用是使我们能够通过观察物理对象对光激发的响应来证明它们的存在。光谱学揭示物质结构和性质的能力则取决于称为光学（或介电）响应函数的数学函数。从技术上讲，这些是张量Green的函数，而不是标量函数。张量形式主义的复杂性有时会导致某些文章和书籍中的混乱。在这里，我们通过尽可能简单和严格地介绍线性和非线性光谱学的物理基础，来澄清这种形式主义。我们专注于数学和实验方面，研究了消光，红外，拉曼光谱和和频谱。因此，在这篇评论中，我们进行了个人介绍，目的是为读者提供线性和非线性光学的连贯视野，并消除参考书和文章中遇到的歧义。