## 2 Theoretical Framework and Micromagnetic Simulations  
The dynamics of the considered magnetic systems, it is governed by the well-known Landau-Lifshitz-Gilbert equation (LLG):  
\[\frac{\partial\mathbf{m}}{\partial t}=-\gamma\,\left(\mathbf{m}\times\mathbf{H}_{eff} \right)+\alpha\,\left(\mathbf{m}\times\frac{\partial\mathbf{m}}{\partial t}\right) \tag{1}\]  
where \(\mathbf{m}(\mathbf{r},t)=\frac{\mathbf{M}}{\mathbf{M}_{s}}\) is the unit vector along the local magnetization, \(\gamma\) is the gyromagnetic ratio, \(\alpha\) is the Gilbert damping constant. \(\mathbf{H}_{eff}\) is the effective field acting on the processing spins, which can be derived from the energy functional according to:  
\[\mathbf{H}_{eff}=-\frac{1}{\mu_{0}}\frac{\partial E_{tot}}{\partial\mathbf{m}} \tag{2}\]  
Therefore, it consists of different contributions, reflecting the energy terms recalled above:  
\[ \begin{split}\mathbf{H}_{eff}=\mathbf{H}_{0}+\mathbf{H}_{exch}+\mathbf{H}_{ani}+ \mathbf{H}_{ms}+\mathbf{H}_{DMI}=\\ =\mathbf{H}_{0}+\frac{2A}{\mu_{0}M_{s}}\
abla^{2}\mathbf{m}+\frac{2K_{0} }{\mu_{0}M_{s}}m_{e}\mathbf{e}_{z}+\mathbf{H}_{ms}+\frac{2D}{\mu_{0}M_{s}}\left[ \frac{\partial m_{z}}{\partial x}\mathbf{e}_{x}+\frac{\partial m_{y}}{\partial y }\mathbf{e}_{y}-\left(\frac{\partial m_{x}}{\partial x}+\frac{\partial m_{y}}{ \partial y}\right)\mathbf{e}_{z}\right]\end{split}  \tag{3}\]  
where \(\mathbf{H}_{0}\) is the external applied field, \(A\) is the exchange stiffness constant, \(K_{u}\) is the uniaxial perpendicular anisotropy constant, and \(D\) the DMI constant. \(\mathbf{e}_{x}\), \(\mathbf{e}_{y}\) and \(\mathbf{e}_{z}\) are the unitary vectors of the reference frame, chosen such that the z axis is perpendicular to the sample plane, as in Figure 1. The magnetostatic dipolar field \(\mathbf{H}_{ms}\) would be uniform in the case of an ellipsoidal body, only, so that for planar nanostructures it is usually calculated numerically, accounting for its non-uniformity and non-locality. The most general method to solve the LLG equation above are based on proper numerical techniques where it is spatially discretized using finite differences or finite elements methods. As a result, a discretized version of the effective field is obtained and the corresponding system of differential equations are solved within suitable time-stepping schemes. The Micromagnetic simulations were performed by the GPU-accelerated software Mumax3 [27]. For the isolated stripe of width w = 100 nm and thickness t = 1.5 nm, the simulated area was divided in 8 \(\times\) 20 \(\times\) 1 cells with a dimension of 5 \(\times\) 5 \(\times\) 1.5 nm\({}^{3}\). The periodic boundary conditions were applied in the x-direction, which represent the easy axis along which it is magnetized, to reproduce the infinite length of the stripe, taking into account that we are looking for stationary eigenmodes, with a uniform profile along the stripe length. (Please note that use of the periodic boundary conditions in the x-direction insures that the static internal field is uniform, avoiding the inhomogeneity that would originate by edges orthogonal to the magnetization. In second place, this is important to reduce the number of cells in the simulation, reducing the computation power and the time of the simulations.) For the elliptical magnetic dots, we kept a thickness t =1.5 nm, but considered three different lateral sizes a \(\times\) b, namely 100 \(\times\) 50 nm\({}^{2}\), 200 \(\times\) 100 nm\({}^{2}\) and 300 \(\times\) 150 nm\({}^{2}\), as sketched in Figure 1. Moreover, in order to better mimic the round shape of the dot borders, the size of the discrete cells was reduced to 2 \(\times\) 2 \(\times\) 1.5 nm\({}^{3}\). Both stripes and dots were saturated along their long axis (coincident with the x-direction) by an external field \(B_{0}\) = \(\mu_{0}H_{0}\) = 0.1 T. Then, spin waves were excited using a spatially uniform sinc-shaped field pulse, with an amplitude \(b_{z}\) = 10 mT directed along the z axis, able to activate modes up 50 GHz.