1. Introduction

Metamaterials (MTMs) are artificially engineering nanostructured metal-dielectric composites with unconventional electromagnetic functionality, which have demonstrated potential benefits in various applications such as superlens [1], cloak [2], hyperinterface [3] and light storage [4]. At optical frequencies, a most popular MTM design is the fishnet structure, stacked multi-layer metallic films patterned with two-dimensional (2D) arrays of subwavelength holes [5–9]. It has been shown that a low loss negative index MTM with a figure of merit up to 3.34 could be achieved in a wide visible spectral range [10]. Geometrically, the fishnet MTM can be viewed as a combination of pairs of short slabs and continuous metal wires [9]. For normal incidence, the fishnet MTM behaves as a magnetic resonator, where magnetic moment is created by antiparallel currents in neighboring conductive layers [11], yielding a virtual current loop when a surface plasmon polariton (SPP) is excited at the inner interfaces. The electric response arises from the interaction of the external electric field with the holes and the external metallic interfaces, when the localized surface plasmon (LSP) is excited [12]. Consequently, a negative refraction index region is achieved over a limited spectral region depending on the resonant excitation of SPP and LSP.

The optical properties of fishnet MTMs have been studied intensively by varying the shape [13] and size [14] of the holes, lattice geometry [15], higher order magnetic resonances [10,16], incidence angle [17] and gain effect [18, 19]. However, the effect of structural asymmetry has not been considered so far. In general, the polarization of incident wave, the reciprocal lattice vectors of periodic holes as well as the symmetric axes of holes determine the optical response of normally incident light. Given the polarization of incident light and the fishnet square array, we can realize two types of asymmetry: one is the inner asymmetry between the symmetric axis of the holes and the lattice vector of the array; the other is the external asymmetry between the symmetric axes of the holes in adjacent layers. The asymmetric fishnet MTMs can be expected to exhibit unique optical activity at optical frequencies.

Recently, chiral MTMs have attracted much interest because of large optical activity [20], circular dichroism [21], and negative index [22, 23]. Many chiral MTM designs based on particles topology have been proposed and experimentally demonstrated in microwave and terahertz region [24, 25]. Due to the additional kinetic inductance of particle geometry at optical frequencies [26], such as twisted split ring resonators and cross stripes, the reported chiral MTM designs can’t be scaled down to the visible wavelengths. Thus the fishnet MTMs with inner and/or external asymmetries provide a practical means to realize optical activity and negative index at optical wavelengths.

The aim of this paper is to investigate how the structural asymmetry affects the optical properties of fishnet MTMs at visible frequencies. For this purpose, we study the spectral and polarization response by varying the azimuth angle ϕ of the elliptical holes in regular mono- and double-layer fishnet structures with respect to the reciprocal lattice vectors. The breakdown of mirror symmetry in such structures leads to spinning LSP modes, i.e., localized surface plasmons revolve around the air holes in one optical cycle, which give rise to strong magnetoelectric coupling and polarization rotation for transmitted light. In addition, the spectral dependence of the magnetic resonances on the orientation, dielectric spacing and aspect ratio of elliptical hole are investigated. For both cases, the large spectral shift associated with minor axis supported LSP mode is observed. Due to the resonant nature of metallic MTMs, the introduction of asymmetry in fishnet MTMs provides a practical channel to engineering the spectral and polarization response of bulk MTMs in optical frequencies. To our best knowledge, our modeling based on asymmetric hole topology is the first optical chiral MTM prototype at visible frequencies.

