1. Introduction

Metamaterials have become an important research field for nano-optical applications, attributed to the intriguing electromagnetic phenomena that do not exist in natural materials, as well as flexible designs of geometric shape and size [1]. Since metamaterials were proposed, many novel structures have been reported ranging from microwave to optical regime, such as I-shaped metamaterials, split-ring resonators, L-shaped metallic antenna and S-shaped holes [2–6]. From then on, many works have paid attention on manipulating the polarization state, phase and direction of electromagnetic waves [7,8]. In 2007, J. Hao et al. showed an I-shape metamaterial to manipulate the electromagnetic polarization and got a complete conversion between two perpendicular linear polarizations [6]. In the same year, unidirectional surface plasmon polariton was realized by nanoslit couplers [9]. In addition, many other unidirectional effects have also been leading-edge topic, in terms of the potential applications in filter, sensor and highly compact integrated nanophotonic devices. For example, the unidirectional reflectionless propagation and asymmetric transmission (AT) effect [10,11]. It is well known that the AT phenomena for linearly and circularly polarized lights have been investigated both in the microwave and optical regimes with kinds of symmetry-broken metastructures [12–14]. Microwave metamaterials are consisted of perfect electric conductors (PEC) at the dimensions of micrometers without Ohmic losses. Here, the micrometer-level dimensions of these metamaterials provide convenience for experiments, and PEC can lead to loss-free electromagnetic responses and thus strong structural resonances [15]. Then the AT effect has been presented excellently in microwave metamaterials. For example, a dual-band AT effect was realized by bilayer chiral metamaterial with the dual transmission differences around 11 GHz and 15 GHz, respectively [16].

Compared with PEC metamaterials for microwave applications, metamaterials at the near-infrared band are designed by nanostructured metals, for which regime the plasmonics merges photonics and electronics at nanoscale dimensions. However, the nanoscale dimensions limit the flexibility of sample fabrication and the plasmon dissipation leads significant Ohmic losses at optical wavelengths [17]. As a result, AT effect usually shows a small value, which restricts the applications of metamaterial at near-infrared band. As we know, near-infrared regime is one of the key operation bands for nanodevice applications, such as polarization rotators and circulators [11]. Therefore, it is important to increase the efficiency of AT at near-infrared band. From 2014, the AT phenomena for linear polarization have been realized by kinds of bi-layer metamaterials, while the maximum $\Delta$ was only about 0.35 resulting from the resonance of these symmetry-broken metastructures [13,18–21]. Recently, highly efficient AT effect of linearly and circularly polarized lights have been realized in trilayer chiral metamaterials, with optical cavity effect involved in [11,22,23]. Though the multi-layer chiral metamaterials can enhance the AT effect, it increases the complexity and difficulty in fabrication process. In addition, the optical cavity has also been proposed to enhance the optical magnetic field localization and optical polarization conversion in bi-layer metamaterials [24–26].

In this paper, we design the infinite-length metal bar to a bi-layer continuous omega-shaped metamaterial, and then both a hybrid resonance and a Fabry–Perot-like cavity mode are induced at near-infrared band. We demonstrate that the AT effect of linearly polarized light is enhanced attributed to strongly coupling of the hybrid resonance with the optical cavity mode. Besides, the bi-layer metamaterial design has a good robustness to misalignment between layers, which facilitates the fabrication process and provides potential applications in the optical communication field.

2. Structure and theory

Figure 1(a) presents the schematic diagram of the bi-layer continuous omega-shaped metamaterial. The perspective view of a unit cell of the metamaterial is shown in Fig. 1(b), which consists of two layers of omega-shaped structure embedded in a silicon-oxide dielectric environment (${\epsilon }_{si{o}_2}=2.1$). In contrast to the first layer, the structure in second layer is rotated clockwise by ${90}^{\circ }$ about the z-axis, and then AT could be realized by linearly polarized incident lights [27,28]. Here, the omega-shaped structure is considered as gold with the Drude-type dispersion, for which the plasma frequency ${\omega }_p=1.367\times {10}^{16}rad/s$ and collision frequency $\gamma =4.0715\times {10}^{13}rad/s$ [29]. The elevation and side views of the unit cell are shown in Figs. 1(c) and 1(d) with period L = 500 nm. Other geometric parameters of the structure are set as: a = 330 nm, b = 330 nm, w = 80 nm, t = 40 nm and d = 70 nm. Here, the continuous design avoids the diffraction of boundaries, which can decrease the losses and thus enhance the resonance [6]. In this work, the CST Microwave Studio is used for numerical simulations, where periodic boundary conditions are applied to $\pm x$ and $\pm y$ directions with $P_x=P_y=L=500\text{nm}$to account for the continuous omega-shaped metamaterial, while open boundary conditions are used in the directions of propagation ($\pm z$ directions). Here, the mesh cells are more than 1.5 millions and the lower and upper distances in –z and + z directions are both 1500 nm, which are longer then a wavelength. All the designs guarantee the accuracy of simulation results. Then, x- and/or y-polarized incidences are set to propagate along forward ( + z) and/or backward (-z) directions, in order to study the asymmetric characteristic of transmittance.

