1. Introduction

Metamaterials, a kind of artificially constructed substances, have peculiar electromagnetic (EM) responses that have not been observed in materials existing in nature. This capability provides an alternative approach to manipulate the propagation of EM radiation, and has already led to many intriguing applications, such as subdiffraction imaging [1,2], EM cloaking [3], and negative refraction [4]. Recently, a lot of efforts have focused on the polarization-sensitive asymmetric transmission effect in metamaterials [5–12], where the partial conversion of the incident EM radiation into the one of opposite handedness is asymmetric for the opposite directions of propagation [5]. This asymmetric transmission phenomenon is irrelevant to the nonreciprocity of the Faraday effect in magneto-optical media [13], while originates from the interaction of EM radiation with sophisticated chiral metamaterial structures [14], including chiral patterns [5,8,9], chiral asymmetrically split rings [6,7,11], and chiral plasmonic meta-molecules [10]. Many unique characteristics of the asymmetric transmission are discussed. For instance, the chiral metamaterials support corotating elliptical polarizations [5,9], associated with polarization-sensitive electric and magnetic dipolar responses [6] or enantiomerically-sensitive plasmon resonance [8,11]. However, due to the mirror symmetry of the metamaterials (i.e., there is a mirror plane perpendicular to the propagation direction) [5–12], the asymmetric transmission is absent for linear polarization in those structures. Very recently Menzel et al. [12] demonstrated the asymmetric EM transmission for both circular and linear polarizations in a hybridized three-dimensional chiral metamaterials. However, to the best of our knowledge, there is no report on the asymmetric transmission for linearly polarized EM radiation only.

In the present article, we investigate the feasibility of realizing asymmetric transmission for linearly polarized EM radiation only in metamaterials, based on our developed criterion on the relationship among the elements of transmission matrix. We show that the broken mirror symmetry in the propagation direction is crucial to achieve this effect. Compared with Ref. [12], from a 4 × 4 matrix analysis based on the classical model of optical activity pioneered by Born and Kuhn [15–17], we reveal that hybridized metamaterials containing two coupled planar metamaterials, with anisotropic or chiral patterns, can satisfy this criterion. We briefly discuss the situation when the hybridized sample rotates around the propagation direction. Based on our developed criterion, we present a simple metamaterial design, with a half-sauwastika and a half-gammadion shaped chiral patterns. Numerical simulations prove our proposed criterion. In addition, we also discuss the dispersions of the eigenstates and the influence of off-angle and non-collimated incidence.

2. Theory

Firstly, let us devote to finding the criterion on asymmetric transmission for linearly polarized EM radiation only. The incident and transmitted EM radiations can be related by a transmission matrix T [5,12]. Considering an incident EM field E ⁰ propagating in the +z direction, the electrical field E ^T of the transmitted radiation can be embodied by E ^T = TE ⁰, where T is the 2 × 2 transmission matrix with elements t_uv. The suffixes u and v correspond to the states of polarization of the transmitted and incident fields, respectively. Here, the states of polarization could be either circularly polarized (described by r and l for right- and left-circular polarizations, respectively) or linearly polarized (described by x and y, respectively).

In the Cartesian coordinate system, the transmission matrix in the +z direction for linear polarization basis, ${\mathbf{\text{T}}}_{\text{lin}}^{+}$, can be expressed as

A parameter Δ, defined as the difference between the transmittances in the two opposite propagation directions (+z and −z) [5,11,12], can be used to represent the degree of asymmetric transmission. As is well known, when the propagation direction is reversed, for a system satisfying reciprocity theorem [12,15], the off-diagonal elements t_xy and t_yx not only interchange their values, but also get an additional π phase shift [12,18]. Thus for the linear polarization case, we have

While for the circular polarization case, the parameter Δ is in the form of

From Eqs. (3) –(5), we can easily find that the asymmetric transmission for linear polarization only (i.e. Δ_circ = 0 and Δ_lin ≠ 0) can be realized, so long as the following condition as a criterion is satisfied

We have to emphasize here that the asymmetric transmission for the linear polarization case depends strongly on the orientation of the sample around the propagation direction. At certain orientation angles, the asymmetric transmission disappears [12]. More explicitly, for a metamaterial satisfying the criterion of Eq. (6), if we rotate it by an orientation angle of θ around the z direction, Δ_lin will become to