2. Model and method

A sketch of the proposed double-layer fishnet MTM is presented in Fig. 1. The structure consists of a pair of silver films perforated with square arrays of elliptical holes. The metal films with thickness t = 43 nm are separated by a flat layer of homogeneous dielectric medium with height h = 45 nm. The unit cell parameters are: in-plane periodicities D_x = D_y = 320 nm, the minor and major radius of the ellipse are a and b, respectively. The air holes are partially overlapping and oriented at a twist angle ϕ to each other. Obviously this modified structure is a regular fishnet structure for ϕ = 0°, and it is a chiral structure when the orientation is nontrivial, i.e. ϕ ≠ {0°, 90°}. The numerical simulations of the light propagation were performed with the frequency solver of CST Microwave Studio and unit cell boundary condition in xy plane is applied. We neglect the substrate and assume the spacing medium between the metallic films is air for simplicity. Drude model $\epsilon (\omega )=1-{\omega }_p^2/\left({\omega }^2+i\gamma \omega \right)$ is used to describe the permittivity of silver, with the plasma frequency ω_p = 1.37 × 10¹⁶s⁻¹ and collision frequency γ = 8.5 × 10¹³s⁻¹ fitting with the experimental value [27]. For all the calculations, we have restricted ourselves to normal incidence, with the polarization of incident light along y direction, which is the (0,1) direction of the square lattice. The proposed asymmetric structures can be fabricated by a standard nanolithography procedure, such as electron beam lithography and/or direct laser writing, followed by electron beam evaporation of the metal films.

 

Fig. 1 Sketch of the proposed asymmetric double-layer elliptical holes fishnet MTM. The silver films are in xy plane and the y polarized incident light propagate in z direction.

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It’s well-known that subwavelength hole arrays in a single silver film enable surface plasmons to be coupled with the incident light with appropriate wavelength, giving rise to extraordinary optical transmission (EOT) [28]. As a result, optical property of multi-layer fishnet MTM is result from the interplay between the transversal channel of gap-SPP formed in the spacing dielectric layer and the longitudinal channel of EOT associated with the excitation of LSP in the air holes [19]. For the fishnet MTM with elliptical hole, the scattering of incident light (or the formation of SPP modes) in the metallic films exhibit optical anisotropy due to the twofold symmetry of ellipse. If ϕ ≠ 0° and 90°, the breakdown of mirror symmetry will lead to magnetoelectric coupling and change the polarization of reflected and transmitted light. In general, the resulting effective MTM is a bi-anisotropic material. Consequently, a linearly polarized incident light will be converted into elliptically polarized light in transmission. This polarization effect can be expressed as the Stokes matrix form:

3. Monolayer fishnet MTMs and the fundamental LSPs

We begin with the study of optical properties of linearly polarized light through monolayer fishnet film with respect to the orientation of the holes. A schematic view of the unit cell of monolayer fishnet structure together with the polarization of incident light is shown in Fig. 2(a), where ϕ is defined as the twisted angle between the major axis of the ellipse with respect to the x axis of the lattice. The optical transmission through a metal film with subwavelength non-circle holes is mediated by both the localized modes due to the enlarged hole shape and the collective modes from periodicity of the hole array [28,29]. Different from earlier works on EOT in elliptical holes [30–32], where the optical response for light with different polarizations can be expressed as linear superposition of two orthogonal polarized states with the aid of Eq. (1), all the lattice symmetry, the geometry of hole, and the polarization of incident light in our case will strongly affect the spectral and polarization response of the transmitted light.

 

Fig. 2 (a). Scheme of the unit cell of the monolayer fishnet MTM with twisted angle ϕ. (b). The reflection spectra for ϕ = 0° and 90°. And the corresponding H_z distribution in the middle plane of the metallic film with the major axis of the elliptic hole is (c) vertical (ϕ = 0°) (d) parallel (ϕ = 90°) to the polarization.

