## Abstract

Circular polarizers with left-handed helical metamaterials can transmit right-handed circularly polarized (RCP) light with few losses. But a certain amount of left-handed circularly polarized (LCP) light will occur in the transmitted light, which is the noise of the circular polarizer. Therefore, we defined the ratio of the RCP light intensity to the LCP light intensity as the signal-to-noise (S/N) ratio. In our previous work, it’s found that circular polarizers with multi-helical metamaterials have two orders higher S/N ratios than that of single-helical metamaterials. However, it has been a great challenge to fabricate such multi-helical structures with micron or sub-micron feature sizes. Is it possible for the single-helical metamaterials to obtain equally high S/N ratios as the multi-helical ones? To answer this question, we systematically investigated the influences of structure parameters of single-helical metamaterials on the S/N ratios using the finite-different time-domain (FDTD) method. It was found that the single-helical metamaterials can also reach about 30dB S/N ratios, which are equal to the multi-helical ones. Furthermore, we explained the phenomenon by the antenna theory and optimized the performances of the single-helical circular polarizers.

© 2012 OSA

## 1. Introduction

In the last few years, there has been growing interest in the study of the chiral metamaterials both theoretically and experimentally due to the exciting potential applications such as giant optical activity [1,2], negative index of refraction [3–5] and circular dichroism [6–8]. In the recent past, Gansel [9,10] succeeded in developing a circular polarizer using gold single-helical metamaterials. Compared with other familiar methods, this polarizer has advantages of broad wavelength ranges and compact structures which are convenient to be integrated with other optical devices. However, due to the high intensity conversions between left-handed circualrly polarized (LCP) light and right-handed circualrly polarized (RCP) light, when the incident light is RCP light, the transmitted light contains both RCP (regarded as signal) and LCP (regarded as noise) light. Therefore, we defined the ratio of the RCP light intensity to the LCP light intensity as the signal-to-noise (S/N) ratio. The single-helixes, mentioned in the Gansel’s reports, have rather low S/N ratios (~10 dB), which means the ellipticity of the transmitted wave will deviate from 1. In consequence, the low S/N ratios will make the transmitted light not a perfect circularly polarized light and restrict the potential applications of the devices. Recently, we proposed a new type of circular polarizers with multi-helical metamaterials which have about 30dB S/N ratios [11]. But it’s a great challenge to fabricate such multi-helical structures with micron or sub-micron feature sizes. It seems that there is a contradiction between the improvement of the S/N ratio and the simplification of the structure. So there should be one question raised easily: Is it possible for the single-helical metamaterials to obtain equally high S/N ratios as the multi-helical ones?

In this work, we systematically investigated the influences of structure parameters on the S/N ratios of the single-helical metamaterials using the finite-different time-domain (FDTD) method. From the simulation results, we found that the S/N ratios do have significant relationship with the length of the helix (LH), the diameter of the helix (DH) and the spacing of the grid (SG). We also explained the phenomena by the antenna theory [12]. Through the optimization, these single-helical metamaterials can reach the same S/N ratios as the multi-helical ones (~30dB). There are some performance parameters used in this paper. To make them clear, Table 1 summarizes the definitions for each of them. The parameters S/N ratio and axial ratio indicate how perfect the transmitted circularly polarized light is. The higher S/N ratios are and the closer to 1 the axial ratios are, the more perfect or “purer” the transmitted circularly polarized light will be. The extinction ratios indicate the differences between the transmittances of RCP and LCP light. With higher extinction ratios, the helical metamaterials exhibit stronger circular dichroism.

## 2. Simulation model

Circular polarizers consisting of aluminum (Al) single-helical metamaterials were simulated using the FDTD method. Figure 1(a)
shows the schematic diagrams of the circular polarizer. The helical nanowire structure is supported by the silica substrate. The refractive index of silica is configured for 1.45. The dielectric function of the Al materials is described by the Lorentz-Drude model [13]. The broadband Gaussian-modulated pulsed left-handed circualrly polarized light and right-handed circualrly polarized light are used as the excitation source to irradiate the polarizers along the negtive *Z* direction respectively. The perfectly matched layers (PML) [14] is used as the boundary conditions along the *Z* direction. The boundaries along *X* and *Y* directions are confined with the periodic boundary conditions [15] due to the periodicity of the structure.

