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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 26268–26273
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Electric and magnetic excitation of coherent magnetic plasmon waves in a one-dimensional meta-chain

C. Zhu, H. Liu, S. M. Wang, T. Li, J. X. Cao, Y. J. Zheng, L. Li, Y. Wang, S. N. Zhu, and X. Zhang  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 26268-26273 (2010)
http://dx.doi.org/10.1364/OE.18.026268


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Abstract

A one-dimensional diatomic meta-chain with equal-size holes and different-length slits is designed. Broadband coherent magnetic plasmon waves (MPW) are formed in such a system, excited by both the electric resonance in the slits and the magnetic resonance in the holes in a wide range of incidence angles ( 0 ° 40 ° ) and broad frequency bands (200–230 THz). The dispersion properties of the MPW measured in our experiments agree with the theoretical calculation based on the Lagrange model. The coherent MPWs reported in this paper may have applications in subwavelength integrated nanocircuits.

© 2010 OSA

1. Introduction

Although the invention of metamaterials has stimulated the interest of many researchers and has important applications in negative refraction [1

1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

6

6. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]

], invisible cloaking [7

7. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

9

9. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

], and many other transform optical designs, the basic design idea is very simple: composing effective media from many small structured elements and controlling their artificial electromagnetic (EM) properties. According to the effective media model, the coupling interactions between the elements in metamaterials are somewhat ignored; therefore, the effective properties of metamaterials can be viewed as the “averaged effect” of the resonance property of the individual elements. However, the coupling interaction between elements should always exist when they are arranged into metamaterials. Occasionally, especially when the elements are very close, this coupling effect is not negligible and will have a substantial effect on the properties of the metamaterials. Recent studies have shown that the resonance coupling effect between split-ring resonators (SRR) can introduce magnetoinductive waves [10

10. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92(10), 6252–6261 (2002). [CrossRef]

16

16. M. Decker, S. Linden, and M. Wegener, “Coupling effects in low-symmetry planar split-ring resonator arrays,” Opt. Lett. 34(10), 1579–1581 (2009). [CrossRef] [PubMed]

], electroinductive waves [17

17. M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, “Electroinductive waves in chains of complementary metamaterial elements,” Appl. Phys. Lett. 88(8), 083503 (2006). [CrossRef]

,18

18. N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules,” Adv. Mater. 20(23), 4521–4525 (2008). [CrossRef]

], planar transmission lines [19

19. J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech. 53(4), 1451–1461 (2005). [CrossRef]

], stereometamaterials [20

20. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

22

22. H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrange model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81(24), 241403 (2010). [CrossRef]

] and radiation suppression [23

23. T. Q. Li, H. Liu, T. Li, S. M. Wang, J. X. Cao, Z. H. Zhu, Z. G. Dong, S. N. Zhu, and X. Zhang, “Suppression of radiation loss by hybridization effect in two coupled split-ring resonators,” Phys. Rev. B 80(11), 115113 (2009). [CrossRef]

]. Besides SRRs, the near-field coupling between other structures, such as nano-rods [24

24. J. Cao, H. Liu, T. Li, S. Wang, T. Li, S. Zhu, and X. Zhang, “Steering polarization of infrared light through hybridization effect in a tri-rod structure,” J. Opt. Soc. Am. B 26(12), B96–B101 (2009). [CrossRef]

], fish-net [25

25. T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, “Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission,” Opt. Express 14(23), 11155–11163 (2006). [CrossRef] [PubMed]

] and nanosanwiches [26

26. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Magnetic plasmon modes in periodic chains of nanosandwiches,” Opt. Express 16(6), 3560–3565 (2008). [CrossRef] [PubMed]

] are also reported.

In this letter, we propose a new design for the meta-chain of SHRs. Here, the unit cell includes two SHRs with different-length slits and equal-sized holes, which is different from our former work. The advantage of this new design is that the coherent magnetic plasmon wave can be excited both by the magnetic resonance in holes and the electric resonance in slits. Due to the strong electric resonance in the slits, the coherent magnetic plasmon in meta-chain can be excited much more efficiently. The excitation can also be realized under normal incidence. The incidence excitation angle can then be tuned in a wide range, from normal incidence to 40°, and a continuous wide excitation frequency band can be obtained through tuning the incidence angle. The measured dispersion of the coherent magnetic plasmon waves agrees with our theoretical calculated results.

