## Abstract

Nanostructured magneto-optical materials that sustain optical resonances provide an efficient way to control light via the magnetic field, which is of prime importance for telecommunication and sensing applications. However, their response is usually narrowband due to their resonance character. Here, we demonstrate and investigate a type of magnetoplasmonic structure, or quasicrystal, that demonstrates a unique magneto-optical response. It consists of a magnetic dielectric film covered by a thin gold layer perforated by slits, forming a Fibonacci-like binary sequence. The transverse magneto-optical Kerr effect (TMOKE) acquires controllable multiple plasmon-related resonances, resulting in a magneto-optical response over a wide frequency range. In particular, for the experimentally studied samples, the TMOKE resonances are observed in the wavelength range from $\lambda =780\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ to $\lambda =970\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$. Multiband TMOKE is valuable for numerous nanophotonic applications, including optical sensing, control of light, all-optical control of magnetization, etc. TMOKE spectroscopy is also an efficient tool for investigating the peculiarities of plasmonic quasicrystals.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Since the 1970s, quasicrystalline structures have attracted the attention of researchers. Their experimental discovery by Shechtman *et al.* [1] was recognized with the Nobel Prize in Chemistry in 2011. Quasicrystals are non-periodic but ordered structures [2]. In contrast to periodic crystalline structures, they do not possess translational symmetry but have long-range ordering. For periodic structures, the reciprocal lattice is given by an equidistant set of reciprocal vectors. Thus, for one-dimensional periodicity of period $d$, the reciprocal vectors $\mathbf{G}$ are given by $\mathbf{G}=(2\pi m/d)\mathbf{e}$, where $\mathbf{e}$ is the unit vector along the periodicity direction and $m$ is an integer. Unlike periodic structures, the quasiperiodic ones are characterized by discrete but non-equidistant reciprocal lattices and corresponding unusual diffraction patterns. An example of a one-dimensional quasicrystalline pattern is the Fibonacci binary sequence [3]. It is constructed by the infinite specific sequence of “0” and “1” symbols by the following rules: let us take ${S}_{0}=\u20180\u2019$, ${S}_{1}=\u20181\u2019$, and ${S}_{n}$ is the concatenation of ${S}_{n-1}$ and ${S}_{n-2}$ for $n>1$ (${S}_{n}={S}_{n-1}{S}_{n-2}$). The resulting sequence ${S}_{\infty}$ is the Fibonacci binary sequence and has the following form:

The formation of the Fibonacci binary sequence is similar to the formation of the well-known Fibonacci numerical sequence where each number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34….

Two-dimensional quasicrystalline structures can possess rotational symmetry ${C}_{n}$ of orders that are prohibited for periodic structures, such as $n=5$, 7, and higher orders. An example of that is Penrose tiling, which possesses ${C}_{5}$ rotational symmetry [4]. Quasicrystalline substances are usually metallic alloys, and they provide high hardness, low coefficient of friction, low thermal conductivity, and wide optical absorption bands. Quasicrystals, such as two-phase materials containing steel, have potential application as materials having high strength and plasticity at high temperatures and also as coatings, e.g., for frying pans, solar energy absorbers, and reflectors [2].

The concept of quasicrystals has been implemented in plasmonics as well [4–12]. Study of plasmonic structures has been of prime interest for the last decade regarding control of plasmonic modes and near-field manipulation [13,14], design of optical properties [15], high sensitivity to surrounding media [16,17], all-optical light control [18–20], and ultrafast magnetism [21,22]. Plasmonic quasicrystals in the form of metal–dielectric nanostructures with quasicrystalline pattern that support excitation of surface plasmon polaritons (SPPs) were designed. Due to the discrete non-equidistant reciprocal space and the rotational symmetry of quasicrystals, a larger number of resonances associated with SPP modes appear, and therefore such structures demonstrate a broadband optical response. Moreover, it is polarization independent for the 2D structures [7]. The advantage of quasicrystalline structures over periodic and non-periodic ones is that they possess rich and designable reciprocal lattice that governs the optical diffraction and dispersion of the eigenmodes, and therefore offer designable broadband optical response [23]. It comes from the fact that the reciprocal lattice is strongly dependent on the geometrical parameters of the structure, while for the periodic structures, it is defined solely by the period.

