Non-reciprocal diffraction in magnetoplasmonic gratings

: Phase-matching conditions—used to bridge the wave vector mismatch between light and surface plasmon polaritons (SPPs)—have been exploited recently to enable nonreciprocal optical propagation and enhanced magneto-optic responses in magnetoplasmonic systems. Here we show that using diffraction in conjunction with plasmon excitations leads to a photonic system with a more versatile and flexible response. As a testbed, we analyzed diffracted magneto-optical effects in magnetoplasmonic gratings, where broken time-reversal symmetry induces frequency shifts in the energy and angular spectra of plasmon resonance. These result in exceptionally large responses in the diffracted magnetooptical effect. The concepts presented here can be used to develop non-reciprocal optical devices that exploit diffraction, in order to achieve tailored electromagnetic responses.

All things considered, we can say that up till now the mutual interplay of three important phenomena, i.e., plasmonics, magneto-optics and diffraction, has not been investigated in a single system. We show here that combining all the aforementioned properties provides new paths for flexible design of functional nanophotonic systems. To corroborate this point, we study the interplay of plasmon resonances and magneto-optics and their influence on diffraction properties of a simple diffraction grating. Despite its simplicity, it supports surface plasmon polaritons propagating both into backward and forward directions, enabling us to easily assess how breaking the time-reversal symmetry and presence of SPPs influences the diffracted beams.

Magnetoplasmonic gratings
To study how SPPs and magnetization influence diffraction, we fabricated magnetoplasmonic diffraction gratings by ion milling into cobalt/gold multilayer thin films. A multilayer of composition Cr (4nm)/Au (16 nm)/[Co (14 nm) / Au (16nm)]4/Co (14 nm)/Au (7 nm) was grown by e-beam evaporation on a commercial single crystal strontium titanate substrate (STO). The multilayer parameters were obtained by running an optimization script in a commercially available Lumerical FDTD software to find a multilayer configuration with good balance of plasmonic and magneto-optical properties. After metallic layer deposition an e-beam lithography process was performed to create an etch mask. The revealed part of the metallic layer was etched away by ion-milling, at the rate of 60 nm/minute, during 2.5 minutes. The final grating structure had a periodicity of 1000 nm, with grooves of 200 nm and a depth of 150 nm. Figures 1(a) and 1(c) show, respectively, a schematic and a SEM micrograph of the grating. Light incident on the grating at an angle of θ 0 gives rise to diffraction maxima at θ m , given by sinθ m = sinθ 0 + mλ/nd, where λ is the wavelength of the incident light, d the period of the grating, n the refractive index of the surrounding medium and m is an integer denoting the order of the diffraction maximum. The grating was immersed in index matching oil with n = 1.5. The experimental setup used to study the optical and magneto-optical properties of the sample is presented in Fig. 1(b). A xenon lamp was used to provide a continuous white light source that was sent through a monochromator. The monochromatic beam was then polarized and collimated through an objective lens 10X N.A 0.4, which was placed over a movable platform, allowing modifications in the beam size. The collimated beam had a Gaussian profile. The incoming beam was reduced by moving the collimating objective so that the beam was limited to impinging angles between −5° and 5°, which resulted in a spot size of 10 և m of radius. The beam was then projected at the rear aperture of an oil immersion objective lens 63X, N.A 1.4. responsible for focusing and collecting the full angular spectrum of the reflected or diffracted light. All experiments were carried out at room temperature (20 C°).  (1 ) β ε ε ε ε = − . In Eq. (2), the signs of the plasmon wave vector k SPP and the magnetization M are determined by the symmetry of the system and can be understood as follows: The magnetization of the grating breaks the time-reversal symmetry of the system. Consequently, a reversal of the SPP propagation direction is equivalent to a reversal of the magnetization direction. As a consequence, the degeneracy between forward and backward propagating modes is lifted and the wave vectors for the forward and backward propagating modes, shown in red and blue in Fig. 1(a), become distinct, which results in changes in the diffracted light intensity as the function of magnetic field. To quantify these changes, the magnetization of the sample was cycled to record a hysteresis loop from which variation of the reflected and diffracted intensities as function of magnetization could be extracted. This is schematically illustrated in the diffracted beams in the lower part of Fig. 1(a). For magnetization in the positive y-direction (blue curves), the diffracted intensity is reduced for m > 0 diffracted orders, while it is increased for m < 0. The opposite is true when the magnetization is reversed (red curves). A more detailed description of the magneto-optical measurement procedure has been included in Appendix 1. However, before we can assess how the magnetization and SPPs influence the diffracted intensity, we first should examine how SPPs influence the diffraction characteristics of our array in the absence of magnetic fields, which is discussed in the following.

