1. Introduction

Magneto-optic (MO) effects make it possible to engineer the polarization and phase of electromagnetic waves by spin–orbit interactions in magnetized media [1] and have been used for magnetic detection [2], optical isolation [3], information recording [4], and optical modulation [5]. On the basis of their plasmonic resonances, metal nanostructures have recently been employed to modify or boost the MO effects [6,7]. Periodic metal gratings combined with a thin magnetic film yielded enhancement of the Faraday rotation [8], MO Kerr effect [9], and MO transparency [10]. Metal–ferromagnet–metal waveguides can modulate the wave vector of guided surface plasmon polaritons (SPPs) and can be used to construct an active SPP interferometer [11]. In such studies, magnetic media have been employed just as an assistant to add MO functions to plasmonic nanostructures. On the other hand, magnetic metal materials such as Ni, Fe, and Co can support plasmonic resonances and MO activity simultaneously, and intermediate their direct interplay [12–14]. Recently, the magneto-plasmonic responses of Ni nanodisks have received wide attention [15,16], and their periodic or random arrangements have been applied to demonstrate resonant MO Kerr effects [17] and phase-sensitive molecule-level sensing [18].

To obtain general guidelines for engineering the MO effects of magnetic metal nanostructures, a systematic understanding of the coupling between the MO activity and the plasmonic resonances is required. However, most theoretical analyses are based on simplified models using quasi-static analysis or the effective medium theory [16–21]. The quasi-static model assumes uniformly distributed internal fields and fails to fully include the modal characteristics of nanostructures with a nonspherical, complex geometry or a size exceeding the Rayleigh criterion (~λ/10). Modified analytical methods, such as the use of the polarizability tensor considering the retarded depolarization to predict the MO properties of ellipsoidal particles [22], have only been suggested. In this work, we systematically investigate the polar MO Kerr effect of a single Ni nanorod in the Mie regime depending on its geometry using the finite-difference time-domain (FDTD) method, and present a universal explanation of the resonant MO Kerr effects from the viewpoint of first-order perturbation analysis.

2. Schematics

In the presence of a magnetic field, either external or internal, two orthogonal electric polarizations in the magnetized medium are coupled by spin–orbit interactions. Because the MO activity changes the permittivity, typically by on the order of 1% or less [23], the MO permittivity can be approximated by adding the perturbation of an off-diagonal tensor ${\overline{\epsilon }}^{\prime }$ to the nonmagnetic permittivity $\epsilon {\rm I}$. When the direction of the magnetization is along the z axis, the permittivity of the Ni medium can be formulated as a Hermitian tensor,

Figures 1(a) and 1(b) show schematics of the MO activity in a single Ni nanorod under a normally incident planewave in the quasi-static and Mie regimes, respectively. The incident light is linearly polarized along the y axis, and a static external magnetic field is applied along the z axis. For simplicity, the surrounding medium is considered to be air. In the quasi-static regime, electric fields are uniformly induced inside the Ni nanorod, and the MO perturbations to the electric fields, ΔE_x, must also be uniform. On the other hand, in the Mie regime, the electric fields and MO perturbations are not uniform due to spatial retardation effects. The incident light polarized along the nanorod length excites SPPs of the transverse-magnetic (TM) mode, and the MO activity interfaces the TM mode to the transverse-electric (TE) mode. We define the TM and TE polarization states according to the length direction of the Ni nanorod. The two ends of the nanorod reflect propagating SPPs and support longitudinal Fabry–Pérot (FP) resonances of the TM and TE modes. Because of their different dispersion relations, the TM and TE modes exhibit FP resonances at different nanorod lengths for a given wavelength of light.

 

Fig. 1 Schematics of the polar MO Kerr effects in the quasi-static and Mie regimes. Red and purple arrows indicate the incident-light-induced E_y field and MO perturbation E_x, respectively. (a) In the quasi-static regime, the incident-light-induced field and the resulting MO perturbation are approximately uniform inside the Ni nanorod. (b) In the Mie regime, both the induced field and MO perturbation become complicated depending on the modal characteristics of the resonances.

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In the FDTD simulations, we suppose that, to support the polar MO Kerr effects, the magnetization of the Ni medium is saturated along the + z direction and a 660-nm-wavelength planewave is normally incident from the top. At the wavelength of 660 nm, the nonmagnetic permittivity of Ni is −10.726 + 15.842i [25], and the Voigt parameter Q in the saturation condition is 0.00588 + 0.0039i [26]. To apply the Hermitian permittivity tensor, we used the grid attribute technique of the FDTD calculator (Lumerical Solutions, Inc.). In the magnetized Ni medium, the electric fields are first calculated in the unitarily transformed coordinates by diagonalizing the Hermitian permittivity tensor and then converted to those in the Cartesian coordinates again. Because the field components in the Yee grid are spatially offset by a half-step and the unitary coordinate transformation is valid only in the magnetized medium, a crucial error occurs at the interface between the magnetized and nonmagnetized media. To avoid this interface error, we suppose that the outermost grids at the boundary of the Ni medium are not magnetized. The artificial effects of the nonmagnetized outer layer can be suppressed sufficiently by reducing the grid size in the Ni medium. We set the spatial grid size inside the Ni medium to 2 nm and gradually increase the grid size in the other region from 2 to 16 nm.

