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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 16050–16058
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Multi-band circular polarizer using planar spiral metamaterial structure

Xiaoliang Ma, Cheng Huang, Mingbo Pu, Chenggang Hu, Qin Feng, and Xiangang Luo  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 16050-16058 (2012)
http://dx.doi.org/10.1364/OE.20.016050


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Abstract

A multi-band circular polarizer is proposed by using multi layered planar spiral metamaterial structure in analogy with classic spiral antenna. At three distinct resonant frequencies, the incident linearly polarized wave with electric field paralleling to one specific direction is transformed into left/right-handed circularly polarized waves through electric field coupling. Measured and simulated results show that right-handed circularly polarized wave is produced at 13.33 GHz and 16.75 GHz while left-handed circularly polarized wave is obtained at 15.56 GHz. The surface current distributions are studied to investigate the transformation behavior for both circular polarizations. The relationship between the resonant positions and the structure parameters is discussed as well.

© 2012 OSA

1. Introduction

During the last decades, metamaterials (MMs) have attracted great interest for their special electromagnetic properties which are unattainable in natural materials. Arbitrary values of effective permittivity and permeability can be designed [1

1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]

, 2

2. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]

] and provide the possibility to design some amazing function device such as perfect lens [3

3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

] and invisible cloaking [4

4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

].

Except from these novel applications, MMs have been recently proposed to manipulate polarization states of electromagnetic waves for achieving polarization rotation [5

5. W. Sun, Q. He, J. Hao, and L. Zhou, “A transparent metamaterial to manipulate electromagnetic wave polarizations,” Opt. Lett. 36(6), 927–929 (2011). [CrossRef] [PubMed]

8

8. D. H. Kwon, P. L. Werner, and D. H. Werner, “Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation,” Opt. Express 16(16), 11802–11807 (2008). [CrossRef] [PubMed]

]. The key of polarization rotation in MMs is believed to be the artificial chirality. For example, twisted U shape split ring resonators, or twisted complementary split ring resonators are typical chiral MMs with circularly polarized eigenmodes [9

9. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009). [CrossRef] [PubMed]

14

14. J. Zhou, J. Dong, B. Wang, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79(12), 121104 (2009). [CrossRef]

]. However, one disadvantage of MMs-based polarization rotator is its narrow working bandwidth. As an attempt to overcome such restriction, J. K. Gansel et al. [9

9. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009). [CrossRef] [PubMed]

] have proposed an infrared broadband circular polarizer composed of periodic gold helix structure, which has also been scaled to terahertz [10

10. S. X. Li, Z. Y. Yang, J. Wang, and M. Zhao, “Broadband terahertz circular polarizers with single- and double-helical array metamaterials,” J. Opt. Soc. Am. A 28(1), 19–23 (2011). [CrossRef] [PubMed]

] and microwave region afterwards [11

11. C. Wu, H. Li, X. Yu, F. Li, H. Chen, and C. T. Chan, “Metallic helix array as a broadband wave plate,” Phys. Rev. Lett. 107(17), 177401 (2011). [CrossRef] [PubMed]

]. Despite of its relatively wide frequency band, this polarizer has high thickness that almost equals to resonant wavelength and is also hard to fabricate and integrate into present systems.

2. Design principle

2.1 Dual-band circular polarizer

Figure 1
Fig. 1 Schematic structure of the dual-band circular polarizer unit cell.
shows the unit cell of the dual-band circular polarizer, which is composed of two layers of planar metallic arcs etched on the two sides of a printed circuit board (PCB) F4B with thickness of h, relative permittivity of εr = 2.55 and loss tangent of 0.0007. The metallic arcs of each layer are concentric in XOY plane with the same radius of R and width of w. The other parameters include the structure periodic P, the central angle of each arc θ, and the angle of the gap between the two metallic arcs Φ1. The optimized geometrical parameters are P = 11.5 mm, R = 4.75 mm, w = 0.85 mm, h = 3 mm, θ = 80°, Φ1 = 40°.

The method we utilize to calculate the transformation behavior of the design polarizer should be firstly discussed, before the simulation process. Assuming the electric field of the incident linearly polarized wave is along y direction, the electric field of the transmitted wave can be stated as [16

16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 205–207.

]:
E=(Txyx±Tyyy)E0,
(1)
where E0 is amplitude of the incident electric field, Txy and Tyy are transmissions of the linearly polarized wave, and the first subscript indicates the transmitted polarization (x- or y-polarized), while the second indicates the incident polarization. The transmitted wave would be LCP (RCP) if the amplitudes of Txy and Tyy equal to each other, and the phase difference Φ(|Txy|) - Φ(|Tyy|) equal to −90° (90°).