However, the potential of quasicrystalline structures for magneto-optics has not been revealed yet. Magneto-optics is a powerful tool for the manipulation as well as to probe the optical properties of different materials and structures. Vast research has been carried out during the recent decade on the enhancement of magneto-optical effects in plasmonic structures [24–41]. Combining magneto-optics with plasmonics provides significant enhancement in magneto-optical properties. It was demonstrated that in systems containing bismuth iron-garnet films [42,43], conventional magneto-optical effects such as the transverse magneto-optical Kerr effect (TMOKE) are resonantly enhanced in magnetoplasmonic crystals [37,38], and also novel promising effects arise [39]. However, these effects are of resonant nature, related to the excitation of eigenmodes of the structure, such as the cavity modes, SPPs, waveguide and quasi-waveguide modes, etc., that leads to narrow spectral range of the magneto-optical response with the typical width of the resonance less than 100 nm. Multiplicity of magneto-optical resonances can be achieved by employing waveguide modes; however, the TMOKE resonances related to them are weaker than those related to SPPs [44].

In the present work, we propose and demonstrate an approach for forming a designable multiband magneto-optical response in wide wavelength range using one-dimensional magnetoplasmonic quasicrystals. We design and fabricate the magnetoplasmonic quasicrystal and measure its optical transmittance and the TMOKE and compare them to the case of periodic structure. We find that the TMOKE in a quasicrystalline structure has multiply resonant SPP-assisted character and, moreover, it is a sensitive tool for the investigation of quasicrystal spectral properties.

## 2. MAGNETOPLASMONIC QUASICRYSTAL SAMPLES AND EXPERIMENTAL SETUP

The considered 1D magnetoplasmonic quasicrystalline structure is formed by a metallic quasicrystal grating on top of a smooth magnetic dielectric layer on a substrate [Fig. 1(a)]. The sequence of metal stripes and air slits of the grating can be described by symbols “1” and “0.” Our structure is based on the 1D binary Fibonacci sequence [Eq. (1)], where each “0” is substituted by “010.” The final formula of the structure is thus the following:

Such structure was chosen to achieve rather large spectral density of resonances.

The schematic of a part of the structure, corresponding to the underlined blocks in Eq. (2), is shown in Fig. 1(a). For comparison, we also consider a periodic crystal that is described by the sequence ‘10101010101…’ [Fig. 1(b)].

The metal grating of the experimentally studied samples was fabricated by electron beam lithography of thermally deposited 80-nm-thick gold layer. The air slit width corresponding to single “0” in the binary sequence is 80 nm, and the metal stripe width corresponding to single “1” in the sequence is 600 nm. These values were chosen to ensure that the SPPs can propagate without significant losses as well as to adjust the Fourier spectra of the structure (see Section 3). The magnetic dielectric is a bismuth-substituted iron-garnet film of composition ${\mathrm{Bi}}_{1.5}{\mathrm{Gd}}_{1.5}{\mathrm{Fe}}_{4.5}{\mathrm{Al}}_{0.5}{\mathrm{O}}_{12}$ [45]. It was deposited by reactive ion beam sputtering on (111) gadolinium gallium garnet substrate. The thickness of the magnetic film was made rather small, 80 nm, to exclude the waveguide mode excitation in the considered frequency range. This is confirmed by the estimation of the cutoff thickness for the fundamental transverse magnetic (TM) mode [46] of a planar waveguide. For the fundamental TM mode to be at a wavelength of 800 nm, the film thickness has to be 230 nm (the refractive indices for the magnetic dielectric and the substrate were taken as 2.3 and 1.9, respectively, and the dielectric constant for gold was taken from [47]). Thus, only SPPs can be excited in the designed structure.