We now briefly discuss the rule satisfying the criterion of Eq. (6) in metamaterial. For any single planar metamaterial, it has always a mirror symmetry in the propagation direction, so the elements of the transmission matrix always support t_xy = t_yx, implying that the transmission is symmetric for linear polarization. Therefore, using hybridized metamaterials to break the mirror symmetry of the structure in the propagation direction is the unique route for achieving the asymmetric transmission for linear polarization. A good example is the three-dimensional chiral meta-atom proposed in Ref. [12], which is composed of planar L-shaped and cut-line shaped elements. It supports t_xy ≠ t_yx, so the transmission for linear polarization is asymmetric. But since t_xx ≠ t_yy, the transmission for circular polarization is also asymmetric [12].

To understand the underlying physics behind the asymmetric transmission in hybridized metamaterials, we can resort to the classical model of optical activity pioneered by Born and Kuhn [15–19]. In this model, each planar metamaterial could be visualized by two spatially separated charged harmonic oscillators moving along the two orthogonal directions of x and y, with effective displacements of d_x and d_y, respectively. Energy of an oscillator excited by the incident EM radiation could be transferred to the other by an elastic coupling between them. The induced oscillation of charges then re-emit the EM radiation to ensure the optical activity. To describe the hybridized metamaterial composed of two planar metamaterial components A and B, we can resort to the operation below [20]

We should pointed out that Eq. (9) has taken into account the mutual coupling effect between the two planar metamaterials. We can directly find the strength of the output EM radiation E_u^j from the dipole oscillations induced in the planar metamaterial j when the EM radiation is incident from the planar metamaterial i, which is proportional to the corresponding effective displacements d_u^j, by solving Eq. (8). Thus the transmission matrix elements can be evaluated. To be more explicitly, under the initial excitation of E = [1,0,0,0], t_xx and t_yx can be found by solving Eq. (8), as follows

With the above consideration, we easily give the elements of ${\mathbf{\text{T}}}_{\text{lin}}^{+}$ for the linear polarization case

Referencing Eq. (9), if the hybridized metamaterial exhibits the mirror symmetry in the propagation direction z, the following conditions are simultaneously satisfied,

Indeed, hybridization can considerably enrich the functionalities of metamaterials, especially in the symmetry-broken metamaterials. Usually, it is quite difficult to find the general solution of Eq. (8). However, we can find some particular solutions satisfying the criterion of Eq. (6). To the best of our knowledge, there has no report on this criterion and the realization of asymmetric transmission for linear polarization only. Here we propose a solution, as follows

3. Numerical simulations

To validate the feasibility of the above discussion, we adopt the CGN500-NF-3006 commercial printed circuit board (PCB) [20,21] to design the metamaterial structure. The PCB is composed of a dielectric substrate sandwiched by two copper layers. The substrate has a 2.3 in permittivity, 0.0008 in dissipation factor, and d_s = 0.5 mm in thickness. The two copper layers have the same thicknesses of t_s = 18 μ m and the conductivity is 5.8×10⁷ S/m. In the hybridized metamaterial structure we designed, as shown in Fig. 1, the patterns in the two cooper layers are easily fabricated by using the commercial photolithography technique. Each unit cell has a dimension of d × d = 6 × 6 mm², which ensures that under normal incidence the metamaterial structure does not produce high-order radiative diffraction for frequency below 50 GHz. The unit cell contains two simple structures in the form of a half-sauwastika in the top layer and a half-gammadion in the bottom layer, with parameters b = 4 mm, a = 2 mm, and w = 0.2 mm. As shown in Fig. 1, the unit cell of the whole hybridized metamaterial consists of two cooper layers separated by the isotropic dielectric substrate. The cooper pattern in any layer can be considered as a chiral planar metamaterial. In fact, to realize the requirements of Eq. (14) , the pattern in the bottom layer can be simply designed from that in the top layer (as a half-gammadion) by rotating 90° around the center z-axis. And then the pattern in the bottom layer is further performed a mirror operation about y axis again, to form the shape of a half-sauwastika. This operation does not violate the requirements of Eq. (14) , while can facilitate the achievement of an efficient coupling between the two chiral planar metamaterials and to enhance the EM response. Obviously, if the two patterns are plotted together in forming a single planar metamaterial, the structure will possess two additional mirror planes forming angles of ±π/4 with respect to the yz plane, respectively. Since the two cooper patterns are segregated by the middle dielectric substrate, the mirror symmetry of the whole structure is broken in the propagation z direction and in the xy plane.

