## 2 Materials and Method
### Lateral and Axial Resolution  
The spatial resolution is an essential factor for interpreting hyperspectral Raman images. Unfortunately, only a few studies have dealt with this topic until now [5; 55; 56; 57; 58; 59; 60]. The spatial resolution of Raman microspectroscopy is governed mainly by the diffraction limit of light, and therefore dependents on the laser wavelength (\(\lambda\)) as well as on the numerical aperture (\(NA\)) of the objective [5]. In a first approximation, the theoretical lateral (\(d_{l}\)) and axial (\(d_{a}\)) resolution is linearly dependent on the wavelength of the incident laser and inversely proportional to the numerical aperture of the objective. It follows that a high spatial resolution can be achieved with lasers at shorter wavelengths and a high-magnification optics. With the 50\(\times\) long-working distance objective (\(NA\) = 0.5) used in this study, the diffraction limited theoretical lateral and axial resolution at the sample surface is 1.3 and 8.5 \(\upmu\)m, as given by \(d_{l}\)\(\approx\) 1.22 \(\lambda/NA\) and \(d_{a}\)\(\approx\) 4\(\lambda/NA^{2}\), respectively, where \(\lambda\) is the excitation wavelength [61]. In practice, the real resolution is certainly worse than the theoretically calculated values due to (i) an imperfect optics, (ii) light scattering at the surface and interfaces of the sample, and (iii) light refraction at the top window of the heating stage. The axial resolution can principally be improved by using a confocal aperture, reducing the volume from which the scattered light is collected by blocking radiation generated from the surrounding volume [55]. A convenient side effect is the reduction of the black body radiation, which is emitted by the sample at elevated temperatures by reducing the confocal  
Figure 8: A series of three hyperspectral Raman images taken between 838 \({}^{\circ}\)C and 927 \({}^{\circ}\)C which are generated with the classical least-squares (CLS) fitting method using in-house reference spectra that are (**a**) normalized and (**b**) not normalized before loading. The reference spectra in (b) are recorded at the same measurement setup and thereby the Raman cross-sections of the individual mineral phases are considered. Note that the normalization of the reference spectra causes mineral phases with low Raman scattering cross-sections such as CaO to be underrepresented in the image. aperture. A drawback of narrowing the confocal hole is that the overall intensity becomes lower (Figure 5). To interpret hyperspectral Raman images and especially to interpret the spectrum arising from a specific point in the map (for example grain boundaries), it is essential to know how much of the observed Raman signal originates within the focal volume and how the response tails off with depth [57]. Two factors are primarily important for the depth resolution, the volume of the laser focus, and how Raman photons generated within this volume are relayed back into the spectrometer via the confocal aperture [5]. The axial resolution obtainable in confocal microscopy can be calculated by the following equation [62]:  
\[d_{a}=\frac{2.2n\lambda}{2\pi NA^{2}} \tag{1}\]  
where \(n\) is the refractive index of the immersion medium. This equation is only valid with the optical focus at the surface of the sample. Based on the theoretic consideration, [PERSON] [5] showed that the true laser focus is always deeper within a transparent sample than the so-called nominal depth (\(\Delta\)), i.e., the distance below surface given by the \(z\) drive of the stage. The axial laser focus also broadens upon moving deeper into the sample. The depth resolution gets worse linearly with the nominal depth (distance below the surface) [55] and can be estimated from:  
\[d_{a}=\Delta\left[\left[\frac{NA^{2}(n^{2}-1)}{(1-NA^{2})}+n^{2}\right]^{\frac {1}{2}}-n\right] \tag{2}\]  
Apart from the point of focus also the depth of focus increases with increasing nominal depth, so it becomes impossible to obtain \"pure\" spectra at interfaces [61].