## Abstract

Spatial symmetry breaking in the magnetoplasmonic system results in magneto-optical effects prohibited in symmetric structures. The transverse magneto-photonic transmission effect (TMPTE), occurs thereby, which is characterized by a non-zero transmission dependency on magnetism at normal incidence of light. This phenomenon is due to inequality of the forward and backward surface plasmon polariton (SPP) waves at the metal/dielectric interface in non-symmetric plasmonic grating. Here we investigate the TMPTE in magnetoplasmonic nanostructures with comb-like gratings and explore how to optimize its parameters for the magnitude of the effect.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

The magnetoplasmonics exploits the combination of the nanostructuring and the magnetic materials to increase or modify the well-known optical and magneto-optical (MO) effects [1–16]. With the development in recent decades of technologies for micro- and nano- surface patterning, lithography, etc. [17–19] completely new possibilities have opened up for controlling optical signals in media with a magnetic response, that can be employed in the devices based on MO interactions to improve their properties [20–22]. On the other hand, the mutual influence of the excited optical modes and the magnetic properties of materials was investigated in detail. For instance, it was shown that the excitation of the optical modes in the magnetoplasmonic nanostructures allows for resonant enhancement of the MO Faraday effect [5–7], the transverse magneto-optical Kerr effect [8–15] and etc. The excitation of optical modes in nanostructures increases the spatial localization of light in it that leads to the higher MO response even for ferrimagnetic dielectrics that are well-known to reveal small magnetic effect. In its turn, the excited optical modes affect the properties of the observed MO effects.

The connection between the excitation of optical modes and the optical and magneto-optical response of a structure can be understood in the following way. Optical modes produce resonances in optical transmission or reflection spectra. In magnetic materials the polarization and dispersion of optical modes can be varied via change of magnetization, that leads to emerging of resonances in spectra of magneto-optical effects [23]. In particular, if magnetization is directed in-plane perpendicular to the mode propagation, the change of mode dispersion leads to the spectral shift of the optical resonance, which implies resonant enhancement of the transverse magneto-optical effect. The essential point is the presence of a propagating mode, as the effect relies on the dispersion properties. This resonant enhancement is mostly pronounced for the case of the surface plasmon polariton excitation, since the Kerr-type magneto-optical effects are boundary rather than bulk effects [2].

For a long time the MO effects have been addressed just in the magnetic nanostructures with the spatially symmetric gratings of different types [1–16]. However, the breaking of the spatial symmetry in the unit cells of the plasmonic grating affects the properties of the plasmonic waves and, as a result, modifies the transmission properties as well as the MO response of the entire nanostructure.

Recently, the novel MO effect has been demonstrated in the magnetoplasmonic nanostructures with spatial symmetry breaking [24]. The transverse magneto-photonic transmission effect (TMPTE) appears when the plasmonic waves excited at the interface of metal and dielectric counterparts of the nanostructure have different effectiveness for the propagation in opposite directions due to the spatial asymmetry of the plasmonic grating. Therefore, the effect can be observed just in the non-symmetric nanostructures and disappears in the symmetric ones.

The distinctive feature of the TMPTE is that it is non-zero even at the normal incidence of the input light. In this case the SPP waves propagating forward and backward in the transverse direction of the plasmonic grating are inequivalent due to the asymmetry of the grating. Moreover, the asymmetry ensures the prevailed excitation of one SPP mode than the other one that affects also the transmission properties of the nanostructure.

At the same time in case of the oblique light the inequivalence of the forward and backward SPP waves occurs also due to the difference of the input wave vector. Therefore, pure impact of the asymmetry on the transmission properties and the MO effect can be observed at the normal incidence of light when the contribution of the input light angle is absent.

