Tunable Faraday rotation of ferromagnet thin film in whole visible region coupled with aluminum plasmonic arrays

1 Introduction

As the main component of the modern optical telecommunication system, optical isolator/rotator has the ability to tune the polarization state of light, and the non-reciprocal behaviors are utilized to prevent the unwanted feedback in the laser cavity. The behind mechanism depends on the magneto-optical (MO) effects of magnetic materials such as Fe, Co, and Ni [1], [2], [3], [4]. Fundamentally, MO effects also provide physical information on the electronic and spin structures of materials [5, 6]. Depending on the transmitted or reflected configuration with different direction of magnetizations, various types of MO effects are distinguished like Faraday and Kerr effects. In this research, we will mainly concentrate on the Faraday rotation (FR) that refers the polarization rotation of transmitted light under the normal incident illumination with an external static magnetic field along the light wave vector, as shown by the schematic illustration in Figure 1.

Figure 1:

Schematic of Faraday rotation (FR) under the external magnetic field B along z direction on the Al NP hexagonal array with 7.5 nm Fe film on the top with different superstrates of refractive index n _super. Left-top inset shows the unit cell of hexagonal array with pitch a in the x-y plane used in the calculation. The sketch of the Al nanostructure used in the FDTD calculations is provided in Appendix Figure S1.

Strong demands to enhance the MO effects exist for miniaturizing the MO devices like optical isolators and magnetic field sensors, however the typical FRs produced by the MO materials especially the traditional ferromagnetic metals are usually very small with high losses [2], [3], [4]. Metallic nanoparticles (NPs) have the ability to confine the light energy into its nanoscale vicinity with an enlarged intensity due to the collective oscillation of free electrons, therefore they show the appealing ability to enhance diversiform properties. In particular, many efforts have also been devoted to amplify the MO effect by plasmonic systems (so-called magnetoplasmonics) with the help of remarkably enhanced light–matter interaction. More advantages include the possibility to tune MO effects by exploring the sensitivity of plasmonic resonances to various factors like the geometry, arrangement, and dielectric environment, thus different plasmonic systems have been proposed to obtain giant MO effects. For instance, the combination of magnetic and plasmonic thin films that make use of propagating surface plasmon polaritons (SPPs) excited on the interface [7], [8], [9], [10], and plasmonic NPs that make use of localized surface plasmon resonances (LSPRs) [11], [12], [13], [14], [15] are two widely-adopted approaches. It has also been demonstrated that diffractive plasmonic nanostructures can largely enhance MO effects [16], [17], [18], [19], [20], [21], [22]. Such systems include periodic one-dimensional gratings and two-dimensional nanostructure arrays that support surface lattice resonances (SLRs), i.e., hybridized modes between LSPRs and in-plane light diffraction [23], [24], [25]. When additionally combined with magnetic materials, these systems utilize two kinds of resonances for MO enhancement: the LSPRs that accumulate light field near the metallic surface enhance the MO effect locally, and the light diffraction that traps light inside the MO-active region further enhances the light–matter interaction. In this sense, diffractive plasmonic nanostructures possess the combined advantages of both diffraction-based, i.e., magneto-photonic crystals and LSPR-based systems. In addition, these systems enjoy advanced properties including wavelength-selective MO enhancement by the periodicity and incident angles [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. In spite of a high degree of freedom in engineering the MO effect, the plasmonic material used in magnetoplasmonics has been thus far limited to Au and Ag [2], [3], [4]. Meanwhile, to date the FR enhancement in the experiment is typically obtained at the optical wavelengths of λ > 600 nm. Al has been proposed as a plasmonic material that can sustain LSPRs from the deep ultra-violet (UV) to the whole visible regions owing to its large carrier density [38], [39], [40], [41], [42]. Meanwhile, Al is especially advantageous for commercial applications because it is abundant in the Earth’s crust and compatible with the complementary metal–oxide–semiconductor (CMOS) process for the industrial production. Moreover, it is worthy to mention that dielectric nanostructures with high refractive index such as the representative material Si recently have also attracted enormous attention in the application of enhanced MO effects due to their reduced dissipative losses comparing to metallic materials [43], [44], [45], [46], [47]. The particular interest on the UV-blue MO effect is valuable in applications like the optical lithography system or other laser systems with short-wavelength light sources. Therefore, one major goal of this research is pushing the enhanced MO effects into UV-blue range. To date, most reported experimental results in magnetoplasmonics show a continuous MO enhancement within a 100 nm wavelength range due to the narrow Fano features utilized. And by exploring the lattice period and incident angles, the spectral tunability around 200 nm can be achieved [2], [3], [4]. Thus, we are also devoted to realize a broad spectral response and high tunability.

