*2.2. Total Internal Reflection*

Analogously to the case of electromagnetic waves in photonics or acoustic waves in phononics, spin waves are also expected to be completely reflected by the interface when a spin-wave travels from a *denser* medium with a higher refractive index to a *thinner* medium with a lower refractive index known as TIR. TIR occurs when the incident angle *θ<sup>i</sup>* ≥ *θc*, where *θ<sup>c</sup>* is often called the critical angle. When *θ<sup>i</sup>* = *θc*, the refracted spin-wave travels along the interface between the two media or the angle of refraction *θ<sup>t</sup>* is *π*/2. According to the magnonic Snell's law in Equation (4), the critical angle can be expressed as

$$\theta\_{\mathcal{E}} = \arcsin\left(\frac{\mathbf{k}\_{\mathbb{g},t} - \delta}{\mathbf{k}\_{\mathbb{g},i}}\right),\tag{5}$$

where *δ* = *δ<sup>i</sup>* − *δt*. Specifically, Equation (5) shows that *θ<sup>c</sup>* equals *π*/2 when the DMI is homogenous (*D*<sup>1</sup> = *D*2), i.e., all incident spin waves are fully transmitted and no reflection occurs. Furthermore, when *δ* (the difference between DMI in two regions) is chosen to be large enough, a gap falls in between the two isofrequency circles and TIR occurs at all incident angles (i.e., *θ<sup>c</sup>* = 0). Equations (4) and (5) are the main analytical results in our paper.

### **3. Micromagnetic Simulations**

To test the validity of these analytical findings in realistic situations, micromagnetic simulations have been proven to be an efficient tool for the investigation of spin-wave dynamics in various magnetic textures and geometries. The simulations here are performed in the GPU-accelerated micromagnetic simulations program MuMax3 [73], which solves the time-dependent LLG Equation (1) based on the finite difference method. In our simulations, we used typical magnetic parameters for YIG at zero temperature [74]: *Ms* = 0.194 × <sup>10</sup><sup>5</sup> A/m, *<sup>A</sup>* = 3.8 pJ/m and *<sup>K</sup>* = <sup>10</sup><sup>4</sup> J/m3.

All simulations presented here were performed for a thin film of size *Lx* × *Ly* × *Lz*, which discretized with cuboid meshes of dimensions *lx* × *ly* × *lz*. The lateral dimensions of unit mesh (*lx* and *ly*) and the thickness of the film *Lz* are all smaller than the exchange length of YIG [3]. The simulations were implemented with the mesh size 2 × <sup>2</sup> × 2 nm3. The simulations were split into two stages: the static and dynamic stage. In the first stage, the static stage, the magnetic configuration is stabilized by minimization of the total energy starting from the random magnetic configuration with a high value of damping (*α* = 0.5).

In the dynamic stage of the simulations, the equilibrium magnetic configuration was used to excite a spin-wave beam that propagates through the film with a small damping parameter (*α* = 0.0005) to ensure long-distance propagation. During this step, a Gaussian type spin-wave beam was continuously generated by a harmonic dynamic external magnetic field following a Gaussian distribution function in a small rectangular region (red double-headed arrow shown in Figure 1). The detailed description of the Gaussian spin-wave beam generation procedure can be found in Ref. [75–77].

The Gaussian spin-wave beam is clearly visible and does not change with time after continuously exciting a sufficiently long time, which corresponds to a steady spin-wave propagation. Moreover, to avoid spin-wave reflection at the boundaries of the film, absorbing boundary conditions are applied on all boundaries by assigning a large damping constant (*α* = 1) near the edges.

In order to verify the magnonic Snell's law in Equation (4) for the spin-wave propagation through a DMI step interface, we focus on a 4 μm × 4 μm × 2 nm nanowire. The spin-wave beams presented here are all exchange-dominated spin waves with 200 nm beam width and 30 nm wavelength generated by an external AC magnetic field with frequency *f* = 100 GHz. The phase diagram of the critical angle *θ<sup>c</sup>* in the *D*<sup>1</sup> − *D*<sup>2</sup> plane is shown in Figure 2b. As *D*<sup>1</sup> < *D*<sup>2</sup> in the white region I, spin waves transmit from a thinner medium with a lower refractive index to a denser medium with a higher refractive index, and thus no TIR happens.