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Firstly, we study the fundamental resonances for trivial orientations, i. e. ϕ = {0°, 90°}, where the magnetoelectric coupling is absent. In Fig. 2(b), the reflection spectra are compared for structures with ϕ = 0° and 90°. In the spectra, well-resolved resonances L and S, which are accompanied by the reflection minima, are observed at 510 THz and 668 THz. From the physical intuition, the observed resonances result from the fundamental (0,1) LSP modes supported by the elliptical holes due to their twofold symmetry, which can be verified by the distribution of the electromagnetic fields (Fig. 2(c, d)) at the resonant frequencies L and S. The magnetic field H_z indicates the local charge accumulation from the excitation of LSPs, since the incident wave propagates along z direction. Supposing the principal axes of the ellipse are along the coordinate axes, the electric field of incident light will excite dipole LSPs with charge oscillating along the major or minor axis of the elliptical holes. Therefore, the currents at the ’neck’ region are opposite to those of the slab areas and the charge accumulation is produced at the areas where the opposing flowing currents meet each other [9]. When the hole is enlarged from circle to ellipse, the different oscillating length and strength of the localized SPPs along the principle axis of ellipse lead to the separation and different band width of the two resonances.

Next, let’s consider the optical properties of monolayer fishnet structure with twisted elliptical holes. The dependence of spectral response on the orientation of the holes ϕ is shown in Fig. 3. One of the most interesting feature in Fig. 3(a) is that there are dual band resonances for each ϕ ∈ (0,π/2). Obviously, both resonances are strongly related with above mentioned fundamental LSP modes L and S. With the increase of ϕ from 0° to 90°, the lower L like resonance becomes weaker and the resonant frequency undergoes a large blue shift, while the higher frequency S-like resonance becomes stronger and shifts towards high frequency until it approaches to S mode. Another difference between these modes is that the bandwidth of L-like mode is much broader than that of S-like mode. The bandwidths of both modes come from the different curvatures of the charged air-metal interface of the elliptical holes at the resonant frequencies, and similar spectral behavior was also found in EOT through 2D array of enlarged holes [29].

 

Fig. 3 Normal incident reflection and transmission spectra of monolayer fishnet MTM with respect to different ϕ, where ϕ is defined by the azimuth angle of the ellipse with respect to x axis. Curves are shifted upward for clarity.

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The other characteristic feature of the asymmetric fishnet monolayer is the strong capability for rotating the polarization of the transmitted light. In Fig. 3(c), we show the calculated cross-polar transmission t_xy for different ellipse orientations. When ϕ is increased from 0° to 45°, a broadly flat band of polarization reversal emerges in a wide spectral range, which indicates a broadband magnetoelectric coupling effect. With a further increase of ϕ to 90°, the cross coupling becomes weaker. The maximum cross coupling between electric and magnetic field occurs at ϕ = 45° which corresponding to the maximum asymmetry between the orientation of ellipse and the reciprocal lattice vectors. We calculate the azimuth angle ψ of the transmitted elliptical polarized light and the corresponding depolarization (|E_x|²/|E_y|²) for ϕ = 45° as shown in Fig. 4. The depolarization of the transmitted light have two peaks near 590 and 690 THz. The reason for both peaks can be observed in Fig. 3(b) and Fig. 3(c): for the first peak near 590 THz, there is a dip between the S- and L-like mode in co-polar transmission, while the cross-polar transmission spectrum for ϕ = 45° is rather flat between the resonances; The peak near 690 THz arises from the faster decline of the co-polar transmission than the cross-polar transmission. Meanwhile the azimuth angle ψ of the polarization ellipse is increased monotonously from −75° to 47° for frequency below 670 THz, then decreased. Since ψ is defined as the angle between the major axis of the polarization ellipse and x–axis, zero ϕ at 593 THz indicates a 90° polarization rotation. This demonstrates that the L and S resonances are responsible for the polarization rotation of the transmitted light from arrays of oriented elliptical holes.

 

Fig. 4 ψ (a) and depolarization (|E_x|²/|E_y|²) (b) of the transmitted elliptical polarized light for single layer fishnet MTM with ϕ = 45°.

