1. Introduction

The advances of metamaterials enable people to manipulate optical resonances in artificial nanostructures toward various practical applications. Although the optical property of a metamaterial can be controlled by changing the shapes, positions and compositions of its constituent elements, it is still highly desired that it can be manipulated by a fast and versatile method. Magneto-optical (MO) effect is then utilized for this purpose, in which the permittivity tensor of a MO medium is tunable by applying an external magnetic field. Great efforts have been devoted to the novel optical phenomena in subwavelength nanostructures with MO constituent [1–8]. It is shown that the MO effect can manipulate the characteristics of the extraordinary optical transmission (EOT) in metamaterials [1–6], not only change the wavelengthes of resonant transmission peaks, but also enhance the MO Faraday and Kerr effects. Nonreciprocal properties, including nonreciprocal dispersion and one-way EOT, have also been discussed [7, 8].

The emergence of these unique MO characteristics is closely related to the nonzero off-diagonal elements in the permittivity tensor under the action of an external static magnetic B[8–10]. On one aspect, nonzero off-diagonal elements imply that the MO medium is anisotropic. Left- and right-handed circular polarizations can propagate at different velocities, leading to the Faraday rotation effect [2, 3]. By using a polarizer element to select the finally transmitted polarization, the transmittance of a incident light wave is determined by the direction and magnitude of B. On the other aspect, the MO effect breaks the time-reversal symmetry and leads to the emergency of nonreciprocal effects [8–10]. For example, for a surface light wave localized at a MO-metal interface [9], its propagation is determined by the right-handed orthogonal vector triplet of (n, B, k), where n is the normal of the interface (from the metal toward the MO medium), B is the direction of the static magnetic field, and k is the wavevector, see Fig. 1(a). Different phase shifts or field profiles can be achieved for two counter-propagating surface light modes. Because the magnitude of B determines the value of the wavevector, the phase difference, or the loss related to the field profile, the MO effect can be used to achieve optical isolation action in short nano-plasmonic guides with low insertion loss [9], and one-way modes in MO photonic crystals [10, 11]. With the electro-magnetic democracy, similar effect can be observed in gyromagnetic media with anisotropic permeability tensors [12].

 

Fig. 1 (a) A single MO-metal interface, and (b) a metal-MO-metal waveguide. With a y-direction applied magnetic field B, the right-handed orthogonal vector triplet frames of (n_±, B, k_±) of the two MO-metal interfaces are plotted.

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Comparing to a single MO-metal interface, the configuration of a metal-MO-metal waveguide has not attracted much attention. This configuration has widely applications, for example, in EOT when the subwavelength waveguides support guided mode resonance [13–15]. A metal-MO-metal waveguide contains two metal-MO interfaces, see Fig. 1(b). Assuming that the static magnetic field B is applied in the y direction, a nonzero B then breaks the degeneracy of these two interfaces via the induced non-reciprocality. To understand this effect, let us label the two interfaces by ′+′ for x = +d/2 and ′−′ for x = −d/2, respectively, and still assume that the normal n of each MO-metal interface is from the metal to the MO medium. We can see now the normals n_± of the two MO-metal interfaces are antiparallel with each other, i.e. n₊ = −n₋. With the requirement of forming right-handed orthogonal vector triplet, now the wavevector k₊ in the triplet frame (n₊, B, k₊) of the upper interface at x = +d/2 is antiparallel to the wavevector k₋ in the triplet frame (n₋, B, k₋) of the lower interface at x = −d/2, i.e. k₊ = −k₋, see Fig. 1(b). For a forwardly propagating guided mode with +z-directed wavevector k, the light properties of the independent upper and lower metal-MO interfaces, if we neglect the coupling effect now, are just identical to these two counter-propagating light modes along a single MO-metal interface. It is well known that the MO-induced nonreciprocal effect renders different field profiles to the two counter-propagating light modes [9]. Consequently, the field profiles of the guided modes in the metal-MO-metal configuration, by considering the mutual coupling between the lower and upper interfaces, are neither symmetric nor anti-symmetric, but asymmetric. In other words, due to the MO-induced non-reciprocality, the field intensities I at the two metal-MO interfaces are not equal, that I₊ ≠ I₋.

Since the waveguide resonance by the interference between the forwardly and backwardly propagating guided modes is of great importance for EOT in subwavelength metal gratings [13–15], it is expected that above mentioned broken symmetry of field distribution would, in principle, influence the resonant transmission spectrum, forming some interesting nonreciprocal phenomena such as MO-induced light transparency and circling energy flux.

