We investigated the generation and control of vectorial and multilevel magnetization states by the interaction of the vortex-encoded vectorial beam with circular polarization and the anisotropic magneto-optical material. The configuration of this idea is shown in Figure 1. In isotropic medium 1, such as the materials in the solid immersion lens system, an incident structured light is first collected by a high numerical aperture (NA) objective lens and then tightly focused towards the interface at *z*=−*d*, refracting into the anisotropic nonabsorbing magneto-optical medium 2, at the focal plane *z*=0, in which there will be optically stimulated magnetization excitation via the IFE. In particular, the structured light is circularly polarized with helical phase wavefront described by Ψ(*ϕ*)=exp(*im**ϕ*), where *ϕ* is the azimuthal coordinate in the transverse plane and *m* is the topological charge. Light fields in media 1 and 2 are parameterized by wave vectors ${\widehat{k}}_{\mathrm{1,2}}$ and polarization vectors ${\widehat{s}}_{\mathrm{1,2}}$ and ${\widehat{p}}_{\mathrm{1,2}}.$ These unit vectors form orthonormal sets, i.e. ${\widehat{p}}_{\mathrm{1,2}}\times {\widehat{s}}_{\mathrm{1,2}}={\widehat{k}}_{\mathrm{1,2}}.$ The high NA objective focusing geometry allows achieving the smallest possible focal spot, thus yielding a high-resolution magnetization pattern.

Figure 1: (Color online) Scheme for the optomagnetic recording via the IFE under tight focusing condition, with refraction at the interface in *z*=−*d* and focal plane in *z*=0; ${\widehat{k}}_{1}$ and ${\widehat{k}}_{2}$ are unit wave vectors in media 1 and 2, respectively; ${\widehat{s}}_{1},\text{\hspace{0.17em}}{\widehat{p}}_{1}$ and ${\widehat{s}}_{2},\text{\hspace{0.17em}}{\widehat{p}}_{2}$ are the associated unit polarization vectors perpendicular and parallel to the plane of incidence, respectively. Medium 2 is characterized by the uniaxial symmetry axis $\widehat{a}$ perpendicular to the interface.

Here, we consider the axially birefringent medium with uniaxial symmetry axis $\widehat{a}$ along the optical axis (*z*-axis). In the principal axes frame, this medium is characterized by the relative permittivity as $\widehat{\epsilon}=\text{diag\hspace{0.17em}}\mathrm{(}{n}_{o}^{2}\mathrm{,}\text{\hspace{0.17em}}{n}_{o}^{2}\mathrm{,}\text{\hspace{0.17em}}{n}_{e}^{2}\mathrm{)},$ where *n*_{o} and *n*_{e} denote the ordinary and extraordinary refractive indices, respectively. The difference between them is referred to as birefringence, Δ*n*=|*n*_{e} – *n*_{o}|.

Generally, intensive light with circular polarization travelling into a nonabsorbing material induces an effective magnetic field, thereby leading to the corresponding magnetization ${M}_{k}\mathrm{(}0\mathrm{)}={\chi}_{ijk}{E}_{i}\mathrm{(}\omega \mathrm{)}{E}_{j}^{\mathrm{*}}\mathrm{(}\omega \mathrm{)}=i{e}_{ijl}{\gamma}_{lk}{E}_{i}\mathrm{(}\omega \mathrm{)}{E}_{j}^{\mathrm{*}}\mathrm{(}\omega \mathrm{)}$ [22], where *e*_{ijl} is the 3D Levi-Civita symbol, third-rank tensor *χ*_{ijk} and second-rank tensor *γ*_{lk} are the optomagnetic susceptibilities, **E**(ω) is the electric field of the light, and asterisks are the complex conjugates. This nonthermal optomagnetic effect, IFE, is a nonlinear optical effect or a linear inverse magneto-optical phenomenon. Microscopically, the IFE is a stimulated Raman-like coherent optical scattering process. High-intensity light fields of elliptical or circular polarization shift the magnetic ground states and combine these ground states with some excited states, eventually giving rise to the light-induced magnetization [23], [24], [25]. The IFE has been investigated in various natural and artificial materials, including diamagnetic, paramagnetic, ferromagnetic, ferromagnetic, and antiferromagnetic systems in dielectrics, semiconductors, and metals, covering both amorphous and crystalline structures [11], [13], [14], [26], [27], [28]. Here, we consider the IFE in the anisotropic dielectric medium, which can be paramagnetic crystals [29] or magnetically ordered crystals in their paramagnetic or diamagnetic phases [13], [26], [30], [31], [32]. Dielectric medium is the promising candidate for the applications via ultrafast optomagnetic process as the thermal effects caused by an intense laser irradiation occur only in the nanosecond range [13]; thus, the thermal effects can be neglected in our discussion. It is worth mentioning that in anisotropic medium *γ*_{lk} is generally a second-order tensor. That is, the optomagnetic response is generally anisotropic as well [33], [34], [35], [36].