 

Fig. 1 (a) Schematic diagram of bi-layer continuous metamaterial embedded in a silicon-oxide dielectric environment. (b) A perspective view of the unit cell. (c) and (d) The elevation and side views of the unit cell.

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As we know, the incident and transmitted electric fields for x- and y-polarized electromagnetic waves can be described as follows, when they propagate along + z direction [28,30,31]:

3. Result and discussion

In our numerical calculations, the transmittance is determined by $t_{ij}={\left|T_{ij}\right|}^2$. Figure 2 shows the simulated transmittances $t_{ij}$ of our continuous omega-shaped metamaterial for linearly polarized incident lights propagating along forward and backward directions. For forward propagations [Fig. 2(a)], both the co-polarization transmittance $t_{xx}^f/t_{yy}^f$ and the cross-polarization transmittance $t_{xy}^f$ keep values lower than 0.1 in the wavelength range from 985 nm to 1530 nm, while the other cross-polarization transmittance $t_{yx}^f$ increases to a resonant peak with a maximum of $t_{yx}^f=0.8$ at around 1364 nm. It means that the x-polarized incident light is completely transformed to y-polarized transmittance, while the y-polarized incident light is almost prevented by the bi-layer continuous omega-shaped metamaterial. For backward propagations the conditions are reversed, i.e. the y-polarized incident light is transformed to an x-polarized transmittance with $t_{xy}^b=0.8$ around 1364 nm, but the x-polarized incident light is prevented when it propagates along the backward direction [Fig. 2(b)]. Take x-polarized incident light along the forward propagation for example, the polarization conversion rate (PCR) can be defined as $PCR=t_{yx}^f/\left(t_{yx}^f+t_{xy}^f\right)$ [5,21,32]. Around 1364 nm, the transmittances satisfy $t_{yx}^f=0.8\text{and}{t}_{xx}^f=0.016$, therefore, we can get the PCR as high as 0.98. Analogously, the PCR is also 0.98 for y-polarized incident along the backward propagation. As a result, we get the high-efficiency asymmetric polarization conversions (i.e. the continuous omega-shaped metamaterial can act as a polarization converter converting x/y-polarized light to y/x-polarized one along forward/backward direction.).

 

Fig. 2 Simulated transmittances for bi-layer continuous omega-shaped metamaterial, when x- and y-polarized lights incident along (a) forward and (b) backward directions, respectively.

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Based on above discussions, we get that the co-polarization and cross-polarization transmittances satisfy $t_{xx}=t_{yy}$and $t_{xy}\ne t_{yx}$. Figures 3(a) and 3(b) present the total transmittances along the forward and backward propagations, calculated by $t=t_{ij}+t_{jj}$. The result shows clearly that $t_x^f=t_y^b$ and it reaches a maximum of 0.82 around 1364 nm, but for $t_y^f(=t_x^b)$ the value decreases to 0.02. Such a high intensity contrast between forward and backward directions has not been reported in previous bi-layer metamaterials [1,18,19]. As a result, the x-polarized incident light is mostly transmitted in forward propagation, while the y-polarized incident light is significantly prevented. Inversely, the y-polarized incident light could be transmitted in the backward direction. Mathematically, AT parameters (${\Delta }_{lin}$) for linearly polarized incident waves are presented as follows [3]:

 

Fig. 3 The calculated total transmittances for forward and backward propagation directions with (a) x- and (b) y-polarized incident lights, respectively. (c) AT effect for the designed bi-layer metamaterial. Blue solid and green bashed lines correspond to AT of x- and y-polarized lights, respectively.

