Observation of a Goos-Hänchen-like Phase Shift for Magnetostatic Spin Waves

In optics, a light beam experiences a spatial shift in the beam plane upon total internal reflection. This shift is usually referred to as the Goos-Hänchen shift. When dealing with plane waves, it manifests itself as a phase shift between an incoming and reflected wave that depends on the wave vector component along the interface. In the experiments presented here, plane spin waves are excited in a 60-nm-thick Permalloy film and propagate towards the edge of the film. By means of time-resolved scanning Kerr microscopy, we are able to directly detect a phase shift between the incoming and reflected wave. With the help of a numerical model, we show that this phase shift naturally occurs for spin waves in the dipolar regime.

Figure 1

(a) Sketch of the experiment: Waves with ${\mathbf{k}}_{\mathrm{in}}$ are excited by the CPW. Upon reflection, the side the spin waves are traveling on is switched, and a phase shift $\mathrm{\Delta }$ is observed. Brown and green curves mark the cut lines with the sample edge. A static external field is applied along the edge in the $y$ direction. (b) shows the reflectivity of the 30° sample (green $k$). All other angles are shown as blue, orange, and violet arrows for 10°, 20°, and 60°. A line scan at $y=20\mu \mathrm{m}$ [white line in (b)] is shown in (c), and its gradient (d) is fitted with a Gaussian (transparent yellow line) to determine the position of the step [red line in (b)–(d)].

Figure 2

(a) shows typical experimental data, recorded at ${\mu }_{0}H=+42.5\mathrm{mT}$ with an incident angle of 30°. The color code provides a scale for the dynamic out-of-plane component. We fit the interference pattern in the white box as described in the main text. Green and brown lines correspond to the phase fronts of incoming and reflected waves, respectively. (b) shows the fit evaluated along the interface. The phase difference in real space $\mathrm{\Delta }$ is evaluated as the distance between the horizontal red lines. Its dependence on the static external field is depicted in (c). $R$, shown in (d), is defined as the ratio of the amplitudes of both waves, corresponding to the dashed lines in (b).

Figure 3

(a) and (b) show reflection coefficient $R$ and phase shift $\varphi$, respectively, recorded on the 30° sample for different external fields. Both field directions ($+y$ and $-y$) together with their geometric mean are shown. (c) and (d) summarize the results on the phase shift for all samples. Here, the external field is along $+y$. The solid lines are numerical calculations. (d) magnifies the gray rectangle of (c).

Figure 4

A part of the simulated profile of $m_x$ for $k_y=-1\mu {\mathrm{m}}^{-1}$ is shown in (a). $m_x$ points to the left (right) as indicated by the arrows below; in between, positive (negative) dynamic magnetic charges occur. The edge is located at $x=50\mu \mathrm{m}$. The inset magnifies the region close to the edge. A plane wave is fitted far away from the edge and is shown in red. By comparing to the simulation, an evanescent wave, shown in orange, is revealed that accounts for the uncompensated dynamic charges at the edge (red “$-$”). (b) shows the phase shift $\varphi$ as a map of wave vector components $k_x$ and $k_y$. Blue, orange, green, and violet lines correspond to the respective lines in Fig. 3. Teal, pink, gray, and yellow lines have a constant wave vector magnitude $k$. Their line cuts are shown versus $k_y$ in (c).