Electromagnetic theory of helicoidal dichroism in reflection from magnetic structures

We present the classical electromagnetic theory framework of reflection of a light beam carrying orbital angular momentum (OAM) by a magnetic structure with generic symmetry. Depending on the magnetization symmetry, we find a change in the OAM content of the reflected beam due to magneto-optic interaction and an asymmetric far-field intensity profile. This leads to three types of magnetic helicoidal dichroism (MHD), observed when switching the OAM of the incoming beam, the magnetization sign, or both. In cases of sufficient symmetries, such as domain walls and magnetic vortices, we establish analytical formulas that link an experimentally accessible MHD signal to the magneto-optical Kerr effect (MOKE) constants.

Figure 1

Sketch of the beam path for the reflection on the magnetic target.

Figure 2

Examples of magnetic samples with cartesian and polar coordinate systems: (a) two homogeneous and antiparallel magnetic domains and (b) counterclockwise magnetic vortex.

Figure 3

(Left) Near field intensity profiles and (right) decomposition on the LG-mode basis of a incoming beam with $\ell =1$ reflected by a magnetic dot of radius 500 nm in four different cases: [(a) and (b)] constant magnetization with incidence angle $\theta =0^{\circ }$ and [(c) and (d)] with $\theta ={45}^{\circ }$; [(e) and (f)] two antiparallel magnetic domains as in Fig. 2 with incidence angle $\theta =0^{\circ }$ and [(g) and (h)] with $\theta ={45}^{\circ }$. The focal spot has a size comparable to that of the magnetic dot. The bar plots values below 4% of the maximum have been forced to 0. Details of the numerical calculations are reported in Appendix pp1.

Figure 4

Computation of MHD for a magnetic dot of 500 nm with two antiparallel domains as in Fig. 2. (a) MHD-$\ell$ for $m=1$ and (b) for $m=-1$; (c) MHD-$m$ for $\ell =1$ and (d) for $\ell =-1$; (e) MHD-$\ell m$ as a difference of intensity maps corresponding to $\ell =1$ and $m=1$ and $\ell =-1$ and $m=-1$, and (f), MHD-$\ell m$, difference of intensity maps corresponding to $\ell =1$ and $m=-1$ and $\ell =-1$ and $m=1$. The MHD plots are normalized to the global maximum of their corresponding far-field intensity profiles. The incoming polarization is P and the angle of incidence is $\theta =5^{\circ }$. The computational details are given in Appendix pp1.

Figure 5

(a) MHD-$\ell$ and (b) MHD-$\ell m$ for $\theta =5^{\circ }$ and $h\nu =50.1$ eV. (c) Lineouts along the circles drawn in panel (a) (purple dots) and (b) (green squares), together with the fits from Eq. (37a)–(37c). Purple crosses and green diamonds correspond to the same lineouts obtained from the $\theta =0^{\circ }$ case. (d) $h\nu$ dependence of amplitude (blue, left axis) and phase (red, right axis) of the retrieved magneto-optical constant $r_0^t$ for $\theta =5^{\circ }$ (full symbols) and $\theta =0^{\circ }$ (three-branch crosses). The dashed lines are the corresponding input $r_0^t$ values. Amplitudes are normalized to 1, with same normalization constant for tilted and not tilted cases. Phases are set equal at resonance.

Figure 6

. Reflectivity coefficients maps used for the computation. $\lambda =23.5$ nm, angle of incidence $\theta =5^{\circ }$. From (a) to (d): $|r_{\mathit{PP}}|$, $|r_{\mathit{SS}}|$, $|r_{\mathit{PP}}{r}_0^t{m}_y|$, and $arg\left(r_{\mathit{PP}}{r}_0^t{m}_y\right)$. For (d), all points with magnitudes of less than 1% of the value at center (maximum), have been set to NaN. (e) Modulus and (f) phase of the following coefficients along the line at $y=0$. Plum cross: $r_{\mathit{PP}}$; blue dotted line: $r_{\mathit{SS}}$; purple plus: $r_0^t$.

Figure 7

(a) Numerical simulation of the relative position of an incident beam and a magnetic vortex (diameter of $10\mu \mathrm{m}$ and placed at $\theta ={48}^{\circ }$ for this simulation on a square of $17.2\mu \mathrm{m}$) and (b) the resulting reflected intensity in the far field (at 10 cm from the sample here, square of 15 cm).