 

Fig. 1 Geometry of the unit cell of the hybridized metamaterial we designed. (a) top layer with a half-sauwastika that can have a lateral displacement g, (b) bottom layer with a half-gammadion, (c) profile view, and (d) the mutual coupling diagram in the whole structure, A and B indicate the top and bottom layers, the thick red and blue arrows (other two thick arrows) stand for the induced orthogonal dipole oscillators in the top (bottom) copper pattern, and the dashed lines represent the coupling processes among induced dipole oscillators.

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To understand the physics of the interaction of the EM radiation with the structure from the Born-Kuhn model, we give a diagram in Fig. 1(d). The pattern structure in each layer could be regarded as two orthogonal dipole oscillators. There have six types of the coupling processes among the induced dipole oscillators, as shown by the dashed lines. The EM response in such a metamaterial structure with the peculiar arrangement can satisfy the requirement of Eq. (14) , thereby the asymmetric transmission for linear polarization only can be realized in the proposed metamaterial.

We simulate the transmission behavior of the hybridized metamaterial under normal incidence over a broad frequency range, by using the finite-difference time-domain (FDTD) method [20,21]. The perfect-matched-layer boundary condition in the propagation z direction and the periodical boundary condition in the transverse xy plane are adopted.

We first investigate the single planar metamaterial with the half-sauwastika pattern. The FDTD simulation results shown in Fig. 2 reveal that the matrix elements are the maximum amplitudes at a resonant frequency of 16.57 GHz. It can be found that the metamaterial supports the relations of t_xx ≠ t_yy and t_xy = t_yx, implying that this structure exhibits the mirror symmetry in the propagation z direction, that is to say, the transmission for linear polarizations is symmetric.

 

Fig. 2 Amplitude of the transmission matrix elements for single planar metamaterial in the forward (+z) direction.

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We then focus on the hybridized metamaterial. From the simulation results shown in Fig. 3(a), we can now see the resonant frequency is shifted to 14.24 GHz, which originates from the coupling effect between the top and bottom planar metamaterials. The most interesting characteristics, as we expected, is that the diagonal elements of the transmission matrix now become identical (including their phases, not shown here), i.e., t_xx = t_yy. The off-diagonal elements are no longer the same, i.e., |t_yx| ≠ |t_xy|. These characteristics are in agreement with our analysis in Eq. (14) based on the Born-Kuhn model. To confirm the criterion of Eq. (6) and the presence of asymmetric (symmetric) transmission for linear (circular) polarizations, we calculate the parameter Δ by the FDTD simulation as shown in Fig. 3(b). We can see that ${\Delta }_{\text{lin}}^{(x)}$ reaches the maximum value of 45% at the resonant frequency, ${\Delta }_{\text{lin}}^{(y)}$ is opposite to ${\Delta }_{\text{lin}}^{(x)}$, and ${\Delta }_{\text{cir}}^{(l,r)}$ are always zero. The FDTD simulation results validate that the Born-Kuhn model can indeed predict the transmission properties of the hybridized planar metamaterials well. To further confirm this feature, the inset of Fig. 3(b) plots the dependence of ${\Delta }_{\text{lin}}^{(x)}$ and ${\Delta }_{\text{lin}}^{(y)}$ on the orientation angle θ for linear polarization at 14.24 GHz. Evidently, ${\Delta }_{\text{lin}}^{(x)}$ and ${\Delta }_{\text{lin}}^{(y)}$ have both the same oscillation period of π. At the critical angles of θ_c = π (2n+1)/4 (where n = 0, 1, 2 and 3), ${\Delta }_{\text{lin}}^{(x)}$ and ${\Delta }_{\text{lin}}^{(y)}$ become zero, as expected from Eq. (7).

 

Fig. 3 (a) Amplitude of the transmission matrix elements for the hybridized metamaterial we designed in the forward (+z) direction. (b) The degree of asymmetric transmission in the linear and circular polarization bases. Inset plots the variation of ${\Delta }_{\text{lin}}^{(x,y)}$ versus the orientation angle θ at 14.24 GHz.