Above we briefly discuss the physical mechanism of the asymmetric field profile of a guided mode in a metal-MO-metal waveguide due to the MO-induced non-reciprocality, and its potential role in manipulating the EOT effect. These interesting phenomena have not been discussed in any literature, to the best of our knowledge. In this paper we would like to demonstrate these mechanisms and phenomena that could contribute to the advances of actively-tunable metamaterials. In Section II, we discuss the guided modes in metal-MO-metal waveguides by rigorously solving the Maxwell’s equations, and prove that their field profiles are indeed asymmetric. Then, in Section III, we demonstrate the excitation of the asymmetric field distributions in EOT thought thin metal films with subwavelength MO slits. In the MO effect, usually the diagonal elements are much larger than the off-diagonal ones. Here special interesting is paid to the situation when the off-diagonal elements are much larger than the diagonal ones, i.e. the MO medium is extremely anisotropic.

2. Guided modes in magneto-optical waveguides

Let us discuss the dispersion and field distribution of a guided mode in a metal-MO-metal waveguide. A schematic of the waveguide structure is shown in Fig. 1(b). Assume the MO medium occupies the region of −d/2 < x < +d/2, and consider a transverse magnetic mode.

The field component H_y can be expressed as

Equation (1) can be solved by using the Maxwell’s equations. Assume the dielectric constant of the metal is ε_metal, and the permittivity tensor of the MO medium reads [9, 10]

From the boundary continuum conditions of H_y, as

Equation (9) generally gives

However, the asymmetric degree of the field distribution, indicated by the ratio of field intensity I ∝ |H_y|² at the two MO-metal interfaces at x = −d/2 and x = +d/2, as

From above analysis we prove that, in general, the field profile of the guided mode in a metal-MO-metal waveguide is indeed asymmetric, that I₊ ≠ I₋. Further more, when a static magnetic field B is applied in the y direction, when k reverses its direction to −k, the profile of the field distribution also reverses with respect to x = 0. Because the asymmetric profile of the guided mode depends on the propagating wavevector k, this effect can also be understood as a MO-induced nonreciprocal characteristic.

In realistic applications, a MO response can be generally achieved at GHz frequency region [10, 12]. This frequency is much lower than that of free electron resonance in metal, usually at 3×10¹⁴ Hz (around 5 orders larger). Consequently, it is a good approximation to treat the metal as a perfect electron conductor (PEC). In this approximation, because ε_metal is much smaller than zero, i.e. ε_metal → −∞, we can approximately write $\alpha =\sqrt{-{\epsilon }_{\mathit{metal}}}{k}_0$. With A = B, Eq. (9) can give two seemly-independent nontrivial solutions, as

Above we discuss how a magnetic field induces anisotropy into the MO media, thus modifies the properties of a guided mode. It is shown that although the mode dispersion (ω, k) does not change when the wavevector k reverses its direction, the field distribution H_y(x) is k-dependent and becomes asymmetric. This phenomenon can be considered to of nonreciprocal nature. Note that magnetic field has also been shown to change the symmetry of the wave profile as well as the symmetry of the current distribution in the magneto-transport problems [16, 17]. The possibility in breaking the symmetry of the guided mode profile and its role in EOT have not been noticed before. Below, let us show how this phenomenon influences EOT in subwavelength MO gratings.

3. Influence on extraordinary optical transmission

Let us consider a MO subwavelength grating as shown in the inset of Fig. 2, which contains a periodically subwavelength slit array in a thin metal (Au) film. With an applied magnetic field B in the y direction, we assume that the permittivity tensor of the MO medium has the form of [4, 5, 10]

 

Fig. 2 Transmission spectra at different magnetic field B, for ω_B = 0 (solid line), 0.1ω_p (short dashed line) and 0.2ω_p (short dotted line), respectively. Inset is the structure under investigation.

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Consider the situation with d = 0.6 cm, h = 2.4 cm, a = 2 cm, and ω_p = 2π × 12.5 GHz, i.e. λ_p = 2.4 cm. By using the standard parameter for carriers in semiconductors, as m^* = 0.2 ∼ 0.5m_e, we can find that it is relatively easy to achieve the required magnetic field B of 0.05 ∼ 0.02 tesla for ω_B = 0.2ω_p in experiments [5, 12]. Because the frequency in our interests is around 12.5 GHz, the metal Au can be assumed to be PEC for simplification. Although here we are interested in an in-principle demonstration of the new mechanism in manipulating EOT by using the asymmetric field distribution in the subwavelength metal-MO-metal resonator, to represent the realistic situation we choose Γ = 0.001 for all the simulations throughout this article. The transmission properties of the structure are investigated by full-field three-dimensional finite element optical simulations (COMSOL Multiphysics 4.3a).

