Magnetoplasmonic nanograting geometry enables optical nonreciprocity sign control

: 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 bare substrate. 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


Introduction
Magnetoplasmonics [1] -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 [2] has been proven a compact, reliable and fast alternative [3][4][5][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 [9].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 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][11][12][13][14], nanogratings of pure ferromagnetic (FM) metal thin films [15,16], FM nanoparticles [17,18] and nanowires [19] on dielectric substrates, hybrid ferromagnetic/noble metal sandwiches both in nanoparticle or nanowires [20][21][22][23], systems of coupled plasmonic and ferromagnetic nanoparticles [24], inverted Babinet nanohole layouts [25,26] and propagating SPP layout [27,28], and magnetic/plasmonic core-shell nanoparticles [29][30][31].
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 [32,33].By themselves, lossless MO materials (such as iron garnets in the near infrared region) can fundamentally not exhibit nonreciprocal TMOKE intensity effects [34,Eq. (1.138)].However by coupling a transparent MO garnet to the Wood anomalies of a gold extraordinary transmission 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 [35,36], to optimize the MO interaction with the symmetries of the modes at play [37,38] or to achieve a more powerful push-pull configuration for the nonreciprocal effect [39,40].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.
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 [41].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 [11].We demonstrate additional degree of freedom in controlling of the magnetooptical nonreciprocity based on tuning of the coupling between resonances without changes of the electromagnetic wave properties.

Magnetoplasmonic grating design, fabrication, and characterization
To investigate magnetoplasmonic sign control and enhancement, we have designed and fabricated one-dimensional (1D) lamellar gold nanogratings on the 4 µm thick, compositionnally optimized, Bi-substituted gadolinium iron garnet (Bi:GIG), Gd The first set of 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 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 have been probed by using a spectroscopic Mueller matrix ellipsometer Wooollam RC2-DI (J.A. Woollam Co.) operating in spectral range from 0.74 eV to 6.42 eV.To measure magnetooptical response of the samples, the ellipsometer was extended with in-house design computer controlled in-plane permanent magnet circuit.See Ref. [42] for more details about parametrized model of the studied plasmonic gratings and used optical optical and magnetooptical functions of materials.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 reciprocity sign is inverted by changing only the slit width.

Measured magnetooptical response of plasmonic nanogratings with different thickness of the gold gratings and comparable slit width
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 3 shows detail on MO response δR observed for both mentioned samples at position of the 2 nd 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. 3) 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 2 nd SPP peak, the same SPP peak as used for the switching demonstration by the grating nanoslit width.In addition in Fig. 3 calculated MO response is shown with red and blue solid lines.The MO response was calculated from model used for determination of the geometry of individual patches.Very good agreement between measured and calculated data proves validity of the used model.In order to understand the origin of this property and to be able to control it, we have analyzed more deeply the reflectivity spectrum of such samples, firstly without magnetization.In the next step, the influence of the magnetooptical activity is introduced in model of the SPPs EOT resonances.