The TM-polarized light hits the sample from the top side, and the plane of incidence is perpendicular to the slits [Fig. 1(c)].

Optical and magneto-optical spectra of the plasmonic quasicrystal samples were measured by the following experimental setup. A tungsten halogen lamp is used as a light source. After the lamp light passes through the fiber, in order to obtain a homogeneous point-like light source, the fiber output is collimated with an achromatic 75 mm lens and focused onto the sample with another achromatic 35 mm lens to a spot of about 200 μm in diameter. To perform the TMOKE measurements, the sample is placed in a uniform external magnetic field of 200 mT generated by an electromagnet in the direction along the gold stripes [Fig. 1(c)]. After the sample, the light is collimated with a $20\times $ microscope objective and detected with a spectrometer. The spectrometer slit was oriented perpendicular to the gold stripes, so only light perpendicular to the gold stripes incidence plane was detected. A spectrally and angularly decomposed light signal is detected by the spectrometer with a charge-coupled device camera. The experimental scheme is shown in Fig. 1(c).

## 3. FOURIER SPECTRA OF THE PLASMONIC STRUCTURES

The spectrum of the SPPs excited by the incident light in a plasmonic grating structure is determined by the reciprocal lattice vector $\mathbf{G}$, which enters the phase-matching condition:

where $\mathit{\beta}$ is the SPP propagation vector, and ${\mathbf{k}}_{\parallel}$ is the in-plane component of the incident light wave vector. The reciprocal lattice vectors $\mathbf{G}$ are found from the Fourier transform of the metal pattern, as the coordinates in the reciprocal space appear as peaks in the Fourier transform.Figure 2(a) depicts the calculated absolute values of the Fourier transform of the considered Fibonacci-like quasicrystal (shown by red curve) and the periodic crystal (shown by black curve) patterns. The lowest peaks in the quasicrystal Fourier transform with amplitudes smaller than 0.01 a.u. can be neglected, as they appear due to numerical calculation error. Moreover, low peaks do not contribute significantly to the excitation of eigenmodes, as the corresponding excitation efficiency is small. The reciprocal lattice for the quasicrystal is discrete, but it is far denser compared to that of the periodic crystal. The discrete non-equidistant reciprocal lattice confirms that the structure is non-periodic but has some long-range order. The reciprocal vectors for the periodic crystal form a discrete equidistant lattice with their values ${G}_{m}^{(\text{per})}=m\xb72\pi /d$, where $d$ is the period and $m$ is an integer. For the considered parameters of the grating, ${G}_{m}^{(\text{per})}=m\xb79.24\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$. Due to the use of same width gold stripes in both structures, the reciprocal lattices correlate with each other. That is, the reciprocal vectors of the quasicrystal lattice form bands located in the vicinity of ${G}_{m}^{(\text{per})}$.

The reciprocal lattice of quasicrystals can be tuned by adjusting their sequence type. Figures 2(a) and 2(b) show the Fourier spectra for both the experimentally considered Fibonacci-like structure defined by Eq. (2) [shown by red curve in (a) and (b)] and the pure Fibonacci sequence defined by Eq. (1) [shown by blue curve in (b)]. Spectral positions of the peaks and their amplitudes are different. To discuss in detail, for the $\mathbf{G}$ vector range covering $15\u201320\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$ (wavelength range covered is 750–1000 nm), the experimental sample has three peaks higher than the 25% level of the main peak. At the same time, the pattern based on the pure Fibonacci sequence provides only two such peaks [Fig. 2(b)]. Another advantage of the experimentally considered structure is that the highest peaks are located closer to each other, so their spectral density is larger than the one for the Fibonacci structure. Consequently, the quasicrystals formed by different sequences have quite different Fourier transforms and therefore the SPP spectra.

