Surface lattice resonances and magneto-optical response in magnetic nanoparticle arrays

Structuring metallic and magnetic materials on subwavelength scales allows for extreme confinement and a versatile design of electromagnetic field modes. This may be used, for example, to enhance magneto-optical responses, to control plasmonic systems using a magnetic field, or to tailor magneto-optical properties of individual nanostructures. Here we show that periodic rectangular arrays of magnetic nanoparticles display surface plasmon modes in which the two directions of the lattice are coupled by the magnetic field-controllable spin–orbit coupling in the nanoparticles. When breaking the symmetry of the lattice, we find that the optical response shows Fano-type surface lattice resonances whose frequency is determined by the periodicity orthogonal to the polarization of the incident field. In striking contrast, the magneto-optical Kerr response is controlled by the period in the parallel direction. The spectral separation of the response for longitudinal and orthogonal excitations provides versatile tuning of narrow and intense magneto-optical resonances.

were measured with two polarizations E x (left column) and E y (right column). Periodicity p x = 400 nm and p y = 400 nm (a, b), 460 nm (c, d), 480 nm (e, f), and 500 nm (g, h). Note the different wavelength scale in (g).The linear dispersions of the diffracted <+1, 0> and <-1, 0> orders of the lattice are clearly visible in each plot. Note that with E x polarization, the crossing point gradually redshifts from 600 nm to 750 nm with increasing periodicity p y (figures a, c, e and g) but with E y polarization the crossing point remains at 600 nm (figures b, d, f and h).

Supplementary Figure 4
Normal incidence extinction curves (defined as 1-T) for samples with different periodicities a) p x = 400 nm, p y = 400 nm, b) 460 nm, c) 480 nm, and d) 500 nm. The blue (red) lines correspond to the E x (E y ) polarization in all the figures. The solid lines correspond to measured extinction, while the dashed lines correspond to extinction obtained from the FDTD simulations. The FDTD simulations were carried out under similar conditions as the experiments: we used cylindrical particles with diameter 120 nm and height 90 nm with optical parameters of nickel (Ni) obtained from [1]. The refractive index of n = 1.5 was used. Importantly, in the simulation only one Ni particle was used with periodic boundary conditions in x and y direction that were matched with the periodicities p x and p y . The good agreement between experiments and simulations further supports our conclusion that the optical response is indeed dominated by the periodicity perpendicular to the incident polarization.

Supplementary Figure 5
The measured zero-order reflectivities and the corresponding simulations a) The measured zero-order reflectivity with polarization E x for samples having p x = 400 nm and p y = 400 nm, 460 nm, 480 nm, and 500 nm (black, red green and blue lines, respectively), and b) the corresponding FDTD simulation. c) The measured zero order reflectivity with polarization E y and d) the corresponding FDTD simulation. The color coding and the y-axis scale is the same in all the figures. Figure 6 Determining the coupling strength between the localized surface plasmon resonance (LSPR) and the diffracted orders of the Ni nanoparticle lattice

Supplementary
The normal incidence extinction simulated using the FDTD method is plotted as functions of energy and the inverse lattice constant 2π/p y . The polarization was E x and sample periodicity p x was maintained at 400 nm while p y was varied from 300 nm to 550 nm. The black horizontal line corresponds to Ni particle (diameter 120 nm, height 90 nm) LSPR, and the tilted line to the crossing of the <+1, 0> and <-1, 0> diffracted orders of the lattice. From the above simulation, one can estimate the coupling strength between the localized surface plasmon resonance (LSPR) and the diffracted orders of the periodic Ni arrays under normal incidence illumination. The horizontal black line corresponds to uncoupled LSPR of the Ni particle, and the tilted black line to the crossing point of the <+1, 0> and <-1, 0> diffracted orders of the lattice at a zero incidence angle. As can be seen, the maxima of the extinction clearly deviate from the uncoupled LSPR. The splitting is on the order of 400 meV. responses, we also studied how small deviations from an ideal input polarization of the driving field can affect the measured responses. In the ideal case, the input polarizations for the driving field are along the array periodicities (in x-and y-directions). But in experimental conditions, the incident polarization is never purely linear and oriented along the periodicities. Therefore, we measured the extinction ratio between the unwanted and wanted polarization components, which was found to be in the order of 15 000:1 corresponding to a ~0.04 degree offset from the ideal direction.
We then simulated the effect of this offset to the magneto-optical responses by performing DDA simulations while rotating the incident polarization a bit. Example results for sample with p x = 400 nm p y = 500 nm for the ideal case and for angle offsets of 0.02 deg and 0.04 deg are shown in Supplementary Figure   7. Interestingly for the ideal case, the results are almost entirely governed by the SLR conditions related to the period parallel to the driving field. For the cases with small deviations in the input polarizations, the cross-coupling effects seen in experimental data (see Figure 2 of the manuscript) appear and increase as a function of the deviation angle giving better agreement between the simulations and the measurements.
Therefore, we conclude that the finer details of the measured magneto-optical responses are due to a small input polarization impurity and a possible slight misalignment in the polarization.  Supplementary Table 1 The parameters used in the coupled oscillator model The model parameters have been obtained from the experimental data. In particular, resonance frequencies (as defined by √ ) and linewidths () of single particle resonances and diffractive orders have been chosen to match the experimentally obtained frequencies and linewidths. The single particle properties are obtained from measurements of the samples with random particle positions. The parameters related to the diffractive orders are obtained from optical transmission measurements. In our case of cylindrical particles, the two single particle resonances in x-and y-direction are equal, while the resonance frequencies of the diffractive modes in x-and y-direction can be different. Further, the linewidths of the diffractive orders have been chosen to be significantly smaller than the single particle resonances to account for the fact that these modes suffer much less from ohmic losses. The coupling strength of the diffractive orders to single particle resonances (i.e., the spring constants k RADy and k RADx ) have been estimated from the extinction data obtained from the FDTD simulations under normal incidence with equal particle size and geometry with the experiments (see also Supplementary Figure 6). Finally, the coupling strength of the optically driven dipole to the magneto-optical one (value for K SO ) has been determined from the existing experimental off-diagonal polarizability elements for Ni [2].