Operational principle of resonant modes in magnetoplasmonic grating
Optical and magnetooptical activity of samples were studied in planar diffraction configuration.
With the plane of incidence perpendicular to the grating direction, the ellipsometric specular intensity ratio R p R s obtained in such measurements will contain the signatures of the surface plasmon polariton (SPP) anomalies of the EOT nanoslit grating.In particular, due to the strong subwavelength nature of the slits the structure has a smooth and overall strong spectral s-reflectivity R s , while the p-reflectivity R p exhibits sharp peaks at the spectral position of the diffractively folded surface plasmon polaritons, Λ (m = ±1, ±2, . . .).The SPP EOT transmission maxima occur slightly blueshifted with respect to these peaks.[43,44].Due to the overall smooth R s , the sharp Fano-like minima in R p R s therefore correspond to the detection of the Wood plasmon transmission anomalies of the gold grating.
The measured ellipsometric spectrum for an incidence angle of ϕ = 20°on a 100 nm thick gold grating formed by a periodic arrangement of strong sub-λ slits (r = 63 nm and Λ = 500 nm) show systematic shift of the reflectivity R dip from 1.68 eV to 1.7 eV for the nanoslit width change from 63 nm to 130 nm (Fig. 4b).As the inset of the Fig. 4a shows, the observed Fano resonances indeed appear close to the crossings of the 20°−light line (solid black) with the theoretically calculated dispersion of the folded SPPs on a flat Au/garnet interface (solid blue).In addition, experimental data were measure for configuration in witch SPPs modes are separates in spectrum.Therefore, the interaction between different SPPs in minimal and their interaction with FP resonant modes can be studied directly.
Interestingly, when zooming in on the measured reflectivity spectra close the +2 nd order garnet/Au SPP, a pronounced redshift of the extraordinary reflection resonance is observed for gratings with decreasing width of nanoslit (Fig. 4b).The Rigorous Coupled Wave Algorithm (RCWA) was used for optical modeling of the plasmonic grating.The widths of nanoslits have been obtained by a single global least-squares fit of model to measured ellipsometric data.The measured spectra for 15 patches with different nanoslit have been processed in a single step on the same garnet substrate.The fitting procedure accounted for the presence of a top roughness layer on the fabricated gold gratings and for depolarization effects due to an angular spread of the incident Gaussian beam [45].It must be noted that the fitted nanoslit widths r follow the trend of the values estimated by SEM observation r SEM (as shown on Fig. 1b).

Optical observation of the effect of coupling between SPP and FP resonances
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.In a first approximation, this resonance shift as a function of nanoslit width r cannot be explained by the diffractive folding of the Au/garnet SPP dispersion, as this latter is only parametrically dependent on the grating's period Λ, the incidence angle ϕ, and the Au and Bi:GIG permittivity, ε Au and ε Bi:GIG .In previous theoretical work we demonstrated how variations of the grating's geometry (in particular its thickness h) causes 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.[46] Increasing thickness leads to an increasing geometric phase of the fundamental TM slit mode, ω c n eff,TM 0 h, thereby redshifting the FP EOT resonances.Due to anticrossing the SPP reflection anomalies will therefore eventually also redshift 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, andΛ), can be obtained by solving the following dispersion equations for the photon energy E: FP : Here n eff,FP is the effective index of the fundamental TM mode of the Au/air/Au plasmonic waveguide formed by the nanoslit and obtained by solving [46] tanh The reflection phase shifts, φ r i , 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 ϕ inc , 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.Fig. 5b illustrates this for the case of ϕ inc = 20°and h = 100 nm, corresponding to the experimental conditions of Fig. 2a.From Eq. (3) one can deduce how the decreasing slit width r will cause an increase of n eff,FP , which in turn by Eq. (2) will redshift the FP resonance.The FP slit resonance (blue) is seen to cross the 2 nd −order Au/garnet SPPs (red) for nanoslits widths between 20 and 40 nm.Due to anticrossing mode coupling the otherwise fixed energy of the 2 nd −order SPPs will be perturbed even for slit widths values beyond this range.This is demonstrated in Fig. 5c where we have numerically calculated the specular reflectivity spectrum R p for a range of slit widths (at ϕ inc = 20°) using an extended coupled-mode formalism [(ECMF), see Ref. [47]].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. 5b is convincingly observed.Fig. 5a zooms in on the redshift of the 2 nd −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 of one geometrical parameter [41,48].

