1. Introduction

The functionality to control the polarization state of electromagnetic (EM) wave is highly required in modern optics since it provides high degree of freedom for manipulating EM wave and expanding optical communications devices. Conventional wave retarders are usually produced by birefringent materials, such as calcite, which can induce a phase difference between two orthogonal axes in consideration of their different refractive indexes for optical transmission. This approach generally requires optical distance long enough to accumulate phase difference, which means the conventional wave retarders are normally bulky in size and inapplicable for advanced applications which require miniaturized optical elements. In addition, the operating wavelength of the conventional wave retarders is inherently narrow bandwidth due to the constraint of material, which also restrains the development of modern optical devices. For these reasons, metamaterial, a kind of ultrathin artificial subwavelength material, has been extensively investigated and successfully employed to tailor the basic characteristics of EM wave [1–7] including phase [8], intensity [9], and polarization [3]. In this development process, numbers of new physical phenomena, such as plasmon-induced-transparency [10], cloaking [11], and negative refractive index [12,13] have been explored in compact, integrated, and multiband designs.

To achieve new-type polarization convertors, a tremendous amount of plasmonic metamaterials [14–17] have been proposed and demonstrated to manipulate the polarization of light. For example, an ultrathin 90° polarization rotator has been realized by bilayered metallic wire pairs, expressing relative high-efficiency and giant optical activity [18]. Yang et al. proposed a planiform plasmonic quarter-wave plate using a periodic array of symmetrical L-shaped antennas [19]. Various possible polarizations (linear, elliptic, and circular) conversions [20,21] have also been realized recently in compact metamaterials, using metallic-graphene resonators [22], L-shaped periodic array supercell [23], and anisotropic metamaterial plate [24]. In some very recent studies [25,26], vector vortex beam has been obtained by periodic array units which can induce discontinuous phase distribution. However, the modulation efficiencies of these convertors are intrinsically unpersuasive since the physical principle of them is the arising of cross-polarization component. In addition, the operating bandwidth is still narrow due to the limitation of material dispersion.

In this paper, we theoretically and numerically demonstrate that broad-band and high-efficiency 90° polarization rotator and quarter-wave plate around optical communication wavelength of 1550 nm can be realized using ultrathin and geometry optimized composite metamaterials. The polarization characteristics of the reflected wave can be manipulated by adjusting the geometrical parameters, enabling artful tailoring of the amplitudes and phases along two orthogonal axes. A high-efficiency (≈80%) and broad-band (≈300 nm) 90° polarization rotator is realized in the wavelength range from 1400 to 1700 nm by rotating the direction of incident electric vector 90 degrees. The resonance modes of 90° polarization rotator are illustrated by investigating the surface electric distributions. Moreover, by focusing on phase modulation, a high-efficiency (≈94%) and broad-band (≈110 nm) quarter-wave plate is also generated in the wavelength range from 1535 to 1645 nm. Furthermore, the ellipticity property and polarization rotation are investigated to further understand the broad-band effect of the proposed polarization convertors. The results of the proposed polarization converters appear to hold the promise for polarization conversion devices and communication systems.

2. Structure and methods

Figure 1(a) depicts the schematic diagram of the design which consists of a proposed metal metasurface, a dielectric layer (n₂, Z₂), and an ideally conducting silver plane (n₃, Z₃). Here n_i and Z_i (i = 1, 2, 3) are the effective refractive index and wave impedance, respectively. Experimental data measured by Johnson and Christy [27] are employed as the frequency-dependent effective permittivity of metal (silver), and the effective refractive index of dielectric (n₂) is considered as 1.9 for guaranteeing the target wavelength (1550 nm). Figure 1(b) shows a unit cell of the proposed design with periods P_x = P_y = 900 nm, and the unit cell of metal metasurface consists of an inner nanocube, an outer ring resonator, and a nanostrip with the symmetrical axis's orientation angle of 45°. The width of both the outer ring and nanostrip is w = 100 nm, and the thickness of all compositions of the metasurface is t₁ = 300 nm. The other parameters are l₂ = 700 nm, t₂ = 410 nm, t₃ = 300 nm.

 

Fig. 1 (a) The schematic diagram of the proposed design. (b) A unit cell of this design.