The discussed TMPTE is determined as a relative change of the transmitted light intensity $T(\bf {M})$ when the nanostructure is re-magnetized. In our case it means that the effect is measured for two opposite directions of magnetization $\bf {M}$ [24]

In this paper we address theoretically and experimentally the TMPTE in the non-symmetric magnetoplasmonic nanostructure with the comb-like plasmonic grating on top of the ferrimagnetic material. In [24] the asymmetry of the grating appeared in two neighbour periods of the subwavelength grating. In contrast to the nanostructure addressed in [24], the comb-like grating has the spatial symmetry breaking within the single period of the plasmonic structure. The asymmetry of the nanostructure is determined by the magnitude of the lateral modulation, i.e. the depth of the teeth of the ’comb’. The dependence of the TMPTE on the parameters of the plasmonic grating opens up the opportunities for the optimization of the nanostructure to maximize the observed MO effect.## 2. Materials and methods

#### 2.1 Sample description and fabrication

The scheme of the addressed comb-like magnetoplasmonic nanostructure with spatial symmetry breaking illuminated by the oblique light is given in Fig. 1. As one can see each gold stripe of the grating looks like a comb with the rectangle teeth turned in the direction coinciding with $x$-axis in the figure. The stripes are separated from each other by air slits as well as each tooth has an air gaps with the neighboring teeth. The comb-like stripes appear with periodicity $P$. Parameter $d_1$ refers to width of the gold stripes in the degenerate case when there is no lateral modulation and is equal to the sum of the comb’s base width and the teeth width in non-degenerate case.

In the paper we assume that the values of periodicity $P=285$ nm and parameter $d_1=205$nm are fixed, but we have verified that the results remain the same for other values of these parameters. The modulation depth of the comb (or the teeth width) is determined by parameter $d_2$. Varying parameter $d_2$ from 0 to some value less than $d_1$ one can obtain the change of the symmetry properties of the nanostructure and, as a result, its magneto-optical properties as well. When $d_2=0$ the modulation of the stripe is absent and the plasmonic grating becomes symmetric. With the increase of parameter $d_2$ one can talk about spatial symmetry breaking of the nanostructure. The magnetization of the sample is in-plane and points out along the comb’s base, i.e. perpendicular to the teeth. The structure is illuminated by the $p$-polarized light falling down to the nanostructure at the incidence angle $\theta$ as it is shown in Fig. 1.

It should be noted that although the measurements in this work were carried out in the angular range from $-10^\circ$ to $10^\circ$, the main interest in this study is focused on the normal incidence of light. In this case, there is no contribution to the wave number of SPP modes traveling across the plasmonic nanostructure in opposite directions, associated with the oblique incidence of the light beam. Consequently, in this case, the MO effect and its modulation, will be caused just by the properties of the asymmetry the nanostructure.

To study the TMPTE in such non-symmetric nanostructure samples of magnetoplasmonics comb-like nanostructures were fabricated. The $100$-nm-thick ferrimagnetic layer of bismuth-substituted iron-garnet (Bi$_{1.5}$Gd$_{1.5}$Fe$_{4.5}$Al$_{0.5}$O$_{12}$) was deposited by reactive ion beam sputtering on non-magnetic (111) gadolinium gallium garnet (GGG) substrate [25]. Then, the 80-nm-thick gold grating with the comb-like stripes was fabricated on top surface of the ferrimagnetic layer by the electron-beam lithography.

Scanning electron microscope images of the obtained gratings are given in Fig. 2. The period of all gratings was $285$ nm, parameter $d_1=205$ nm, and the modulation depth increased from $d_2=0$ (a degenerate case of the symmetric grating) to $d_2=155$ nm. The first plasmonic grating in Fig. 2(a) is a spatially symmetric with $d_2=0$. In Fig. 2(b) the modulation parameter becomes $d_2=75$ nm and the asymmetry of the plasmonic grating appears. In Figs. 2(c,d) the asymmetry properties of the nanostructure become stronger with the growth of the modulation depth of the comb-like grating and become $d_2=125$ nm and $d_2=155$ nm, correspondingly. The parameters of the obtained gratings were also verified by the numerical simulation of the optical and magneto-optical properties of the corresponding magnetoplasmonic structures.

