We experimentally demonstrate a disruptive approach to control magnetooptical nonreciprocal effects. It has been known that the combination of a magneto-optically (MO) active substrate and extraordinary transmission (EOT) effects through deep-subwavelength nanoslits of a noble metal grating, leads to giant enhancements of the magnitude of the MO effects that would normally be obtained on just the bar substrate. This was demonstrated both in the transmission configuration, where the OET is directly observed, as well as in reflection configuration, where an increase of a transmitted power results in a decrease in reflected power. We show here that even more than just an enhancement, the MO effects can also undergo a sign reversal by achieving a hybridization of the different types of resonances at play in these EOT nanogratings. By tuning the geometrical profile of the grating’s slits, one can engineer — for a fixed wavelength and fixed magnetization — the transverse MO Kerr effect (TMOKE) reflectivity of such a magnetoplasmonic system to be enhanced, extinguished or inversely enhanced. We have fabricated gold gratings with varying nanoslit widths on a Bi-substituted gadolinium iron garnet and experimentally confirmed such a behavior using a customized magneto-optic Mueller matrix ellipsometer. This demonstration allows new design paradigms for integrated nonreciprocal circuits and biochemical sensors with increased sensitivity and reduced footprint.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Magnetoplasmonics  — a recently coined term referring to any physical system exploiting the combined effect of a magnetic and a plasmonic functionality — is attracting a surge of interest in recent years. This research domain knows two main driving forces. On the one hand, the use of a magnetic field to add an active control, such as switching and modulation, to plasmonic-based nanophotonic circuitry  has been proven a compact, reliable and fast alternative [3–6] to more complex control agents such as nonlinearities [7, 8]. Secondly and more importantly, plasmonic field concentration has been exploited to enhance the unique time reversal breaking properties of magneto-optic (MO) materials . In recent years numerous experimental reports have demonstrated the strong enhancement of all traditional nonreciprocal MO phenomena — complex Kerr and Faraday rotation and complex transverse Kerr phase shift — due to coupling with propagating surface plasmon polaritons (SPP) or excitation of localised plasmon resonances. A wide range of magnetoplasmonic configurations has been investigated: semi-transparent dielectric magnetic iron garnet films coupled to noble metal nanogratings [6, 10–14], 2D arrays of noble metal nanoparticles or nanoholes in noble metal sheet [15, 16], nanogratings of pure ferromagnetic (FM) metal thin films [17, 18], FM nanoparticles [19, 20] and nanowires  on dielectric substrates, hybrid ferromagnetic/noble metal sandwiches both in nanoparticle or nanowires [22–24], systems of coupled plasmonic and ferromagnetic nanoparticles , inverted Babinet nanohole layouts [26, 27] and propagating SPP layout [28, 29], and magnetic/plasmonic core-shell nanoparticles [30–32].
All these demonstrations share the same conceptual approach to boost the magneto-optic properties of the system: by operating closely to a high quality factor plasmonic resonance the impact of an otherwise weak MO phenomenon is sharply enhanced. A particularly convincing example of plasmonic enhanced magneto-optics is the demonstration of giant transverse MO Kerr reflectivity (TMOKE) when applying a one-dimensional plasmonic slit grating on a transparent magneto-optic iron garnet substrate (which can be possibly used as an optical waveguide) that is magnetized in-plane and parallel to the grating’s slits [33, 34]. By themselves, lossless MO materials (such as iron garnets in the near infrared region) can fundamentally not exhibit nonreciprocal TMOKE intensity effects [35, Eq. (1.138)]. However by coupling a transparent MO garnet to the Wood anomalies of a gold extraordinary optical transmission (EOT) grating, nonreciprocal intensity reflectivities are obtained that even exceed those of strongest lossy FM metals.
In many optimized layouts for nonreciprocal circuits (such as isolators and circulators) it is inevitable to have antisymmetric magnetized regions in order to break also the spatial inversion symmetries [36, 37], to optimize the MO interaction with the symmetries of the modes at play [38, 39] or to achieve a more powerful push-pull configuration for the nonreciprocal effect [40, 41]. Gaining control over the sign of the nonreciprocity by plasmonic engineering would allow realizing such layouts without the need for complicated inverted magnetic domains and just having a uniform magnetization in the whole structure.