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Let us model the proposed metamaterial as regular arrays of small particles which can be described by dipole effective polarizability tensor ${\alpha }^{eff}$, defined as ${\alpha }^{eff}={({\alpha }^{-1}-C)}^{-1}=(\begin{array}{cc}{\alpha }_{xx}^{eff}& {\alpha }_{xy}^{eff}\\ {\alpha }_{yx}^{eff}& {\alpha }_{yy}^{eff}\end{array})$. Here $\alpha$ is the electric polarizability tensor and C is interaction dyadic. According to the local-field approach and dipole approximation [28], the dipole moment of a unit cell is excited by the incident light and the radiation field of the other dipoles in the design. Following the transmission-line model [29,30], the boundary conditions for the metasurfaces can be written as

3. Results and discussions

3.1 Broad-band and high-efficiency 90° polarization rotator

In our work, an x polarized plane wave [see the x-pol. in Fig. 1(a)] is considered to normally irradiate the proposed design. Firstly, the l₁ is assumed as 300 nm. In this situation, the linear reflectivities $R_{yx}$ and $R_{xx}$ ($R_{\text{ij}}={|r_{\text{ij}}|}^2$) are calculated and depicted in Fig. 2(a). It can be found that the cross-polarization reflectivity $R_{yx}$ is larger than the co-polarization reflectivity $R_{xx}$ from 1370 to 1720 nm. There are three resonance peaks for cross-polarization reflectivity at λ₁ = 1415 nm, λ₂ = 1550 nm, and λ₃ = 1679 nm, and the values of which are almost equal to 90%. In addition, from 1400 to 1700 nm, the cross-polarization reflectivity $R_{yx}$ is more than 75%, while the co-polarization reflectivity $R_{xx}$ is lower than 20%. It illustrates that the co-polarization power is suppressed in the reflected light, while the cross-polarization power is connived at above wavelength range. In other words, the power of x polarized plane wave is converted to y polarized light after being reflected by the proposed design. For more preciseness, the calculated phase difference $\Delta \phi ={\phi }_{yx}-{\phi }_{xx}$ is also shown in Fig. 2(a). It can be obviously seen that the $\Delta \phi$ is nearly equal to 0° at 1550 nm, which means a strict 90° polarization rotator. Alongside 1550 nm, the $\Delta \phi$ is $\pm 90°$ from 1400 to 1700 nm, which is an essential condition for realizing a quarter-wave plate. Then we can only define it as the pseudo 90° polarization rotator at this wavelength range. To better understand this 90° polarization rotator, the polarization conversion ratio (PCR), defined as $PCR=R_{yx}/(R_{yx}+R_{xx})$, is illustrated in Fig. 2(a). It is obvious that the PCR can reach close to unity at 1378, 1415, 1550, and 1679 nm. But, it should be noted that the cross-polarization reflectivity $R_{yx}=42\%$ at 1378 nm, which is undesirable for application. Fascinatingly, the PCR is always above 0.8 in the range from 1400 to 1700 nm, which clearly means the 90° polarization rotation effect and further illustrates the functionality of broad-band and high-efficiency.

 

Fig. 2 (a) The co-polarization $R_{xx}$, cross-polarization $R_{yx}$ reflectivity, the phase difference$\Delta \phi$ between ${\phi }_{yx}$ and ${\phi }_{xx}$, and the (PCR) for the proposed 90° polarization rotator. (b) The calculated ellipticity angle $\zeta$ and PRA $\chi$.

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To further understand the broad-band characteristic of this rotator, the ellipticity angle $\zeta$ and polarization rotation angle (PRA) $\chi$ are calculated by the results of Eq. (3) and defined as [31,32]:

To gain insight into the EM resonance modes of this proposed 90° polarization rotator, we demonstrate the electric field $|E|$, real $E_z$ of the upper surface of metasurface, and real $E_z$ of the bottom surface of metasurface distributions corresponding to the three cross-polarization reflectivity maxima (λ₁, λ₂, and λ₃) and λ₄ = 1750 nm in Fig. (3). For 1415 nm, the electric field distribution is principally focused on the bottom-left corner of inner nanocube and the bottom-left and upper-right corners of outer ring resonator [see Fig. 3(a₁)]. Thus, the charge should accumulate at the corresponding corners (along the direction of the electric field). As shown in Fig. 3(b₁), it is obvious that the positive charge mainly locates at the bottom-left corners of nanocube and ring resonator, and the negative charge mainly locates at the upper-right corner of ring resonator in considering the electroconductibility of nanostrip. These electric dipole resonance patterns can forcefully couple with its own electric resonance images [Fig. 3(c₁)], and the antiparallel current flow on the metasurface can be produced. As a result, a strong magnetic resonance can be generated from the combined action of the antiparallel current flow, in which the x component of the magnetic field can induce an electric field radiation along y direction [17,33]. Thus, the power of incident light can be partially converted to y direction by reflecting. Similar results exist when the incident wavelengths are 1550 and 1679 nm [see Fig. 3(a₂)-3(c₂) and Fig. 3(a₃)-3(c₃)]. Actually, the broad-band effect of this proposed rotator originates from the superposition of these three resonance eigenwavelengths. The electric field distributions for 1750 nm are also calculated and depicted in Fig. 3(a₄)-3(c₄), respectively, for comparison. It is clear that electric field distribution is principally focused on the both sides of the outer ring resonator, and the opposite charges mainly locate at both sides of the ring resonator. Thus, the electric field radiation has no y component for reflected light [17], which means the polarization plane has not rotated and it is in good agreement with the results of Fig. 2(a).