The sharp focusing beam in the focal region within the anisotropic medium can be numerically calculated by the Debye diffraction method [37], [38], [39], [40]. Owing to the optical anisotropy, the incident light will split into two eigenmodes in the birefringent medium. We pay attention only to the anisotropic medium with small birefringence so that the radially extraordinary (*p*) and tangentially ordinary (*s*) modes are still combined to form a single focal spot, which is vital to many applications especially for the all-optical magnetic recording. In this situation, the refractive index of the axially birefringent medium, medium 2 in Figure 1, can be approximately represented by *n*_{o}. Therefore, the electric field in the focal region of this axially birefringent dielectric can be expressed as E(r_{2})=E^{p}(r_{2})+E^{s}(r_{2}) [41] (see details of the derivations in Section 1 of Supplementary Material):

$$\begin{array}{c}\left(\begin{array}{l}{E}_{x}^{p}\\ {E}_{y}^{p}\\ {E}_{z}^{p}\end{array}\right)\mathrm{=}{\displaystyle {\int}_{0}^{\alpha}}d{\theta}_{1}\mathrm{sin}{\theta}_{1}\sqrt{\mathrm{cos}{\theta}_{1}}{e}^{i{k}_{2}{z}_{2}\mathrm{cos}{\theta}_{2}}\\ \left(\begin{array}{c}\mathrm{cos}{\theta}_{2}\mathrm{(}-i{I}_{m}^{p}-\frac{i}{2}\mathrm{(}\sigma +1\mathrm{)}{I}_{m+2}^{p}+\frac{i}{2}\mathrm{(}\sigma -1\mathrm{)}{I}_{m-2}^{p}\mathrm{)}\\ \mathrm{cos}{\theta}_{2}\mathrm{(}\sigma {I}_{m}^{p}-\frac{1}{2}\mathrm{(}\sigma +1\mathrm{)}{I}_{m+2}^{p}-\frac{1}{2}\mathrm{(}\sigma -1\mathrm{)}{I}_{m-2}^{p}\mathrm{)}\\ i\mathrm{sin}{\theta}_{2}\mathrm{(}\mathrm{(}1+\sigma \mathrm{)}{I}_{m+1}^{p}+\mathrm{(}1-\sigma \mathrm{)}{I}_{m-1}^{p}\mathrm{)}\end{array}\right)\mathrm{,}\end{array}$$(1)

and

$$\begin{array}{c}\left(\begin{array}{l}{E}_{x}^{s}\\ {E}_{y}^{s}\\ {E}_{z}^{s}\end{array}\right)={\displaystyle {\int}_{0}^{\alpha}}d{\theta}_{1}\mathrm{sin}{\theta}_{1}\sqrt{\mathrm{cos}{\theta}_{1}}{e}^{i{k}_{2}{z}_{2}\mathrm{cos}{\theta}_{2}}\\ \left(\begin{array}{c}-i{I}_{m}^{s}+\frac{i}{2}\mathrm{(}\sigma +1\mathrm{)}{I}_{m+2}^{s}-\frac{i}{2}\mathrm{(}\sigma -1\mathrm{)}{I}_{m-2}^{s}\\ \sigma {I}_{m}^{s}+\frac{1}{2}\mathrm{(}\sigma +1\mathrm{)}{I}_{m+2}^{s}+\frac{1}{2}\mathrm{(}\sigma -1\mathrm{)}{I}_{m-2}^{s}\\ 0\end{array}\right),\end{array}$$(2)

where ${I}_{q}^{l}={t}_{l}{e}^{i{W}_{l}}{i}^{q}{e}^{iq\varphi}{J}_{q}\mathrm{(}{k}_{1}{r}_{1}\mathrm{sin}{\theta}_{1}\mathrm{)};\text{\hspace{0.17em}}l=p\mathrm{,}\text{\hspace{0.17em}}s;\text{\hspace{0.17em}}q=m\mathrm{,}\text{\hspace{0.17em}}m\pm \mathrm{1,}\text{\hspace{0.17em}}m\pm 2.$