Despite the fact, that behavior of the EOT resonances and their interactions in noble metal plasmonic structures were studied in details [42, 43], the influence of the possible coupling on the MO response was never deeply analyzed before. Differently and adding to these demonstrations of magnetoplasmonic nonreciprocity enhancement, we report here on the anomalous control of the sign of the nonreciprocal phenomena when exploiting the different resonances in a magnetoplasmonic system . Apart from the obvious interest of investigating controlled tuning of resonances in magnetoplasmonic systems and their impact on the MO enhancement, there is another important motivation for controlling the sign of the nonreciprocity at play. Time reversal breaking (or thus nonreciprocity) is governed by the sense of the magnetization in the system. Up to first order, inverting the magnetization will invert the sign of the nonreciprocal phenomenon at play. Naturally, in design of the magnetoplasmonic gratings and arrays, the optical reciprocity can be tuned by controlling of the wavevector/angle of incidence . In this paper we demonstrate approach of static controlling of the magnetooptical nonreciprocity based on tuning of the coupling between resonances without changes of the electromagnetic wave properties.
2. Magnetoplasmonic grating structure design, fabrication, optical characterization and numerical modeling methods
To investigate magnetoplasmonic sign control and enhancement, we have designed and fabricated one-dimensional (1D) lamellar gold nanogratings on the 4 µm thick, compositionally optimized, Bi-substituted gadolinium iron garnet (Bi:GIG), Gd1.24Pr0.48Bi1.01Lu0.27Fe4.38Al0.6O12 grown by liquid phase epitaxy (LPE) on the substituted Gadolinium Gallium Garnet (CaMg-GGG, lattice parameter as = 12.498 Å) substrate. Fig. 1(a) schematically represents the studied structure. A number of square patches containing 600 periods (Λ = 500 nm) of 300 µm long and 93.7 nm thick Au stripes separated by nanoslits, have been processed by standard lift-off techniques. 
The first set of 15 samples was fabricated by varying the e-beam exposure dose per written patch: the width r of the nanoslit is varied from patch to patch (see Fig. 1b). In this way we expect all developed patches have the same period and thickness of deposited gold. Therefore, the effect of the nanoslit width on the magnetoplasmonic properties of the grating structure is directly comparable. To complete our study in investigation of the sign control, two additional sets of patches with gold thickness of 118 nm and 134 nm were fabricated. The new fabricated sets of patches were analyzed and two patches with similar nanoslit widths were selected. Therefore, MO response of the patches which are different only in the thickness can be directly compared.
The grating’s optical response has been probed by using a Mueller matrix spectroscopic ellipsometer (SE) Woollam RC2-DI (J.A. Woollam Co.) in specular reflection configuration. The ellipsometer uses a combination of a halogen bulb and deuterium lamp as a light source and it operates in the spectral region from 0.74 eV to 6.42 eV (193-1700 nm). The spectral resolution of the setup was 2.5 nm in infrared spectral region and 1 nm in visible and ultraviolet spectral regions. For characterization of the optical response of the developed gratings we have used micro focusing probes with a focal length of 27 mm to reduce the spot size to approximately 150 µm. Focused beam allows reducing the beam spot size to individually measure each grating. To measure magnetooptical response of the samples, the ellipsometer was extended with in-house design computer controlled in-plane permanent magnet circuit.
The Rigorous Coupled Wave Algorithm (RCWA) was used for optical modeling and fitting of experimental SE data measured on the plasmonic grating. From data measured on the first batch of 15 patches the widths of nanoslits have been obtained by a single global least-squares fit of model to all measured SE data. See ref.  for more details about used SE setup, parametrized model of the studied plasmonic gratings and used optical and magnetooptical functions of materials.
3. Operational principle of resonant modes in magnetoplasmonic grating structure
In the following we will explain the phenomena of tuning the plasmon resonances and MO sign conversion by a simple model of coupling SPP and Fabry-Pérot (FP) modes. We firstly describe the case without MO effect and then add the MO perturbation.
3.1. Optical observation of the effect of coupling between different grating resonances (without MO effect)
In a first approximation, the plasmonic resonance shift as a function of the nanoslit width r cannot be explained by the diffractive folding of the Au/garnet SPP dispersion, as this latter is parametrically dependent only on the grating’s period Λ, the incidence angle φ, and the Au and Bi:GIG permittivity, εAu and εBi:GIG. In previous theoretical work we demonstrated how variations of the grating’s geometry (in particular its thickness h) cause an anticrossing interaction of the diffractively coupled SPP resonances and the FP slit resonances undergone by the guided TM mode in the subwavelength metal/insulator/metal slit.  Increasing thickness leads to an increasing geometric phase of the fundamental TM slit mode, , thereby redshifting the FP EOT resonances. Due to anticrossing the SPP reflection anomalies therefore eventually also redshifts with the grating thickness h.