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To understand the physical mechanism of the enhanced AT effect, we start studying the resonance induced by the bi-layer continuous omega-shaped metamaterial. We present the electric-field distributions on the first and second layers, at the resonant wavelength of 1364 nm for x- and y-polarized incident wave along the + z direction, respectively (Fig. 4). Figure 4(a) show that, when x-polarized wave incident normally on the first layer, a hybrid resonance mode is induced, including the parallel electric-field distributions in x-axis arms and antiparallel electric-field distributions in y-axis arms. Note that this hybrid resonance in such folded metal bars is totally different from any mode existed in straight metal bars. More importantly, for our bilayer omega-shaped design most of the electromagnetic wave is coupled into the second layer and the similar hybrid resonant mode can be observed on the second layer [Fig. 4(b)] [18]. Consequently, the chiral arrangement of this bi-layer metamaterial and strongly coupling effect result in a high intensity cross-polarized resonance with t_yx = 0.8, and thus the x-polarized incident light is transformed to y-polarized transmittance. Meanwhile, Figs. 4(c) and 4(d) show that there are hardly any resonances induced by the continuous omega-shaped metamaterial, for y-polarized incidences. Besides, the wavelength (1364 nm) here is much larger than the structural gaps, then the y-polarized incident light cannot pass through the metamaterial directly. As a result, there is nearly no transmittance for y-polarized incident wave. However, the inverse phenomenon can be observed when x- and y-polarized lights are incident along the -z direction. At the last, we can conclude that the enhanced cross-polarization conversion and AT effect are realized at near-infrared band.

 

Fig. 4 Electric-field magnitude distributions at resonant wavelength for continuous omega-shaped metamaterial. (a) and (b) correspond to the x-polarized wave incident along the + z direction, (c) and (d) correspond to the y-polarized wave incident along the + z direction.

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Figure 5(a) shows that the continuous omega-shaped metamaterial is separated by a gap (g = 60 nm), which breaks the hybrid resonance in continuous metamaterial. Interestingly, the forward transmittances in Fig. 5(a) under x- and y-polarized incident lights is completely different from Fig. 2(a), which implies that the gap plays an important role in the structural resonance of this uncontinuous omega-shaped metamaterial. The transmittances in Fig. 5(a) show that the enhanced cross-polarized resonance around 1364 nm is lost. Moreover, different resonant modes are induced and the values of cross-polarization t_yx can only reach 0.3. Inversely, the co-polarizations t_xx and t_yy show a relatively large transmittance. For the uncontinuous omega-shaped metamaterial, the co-polarization transmittances satisfy $t_{xx}=t_{yy}$with a maximum value about 0.47, and for the cross-polarization transmittances ($t_{xy}\ne t_{yx}$), it is obvious that t_yx can only reach up to 0.3 and t_xy is approximately 0. Accordingly, the AT parameter is not more than 0.3, as presented in Fig. 5(b).

 

Fig. 5 (a) Simulated transmittance when the continuous omega-shaped metamaterial is separated (not connected with each other) with a gap g = 60 nm. Inset: Schematic diagram of bi-layer uncontinuous omega-shaped metamaterial. (b) The corresponding AT parameter ${\Delta }_x$.

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Then, we studied the cavity coupling effect by adjust the cavity length d. Figure 6(a) shows that the enhanced AT effect as a function of cavity length d and the AT value can reach up to a maximum at d = 70 nm, when x-polarized incident light propagates along the forward direction. Figures 6(b) and 6(c) show the electric vector distributions for x- and y-polarized light incident along the + z direction. Figure 6(b) shows clearly that, for x-polarized incidence, the hybrid resonance is strongly coupled to the bilayer cavity and enhanced the transmittance of t_yx, when d = 70 nm. At this case, the x-polarized light almost completely passes through the bi-layer metamaterial, with its polarization transformed to the y-direction. While for y-polarized incident light, the hybrid resonance cannot be induced, and there is no cavity coupling phenomenon [Fig. 6(c)]. Then, the y-polarized incident light almost completely reflected by the continuous omega-shaped metamaterial.

 

Fig. 6 (a) The AT parameter as a function of cavity length d. (b) and (c) Electric vector distributions of the bi-layer omega-shaped metamaterial, corresponding to d = 70 nm.