Based on the above discussion, numerical simulation is carried out to analyze the transformation behavior of the polarizer by adopting finite difference time domain (FDTD) method. A broadband Gaussian-modulated pulse source with y-polarized electric filed is used as the excitation source in the simulation. Thus the information of the frequencies we interest in can be totally covered. In the process of simulation, the boundaries along x and y directions are set to be periodic boundary condition, while perfectly matched layers are set along z direction.

Figure 2(a)
Fig. 2 Transmission of Txy and Tyy (a) amplitude of Txy and Tyy and (b) the phase of Txy and Tyy.
presents the amplitudes of Txy and Tyy while the phases of Txy and Tyy are shown in Fig. 2(b). It can be calculated that the ratio between the amplitude of Txy and Tyy (|Txy|/|Tyy|) equals to 0.93 at 14.25 GHz (denoted as f1) and 1.02 at 16.35 GHz (denoted as f2), and the phase difference Φ(|Txy|) - Φ(|Tyy|) equals to −91.2° and 90.4° at the above two resonant frequencies, respectively. It indicates that a LCP wave is induced at 14.25 GHz, whereas a RCP wave is induced at 16.35 GHz.

Then we define CRCP as the transformation coefficient for RCP wave and CLCP for LCP wave, and the conversion from the linear transmission coefficients to the circular transformation coefficients can be obtained as follows [15

15. Z. Li, R. Zhao, T. Koschny, M. Kafesaki, K. B. Alici, E. Colak, H. Caglayan, E. Ozbay, and C. M. Soukoulis, “Chiral metamaterials with negative refractive index based on four “U” split ring resonators,” Appl. Phys. Lett. 97(8), 081901 (2010). [CrossRef]

]:

CRCP=Txy+iTyy,
(2)
CLCP=TxyiTyy.
(3)

The calculated transformation coefficients CRCP and CLCP are depicted in Fig. 3
Fig. 3 Calculated transformation coefficients from linear transmission coefficients to circular transmission coefficients.
. It can be seen that the transformed field is LCP at 14.25GHz, while is RCP at 16.35GHz. Besides, the transformation coefficient of LCP wave is 29 dB higher than that of RCP wave at 14.25 GHz, whereas is 36 dB lower than that of RCP wave at 16.35 GHz. As a result, relatively pure circularly polarized waves with different rotation are realized at these two frequencies.

To understand the physical origin of the polarization transformation, surface current distributions in the metallic arcs are calculated at the resonant frequencies. The front view of instantaneous induced current at 14.25 GHz is depicted in Fig. 4(a)
Fig. 4 Instantaneous surface current induced in the arcs by the incident y-polarized wave (a) at 14.15 GHz and (b) at 16.70 GHz.
, and the direction of current in each arc is clockwise. The directions of the instantaneous surface current in the two metallic arcs are opposite to each other at 16.35 GHz, as shown in Fig. 4(b). The current distributions illustrate that electric field along x direction of is induced. The electric field induced by metal layer 1 and metal layer 2 can be resolved into x and y components. It is clear that the x component of electric field is along + x direction and the y component is along -y direction at 14.25 GHz, which illustrates that a LCP wave is transformed. However, at 16.35 GHz, the x component of electric field is along -x direction and the y component is along -y direction, which indicates that a RCP wave is realized.

2.2 Triple-band circular polarizer

Subsequently, the concept of planar spiral polarizer is extended to construct a triple-band circular polarizer by adding another metallic arc layer. The photograph of the fabricated circular polarizer is geometrically shown in Fig. 5(a)
Fig. 5 Schematic configuration of the triple-band circular polarizer. (a) Photograph of the circular polarizer and the magnified unit cell. (b) Front view of the unit cell. Φ1 represents the angle between the upper end of metal layer 1 and x axis.
. Three layers of the identical metallic arcs are etched on the two overlapped F4B laminas. Since the dielectric laminas are not transparent, only the metallic arcs on the top layer can be seen in the photograph. The structure parameters of the polarizer are illustrated in Fig. 5(b). The width of each arc is designed to be w = 0.85 mm, and the central angle is θ = 80°.The period and thickness of the polarizer are P = 11.5 mm and 2 × h = 3.048 mm, which equal to about 0.5λ and 0.135λ (λ represents the wavelength) at the lowest operating frequency, respectively. Φ1 represents the angle between the upper end of metal layer 1 and x axis, and angle Φ2 denotes the angle of the gap between two neighboring arcs. When Φ1 is varied, the metal structure of each unit cell is rotated along z axis. In this design, Φ1 is 22.5°, while angle Φ2 is designed to be 40°. The whole optimization process will be shown in the discussion section.

