# Magnonic Metamaterials for Spin-Wave Control with Inhomogeneous Dzyaloshinskii-Moriya Interactions
## 4 Spin-Wave Fiber and Lens  
We now turn to the realization of the spin-wave fiber and spin-wave lens, which are important to manipulate spin waves in spin-wave circuitry. Two kinds of spin-wave fiber have been proposed and designed, one based on the TIR by the magnetic domain wall [37] and the other based on the TIR in the medium with a uniform external magnetic field [51]. Here, utilizing the TIR at the interface with a DMI step, we propose a new type of spin-wave fiber as shown in Figure 4a of system size \(12\)\(\upmu\)m \(\times\)\(1.6\)\(\upmu\)m \(\times\)\(2\) nm. The DMI constant in the core (region II, \(|x|\leq 400\) nm) is \(0.5\times 10^{-3}\) J/m\({}^{2}\) surrounded by transparent cladding FM layers (region I, \(|x|\geq 400\) nm) with a lower index of refraction (\(D=-0.4\times 10^{-3}\) J/m\({}^{2}\)). Two DMI steps are formed with the critical angle \(\theta_{c}=48^{\circ}\).  
Figure 3: (**a**) The refracted angle as a function of the incident angle. Vertical dashed and solid lines correspond to the critical angle \(\theta_{c}\). (**b–f**) The micromagnetic simulations results for spin-wave beam reflection and refraction under different incident angles (**b**) \(\theta_{i}=17^{\circ}\), (**c**) \(\theta_{i}=41.5^{\circ}\), (**d**) \(\theta_{i}=44^{\circ}\), (**e**) \(\theta_{i}=51.2^{\circ}\) and (**f**) \(\theta_{i}=67^{\circ}\). The DMI constants in medium A and B are \(D_{1}=4\times 10^{-3}\) J/m\({}^{2}\) and \(D_{2}=3.5\times 10^{-3}\) J/m\({}^{2}\), respectively. The color map shows the z component of the magnetization in the snapshot of micromagnetic simulations at some selected time. The black solid lines correspond to the rays of the incident and refractive beams. The red rectangular area is the excitation area of the spin-wave, and the exciting field frequency is \(f=100\) GHz.  
The upper one is located at \(x=-400\) nm and the lower one is located at \(x=400\) nm. In Figure 4a, the spin-wave beam born at the middle of the nanowire (blue bar) propagates inside the core with an incident angle of \(52^{\circ}\) greater than \(\theta_{c}\). This is different from the unidirectional spin-wave fiber based on domain walls [37]. The fiber here is fully bidirectional for both right/left-moving spin-wave beams when the incident angle is greater than the critical angle.  
More interestingly, a bound spin-wave mode propagates a long distance inside the DMI step interface as illustrated in the inset of Figure 4a. Similar to the bound spin wave mode inside a domain wall, which acts as a local potential well for spin waves [31; 79], a DMI step also creates an imaginary potential well for the bound spin wave mode [80]. The details will be discussed in our future publications.  
A fundamental building block in spin-wave circuitry is a spin-wave lens that can focus or diverge spin-wave beams. Since the dispersion relation strongly depends on DMI constant, we propose a spin-wave lens by tuning the DMI distribution in the film. Figure 4b illustrates an example of a spin-wave convex lens (region I inside red dotted lines) with a DMI constant inside/outside the lens \(D=0.5/-0.4\times 10^{-3}\,\mathrm{J}/\mathrm{m}^{2}\), respectively. The size of the sample presented here is \(6\ \upmu\ \times\ 6\ \upmu\ \times\ 2\) nm.  
Comparing the solid blue lines along the incident spin-wave beam propagation direction and the spin-wave trajectory (solid red lines with an arrow) passing through the lens, it is easy to observe focusing in the propagation of spin waves. Furthermore, a concave spin-wave lens can be obtained by reversing the DMI constants of regions I and II, which can be used to spit the spin-wave beams. Consequently, we believe that the inhomogeneous DMI can be a good playground to study spin-wave beam propagation [81].