## 4. TRANSMISSION SPECTRA OF PLASMONIC QUASICRYSTALS

Transmission spectra of the plasmonic samples are shown in Figs. 3(a) and 3(b). Transmittance for the quasicrystalline structure is lower than for the periodic one due to the fact that the averaged fraction of slits over the pattern is lower for the former [the number of zeros in Eq. (2) is less than the number of ones].

Transmission spectra demonstrate some resonant features that are dispersive with respect to the angle of incidence [Figs. 3(a) and 3(b)]. They are due to the SPPs propagating at the interface between the magnetic film and the gold grating as is confirmed by the calculated dispersion of the SPP resonances (green dashed lines).

The dispersion was calculated by Eq. (3) in the empty lattice approximation, i.e., with propagation constant ${\beta}_{0}$ taken for a smooth single-interface between two semi-infinite metal and dielectric media: ${\beta}_{0}=2\pi /\lambda \xb7{[({\u03f5}_{m}{\u03f5}_{d})/({\u03f5}_{m}+{\u03f5}_{d})]}^{1/2}$, where ${\u03f5}_{m}$ and ${\u03f5}_{d}$ are the dielectric constants of metal and dielectric, respectively, and $\lambda $ is the vacuum wavelength. The dielectric constants were taken from [47] for the gold and from [42] for the magnetic dielectric. To take into account the finiteness of the magnetic film thickness, the presence of the slits and the difference in magnetic/chemical composition with respect to [42], ${\u03f5}_{d}$ was varied to obtain the best fit with experimentally observed spectral positions of optical resonances in both samples. The best correspondence was achieved when ${\u03f5}_{d}$ was multiplied by 0.95 for the plasmonic crystal and by 0.91 for the plasmonic quasicrystal compared to the values from [42]. The difference in the multiplication factors comes due to difference in the grating patterns; periodic and quasiperiodic crystals have different effective filling factors of metal.

The calculated dispersion curves are located neither along the peaks nor dips of the transmittance. It is due to two reasons. First of all, one should note that excitation of SPPs leads to Fano-shape resonances in transmittance that have asymmetric line shapes, containing both a peak and a dip. This is due to the fact that these spectral features are the result of interference between two contributions, the resonant SPP excitation and the non-resonant diffraction on the grating. Depending on the grating parameters, the SPP eigen frequency is located somewhere between the transmittance peak and dip and might be closer either to the former or to the latter. Moreover, the shown dispersion is calculated in the empty lattice approximation where the grating slits are assumed to be negligibly small. In reality, the slit width does not vanish, and it leads to some deviation from the experimental data.

In the experimentally studied samples, main transmittance resonances are due to SPPs excited by the second diffraction order with $G=18.48\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$ and $G=19.01\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$, in the periodic and quasiperiodic structures, respectively [Fig. 2(a)]. For the normal incidence, it corresponds to the transmittance peaks at $\lambda =820\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ [for the crystal, Fig. 3(a)] and $\lambda =790\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ [for the quasicrystal, Fig. 3(b)]. For oblique incidence, SPP resonances are split into low- and high-frequency ones and demonstrate dispersion, which is seen in Figs. 3(a) and 3(b) as peaks and dips following the SPP dispersion curve (green lines). Though the dispersion lines for the plasmonic quasicrystal predict more resonances in the range from $\lambda =880\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ to $\lambda =950\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, they are hardly seen in the transmittance. As a result, transmittance spectra for the crystal and its quasiperiodic counterpart look rather similar.

## 5. REVEALING PLASMONIC MODES IN QUASICRYSTALS FROM TMOKE SPECTRA

Since magneto-optical effects in plasmonic structures are enhanced by the excitation of eigenmodes, they can be used for investigating the quasicrystal spectra. The impact of the magnetization $\mathbf{M}$ on the SPP dispersion is given by [48]

Experimentally measured TMOKE spectra are shown in Figs. 3(c) and 3(d). In contrast to the transmission spectra, they are quite different for the crystal and quasicrystal samples. It highlights high sensitivity of the TMOKE to the excitation of the eigenmodes of the structure.