*α* determined by NA=*n*_{1}sin*α* is the maximal half-angle of the cone of light that exits the objective lens, where *n*_{1} is the refractive index of medium 1. *θ*_{1} denotes the incident angle and *θ*_{2} denotes the refraction angle. *J*_{q} is the *q*th-order Bessel function of the first kind. *t*_{p} and *t*_{s} are the transmission coefficients for the *p* and *s* polarizations, respectively. σ represents the states of polarization of the circularly polarized light, which is either 1 or −1. The dynamical phases of the *p* and *s* components introduced in this anisotropic medium are *W*_{p} and *W*_{s}, respectively. They are considered as aberrations *W*_{p}=exp(*ik*_{o}(*d*+*z*_{2})Δ*n*) and *W*_{s}=0, where we have assumed that the focusing systems are corrected for spherical aberrations caused by the mismatch in the interface between media 1 and 2. Physically, *W*_{p} accounts for the influence of the birefringence and causes distortion and broadening of the focal spot. In our simulations, NA=1.4, *d* is equal to 10 times of the wavelength, and *n*_{1} is set to 1.514 and *n*_{o} to 1.655; we only consider left-circularly polarized vortex light (σ=1). We emphasize that, in addition to circularly polarized beams, linearly polarized light fields and generalized cylindrical vectorial beams can also be used to achieve comparable effects, as these polarized vortex beams generally result in mixture of both extraordinary (*p*) and ordinary (*s*) modes in the focal region.

Accordingly, light-induced magnetization can be calculated by the focal electric fields via the IFE as

$$\begin{array}{c}M=\left(\begin{array}{l}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right)=i\left(\begin{array}{l}{\gamma}_{xx}\mathrm{(}{E}_{y}{E}_{z}^{\mathrm{*}}-{E}_{z}{E}_{y}^{\mathrm{*}}\mathrm{)}\\ {\gamma}_{yy}\mathrm{(}{E}_{z}{E}_{x}^{\mathrm{*}}-{E}_{x}{E}_{z}^{\mathrm{*}}\mathrm{)}\\ {\gamma}_{zz}\mathrm{(}{E}_{x}{E}_{y}^{\mathrm{*}}-{E}_{y}{E}_{x}^{\mathrm{*}}\mathrm{)}\end{array}\right)\\ \equiv i{\gamma}_{xx}\left(\begin{array}{c}{E}_{y}{E}_{z}^{\mathrm{*}}-{E}_{z}{E}_{y}^{\mathrm{*}}\\ {E}_{z}{E}_{x}^{\mathrm{*}}-{E}_{x}{E}_{z}^{\mathrm{*}}\\ \gamma \mathrm{(}{E}_{x}{E}_{y}^{\mathrm{*}}-{E}_{y}{E}_{x}^{\mathrm{*}}\mathrm{)}\end{array}\right).\end{array}$$(3)

For uniaxial medium with time reversal invariance, there are just two independent elements in *γ*_{lk}: *γ*_{xx}=*γ*_{yy} and *γ*_{zz}, which is uniquely determined by its crystallographic point group [42]. Briefly, we have used the ratio *γ*=*γ*_{zz}/*γ*_{xx} to characterize the relative ratios of strengths of the optomagnetic responses in different directions. Because of the anisotropy, it is anticipated to further manipulate the magnetization via the anisotropic optomagnetic property by selecting well-behaved media or even changing the material structure to enhance or suppress the optomagnetic response for a suitable *γ*. Although the anisotropic optomagnetic process is not relevant to the focused electric diffraction pattern, it has a direct impact on the IFE process.

It should be pointed out that in our calculations the maximal value of Δ*n* is 0.03, in which the focal spot does not form a perfectly symmetric pattern but is still a combined one. In this case, it shows some abnormal effects on the light-induced magnetization compared to the smaller Δ*n* cases, which is caused by the broadening of the focal spot (see Section 2 of Supplementary Material for more details).

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