In this study, we fabricated Al NP hexagonal arrays to obtain the enhanced FR at shorter-wavelength spectral regions. The array was designed to show the hybridized lattice mode at around λ = 500 nm, overlapping with the LSPRs of Al NPs. As a proof of concept, Fe was chosen as the MO-active material in this research, considering its simple fabrication and larger saturation magnetic flux density [1]. A 7.5 nm Fe film was deposited on top of the fabricated Al NP array and the FR effect was examined. We further demonstrated the controllability of enhancement in the spectral region by the modulation of two types of modes, i.e., LSPRs and SLRs resorting to changing the superstrate and lattice pitch. To confirm the experimental results, finite-difference time-domain (FDTD) method was also used to calculate the transmission spectra and FR spectra.

2 Methods

3 Results and discussion

To fully explore the advantage of broadband scatter of Al NP, the geometry of Al NP was optimized aiming to display a broadband spectral tunability [39], [40], [41], [42]. To start with, the forward scattering spectra of single Al NP with Fe film are plotted as a function of n _super in Appendix Figure S4. When n _super = 1, only one broad peak arises around 454 nm in the spectral range from 350 to 800 nm. When switching n _super to 1.46 and 1.71, this peak redshifts to 592 and 636 nm respectively, accompanied with a higher mode at the shorter wavelength side below 450 nm. The color-coded forward scattering spectra demonstrate the tunability to switch the LSPR window by changing the dielectric environment around Al NP. By investigating the charge distribution of these two modes as shown in the right of Figure S4, the complex behaviors far more than dipole and quadrupole are revealed [41, 42]. This is due to the complicated structures that we are considering in the calculation: the Al₂O₃ layer and Fe film on the bottom and top of Al NP. The influence of this nanostructure on the optical properties will be discussed later.

Figure 2a and b show the SEM images of Al NP hexagonal arrays with lattice constant a = 400 and 460 nm after the deposition of 7.5 nm Fe film, while Appendix Figure S2 shows the SEM images of Al NP hexagonal arrays with a = 400 nm before and after the deposition of 7.5 nm thick Fe film. When arranged into periodic array, the lattice coupling with the LSPRs of single NP appears [23], [24], [25, 41, 42]. To verify the wavelength of lattice mode or SLR in the hexagonal array, the experimental zeroth-order transmittance measurements were done as a function of θ ₀, where θ ₀ was varied to give momentum in the x direction (see the inset of Figure 1 for x direction). Based on the conservation of the parallel component of the wave vector in the NP array plane, the Rayleigh anomaly conditions satisfy the relation: k _out =  k _in ±  G , where k _out and k _in are the in-plane component of diffracted and incident wave vectors, respectively. G  = (m₁ b ₁ , m₂ b ₂ ) is the reciprocal lattice vector of the NP array, which can be expressed for the hexagonal lattice as [48]:

where m ₁ and m ₂ are integers defining the diffraction order. When k _in does not have a component in the y-direction, as is true in the present case, k _out can be expressed as:

where k _out = 2πn/λ and k _in = (2πn _quartz/λ)sinθ _in are, respectively, the parallel components of the diffracted and incident wave vectors, n is the refractive index of different diffracted environments either to be n _quartz or n _super, and n _quartz ≈ 1.46 is the refractive index of quartz. Note that we denote the magnitude of k by k. According to Eq. (7), the specific attribution to each lattice mode is provided in Appendix Figure S5. When the substrate and superstrate have a mismatch on the refractive index, a partial SLR through either the superstrate or the substrate exists, which are denoted as dashed and solid lines respectively in Figure S5 [25, 41].