One should note that, due to the fact that SPPs produce not just transmission spectral peaks but also resonances of more complex shapes, the TMOKE resonances are also of asymmetric shape, having quite different amplitudes of positive and negative maxima but always having zero point in the vicinity of the dips in transmittance. As the TMOKE resonances have highly asymmetric form, their maxima of opposite signs are quite different and may not be easily seen in the spectra. Thus, at the incidence angle of 10°, the TMOKE is well seen on both slopes of the transmittance resonance and demonstrates positive and negative maxima for each resonance. However, at some smaller incidence angles, one of the TMOKE maxima becomes much more pronounced with respect to the opposite one.

Let us consider them in detail. For the periodic crystal, the TMOKE spectrum demonstrates pronounced resonances at the slopes of the transmittance peaks and vanishes at normal incidence due to symmetry reasons [Fig. 3(c)]. At the oblique incidence, two pronounced TMOKE peaks of opposite signs are observed at around $\lambda =850\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$. They are related to the low-frequency and high-frequency branches of the plasmonic dispersion, and their opposite signs are due to the propagation of SPPs in forward and backward directions. The TMOKE reaches 2% at the extrema of the spectra. No noticeable TMOKE response is observed for $\lambda >900\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$.

However, for the plasmonic quasicrystal the TMOKE demonstrates a much richer response: six TMOKE peaks with alternating signs are observed. Apart from the two peaks at around $\lambda =820\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ (for small incidence angles), that is quite similar to the resonances for the periodic structure at $\lambda =850\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$; there are two peaks at around $\lambda =890\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ and two more peaks at $\lambda =950\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, corresponding to $G=16.76\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$ and $G=15.39\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$, respectively [see Fig. 2(b)]. The resonance at $\lambda =890\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ is well pronounced and provides TMOKE as large as 0.8%. At $\lambda =950\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, the resonance is rather weak, and the TMOKE does not exceed 0.3%, though it is still detectable above the noise level. The decrease of the TMOKE is related to the decrease of the magneto-optical parameter of the magnetic dielectric [42] as well as the decrease of the amplitudes of the corresponding peaks in the Fourier spectrum of the structure pattern [see Fig. 2(b)].

The Fourier spectrum of the quasicrystal indicates that there are two more reciprocal vectors of $G=17.62\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$ and $G=18.15\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$ in the vicinity of $G=18.48\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\mu m}}^{-1}$, but they have relatively low Fourier amplitudes (0.03 and 0.02, respectively) and, therefore, do not produce any notable resonances even in TMOKE.

Thus, the plasmonic quasicrystals offer multiple plasmonically mediated TMOKE resonances. Figure 4(a) demonstrates it more clearly. The TMOKE spectrum for the range of wavelengths from $\lambda =750\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ to $\lambda =1000\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ is shown for an incident angle of 2 deg for the periodic (red curve) and quasicrystalline (black curve) structures. As the incident angle is rather small, resonances overlap. For the periodic structure, only one resonance (marked by “P”) is observed providing non-vanishing TMOKE in the range from $\lambda =800\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ to $\lambda =870\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, while the plasmonic quasicrystal clearly exhibits two close resonances (“Q1” and “Q2”) and also the third weaker one (“Q3”), expanding the TMOKE range from $\lambda =780\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ to $\lambda =970\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$. It confirms that plasmonic quasicrystals allow much broader magneto-optical response.

As we saw in Figs. 2(a) and 2(b), the quasicrystals formed by different type of sequences have quite different Fourier transforms and therefore SPP spectra. However, most of the resonances do not have large efficiency and are hardly observable in transmission or reflection. At the same time, measuring the TMOKE reveals these resonances and helps identify different quasicrystal patterns.

