## 2 Concept of the Photothermal Approach  
During and after heating of the surface of a solid, a temperature gradient arises between the surface and the bulk, leading to a heat flow into the volume of the solid. This heat flow can be described by the well-known heat equation provided by numerous textbooks (see, for example, [20; 21]). Since it is a partial differential equation of the second grade, its solutions are mainly determined by spatial and temporal boundary conditions. However, for certain conditions and simplifications, analytical solutions have been found and published in textbooks [20; 21]. For the intended application, i.e., the determination of thermal properties in an on-site measurement, a relatively simple equation can be derived:  
\[\Delta\mathrm{T}(\mathrm{t})=\frac{Q}{\sqrt{\pi}\cdot\varepsilon\cdot\sqrt{t}} \tag{1}\]  
Equation (1) describes the surface temperature decrease after heating the surface with a Dirac pulse with the absorbed spatial energy density (\(Q\), given in \(\mathrm{J}/\mathrm{m}^{2}\)) at \(\mathrm{t}=0\). Here, \(\Delta\mathrm{T}(\mathrm{t})\) stands for the temperature change in relation to the initial surface temperature (\(\mathrm{T}0\)) before the pulse. The assumed simplifications are:  
1. The material properties and surface heating are completely isotropic;
2. The initial temperature distribution is isotropic and equal to the ambient temperature;
3. Theoretically, the material must be semi-infinitely thick, but in reality, it is sufficient for the temperature on the rear side to remain unchanged during the observation period;
4. The entire surface is heated, and edges are not considered, which means that heat flows only in one direction, i.e., perpendicular to the surface; this is a one-dimensional consideration of the depth profile of the temperature;
5. The material is opaque, which means that the entire heat energy is absorbed within an infinitesimally thin surface layer;Thermal heat losses due to convection and irradiation are neglected. The thermal properties of the material are included in \(\varepsilon\), which is the thermal effusivity formed by the simple product of density (\(\rho\)), thermal conductivity (\(k\)), and specific heat capacity (\(c\)):  
\[\varepsilon=\sqrt{k\rho c} \tag{2}\]  
Thus, any property of a material that affects one of these material parameters will cause a change in the thermal effusivity. In the case of mixtures of materials, the effusivity of the mixture is affected by the mixture ratio of the respective components (if they have different material properties, which is usually the aim of a mixture). Coming back to a concept for an on-site inspection: Equation (1) can be used to determine the actual material effusivity (\(\varepsilon\)) when \(Q\) is known and \(T\)(t) is correctly recorded. The implementation in practice faces some challenges, which are compiled in Table 1, together with their solutions. Thus, a technical implementation of the photothermal approach for concrete surfaces requires only three steps:  
1. Well-defined heating of a certain area with a short pulse (\(\leq\)1 s);
2. Measurement of the actual reflection coefficient in this area for the applied pulse; and
3. Measurement of the temperature decrease for about 1 min after the pulse with at least 0.1 s temporal resolution. The basic setup is illustrated in Figure 1.  
The final determination of the effusivity value can be achieved by a non-linear regression of Equation (1) or, even more simply, by calculating the value at a certain time. A successful application of this simple model was also demonstrated earlier to determine the absorbed energy of flash lamps [22]. In addition, considering the curve shape of the temperature transient allows for verification of whether the use of Equation (1) is useful for the recorded measurement dataset.