#### 2.2 Experimental setup

To explore an influence of the asymmetry of the addressed comb-like magnetoplasmonic gratings we have measured the optical and MO spectra over the wide range of the wavelength and incidence angles. A tungsten halogen lamp was used as a source operating in visible and near-IR ranges. The source light was focused on the sample surface into a spot of $200$ $\mu$m in diameter by two achromatic lens. The magnetization of the ferrimagnetic layer was provided by a uniform saturating external magnetic field of 2000 Oe generated by an electromagnet. After the sample the light was collimated with a 20x microscope objective and then was detected with the spectrometer. The slits of the spectrometer were oriented perpendicularly to the sample’s gold strips. In the spectrometer the light experiences a spectral decomposition along one axis and the incidence angle decomposition along the perpendicular axis by a 2D CCD camera. We measured the angular and wavelength resolved transmission spectra of all samples for two opposite directions in the magnetic field. Each measurement with alternating opposite directions of the magnetic field was repeated 200 times and then these results were averaged. This regime provides a signal to noise ratio exceeding three orders of magnitude in the spectral range of our interest. Although the measurements included wide angular spectrum, we are mostly interested in the normal and quasi-normal incidence of light due to the peculiarities of the explored effect, the TMPTE. In this case the influence of the asymmetry is manifested best. The period of the plasmonic gratings of the addressed nanostructure is about half of the wavelength of light. It means that we observe just zero diffraction orders in the transmitted light.

## 3. Results

#### 3.1 Experimental results

There were measured the transmission spectra of the addressed symmetric and asymmetric plasmonic gratings for two opposite directions of the magnetization. The corresponding angular and wavelength resolved spectra are given in Section 1 of the Supplementary Materials.

Figure 3 shows the TMPTE, $\delta _T$, spectra of the addressed comb-like magnetoplasmonic nanostructures as observed (left column) and calculated (right column, also see details in Section 3.2) TMPTE. The nanostructures were designed in such a way that SPP resonances could be observed in the optical range (red-blue crossed area in Fig. 3). SPP resonances are well-known to enhance the MO effects [2]. Moreover, here we attempt to modify the SPP propagation direction by structure asymmetry and investigate its impact on the TMPTE effect.

Symmetric structure is characterized by zero intensity changing with magnetic field at normal incidence (Fig. 3(a,b)). However, the TMPTE at normal incidence of light becomes non-zero and undergoes a modulation of its magnitude on the wavelength of the input light when the spatial symmetry is broken (see Fig. 3(c-h)). The most noticeable changes can be detected between 730 and 760 nm.

In the grating with $d_2=75$ nm (c,d) the TMPTE modulation at $\theta =0^\circ$ has narrow resonance with an amplitude of around $0.01$. The gratings with larger asymmetry provide the smaller magnitude of the MO effect modulation, but the observed SPP resonances are wider (see Figs. 3(e-h)). The third row give the transmission and TMPTE spectra in the grating with $d_2=125$ nm (e,f) and the fourth row depicts the spectra in the grating with $d_2=155$ nm (g,h), correspondingly. One can see that the magnitude of the MO effect at the normal incidence of light becomes smaller with the increase of parameter $d_2$.

As a result, measurements of the MO effect in comb-like plasmonic nanostructures revealed that the TMPTE is non-monotonic in relation to the depth of lateral modulation and that there is a modulation depth $d_2$ that provides the highest amplitude of the MO effect. Furthermore, additional structure characteristics such as the period or $d_1$ may influence the maximal MO effect. However, for simplicity of numerical simulation we fix the majority of the structure parameters and analyze only the dependence on the $d_2$.

#### 3.2 Numerical simulations

The pure TMPTE effect observed at the normal incidence of the input light (Figs. 3(d, f, h)) is smaller by magnitude than in case of the oblique light. As one can see from Fig. 3 varying the parameters of the nanostructure one can change and even increase the value of the TMPTE. So, the question arises, is there any optimal parameter that provides the greatest value of the TMPTE?