Figures 1(b) and 1(c) show the parameters of helical structure array, which includes the length of the helix (LH), the diameter of the helix (DH), the number of helix-period (NH), the diameter of wire (DW), and the spacings of grid (SG).

## 3. Simulation results and analyses

#### 3.1 Dependence on the Diameter of Wire and the Number of Helix-Period

The single-helical circular polarizers with different DWs and NHs were simulated respectively. Other parameters are LH = 200 nm, SG = 190 nm, DH = 100 nm. The parameters and the simulation results including the average axial ratios and the average S/N ratios are summarized in Table 2 and shown in Fig. 2 . Figure 2(a) shows the transmittance spectrum for the structure of DW = 30 nm, and NH = 3. In the Figs. 2(b), 2(c), and 2(d), the Poincaré sphere [16] was used to analyze the polarization states of the transmitted RCP light. Five wavelengths (1.25μm, 1.07μm, 0.93μm, 0.83μm and 0.75μm) were chosen to been plotted on the Poincaré sphere (also plotted on the Fig. 2(a)), each point denotes the polarization states of the corresponding transmitted RCP light. Figures 2(b) and 2(c) show the comparison of the structures with same DWs of 30 nm but different NHs of 3 and 6; Figs. 2(b) and 2(d) compare the structures with same NH of 3 but different DWs of 30 nm and 60 nm. In the figures, these red points indicate the polarization states of the five different wavelengths. For a perfect RCP light, its corresponding point ought to be on the pole of the Poincaré sphere. The closer the red pionts are to the pole, the higher the S/N ratios of the polarizers are and the closer the axial ratios are to the value of 1. Comparing all the results, it is indicated that the average S/N ratios have no obvious changes with the different DWs and NHs.

#### 3.2 Dependence on the Spacing of Grid

Both single-helical circular polarizers with SG = 190 nm and 390 nm were simulated. All other parameters are as follow: LH = 200 nm, DW = 30 nm, DH = 100 nm, NH = 3. The parameters and the simulation results including the average axial ratios and the average S/N ratios are summarized in Table 3 and shown in Figs. 3(a) and 3(b). It’s clear that the red points in Fig. 3(b) are much closer to the pole than that in Fig. 3(a), which means that increasing the spacing of grid can improve the average S/N ratio evidently.

This phenomenon can be explained by the antenna theory. When the electromagnetic wave propagates through the helical metamaterials, electrical currents will occur on the surfaces of the helical wires. In our previous studies on elliptical helices [17], it’s found that if the axial ratios of the horizontal projections of the currents’ paths along elliptical helix surfaces are 2:1, the axial ratios of the transmitted elliptical polarized light will also be 2:1. Therefore, we think that there is a certain relationship between the existence of the electrical currents and the polarization states of the transmitted light: the axial ratio of the transmitted light is determined by the shape of horizontal projection of the current’s path. If the horizontal projection is a perfect circle, the axial ratio of the tansmitted light will be 1, which leads to a higher S/N ratio. In contrast, the polarizers will have a lower S/N ratio. Figure 4 shows the schematic diagram of the horizontal projection of the current’s path of the single-helices with different SGs. The red lines in the helices indicate the current occuring on the surfaces of helices. In the helical wire array, the surface currents will interact with the neighbouring helix cells, which will have an impact on the currents’ paths in return. When the SGs are small, the strong interactions will distort the current’s path so that the horizontal projection can’t be a perfect circle any more (shown in Fig. 4(b)). For this reason, the average S/N ratios of the transmitted RCP light are lower. When the SGs are large enough to ignore the interactions, the current’s projection will be a perfect circle (shown in Fig. 4(c)), and the average S/N ratios will reach higher values. Therefore, increasing the spacing of grid can improve the S/N ratios of the single-helical metamaterials.

