## 2 Mathematical Setup
### Basics of Thermal Wave Interference
#### 2.3.1 One-Layered Coatings on a Substrate  
Consider the following system of two layers consisting of media \(M_{1}\) and \(M_{2}\), as illustrated in Figure 4.  
Assume that both media \(M_{1}\) and \(M_{2}\) have homogeneous thermophysical properties and that \(M_{1}\) has thickness \(L_{1}\) and \(M_{2}\) has thickness \(L_{2}\). Furthermore, assume that \(M_{2}\) is thermally thick. That means that \(L_{2}\) is significantly larger than the thermal diffusion length \(\mu_{2}\). This assumption is not unusual, as the substrate is often much thicker than the applied coating. This guarantees that the transmitted thermal waves into the substrate do not have an effect on the surface temperature because they can be neglected due to the large attenuation. Moreover, assume that the whole system is exposed to air (denoted by \(M_{0}\)). Suppose that the surface is illuminated by a plane, normal and periodic heating. The single wave trains that are generated near the surface \(x=0\) propagate towards the interface between the two media and back towards the surface of \(M_{1}\). When meeting any interfaces, the waves are partially reflected and transmitted. Thermal wave interference effects occur, meaning that the surface temperature at \(x=0\) is a sum of all thermal wave trains. In general, when a thermal wave has traveled a distance \(x>0\) the amplitude will be damped by \(\exp(-\sigma x)\), where \(\sigma=(1+i)\frac{1}{\mu}\) is the complex wave number. Hence, for the first reflection order wave train in material \(M_{1}\), the thermal wave has a propagation path of length \(2L_{1}\) leading to an attenuation by \(\exp(-2\sigma_{1}L_{1})\). The following figures include sketches of thermal wave trains and give the impression that there is a lateral diffusion in the material itself. This, however, does not hold true since only one-dimensional propagation is present. The sketches serve only for graphic visualization. Note that no bulk absorption of the absorbed radiation is considered in this article. This means that the photothermal effect provides surface heating only, i.e., thermal waves originate exclusively near \(x=0\). The presented references distinguish between the following two cases:  
1. Waves that are first reflected from the interface between \(M_{0}\) and \(M_{1}\), see Figure 5: Let \(a_{n}\) denote the \(n\)th reflection order wave and \(R_{0}(R_{1})\) the reflection coefficient at the interface between \(M_{0}\) and \(M_{1}\) (\(M_{1}\) and \(M_{2}\), respectively). Furthermore, let \(T_{0}(T_{0}^{\prime})\) denote the transmission coefficients at the interface between \(M_{0}\) and \(M_{1}\) in the upward (downward) direction. Since the single wave trains are reflected at the  
Figure 4: Thermal wave interference in one coating layer on a thermally thick substrate material. interface \"infinitely\" often, one obtains the following series corresponding to the surface temperature: \[\sum_{n=0}^{\infty}a_{n} =T_{0}A_{0}\sum_{n=1}^{\infty}(1+R_{0}^{n}R_{1}^{n}\exp\left(-2n \sigma_{1}L_{1}\right))\] \[=T_{0}A_{0}\sum_{n=0}^{\infty}R_{0}^{n}R_{1}^{n}\exp\left(-2n \sigma_{1}L_{1}\right)\] \[=T_{0}A_{0}\sum_{n=0}^{\infty}[R_{0}R_{1}\exp\left(-2\sigma_{1}L _{1}\right)]^{n}\] (9) \[=T_{0}A_{0}\frac{1}{1-R_{0}R_{1}\exp\left(-2\sigma_{1}L_{1}\right)}\] \[=\frac{T_{0}A_{0}}{1-R_{0}R_{1}\exp\left(-2\sigma_{1}L_{1}\right)},\] where the geometric series formula is used, since \(|R_{0}R_{1}\exp\left(-2\sigma_{1}L_{1}\right)|<1\) holds true. 2. For waves that are first reflected from the interface between \(M_{1}\) and \(M_{2}\), see Figure 6, let \(b_{n}\) denote the \(n\)th reflection order wave.