Figure 2:

The 30°-tilted scanning electron microscopy (SEM) images of Al nanoparticle (NP) hexagonal array after the deposition of 7.5 nm Fe film with lattice constant a = 400 nm (a) and 460 nm (b). The calculated (Sim.) and experimental (Exp.) transmission spectra as a function of θ ₀ for the Al NP array with a 7.5 nm thick Fe film on the top with n _super = 1, 1.46, and 1.71 respectively for a = 400 nm (c, e, g) and 460 nm (d, f, h). White dashed lines indicate the conditions of in-plane diffraction (the specific modes are indicated in Appendix Figure S5), and the color bar is uniformly indicated on the right side of each row.

In Figure 2, the experimental transmission spectra as a function of θ ₀ are displayed in row for varied lattice constant a and in column for varied n _super, and the lattice mode wavelengths are also plotted in white dashed lines. When placed in a uniform dielectric environment, i.e., with n _super = 1.46, the refractive index mismatch between the upper layer and the quartz substrate around the Al NP is reduced, facilitating the in-plane diffraction, therefore the least diffraction orders are shown in Figure 2e and f. More diffraction orders appear with varied n _super from 1.46 to 1 and 1.71 due to the partial SLRs through either the superstrate or the substrate, and the spectral shift is caused by the change of refractive index around the Al NP. According to Eq. (7), the specific attribution to each lattice mode is provided in Appendix Figure S4. As for a = 400 nm and n _super = 1 (Figure 2c), a prominent dip around λ = 410 nm at θ ₀ = 0° corresponds to the excitation of LSPR of the Al NPs. This main dip redshifts with increasing θ ₀, and the shifts follow the dashed lines (0, ±1) mode from the superstrate (Appendix Figure S5), representing the conditions for diffraction in the plane of the array. As for a = 460 nm and n _super = 1 (Figure 2d), the dip is redshifted to 430 nm due to the shifting of lattice coupling with increasing lattice constant a. However, it also follows the (0, ±1) lattice mode from the superstrate. Similarly, the angular profiles of the main dips follow well the different diffraction orders with varying a and n _super in Figure 2e–h, indicating the hybridization of LSPRs with in-plane diffraction, i.e., SLRs. The symmetric behavior between calculated and experimental spectra as plotted in Figure 2c–h confirms the same phenomena, and proves a good accordance with each other. Furthermore, the transmission intensity shows a value below 0.4 even in the calculation, and the relatively higher transmission of array with a = 460 nm (Figure 2d, f, and h) than the one with a = 400 nm (Figure 2c, e, and g) is owing to the reduced NP number density [40]. More detailed analysis on the mode coupling for lattice modes in the Al NP array are available in the previous researches [23], [24], [25, 39], [40], [41], [42]. Complex features on the nature of plasmonic resonances may arise when we play with the polarization and θ ₀ on the hexagonal lattice as pointed out by other groups [36, 37]. With respect to our main purpose on FR, at the normal incidence our hexagonal nanostructure shows very little difference on the transmission spectra by using x- or y-polarized light.

Since main concerns may arise on the possibility of SPPs on the surface of Fe film, here we provide several reasons to simplify our situations without SPPs: Firstly, SPPs more likely happen on the metallic film with free electrons like Au, Ag, and Al. Fe is a ferromagnetic metal, thus it is usually not a good material for SPP [1]. If we check the measured dielectric function of Fe in Appendix Figure S3a, we can see the absolute value of its imaginary part is much larger than the one of its real part. Thus, according to the frequently adopted equation to calculate the SPP wavelength λ _spp = λ ₀ ((ε + ε _m)/εε_m)^1/2, where ε_m is the real part of the dielectric function of Fe as displayed in Figure S3a and ε is the dielectric function of the dielectric medium, i.e., ε = (n_super)², we will most probably obtain an imaginary value of λ _spp in the range of 400–800 nm. Secondly, Fe film is only 7.5 nm here, considering that the height of Al nanostructure is 150 nm. Thus, the influence of SPP will be quite small. Thirdly, according to Eq. (7) the obtained in-plane diffraction wavelengths in Figure S5 already show a good coincidence with the Fano features in both the calculated and experimental transmission spectra as shown in Figure 2. Lastly, we calculate different nanostructure setups to peel off the influence of each part subsequently as shown in Appendix Figure S6. The Fe nanohole array (black box 4) only creates a very small kink in the transmission spectrum (black line spectrum in Figure S6a), which is better shown in the absorption spectrum in Figure S6b. Meanwhile, this kink is also well located at 505.8 nm as denoted by the vertical black dashed line, exactly predicted by Eq. (7) for the in-plane diffraction wavelength. Thus, we can safely exclude the influence of SPP in this research.