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To better understand the origin of the L- and S-like mode, we plot the distribution of instantaneous magnetic field H_z at two moments with π/4 phase difference for each resonant frequency for ϕ = 45° in Fig. 5. It is shown that the charge accumulation together with the current distribution in the films for both resonances evolves between the two fundamental modes L and S in an optical cycle, which is different from the ordinary time harmonic field in EOT [28]. Compared Fig. 5(a) and (b), the dominated H_z distribution for lower resonance is L -like (Fig. 5(b)). While the dominated H_z distribution for the higher resonance is S-like (Fig. 5(d)). Further inspection of the current distribution (not shown) shows that the spinning current mode for nontrivial ϕ gives rise to the observed magnetoelectric coupling as well as the polarization rotation of the transmitted light due to the conversation of angular momentum in scattering process.

 

Fig. 5 The distribution of H_z at resonance (a,b) L -like LSP, 529 THz and (c,d) S -like LSP, 663 THz in the middle plane of the metallic film with ϕ = 45°.

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4. Double-layer fishnet MTMs and the hybridized LSPs

It is known that MTMs with same constituents but different spatial arrangements exhibit exotic electromagnetic properties beyond the averaged effect [12, 33–37], so as to modify the resonant frequency and line shape. The magnetic resonance of regular fishnet MTM is produced by the antiparallel currents in adjacent metal films, which can be viewed as the asymmetric coupling of the LSPs supported by the two layers [11, 12, 19]. The resonant feature of this structure resembles an effective LC circuit model [9]. When our double-layer fishnet MTM is modified by the rotation of the elliptical holes as shown in Fig. 1, LSP resonances supported by adjacent layers hybridize, resulting in the mode coupling and spectral shift. Thus it can be expected that the structural chirality together with the coupling effect between the revolving LSPs supported by monolayer should produce exotic optical response which different from the case with inner asymmetry studied in previous section.

Figure 6 shows the co- and cross-polar response of double-layer fishnet structure with respect to the orientational angle ϕ of the holes in the first layer. For simplicity, the major radius of the ellipse in the second layer is aligned with x axis. For the special orientations ϕ = 0° and 90°, where the major axis of the ellipse in the first layer are parallel and vertical to the ellipse in second layer, there are no magnetoelectric coupling (Fig. 6(c)) due to the absence of structural asymmetry. In Fig. 6(a) we can see that for ϕ = 0°, which is a regular fishnet MTM structure, the lower magnetic resonance lies at 500 THz, and the resonant band is very flat compared with other resonances. For ϕ = 90°, there is a resonant dip near 632 THz. The observed lower frequency resonances of the double-layer MTMs are similar to the S and L modes in monolayer, despite a small red shift. The large spectral difference between the two lower resonances for ϕ = 0° and 90° results from the different fundamental LSP modes supported by holes in the upper layer. The resonance around 710 THz is the (1,1) mode supported by the holes, which does not appear in the spectral region for monolayer. Because the higher symmetry of (1,1) mode [16, 38], it’s spectral position is less sensitive to the relative orientation and aspect ratio of the elliptical holes.

 

Fig. 6 Normal incident reflection and transmission spectra of double-layers fishnet MTM with different ϕ. Curves are shifted upward for clarity.

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When ϕ is increased from 0° to 90°, an additional asymmetric magnetic resonance emerges due to the structural chirality. With the increase of ϕ, the first resonance A with lowest frequency becomes narrower and shifts slightly towards longer frequency. Meanwhile, the new resonance B around 600 THz undergoes a large blue shift with ϕ, which is a similar to the behavior of the S-like mode monolayer. And the mode C is blue shift slightly and approaches the second resonance of fishnet MTM with ϕ = 90°.