 

Fig. 3 Distributions of the electric field $|E|(a_1-a_4)$, real $E_z(b_1-b_4)$ for the upper surface of metasurface, and real $E_z(c_1-c_4)$ for the bottom surface of metasurface at wavelength of 1415 nm (a₁, b₁, and c₁), 1550 nm (a₂, b₂, and c₂), 1679 nm (a₃, b₃, and c₃), and 1750 nm (a₄, b₄, and c₄), respectively.

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3.2 Broad-band and high-efficiency quarter-wave plate

The $\Delta \phi =n\pi /2$ is an essential condition for realizing a quarter-wave, and the another essential condition is $R_{xx}=R_{yx}$. For purposes of these conditions, we optimize the geometrical parameter as l₁ = 250 nm. The linear reflectivities $R_{yx}$, $R_{xx}$, and the phase difference $\Delta \phi$ are calculated and depicted in Fig. 4(a). It can be found that the $\Delta \phi$ is always nearly equal to 270° at the wavelength range from 1450 to 1660 nm. In addition, the cross-polarization reflectivity $R_{yx}$ is equal to its co-polarization reflectivity $R_{xx}$ at 1550 and 1636 nm. These results indicate that the reflected light for this optimized design has the equivalent orthogonal amplitudes and a 270° phase difference at these two wavelengths, meaning that the quarter-wave plate has been obtained with high reflected intensity $I=R_{yx}+R_{xx}=0.94$. What is more, when the value of $R_{yx}/R_{xx}$ for reflected wave is within the range 0.65-1.37 [see the shadowed region in Fig. 4(a)], the functionality as a quarter-wave plate is still acceptable. Furthermore, the ellipticity $\eta$ and ellipticity angle $\zeta$ are calculated and depicted in Fig. 4(b). It can be seen obviously that the $\zeta$ is near $-45°$ at the wavelength range from 1535 to 1645 nm, indicating the reflected wave is a circularly polarized light. And the calculated $\eta$, standing for the ratio between the minor axis and major axis, is more than 0.8 around above wavelength range. These results further validate that the linearly polarized light has been converted to circularly polarized light after being reflected by the proposed design. It is worth noting that the performance of the rotation effect will be significantly influenced by the geometrical parameters of the unit cell structure according with Eq. (3).

 

Fig. 4 (a) The co-polarization $R_{xx}$, cross-polarization $R_{yx}$ reflectivity, the phase difference $\Delta \phi$ between ${\phi }_{yx}$ and ${\phi }_{xx}$ for the proposed quarter-wave plate. (b) The calculated ellipticity $\eta$ and ellipticity angle $\zeta$.

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4. Conclusion

In conclusion, a 3D ultrathin design consisting of a proposed metasurface, a dielectric substrate, and an ideally conducting silver plane has been theoretically and numerically demonstrated for polarizer applications. The polarization characteristics of radiation wave after being reflected by the design can be manipulated by adjusting the geometrical parameters which can artfully tailor the polarizability of design. A 90° polarization rotator with an efficiency of 80% and bandwidth of 300 nm has been obtained in the wavelength range from 1400 to 1700 nm. And the resonance modes of 90° polarization rotator were illustrated by investigating the surface electric distributions. In addition, a quarter-wave plate with an efficiency of 94% and bandwidth of 110 nm has also been generated in the wavelength range from 1535 to 1645 nm. The ellipticity property and polarization rotation have been investigated to further understand the broad-band behavior of the proposed polarization convertors. The results of the proposed polarization converter appear to hold promise for polarization conversion devices and communication systems.