## Abstract

We have investigated extraordinary optical transmission (EOT) with enhanced Faraday effect in one-dimensional metallic-magnetic slit arrays under polar magnetization using the rigorous coupled-wave analysis performed by the Airy-like internal reflection series. The roles of surface plasmon polaritons and quasi-guided waves are studied in which the latter plays a key role. Based on the mechanism of EOT with an enhanced Faraday effect, both enhanced transmittance and enhanced Faraday effect are optimized by adjusting the geometric parameters of slit arrays evaluated by the figure of merit.

©2008 Optical Society of America

## 1. Introduction

Regarding extraordinary optical transmission (EOT) [1], many researchers have carried out extensive theoretical and experimental works on one-dimensional (1D) slit and two-dimensional (2D) hole arrays [2, 3, 4, 5]. EOT is a phenomenon in which light is transmitted with greater efficiency than unity (when normalized to the area of slits or holes) [1, 4, 6]. Furthermore, magneto-optical (MO) Faraday and Kerr effects are of interest for optical readout of magnetically stored information in erasable video and audio disks. Recently, Belotelov et al. exhibited EOT with an enhanced MO Faraday effect in both 1D and 2D bilayer systems consisting of a thin metallic layer perforated with slit and hole arrays and a uniform magnetic layer [7, 8]. However, obtaining enhanced optical transmittance with great Faraday effect is difficult [9], as in the case of reflectance with Kerr effect [10, 11]. Therefore, efforts have been made to achieve both EOT and MO Faraday effect simultaneously [12]. In Refs. 7 and 8, researchers attributed EOT with an enhanced Faraday effect to the surface plasmon polaritons (SPPs) coupled with the quasi-guided waves. So far, the origin of EOT from the novel metallic arrays [13, 14] has not been completely understood, even for the compound structures of the arrays with dielectric films [15, 16]. More importantly, there are only a few works have been done on EOT with an enhanced MO effect.

In this paper, we investigate the physics of EOT with enhanced Faraday effect. The rigorous coupled-wave analysis (RCWA) performed by the Airy-like internal reflection series (AIRS) is utilized to analyze both EOT and MO Faraday effect together in one-dimensional metallicmagnetic slit arrays. The roles of SPPs and quasi-guided waves playing in the system are evaluated. For the mechanism of EOT with an enhanced Faraday effect, we also discuss how to optimize EOT and Faraday effect in order to obtain a larger figure of merit (FOM).

## 2. Model description

Figure 1 shows a bilayer system consisting of a gold periodic array and a uniform magnetic film, Bi-substituted yttrium iron garnet (Bi:YIG) studied in Refs. 8. The bilayer is identified with period *d*, bar width *w*, thickness of array *h _{1}*, and thickness of magnetic film

*h*. We assume that a transverse-magnetic (TM) light is incident on the bilayer system, where the magnetic component of light is polarized along the slits and the incident is placed on the

_{2}*yz*plane, and the magnetic film is magnetized perpendicularly to the surface [17].

Concerning both optical andMOproperties of the bilayer system of Fig.1, the RCWA, implemented by the AIRS is employed. This approach has been validated on both isotropic [18] and anisotropic periodic micro- and nanostructures [19]. In the frame of RCWA, all the components of the electromagnetic fields and the permittivity are expended into the generalized Fourier series with a truncation of *n _{max}*. Substituting them into the time-harmonic Maxwell’s equations, a system composed of ordinary differential equations is obtained. Then the following procedure is different from the conventional transfer-matrix or scattering-matrix implementations. Here we match the boundary conditions at each interface and utilize the multiple reflection with the AIRS. The transmission matrix can be expressed as a function of interfacial (

**R**

*,*

_{J,0}*+1,*

**R**J,J*+1, and*

**T**J,J**T**

_{0,J}) and phase matrices

**P**

*J*:

where *Q _{J}*=

**P**

_{J}**R**

_{J,0}**P**

_{J}**R**

*and the subscripts denote the index of different layers. Thus, the final transmission matrix is always solvable through this recursive algorithm regardless of the number of layers. Finally, these 2(2*

_{J,J+1}*n*

_{max}+1) equations can be solved to acquire the transmitted amplitudes. Accordingly both Faraday rotation and ellipticity can be calculated.