Supplementary Note 1 The mathematical description of the coupled oscillator model
The equations of motion for the coupled oscillator model (see also Manuscript Figure 3) are The above equations can be solved with respect to r SPx and r SPy for both driving conditions (F x ≠ 0 or F y ≠ 0).
The displacement r from the equilibrium position and the driving force F are assumed to be of the form r = r 0 e iωt and F = F 0 e iωt , respectively. Here ω is the angular frequency. While both r SPx and r SPy have extremely complex expressions, it turns out that the ratio is significantly simpler, despite the inherent asymmetry in the system (p x ≠ p y , thus the m DO , k DO and  DO will be different for each direction as well). In the case of F x ≠ 0, the ratio between the displacements of the nondriven oscillator and the driven one can be written in the Here λ is defined as 2πc / ω, where c is the speed of light. The expression for F y ≠ 0 is obtained upon switching the indices x and y. Interestingly, for F x ≠ 0 the ratio r SPy / r SPx defining the magneto-optical response, has no dependence on any of the parameters related to the SLR y , i.e., (m, k, ) SPx , (m, k, ) DOy or k RADy . Note that in our model, all these parameters are on the left side of the central spring K SO, and in the absence of magnetic field (K SO = 0), they are solely responsible for the purely optical response (i.e., the reflectivity and the value of |r SPx | in our model). In contrast, the terms (m, k, ) SPy , (m, k, ) DOx and k RADx (parameters are on the right side of the central spring K SO in Manuscript Figure 3) do affect the ratio r SPy / r SPx and thus the magneto-optical response, but they have no effect on purely optical response (i.e., to |r SPx | when F x ≠ 0 and K so =0).
By neglecting the radiation and periodicity induced effects, that is, by setting k RADx(y) = k DOx(y) = 0, one recovers a result for individual particle (or randomly distributed particles). In the case F x ≠ 0, the ratio becomes ( )

.
The ratio is again independent of the parameters of the driven oscillator m SPX and is in agreement with the earlier studies made for individual Ni particles [3,4]. In this simple case, the physical interpretation is straightforward: The force felt by the non-driven oscillator m SPy is linearly dependent on the displacement of the driven one, r SPx , which in turn depends on the parameters on the left side of the central spring K SO .
Thus r Spy depends also on these parameters, but in the ratio r Spy / r Spx these parameters on the left of the central spring K SO cancel out.
Finally, we note that while the analytical model neglects a number of experimentally relevant parameters, such as the wavelength dependence of both the Ni optical parameters and the spin orbit coupling and the actual radiation pattern of the particles, it is nevertheless able to reproduce the most prominent features of the experimental data, namely 1) the dip in the ellipticity curve and 2) strongly asymmetric line shape in the rotation, both appearing at the resonance wavelength of the non-driven oscillator.

Supplementary Note 2 Jones matrix representation of the magneto-optical measurement setup
The components of the measurement setup (Supplementary Figure 8) The intensity of the beam is measured with fast responsive photodetector (DET-100, Hinds Instruments) and a lock-in amplifier is used to filter the signal at the PEM modulation frequency. Further analysis requires decomposition of the exponential terms into harmonics of the retardation frequency by a Jacobi-Anger series [5]. The intensity can then be grouped into DC, 50 kHz (1 st harmonic), 100 kHz (2 nd harmonic) and higher frequency terms , where ( ) are the Bessel functions of first kind. The last term of the DC signal can be eliminated by setting to zero at . Furthermore, the term proportional to is very small and can be neglected. The DC signal and the first two harmonics are read by a lock-in amplifier. The Kerr rotation  and ellipticity  are extracted from these signal using In the magneto-optical Kerr spectrometer, an electromagnet is used to orient the magnetization of the Ni nanoparticles perpendicular to the sample plane. Full hysteresis loops of the polar Kerr rotation and ellipticity are measured at each wavelength using magnetic fields up to  400 mT. From these