We have addressed the MO properties by means of the numerical simulation of the comb-like magnetoplasmonic nanostructures by the rigorous coupled-wave analysis (RCWA) [26,27]. We consider the parameter $d_2$ as an optimization parameter responsible for the asymmetry of the structure for the determined spatial period of the plasmonic grating. Varying $d_2$ we can change the magnitude of the TMPTE. As soon as at $d_2=0$ and $d_2=d_1$ an asymmetric structure transform into a symmetric one, the TMPTE should have a maximum for some intermediate value of $d_2$.

The obtained spectra of the TMPTE versus the lateral modulation depth $d_2$ at the normal incidence of light is shown in Fig. 4. One can see that the TMPTE achieves its maximum amplitude at $d_2=0.085$ $\mu$m depicted by black line. The red line corresponds to the grating with $d_2=0.075$ $\mu$m shown in Fig. 2(b) and its spectra in Fig. 3(d). It doesn’t provide the maximum possible MO effect in this structure, but it is very close. For the values of $d_2$ greater than $0.085$ $\mu$m the TMPTE gradually decreases. The green and yellow lines in Fig. 4 correspond to Fig. 3(f) and (h), respectively. They have a comparable amplitude of the effect, but for both of them the effect is weaker than for the other curves (except the blue one referring to degenerate case).

#### 3.3 Theoretical analysis

The influence of the optimization parameter, $d_2$, and the geometry of the structure on the TMPTE and underlying physics can also be explored analytically. For this purpose, we consider the SPP modes propagating in the studied comb-like magnetoplasmonic nanostructure at the [metal]/[ferrimagnetic dielectric] interface.

The resonant enhancement of the intensity magneto-optical effects in plasmonic structures is related to the excitation of SPP modes. The enhancement comes from the fact that the SPP dispersion is dependent on magnetization in the transversal configuration, that leads to the spectral shift of the SPP resonances. For the MO resonance it is necessary that the excited mode is a propagating one [2].

Let’s consider the resulting surface wave as a superposition of the modes running forward and backward along the $x$-axis. Also, it should be taken into account that the SPP modes excited at different sides of the comb-like grating have a phase mismatch depending on the modulation depth $d_2$ and the width $d_1$ (see Fig. 5). The single SPP mode [28] at the [metal]/[ferrimagnetic dielectric] interface is $p$-polarized and its magnetic field component is described as

where $\kappa$ and $\gamma _i$ are the longitudinal wavenumber and the surface plasmon polariton localization coefficient [2]. Here $\kappa =\kappa _0 (1+\alpha g)$, where $\kappa _0 = k_0\sqrt {\frac {\epsilon _m \epsilon _d}{\epsilon _m+\epsilon _d}}$, $\alpha = [\sqrt {-\epsilon _m \epsilon _d}(1-\frac {\epsilon _d}{\epsilon _m})]^{-1}$, $\gamma _i=k_0\sqrt {\frac {-\epsilon _i^2}{\epsilon _m+\epsilon _d}}$ with $i$ is $m$ or $d$, and $k_0$ is a wavenumber in the free space. Then, $g$ denotes the gyration of the ferrimagnetic material [29], which is related to magnetization. The magnetization of the material leads to the modification of its dielectric permittivity tensor $\epsilon$, in particular, the non-diagonal components of $\epsilon$ are equal to $\pm i g$. $\epsilon _d$ and $\epsilon _m$ describe the dielectric properties of the metal and dielectric film, correspondingly.We consider the interference of the four waves shown in Fig. 5. Our analytical model employs two approaches. First, there is a phase shift between the SPP modes excited at the “top" of the comb tooth and at its base. This phase shift is described by $\exp (-i\kappa d_2)$. Second, the comb-like plasmonic grating can be represented as a superposition of two unmodulated plasmonic gratings with the same periods but different air gap widths. It leads to the fact that SPP modes excited at the top of the comb tooth, and at its blade have the different field amplitude. The ratios of the light wavelengths to the dimensions of the air grooves addressed in this work allow applying Fraunhofer diffraction approach. Therefore, the SPP mode amplitude is proportional to $\sin (\kappa w/2)$, where $w$ is the width of the air groove, that is equal to $P-d_1$ or $P-d_1+d_2$ for different waves (see Fig. 5) [30].

