## Gold helix photonic metamaterials: A numerical parameter study

Optics Express, Vol. 18, Issue 2, pp. 1059-1069 (2010)

http://dx.doi.org/10.1364/OE.18.001059

Acrobat PDF (1069 KB)

### Abstract

We have recently shown that metamaterials composed of three-dimensional gold helices periodically arranged on a square lattice can be used as compact “thin-film” circular polarizers with one octave bandwidth. The physics of the motif of these artificial crystals is closely related to that of microwave sub-wavelength helical antennas in end-fire geometry. Here, we systematically study the dependence of the metamaterial’s chiral optical properties on helix pitch, helix radius, two-dimensional lattice constant, wire radius, number of helix pitches, and angle of incidence. Our numerical calculations show that the optical properties are governed by resonances of the individual helices, yet modified by interaction effects. Furthermore, our study shows possibilities and limitations regarding performance optimization.

© 2010 OSA

## 1. Introduction

1. E. Hecht, *Optics* (Addison-Wesley, San Francisco, 2002, 4th edition). [PubMed]

2. K. F. Lindman, “Über eine durch ein isotropes System von spiralförmigen Resonatoren erzeugte Rotationspolarisation der elektromagnetischen Wellen,” Ann. Phys. **368**(23), 621–644 (1920). [CrossRef]

3. 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]

4. P. A. Belov, C. R. Simovski, and S. A. Tretyakov, “Example of bianisotropic electromagnetic crystals: the spiral medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(5), 056622 (2003). [CrossRef] [PubMed]

6. R. Abdeddaïm, G. Guida, A. Priou, B. Gallas, and J. Rivory, “Negative permittivity and permeability of gold square nanospirals,” Appl. Phys. Lett. **94**(8), 081907 (2009). [CrossRef]

7. Z.-Y. Zhang and Y.-P. Zhao, “Optical properties of helical and multiring Ag nanostructures: The effect of pitch height,” J. Appl. Phys. **104**(1), 013517 (2008). [CrossRef]

*R*. Notably, under the above conditions, the helical antenna operation wavelength neither depends on the length of the antenna (i.e., on the number of helix pitches) nor on the helix pitch

*p*alone. For infinitely many helix turns

*N*, one obtains perfectly circularly polarized emission. In the language of antenna theory, the on-axis “axis ratio”

*AR*(which is given by the ratio of the major and minor radius of the polarization ellipse) [8] becomes unity. For finite values of

*N*, the axis ratio scales according to

*AR*= (2

*N*+ 1)/(2

*N*) [8].

*a*, will lead to reflection and/or absorption of light incident onto the array if the handedness of the light is identical to that of the helices. If metal losses are negligible, absorption does not occur and all light that is not transmitted will be reflected. In contrast, light with the opposite circular handedness will be transmitted. In this reasoning, we have tacitly assumed that the lateral interaction among the different metal helices in the two-dimensional array is negligible. Indeed, on page 225 of [8], the inventor of the helical antenna, John Kraus, states in boldface: “… Not only does the helix have a nearly uniform resistive input over a wide bandwidth but it also operates as a ‘supergain’ end-fire array over the same bandwidth! Furthermore, it is noncritical with respect to conductor size and turn spacing. It is also easy to use in arrays because of almost negligible mutual impedance …”. We will see below that the latter assumption is not quite correct under our conditions. Nevertheless, the qualitative behavior does remain for arrays of helices.

3. 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]

3. 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]

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

*dielectric*structure [10

10. M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. **32**(17), 2547–2549 (2007). [CrossRef] [PubMed]

11. M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. **19**(2), 207–210 (2007). [CrossRef]

11. M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. **19**(2), 207–210 (2007). [CrossRef]

## 2. Numerical calculations

14. www.jcmwave.com

^{3}and (400 nm)

^{3}.

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

*ω*

_{pl}= 1.37 × 10

^{16}rad/s and the collision frequency is

*ω*

_{col}= 1.2 × 10

^{14}rad/s. All calculations in this work with the notable exception of Fig. 7 assume propagation of light along the helix axis (see Fig. 1). To avoid artificial sharp edges, the ends of the gold wire are terminated by gold half spheres with radius

*r*. All simulated metamaterial structures are located on a glass substrate (half-space geometry) with real refractive index

*n*= 1.5. Light impinges from the air side. We show left- and right-handed circular polarization of the incident light and analyze the emerging polarization in terms of left- and right-handed circular polarization. We explicitly show results for left-handed gold helices. By symmetry, for right-handed helices, for all handedness of the incident and transmitted light, strictly identical results are obtained if “left” and “right” are consistently interchanged.

