1 Introduction

Metamaterials (MMs) are periodical artificial media with a pitch smaller than the wavelength of interest and can exhibit unusual properties not found in their constituent materials [1]. By delicate designs, many fascinating functional devices can be developed based on MMs structures, such as perfect lens [2], invisible cloaking [3] and negative refractive index [4]. The concept of chirality in MMs is defined as the lack of mirror image symmetry [5]. Chiral metamaterials (CMMs) possess strong ability to rotate the plane of polarization of electromagnetic (EM) waves; this ability is called optical activity which is of great interest to many areas of science, for example, analytical chemistry and molecular biology. Recently, many CMMs have been proposed to realize negative refractive index [6], polarization rotators [7–9] and polarization spectrum filters [10]. Three-dimensional CMMs can result in coupled magnetic and electric dipole type responses, due to the currents around the loops and along the structure [11]. However, three-dimensional CMMs are difficult to fabricate, especially when scaled down for use at terahertz and optical frequencies. In contrast to the three-dimensional CMMs (e.g., helices), planar chiral patterns (e.g., flat spirals) have the unique property that their sense of twist is reversed for observation from opposite sides. Not only human observers, but also circularly polarized waves incident on opposite sides of a planar chiral structure, see materials of opposite handedness [12–14]. It was recently discovered that CMMs with strong symmetry breaking could exhibit the intriguing phenomenon of asymmetric transmission (AT) of light for circular [12, 15–20] and linear polarizations [21–23]. Subsequently, experiments verified that the AT effect can also be observed in extrinsically CMMs, if chirality is associated with the mutual orientation of the planar MMs and the wave propagation direction [16, 23]. More recently, great efforts have been put to achieve the AT effect for linearly polarized waves [24–30]. Interestingly, diode like AT of linearly polarized waves based on combining chirality and EM wave tunneling has been suggested, which demonstrated a nearly total intensity transmission at normal incidence in one direction and a small intensity transmission in the opposite direction [22].

In this paper, we propose a new dual-band AT for forward and backward propagating linearly polarized waves using Π-shaped chiral metamaterial (Π-CM). The proposed Π-CM is composed of two planar Π-shaped metallic layers separated by a single dielectric layer. The simulated and experimental results demonstrate that the proposed Π-CM is nearly transparent to incident linearly polarized waves in two separated AT bands, where a linearly polarized wave can be converted to its cross-polarization and then transmitted, while the same linearly polarized wave cannot pass through the Π-CM in the opposite direction. Each AT band can be switched on/off by changing the polarization state of the incident wave. The novel property of the Π-CM is similar to the diode-switching characteristics.

2 Theory, simulation and experiment

CMs with the exhibition of cross-coupling and cross-polarization conversion generally result in AT for forward and backward propagation of EM waves over certain frequency ranges. This AT provides a certain difference between the polarizations of the waves applied in opposite sides. For the theoretical analysis, we consider an incoming plane wave that propagates in +z direction with the time dependence of e −iωt which is suppressed through this work [16, 23]:

$$ \varvec{E}_{i} \left( {\varvec{r},t} \right) = \left( {\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ \end{array} } \right)e^{ikz} $$
(1)

with angular frequency ω, wave vector k and complex amplitudes E x and E y are the complex amplitudes of the normally incident plane wave polarized in x and y directions along the propagating direction, respectively. To understand polarization conversion, transmission matrix (T) can be applied for the given electric field as follows:

$$ \varvec{E}_{t} \left( {\varvec{r},t} \right) = \left( {\begin{array}{*{20}c} {T_{x} } \\ {T_{y} } \\ \end{array} } \right)e^{ikz} $$
(2)

The T matrix relates the complex amplitudes to the incident fields to that of the transmitted fields:

$$ \left( {\begin{array}{*{20}c} {T_{x} } \\ {T_{y} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {t_{xx} } & {t_{xy} } \\ {t_{yx} } & {t_{yy} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ \end{array} } \right) = T_{lin}^{f} \left( {\begin{array}{*{20}c} {E_{x} } \\ {E_{y} } \\ \end{array} } \right) $$
(3)

