Figure 1

Schematic of (a) the linear Faraday rotation (LFR) and (b) the third-order Faraday rotation (TFR). The plane of polarization of the outgoing wave at $\omega$ (LFR) or $3\omega$ (TFR) is rotated with respect to the plane of polarization of the ingoing light wave at $\omega$. The rotation is caused by the spontaneous or field-induced magnetization $M_z$ parallel to the direction of light propagation $z$. (c) TFR in a EuO(001) film for different $M_z$. The measurement shows the intensity of the frequency-tripled light as function of the angular position ${\phi }_A$ of a linear polarization filter. The nonlinear rotation angle is derived from the value of ${\phi }_A$ at the maximum of the intensity of the frequency-tripled light. The data for $\pm M_z$ and $M_z=0$ were obtained in fields of $\pm 3$ T and 0 T applied along the $z$ axis of the EuO(001) film. The lines show sinusoidal fits. The TFR is investigated for $x$-polarized incident light at 10 K and $3\hslash \omega =2.0$ eV.Reuse & Permissions

Figure 2

Temperature dependence of the TFR of a EuO(001) film at $3\hslash \omega =2.0$ eV. (a) Determination of the nonlinear rotation angle for temperatures between 10 K and 200 K. For ${\mu }_0{H}_z=+3$ T (closed circles) and ${\mu }_0{H}_z=-3$ T (open circles) this angle is indicated by the respective triangles. (b) Temperature dependence of the nonlinear susceptibilities ${\chi }_{yxxx}$ and ${\chi }_{xxxx}$ for ${\mu }_0{H}_z=+3$ T. (c) Comparison of the rotation angle derived from (a) (open squares) and (b) (open circles). The hexagons refer to the LFR at 0.66 and 2.0 eV.Reuse & Permissions

Figure 3

Magnetic-field dependence of the TFR of a EuO(001) film at $3\hslash \omega =2.0$ eV. (a) Determination of the nonlinear rotation angle as in Fig. 2 for magnetic fields between 0 T and $\pm 2.5$ T. (b) Magnetic-field dependence of the nonlinear susceptibilities ${\chi }_{yxxx}$ and ${\chi }_{xxxx}$ at 10 K. Data represented by closed (open) symbols were taken with increasing (decreasing) field. (c) Comparison of the rotation angle derived from (a) (open squares) and (b) (closed and open circles). The magnetic-field dependence of the out-of-plane magnetization $M_z$ derived from (b) is shown by triangles. It agrees well with published magnetization measurements.[15]Reuse & Permissions

Figure 4

Thickness dependence of the TFR measured on EuO(001) films at 10 K and $3\hslash \omega =2.0$ eV. Data were corrected by the measured THG contribution from the a-Si cap layer. The expected linear dependence of the LFR at 2.0 eV is also plotted as a reference.Reuse & Permissions

Figure 5

Spectral dependence of nonlinear susceptibilities and TFR of a EuO(001) film. (a) Spectrum of ${\chi }_{yxxx}$ and ${\chi }_{xxxx}$ at 10 K in a magnetic field ${\mu }_0{H}_z=+3$ T. The inset shows schematics of the spin-dependent electronic band structure of ferromagnetic EuO and of the optical transitions of the LFR and the TFR in the depicted spectral range. (b) Spectral dependence of the TFR derived from the data in (a) by $|{\theta }^{\left(3\right)}|\approx \arctan (|{\chi }_{yxxx}|/|{\chi }_{xxxx}\left|\right)$ (see text). Inset: Spectral dependence of the LFR at 5 K for a EuO film of 153 nm, taken from Ref. 18.Reuse & Permissions