## 3 Results and Discussion  
### Surface Topography and Roughness  
Changes of surface topography after different depth etchings and their analysis are presented in Figure 1. The surface roughness was defined using the \(R_{q}\) parameter--the root-mean-square average of the height deviation from the mean image data plane. \(R_{q}\) was calculated as \(R_{q}=\sqrt{\frac{\sum_{i=1}^{N}Z_{i}^{2}}{N}}\), where \(Z_{i}\) is the height of the measured point from the average image plane and N is the number of image data points.  
It is clearly seen that deeper scratches were hidden under the initial relatively smooth polished surface (Figure 1a,g). They also were present after 50 nm (Figure 1b,h) and 100 nm (Figure 1c,i) etchings. The highest increase of surface roughness up to \(R_{q}=1.4\) nm was observed for the latter case. An uneven etching of the polished crystal surface might be explained in the following way. The scratches generated by the grinding and polishing processes remained filled with a redeposited mixture of crystal nanoparticles and remnants of the polishing slurry. This redeposited material had a lower density as it was demonstrated in the case of the polishing of fused silica [26]. It was expected that the etching rate of the redeposited mixture would be higher than for a pure Co:MALO crystal, which  
Figure 1: AFM scans of (**a**) unprocessed; (**b–e**) plasma-etched crystals. (**f**) Surface roughness \(R_{q}\) and (**g–k**) cross-sectional cuts (marked in green, short-dashed line).  
has a high hardness (8.2 on the Mohs scale [27]). Therefore, after initial exposure to the plasma, that lower density mixture was effectively removed, and cracks were uncovered with their almost real depth. During longer etchings of 250 nm and 400 nm depth, the plasma treatment started to reveal its polishing aspect on the crystal surface. The maximum depths of scratches were reduced from 5-6 nm (Figure 1f) to 3-4 nm (Figure 1j) and 2-3 nm (Figure 1k). Correspondingly, the surface roughness decreased down to \(R_{q}\) = 1.1 nm and \(R_{q}\) = 0.89 nm, respectively. The changes of \(R_{q}\) with the etching depth are summarized in Figure 1f. A plasma etching of 400 nm in this case was optimal in order to maintain the initial surface quality of the polished spinel crystal. It is very likely that in this particular case, a deeper etching of \(\sim\)500-600 nm would allow to reach a smooth surface, comparable to a plasma-unprocessed sample. However, this possible trend should be examined by additional investigations. The observed changes in surface roughness and topography might be different if other grinding and polishing recipes are used. This might lead to different depths and surface density of cracks, depending on the chosen polishing slurries, their fraction, number of process steps and their durations, applied forces, pH, temperature, etc. This might be illustrated by previously obtained roughness results of a very similar oxygen plasma etching of single-crystal YAG-polished substrates [28], which have comparable hardness characteristics (8.5 on the Mohs scale) as those of the Co:MALO crystal. There, a 60 nm etching resulted in \(R_{q}\) = 0.27 nm while the nonetched initial surface had a bigger \(R_{q}\) = 0.37 nm.