3. Modal properties of nonmagnetized Ni nanorods

We first investigate the modal properties of the nonmagnetized Ni nanorod by calculating the extinction and scattering efficiencies. The extinction (scattering) efficiency is defined as the ratio between the extinction (scattering) cross section and the physical cross section of the nanorod. Figures 2(a) and 2(b) show two-dimensional (2D) maps of the extinction and scattering efficiencies, respectively, as a function of the length and width of the nanorod under a normally incident TM-polarized planewave. The length and width vary from 80 to 1120 nm, and the thickness is fixed at 40 nm. The FP resonances of the guided modes locally maximize the extinction and scattering efficiencies of single nanorod structures [27,28]. For the 80-nm-wide nanorod, the extinction efficiency is locally maximized at lengths of ~200 and ~720 nm by the first- and third-order longitudinal FP resonances of the TM mode, with extinction efficiencies of ~5.35 and ~2.58, respectively. The even-order FP resonances cannot be excited by the normally incident planewave because of their antisymmetric field distributions. The electric field amplitude and surface charge density distributions in Figs. 2(c)–2(f) clearly show the spatial properties of the first- and third-order FP resonances of the TM mode. The electric field and charge density distributions of the first-order resonance have the features of typical dipolar oscillation. As the width of the nanorod decreases into the quasi-static regime, the absorption efficiency becomes larger than the scattering efficiency, and the extinction efficiency gradually increases. As the nanorod width increases, the phase pickup for reflecting SPPs decreases, and FP resonances appear for slightly longer nanorods. In addition, we note that, even though they are not excited by the TM-polarized incident light, the longitudinal FP resonances of the TE mode also exist and play an important role in the MO Kerr effect, which will be discussed later.

 

Fig. 2 (a, b) Two-dimensional maps of the extinction and scattering efficiencies of nonmagnetized Ni nanorod depending on its width and length. (c) Normalized scattered electric field amplitude and (d) charge density distributions of the 80-nm-wide and 200-nm-long nanorod supporting the first-order longitudinal FP resonance of the TM mode. (e) Normalized scattered electric field amplitude and (f) charge density distributions of the 80-nm-wide and 720-nm-long nanorod supporting the third-order longitudinal FP resonance. The distributions are taken in the xy plane including the center of the nanorod. The charge density is obtained by calculating the divergence of the electric field.

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4. Polar magneto-optic Kerr effects of magnetized Ni nanorods

To investigate the polar MO Kerr effect of the Ni nanorod, we calculated the Kerr rotation ${\theta }_K$ and Kerr ellipticity ${\epsilon }_K$ of the far-field radiation scattered back to the vertical direction ( + z direction):

 

Fig. 3 Polar MO Kerr effect for light scattered back to the vertical direction ( + z direction). (a) Kerr rotation, (b) Kerr ellipticity, (c) amplitude ratio, and (d) phase differenceare plotted as a function of the length and width of the Ni nanorod. (e–h) Normalized profiles of the MO perturbation ΔE_x of the Ni nanorods supporting the local maxima of the amplitude ratio.

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The MO perturbation ΔE_x profile provides a better understanding of the MO Kerr effect of the Ni nanorod. Figures 3(e)–3(h) show the ΔE_x profiles in the xy plane for the four selected nanorods that support the local maxima of the amplitude ratio in Fig. 3(c). The one- or two-dimensional standing wave pattern in the ΔE_x profiles verifies the resonance of the plasmonic TE mode, the polarization state of which is orthogonal to the incident-light-induced TM mode. The TE mode resonances associated with each local maximum of the amplitude ratio/Kerr rotation have different modal characteristics. First, the enhancement of the Kerr effect for short nanorods with a length of 80 nm originates from the transverse FP resonances of the TE mode along the width direction of the nanorod. Rotated by 90°, the transverse FP resonances of the TE mode are identical to the longitudinal FP resonances of the TM mode. As shown in Figs. 3(e) and 3(f), the 200-nm- and 720-nm-wide nanorods support the first- and third-order FP resonances of SPPs propagating along the x axis, respectively. On the other hand, the local maximum of the amplitude ratio in the 680-nm-long and 200-nm-wide nanorod originates from the longitudinal FP resonance of the TE mode, the modal characteristics of which are completely different from those of the longitudinal (transverse) FP resonance of the TM (TE) mode.