2. Numerical Model and experimental results

Figure 1(a)
Fig. 1 Structure (a) and equivalent LC circuit (b) of a single SHR.
shows the structure of our designed slit-hole resonator, which is comprised of a nanohole and a nanoslit connecting the hole and the edge. The bulk material is silver. Compared with Pendry’s SRR structure and other magnetic resonance structures, the SHR structure is much easier to fabricate with the focus ion beam (FIB) technique, and its resonance frequency is easily realized in the infrared range. This structure can be described by an equivalent LC circuit method, as shown as Fig. 1(b). The slit in the SHR can be seen as a capacitor, and the nanohole can be seen as a conductor that connects it. The Lagrangian of SHR structure can be written as =LQ˙2/2Q2/(2C), where L is the inductance of the hole, Q is the total oscillation charge in the SHR, and C is the capacitance of the slit.

3. Simulation results and discussions

The resonance frequency of the two SHRs can also be simulated numerically using the finite-difference time-domain method (FDTD) with a commercial software package (CST Microwave Studio). The longer SHR with a 180 nm slit has a resonance frequency of ω1=135THz. Using the same method, the resonance frequency of the SHR with a 60 nm slit can also be found at ω2=91THz. For the infinite diatomic SHR chain, its equivalent LC circuit could be described as Fig. 2(b), in which the longer SHR has a capacitance of C1 and the shorter SHR has a capacitance of C2. Therefore, the Lagrangian equation of the chain could be expressed as
=m(LQ˙m22+Lq˙m22(Qmqm1)22C1(Qmqm)22C2)
(1)
Here, we define Qm as the oscillation charge of the m-th longer SHR and qm as the oscillation charge of the m-th shorter SHR. Based on the Euler-Lagrangian equation, the dispersion of MPW could be obtained as
ω±2=(ω12+ω22)±(ω14+ω24)+2ω12ω22cos(kd)
(2)
where k is the wave vector, ω is the angular frequency, and d = 740 nm is the period of the chain. In the above simulations, we have already obtainedω1=135THz, ω2=91THz. Substituting them into Eq. (2) we can obtain two different dispersion branches, which are depicted as two separate black curves in Fig. 4(a)
Fig. 4 (a). Dispersion properties of MPW; (b). Current distribution of induced current in one unit cell for the optical mode, in which the arrays represent the direction and the intensity of the incident electric field; (c). Current distribution for the acoustic mode.
. CST simulations show that the two branches stand for two different MPW modes by normal incidence. For the upper branch ω+, the neighboring SHR units oscillate in the opposite phase [see Fig. 4(b)], which can be called the optical branch, as reported by Sydoruk [29

29. O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87(7), 072501–072503 (2005). [CrossRef]

]. For the lower branch ω, the neighboring units oscillate in the same phase [see Fig. 4(c)], which can be called the acoustic branch. The light in the free space is depicted as the blue line in Fig. 4(a). The optical branch of MPW is above the blue line; therefore, it can be excited by the plane wave from the far-field with an oblique incidence angle. This introduces an extraordinary optical transmission in the experiment. To compare our experimental results with the Lagrange theory, the dispersion of MPW under different incidence angles was calculated from Eq. (2) based on the wave vector matching condition k=k0sinθ. The theoretical results are shown as the black solid curve in Fig. 3 (b), which agree with our measurements.

In our former work, the SHR chain had equal-length slits and different-sized holes, and its EOT can only be excited for the incidence angle θ60. In this work, for the normal incidence wave (θ=00), an EOT resonance peak is observed at 230 THz [see Fig. 3(a)]. This is caused by the electric resonance in the SHR, which was not reported in our former work. When the light is normally incident on the sample, no magnetic field penetrates the hole; therefore, the magnetic resonance does not contribute to the MPW. Only the electric field, which is always in the y direction, excites an electric resonance in the slits. If the neighboring slits are of equal lengths, as in our former work, the electric resonances in the two slits will have equal strengths and cancel out each other. Then, the electric resonance cannot contribute to the excitation of MPW for the normal incidence wave. For the design structure, the two neighboring slits have different lengths; therefore, the electric resonances have different strengths and do not cancel out each other [see Fig. 4 (b)]. The final contribution of electric resonance to MPW should be that the electric resonance of the longer slit subtracts that of the shorter slit. When the incidence angle increases, increasing magnetic flux will pass through the hole, and the magnetic resonance in the holes will become stronger. As a result, the EOT peak becomes stronger when the incidence angle increases from 0° to 24°, as shown in Fig. 3(a). For θ24°, the EOT peak will begin to decrease because the projected area of the fabricated pattern in the incidence direction decreases and the whole transmitted energy is reduced.