For the numerical simulations, the same parameters in Refs. [7] and [8]are used. The permittivity of gold film is fitted into the Drude model ε_{1}=ε_{∞}-*ω ^{2}_{p}*/(

*ω*-

^{2}*iγω*), where

*ε*

_{∞}=7.9,

*ω*=8.77 eV, and

_{p}*γ*=0.075 eV. For polar MO configuration, the permittivity components of Bi:YIG are as following:

*ε*=

_{xx}*ε*=

_{yy}*ε*=5.5-

_{zz}*i*0.0025 and

*ε*=-

_{xy}*ε*=(0.15-

_{yx}*i*)×10

^{-2}.

## 3. Results and discussion

In comparison with the transmittance and the MO Faraday spectra in Ref. 8 the results are similar. In the bilayer structure consisting of the gold slits arrays and the Bi:YIG film, Fig. 2 shows that the transmittance and the Faraday peaks are located at 894 nm with the amplitudes of 0.3 and 1.7°, respectively. Considering various combinations of the gold film and magnetic layer we analyze the EOT and MO Faraday effect as follows. For the first case, we set only a gold film without any slits on the top of the Bi:YIG layer. Both the transmittance and the Faraday effect (dash lines) are insignificant compared with those of the structure of Fig. 1 (solid lines). For the second case, we set only a single Bi:YIG layer with the same thickness of 547 nm. The Faraday rotation is around 0.5 degree much less than that of the bilayer structure; no distinct Faraday peak is observed even though the transmittance is fairly high, exceeding the scale of Fig. 2. For the third case, we set only the gold slits array, which is nearly opaque with the transmittance less than 0.02. Obviously, the electromagnetic waves are unable to be transmitted in the array without the Bi:YIG layer, since a critical thickness is needed for a single metallic array to realize EOT as well as the width of slits [2].

Concerning the origins of EOT with an enhancement of Faraday effect in both 1D and 2D arrays, Belotelov et al. attributed these origins to the SPPs coupling with quasi-guided waves [7, 8]. In order to elucidate the roles which SPPs play in 1D arrays, we investigate the SPP wavelengths λSP of a flat interface [13, 20, 21], which is given by

where *m*, *θ _{i}*, and ε

_{1}are a nonzero relative integer, the incident angle, and the permittivity of metallic materials, respectively. The

*ε*is the permittivity of air or magnetic medium and

_{d}*θ*=0 for normal incidence. Having the period

_{i}*d*=750 nm, calculation shows that λ

_{SP}=960 nm at the metallic-magnetic interface, while λ

_{SP}=760 nm at the air-metallic interface in the range of interest. Notably, no resonant peak is located nearbyλ

_{SP}, as shown by the arrow in Fig. 2, except the tiny peaks in both the transmittance and Faraday spectra.

Generally, the dependence of the peak position on the period is a usual way to judge whether the origin of EOT peaks is from SPPs [4], since λ_{SP} is sensitive to the period *d* according to Eq. (2). The relation between the resonant peaks and the period *d* is illustrated in Fig. 3. while the period decreases, the resonant peaks are blueshifted and vice versa, similar to the result of Ref. 4. However SPPs do not play a key role here, because of the discrepancy between the EOT and the calculated λ_{SP}. The incident beam irradiates at the gold array, which imposes an additional momentum on the incident wave through the grating momentum wave vector. Hence, we have

where *β*, *k _{0}* and

*n*are the transverse wavenumber, the incident wavenumber, and a coupling integer, respectively. Subsequently, a resonant peak might be excited if the transverse wavenumber satisfies Eq. (4), responsible for the guided waves [14, 22] as shown below,

where *α _{i}*=

*k*/

_{zi}*ε*for TM polarization, while

_{i}*α*=

_{i}*k*for TE (transverse electric) polarization; ${k}_{\mathrm{zi}}=\sqrt{{\epsilon}_{i}{k}_{0}^{2}-{\beta}^{2}}$, for

_{zi}*i*=1,2,3 denoting the metallic, the magnetic, and the surrounding medium, respectively. From Eq. (4), we can derive the relation: the increase of

*d*gives rise to the decrease of

*β*, Without varying the thickness

*h*

_{2}, in order to satisfy this equation,

*k*

_{0}should be reduced,(i.e., to make the wavelength larger), which leads to the redshift. Therefore, in this bilayer system, the dependence of resonant peaks on the period is ascribed to the influence of the additional momentum from the grating on the guided waves rather than the coupling of SPPs.