The superscript f and subscript lin indicate the propagating EM wave along +z (forward) direction and the specific linear base. According to the reciprocity theorem, the −z (backward) propagation can be derived as:

$$ T_{lin}^{b} = \left( {\begin{array}{*{20}c} {t_{xx} } & { - t_{yx} } \\ { - t_{xy} } & {t_{yy} } \\ \end{array} } \right) $$
(4)

The superscript b indicates the propagating EM wave along −z (backward) direction. In an ideal AT, in one direction the transmission is unity while in the opposite direction the transmission is zero. This requires the diagonal components of the T matrix to be zero, and one of the off-diagonal components to be zero while the other is unity, such that:

$$ T_{lin}^{f} = \left( {\begin{array}{*{20}c} 0 & 1 \\ 0 & 0 \\ \end{array} } \right)\quad T_{lin}^{b} = \left( {\begin{array}{*{20}c} 0 & 0 \\ { - 1} & 0 \\ \end{array} } \right) $$
(5)

The AT of the linearly polarized waves is usually characterized by the AT parameter Δ, which is defined as the difference between the transmitted coefficients of cross-polarized waves:

$$ \varDelta_{lin}^{x} = \left| {t_{yx}^{f} } \right|^{2} - \left| {t_{yx}^{b} } \right|^{2} = \left| {t_{yx}^{f} } \right|^{2} - \left| {t_{xy}^{f} } \right|^{2} = - \varDelta_{lin }^{y} $$
(6)

Therefore, the AT for linearly polarized waves is rigorously judged by Eqs. (5) and (6). It has been pointed out that a one-layered metallic structure (a planar structure) is not enough to achieve the AT of linearly polarized waves because the off-diagonal components of its T matrix are equal (t xy  = t yx ). Thus, we must use multilayered metallic structures to accomplish this task. According to this idea, we have designed the Π-shaped metallic layers on both sides of a dielectric layer rotated relative to each other at an angle of 90° around the z-axis [30, 31].

Figure 1a, b shows a part of the photograph of the sample used in the experiment and the schematic diagram of one unit cell, respectively. The metallic patterns on both sides of a dielectric layer are identical, but are rotated relative to each other at an angle of 90° around the z-axis, as shown in Fig. 1(b). It can be seen that the Π-CM structure lacks fourfold rotational symmetry, and the mirror symmetry in z direction is broken. Therefore, the Π-CM will exhibit an AT effect for a linearly polarized wave due to chiral property [32, 33]. The Π-shaped structures are placed on both sides of the Rogers RO4003 substrate with a relative permittivity of 3.38 and a dielectric loss tangent of 0.0027. The metallic patterns cladding on the Rogers RO4003 substrate are copper with the thickness of 30 μm. The optimal parameters are as follows: a = 12 mm, t = 0.813 mm, W = 1.2 mm, L = 4.8 mm, h = 6.8 mm, g = 0.4 mm, d 1  = 1.4 mm and d 2  = 2.4 mm, which is shown in Fig. 1b and c.

Fig. 1
figure 1

a A part of the photograph of the sample. b The unit cell in stereogram. c The front planar structure of unit cell

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We started the analysis with numerical simulations using CST MICROWAVE STUDIO 2012 (High-frequency electromagnetic simulation software which is based on finite-integration time-domain method). In the simulations, the transmittances from a single unit cell with periodic boundary condition are obtained; EM waves propagate along the z direction (The linearly polarized wave incident normal to the sample surface). The electric field polarization is kept along the x-axis, and magnetic field polarization is kept along the y-axis. For experiments, we fabricated the structure with the dimension of 15 by 15 unit cells by the conventional printed circuit board (PCB) process with the same structural parameters as the simulation model, and a part photography of the fabricated Π-CM slab is shown in Fig. 1a. Agilent E8362B vector network analyzer connected to the two standard gain broadband linearly polarized antenna that produced microwaves in the range of 9–16 GHz was employed to measure the Π-CM slab in an EM anechoic chamber. We could obtain all components of the EM wave transmittances for different polarizations by changing the orientation of the two antennas.