In a first approximation the spectral position of the grating’s resonances as a function of its geometrical parameters (h, r, and Λ), can be obtained by solving the following dispersion equations for the photon energy E:
Here neff,FP is the effective index of the fundamental TM mode of the Au/air/Au plasmonic slot/slit waveguide formed by the nanoslit and obtained by solving 
The reflection phase shifts, , at both ends of the slit cavity can in first approximation be those of normal plane wave incidence, . Beside the SPP resonances depend only on Λ and φinc, and are independent of the grating’s geometrical parameters governing the FP spectral location, h and r, as seen from Eqs. 1–3. Varying therefore only, for instance, the nanoslit width r at fixed incidence angle and grating periodicity may lead to SPP-FP resonance coupling.
Figure 2(b) illustrates this for the case of φinc = 20° and h = 100 nm. In order to numerically observe the dispersion of the resonant modes, the structure was simulated with a semi-infinite Bi:GIG substrate. From Eq. (3) one can deduce how the decreasing slit width r will cause an increase of neff,FP, which in turn by Eq. (2) will redshift the FP resonance. The FP slit resonance (blue) is seen to cross the 2nd−order Au/garnet SPPs (red) for nanoslits widths between 20 and 40 nm. Due to anticrossing mode coupling the otherwise fixed energy of the 2nd−order SPPs will be perturbed even for slit widths values beyond this range. This is demonstrated in Fig. 2(c) where we have numerically calculated the specular reflectivity spectrum Rp for a range of slit widths (at φinc = 20°) using an extended coupled-mode formalism [(ECMF), see Ref. ]. The reflection minima corresponding to the different mentioned grating resonances can be distinguished, but more importantly their coupling and anticrossing in the region predicted in Fig. 2(b) are convincingly observed. Fig. 2(a) zooms in on the redshift of the 2nd−order Au/garnet SPP. The experimentally observed redshift of the plasmonic grating reflection anomaly can therefore be correctly attributed to a coupling between FP and SPP resonances. This experimental observation confirms for the first time the previously suggested theoretical possibility of tuning the strength of the EOT effects only by tuning one geometrical parameter [44, 51].
3.2. Study of magnetooptical perturbation of the SPPs EOT modes
Turning our attention to the magnetic character of the garnet substrate of the studied plasmonic nanograting, we explore the impact of the observed coupling between the grating’s resonances on the magneto-optical activity of the system. In MO materials the presence of a magnetization creates antisymmetric off-diagonal components in the permittivity tensor, , where in first order the gyration vector g = g1M is parallel to the magnetization and its magnitude is linearly proportional to that of M, g ∼|M|. For lossless MO media must be hermitian, so that both ε and g must be real. In the most general case this gives rise to a nonreciprocal polarization change of a polarized beam  when reflected from a surface of a MO layer. The fundamental process induced by the magnetization is the Lorentz force on electrons. Therefore, in a unique configuration of a magnetization perfectly perpendicular to the incidence plane, the s-wave having its E-field perfectly parallel to the magnetization, will not undergo any MO effect. In this so-called transverse configuration only the p-polarization will undergo a nonreciprocal correction. The ensuing transverse MO Kerr effect leads to a nonreciprocal first-order correction on the Fresnel reflection coefficient, , where is the isotropic conventional reciprocal Fresnel coefficient (for M = 0) and the small effect correction . On a lossless MO material and below its critical incidence angle, TMOKE is then solely a reflection phase shift since Im(gδrp) = 0, and the power reflectivity Rp remains perfectly reciprocal, ΔRp = |rp(M)|2 −|rp(−M)|2 = 0. Nonreciprocal power reflectivity can only be realized on absorbing media. However on bare lossy ferromagnetic metals such as Co or Fe, ΔRp doesn’t exceed 10−3. 
Alternatively, TMOKE power nonreciprocity on a lossless MO garnet can be obtained by cladding it with a plasmonic grating . The gold/garnet SPP dispersion gets a small nonreciprocal correction due to the nonreciprocal phase shift by the transverse MO garnet substrate and the intrinsic relation between the real and imaginary parts of the complex SPP effective index (Kramers-Kroenig relation) leads to the nonreciprocal power shift. In combination with the sharp reflection resonances of the grating’s FP, this leads to a huge enhancement. 