### 2.1 Varying the number of pitches N

*N*. The incident light has left-handed circular polarization (LCP) or right-handed circular polarization (RCP). Precisely, “transmittance” refers to the ratio of LCP (RCP) transmitted light intensity and LCP (RCP) incident light intensity. Similarly, “conversion” refers to the ratio of LCP (RCP) transmitted light intensity and RCP (LCP) incident light intensity. In Fig. 2, the parameters

*R*= 0.6 µm,

*r*= 0.1 µm,

*a*= 2 µm, and

*p*= 2 µm (compare Fig. 1) are fixed and intentionally identical to those in Fig. 1 of our previous work [3

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

*n*= 1.5 times the lattice constant

*a*= 2 µm, i.e., to 3 µm wavelength. Hence, for wavelengths shorter than this critical value, the structures can certainly no longer be viewed as an effective material and, furthermore, they can no longer be really used as a circular polarizer. Thus, throughout this article, we restrict ourselves to the spectral window of Fig. 2 (3 µm to 12 µm wavelength, equivalent to two octaves). For

*N*= 1, two distinct and sharp resonances are found [3

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

*N*= 1, a broad and more or less unstructured transmittance minimum evolves with increasing

*N*. This aspect, which has been discussed qualitatively above, connects to our previous work [3

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

*N*than in [3

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

*N*. This means that none of the resonances shifts proportional to the total length of the metal wire or proportional to the height of the structure, i.e., proportional to

*N*×

*p*.

*N*= 1.

### 2.2 Varying the pitch height p

*p*for

*N*= 1. The other parameters are fixed to their respective values in Fig. 2. It becomes obvious that the resonance positions do shift to some extent. However, while the pitch

*p*varies by more than a factor of three, the resonance positions merely shift by some tens of percent. We can conclude that the physics of metal helices is quite distinct from that of dielectric helices with closely similar geometry [11

11. M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. **19**(2), 207–210 (2007). [CrossRef]

**19**(2), 207–210 (2007). [CrossRef]

*p*. This means that the response of dielectric helices is dominated by the Bragg resonance, whereas that of the metal helices is determined by the interplay of pronounced internal resonances and their mutual coupling [3

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

### 2.3 Varying the lattice constant a

*N*= 1. Here, the square lattice constant,

*a*, is varied. All other parameters are fixed to their respective values in Fig. 2. The long-wavelength resonance shifts towards shorter wavelengths with decreasing lattice constant

*a*. For this resonance [3

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

**325**(5947), 1513–1515 (2009). [CrossRef] [PubMed]

*a*being only slightly larger than the helix diameter 2

*R*. Nevertheless, a direct mechanical/electrical contact of adjacent metal helices should be avoided as this is expected to lead to very large deviations. For direct contact, the picture of separated antennas is definitely no longer applicable.

### 2.4 Varying the helix radius R

*R*. Figure 5 investigates this aspect. Indeed, we find that the resonance wavelengths shift more or less proportional to

*R*. Even the absolute positions of the resonance wavelengths roughly coincide with the prediction of antenna theory outlined in the introduction. Clearly, decreasing

*R*while keeping the lattice constant

*a*fixed dilutes the array, leading to more shallow resonances (compare Fig. 4).

### 2.5 Varying the wire radius r

*r*, is varied. All other parameters are fixed to their respective values in Fig. 2. With increasing

*r*, the resonances experience a slight shift towards shorter wavelengths. This trend can be understood by appreciating that the electrical current follows the shortest possible path, i.e., it mainly flows at the inner radius of the helix. Thus, increasing

*r*effectively approximately reduces the helix radius

*R*according to

*R*→ (

*R*-

*r*), the net effect of which corresponds to that shown in Fig. 5.

### 2.6 Dependence on angle of incidence α

### 2.7 Conversion and reflectance

*p*= 0 µm in Fig. 3 which is not chiral, hence not relevant). For suitable parameter combinations, the intensity conversion stays below a few percent for the entire bandwidth of one octave.

