## 2 Materials and Methods
### Measuring Absolute Temperatures with Small-Band Pyrometers over Large Temperature Ranges  
In order to calculate the absolute temperature values from the measured voltage signal received from the small-band pyrometer, the emissivity of the material to be probed must be considered. The emissivity coefficient \(\varepsilon\) is not constant but depends on the temperature \(T\), as well as on the wavelength \(\lambda\) at which the pyrometer is operated. In the most general case, the emissivity depends on both quantities:  
\[\varepsilon=\varepsilon(T,\lambda). \tag{1}\]  
Figure 4: Experimental setup for measuring the temperature near the keyhole front wall. FPGA: field programmable gate array; CNC: computerized numerical control. The emissivity is a factor that determines the share of emitted radiation compared to a black body with an ideal emissivity of \(\varepsilon=1\). Considering a measuring spot with an area \(A_{\text{pyro}}\) and an effective spectrum [\(\lambda_{1}\), \(\lambda_{2}\)] for the pyrometer, the radiant power to be measured at a given temperature is described by the wavelength integral of [PERSON]'s law:  
\[I(T)=\int_{\lambda_{1}}^{\lambda_{2}}\varepsilon(T,\lambda)\;S(T,\lambda)\;d \lambda\text{ with }S(T,\lambda)=\frac{2\pi hc}{\lambda^{5}}\frac{1}{e^{\frac{h \varepsilon}{h\text{B}}}-1}, \tag{2}\]  
where \(h\) describes the [PERSON] constant, \(c\) represents the vacuum speed of light, and \(k_{\text{B}}\) is the Boltzmann constant. From Figure 5b, it becomes clear that the analytical approximation of the numerically calculated integral is more accurate for high temperatures. Therefore, the following temperature calculations must be considered more reliable for temperatures exceeding 2000 K, which corresponds to the range of expected keyhole temperatures. As there is no analytical solution for a finite integral of \(S(T,\lambda)\), the solution can only be acquired numerically. Considering the emissivity to only have a weak dependency on the wavelength within the given operational spectrum, the radiant power can be calculated numerically for each temperature within the expected temperature range and fitted with a potency function with the fitting parameters \(a\) and \(b\). It should be noted that this crucial assumption of the spectral independence of the emissivity cannot be assumed in general but only for specific intervals for which such a behavior is documented. \[P(T)=A_{\text{pyro}}\cdot I(T)\approx A_{\text{pyro}}\cdot a\cdot T^{b} \tag{3}\]  
As during the experiments, all temperature values are recorded with an emissivity preset of \(\varepsilon=1\), it is possible to define a calibration coefficient \(\xi\) for the temperature. \[\xi(T)=\frac{T(P,\varepsilon)}{T(P,1)}=\varepsilon(T)^{\left(-1/b\right)} \tag{4}\]  
This calibration coefficient enables the calculation of the absolute temperatures from a noncalibrated pyrometric voltage signal \(U_{\text{pyro}}\). Although, due to the neglection of detailed physics when evaluating the emissivity, this postmeasurement evaluation of the temperature \(T\) cannot be assumed to be as accurate as a material-specific calibration, it can be very helpful in investigating large temperature  
Figure 5: Theoretical heat-induced radiation: (**a**) array of curves showing the black body radiation for different temperatures, and (**b**) corresponding numerical wavelength integral for the operating spectrum of the pyrometer. ranges, as well as temperatures that may be too high for sophisticated calibration under conventional laboratory conditions. The Equations (3) and (4) would hold for an ideal pyrometer covering the whole temperature scale starting at \(T_{\text{min}}=0\) K. Due to technical constraints, the detectable temperatures have a lower and an upper boundary, \(T_{\text{min}}\) and \(T_{\text{max}}\), each corresponding to temperature values greater than 0 K. In this case, the calibration coefficient only acts upon the temperature difference within the measuring interval \(\Delta T=T-T_{\text{min}}\).