Based on the aforementioned approaches at normal incidence of light the sum of the four waves shown in Fig. 5 can be written as

One can derive the following expression for the amplitude of the resulting wave

From Eq. (4) one can see that the resulting wave is the wave with the spatially varying phase depending on the lateral modulation depth, $d_2$, as well. From the experimental measurements and the numerical simulations it was found out that this feature leads to the modulation of the magneto-optical effect at the normal incidence of light.

The question arises whether the wave described by Eq. (4) with the spatially varying phase is a standing wave or the propagating one. To answer this question one should address the Poynting vector of the wave, $\vec {S} = Re [\vec {E} \times \vec {H}^*]$. Moreover, we are interested in the $x$-component of the Poynting vector, which is calculated as

The non-zero magnitude of the Poynting vector brings an evidence that there is an uncompensated surface plasmonic wave at the interface of [plasmonic grating]/[ferrimagnetic dielectric]. It leads to the modulation of the magneto-optical effect even at the normal incidence of light.

The dependence of the Poynting vector magnitude on the optimization parameter, i.e., lateral modulation depth, is non-monotonic, that leads to the existence of the maximum value of the Poynting vector magnitude at some optimal value of $d_2$. In Fig. 6 the normalized dependence of the magnitude $S_x$ versus $d_2$ for the fixed set of parameters $P$ and $d_1$ and the wavelength $\lambda =0.750$$\mu$m is given. One can see that there is a value of $d_2$, when $S_x$ achieves its maximum. Besides that, when $d_2$ is high enough, the sign of $S_x$ changes. That means that the propagation direction of the resulting SPP wave changes to the opposite one.

The non-monotony of the Poynting vector dependence on the lateral modulation depth meets the non-monotonic dependence of the TMPTE on $d_2$ shown above experimentally and numerically. As soon as the SPP modes provide the resonant enhancement of the MO effects due to the dispersion modification, the optimization of the comb-like plasmonic nanostructure for the maximum of the resulting Poynting vector, the MO effect achieves its maximum as well.

So, the measured spectra, numerical simulations and the analytical dependence given in Fig. 6 agree with each other. It allows us to conclude that, indeed, the optimization of the parameters of the plasmonic grating makes it possible to achieve an increase in the TMPTE.

## 4. Conclusion

To sum up, the properties of the TMPTE in the non-symmetric magnetoplasmonic nanostructures with comb-like grating have been addressed. This effect is only seen in the magnetoplasmonic nanostructures with the spatial symmetry breaking. The asymmetry of the design subwavelength comb-like grating is determined by the lateral modulation depth that serves as an optimization parameter of the setting. By varying it, one can control the magnitude of the observed MO effect and achieve the highest possible value of it for the certain magnetoplasmonic nanostructure.

## Funding

Russian Science Foundation (19-72-10139); Russian Foundation for Basic Research (19-02-00856).

## Acknowledgments

The theoretical study was supported by the Russian Foundation for Basic Research (project No. 19-02-00856). The experimental research was supported by the Russian Science Foundation (project No. 19-72-10139).

The authors thank A.N. Kuzmichev for his support with the numerical simulations. O.V.B. thank the personal support of the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS. The equipment of the “Educational and Methodical Center of Lithography and Microscopy", M.V. Lomonosov Moscow State University Research Facilities Sharing Centre was used. H.H. gratefully acknowledge the 5 top 100 Russian Academic Excellence Project at the Immanuel Kant Baltic Federal University. A.N.K. and V.I.B. are members of the Interdisciplinary Scientific and Educational School of Moscow University “Photonic and Quantum technologies. Digital medicine."

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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