Total Angular Momentum Dichroism of the Terahertz Vortex Beams at the Antiferromagnetic Resonances

Terahertz vortex beams with different superposition of the orbital angular momentum $l=\pm 1$, $\pm 2$, $\pm 3$, and $\pm 4$ and spin angular momentum $\sigma =\pm 1$ were used to study antiferromagnetic (AFM) resonances in ${\mathrm{TbFe}}_3{({\mathrm{BO}}_3)}_4$ and ${\mathrm{Ni}}_3{\mathrm{TeO}}_6$ single crystals. In both materials we observed a strong vortex beam dichroism for the AFM resonances that are split in external magnetic field. The magnitude of the vortex dichroism is comparable to that for conventional circular dichroism due to $\sigma$. The selection rules at the AFM resonances are governed by the total angular momentum of the vortex beam: $j=\sigma +l$. In particular, for $l=\pm 2$, $\pm 3$, and $\pm 4$ the sign of $l$ is shown to dominate over that for conventional circular polarization $\sigma$.

Figure 1

Magnetic dichroism for circularly polarized light and vortex beams with $l=\pm 1$ in NTO. (a),(b) Normalized transmittance for circularly polarized light, ${\overrightarrow{e}}_L$ and ${\overrightarrow{e}}_R$, and for conventional linear polarization ${\overrightarrow{e}}_y$. (c),(d) Normalized transmittance for broadband vortex beams with ${\overrightarrow{e}}_{+1}$ (blue spectra) and ${\overrightarrow{e}}_{-1}$ (red spectra). All experimental data in (a)–(d) are taken at $B=\pm 0.47\mathrm{T}$ and $T=10\mathrm{K}$ and are normalized to that measured at $T=60\mathrm{K}$ in the same field: $I(\nu ,T)/I(\nu ,T=60\mathrm{K})$. (e) Schematics of the experimentally determined selection rules for the AFM resonances using both circularly polarized light and vortex beams with $l$ and $j$. The positive $\overrightarrow{B}$ is parallel with $\overrightarrow{k}$.

Figure 2

Vortex dichroism for higher orders of the OAM for (a) NTO at $T=30\mathrm{K}$ and (b) Tb FB at $T=19\mathrm{K}$. Normalized transmittance spectra for two linearly polarized LG vortex beams ${\overrightarrow{e}}_{l>0}$ (blue spectra) and ${\overrightarrow{e}}_{l<0}$ (red spectra) were taken at $B=-0.47\mathrm{T}$ that brings the AFM resonances close to ${\nu }_{l=\pm 3}$. The corresponding AFM resonances $\hslash {\mathrm{\Omega }}_M^{\pm }(B,T)$ are marked with vertical olive and magenta lines. The frequencies that correspond to ${\nu }_{l=\pm 2,\pm 3,\pm 4}$ are marked with bold dark yellow arrows. Note the absorption peak asymmetry for the ${\overrightarrow{e}}_{l=+3}$ and ${\overrightarrow{e}}_{l=-3}$ AFM resonances in (b). (c) Schematics for the measurement geometry with linear polarization ${\overrightarrow{e}}_y$ for the terahertz beam input. Sample (green hexagon) is placed between a pair of identical spiral plates with step height $h$.

Figure 3

Normalized transmittance spectra for the LG vortex beams for (a) $j>0$ and (b) $j<0$ in Tb FB at $B=-0.47\mathrm{T}$. The resonance frequencies for ${\nu }_{l=\pm 2}$, ${\nu }_{l=\pm 3}$, and ${\nu }_{l=\pm 4}$ are marked with bold dark yellow arrows. The temperatures were chosen to bring the AFM resonances $\hslash {\mathrm{\Omega }}_M^{\pm }(B,T)$ shown with vertical olive and magenta lines close to ${\nu }_{l=\pm 3}$ in both (a) and (b). Note that the corresponding AFM resonances coincide for $j=+2$ ($l=+3$, $\sigma =-1$), $j=+3$ ($l=+3$, $\sigma =0$), and $j=+4$ ($l=+3$, $\sigma =+1$) in (a) and for $j=-2$ ($l=-3$, $\sigma =+1$), $j=-3$ ($l=-3$, $\sigma =0$), and $j=-4$ ($l=-3$, $\sigma =-1$) in (b). The measurement geometry is the same as in Fig. 2 but with various input polarizations ${\overrightarrow{e}}_y$ for black, ${\overrightarrow{e}}_L$ for blue, and ${\overrightarrow{e}}_R$ for red spectra.

Figure 4

Total angular momentum dichroism in Tb FB revealed in the temperature dependencies of the AFM resonance frequencies $\hslash {\mathrm{\Omega }}_M^{\pm }(B,T)$ for different combinations of the $\overrightarrow{B}$ direction and the sign of $j$ for the vortex beams. (a) $j>0$, $B<0$, (b) $j<0$, $B<0$, and (c)$j<0$, $B>0$. Solid curves correspond to the AFM frequencies $\hslash {\mathrm{\Omega }}_M^{\pm }(B,T)$ measured using conventional circularly polarized light. The color code for the curves (olive and magenta) is the same as for $\hslash {\mathrm{\Omega }}_M^{\pm }(B)$ lines in Fig. 1. The measurement geometry is the same as shown in Fig. 2 but with various input polarizations ${\overrightarrow{e}}_y$ for black, ${\overrightarrow{e}}_L$ for blue, and ${\overrightarrow{e}}_R$ for red symbols.