## Wave propagation retrieval method for chiral metamaterials |

Optics Express, Vol. 18, Issue 15, pp. 15498-15503 (2010)

http://dx.doi.org/10.1364/OE.18.015498

Acrobat PDF (1055 KB)

### Abstract

In this paper we present the wave propagation method for the retrieving of effective properties of media with circularly polarized eigenwaves, in particularly for chiral metamaterials. The method is applied for thick slabs and provides bulk effective parameters. Its strong sides are the absence of artificial branches of the refractive index and simplicity in implementation. We prove the validity of the method on three case studies of homogeneous magnetized plasma, bi-cross and U-shaped metamaterials.

© 2010 OSA

## 1. Introduction

1. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. **97**(17), 177401 (2006). [CrossRef] [PubMed]

3. M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. **34**(16), 2501–2503 (2009). [CrossRef] [PubMed]

4. M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. **32**(7), 856–858 (2007). [CrossRef] [PubMed]

6. 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]

7. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and Energy in Chiral Nihility,” J. Electromagn. Waves Appl. **17**(5), 695–706 (2003). [CrossRef]

12. 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]

13. C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett. **93**(23), 233106 (2008). [CrossRef]

15. E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B **79**(3), 035407 (2009). [CrossRef]

*n*is retrieved. Different branches of

*n*can be very close to each other thus creating difficulties in choosing the correct refractive index. The SM is applicable only to thin MTM slabs consisting of few MTM monolayers as it relies on the transmission simulations. In a thick slab transmission can be at the noise level that distorts the restored parameters severely. At the same time few layers of MTM often do not show bulk effective parameters [16

16. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) **1**(2), 62–80 (2007). [CrossRef]

17. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B **80**(19), 193101 (2009). [CrossRef]

17. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B **80**(19), 193101 (2009). [CrossRef]

## 2. Wave propagation retrieval methodology

17. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B **80**(19), 193101 (2009). [CrossRef]

*x*-polarized wave propagating along the

*z*-axis in vacuum:It normally impinges onto the flat interface of an isotropic chiral material with the RCP and LCP eigenwaves:where

*k*

_{0}is the vacuum wavenumber,

*n*

_{R}and

*n*

_{L}are the RCP and LCP refractive indices correspondingly. The linear polarized wave can be decomposed in a sum of RCP and LCP waves as read in formula (1).

*t*

_{R}=

*t*

_{L}=

*t*, so the electric field of the wave inside the chiral material is: By simple algebraic manipulations, from Eq. (4) and (5) we derive

*Z*

_{B}(see the relevant discussion in review [18

18. C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. **107**(5), 726–753 (2009). [CrossRef]

*R*on the vacuum–chiral MTM interface

*m*, where

*m*is an integer. In order to apply a linear fit while using Eqs. (6) and (7), the dependence of

*z*

_{UC}should be made continuous. If the metamaterial is quasi-homogeneous the difference of the wave phase between neighboring unit cells should be less than π/2, thus aforementioned continuity is easy to persuade.

**80**(19), 193101 (2009). [CrossRef]

*d*is the slab thickness. For the case of low absorption, one can always increase

*d*. Another option is to use a time-domain method and to terminate the simulation when the excitation pulse reaches the rear interface of the slab. Second, the accuracy of the simulations should be sufficient to observe the linear dependence of

*z*, for that the electric field doesn’t have to drop down to the noise level within less than three unit cells.

## 3. Case studies

19. “CST Computer Simulation Technology AG,” http://cst.com/

### 3.1 Homogeneous magnetized plasma

*z*. We should note that such medium is not a chiral medium, but its effective refractive index can be calculated analytically, so it is a reliable reference. Its permittivity tensor is

_{∞}= 3. For the simulation the slab of the plasma medium was divided into 100 layers. The WPRMC restores EPs in perfect agreement with the analytical ones (Fig. 1 ).

### 3.2 Bi-cross metamaterial

21. J. Dong, J. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express **17**(16), 14172–14179 (2009). [CrossRef] [PubMed]

22. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. **31**(12), 1800–1802 (2006). [CrossRef] [PubMed]

*n*= 1.44).