In this way, including the effect of the gyrotropy in the boundary conditions at the interface between gold and a transversely magnetized garnet, the following nonreciprocal dispersion equation is obtained for the surface magnetoplasmon polaritons:
The gyrotropy clearly breaks the inversion symmetry neff,SPP →−neff,SPP. Linearizing this equation with respect to g — for the considered garnet  in the near infrared g ≈ 0.003 (g = 0.0096 @ 1.67 eV) — and solving to first order of g:
Similarly with non-MO configuration described by Eqs. (1–3) FP-SPP coupling occurs. The dependence of the imaginary part of neff,SPP with g induces non-reciprocal response of coupled FP-SPP modes. The model of MO SPP describes the spectral shift of the SPP mode by MO effect by a perturbation linearly dependent on gyrotropy g. It could thus be used to predict the effect and to design devices based on non-reciprocal reflection or transmission with locally inverted sign.
4. Experimental demonstration of control of plasmonic and magnetoplasmonic peak position
Optical and magnetooptical activity of samples were studied in planar diffraction configuration. For straightforward comparison of measured and calculated optical data the following ratio of reflected intensity of p-and s-polarized light and quantification of the MO response is used in this paper:
Traditionally, the TMOKE is normalized by the intensity of the reflected light from structure without applied magnetic field, because of the relative high reflectivity from metallic flat surfaces. We have used modified definition of the MO effect due to the fact, that MO activity in magnetoplasmonic structures presents as a shift of a dip in p-polarized light intensity. Therefore quantity δℛ directly shows change of the reflected light intensity without artificial numerical enhancement by division of small numbers.
We firstly analyse the experimental plasmonic grating reflection, and secondly its magneto-optical response, for different slit widths. The third paragraph is dedicated to the experimental magneto-optical gold grating reflection for different slit heights.
4.1. Measured optical response of plasmonic nanogratings with different nanoslit widths
Figure 3(a) shows measured optical data for the angle of incidence φ = 20° for grating with nanoslit width r = 63 nm, and the grating thickness h = 93.7 nm.
The red curve on subplot 3(a) shows flat profile of the spectral dependence of the s-polarized intensity. On the other hand, profile of the blue curve representing the measured ratio ℛ [defined by Eq. (6)] shows drops in reflected intensity. Therefore observed dips are related to the decrease of the p-polarized intensity which is the result of the light absorption by excitation of surface plasmon polaritons. The vertical arrows point on spectral positions of excitation of ±1st and +2nd SPPs modes described by Eq. (1) (with m1 = ±1, +2 and i= Bi:GIG). The SPP minima occur slightly blueshifted with respect to these peaks [46, 47]. Due to the overall smooth Rs, the sharp Fano-like minima in ℛ therefore correspond to the detection of the Wood plasmon transmission anomalies of the gold grating. For clear understanding of the role of EOT and grating resonances, the reflection, transmission and TMOKE spectra and corresponding field distribution plots are introduced in Appendix A. The observed fringes in measured data are caused by interferences of the transmitted light in the Bi:GIG layer.
The inset of the Figure 3(a) shows part of the eV–K y dispersion diagram. Blue curves represent dispersion curves of the SPPs modes on Au-Bi:GIG interface calculated from Equations 1. Solid black line represents the 20°−light line with black dashed lines to highlight spectral position of the crossing points related to the observed dips in experimental data. The red line represents SPP mode on the air-Au interface.
By the analysis of SE data measured on the set of 15 samples we have determined widths of nanoslits of different patches. Figure 3(b) shows detail on the spectral region of the +2nd Au/Bi:GIG SPP peak. The number of curves was limited to 6 cases and legend with fitted widths of the nanoslits were added. Interestingly, when zooming in on the measured reflectivity spectra the redshift of the extraordinary reflection resonance is observed for gratings with decreasing width of nanoslit.
4.2. Measured magnetooptical response of plasmonic nanogratings with different nanoslit widths
Figures 4(a) and 4(b) directly compare measured MO data on 15 samples having different width of nanoslit with MO numerical simulation data. The other parameters are the same as in simulations of Fig. 2. Experimental data were measured at the nominal angle of incidence of 20°. Numerical simulations were calculated for the incident angle of 20°. The systematic shift of the peak position in model proves observed trend in the experiment. In Fig. 4(b) the neff,SPP gyrotropy dependence (from Eq. (5)) introduces spectral shift of typically ≈ 30 meV whatever the slit. Clearly at a given photon energy, for example 1.68 eV, the TMOKE response can be positive (nanoslit width r=120 nm) or negative (r=70 nm). In other words, for a given optical signal and a given magnetization the non-reciprocity sign is inverted by changing only the slit width.
4.3. Measured magnetooptical response of plasmonic nanogratings structure with different thicknesses and comparable slit widths
For complete analysis of the FP and SPP modes coupling here we also demonstrate tuning of TMOKE via plasmonic grating thickness. According to Eq. (2) the dispersion of the FP modes is strongly affected by the length of the resonant cavity, which is in our case presented by thickness of the gold grating h. For direct demonstration of the induced coupling, two sets of the grating samples with different thicknesses were fabricated by the same procedure of electron beam dose variation as was described above. Samples were measured and characterized with the Mueller matrix ellipsometry. By the detailed analysis of all fabricated patches we have determined thickness and nanoslits width of all samples.