The multiplicity of the excited plasmonic modes attracts attention also because they possess different values of the penetration depth. The penetration depth is found from approximation of a smooth single interface between two semi-infinite metal and dielectric media as

[blue dashed line in Figs. 4(b) and 4(c)]. The SPP field distribution at different wavelengths in the magnetic film is schematically shown by the blue color map in Figs. 4(b) and 4(c) as square of the field amplitudeIt is shown that at the observed plasmonic resonances, the SPP penetration depth in the magnetic dielectric varies with wavelength approximately from 70 nm to 100 nm. This fact opens new possibilities for manipulation of the optical near field, 3D sensing, control of the inverse magneto-optical effects, and optically induced magnetization.

After demonstration of the concept of magnetoplasmonic quasicrystals, the next step is the design of plasmonic quasicrystalline structures in order to obtain the desired optical response. Quasicrystalline structures provide designable reciprocal lattice, i.e., the set of reciprocal vectors and therefore dispersion of eigenmodes, by means of adjusting geometrical parameters. High spectral density leads to overlapping resonances that result in broadband response. In particular, plasmonic quasicrystals offer a designable spectrum of magneto-optical response for light modulation, which is prosperous for parallel light information processing at several frequencies.

Furthermore, the plasmonic quasicrystals are promising for achieving other broadband magneto-optical effects related to the excitation of eigenmodes. If the structure supports waveguide modes, then there are many resonances for transverse electric (TE) and TM modes with the resonant wavelengths close to each other. This condition is favorable for the enhancement of the Faraday effect as the TE–TM conversion is the most effective. Besides that, the magnetization affects the waveguide mode polarization, which usually leads to the longitudinal magnetophotonic intensity effect (LMPIE) in plasmonic structures. The dense dispersion of the waveguide modes might lead to multiple LMPIE resonances.

## 6. CONCLUSION

We have proposed and demonstrated a novel structure—magnetoplasmonic quasicrystal—for getting significant magneto-optical effects in the wide wavelength range. It is based on magnetic dielectric film and the gold film perforated with the subwavelength slits forming a Fibonacci-like binary sequence. While transmission spectra of the periodic and quasiperiodic patterns are quite similar, the TMOKE spectra demonstrate significant difference. Namely, for the quasicrystal, the magneto-optical response is much more abundant. It shows that TMOKE spectroscopy is an efficient tool for investigation of the peculiarities of plasmonic quasicrystals.

Due to the larger density of the discrete peaks in the Fourier transform of the quasicrystal, the additional plasmonic resonances appear, leading to multiple TMOKE resonances. In particular, instead of one resonance at $\lambda =850\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ for the plasmonic crystal, three resonances at $\lambda =820\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, $\lambda =890\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, and $\lambda =950\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ appear. The magneto-optical resonances can be controlled by adjusting the parameters of the quasicrystalline pattern. Proper design of the structure leads to the emergence of multiple additional resonances and therefore to the pronounced magneto-optical response in the wide wavelength range from $\lambda =780\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ to $\lambda =970\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$. Moreover, overlapping magneto-optical resonances might potentially lead to broadband response. It makes the proposed approach very promising for numerous nanophotonic applications, including optical sensing, control of light, all-optical control of magnetization, etc. In particular, it can be useful for nonlinear magneto-plasmonics [49–51] for managing the interactions between multiple co-propagating SPP modes.

## Funding

Russian Foundation for Basic Research (RFBR) (16-52-45061); Russian Presidential Grant (MK-2047.2017.2); Department of Science and Technology, India; Ministry of Education and Science of the Russian Federation (Minobrnauka) (3.7126.2017/8.9).

## Acknowledgment

A. N. S., A. R. P., and V. N. B. acknowledge support by the Ministry of Education and Science of the Russian Federation.

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