**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 μm × 1.6 μm × 2 nm. The DMI constant in the core (region II, <sup>|</sup>*x*<sup>|</sup> <sup>≤</sup> 400 nm) is 0.5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> J/m<sup>2</sup> surrounded by transparent cladding FM layers (region I, <sup>|</sup>*x*<sup>|</sup> <sup>≥</sup> 400 nm) with a lower index of refraction (*<sup>D</sup>* <sup>=</sup> <sup>−</sup>0.4 <sup>×</sup> <sup>10</sup>−<sup>3</sup> J/m2). Two DMI steps are formed with the critical angle *θ<sup>c</sup>* = 48◦.

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◦ greater than *θc*. 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.

**Figure 4.** (**a**) Schematic illustration of a spin-wave fiber. The inset shows the enlarged figure at the interface. (**b**) Schematic illustration of a spin-wave convex lens. In all of the above figures, the color map shows z component of the magnetization in the snapshot of micromagnetic simulations at some selected time. The spin-wave trajectories are represented by solid red lines with an arrow. The simulated propagation of the spin wave excited by a AC source in blue bars with an exciting frequency *f* = 100 GHz.

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 *<sup>D</sup>* = 0.5/−0.4 × <sup>10</sup>−<sup>3</sup> J/m2, respectively. The size of the sample presented here is 6 μm × 6 μm × 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].