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For the transmission properties of metamaterials, the linearly and circularly polarized EM radiations are frequently investigated. In fact, the metamaterials also support the EM eigenstates with the special states of polarizations [5,12], such as the elliptical polarization [5]. For the hybridized metamaterial investigated here, the two eigenstates can be represented by (1 + A ²)^−1/2[1,±Ae^iδ ] in the linear polarization basis, where A = (|t_yx|/|t_xy|)^1/2 is the amplitude and δ = (φ_yx − φ_xy)/2 is the phase difference. Figure 4 plots the dependence of A and δ on the frequency. We can see that both A and δ oscillate with the frequency. At the resonant frequency of 14.24 GHz, the eigenstates are indeed elliptically polarized, where the main axes of the two eigenstates form an angle of about 60.45° with each other.

 

Fig. 4 Dispersion of the eigenstates of the metamaterials, including the amplitude A (red solid) and the phase δ (blue dot).

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To provide more information about the influence of structural symmetry on the performance of transmission, we introduce a geometric parameter g characterizing the shift of the longer arm in the top planar metamaterial structure with respect to the center line, as defined in Fig. 1(a). Correspondingly, the lengths of the other two shorter arms changes by ±g in order to ensure the strong coupling. As g increases from zero, as illustrated in Fig. 5, ${\Delta }_{\text{lin}}^{(x)}$ decreases for linear polarization, whereas ${\Delta }_{\text{circ}}^{(l)}$ increases for circular polarization (of course, its value is still much smaller than that of ${\Delta }_{\text{lin}}^{(x)}$). The influence of the parameter g can be traced to the anisotropic behavior in the diagonal elements of the transmission matrix, that when g is nonzero, requirements of Eq. (14) are no longer hold, resulting in t_xx ≠ t_yy. The breaking of the geometric symmetry in the hybridized metamaterial we designed (g ≠ 0) results in indeed the degradation of the performance of the asymmetric transmission for the linear polarization and the weak asymmetric transmission for the circular polarization. However, the influence is not very strong even if when g/a = 7.5% .

 

Fig. 5 Influence of the geometric parameter g on the transmission performance of the hybridized metamaterial. (a) ${\Delta }_{\text{lin}}^{(x)}$ for linear polarization and (b) ${\Delta }_{\text{circ}}^{(l)}$ for circular polarization.

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Considering the experimental verification or practical applications, importantly, we should discuss the oblique and non-collimated incidence cases. We give the FDTD simulation results of t_uv under the oblique incidence with an incident angle of 5°, for the linearly polarized plane wave, in Fig. 6(a). For the non-collimated incidence case, as is well known, the finite-aperture incident EM radiation can be considered to be non-collimated due to the intrinsic diffraction effect, As an example, we treat the finite-aperture EM radiation with a diameter of 35d, as shown in Fig. 6(b). Evidently, the simulation results shown in Fig. 6 have no distinct difference from the results of the plane wave under for the normal incidence shown in Fig. 3(a), implying that the hybridized metamaterial we proposed has a good tolerance for the practical experiment and application. As expected, t_uv has the tiny variation, as shown in Fig. 6. The fluctuation of Δ_lin is only about 5%, while Δ_circ maintains zero.

 

Fig. 6 Simulation results for the oblique and non-collimated incidence cases. (a) The oblique incidence case with an incident angle of 5° and (b) The non-collimated incidence case with the finite-aperture EM radiation with a diameter of 35d.

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When this article has been finished, we noticed the analysis based on advanced Jones calculus by Menzel et al. [22]. Although the general form of the Jones matrix can be determined rather easily by the symmetry consideration [22], the analysis based on the Born-Kuhn model used in the present article can explicitly reveal what a kind of hybridized metamaterial configuration is needed, and can characterize the value of asymmetric transmission factor. Further, the detailed effort can be used to the fitting of the parameters in the matrix Eq. (9) from the Born-Kuhn model by using the FDTD simulation results [12,22], where at least 7 independent parameters are required.

4. Conclusion

In conclusion, we propose a criterion on the transmission matrix elements for realizing the asymmetric transmission for linearly polarized EM radiation only. We discuss how to realize this criterion in metamaterials and give a clear demonstration on this intriguing phenomenon by a simply hybridized metamaterial design. Our study enriches the asymmetric transmission effect and allows the engineering of the EM properties through a metamaterial design. We believe that our proposal can be realistically achieved experimentally, which has the promising applications such as in optical isolation [23,24] and is highly valuable for the development of nanophotonic devices.