In the experiment, the fabricated triple-band circular polarizer is composed of 26 × 26 unit cells. The transformation characteristic of the sample is measured by vector network analyzer R&S ZVA40. A standard linearly polarized horn antenna is utilized as transmitter, and the receiver adopts the LCP and RCP antennas in turn. In the process of measurement, the sample is adjusted in the middle position between the two horn antennas.

The simulated results of the transformation responses for this triple-band polarizer are depicted in Fig. 6(a)
Fig. 6 The transformation levels for the fabricated circular polarizer (a) the simulated results of the polarizer and (b) measured results for transformation.
by adopting the similar method described before. There are three obviously resonant frequencies occurring at 13.33 GHz, 15.56 GHz, and 16.75 GHz. The transmitted waves are RCP at 13.33 GHz and 16.75 GHz, while is LCP at 15.56 GHz. Figure 6(b) presents the measured transformation results of the fabricated sample. It is shown that RCP waves are produced at 12.75 GHz and 16.20 GHz, and LCP wave is transformed at 15.25 GHz. The whole three resonant frequencies shift towards lower frequency by about 0.5 GHz, which is mainly attributed to the utilization of prepreg between PCB layers in the fabrication that is not considered in the simulation.

Surface current distributions in the metallic arcs are investigated at each resonant frequency to analyze the polarization transformation. For the lowest resonant frequency f = 13.33 GHz, the instantaneous surface current distributions are illustrated in Fig. 7(a)
Fig. 7 Instantaneous surface current induced in the arcs (a) surface current at 13.33 GHz, (b) current at 15.56 GHz, (c) current at 16.75 GHz, (d) (e) and (f) current vectors in the middle of each arc at 13.33 GHz, 15.56 GHz, and 16.75 GHz.
. It shows that the currents rotate clockwise at these three metal arcs. Then we take the current vectors from the surface current distributions to explain the rotation of the transmitted circularly polarized wave, as seen in Fig. 7(d). The numbers 1, 2, and 3 represent the current vectors in the middle of the corresponding metal layer. Obviously, the vectors from metallic layer 1 to layer 3 form a counter-clockwise rotation, which is coincident with the rotation of electric field for RCP wave. Therefore, it is well understood that the linearly polarized incident wave is transformed to RCP wave.

Figure 7(b) shows the instantaneous current distributions in each arc at 15.56GHz. The rotating direction of current in metal layer 2 is the same with that in metal layer 3, but is opposite to the current in metal layer 1. Then the current vectors in the middle of each arc are also taken to investigate the transformation behavior, as shown in Fig. 7(e). The three current vectors in the middle of each arc rotate clockwise by the order from 1 to 3, which illustrates that a LCP wave is produced at 15.56 GHz. Figures 7(c) and 7(f) present the induced current distributions and the current vectors in the arcs at 16.75 GHz. It is worth to note that the current vector in metallic layer 2 at 16.75 GHz is perpendicular to metal layer 2, which is denoted by a dashed arrow. In addition, the current in metal layer 2 is much weaker in comparison with that in the other two arcs. It is also considered that the vectors rotate counter-clockwise from layer 1 to layer 3 in this case, resulting in the transformation of RCP wave.

The above results are suitable for y-polarized electric field. When the polarization of the electric field is changed from y to x-direction, different conversion coefficients for LCP and RCP waves can be obtained, as shown in Fig. 8
Fig. 8 Transformation behavior of the triple-band circular polarizer illuminated by an x-polarized wave.
. The three resonances are at 12.96 GHz, 15.26 GHz, and 17.17 GHz, respectively. This difference is due to the lack of C4 symmetric for this triple-band polarizer, that is to say, the sample cannot be superimposed on itself when it is rotated by 90° around its center in XOY plane.