3 Results and discussions

Figure 2 shows the simulated and measured transmissions in the Π-CM for the forward and backward propagating waves. The co-polarization transmission coefficient t xx of x-polarized wave coincides well with t yy of y-polarized wave when an orthogonal arrangement of two subwavelength resonators ensures that the whole structure is isotropic for observations at normal incidence [26]. Compared to the co-polarization transmission, the cross-polarization coefficient t xy is extremely different from t yx at all frequencies. The two above conditions indicate the presence of the AT effect for linearly polarized waves and the absence of the AT effect for circularly polarized waves in the Π-CM [18]. Moreover, there are two individual passbands showing strong optical activity in the Π-CM with a total thickness of about λ/31, one for x-to-y polarization conversion and the other for y-to-x polarization conversion.

Fig. 2
figure 2

Transmission spectra for propagating waves a cross-polarization along forward direction b co-polarization along forward and backward directions c cross-polarization along backward direction

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For the forward propagation, the simulated cross-polarization transmission t yx reaches a maximum of 0.93 at around 10.82 GHz, and both the co-polarization transmissions t xx and t yy are below 0.103, which is shown in Fig. 2a and b, In this passband, incident x-polarized wave is almost completely transmitted to y-polarized wave while incident y-polarized wave is completely blocked through the metamaterial slab. On the contrary, an obvious resonant peak in t xy can be observed with a maximum larger than 0.90 at around 14.1 GHz, where incident y-polarized wave is almost completely transmitted to x-polarized wave while incident x-polarized wave cannot pass through the metamaterial slab. In addition, t yx and t xy interchange with each other shown in Fig. 2c when the propagation direction is reversed. The measured cross-polarization transmission t xy reaches a maximum of 0.91 at the resonant frequency of 10.86 GHz, and t yx can be observed with a maximum larger than 0.90 at the resonant frequency of 14.03 GHz along the forward direction.

For the backward, the simulated cross-polarization transmission of t yx is 0.91 at the resonant frequency of 14.03 GHz, and of t xy is 0.90 at the resonant frequency of 10.86 GHz along the forward direction. The measured results are in good agreement with the simulated ones. The measured and simulated co-polarization transmissions t xx and t yy basically agree with each other shown in Fig. 2b, except for a slight shift. The slight frequency discrepancy may be due to the fabrication tolerance as well as the dielectric board material whose actual dielectric constant is slightly different from the value used in the simulations. In addition, these discrepancies may be due to the following two reasons: Firstly, in the simulations by using CST microwave studio, periodic boundary conditions are set, which means the physical size of the proposed absorbers is infinite. However, the dimension of the fabricated sample in the experiments is finite, thereby resulting in the edge diffraction. Hence, edge diffraction can create discrepancies between the simulated and experimental results. Secondly, experimental conditions can also induce discrepancies such as background interference and so on. Therefore, we cannot ignore these reasons leading to the discrepancies between the simulated and measured spectra. In spite of these discrepancies, the experimental measurements agree well with the simulation results, we can conclude that the agreement between the numerical and experiment results is good and the operation of the Π-CM is verified experimentally.

In order to visualize the AT effect, Fig. 3 presents the simulated and measured AT parameter for forward propagating x and y-polarized waves according to Eq. (6). It can be seen clearly that both the asymmetry factor curves ∆x and ∆y show two opposite peaks locating at about 10.82 and 14.1 GHz. Two values of 0.86/−0.86 and −0.85/0.84 in the curves, ∆x and ∆y imply that the x/y-polarized wave in the forward direction is mostly forbidden/allowed at about 10.82 GHz and allowed/forbidden at 14.1 GHz, respectively. The measured ATs agree well with the simulated ones, which are currently 0.84 at 10.89 GHz and 0.86 at 14.03 GHz. It is clearly demonstrated that two curves of ∆x and ∆y are exactly contrary to each other. Apparently, the Π-CM reveals a dual-band transmission for linearly polarized waves, one for x-polarized wave and the other for y-polarized wave [23].