15. E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B **79**(3), 035407 (2009). [CrossRef]

### 3.3 U-shaped metamaterial

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

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

24. Z. Li, H. Caglayan, E. Colak, J. Zhou, C. M. Soukoulis, and E. Ozbay, “Coupling effect between two adjacent chiral structure layers,” Opt. Express **18**(6), 5375–5383 (2010). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. |

2. | E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen, “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. |

3. | M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. |

4. | M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. |

5. | D. H. Kwon, P. L. Werner, and D. H. Werner, “Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation,” Opt. Express |

6. | 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 |

7. | S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and Energy in Chiral Nihility,” J. Electromagn. Waves Appl. |

8. | J. B. Pendry, “A chiral route to negative refraction,” Science |

9. | S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. |

10. | E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B |

11. | B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. |

12. | J. Zhou, J. Dong, B. Wang, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B |

13. | C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett. |

14. | 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 |

15. | E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B |

16. | C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) |

17. | A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B |

18. | C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. |

19. | “CST Computer Simulation Technology AG,” http://cst.com/ |

20. | A. Akhiezer, |

21. | J. Dong, J. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express |

22. | G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. |

23. | X. Xiong, W. Sun, Y. Bao, M. Wang, R. Peng, C. Sun, X. Lu, J. Shao, Z. Li, and N. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B |

24. | Z. Li, H. Caglayan, E. Colak, J. Zhou, C. M. Soukoulis, and E. Ozbay, “Coupling effect between two adjacent chiral structure layers,” Opt. Express |

**OCIS Codes**

(160.4760) Materials : Optical properties

(260.5430) Physical optics : Polarization

(160.1585) Materials : Chiral media

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 22, 2010

Revised Manuscript: May 19, 2010

Manuscript Accepted: May 20, 2010

Published: July 7, 2010

**Citation**

Andrei Andryieuski, Radu Malureanu, and Andrei V. Lavrinenko, "Wave propagation retrieval method for chiral metamaterials," Opt. Express **18**, 15498-15503 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15498

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### References

- A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke, and N. I. Zheludev, “Giant gyrotropy due to electromagnetic-field coupling in a bilayered chiral structure,” Phys. Rev. Lett. 97(17), 177401 (2006). [CrossRef] [PubMed]
- E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen, “Giant optical gyrotropy due to electromagnetic coupling,” Appl. Phys. Lett. 90(22), 223113 (2007). [CrossRef]
- M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34(16), 2501–2503 (2009). [CrossRef] [PubMed]
- M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32(7), 856–858 (2007). [CrossRef] [PubMed]
- 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]
- 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]
- S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and Energy in Chiral Nihility,” J. Electromagn. Waves Appl. 17(5), 695–706 (2003). [CrossRef]
- J. B. Pendry, “A chiral route to negative refraction,” Science 306(5700), 1353–1355 (2004). [CrossRef] [PubMed]
- S. Zhang, Y. S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009). [CrossRef] [PubMed]
- E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009). [CrossRef]
- B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94(15), 151112 (2009). [CrossRef]
- 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]
- C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett. 93(23), 233106 (2008). [CrossRef]
- 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]
- E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009). [CrossRef]
- C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) 1(2), 62–80 (2007). [CrossRef]
- A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B 80(19), 193101 (2009). [CrossRef]
- C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. 107(5), 726–753 (2009). [CrossRef]
- “CST Computer Simulation Technology AG,” http://cst.com/
- A. Akhiezer, Plasma Electrodynamics, Volume One: Linear Theory (Monographs in Natural Philosophy), 1st ed. (Pergamon, 1975).
- J. Dong, J. Zhou, T. Koschny, and C. Soukoulis, “Bi-layer cross chiral structure with strong optical activity and negative refractive index,” Opt. Express 17(16), 14172–14179 (2009). [CrossRef] [PubMed]
- G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31(12), 1800–1802 (2006). [CrossRef] [PubMed]
- X. Xiong, W. Sun, Y. Bao, M. Wang, R. Peng, C. Sun, X. Lu, J. Shao, Z. Li, and N. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010). [CrossRef]
- Z. Li, H. Caglayan, E. Colak, J. Zhou, C. M. Soukoulis, and E. Ozbay, “Coupling effect between two adjacent chiral structure layers,” Opt. Express 18(6), 5375–5383 (2010). [CrossRef] [PubMed]

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