The calculated and experimental transmission spectra for the array of a = 400 nm at the normal incidence (θ ₀ = 0°) with varying n _super are plotted in Figure 3a. For the Fe film (black solid line spectra in Figure 3a), only a shallow dip shows on the transmission spectra around 480 nm, while distinct dips appear with Al NP array (colored solid line spectra in Figure 3a) because of the strong LSPRs. For n _super = 1 (blue line spectra), beside a main dip around 410 nm in the experiment (blue square marker), another small dip at around λ = 528 nm appears (blue circular marker). This is because that the diffraction in the substrate side at 505.8 nm (vertical black dashed line) split the LSPR of single Al nanostructure about 454 nm (Figure S4) into two dips. The typical Fano lineshapes exist around 505.8 nm for all the three spectra due to the lattice mode coupling, while the main dips move from 410 to 580 (green circular marker) and 656 nm (red circular marker) when n _super increases from 1 to 1.46 and 1.71 owing to the redshifted LSPR of single Al nanostructure with increasing refractive index of background, as shown in Figure S4. As for n _super = 1.71, additional lattice mode (superimposed (0, 1), (−1, 0) and (−1, 1) modes from the superstrate side) appears at 592.4 nm, shown by the red vertical dashed line in Figure 3a. Both calculation and experiment confirm the distinct grating effect at the same wavelength, which is easily confirmed by the vertical dashed lines. In general, the calculated transmission spectra agree well with the experimental ones, although the LSPR positions marked by square and circular markers display a mismatch with each other. This difference can be attributed to the nanofabrication imperfections in the experiment, which cause internal defects inside the Al NPs and various disorder effects in the periodic array [39], [40], [41], [42]. Moreover, the highly damped higher mode below 500 nm in the case of n _super = 1.46 and 1.71 can be additionally ascribed to the surface roughness of NP [41]. It is worthy to note that the transmittance of 7.5 nm Fe film already shows a low value below 0.3 or 30% in the experiment, which can hinder the further application devoted to MO devices. Furthermore, the LSPRs and SPPs enhanced MO effects often accompany with a transmission damp, which has been recently reported to be improved resorting to the extraordinary transmission of nanoholes [35].

Figure 3:

Calculated (upper panel, Sim.) and experimental (lower panel, Exp.) spectra of (a) optical transmission and (b) Faraday rotation (FR) angle θ _F of the Al nanoparticle (NP) array with a = 400 nm and 7.5 nm Fe film on the top at θ ₀ = 0° with different superstrates. Calculated absorption spectra are shown in (c), while the nearfield electric fields in x-z plane are plotted on the right side in colored boxes with the wavelengths noted on the top, and the colors of the boxes are in accordance with the colors of the line spectra. The direction and tick scale of x-z axis for the nearfield profiles are uniformly shown by one profile in the green box for the sake of the concision, and the white lines depict the nanostructure in conformity to Figure S1. The colors of spectra indicate n _super = 1 (blue), 1.46 (green), and 1.71 (red), while the black line spectra are the reference spectra of the Fe film with n _super = 1. The black vertical dashed line at λ = 505.8 nm denotes the condition of in-plane diffraction for all three different n _super (=1, 1.46, and 1.71), and the red vertical dashed line at λ = 592.4 nm denotes the in-plane diffraction for n _super = 1.71 for the red line spectra. The wavelengths of dips in the experimental transmission spectra (a, lower panel) are marked by square (at shorter wavelength) and circular marker (at longer wavelength) with their colors in accordance with the line colors of spectra. The corresponding wavelengths are indicated nearby the markers. To keep the uniformity, the same wavelength positions are also indicated by the same markers in the calculated transmission spectra (a, upper panel), FR spectra (b), and absorption spectra (c).