To interpret the observed spectral behavior, we analysis the characteristic of the resonant modes for a specified ϕ = 45°. It is found that all the modes result from asymmetric hybridization of the LSPs supported in each individual layer, and the charge accumulation in each layers evolves between these LSPs in an optical cycle. These can be verified by inspecting the magnetic field distribution in a xy-plane half away between two metal layers at resonant frequencies for ϕ = 45° in Fig. 7. Specifically, the resonance A at 511 THz is hybridized by the L-like mode in the first layer and the L mode in the second layer, since we suppose only the holes in upper layer are rotated. Analogously, the resonance B at 585 THz comes from the coupling effect between S-like LSP mode in the first layer and the L mode in the second layer. Thus, the larger spectral shift of the mode B is produced by the effect of L-like mode supported by the rotated holes in the upper layer, since the spectral interval between L and S mode is very large. While the higher mode C at 711 THz results from the asymmetric coupling between (1,1) modes in the layers. Note that (1,1) mode of single layer is absent in the spectral range that we studied (seen in Fig. 3), because the frequency of asymmetric mode lies in the lower branch when two eigenmodes hybridized.

 

Fig. 7 Simulated H_z distribution at hybridized mode (a,b) A (511 THz), (c,d) B (585 THz) and (e,f) C (711 THz) in the middle plane of the dielectric spacer with ϕ = 45°.

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Strong magnetoelectric effect is observed near the mode C for nontrivial orientations in double-layer fishnet structure. Figure 8(b) shows the depolarization, |E_x|²/|E_y|² ∼ 23, near 710 THz for ϕ = 45°, which is one order of magnitude bigger than that near 480 and 555 THz. At the same time, the polarization rotation of the transmitted light reaches 90° at 710 THz, while the azimuth angle ψ varied slowly for the first two resonances as shown in Fig. 8(a). As a result, fully control of the polarization rotation of transmitted light and depolarization between two orthogonal linearly polarized light could be accomplished via the (1,1) hybridized mode.

 

Fig. 8 ψ (a) and depolarization (b) of the transmitted elliptical polarized light for double layers FMM with ϕ = 45°.

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5. Effects of dielectric spacing, aspect ratio and background material

In what follows, we consider the effect of the other important geometry parameters on the spectral response of double-layer fishnet MTMs by tuning the dielectric layer spacing and the aspect ratio of elliptical holes while keeping the unit cell and twist angle ϕ = 45°. Fig. 9(a) presents the frequency dependence of the hybridized modes on the spacing between the metal layers from 40 nm to 145 nm. When the dielectric spacing h is increased from 40 nm, both resonant frequencies are blue shifted. Note that interestingly, the resonant frequencies of the lower two modes A and B are decreased slightly with the spacing for h exceed about 95 nm. In general, the coupling effect between the two layers becomes weaker when the spacing is increased, thus, the resonant frequency of the coupled mode will shift monotonously towards the resonant frequency of an individual mode. To interpret the blue shift with h, we note that the individual mode supported by the two layers are not degenerate modes for ϕ ≠ 0. Hence, with the increase of spacing h, L–like mode supported in the upper layer will dominate the reflection response and give rise to the observed blue shift.

 

Fig. 9 The resonant frequency is plotted as a function of the spacing between metal films (a) and aspect ratio of the elliptical holes (b) for double fishnet MTM geometry with ϕ = 45° while keeping the other parameters. In Fig. 9(b), the minor axis of the hole is set to 60 nm. Dashed green lines indicates the parameters adapted in previous section.