In this work, the experimental results show that the optical branch of MPW can be excited both by electric and magnetic resonance from far-field incidence waves. However, the acoustic MPW mode cannot be coupled to the far-field because its dispersion curve is below the light line [see Fig. 4 (a)]. Some near-field techniques have to be used to excite this mode, such as scanning near-field optical microscopy (SNOM). However, without coupling with far-field waves, the radiation loss of this mode could be very low, making it useful for subwavelength energy transport. Further experimental investigation on the SHR chain should be conducted in the future.

4. Conclusion

In summary, we have proposed and studied the coherent magnetic plasmon mode in a one-dimensional meta-chain. Our theoretical calculations and experimental results have proven that the coherent MPW in such a structure can be excited not only by the magnetic resonance, but also by the electric resonance; therefore, it has better properties compared with our former structure. The resonance frequency of the excited MPW can be easily tuned in a broad width by directly changing the incidence angle. A Lagrange model was employed to describe the coherent mode, and the calculated results agree with the experimental results. The coherent MPWs reported in this paper may have applications in subwavelength integrated nanocircuits.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No.10704036, No.10874081, No.60907009, No.10904012, No.10974090 and No. 60990320), and by the National Key Projects for Basic Researches of China (No. 2006CB921804, No. 2009CB930501 and No. 2010CB630703).

References and links

1.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

2.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004). [CrossRef] [PubMed]

3.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]

4.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]

5.

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]

6.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]

7.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

8.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]

9.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]

10.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92(10), 6252–6261 (2002). [CrossRef]

11.

O. Sydoruk, A. Radkovskaya, O. Zhuromskyy, E. Shamonina, M. Shamonin, C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, “Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell,” Phys. Rev. B 73(22), 224406 (2006). [CrossRef]

12.

A. Radkovskaya, O. Sydoruk, M. Shamonin, E. Shamonina, C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, “Experimental study of a bi-periodic magnetoinductive waveguide: comparison with theory,” IET Proc. Microwaves Antennas Propag. 1, 80–83 (2007). [CrossRef]

13.

I. V. Shadrivov, A. N. Reznik, and Y. S. Kivshar, “Magnetoinductive waves in arrays of split-ring resonators,” Physica B 394(2), 180–183 (2007). [CrossRef]

14.

N. Liu and H. Giessen, “Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling,” Opt. Express 16(26), 21233–21238 (2008). [CrossRef] [PubMed]

15.

M. Decker, S. Burger, S. Linden, and M. Wegener, “Magnetization waves in split-ring-resonator arrays: Evidence for retardation effects,” Phys. Rev. B 80(19), 193102 (2009). [CrossRef]

16.

M. Decker, S. Linden, and M. Wegener, “Coupling effects in low-symmetry planar split-ring resonator arrays,” Opt. Lett. 34(10), 1579–1581 (2009). [CrossRef] [PubMed]

17.

M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, “Electroinductive waves in chains of complementary metamaterial elements,” Appl. Phys. Lett. 88(8), 083503 (2006). [CrossRef]

18.

N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules,” Adv. Mater. 20(23), 4521–4525 (2008). [CrossRef]

19.

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech. 53(4), 1451–1461 (2005). [CrossRef]

20.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]

21.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007). [CrossRef]

22.

H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrange model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81(24), 241403 (2010). [CrossRef]

23.

T. Q. Li, H. Liu, T. Li, S. M. Wang, J. X. Cao, Z. H. Zhu, Z. G. Dong, S. N. Zhu, and X. Zhang, “Suppression of radiation loss by hybridization effect in two coupled split-ring resonators,” Phys. Rev. B 80(11), 115113 (2009). [CrossRef]

24.

J. Cao, H. Liu, T. Li, S. Wang, T. Li, S. Zhu, and X. Zhang, “Steering polarization of infrared light through hybridization effect in a tri-rod structure,” J. Opt. Soc. Am. B 26(12), B96–B101 (2009). [CrossRef]

25.

T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, “Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission,” Opt. Express 14(23), 11155–11163 (2006). [CrossRef] [PubMed]

26.

S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Magnetic plasmon modes in periodic chains of nanosandwiches,” Opt. Express 16(6), 3560–3565 (2008). [CrossRef] [PubMed]

27.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006). [CrossRef]

28.