The gold grating thickness of new set of samples was determined to be h = 118 nm and h = 142 nm, respectively. For direct demonstration of impact of SPP and FP modes coupling on MO response, the two patches with comparable nanoslit widths r = 148 nm and r = 137 nm were selected. Figure 5 shows detail on MO response δℛ observed for both mentioned samples at position of the 2nd SPP peak. In our case it is clearly visible that measured MO response (blue and red circles) at 1.7 eV (black line in Fig. 5) is positive for the patch with thickness h = 118 nm and negative for the patch with grating thickness h = 142 nm. The MO sign inverting phenomena by the grating thickness is observed for the 2nd SPP peak, the same SPP peak as used for the switching demonstration by the grating nanoslit width. In addition in Fig. 5 calculated MO response is shown with red and blue solid lines. The MO response was calculated from the model used for the determination of the geometry of individual patches. Very good agreement between measured and calculated data proves validity of the used model. Therefore our simplified model of the developed structure can be used for further design and optimization of magnetoplasmonic structures. Comparing to the previous case, Fig. 4, the MO effect peak-to-peak width increases from approximately 34 meV to 51 meV and 92 meV for grating thicknesses of 93 nm, 118 nm and 148 nm, respectively. This spectral spreading of the MO response of the plasmonic grating was previously observed and discussed [ , Fig. 8 ].
In this paper we have demonstrated unique approach for fine local tuning and switching of the magnetooptical effect in 1D periodic plasmonic gratings. By tuning geometrical parameters of the grating, the non-reciprocal reflectivity not only can be enhanced but also tuned in terms of sign and magnitude. The MO tuning was experimentally and numerically demonstrated for two cases: variation of the grating nanoslit width r and variation of the grating thickness h. We have proved correctness of our models used for determination of individual gratings geometry by direct comparison of measured and calculated MO response. We have also demonstrated how these significant effects can be correctly modeled with a linear analytic approach, giving easy design tool. The presented study offers possibility of additional enhancement of the optical nonreciprocity in the magnetoplasmonic structure. In addition, the preferred sigh of the MO effect (under the static magnetic field) can be adjusted in the design by playing on technologically realistic combination of grating of the same thickness and different slits for advanced circuits including non-reciprocal functions.
Appendix A Resonant modes in magnetoplasmonic gratings
In order to establish the main ideas of the magnetoplasmonic gratings principle Fig. 6 shows the specular reflectivity of the structure (middle red line) and the corresponding TMOKE spectrum (upper, blue curve) for p-polarized light impinging at φ0 = 10° on a typical EOT grating configuration (Λ = 500 nm, h1 = 150 nm, r = 20 nm). This configuration of the model was chosen because it provides all discussed modes separated in the spectral domain. Therefore each resonant mode can be observed separately. From the bottom green line one can observe extra-ordinary optical transmission resonances as pronounced dips in the specular reflection peaks in the specular transmission. It should also be noted that these reflection dips are not related to Wood-Rayleigh (WR) anomalies. The observed dips in the reflection curve originate therefore from the grating’s own resonant modes, or in other words its Bloch modes. Depending on their nature these resonances might experience a more or less important MO response (nonreciprocal spectral shift) upon magnetization reversal, as can be seen from the magnitude of the corresponding TMOKE signature for each EOT resonance (see top subplot in Fig. 6).
In Fig. 7 we have plotted field distributions in the grating (in particular |Hx|2) that have been calculated at the different indicated resonances in the EOT spectrum of Fig. 6. This confirms that the high−Q resonances with strong TMOKE effect (A) and (B) are indeed Au/MO substrate SPPs coupled to ±1 diffraction orders. The proximity of the first order substrate Rayleigh anomalies also reveals their origin. It is also confirmed that (C) is indeed a low Q FP slit resonance.
IT4Innovations national supercomputing center (CZ.02.1.01/0.0/0.0/16_013/0001791); French RENATECH network; Grant Agency of the Czech Republic (18-22102S)
This work was supported by the European Regional Development Fund in the IT4Innovations national supercomputing center -path to exascale project, CZ.02.1.01/0.0/0.0/16_013/0001791 within the Operational Programme Research, Development and Education. The authors gratefully acknowledge the support by the French RENATECH network and by the Grant Agency of the Czech Republic (Project 18-22102S).
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