## 1 Introduction
4 Finally, a computer processes and evaluates the data to provide certain properties (or even defects) of the considered test object. It should be mentioned that the photothermal method requires only two conditions:  
* The investigated coatings must be susceptible to optical radiation in a certain wavelength regime, i.e., for near-infrared, visible, or UV light. * There must be a thermal contrast between two adjacent layers. Otherwise, a transparent interface would be created without significant reflection. There are mainly two classical optical excitation types in thermographic processes, i.e., the lock-in thermography and pulsed thermography; see e.g., [5; 6; 7]. The latter method analyses the transient response and propagation of heat pulses in the test object. A prominent processing technique in pulsed thermography is the TSR (Thermographic Signal Reconstruction) method, where the logarithmic time derivatives of the signal are examined, cf. [8; 9; 10]. However, pulsed thermography has been used mainly for one-layered coating systems so far or simple defects. Since the intensity of the pulse decreases strongly in almost all materials, high energies are sometimes necessary to cause significant heating of the test object. In addition, this method is error-prone in case of surface irregularities, and the infrared camera would need a very high frame rate in order to detect reflections from multiple-layer interfaces in multi-layered coating systems. Lock-in thermography, whose principle the article is based on, means that the surface of the test object is periodically modulated using a planar sinusoidal waveform with a fixed frequency. Since the generated thermal wave has the same frequency as the excitation, especially the phase angle (or phase difference) as a result of thermal wave interference carries a lot of information. The goal is to determine \(\mathbf{L}\in\mathbb{R}_{+}^{n}\) from phase angle data for different modulation frequencies measured by an infrared camera. Since thermal  
Figure 2: Measurement principle of infrared thermography. waves show the same behavior as optical waves in the visible spectral range, thermal reflection, and transmission coefficients can be derived; see e.g., [11; 12]. Hence, thickness estimation from phase angle measurements represents an inverse problem; see (cf. [13; 14; 15; 16]) for corresponding literature. The overview and contributions of this article are summarized as follows:  
* Section 2 discusses the basics of thermal wave generation and thermal wave interference in the well-established one-layered and two-layered coating cases. Subsequently, these principles are used in order to obtain a new thermal wave interference formula in multi-layered coating systems with \(n\in\mathbb{N}\) coating layers on a thermally thick substrate. * Section 3 defines the forward operator \(F_{n}\) for a known number \(n\in\mathbb{N}\) of coating layers, which maps \(\mathbf{L}\in\mathbb{R}_{+}^{n}\) to the phase angle data \(\boldsymbol{\varphi}_{T}\in\mathbb{R}^{m}\) with \(m\) being the number of used modulation frequencies. The inverse operator \(G_{n}\) is then defined, which approximates an inversion of \(F_{n}\) by solving a nonlinear least squares problem. Finally, since \((\alpha_{i},e_{i}),\quad i=1,\ldots,n+1\) are often partially or even completely unknown, a concept is additionally presented, which utilizes the generated data of multiple samples in a clever way, where the unknown thermal properties can also be determined and used to calibrate a model for the layer thickness determination. * Section 4 concludes with numerical experiments that demonstrate the performance of the developed layer thickness determination procedure.