The dependency of the transmission spectra on the applied magnetic field B under a normal incidence is shown in Fig. 2. We can see in the absence of external magnetic field B, there exists only one transmission peak around 2.065 cm. As the magnetic field B increases to 0.2ω_p, this transmission peak slightly shifts to shorter wavelength. The transmittance T could not reach 1 due to the presence of absorption. In particular, a new MO-induced transparent peak appears in the long-wavelength region, around 2.360 cm, which also blue-shifts with the increased magnitude of B. By studying how the distributions of field intensity and energy flux vary at these transmission peaks, we can briefly understand the physical mechanism of their appearance and variation with the magnetic field B.

First, we pay attention to the transmission peak with a shorter wavelength, at 2.065 cm and 2.051 cm when ω_B = 0 and 0.2ω_p, respectively. The distributions of field intensity and energy flux (Poynting vectors) are shown in Fig. 3.

 

Fig. 3 Distributions of field intensity and energy flux (Poynting vectors) at the transmission peaks of (a) 2.065 cm when ω_B = 0, and (b) 2.051 cm when ω_B = 0.2ω_p, respectively.

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From Fig. 3(a) we can see when no magnetic field is applied, the field pattern is uniformly distributed inside the MO slits, and does not vary with x. Obviously, this EOT peak is associated with a guided mode resonance [13]. The MO medium is isotropic, with ε = 0.259 − 0.741 × 10⁻³i and γ = 0. According to Eqs. (15) to (18), now the profile of the guided mode is uniformly distributed in the x direction because L approaches infinite. Interference between the forwardly and backwardly propagating guided modes, with wavevectors k and −k, leads to the intensity node at the middle of the structure. The energy flux inside the MO medium is also uniformly distributed, and is z directed. Outside the slit apertures, energy flux forms two counter-rotating vortices at each opening, which can be explained by considering the excitation of high diffractive orders that are evanescent in nature [13].

When a nonzero magnetic field B is applied, the transmission wavelength and its field profile are modified. They can be understood as a manifestation of the nonreciprocal characteristic from the non-diagonal elements in the permittivity tensor. The larger the applied magnetic field B be, the stronger distortion the field distribution is, which can be easily understood by considering the B-dependent values of tensor elements, see Eqs. (17) to (21). For the transmission peak at 2.062 cm when ω_B = 0.1ω_p, ε = 0.256−0.749×10⁻³i and γ = 0.064+0.644×10⁻⁴i. The characteristic length L equals 2.602 cm. For the transmission peak at 2.051 cm when ω_B = 0.2ω_p, ε = 0.248 − 0.775 × 10⁻³i and γ = 0.129 + 0.133 × 10⁻³i, and L equals 1.263 cm.

From the distributions of field intensity and energy flux shown in Fig. 3(b), we can see they are all asymmetrically distributed inside the MO medium, unlike that in Fig. 3(a). Now they become to concentrate to one side of the metal-MO-metal waveguide. This phenomenon can be understood by considering the asymmetric field distribution following Eq. (17), and the non-100% reflection of guided wave at the rear aperture. To be more explicitly, because now L is positive, according to Eq. (17) the field of forwardly (backwardly) guided mode with positive (negative) k prefers the upper (lower) MO-metal interface at x = d/2 (x = −d/2). The forwardly and backwardly guided modes thus dominates different MO-metal interfaces. Interference between these two guided modes produces an elliptically sharped node in the middle of the structure, because their spatial overlapping is x-dependent. The energy flux S is proportional to the local field intensity and directs to the same direction as that of the wavevector k, thus clockwisely rotated energy flux can be observed inside the MO resonator. Because the reflection from the rear aperture is not 100%, the field intensity and energy flux at x = −d/2 is relatively weaker than these at x = d/2.

