## 1 Introduction  
In this article, general multi-layered coating systems of \(n+1\) total layers, where \(n\in\mathbb{N}\) coating layers of different materials \(M_{1},\ldots,M_{n}\) are applied on top of each other on a substrate material \(M_{n+1}\) are investigated, see Figure 1. For \(i=1,\ldots,n+1\), \(L_{i}\in\mathbb{R}_{+}\) denotes the thickness of the associated layer. The thermal properties of each material are characterized by the thermal diffusivity \(\alpha_{i}\in\mathbb{R}_{+}\) and the thermal effusivity \(e_{i}\in\mathbb{R}_{+}\). The determination of  
\[\mathbf{L}:=(L_{1},\ldots,L_{n})^{T}\in\mathbb{R}_{+}^{n}\]  
using a method of nondestructive testing is of great interest in the manufacturing process and quality control of such systems. In the following, a few application examples for a varying number of coating layers are listed. \(n=1\):
* The manufacturing process of electrode coatings of Lithium-ion batteries, where a Lithium cobalt oxide coating of 50 to 100 \(\upmu\)m is applied to a thin (\(\pm\)10 \(\upmu\)m) aluminum substrate. Inaccurate coating thicknesses lead to performance issues of the battery or higher production costs. * The application of non-electrolyte zinc-aluminum flakes to protect metallic surfaces by increasing their corrosion resistance. \(n=2\):
* Production of plastic housings of laptops or smartphones. For example, the substrate might be some rigid ABS plastic of 1 to 2 mm, the first thin coating layer (the so-called basecoat) might be some polyurethane (\(\pm\)20 \(\upmu\)m) that acts like a thermal insulator against the increasing heat of the electronics, and the second coating layer (the so-called topcoat of \(\pm\)20 \(\upmu\)m) might be some UV hardened resin acting like a visually appealing surface finish. \(n=4\):
* In the automotive and aviation industries [1; 2], where metallic or CFRP substrates are coated with different paints, e.g., a primer, basecoat, topcoat, and clearcoat. Here again, each layer fulfills its own function, and it is, therefore, necessary to control the individual layer thicknesses. There are some issues to be taken into account when choosing an appropriate measuring process (cf. [3]). Eddy current or inductive-based measurement methods only work on metallic substrates, and X-ray or beta backscattering methods only work with certain metal groups and also require compliance with strict occupational health and safety, radiation protection, and disposal measures. Furthermore, ultrasonic and capacitive methods need contact with the test specimen and are therefore not suitable for measuring wet coatings (e.g., for the automotive paint process line) or uncured powder coatings. A very promising method, which is contact-free and uses harmless and non-invasive electromagnetic radiation, is presented by the terahertz technology; see [2]. Unfortunately, the usage of highly sensitive devices is needed, such as a femtosecond laser, which is often prohibitively expensive for medium-sized companies. Figure 1: System of \(n\) coating layer materials \(M_{1}\),..., \(M_{n}\) on a substrate material \(M_{n+1}\). In this article, it is suggested that photothermal methods, such as infrared thermography, are especially suited for the measurement of multi-layered coating systems. Such methods are contact-free and have manageable costs. A typical setting is presented in Figure 2, where the components and the operating principle are as follows:  
1 Optical excitation sources, such as lamps or lasers (cf. [4], [Chapter 4]), irradiate an object under investigation with optical (i.e., visible) light. Its task is to produce the photothermal effect, i.e.,
2 The test object absorbs the optical energy (light) and converts it into thermal energy (heat). Heat transfer inside the test object occurs, which is dependent on the characteristics of the object geometry, its material composition, and thermal properties. 3 The thermal response of the test object is recorded by a measuring device, such as an infrared camera.