Magnetic hot-spots in hollow silicon cylinders

Silicon nanoparticles can possess magnetic Mie-resonant response in the visible and near infrared wavelength ranges. In this paper, we consider numerically the features of magnetic hot-spots realized inside silicon nanocylinders at the conditions of the optical magnetic resonances, and show that the intensity of the magnetic field inside nanoparticles with a coaxial through hole can be much stronger than the intensity of incident light waves.

(a) Magnetic field distribution (a), and displacement currents distribution (b) in the dielectric cylinder on the TE0mn eigen modes, m=1, 2, 3. (c) Magnetic field distribution in the hollow dielectric cylinder on the TE0mn eigen modes, m=1, 2, 3. White contours highlight boundaries of cylinder (a-c) and hollow (c).
In order to design electromagnetic response of silicon nanoparticles, their modal structure should be analyzed. Among the number of high-index cylinder eigen modes there are some of them which correspond to the magnetic field concentration inside the cylinder. At the figure 1a magnetic field distributions on the TE0mn modes, m=1, 2, 3, n=1, are shown. These distributions, for cylinder with height of 100 nm and radius of 200 nm and dielectric permittivity equals to 16, were obtained numerically (with help of Comsol Multiphysics [16]). Here modes with n>1 are not shown because for n>1 magnetic field distribution at the frontal cross section is repeated manually this for TE0m1 mode but number of magnetic field maxima on the cylinder axis are increasing (1 for TE0m1 mode, 2 for TE0m2 mode and so on). TE0mn modes are of highest interest, because special circular distribution of displacement currents (see figure 1b), slightly depends on the defects on the cylinder axis.
Actually, in coaxial cylinders with a cavity the magnetic field concentration stays quite high for low-order modes but disappears for high-order modes (see figure 1c) because of overlapping between the cavity and the area of displacement currents maxima. In order to estimate the magnetic field enhancement in the cavity a full-wave simulations should be done. Results are presented in the next section.

Magnetic hot-spots in hollow cylinders under plane wave irradiation
In figure 2 results of numerical simulations for cylinder (radius is 200 nm, height is 100 nm) irradiated by linearly polarized plane wave are presented. Here we take into account dispersion of silicon dielectric function [3]. We consider two different propagation directions: along and perpendicular to the cylinder axis (see figure 2a, b). Results for third case, when propagation vector is perpendicular to  Distribution of electromagnetic field for the first resonance (corresponding to the TE011 mode) in particles with relatively small cavity and without the cavity is similar (figure 2ci-fi). Inside the cylinders without cavity magnetic field is quite homogeneous and have a Lorentz-like maximum near the middle of cylinder axis, and for cylinders with void core maximum is shifted to the cavity boundary. Maximal magnetic field in the cavity decreases when cavity size enlarges. Interestingly, for lateral irradiation case (figure 2b) the magnetic field in the cavity is 2 times higher than the magnetic field of the incident wave even in the case of very thin silicon shell. For all the resonances the lateral excitation of cylinder is more prospective than the frontal one from the application point of view. This is a consequence of the circular character of displacement currents, which are not changed significantly for the lateral irradiation. The wavelength of resonance decreases monotonically when the size of cavity goes up in the both cases of excitation ( figure 2g, h). Note the following important feature of the MHSs for the first resonance: it is a large region (up to 80% of overall cylinder radius) where the enhancement coefficient exceeds the factor 4. The obtained dependencies of frequency and enhancement factor on the cavity size can be used allow for the construction of systems with required level of magnetic field enhancement at the certain frequency.

Eigen modes of nanocylinders
Higher-order hybrid modes excited in the cylinder can also provide the magnetic field enhancement. In figure 2cii-fii, ciii-fiii distributions of the magnetic field for the second and third resonance are shown for both irradiation cases, respectively. Note that the high-order resonances give stronger magnetic field enhancement (see figure 2i, j). However, the influence of cavity size on the high-order resonances is more considerable because the regions of magnetic field concentration decrease with the increase of the resonant order.
As a conclusion, we investigated the magnetic field distributions inside the silicon cylinders with and without coaxial through holes (cavities) at the resonant optical conditions using the full-wave numerical simulations. Obtained results can be used for development of new optical components and devices for management, trapping and detection of magnetic nanoparticles and molecules with magnetic transitions.