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For this work, we consider the proposed bilayer structure as a Fabry-Perot-like cavity to explain the enhanced AT effect further [3,11]. Here, the hybrid plasmonic resonance in the folded bar structure is coupled to the bilayer cavity in deep subwavelength, which contributes to the novelty of our cavity as compared with a traditional Fabry-Perot-type cavity. Figures 7(a) and 7(b) present the schematic of the Fabry-Perot-like resonance. It shows clearly that when x-polarized wave incident normally on the metamaterial, the transmission, reflection, and coupling at resonant wavelength can be enhanced in the cavity. Here, the comparison between reflectance and transmittance are presented in Fig. 7(c), and it clearly demonstrates that the reflectances for cross-polarization (r_yx) and co-polarization (r_xx) keep lower than 0.02 around 1364 nm. However, the transmittance for cross-polarization (t_yx) reaches up to 0.8 in contrast to a low co-polarization transmittance (t_xx). Then, most of the x-polarized lights are transformed to y-polarization and pass through the second layer while nearly none of the x-polarized lights are reflected by this metamaterial. In addition, the y-polarized light is almost completely reflected when it is incident on the first layer of the metamaterial, which results in the weak interaction in cavity and nearly zero transmittance [Fig. 7(b)]. Figure 7(d) shows that co-polarization reflectance (r_yy) keeps about 0.9 in a broad band covering the resonant range, and the other components both for reflectance and transmittance can be considered as 0. As a result, the bi-layer continuous omega-shaped metamaterial just acts as a reflected mirror [Fig. 7(b)].

 

Fig. 7 (a) and (b) Schematic diagram of Fabry–Perot-like resonance based on the bi-layer omega-shaped metamaterial. Note the solid lines mean strong transmission/reflection processes, while the dotted lines mean weak ones. (c) and (d) the transmittance and reflectance for the bi-layer metamaterial, when x- and y-polarized lights incident along the + z direction.

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As we know, for a bi-layer metamaterial, there are kinds of fabrication errors and misalignments to influence the AT efficiency. Then we modulate the geometric parameters and bilayer misalignments, for which the results are presented in Fig. 8. When parameter a increases from 280 nm to 450 nm, the capacitance increases, and thus the resonant frequencies suffer from a large redshift while keeping the high-efficiency AT. However, when b increases from 280 nm to 450 nm, the period of the infinite-length omega-shaped metamaterial is changed, which influenced the intensity of hybrid resonance. Then resonant peaks reach a maximum at b = 330 nm, but the resonant frequencies change moderately. In addition, the resonant peaks reach a maximum at w = 80 nm with w increasing from 40 nm to 140 nm. Figure 8(d) shows that when the transversal and longitudinal misalignments are increased from 25 nm to 50 nm, the maximum AT values have a slightly decrease from 0.8 to 0.75. And, when the transversal and longitudinal misalignments reach $\delta x=\delta y=50\text{nm}$, the AT parameter still has 0.7. Therefore, it is safe to evaluate that the AT effect is insensitive to transversal and longitudinal misalignments. To the last, the collision frequency $\gamma$ of the considered metal (gold) is modulated to account for the unexpected material absorption in experiments. We increase $\gamma$ from $4.1\times {10}^{13}$to $12.2\times {10}^{13}$and find that the AT parameter occurs a decrease from originally 0.8 to 0.56 without resonant frequencies changing [Fig. 8(e)]. At near-infrared region, 0.56 can also be considered as an efficient AT parameter comparing with previous reports [13,18–21].

 

Fig. 8 Calculated AT parameter ${\Delta }_x$ of the bilayer chiral metamaterial for different a, b, w, the misalignments $\delta x$, $\delta y$ and the collision frequency $\gamma$.

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4. Conclusion

In this paper, we proposed a bi-layer continuous omega-shaped metamaterial, which highly enhanced the AT effect for linearly polarized light at near-infrared communication band. Here, the continuous omega-shaped metamaterial provides a hybrid resonant mode and the bi-layer design presents a Fabry–Perot-like cavity. By considering the strongly coupling between the hybrid resonance and Fabry–Perot-like cavity, we optimize the parameters of our structure. as a result the maximum parameter of AT is up to 0.8, which destroys the limitation that only tri-layer or multi-layer designs could realize such high AT effect at the near-infrared band. Moreover, the bi-layer continuous omega-shaped metamaterial provides a great convenience in experiments, and then it has potential applications in designing polarization rotators and switches in optical communication systems.