#### 3.3 Dependence on the Length of Helix and the Diameter of Helix

In this section, the helical circular polarizers with different LHs and DHs were simulated respectively. Other parameters are as follow: DW = 30 nm, NH = 3, SG = 190 nm. The parameters and the simulation results including the average axial ratios and the average S/N ratios are listed in Table 4 and shown in Figs. 5(a) , 5(b) and 5(c). These figures show the polarization states of transmitted RCP lights on Poincaré sphere. Figures 5(a) and 5(b) are the comparison of LH = 200 nm and LH = 600 nm. Figures 5(a) and 5(c) are the comparison of DH = 100 nm and DH = 50 nm. It’s obvious that the red points in Fig. 5(b) and 5(c) are closer to the pole than that in Fig. 5(a). From the results, it is clear that increasing the LH and decreasing the DH can both greatly improve the S/N ratios.

Figure 6 shows the schematic diagram of the currents’ paths on the surfaces of the three single-helix with different LHs or DHs. Comparing the red lines in Figs. 6(a), 6(b) and 6(c), the current flowing along the helixes with longer helixes or shorter diameters of helixes is much steeper. For the current in Fig. 6(a), its horizontal components are predominant, so the interactions between the helix cells are strong, which will result in that the horizontal projection can’t be a perfect circle (shown in Fig. 6(d)). Whereas for the current in Figs. 6(b) and 6(c), its horizontal components are not predominant any more, so the currents don’t have much impact on neighbouring currents, which lead to a perfect circular horizontal projection of the current’s path (shown in Figs. 6(e) and (f)). Therefore, increasing the LH and decreasing the DH can also great improve the S/N ratios.

#### 3.4 Optimized helical circular polarizers

Summarizing the above simulations, there are three methods to improve the S/N ratios of single-helical metamaterials: increasing the spacing of grid, increasing the length of helix and decreasing the diameter of helix. However, according to our previous works [18], increasing the spacing of grid will reduce the extinction ratio, and decreasing the diameter of helix will narrow the operation region. Therefore, to improve the S/N ratio and also maintain a fairly good extinction ratio and operation region at the same time, we increased the number of helix-period and the spacing of grid along only one axis, and kept the diameter of helix constant. This is a trade-off between the high S/N ratios and the extinction ratios. Finally, by properly selecting the parameters as follow: DW = 30 nm, NH = 6, LH = 200 nm, DH = 100 nm, SG along *X*-axis = 480 nm, SG along *Y*-axis = 220 nm, we proposed a single-helical metamaterials with a high S/N ratio. As shown in Figs. 7(a)
and 7(b), it can reach an average extinction ratio of 35:1 and an average S/N ratio of 28.2dB in the operation region from 0.68 μm to 1.06 μm.

## 4. Conclusion

In summary, we investigated the influences of the single-helical metamaterals’ structure parameters on the S/N ratios systematically and explained the phenomena in the language of antenna theory. The influences of the structure parameters on the S/N ratios are summarized in Table 5 . It was found that by properly designing the structure these single-helical circular polarizers can also reach a high S/N ratio of about 30 dB. Comparing with the multi-helical metamaterials, these single-helical ones have great advantages in fabrication. Although the extinction ratios can’t keep up with the multi-helical ones yet, it’s a sure fire way to fabricate high performance circular polarizers with helical metamaterials.

## Acknowledgment

We acknowledge support by the Natural Science Foundation of China (NSFC) (Nos. 11104094, 61007019 and 50735007), Doctoral Fund of Ministry of Education of China (No. 200804871147), the Natural Science Foundation of Hubei Province of China (No. 2008CDB004), and the Fundamental Research Funds for the Central Universities (HUST:Nos. 2010MS063 and 2011TS060).

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