1. Introduction

Polarization of light has been of interest for over a century, in applications across many fields of science and technology. Polarized lights can be classified into the following three types: linearly polarized light, circularly polarized light, and elliptically polarized light [1]. In the last few years, the metallic structure with sub-wavelength metamaterials has been proved to achieve the linearly [2–7] and the circularly polarized light [8–12]. Compared with traditional crystal and cube polarizers, the polarizers with the sub-wavelength metamaterials have advantages of broad wavelength ranges, high temperature durability, and compact structures which are convenient to be integrated with other optical devices [3, 7].

From the view of the mathematical description, both the linearly and the circularly polarized light are special cases of the elliptically polarized light [1]. There should be one question raised easily: Is it possible for the metallic structure with sub-wavelength metamaterials to generate the elliptically polarized light? If it is, this will create much broader potential applications, such as two-dimensional polarization encoding in quantum information processing [13], coherent population trapping (CPT) of atoms driven by elliptically polarized light [14] and researches on nonlinear polarization rotation in some cubic crystals [15].

For this purpose, we studied the possibility of the metallic structure generating elliptically polarized light using the finite-difference time-domain (FDTD) method. From the simulation results, it is confirmed that the elliptically polarized light can be realized through such metallic structure with elliptical helixes. And we also made careful analysis of the polarization states of the output light through different elliptically helical structures.

2. Simulation models

Figure 1 shows the schematic diagram of the metallic structure with elliptically single-helical nanowires. The parameters of the elliptically helical nanowire include diameters of wires (DW), numbers of helix-period (NH), lengths of helix-period (LH), long-axis of elliptical helix (LEH), short-axis of elliptical helix (SEH), and spacing of grids in long-axis direction (SGL) and in short-axis direction (SGS). The axial ratio (ratio of the length of the major semiaxis to that of the minor semiaxis) is 2:1. The metal is aluminum (Al). The elliptically helical nanowires are supported by a silica substrate with refractive index of 1.5. The excitation sources are orthogonal left-elliptically polarized (LEP) light and right-elliptically polarized (REP) light propagating along z axis in free space. They are represented with the following Jones vectors [1]:$\text{LEP}:\frac{1}{\sqrt{5}}\left[\begin{array}{c}1\\ -2i\end{array}\right] \text{REP}:\frac{1}{\sqrt{5}}\left[\begin{array}{c}2\\ i\end{array}\right]$

 

Fig. 1 Schematic diagram of the metallic structure consisting of elliptically helical nanowires.

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A broadband Gaussian-modulated pulsed light source is used as the excitation source. The perfectly matched layers (PML) [16] are along the z axis. The boundaries along x and y axes are confined with the periodic boundary conditions [17] owing to the periodicity of the structure. During the calculation, the dielectric function of the metal Al is described by the Lorentz-Drude model [18].

There are some performance parameters used in this paper. To make them clear, Table 1 summarizes the definitions for each of them.

Table 1. Definitions of the metallic structure’s performance parameters

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3. Simulation results and analyses

Firstly, we simulated an elliptically single-helical metamaterials with the parameters’ values: DW = 30 nm, NH = 3, LH = 200 nm, LEH = 80 nm, SEH = 40 nm, SGL = 310 nm, and SGS = 170 nm. Figure 2(a) shows the transmittance and the extinction ratio of the structure. According to the definitions in Table 1, the operation region is 680-1100 nm. In the region, the average transmittance of LEP light is 74% and the average extinction ratio is 11.1 dB. It is obvious that this metamaterials has a giant elliptical dichroism. To analyze the polarization states of the transmitted LEP light in detail, the Poincaré sphere [19] is plotted in Fig. 2(b), in which these red points refer to different wavelengths. The coordinates of the points on the Poincaré sphere, their corresponding polarization states and the conversions of LEP are listed in Table 2 . From the calculation results, it is clear that the axial ratios of the transmitted LEP light are all around 2:1, which is the same as that of the incident LEP light.

 

Fig. 2 (a) Optical performance of the elliptically single-helical metamaterials; (b) The polarization state of the transmitted LEP light represented on the Poincaré sphere.

