Snell’s Law for Spin Waves

We report the experimental observation of Snell’s law for magnetostatic spin waves in thin ferromagnetic Permalloy films by imaging incident, refracted, and reflected waves. We use a thickness step as the interface between two media with different dispersion relations. Since the dispersion relation for magnetostatic waves in thin ferromagnetic films is anisotropic, deviations from the isotropic Snell’s law known in optics are observed for incidence angles larger than $25°$ with respect to the interface normal between the two magnetic media. Furthermore, we can show that the thickness step modifies the wavelength and the amplitude of the incident waves. Our findings open up a new way of spin wave steering for magnonic applications.

Figure 1

(a) Sketch of the sample with the $z$ axis not drawn to scale. The red arrow indicates the direction of the externally applied magnetic field which is aligned parallel to the coplanar wave guide (yellow). The latter is used to excite spin waves which propagate perpendicular to it. (b) Top view of (a) with exemplary data acquired by TRMOKE. The green arrows show the wave vectors $k_1$, $k_2$, and $k_3$ relevant for the analysis. ${\phi }_{1-3}$ denote the angles of the wave vectors with respect to the external field, while ${\theta }_{1-3}$ denote the angles with respect to the interface normal. The indices 1–3 correspond to the incident, refracted, and reflected wave, respectively.

Figure 2

Experimental results for two samples. In (a), the incoming wave has an angle of ${\theta }_1=20°$ with respect to the interface normal, and in (b), the angle is ${\theta }_1=40°$. (c) and (d) show the corresponding plane wave fits. The $x$ and $y$ axes give the dimensions of the images while the color code provides a scale for the dynamic magnetization component in arbitrary units. The gray line marks the step between thick (on the left) and thin (on the right) Permalloy films and the white boxes indicate the area of the fit. The images are recorded at a fixed frequency of $\omega =2\pi \times 8\mathrm{GHz}$ and an external field of ${\mu }_{0}H=54\mathrm{mT}$ along the wave fronts of the incoming wave. The color scale is cropped in order to enhance the contrast in the areas with lower signal. To emphasize the reflected waves, (e) and (f) show line scans along the blue lines in (a) and (b). The blue dots are interpolated from the data; the red lines are fits extracted from (c) and (d), see main text. “Distance” indicates the distance from the lower left to the upper right of the blue lines.

Figure 3

(a) Refracted angle ${\theta }_2$, (b) refracted wave vector $k_2$, (c) reflected angle ${\theta }_3$, and (d) reflected wave vector $k_3$, all shown versus incident angle ${\theta }_1$. In all graphs, the blue dots are experimental values measured with TRMOKE, while red dots are measured with $\mu$-BLS. The orange line shows Snell’s law for an isotropic dispersion relation, and the green and purple curves show Snell’s law for spin waves. The latter are calculated with the help of the anisotropic dispersion relation, Eq. (2), reflecting the different experimental conditions: The $\mu$-BLS data are measured at an external field of ${\mu }_{0}H=41\mathrm{mT}$ and an excitation frequency of 8.1 GHz, while TRMOKE data were recorded at an external field of ${\mu }_{0}H=54\mathrm{mT}$ and an excitation frequency of 8.0 GHz. The purple curve is not shown in (a), since it overlaps with the green curve. The errors are the result from least square fitting.