3. Discussion

Leff2πθ120×(Rw2)+2h
(4)

The unit of parameter θ is degree, and the value of θ is optimized to be 80° in this triple-band circular polarizer. According to the above formula, the structure parameters can influence the value of Leff. Since the operating wavelength is directly proportional to Leff, the resonant frequencies would shift towards lower (higher) frequency when R or h increases (decreases). In order to validate the assumption, the analysis of the parameters is carried out by varying one parameter while keeping the other parameters fixed. Figures 9(a)
Fig. 9 Transformation spectra for different radius R (a) transformation of LCP wave (b) transformation of RCP wave.
and 9(b) present the transformation for LCP and RCP waves with different values of R. We can see that the positions of transformation frequencies for LCP and RCP waves all shift towards lower frequency when R increases. Since the effective wavelength is directly proportional to Leff, the increase of effective length Leff of the arcs would result in red shift of the resonant frequency when R varies from 4.5 mm to 5.0 mm.

Figure 10
Fig. 10 Transformation spectra for different width w (a) transformation of LCP wave (b) transformation of RCP wave.
depicts the transformation spectra when the width w of arc is varied from 0.75 mm to 0.95 mm. The resonant positions shift towards higher frequency while w increases. As the current would flow along the possible shortest path, so the increase of w reduces the effective radius R of the arc, resulting in the decrease of effective wavelength according to the Eq. (4) above. Transformation behavior for different thickness h of dielectric lamina from 1.27 mm to 1.778 mm with a step of 0.254 mm is depicted in Fig. 11
Fig. 11 Transformation spectra for different thickness h (a) transformation of LCP wave (b) transformation of RCP wave.
. Similar to the shift trend of changing radius R, the resonant positions shift towards lower frequency when h increases. The thickness h analogously acts as the helix pitch of the metallic helix antenna, so the resonant frequencies red shift when h increases according to the antenna theory.

It can be seen that the shift of the resonant frequency is especially obvious at high frequencies in the above three simulations. It could be easily explained that since the wavelength is shorter at higher frequency, the same variation of geometrical parameters would cause larger frequency shift. It should be noted that the dependence of the resonant positions on the radius R is stronger than that on w and h, thus R can be taken as the main parameter to determine the resonant positions while w and h are suitable for slight tuning.

At last, it is necessary to discuss the effect of varying angle Φ1 on the transformation performance. The value of Φ1 determines the relative angle of each arc to x axis. So changing Φ1 would result in the variation of the x and y component of induced electric field, that is to say, the variation of Φ1 can influence the phase differences between the transmission coefficients of |Txy| and |Tyy|, and its ratio |Txy|/|Tyy|. As shown in Fig. 12(a)
Fig. 12 Transformation spectra for different Φ1 (a) transformation of LCP wave (b) transformation of RCP wave.
and 12(b), the resonant frequencies are shifted and transformation values are varied as well when Φ1 changes from 10° to 30°, proving the important role of Φ1 in the design of the circular polarizer stated above.

4. Conclusion

In conclusion, a method of constructing multi-band circular polarizer is proposed. By adopting multi-layered planar spiral structure with a certain twist angle between neighboring layers, obvious different transformation responses for the LCP and RCP waves have been demonstrated around three frequency bands by illumination of a linearly polarized wave. In addition, the circular polarizer with more resonant frequencies could be expected by adopting more dielectric layers and properly designing the parameters of metallic arcs. The design process has conceptually provided an approach to realize the multi-band planar metamaterial in microwave regime, and it is still feasible to be scaled to higher frequency range, such as terahertz and even optical region.

Acknowledgment

This work was supported by 973 Program of China (No.2011CB301800) and National Natural Science Funds for Distinguished Young Scholar (No.60825405).

References and links

1.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]

2.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]

3.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

4.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

5.

W. Sun, Q. He, J. Hao, and L. Zhou, “A transparent metamaterial to manipulate electromagnetic wave polarizations,” Opt. Lett. 36(6), 927–929 (2011). [CrossRef] [PubMed]

6.

M. Euler, V. Fusco, R. Cahill, and R. Dickie, “325 GHz single layer sub-millimeter wave FSS based split slot ring linear to circular polarization convertor,” IEEE Trans. Antenn. Propag. 58(7), 2457–2459 (2010). [CrossRef]

7.

M. G. Silveirinha, “Design of linear-to-circular polarization transformers made of long densely packed metallic helices,” IEEE Trans. Antenn. Propag. 56(2), 390–401 (2008). [CrossRef]

8.

D. H. Kwon, P. L. Werner, and D. H. Werner, “Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation,” Opt. Express 16(16), 11802–11807 (2008). [CrossRef] [PubMed]

9.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009). [CrossRef] [PubMed]

10.