Fig. 3
figure 3

Asymmetric transmission parameters of x and y linearly polarized waves in the forward propagation direction

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The polarization conversion results from strong electric and magnetic response, which creates a component of a retarded wave with a polarization perpendicular to that of the incident wave and then rotates the polarization plane of the transmitted wave [34, 35]. We present the surface current distributions of two metallic layers at two resonant frequencies to look into the fundamental principles, when propagating through the structure around the resonant frequency. Figure 4 shows the instantaneous induced surface current distributions of the top and bottom metallic layers for the waves passing through the structure along the +z direction at the resonant frequencies of 10.82 and 14.1 GHz, respectively. Figure 4a depicts the current distribution of the top and bottom layers of Π-CM at 10.82 GHz, it can be seen that the surface currents are antiphase between the top and bottom metallic layers. The incident wave excites two current loops (blue and purple arrows) at the top and bottom metallic layers and thus results in two magnetic dipole moments m 1 and m 2 . Meanwhile, the other two electric dipoles p 1 and p 2 (black and red arrows) at the top and bottom metallic layers are excited, which also can result in polarization conversions. The induced magnetic field H 1 (occurred by m 1 ) is perpendicular to the incident electric field E; thus, there is no cross-coupling between H 1 and the electric field E, and H 1 never contributes to the polarization conversion. On the contrary, the induced magnetic field H 2 (occurred by m 2 ) is along the x-axis, which is paralleled to the electric field E; thus, the cross-coupling between the electric field E and the induced magnetic field H 2 leads to a cross-polarization with x-to-y polarization conversion. The electric dipoles p 1 and p 2 can induce the electric field E 1 and E 2 , respectively. The electric field E 1 is perpendicular to the incident electric field E, and the electric field E 2 parallels to the incident electric field E. Thus, the cross-coupling between the electric field E and the induced electric field E 1 leads to a cross-polarization with x-to-y polarization conversion while no cross-coupling between E 2 and the electric field E. At the frequency of 14.1 GHz, electric dipole moments are excited in the same way shown in Fig. 4b. However, the polarization conversions are mainly excited by electric dipole moments p II ; the direction of the induced electric field E II perpendicular to the electric field E, and the cross-coupling between the electric field E and induced electric field E II leads to a cross-polarization with y-to-x polarization conversion. There is no cross-coupling between E and the induced field E I, which parallels to the incident electric field E.

Fig. 4
figure 4

Surface current distributions a incident x-polarized waves at 10.82 GHz b incident y-polarized waves at 14.1 GHz along +z direction

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On the other hand, a direct way to observe the evolution of polarization conversions inside the structure is the electric field distribution for x-polarization and y-polarization waves passing through the designed slab forward. Figure 5 illustrates the snapshots correspond to the electric field distributions for the EM wave in the incoming and outgoing regimes as well as inside the designed Π-CM. From Fig. 5a, it can be seen clearly that the field patterns are twisted inside the slab as a consequence of interlayer coupling for forward transmission at 10.82 GHz, the outgoing wave is y-polarized even when the incident wave is x-polarized. This is because only the y-polarized wave component is allowed to pass through the x-oriented Π-CM at the outgoing surface at 10.82 GHz. On the contrary, the outgoing wave is x-polarized when the incident wave is y-polarized at the resonant frequency of 14.1 GHz, which is shown in Fig. 5b. This observation supports that the AT is caused by the resonant polarization convention together with the bianisotropic Π-shaped structures with a different coupling to orthogonal polarizations.

Fig. 5
figure 5

Calculated electric field distributions a at 10.82 GHz for an x-polarized wave incident b at 14.1 GHz for an y-polarized wave incident along forward directions

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4 Conclusion

In conclusion, we have proposed a bilayered chiral metamaterial that is composed of Π-CM arranged by a twist angle of 90°. The Π-CM has been theoretically and experimentally demonstrated to reveal a dual-band AT effect for two orthogonal linearly polarized waves. AT effect of linearly polarized waves along two opposite propagation directions occurs, and the Π-CM can reveal a reasonable polarization conversion with over 90 % conversion efficiency for both x- and y-polarized waves. It can also be predicted that the Π-CM can be functionalized as a dual-band polarization spectral filter for incident unpolarized light sandwiched between two orthogonal polarizers, underpinned by optical activity phenomenon. The ideas of the suggested design can be adapted in future research for terahertz and optical applications.