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79(2), 024304 (2009). [CrossRef]

29.

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87(7), 072501–072503 (2005). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.5740) Physical optics : Resonance
(220.4241) Optical design and fabrication : Nanostructure fabrication

ToC Category:
Optics at Surfaces

History
Original Manuscript: September 13, 2010
Revised Manuscript: November 13, 2010
Manuscript Accepted: November 14, 2010
Published: December 1, 2010

Citation
C. Zhu, H. Liu, S. M. Wang, T. Li, J. X. Cao, Y. J. Zheng, L. Li, Y. Wang, S. N. Zhu, and X. Zhang, "Electric and magnetic excitation of coherent magnetic plasmon waves in a one-dimensional meta-chain," Opt. Express 18, 26268-26273 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-26268


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References

  1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
  2. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004). [CrossRef] [PubMed]
  3. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]
  4. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]
  5. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]
  6. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]
  7. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  8. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
  9. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
  10. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92(10), 6252–6261 (2002). [CrossRef]
  11. O. Sydoruk, A. Radkovskaya, O. Zhuromskyy, E. Shamonina, M. Shamonin, C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, “Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell,” Phys. Rev. B 73(22), 224406 (2006). [CrossRef]
  12. A. Radkovskaya, O. Sydoruk, M. Shamonin, E. Shamonina, C. J. Stevens, G. Faulkner, D. J. Edwards, and L. Solymar, “Experimental study of a bi-periodic magnetoinductive waveguide: comparison with theory,” IET Proc. Microwaves Antennas Propag. 1, 80–83 (2007). [CrossRef]
  13. I. V. Shadrivov, A. N. Reznik, and Y. S. Kivshar, “Magnetoinductive waves in arrays of split-ring resonators,” Physica B 394(2), 180–183 (2007). [CrossRef]
  14. N. Liu and H. Giessen, “Three-dimensional optical metamaterials as model systems for longitudinal and transverse magnetic coupling,” Opt. Express 16(26), 21233–21238 (2008). [CrossRef] [PubMed]
  15. M. Decker, S. Burger, S. Linden, and M. Wegener, “Magnetization waves in split-ring-resonator arrays: Evidence for retardation effects,” Phys. Rev. B 80(19), 193102 (2009). [CrossRef]
  16. M. Decker, S. Linden, and M. Wegener, “Coupling effects in low-symmetry planar split-ring resonator arrays,” Opt. Lett. 34(10), 1579–1581 (2009). [CrossRef] [PubMed]
  17. M. Beruete, F. Falcone, M. J. Freire, R. Marques, and J. D. Baena, “Electroinductive waves in chains of complementary metamaterial elements,” Appl. Phys. Lett. 88(8), 083503 (2006). [CrossRef]
  18. N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and Electroinductive Coupling in Plasmonic Metamaterial Molecules,” Adv. Mater. 20(23), 4521–4525 (2008). [CrossRef]
  19. J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microw. Theory Tech. 53(4), 1451–1461 (2005). [CrossRef]
  20. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). [CrossRef]
  21. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007). [CrossRef]
  22. H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrange model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81(24), 241403 (2010). [CrossRef]
  23. T. Q. Li, H. Liu, T. Li, S. M. Wang, J. X. Cao, Z. H. Zhu, Z. G. Dong, S. N. Zhu, and X. Zhang, “Suppression of radiation loss by hybridization effect in two coupled split-ring resonators,” Phys. Rev. B 80(11), 115113 (2009). [CrossRef]
  24. J. Cao, H. Liu, T. Li, S. Wang, T. Li, S. Zhu, and X. Zhang, “Steering polarization of infrared light through hybridization effect in a tri-rod structure,” J. Opt. Soc. Am. B 26(12), B96–B101 (2009). [CrossRef]
  25. T. Li, H. Liu, F. M. Wang, Z. G. Dong, S. N. Zhu, and X. Zhang, “Coupling effect of magnetic polariton in perforated metal/dielectric layered metamaterials and its influence on negative refraction transmission,” Opt. Express 14(23), 11155–11163 (2006). [CrossRef] [PubMed]
  26. S. M. Wang, T. Li, H. Liu, F. M. Wang, S. N. Zhu, and X. Zhang, “Magnetic plasmon modes in periodic chains of nanosandwiches,” Opt. Express 16(6), 3560–3565 (2008). [CrossRef] [PubMed]
  27. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006). [CrossRef]
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