Results and discussion
Throughout most of the measured range of wavelengths, three emerging beams can be observed in our setup [ Fig. 2(b)], corresponding to the zero-order specular reflection and the m = ± 1 diffracted orders. At shorter wavelength range also the m = ± 2 diffracted orders could be detected by our setup. Figure 2(a) shows the experimental results in the intensity of light scattered to a range of far-field angles throughout the visible wavelength regime. We used a normally incident Gaussian beam to probe the sample, and as a consequence, there is an angular spread in the emergent diffracted beams. As the SPP excitation is sensitive to the angle of incidence, the spread of angles facilitates assessment of the effects arising from SPPs. The SPP excitation angles calculated from Eq. (1) are superimposed on Fig. 2(a). We observe that in this far-field map the SPP excitation bands can be seen as distinctive crossshaped minima, distinguished by a dip in reflectance in the angular spectrum of the light reflected or diffracted by the sample. A crossing of SPP bands occurs in the wavelength range of 700-850 nm, centred at 785 nm. A weaker SPP excitation can be also observed with crossing around 550 nm that also results in reduced diffraction efficiency. Figure 2(c) presents cross sections of the emitted intensity at selected wavelengths. We used Eq. (1) to calculate the angular position of the SPP excitations and indicate them in Fig. 2(c) as vertical dotted lines and arrows indicating the propagation direction of the SPP mode. The calculated SPP modes correspond with the observed dips in the emitted light intensity. We conclude that the excitation of SPPs travelling along forward/backward directions can be fingerprinted in the far-field angle-resolved reflectance scans, where at particular conditions of wavelengths and angles the intensity of diffractive beams is attenuated due to the transfer of energy and momentum to the propagating SPPs.   Fig. 4 in Appendix 1. In Fig. 3(b) we highlight the spectral profile of the TMOKE activity for the part of the beam with positive incidence angles, where the magneto-optical activity is averaged at each wavelength. Namely, only the black dotted rectangular areas of Fig. 3(a) -corresponding to scattering to negative angles -have been considered, as summing over the whole range of angles would result in nulling the magneto-optical activity near the SPPs. A complementary plot for positive reflection angles is shown in Fig. 5. In the region between 700 and 850 nm we can observe a modulation of TMOKE around the plasmon resonance. As a consequence of this modulation, the magneto-optical activity abruptly increases, changing sign at the backward propagating SPP mode (η = + 1, for m = −1 and η = + 3, for m = + 1) and again reversing at the forward propagating mode (η = −3, for m = −1 and η = −1, for m = + 1). These modes are indicated by dotted lines in Fig. 3(b). We note that the signs of TMOKE are opposite for the specularly reflected (m = 0) and diffracted (m = ± 1) beams. Hence, we can state that magnetoplasmonic gratings exhibit tuneable diffraction efficiency that can be either enhanced or reduced by choosing the magnetization state.
In order to have a broader view of the interplay between magnetism, plasmonics and diffraction, we present in Fig. 3(c) the far field profile of the magneto-optical activity for 3 different wavelengths around the surface plasmon resonance. The SPP excitation angles are indicated by vertical lines with arrows showing their propagation direction. Here, too, we can indeed observe the distinctive derivative line shape that results from magnetization shifting the SPP resonance condition, thus confirming that the magnetization doesn't just shift the plasmon resonance in energy but also in the angular space. As pointed out earlier, Eq. (1) couples the incident angle with the wave vector of the SPP, hence these two quantities depend on each other and both can be adjusted by applying magnetic fields. As a consequence of the non-reciprocal nature of the MO effect, the sign of the MO effect is reversed by inversion of momentum k → -k (i.e., for reversal of the sign of the emission angle).
We make an additional observation on the specularly reflected beam in the center of Fig.  3(a). No magneto-optical effect is observed for strictly zero incidence/emission angle. For non-zero angles in the m = 0 beam, a weak TMOKE signal, similar to those in the diffracted beams, appears at the SPP crossing centred at λ = 785 nm. This can be contrasted with the m = ± 1 and ± 2 diffracted beams, plotted at the sides of Fig. 3(a) at their respective emission angles, where a noticeable TMOKE response is measured. This is due to the fact that the diffracted beams contain a longitudinal component that gives rise to ordinary diffracted magneto-optical Kerr effects (DMOKE) even in the absence of SPP excitations. In the wavelength range where SPP resonances are not present, this results in a magneto-optical effect whose sign is determined by the far-field angle of the diffracted beam. We have chosen the sign convention of the magnetization of the sample M so that the sign of this "ordinary DMOKE" is positive for positive angles of emission and negative for negative angles of emission. This effect is more pronounced for the m = ± 2 diffracted orders, only present in shorter wavelengths at high angles, because of the larger angle at which these beams are diffracted to [ Fig. 3(a)].

