**5. Conclusions**

In conclusion, we both theoretically and numerically studied spin-wave beam propagation in a two-dimensional ferromagnetic film with an inhomogeneous interfacial DMI. Utilizing a spatially varied magnonic refractive index introduced by the variation of DMI, a magnonic metamaterial or graded-index magnonic material can be realized. Snell's law

and TIR for spin waves were predicted with a DMI step interface. Moreover, we designed and studied spin-wave fibers and spin-wave lenses via micromagnetic simulations. We believe that our findings shall open up alternative directions for building reconfigurable, stabilized and scalable spin-wave circuitry in magnon introspection devices.

However, the parameters that we adopted in our simulations to investigate spin-wave propagation in the presence of spatially modulated DMI are not meant to represent a specific material but rather to explore the physical conditions under which the spin-wave total reflection occurs. From the materials standpoint, we acknowledge that the dual requirements of low damping and large DMI may seem incompatible since spin–orbit coupling originating from the adjacent heavy metal layer is detrimental to the former but central to the latter.

The excitation of short-wavelength propagating spin waves with a wavelength of 45 nm in a YIG thin film covered by Co25Fe75 nanowires was reported in a recent experiment [82], where the effective damping was only enhanced to about 10<sup>−</sup>3. Recent progress in materials science has proven that certain magnetic insulators do possess sizable DMIs either in their bulk [83–85] or at the interface [86–88]. Although these values remain small (typically ∼ <sup>10</sup>−3–10−2mJ/m2), these results open interesting perspectives for the achievement of large DMIs in magnetic insulators.

**Author Contributions:** Conceptualization, F.Z. and A.M.; methodology, F.Z.; software, F.Z.; validation, F.Z. and A.M.; formal analysis, F.Z., H.L., Z.C. and A.M.; investigation, F.Z.; resources, H.L. and A.M.; data curation, F.Z.; writing—original draft preparation, F.Z.; writing—review and editing, H.L., Z.C. and A.M.; visualization, H.L., Z.C. and A.M.; supervision, A.M.; project administration, A.M.; funding acquisition, H.L., Z.C. and A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** F.Z. and H.L. acknowledge the support from National Key R&D Program of China (No. 2018YFB0407600), Henan University (No. CJ3050A0240050) and National Natural Science Foundation of China (No. 11804078). F.Z. was supported by King Abdullah University of Science and Technology (KAUST). A.M. acknowledges support from the Excellence Initiative of Aix-Marseille Université— A\*Midex, a French "Investissements d'Avenir" program. Z.C. acknowledges the support from Grant No. ARC (DP190100150).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available upon reasonable request from the authors.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