The comparison of calculated and experimental FR spectra for the Al NP array of a = 400 nm can be examined in Figure 3b. The reference of thin Fe film displays an almost linear increment of θ _F as a function of wavelength without any featured resonant behavior (increase from 0° at 400 nm to 0.44° at 800 nm in the experiment), as denoted by the black line spectra in Figure 3b. In contrast, θ _F of the Fe film on the Al NP arrays displays various resonant features. Therefore, we can readily trace the correlation between enhanced θ _F and the LSPRs or lattice modes due to the appearance of Al NP. The corresponding wavelengths of LSPRs and lattice mode from the experimental transmission spectra in Figure 3a are also indicated by the square/circular markers and vertical dashed lines respectively in Figure 3b. With the appearance of Al NP, the MO response shows a resonant modulation at the spectral vicinity of LSPRs where the minimum transmittance occurs approximately. In detail, the excitations of LSPRs of Al NPs produce the prominent peaks and dips in the experimental FR spectra: peak 0.57° at λ = 410 nm and dip 0.26° at λ = 534 nm for n _super = 1; dip −0.05° at λ = 529 nm and peak 0.81° at λ = 580 nm for n _super = 1.46; a boosting region around 0.5° from 400 to 500 nm due to the higher mode (red square marker), dip −0.07° at λ = 589 nm and peak 1.66° λ = 638 nm for n _super = 1.71. The spectral peak/dip shift of θ _F in Figure 3b comparing to the marker position in Figure 3a are mainly due to the varied dielectric function under the magnetic field, as shown by the variation on the dielectric function related to the external magnetic field (Eqs. (4) and (5)). Meanwhile, the obvious Fano lineshapes in the FR spectra around the markers’ wavelengths in the calculation and experiment are possibly associated with the quick phase transition across the LSPRs [34].

In the condition of in-plane diffraction denoted by the vertical lines at λ = 505.8 (black dashed vertical line) and 592.4 nm (red dashed vertical line), a modulation with Fano lineshape is found for the value of θ _F. Bumps around λ = 505.8 nm of θ _F appears as 0.32° for n _super = 1, as 0.10° for n _super = 1.46, and as 0.39° for n _super = 1.71 in the experimental FR spectra. Meanwhile, the dip around λ = 592.4 nm in experimental θ _F for n _super = 1.71 can be also partially ascribed to the grating diffraction except for the nearby LSPR, while this wavelength is located between two small dips in the MO calculation (red line spectrum in the upper panel of Figure 3b). However, in the MO calculation distinct bumps are produced due to the SLRs at the wavelengths denoted by the vertical dashed lines in Figure 3b. The controllability of enhanced θ _F is demonstrated by shifting the enhanced region resorting to switching the immersion oil: with increasing n _super the enhanced region is redshifted. Particularly, in experiment we obtain enhanced θ _F to 0.57° at λ = 410 nm for n _super = 1, to 0.81° at λ = 580 nm for n _super = 1.46, to 1.66° at λ = 638 nm for n _super = 1.71, which indicates a stronger interaction of LSPRs during FR with the thin 7.5 nm Fe film. The UV and blue region have been rarely reported for enhanced MO effect, however with the Al NP array and the thin 7.5 nm Fe film we push the enhanced FR into the blue region. In spite of the different value of θ _F, the calculated FR (Figure 3b, upper panel) shows similar features with experimental results. Both enhanced θ _F due to LSPRs and lattice modes are distinct, as indicated by spectral region close to the markers and vertical dashed lines. However, θ _F at the cutting-off wavelength of lattice mode (vertical dashed lines) seems to display weaker enhancement comparing to the spectral range of LSPRs (around square and circular markers). Beside the above-mentioned reasons for the mismatched transmission spectra between calculation and experiment, the mismatch in the FR spectra additionally comes from the deviations in the permittivity tensor between calculation and experiment, because it is significantly difficult to experimentally obtain an accurate estimation for the ferromagnetic material. Additional errors may come from the multiple reflections produced by the coverslip and immersion oil in the experiment to switch n _super, which is impossible to consider in the FDTD calculations due to the unknown thickness of the oil film and the heavy need for computation memory.