Although nonreciprocal property of the guided mode in the metal-MO-metal waveguide plays a role in the modification of the short-wavelength transmission peak, the contribution from the nonzero off-diagonal element is limited, because the characteristic decay length L is usually much greater than the width d of the MO medium. On the contrary, the MO-induced transparent window in the long-wavelength regime belongs to the situation of extreme anisotropy, with a much smaller L, i.e. a greater degree of asymmetry. For example, at the transmission peak of 2.387 cm for ω_B = 0.1ω_p, ε = 10⁻² × (0.124 − 0.101i), γ = 0.099 + 0.100 × 10⁻³i, and L =0.144 cm. At the transmission peak of 2.350 cm for ω_B = 0.2ω_p, ε = 10⁻² × (0.284 − 0.104i), γ = 0.195 + 0.203 × 10⁻³i, and L =0.104 cm. Now the MO medium is extremely anisotropic. The diagonal element ε is very small, thus the wavevector $k=\sqrt{\epsilon }{k}_0$ is much smaller than k₀. On the other hand, the off-diagonal element γ becomes much larger. As a result of this extreme anisotropy, the characteristics decay length L is much smaller than d. The spatial overlapping of the forwardly and backwardly propagating guided modes is thus very poor.

Figure 4 plots the distributions of field intensity and energy flux for this transmission peak, where the MO medium is extremely anisotropic. Because the energy flux inside the MO medium is much larger than that outsides, here the size of arrow is logarithmically proportional to the magnitude of the energy flux. We can see that, in sharp contrast with Fig. 3, the H field and the energy flux is strongly localized to the metal-MO interfaces. This is in consistence with above numerical analysis about the value of L that L ≪ d. The forwardly propagating energy flux is mainly concentrated at the upper metal-MO interface at x = d/2, while the backwardly propagating one is localized at the lower x = −d/2 interface, forming a obvious vortex-like circling flux loop. Now the metal-MO-metal waveguide is more likely to be a traveling-wave resonator. Longitudinal resonance that contributes to the node in Fig. 3(a) is destroyed. On the other hand, a huge void of energy intensity and field intensity in the middle of the MO media is formed, due to the concentration of energy flux to the boundaries. Obviously, it is the extreme anisotropy that contributes to this MO-induced transparent window.

 

Fig. 4 Distributions of field intensity and energy flux at the MO-induced transparent peak of 2.350 cm, for ω_B = 0.2ω_p.

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Recently, epsilon-near-zero metamaterials have attracted much attention, see [18, 19] and references therein. The MO-induced extreme anisotropy, in fact, represents an approach toward the realization of epsilon-near-zero media. Now the guided mode has a effective refractive index of $\sqrt{\epsilon }$. From Eq. (19) it is evident that the diagonal element ε approaches zero when ω is around ${\left({\omega }_B^2+{\omega }_p^2\right)}^{0.5}$, if we can neglect the damping. For ω_B = 0.1ω_p and 0.2ω_p, the corresponding wavelengthes are around 2.388 cm and 2.353 cm, respectively, close to the wavelengthes of the MO-induced new transparent peaks shown in Fig. 2 and Fig. 4. As a future potential subject of investigation, it might be of great interest to investigate the relevancy of the MO-induced transparency here and the other documented properties of epsilon-near-zero metamaterials [18, 19], with an emphasis on the induced extreme anisotropy.

Before ending this section, we would like to provide a more detailed discussion about the mechanism of the MO-modified resonant transmission. Usually it is believed that the resonance transmission appears due to the existence of the SPP resonance [13–15], which exists also in the absence of the applied magnetic field. However, here the metal is PEC, so all the SPP resonances are forbidden. Although the wavelength variation of the short-wavelength transmission peak (∼2.06 cm) can be briefly understood as a MO-induced shift of the transparency peaks, the peak with a longer wavelength (∼2.36 cm), which is the emphasis of this investigation, must be associated with the unique anisotropic properties of the MO medium. In the absence of the applied magnetic field, this transmission feature could not be observed. When a magnetic field is applied, the MO medium becomes anisotropic, which, in turn, enables the existence of asymmetric field distribution. This effect is especially strong near the resonance at ${\left({\omega }_B^2+{\omega }_p^2\right)}^{0.5}$, which can be termed the effective cyclotron resonance [4]. Although now the field profile is localized at the MO-metal boundaries, see Fig. 4, it is not associated with any SPP resonances, and a strong MO-induced anisotropy is evidently the deeply physical mechanism of this transmission feature.

4. Conclusion

In summary, we provide a theoretic interpretation and analysis about the existence of asymmetric field profiles for guided modes in metal-MO-metal waveguides, due to the MO-induced non-reciprocality. We study the resonant optical transmission through thin metal film with sub-wavelength MO slits, and show that magnetic field changes the transmission spectra of the structure. MO-induced transparent windows are observed, where the MO medium becomes extremely anisotropic. From the analysis of field distribution, we prove that the resonant transmission is associated with an asymmetric field distribution and a circling energy flux, which implies that the mechanism discussed in this paper is experimentally observable.