In a thicker metallic layer the light transmittance may be suppressed or lead to the extra resonances, as demonstrated in Refs. 2 and 8. A way to optimize the EOT with enhanced Faraday effect in this bilayer is to adjust the bar width w. At a point of view from the effective refractive index, the variation of w gives rise to the changes in the effective refractive index of metallic gratings[23]. Correspondingly, the optical and the MO properties might be improved. For the MO Faraday effect, the FOM is employed to evaluate the trade-off between them, defined as the product of the modulus of the Faraday angle and the square root of the transmittance (*θ _{F}*·√T) [24]. The relation of the FOM with the bar width wis shown in Fig. 4(a), with an optimal bar width around

*w*=600 nm, where the FOM is improved from 0.96 in Ref. 8 to 2.1 here. The transmittance and the Faraday spectra of the bilayer system with the suboptimal and the optimal bar width are shown in Fig. 4(b). Explicitly, the small w benefits the interaction between the quasi-guided waves and the magnetic layer while it is likely to suppress the localization of light leading to the broadening of peaks [25], shown in 4(c). However, the further decrease of w does not contribute to enhance the Faraday effect any more even though the transmittance increases. And it might also complicate this study due to the introduction of more modes, such as propagated mode [26]. When

*w*is set to 600 nm, the transmittance and the Faraday angle nearly reach 0.6 and 3° respectively. At the same time, the thickness of the magnetic layer goes down to 515 nm from 547 nm, compared with the case of

*w*=675 nm in Ref. 8.

Interestingly, the transmittance of TE-polarized incidence is up to 0.35 at *w*=600 nm (not shown here), but still less than the threshold to excite the cavity resonant mode for TE-polarized light [25]. It is well known that only TM-polarized light can excite the coupled SPPs [27, 28], because the SPPs have a TM-wave like character [14, 20]. Therefore, the appearance of TE-polarized transmission has a further negative effect on the roles of SPPs in the bilayer system. Figure. 5 exhibits that the resonant transmission appears discontinuously with the thickness of magnetic layer, which is exactly the character of quasi-guided waves. According to the transcendent Eq. (4), certain wavelength have a certain corresponding thickness given by the graphical solution [22]. However, Eq. (4) can be used only to estimate the situation qualitatively, because the effective refractive index of metallic layer depends on the transverse geometrical parameters. Furthermore, the resonant peaks are blueshifted with the decrease of *h*
_{2} (see Fig. 4), even with different width of slits, which is also illustrated in Fig. 5.

Additionally, the line shapes are asymmetric in that 2D system [7], attributed to a character of SPPs [29, 30], while all line shapes are symmetric in 1D system here. More important, the resonant peak occurs at 963nm in that 2D structure which is fairly close to the SPP resonance at λ_{SP}=960 nm expected from Eq. (2) with a period of 750nm. Therefore, we strongly believe that the physical origin of EOT with the enhancement of Faraday effect in 1D systems is different from that in 2D systems. In 2D hole arrays, the physical origin might be induced by SPPs coupling with quasi-guided waves, whereas the latter plays much more crucial roles than SPPs in 1D slit arrays.

## 4. Conclusion

The rigorous coupled-wave analysis performed by the Airy-like internal reflection series is used to study the optical and MO properties in 1D bilayer system, where EOT with an enhanced Faraday effect is presented. In order to understand the origin, the investigation is carried out mainly for the following five aspects: the peak of SPPs λ _{SP}; the dependence of resonant peaks on the period; the relation of those peaks to the thickness of magnetic layers; the TE-polarized transmission; and line shapes which definitely reveal that the quasi-guided wave plays a more crucial role in this system than SPPs. Moreover, the FOM is improved from 0.96 to 2.1 by changing the width of slits owing to the variation of the effective refractive index of the gold slit array. For this enhancement, we have a thinner magnetic layer compared to the previously reported one by Belotelov et al.[8].

## Acknowledgments

This work was supported by the Creative Research Initiative Program (Center for Photon Information Processing) by MEST via KOSEF, S. Korea.

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