The ΔE_x profile in Fig. 3(g) clearly shows the longitudinal FP resonance of the TE mode propagating along the length direction. Three equally spaced lobes exhibit the typical characteristics of the third-order longitudinal FP resonance. We calculated the propagation wavelength of the TE-polarized SPPs and the reflection pickup at the end facet for the Ni nanorod with a 40-nm-thick and 200-nm-wide cross section. From the calculated propagation wavelength and reflection phase pickup of ~650 nm and ~166°, respectively, the nanorod length supporting the third-order longitudinal FP resonance is estimated to be ~676 nm, in excellent agreement with the length maximizing the Kerr effect. In contrast to the TM mode resonances, the TE mode resonances cannot be effectively excited by the normally incident planewave because of their low radiation efficiencies in the vertical direction. However, the internal electric fields induced inside the Ni nanorod cause the MO electric polarizations acting as point sources to excite the TE mode resonances. Two-dimensional SPP resonances can also be allowed with increasing length and width of the nanorod. Figure 3(h) shows the 2D standing wave pattern of the ΔE_x field in the 680-nm-long and 720-nm-wide nanorod. The locally maximized MO Kerr rotation values of the third-order longitudinal FP resonance of the TE mode and the 2D SPP resonance are ~2.20 and ~1.56 mrad, respectively. The Kerr rotations maximized by the first- and third-order transverse FP resonances of the TE mode are ~3.18 and ~1.48 mrad, respectively.

To support a local maximum of the first-order perturbation integral of the MO Kerr effect, it is necessary to optimize the spatial overlap and phase matching between the incident-light-induced electric field and the Green function, in particular, in the resonance condition of the final-state mode, where the Green function has large amplitudes and is accompanied by a high LDOS. In the numerical simulations, the optimization condition can be identified by examining whether the amplitude and phase distributions of the incident-light-induced field inside the MO medium can promote the resonance condition of the final-state mode with an orthogonal polarization (Fig. 4(a)). This mechanism is analogous to that of the Purcell effect, in which the emission rate enhancement is proportional to the LDOS at the position of the emitter. In Fig. 4, we calculate the amplitude and relative phase distributions of the y component of the incident-light-induced field E_y inside the Ni nanorod and the x component of the resulting MO perturbation ΔE_x. Here, the lengths of the nanorods are 480, 680, and 760 nm, respectively, and the widths are fixed at 200 nm.

 

Fig. 4 (a) Schematic of the spatial overlap and phase matching of the incident-light-induced electric field with the FP resonance of the TE mode. The spatial overlap maximizes the MO Kerr rotation and amplitude ratio. Profiles of normalized amplitude (left) and relative phase (middle) of the incident-light-induced E_y fields and normalized amplitude (right) of the MO perturbation ΔE_x in the xy plane are plotted for Ni nanorods having lengths of (b) 760 nm, (c) 680 nm, and (d) 480 nm. We set the reference phase at the end of the nanorod to −π/2.

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First, we focus on the 680-nm-long Ni nanorod showing the local maximum of the amplitude ratio of the MO Kerr effect. The 680-nm-long nanorod does not support a resonance of the TM mode, whereas the 480-nm- and 760-nm-long nanorods support the second- and third-order longitudinal FP resonances of the TM mode, respectively. However, the 680-nm-long nanorod supports the third-order longitudinal FP resonance of the TE mode, which is the final-state mode in the first-order perturbation integral of the MO Kerr effect. As shown in Fig. 4(c), the incident-light-induced E_y field has three evenly spaced amplitude antinodes along the longitudinal direction. Neighboring antinodes with comparable magnitudes oscillate with opposite phases to each other. In the transverse direction, the amplitude antinodes of the E_y field are positioned at the sides of the nanorod, where the E_x field of the TE mode is maximized. This profile of the E_y field causes the MO perturbation ΔE_x to seamlessly couple to the third-order FP resonance of the TE mode and thus the electric field amplitude ratio of the MO Kerr effect is locally maximized.

On the other hand, the amplitude and phase distributions of the E_y field of the 760-nm- and 480-nm-long nanorods are far from those of a standing wave pattern (Figs. 4(b) and 4(d)). Despite its length, the 760-nm-long nanorod has only two antinodes of the E_y field along the longitudinal direction. Further, the antinodes oscillate in the same phase. The E_y field profile of the 480-nm-long nanorod has a single antinode in the middle along the longitudinal direction, but the nanorod is too long to support the first-order FP resonance of the TE mode. The ΔE_x profiles also show features of the transitional stages between the FP resonance conditions. Indeed, the 480-nm- and 760-nm-long nanorods cannot support a high LDOS in the TE mode or enhance MO Kerr activity, even though they support the FP resonances for TM-polarized incidence.

5. Summary

We presented a systematic study of the coupling between the polar MO Kerr activity and the plasmonic resonances of a single Ni nanorod in the Mie regime. The electric field amplitude ratio of the MO Kerr effect is locally maximized when the Ni nanorod supports a plasmonic resonance in the polarization state orthogonal to the incident light. The plasmonic resonances directly induced by the incident light do not contribute to the maximization of the amplitude ratio. From the viewpoint of first-order perturbation analysis, the spatial overlap and phase matching between the incident-light-induced electric field and the Green function determine the MO Kerr effect. Our understanding of the MO effects of optical nanostructures will be useful for various applications such as MO sensors, optical logic devices, and MO metamaterials.