# Tailored Magnetic Linear Birefringence in Wedge-Shaped Co Nanocluster Assemblies
## 4 Discussion  
In the light of results shown in Section 3.4, the detailed investigation of MLB and the origin of M\({}_{z}\) in our Co cluster assemblies requires an additional analysis of those \(\delta\) raw data, shown in Figure 8. In general terms, for an electromagnetic wave propagating through a ferromagnetic medium, its dielectric tensor can be expanded in powers of \(H\). Limiting ourselves to first- and second-order terms, MOVE depends on M\({}^{2}\), while MOFE is linear with the magnetization [7,27]. Thus, these even and odd contributions can be extracted from to the \(\delta\) raw data of Figure 8 after applying the following symmetry relations:  
\[\delta_{(F/V)}=[\delta_{inc}(H)\mp\delta_{dec}(-H)]/2 \tag{1}\]  
where \(\delta_{(inc/dec)}\) denotes the loop branch of \(\delta\) when \(H\) is increasing or decreasing, respectively [28,29]. The resulting \(\delta_{V}\) and \(\delta_{F}\) loops for P6, P4, and P3 are shown in respective graphs of Figure 9. Summarized data for P5 and P2 are included in the Supplementary Material. First, the predicted two intensity peaks in MOVE are clearly seen now, their magnitude increasing as both \(H\) and TGD tend to be coincident (\(\Phi=0^{\circ}\)). That turns that direction into an effective anisotropy axis, where M reversal is conducted perpendicular to the Co nanocluster assembly and completely different to the easy axis (\(\Phi=90^{\circ}\)) [10,30]. The progressive approach of those peaks indicates an increasing dominance of the coherent rotation to reverse M, as \(H\) is parallel to TGD. For P3, they are merged in a single peak at \(\Phi=0^{\circ}\), showing an _ideal_ magnetic linear birefringence [8]. Similar results can also be observed in P2. In these cases, both easy and hard axes are univocally related to respective perpendicular and parallel directions to grad. Thus, the existence of MLB and UMA in our ultrathin Co nanoclusters assemblies, and its close relation with that thickness gradient, can be definitively settled after measuring and correlating their MOKE and MOVE. Additionally, these MOFE results show a novel evidence of emerging M\({}_{z}\) under in-plane magnetic field for disordered 2D magnetic media, not reported previously in the literature concerning ordered two-dimensional systems or Co cluster assemblies, and should require an additional discussion to explain those Faraday loops. [PERSON] found the possibility to tune the magnetization orientation in multilayers, by means of a graded magnetic anisotropy among alternate layers along z-axis [20]. Later, that method was extended to continuous films, where that graded anisotropy could be achieved with growing samples with a compositional gradient, and subsequently a graded magnetization [31,32]. Basically, the magnetization orientation resulted from a competition between soft and hard magnetic layers, whose magnetic exchange could be tuned with some of those grading methods. In our work, the previous discussions of MOKE and MOVE data showed the thickness gradient to be a source of magnetic linear birefringence and UMA. Particularly,  
Figure 9: Dependence of Voigt and [PERSON] rotation angle on the magnetic field and \(\Phi\) angle for three locations on the Co nanocluster assembly: Panels (**a**,**b**) correspond to P6, panels (**c**,**d**) to P4, and panels (**e**,**f**) to P3, respectively. In all cases, \(\Phi\) value is indicated in each panel. the magnitude of both effects has been proved to increase strongly upon decreasing thickness along TGD. Thus, the thickness gradient also yields a graded magnetization in z-axis, weakening the in-plane magnetic anisotropy and allowing the occurrence of a subtle M\({}_{z}\) contribution. This small out-of-plane magnetization is enough to give a detectable Faraday rotation, as seen in Figure 9. We have measured M\({}_{z}\) with MOFE only for P3 and P2 samples of 2.3 and 5.4 nm average thickness, respectively, i.e., for intermediate values of thickness.