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Fig. 9(b) shows the resonant frequency when the aspect ratio b/a of the holes varied from 1 to 2, where a is 60 nm. Except a bifurcation of lower frequency mode A in a limited range, all of the resonant frequencies are deceased with the aspect ratio. The red shift is in accordance with the anticipation of the LC circuit model [9], since the effective inductance L and capacitance C will increase with the aspect ratio, thus the resonant frequency shifts to lower frequencies. Surprisingly, the lower frequency resonance splits into two branches in the range of b/a from 1.17 to 1.58 as shown in Fig. 9(b). Note that when b/a is increased from 1, the rotation symmetry of circle is broken, resulting in splitting and evolving of LSP modes for monolayer with oriented holes. When the aspect ratio of the ellipse lies in the limited range, there is strongly coupling between the L mode in the lower layer and L–like mode in the upper layer, which give rise to the bifurcation. Numerical simulations (not shown) indicate that the instantaneous distribution of the maximum currents for the lower branch flow in y direction. In contrast, the direction of current for the upper branch is in x direction. Furthermore, a π/2 phase difference between them is observed. This indicates that the bifurcated modes is excited by the incident electric and magnetic field, respectively. Another characteristic feature should be noted is the sensitivity of resonant frequency on the geometric parameters, i.e. the spacing and the aspect ratio. In Fig. 9(a,b), we can see that the higher frequency (1,1) hybridized mode is sensitive to the spacing, and the lower frequency mode is more sensitive to the aspect ratio, while the S dominated mode is sensitive to both the spacing and aspect ratio.

Finally, let us examine the influence of different background materials on the spectral response of asymmetric fishnet MTMs for the real sample. Figure 10(a,b,c) show the reflection spectra of monolayer fishnet structures with ϕ = 45°, which embedded in air, Al₂O₃ and glass, respectively. It is shown that the resonant frequencies of both L and S modes shift towards lower frequencies. Specifically, the normalized frequency shift Δf / f₀ of L mode is increased from 0.256 to 0.32 when the index of embedding media is changed from 1.38 to 1.52, where f₀ is the resonant frequency in air. At the same time, the normalized frequency shift of S mode is increased from 0.24 to 0.257, which indicates that L mode is more sensitive to the embedding materials than S. The observed red shift of resonant frequency has two physical origins: firstly, the wavelength of light wave in dielectric materials is linearly reduced by a factor of 1/n; secondly, the resonant frequency of LSP is nonlinearly reduced according to the dispersion relation. Similar red shift is observed in Fig. 10(d,e,f) for double layer fishnet MTM with ϕ = 45°. While interestingly, the normalized frequency shifts of mode A and B have almost the same value 0.255 and 0.32 for index 1.38 and 1.52, which indicate the effect of L-like LSP mode in the lower layer is more important than the S-component of the spinning mode supported in the upper layer. Because of the higher symmetry of (1,1) mode, resonance C is less sensitive to the background materials compared with mode A and B.

 

Fig. 10 Reflection spectra r_yy of monolayer fishnet structures (a,b,c) and double layer fishnet structures with ϕ = 45° (d,e,f) for different background materials: (a,d) air, n=1; (b,e) Al₂O₃, n=1.38; and (c,f) glass, n=1.52.

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6. Summary

To conclude, we have demonstrated theoretically strong polarization rotation at visible frequencies associated with large spectral tunability of LSP resonances of asymmetrical fishnet MTMs. This effect is achieved by tuning the relative orientation of elliptical holes with respect to the reciprocal lattice vectors of periodic structure. For monolayer structure, dual band of resonances arises due to the excitation of spinning LSP modes between L and S mode supported by elliptical hole. For double-layer fishnet MTM, triple magnetic resonances appear for nontrivial orientated holes due to the spinning hybridized LSPs between adjacent layers. The spectral dependence on orientation of the ellipse would be useful for practical purpose for engineering the bandwidth of multi-layer fishnet MTMs. Furthermore, the dependence of these hybridized modes on the parameters, dielectric spacing h and aspect ratio of the ellipse b/a in investigated. It is shown that the S dominated mode are sensitive to both h and b/a, while large depolarization and strong polarization rotation can be easily accompanied by using the (1,1) hybridized mode. Finally, the red shifts of resonant frequencies due to the background materials are analyzed for single and double layer structures. Our observations is another stepping stone in the path to enrich the optical properties of fishnet MTM. An experimental verification of our findings in chiral fishnet MTM is expected with modern semiconductor nanofabrication technology. It would be useful for the design of bulk chiral MTMs in the visible frequencies and tailoring the polarization behavior of the metallic nanostructures.