A gap falls in between the two isofrequency circles—in other words, TIR occurs in all incident angles when *δ* is chosen to be large enough as shown in the white region II. Figure 2c shows the critical angle as a function of the DMI constant *D*<sup>2</sup> in medium B, where the DMI constant of medium A is fixed at *<sup>D</sup>*<sup>1</sup> = <sup>4</sup> × <sup>10</sup>−<sup>3</sup> J/m2. All incident angle spin waves are totally reflected at a small *D*<sup>2</sup> corresponding to Region II in Figure 2b. After that, the critical angle increases monotonically with *D*<sup>2</sup> and shows a good agreement with the analytical results.

In Figure 3a, we show the refracted angle *θ<sup>t</sup>* as a function of the incident angle *θ<sup>i</sup>* from micromagnetic simulations (red triangle) and the prediction from Equation (4) (blue curve) with DMI constants *<sup>D</sup>*<sup>1</sup> = <sup>4</sup> × <sup>10</sup>−<sup>3</sup> J/m<sup>2</sup> and *<sup>D</sup>*<sup>2</sup> = 3.5 × <sup>10</sup>−<sup>3</sup> J/m2, respectively. The micromagnetic simulation for the five different incident angles, *θ<sup>i</sup>* = 17◦, 41.5◦, 44◦, 51.2◦ and 67◦, are displayed in Figure 3b–f. Figure 3b–d correspond to the refraction mode, and Figure 3e,f are the total reflection mode. Vertical dashed lines correspond to the interface at *x* = 2000 nm between medium A (left) and medium B (right).

The critical angle observed in our simulation is estimated to be *θ<sup>c</sup>* = 51.2◦ as shown in Figure 3e. It is important to comment that the spin-wave propagation direction is not strictly perpendicular to the spin-wave wavefronts in our simulation. That is to say, it is easy to observe strong anisotropy in the propagation of spin waves. Typically, for in-plane magnetized films, spin waves dynamics are anisotropic. This means that iso-frequency dispersion relation lines (IFDRLs, slices of dispersion relations for particular frequencies) are not circular. Therefore, the group velocity and phase velocities (parallel to the wavevector) are not parallel to each other, since the group velocity direction should be normal to the IFDRLs [78].

Such an intrinsic anisotropy called spin-wave collimation effect is common in ferromagnetic films with the magnetization fixed in the plane of the film by an external magnetic field or a strong in-plane anisotropy [75–77]. However, in the present case, the symmetry of the spin-wave dispersion relation is broken due to the presence of DMI, and the wave vector can be shifted from the direction perpendicular to spin-wave wavefronts [70]. This anisotropy decreases with the increasing frequency of the spin-wave but it is still present at the high frequency *f* = 100 GHz assumed in our simulations.

Moreover, a lateral shift *GH* of the spin-wave beam is observed at the interface between the reflected and the incident beams, which is called the Goos–H*a*¨nchen (GH) effect. The GH effect for spin-waves was reported in Refs. [75–78]. Furthermore, detailed investigations elucidating the role of inhomogeneous DMI on the GH shift in the reflection of the spin-wave at the interface were discussed in Ref. [22].

**Figure 3.** (**a**) The refracted angle as a function of the incident angle. Vertical dashed and solid lines correspond to the critical angle *θc*. (**b**–**f**) The micromagnetic simulations results for spin-wave beam reflection and refraction under different incident angles (**b**) *θ<sup>i</sup>* = 17◦, (**c**) *θ<sup>i</sup>* = 41.5◦, (**d**) *θ<sup>i</sup>* = 44◦, (**e**) *<sup>θ</sup><sup>i</sup>* <sup>=</sup> 51.2◦ and (**f**) *<sup>θ</sup><sup>i</sup>* <sup>=</sup> <sup>67</sup>◦. The DMI constants in medium A and B are *<sup>D</sup>*<sup>1</sup> <sup>=</sup> <sup>4</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> J/m2 and *<sup>D</sup>*<sup>2</sup> <sup>=</sup> 3.5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> J/m2, 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.