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Table 2. The polarization states of the transmitted LEP light

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In previous works of the helical nanowire [8–12], they concentrated on structures of circular helixes. This kind of structure is similar to the helical antennas with the so-called end-fire geometry, which are widely used in microwave wireless local-area network (WLAN) applications [20]. According to the results of these research works, a conclusion for the polarization property of the circular helix is drawn as follows: The circularly polarized light with the opposite handedness to the helical structure can transmitted, but the light with the same handedness of the helical structure cannot [8]. In other words, the circularly helical metamaterials have giant circular dichroism. Based on the results of the elliptically helical nanowire above, we think that the dichroism property can be expanded from the circular polarizations to the whole elliptical polarizations for the helical metamaterials provided that a little change is made on the conclusion above: The elliptically polarized light with the opposite and orthogonal handedness to the helical structure can transmit, but the light with the same handedness of the helical structure cannot.

Next, the structures with elliptically double-, three- and four-helical nanowires were simulated. Figure 3(a) , 3(b), and 3(c) show the schematic diagrams of structures with elliptically double-, three-, and four-helical nanowires, respectively. The parameters of the elliptically multi-helical nanowires are the same as those of the single-helical ones above. Figures 3(d), 3(e) and 3(f) are optical performances of such elliptically double-, three- and four-helical structures, respectively. It is obvious that these multi-helical structures do also have giant elliptical dichroism, which is similar to the single-helical ones.

 

Fig. 3 Comparison of the optical performances of the elliptically double-, three- and four- helical nanowires metamaterials.

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However, along with studying the polarization state of the transmitted light in detail, different characteristics occur. Figures 4(a) , 4(b) and 4(c) are the polarization states of the transmitted LEP light represented on the Poincaré sphere for the elliptically double-, three- and four-helical nanowires, respectively. The coordinates of the points on the Poincaré sphere, their corresponding polarization states and the conversions of LEP are listed in Table 3 . From the results, it is found that the axial ratios of the transmitted LEP light decrease with the increasing of the number of wires in one helix.

 

Fig. 4 The polarization state of the transmitted LEP light represented on the Poincaré sphere. (a) Double-helix; (b) Three-helix; (c) Four-helix.

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Table 3. The polarization states of the transmitted light of the the structures with elliptically double-, three- and four-helical nanowires

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These phenomena can be explained in the language of the antenna theory. When the electromagnetic wave propagates through the helical metamaterials, electrical currents will occur in the surfaces of the helical wires. In our case, when the LEP light source propagates through the elliptically helical nanowires, the electrical currents will be on the surfaces of them. We think that there is a certain relationship between the existence of the current and the transmitted light’s polarization states: The handedness of the transmitted light is orthogonal to that of the current’s path. Generally, the path of the current depends on two factors: One is the interaction between the neighbouring helical structures, which we call outer interactions; the other is that among the different helical wires within one multi-helix, which we call inner interactions. With the increase of the number of wires in one helix, the currents’ paths will be more and more dominated by the inner interaction. Therefore, we pay more attention to the inner interaction here. Figure 5 shows the sketch map of the inner interaction for the elliptically multi-helix and the paths of the currents. We take an elliptical double-helix as an example here. The inclination angles of different position on the elliptical helix are not uniform: On both sides of the major axis, the inclination angles (θ) reach minimum (shown in Fig. 5(a)); but on both sides of the minor axis, the inclination angles (Φ) reach maximum (shown in Fig. 5(b)). It’s obvious that the angle Φ is much bigger than θ. Therefore, the paths of the current flowing through the sides of minor axis are much steeper than that flowing through the sides of major axis (shown in Fig. 5(c)).

 

Fig. 5 The schematic diagram of the inner interaction for elliptical multi-helix and the paths of currents

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For the currents flowing through the sides of minor axis, their vertical components are predominant, which leads to repulsive forces between them. And for the currents flowing through the sides of major axis, their horizontal components are predominant, which leads to attractive forces between them. These two forces make the currents’ paths have the following properties: On both sides of the major axis, the current has a trend flowing along the inner surface of the elliptical helix; on both sides of the minor axis, the current has a trend flowing along the outer surface (shown in Fig. 5(d)). Therefore, when the number of wires in one helix increases, the inner interactions will become stronger, and the axial ratio of the transmitted light decreased.

4. Conclusions

In summary, this paper starts with such a question: Is it possible for the metallic structure with sub-wavelength metamaterials to generate the elliptically polarized light? To answer this question, we proposed a metallic structure with elliptically helical nanowires, and analyzed the polarization states of the transmitted light using FDTD method. It is confirmed that this metallic structure does have a giant elliptical dichroism. Furthermore, we also compared the distinct optical performances of elliptically single-, double-, three-, and four-helixes, and made a qualitative explanation for them based on the antenna theory. Finally, it can be expected that with the affirmative answer of our question the metallic structure with sub-wavelength metamaterials will have applications in much broader potential areas such as optical communications, optical computings, and optical sensors.