To better understand the enhancement mechanism, the calculated absorption spectra are also plotted in Figure 3c. The calculated reference spectrum of Fe film only displays a slight bump around 500 nm. As the Al NP lattice appears, the kinks due to the occurrences of Rayleigh anomalies are distinctly visible as shown by the vertical lines, which is in the same manner as for the transmission curves in Figure 3a. As for the positions of markers in the calculated absorption spectra, they are well located in the vicinities of absorption peaks. The absorption is associated with the imaginary part of the dielectric function and electric field intensity [41], thus the strong absorption intensity usually means the strong localization of light field nearby the plasmonic nanostructures. This can be proved by the plotted electric field profiles on the right side that with a larger absorption value in the absorption spectra in Figure 3c a higher maximum value of the electric field intensity appears. The tiny hot spots near the Fe material on the top of Al NP corroborate the physical picture described above of the plasmon-assisted strengthened FR. We can further investigate the role of Fe material on the top of Al NP comparing to the Fe material on the substrate. Without the Fe material on the top of Al NP, the transmission dips (yellow line spectra in Figure S6a) become much shallower than the original one (blue line spectra in Figure S6a), and the absorption intensity decreases as shown in Figure S6b. Therefore, without the Fe material on the top of Al NP the plasmon enhanced MO effect is expected to be decreased because of the less localization of light energy. This can be also illuminated by the hot spots in the near field profiles in Figure 3c that strong hot spots are located on the top of Al NP, which is in strong contrast to the Fe material near the bottom of Al NP.

Figure 4 compares the calculated (upper panel) and experimental (lower panel) wavelength dependence of transmission (a) and FR (b) at θ ₀ = 0° for the Fe film on the Al NP array with a = 460 nm. The transmission spectra with a = 460 nm in Figure 4a are very similar to the ones with a = 400 nm in Figure 3a, except for the different diffraction wavelengths and redshifted transmission dips because of increasing lattice constant a. Similarly, we obtained the peaks and dips in the experimental FR spectra due to the lattice mode and LSPRs: peak 0.24° at λ = 450 nm, peak 0.36° at λ = 575 nm, dip 0.34° at λ = 595 nm, and peak 0.60° at λ = 681 nm for n _super = 1; peak 0.08° at λ = 435 nm, peak 0.16° at λ = 580 nm, dip −0.09° at λ = 604 nm, peak 0.69° at λ = 735 nm for n _super = 1.46; peak 0.08° at λ = 433 nm, peak 0.11° at λ = 580 nm, dip 0.06° at λ = 604 nm, peak 0.25° at λ = 687 nm, dip 0.21° at λ = 708 nm, peak 0.63° at λ = 800 nm for n _super = 1.71. The comparison between Figures 3b and 4b clearly shows that the Al NP array is better able to enhance FR with smaller lattice constant in the region 400–500 nm. As for longer wavelengths, notable enhancement in θ _F comparing to Fe film is still observed although the enhancement is smaller than the array with a = 400 nm. The reduced enhancement is linked to the decreased localized energy in the array with a larger lattice constant a by decreased NP density, although less influence is brought to the transmission spectra [40]. With respect to the calculated absorption spectra and plotted nearfield distribution at selected wavelengths in Figure 4c, a similar observation can be acquired as in Figure 3. And the enhanced MO phenomenon in Figure 4b can be well traced back to the resonant excitation of plasmonic modes shown by the markers for LSPRs and vertical dashed lines for SLRs in Figure 4c as explained above.

Figure 4:

Calculated (upper panel, Sim.) and experimental (lower panel, Exp.) spectra of (a) optical transmission and (b) Faraday rotation (FR) angle θ _F of the Al nanoparticle (NP) array with a = 460 nm and 7.5 nm Fe film on the top at θ ₀  = 0° with different superstrates. Calculated absorption spectra are shown in (c), while the nearfield electric fields in x-z plane are plotted on the right side in colored boxes with the wavelengths noted on the top, and the colors of the boxes are in accordance with the colors of the line spectra. The direction and tick scale of x-z axis for the nearfield profiles are uniformly shown by one profile in the green box for the sake of the concision, and the white lines depict the nanostructure in conformity to Figure S1. The colors of spectra indicate the n _super = 1 (blue), 1.46 (green), and 1.71 (red), while the black line spectra are the reference spectra of the Fe film with n _super = 1. The black vertical dashed line at λ = 581.6 nm denotes the condition of in-plane diffraction for all three different n _super (=1, 1.46, and 1.71), the blue vertical dashed line at λ = 398.4 nm denotes the in-plane diffraction for n _super = 1.46 for the blue line spectra, and the red vertical dashed lines at λ = 393.3 and 681.2 nm denote the in-plane diffraction for n _super = 1.71 for the red line spectra. The wavelengths of dips in the experimental transmission spectra (a, lower panel) are marked by square (at shorter wavelength) and circular marker (at longer wavelength) with their colors in accordance with the line colors of spectra. The corresponding wavelengths are indicated nearby the markers. To keep the uniformity, the same wavelength positions are also indicated by the same markers in the calculated transmission spectra (a, upper panel), FR spectra (b), and absorption spectra (c).

To evaluate the values of θ _F from the point of view of applications, we calculated the figure of merit (FOM) at certain wavelengths, defined as |θ _F |T ^0.5, where T stands for the transmission value. The higher the FOM, the better it is in terms of the applicability to MO devices. Fe films on the arrays tend to show a higher FOM in the shorter wavelengths. For instance, the Fe film on the Al NP array shows a local maximum of FOM = 0.11° at λ = 410 nm, while that of a thin Fe film is very small (<0.01°). From the FOMs, it is manifest that the existence of Al NP arrays is beneficial for the enhancement of θ _F by the thin Fe film. In Table 1, we summarize the experimental FOMs of various plasmonic-magnetic systems in preceding works along with those in our study. In a random dispersion of Au NPs on Bi:YIG thin films, the enhancement was spectrally broad reflecting the width of the LSPPs on each Au NP (FOM is around 0.08°) [9]. In the periodic array of Au NPs, FOM becomes larger reaching a value of 0.11° at λ = 780 nm [16]. On the other hand, our results with Al NP arrays manifest the enhancement at the shortest wavelengths among the materials listed in the table because of the larger carrier density of Al and designed hybridized lattice mode around λ = 500 nm. The FOM is 0.11° at λ = 410 nm which is comparable to other experimental values for periodic plasmonic structures. By tuning the background, even higher FOM values are obtained at longer wavelength: 0.14° at λ = 580 nm for n _super = 1.46, and 0.39° at λ = 638 nm for n _super = 1.71. The exceptions to the previous statement are Au nanowire arrays on the Bi:YIG thin film [19] and Au nanowire arrays embedded in EuS thin film [20], where the working wavelength is beyond 700 nm. In these cases, the thin films act as a slab waveguide through which light–matter interaction was accelerated by additional round-trips propagation inside the MO materials.

4 Conclusions

Enhanced FR is potentially important on the development of miniatured optical devices such as isolators, circulators, sensors, and so forth. We designed a diffractive hexagonal array composed of Al NPs and enhanced the FR effect of a thin Fe film deposited on the top of the array. The tunable FR effects in the whole visible range from 400 to 800 nm were demonstrated by changing the lattice constant and the dielectric environment. The maxima appeared in the FR spectra are associated with the hybridized lattice modes. The varying dielectric environment around the Al NP can further tune the intensities and spectral regions of enhanced FR from blue to red region, which is ascribed to the shifting of LSPR of single Al NP in different backgrounds. By increasing the lattice constant, a weaker enhancement was observed due to the reduced energy confined in the sparse NP array. Our study clarified the advantages of Al NP array to drive the boosted MO effect from the widely studied green region into the UV-blue range that cannot be achieved by Au nanostructures reported thus far. Lastly, the hexagonal/honeycomb lattice is a complicated nanostructure, and different nature (collinear or non-collinear) of the diffraction modes can be excited depending on the mutual orientation between the lattice and the polarization of the incoming light [36, 37]. Here, we mainly pay our attention on FR at normal incidence, but further investigation on this feature will be physically important on unveiling the enhancement mechanism in other MO systems for polar and longitudinal MO Kerr effect.