S. X. Li, Z. Y. Yang, J. Wang, and M. Zhao, “Broadband terahertz circular polarizers with single- and double-helical array metamaterials,” J. Opt. Soc. Am. A 28(1), 19–23 (2011). [CrossRef] [PubMed]

11.

C. Wu, H. Li, X. Yu, F. Li, H. Chen, and C. T. Chan, “Metallic helix array as a broadband wave plate,” Phys. Rev. Lett. 107(17), 177401 (2011). [CrossRef] [PubMed]

12.

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric chiral metamaterial circular polarizer based on four U-shaped split ring resonators,” Opt. Lett. 36(9), 1653–1655 (2011). [CrossRef] [PubMed]

13.

X. Xiong, W. H. Sun, Y. J. Bao, M. Wang, R. W. Peng, C. Sun, X. Lu, J. Shao, Z. F. Li, and N. B. Ming, “Construction of chiral metamaterial with U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010). [CrossRef]

14.

J. Zhou, J. Dong, B. Wang, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79(12), 121104 (2009). [CrossRef]

15.

Z. Li, R. Zhao, T. Koschny, M. Kafesaki, K. B. Alici, E. Colak, H. Caglayan, E. Ozbay, and C. M. Soukoulis, “Chiral metamaterials with negative refractive index based on four “U” split ring resonators,” Appl. Phys. Lett. 97(8), 081901 (2010). [CrossRef]

16.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 205–207.

OCIS Codes
(260.5430) Physical optics : Polarization
(260.5740) Physical optics : Resonance
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: April 16, 2012
Manuscript Accepted: June 19, 2012
Published: June 29, 2012

Citation
Xiaoliang Ma, Cheng Huang, Mingbo Pu, Chenggang Hu, Qin Feng, and Xiangang Luo, "Multi-band circular polarizer using planar spiral metamaterial structure," Opt. Express 20, 16050-16058 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-16050


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References

  1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett.76(25), 4773–4776 (1996). [CrossRef] [PubMed]
  2. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
  5. W. Sun, Q. He, J. Hao, and L. Zhou, “A transparent metamaterial to manipulate electromagnetic wave polarizations,” Opt. Lett.36(6), 927–929 (2011). [CrossRef] [PubMed]
  6. M. Euler, V. Fusco, R. Cahill, and R. Dickie, “325 GHz single layer sub-millimeter wave FSS based split slot ring linear to circular polarization convertor,” IEEE Trans. Antenn. Propag.58(7), 2457–2459 (2010). [CrossRef]
  7. M. G. Silveirinha, “Design of linear-to-circular polarization transformers made of long densely packed metallic helices,” IEEE Trans. Antenn. Propag.56(2), 390–401 (2008). [CrossRef]
  8. D. H. Kwon, P. L. Werner, and D. H. Werner, “Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation,” Opt. Express16(16), 11802–11807 (2008). [CrossRef] [PubMed]
  9. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325(5947), 1513–1515 (2009). [CrossRef] [PubMed]
  10. S. X. Li, Z. Y. Yang, J. Wang, and M. Zhao, “Broadband terahertz circular polarizers with single- and double-helical array metamaterials,” J. Opt. Soc. Am. A28(1), 19–23 (2011). [CrossRef] [PubMed]
  11. C. Wu, H. Li, X. Yu, F. Li, H. Chen, and C. T. Chan, “Metallic helix array as a broadband wave plate,” Phys. Rev. Lett.107(17), 177401 (2011). [CrossRef] [PubMed]
  12. M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric chiral metamaterial circular polarizer based on four U-shaped split ring resonators,” Opt. Lett.36(9), 1653–1655 (2011). [CrossRef] [PubMed]
  13. X. Xiong, W. H. Sun, Y. J. Bao, M. Wang, R. W. Peng, C. Sun, X. Lu, J. Shao, Z. F. Li, and N. B. Ming, “Construction of chiral metamaterial with U-shaped resonator assembly,” Phys. Rev. B81(7), 075119 (2010). [CrossRef]
  14. J. Zhou, J. Dong, B. Wang, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B79(12), 121104 (2009). [CrossRef]
  15. Z. Li, R. Zhao, T. Koschny, M. Kafesaki, K. B. Alici, E. Colak, H. Caglayan, E. Ozbay, and C. M. Soukoulis, “Chiral metamaterials with negative refractive index based on four “U” split ring resonators,” Appl. Phys. Lett.97(8), 081901 (2010). [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 205–207.

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