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-optic activity of the system.In MO materials the presence of a magnetization creates antisymmetric off-diagonal components in the permittivity tensor, ← → ε = , where in first order the gyration vector g = g1 M is parallel to the magnetization and its magnitude is linearly proportional to that of M, g ∼ |M|.For lossless MO media ← → ε 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 [49] when reflected from a surface of a MO layer.The fundamental process induced by the magnetization is the Lorentz force on carriers.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, r p = r 0 p (1 + igδr p ), where r 0 p is the isotropic conventional reciprocal Fresnel coefficient (for M = 0) and the small effect correction δr p = r 0 p gk y 2ε MO k z,MO .On a lossless MO material and below its critical incidence angle, TMOKE is then solely a reflection phase shift since Im(gδr p ) = 0, and the power reflectivity R p remains perfectly reciprocal, ∆R p = |r p (M)| 2 − |r p (−M)| 2 = 0. Nonreciprocal power reflectivity can only be realized on absorbing media.However on bare lossy ferromagnetic metals such as Co or Fe, ∆R p doesn't exceed 10 −3 .[49] On the other hand, TMOKE power nonreciprocity on a lossless MO garnet can be obtained by cladding it with a plasmonic grating.[32].The gold/garnet SPP dispersion gets a small nonreciprocal correction due to the nonreciprocal phase shift by the transverse MO garnet substrate.In combination with the sharp reflection resonances of the grating's FP, this leads to a huge enhancement.[41] Including the effect of the gyrotropy in the boundary conditions at the interface between gold and a transversely magnetized garnet, leads to the following nonreciprocal dispersion equation for the surface magnetoplasmon polaritons: The gyrotropy clearly breaks the inversion symmetry n eff,SPP → −n eff,SPP .Linearizing this equation with respect to g -for the considered garnet [42] 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 n eff,SPP with g induce non-reciprocal response of coupled FP-SPP modes.The model of MO SPP describes 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 thus to design devices based on non-reciprocal reflection or transmission with locally inverted sign.In Fig. 2b this n eff,SPP dependence introduces spectral shift of typically ≈ 30 meV whatever the slit.

Conclusion
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 demonstrated experimentally and numerically 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.This offers possibility of additional enhancement thanks to push-pull configuration by playing on technologically realistic combination of grating of the same thickness and different slits for advanced circuits including non-reciprocal functions.

Fig. 1 .
Fig. 1.(a) Definition of parameters in the model describing the magnetoplasmonic structure in planar diffraction configuration, (b) fabricated set of diffraction gratings with SEM images of developed plasmonic gratings showing good quality of the grating lamelas and observed widths of nanoslits r.

2. 2 .
Fig. 2a and 2b directly compare measured MO data on 15 samples having different width of nanoslit with MO numerical simulation data.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.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 reciprocity sign is inverted by changing only the slit width.

Fig. 2 .Fig. 3 .
Fig. 2. Experimentally observed MO response (a) (δR) measured at incident angle of 20 • on 15 different patches of samples with various opening r is compared with numerical model calculated for various nanoslit width r.(b) Black dashed line at photon energy 1.67 eV highlighting the change of the MO peak sign with the grating nanoslit width r .

Fig. 4 .
Fig. 4. Spectra of relative reflectance R = R p /R s measured for the angle of incidence ϕ = 20 • , grating nanoslit width r = 63 nm, and the grating thickness h = 100 nm.In subplot (a) the full spectra is presented with marked position of the 1 st and 2 nd garnet SPP peaks.The inserted detail of light-cone includes dispersion curves of the predicted SPP (solid blue) crossing the 20 • -light line (solid black).The area for air SPP is marked as red line.Subplot (b) detail red/shift of the +2 nd garnet SPP peak measured for increasing width of the nanoslit r.

Fig. 5 .
Fig. 5. Analysis of coupling between FP and SPP modes base on variation of the grating air-gap width r.Subplot (a) shows red-shift of coupled SPP/FP mode at cross-cuts for air-gap width r = 40, 60, 80, and 100 nm.Subplot (b) shows positions of the SPP modes without geometrical dispersion and dispersive FP mode (calculated from Eq. (1, 2)).Subplot (c) shows dispersion map of SPPs and FP modes calculated using extended coupled-mode formalism.