1. E. Hecht, *Optics* (Addison-Wesley, San Francisco, 2002, 4th edition). [PubMed]

16. D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Material parameter retrieval procedure for general bi-isotropic metamaterials and its application to optical chiral negative-index metamaterial design,” Opt. Express **16**(16), 11822–11829 (2008). [CrossRef] [PubMed]

16. D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Material parameter retrieval procedure for general bi-isotropic metamaterials and its application to optical chiral negative-index metamaterial design,” Opt. Express **16**(16), 11822–11829 (2008). [CrossRef] [PubMed]

## 3. Conclusion

## Acknowledgments

## References and links

1. | E. Hecht, |

2. | K. F. Lindman, “Über eine durch ein isotropes System von spiralförmigen Resonatoren erzeugte Rotationspolarisation der elektromagnetischen Wellen,” Ann. Phys. |

3. | 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 |

4. | P. A. Belov, C. R. Simovski, and S. A. Tretyakov, “Example of bianisotropic electromagnetic crystals: the spiral medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | M. G. Silveirinha, “Design of linear-to-circular polarization transformers made of long densely packed metallic helices,” IEEE Trans. Antenn. Propag. |

6. | R. Abdeddaïm, G. Guida, A. Priou, B. Gallas, and J. Rivory, “Negative permittivity and permeability of gold square nanospirals,” Appl. Phys. Lett. |

7. | Z.-Y. Zhang and Y.-P. Zhao, “Optical properties of helical and multiring Ag nanostructures: The effect of pitch height,” J. Appl. Phys. |

8. | J. D. Kraus, and R. J. Marhefka, |

9. | H.-S. Kitzerow, and C. Bahr, eds., |

10. | M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. |

11. | M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. |

12. | |

13. | |

14. | |

15. | I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, |

16. | D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Material parameter retrieval procedure for general bi-isotropic metamaterials and its application to optical chiral negative-index metamaterial design,” Opt. Express |

**OCIS Codes**

(260.5430) Physical optics : Polarization

(160.1585) Materials : Chiral media

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: November 13, 2009

Revised Manuscript: December 22, 2009

Manuscript Accepted: December 28, 2009

Published: January 7, 2010

**Citation**

Justyna K. Gansel, Martin Wegener, Sven Burger, and Stefan Linden, "Gold helix photonic metamaterials: A numerical parameter study," Opt. Express **18**, 1059-1069 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-1059

Sort: Year | Journal | Reset

### References

- E. Hecht, Optics (Addison-Wesley, San Francisco, 2002, 4th edition). [PubMed]
- K. F. Lindman, “Über eine durch ein isotropes System von spiralförmigen Resonatoren erzeugte Rotationspolarisation der elektromagnetischen Wellen,” Ann. Phys. 368(23), 621–644 (1920). [CrossRef]
- 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]
- P. A. Belov, C. R. Simovski, and S. A. Tretyakov, “Example of bianisotropic electromagnetic crystals: the spiral medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056622 (2003). [CrossRef] [PubMed]
- 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]
- R. Abdeddaïm, G. Guida, A. Priou, B. Gallas, and J. Rivory, “Negative permittivity and permeability of gold square nanospirals,” Appl. Phys. Lett. 94(8), 081907 (2009). [CrossRef]
- Z.-Y. Zhang and Y.-P. Zhao, “Optical properties of helical and multiring Ag nanostructures: The effect of pitch height,” J. Appl. Phys. 104(1), 013517 (2008). [CrossRef]
- J. D. Kraus, and R. J. Marhefka, Antennas: For All Applications (McGraw-Hill, New York, 2003, 3rd edition).
- H.-S. Kitzerow, and C. Bahr, eds., Chirality in Liquid Crystals (Springer, Heidelberg, 2001, 1st edition).
- M. Thiel, G. von Freymann, and M. Wegener, “Layer-by-layer three-dimensional chiral photonic crystals,” Opt. Lett. 32(17), 2547–2549 (2007). [CrossRef] [PubMed]
- M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007). [CrossRef]
- www.cst.com/Content/Products/MWS/Overview.aspx
- www.lumerical.com
- www.jcmwave.com
- I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Boston, 1994).
- D. H. Kwon, D. H. Werner, A. V. Kildishev, and V. M. Shalaev, “Material parameter retrieval procedure for general bi-isotropic metamaterials and its application to optical chiral negative-index metamaterial design,” Opt. Express 16(16), 11822–11829 (2008). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.