Conclusio
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ons plored how m that, in combi ards flexible d quences of m at the presence pace [26,27], b in derivative ly understood xcitation angle nd observation changes in the plasmon-enhan ularly reflected ed in Fig. 3 been shown to support even higher modulation, though accompanied with a large losses [27,44]. There is still a third consideration, regarding the exploitation of optical absorption -which always appears at plasmon resonances-for nonreciprocal propagation. By using magnetic fields, we were able to nudge the absorption maximum both in the energy and angular space. Effectively, we have demonstrated how magnetic field modulation of excitation of SPPs can be used to tune diffraction efficiency for a given wavelength and angle. As many recent advances in nanoscale optics, such as metalenses [45] rely on sub-wavelength arrangement of diffraction elements, it is interesting to consider how active properties could be integrated to such elements. Here we have explored a magnetoplasmonic grating and shown how it can be switched between a more diffractive "on" and less diffractive "off" state using an external magnetic field. Given the non-reciprocal nature of the diffraction in magnetoplasmonic gratings, this kind of grating could act as an isolator or an active grating coupler, where external magnetic fields could be used to tune coupling efficiency. Additional applications for magneto-optically active plasmonic devices could be found in bio-sensing, where ultrasensitive magnetoplasmonic sensors have been explored recently [14,15].
Our experiments were performed with a simple metallic grating in order to highlight the interaction between the three phenomena under investigation: plasmons, magneto-optics and diffraction. However, we anticipate that our results can be generalized and used to design more complex diffractive elements, such as many plasmonic metasurfaces, where they could find use in designing non-reciprocal, isolating devices.

Appendix 1
To extract the TMOKE parameters at one wavelength and reflected/diffracted angle, multiple far-field optical emission images of the grating were recorded while the magnetization was cycled from positive to negative saturation and back, enabling us to record a complete magneto-optical hysteresis loop. Two such hysteresis loops are shown in Fig. 4(a) and 4(b). As is evident from their shape, they correspond to areas with negative and positive TMOKE, respectively. TMOKE is calculated from the hysteresis loops by averaging over the data points in positive (I + ) and negative (I -) saturation and subtracting them from each other. The intensity change is then normalized to obtain the relative change in emitted intensity that is described by the definition TMOKE = 2(I + -I -)/(I + +I -). Here, the pre-factor of two accounts for the averaging of the positive and negative saturation intensities. Repeating this procedure for each measured emission angle over a range of wavelengths from 550 to 865 nm with 5 nm steps, resulted in a composite image that is shown in Fig. 3(a).