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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 14280–14292
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A standing-wave interpretation of plasmon resonance excitation in split-ring resonators

Wen-Yu Chen and Chun-Hung Lin  »View Author Affiliations


Optics Express, Vol. 18, Issue 13, pp. 14280-14292 (2010)
http://dx.doi.org/10.1364/OE.18.014280


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Abstract

In this study, we investigated the plasmon resonances of split-ring resonators (SRRs) numerically at incident angles of 0 and 45° under illumination with linearly and circularly polarized waves. At 45° incidence, perpendicular polarized waves excited distinct odd plasmon modes; the difference in the reflections of right and left circularly polarized incident radiation was very large. From simulated near field plots, we found that the parallelism of the incident electric field and the induced plasmon current was the key factor affecting excitation. We propose the use of a parallelism factor (P-factor), based on a standing-wave approach, to characterize the ability of incident fields to excite multiple plasmon resonance currents. The mechanism of the field and current parallelism can explain the resonance behavior of SRRs when considering the polarization state, incident angle, and geometry of the SRR.

© 2010 OSA

1. Introduction

Split-ring resonators (SRRs) were proposed as a negative refractive index material [1

1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

,2

2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]

]. Negative permeability is an extraordinary phenomenon and can be produced with the magnetic resonance of SRRs [3

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

]. Interesting applications of SRRs were investigated, such as infrared cloacking [4

4. B. Kanté, A. de Lustrac, J. M. Lourtioz, and S. N. Burokur, “Infrared cloaking based on the electric response of split ring resonators,” Opt. Express 16(12), 9191–9198 (2008). [CrossRef] [PubMed]

] and superlensing [5

5. M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Negative refraction, surface modes, and superlensing effect via homogenization near resonances for a finite array of split-ring resonators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(4 Pt 2), 046309 (2009). [CrossRef] [PubMed]

]. As a localized plasmonic sensor, multi plasmon modes provide multi-functional detection [6

6. Y. T. Chang, Y. C. Lai, C. T. Li, C. K. Chen, and T. J. Yen, “A multi-functional plasmonic biosensor,” Opt. Express 18(9), 9561–9569 (2010). [CrossRef] [PubMed]

]. The plasmon resonances of SRR enhance Raman scattering for molecule detection [7

7. A. W. Clark, A. Glidle, D. R. S. Cumming, and J. M. Cooper, “Nanophotonic split-ring resonators as dichroics for molecular spectroscopy,” Appl. Phys. Lett. 93(2), 023121 (2008). [CrossRef]

]. Localized surface plasmon resonance (LSPR) is the oscillation of charge density confined in metallic nanostructures excited under direct illumination [8

8. S. A. Maier, “Localized surface plasmons,” in Plasmonics: fundamentals and applications (Springer, 2007), pp. 65–88.

]. The magnetic and electric resonances are the earliest discussed plasmon modes in SRR. There are two methods for exciting a magnetic resonance. One is a magnetic coupling of the magnetic field perpendicular to an SRR plane, the other is an electric coupling of the electric field parallel to an SRR gap. In contrast, the electric resonance couples to the electric field perpendicular to an SRR gap [3

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

,9

9. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84(15), 2943–2945 (2004). [CrossRef]

]. Then, the multiple plasmon modes in SRRs at normal incidence were discovered [10

10. C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express 14(19), 8827–8836 (2006). [CrossRef] [PubMed]

,11

11. A. K. Sheridan, A. W. Clark, A. Glidle, J. M. Cooper, and D. R. S. Cumming, “Multiple plasmon resonances from gold nanostructures,” Appl. Phys. Lett. 90(14), 143105 (2007). [CrossRef]

]. The plasmon modes can be recognized by counting the number of nodes in the near field plot. The LC resonance can be treated as the fundamental mode of an SRR and the electric resonance the second mode. For the polarization state parallel to the gap, the charge distributions of the plasmon modes are in odd symmetry; they are the odd modes. For perpendicular polarization, the plasmon resonance modes are even modes. The antenna resonances [12

12. K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94(7), 4632–4642 (2003). [CrossRef]

15

15. F. Neubrech, T. Kolb, R. Lovrincic, G. Fahsold, A. Pucci, J. Aizpurua, T. W. Cornelius, M. E. Toimil-Molares, R. Neumann, and S. Karim, “Resonances of individual metal nanowires in the infrared,” Appl. Phys. Lett. 89(25), 253104 (2006). [CrossRef]

] found in nanowires suggested that the multiple resonances of an SRR could be interpreted using the standing-wave model [13

13. G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher, A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu, B. Monacelli, and G. Boreman, “Plasmon dispersion relation of Au and Ag nanowires,” Phys. Rev. B 68(15), 155427 (2003). [CrossRef]

,15

15. F. Neubrech, T. Kolb, R. Lovrincic, G. Fahsold, A. Pucci, J. Aizpurua, T. W. Cornelius, M. E. Toimil-Molares, R. Neumann, and S. Karim, “Resonances of individual metal nanowires in the infrared,” Appl. Phys. Lett. 89(25), 253104 (2006). [CrossRef]

,16

16. C. Y. Chen, S. C. Wu, and T. J. Yen, “Experimental verification of standing-wave plasmonic resonances in split-ring resonators,” Appl. Phys. Lett. 93(3), 034110 (2008). [CrossRef]

]. The validity of this model was confirmed by the existence of induced plasmon current nodes in the simulation and by the linear relationship between the resonance wavelength and total length divided by the mode number (L/m) [14

14. G. Laurent, N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Evidence of multipolar excitations in surface enhanced Raman scattering,” Phys. Rev. B 71(4), 045430 (2005). [CrossRef]

,16

16. C. Y. Chen, S. C. Wu, and T. J. Yen, “Experimental verification of standing-wave plasmonic resonances in split-ring resonators,” Appl. Phys. Lett. 93(3), 034110 (2008). [CrossRef]

].

Chiral metamaterials are metamaterials that lack mirror symmetry; they are said to possess intrinsic chirality [17

17. B. Bai, Y. Svirko, J. Turunen, and T. Vallius, “Optical activity in planar chiral metamaterials: Theoretical study,” Phys. Rev. A 76(2), 023811 (2007). [CrossRef]

,18

18. 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]

]. An optical system is said to be an extrinsic chiral system if the structure of the metamaterial is symmetric but the arrangement of the system is not identical to its mirror image [19

19. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Extrinsic electromagnetic chirality in metamaterials,” J. Opt. A, Pure Appl. Opt. 11(7), 074009 (2009). [CrossRef]

,20

20. E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009). [CrossRef] [PubMed]

]. One such example, described herein, is that of SRRs measured at oblique incidence. The distinguishing feature of a chiral optical system is different optical phenomena resulting from irradiation with right- (RCP) and left-circularly-polarized (LCP) electromagnetic (EM) waves—for example, optical activity [21

21. M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95(22), 227401 (2005). [CrossRef] [PubMed]

,22

22. B. F. Bai, K. Konishi, X. F. Meng, P. Karvinen, A. Lehmuskero, M. Kuwata-Gonokami, Y. Svirko, and J. Turunen, “Mechanism of the large polarization rotation effect in the all-dielectric artificially chiral nanogratings,” Opt. Express 17(2), 688–696 (2009). [CrossRef] [PubMed]

], circular dichroism [23

23. 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]

], and asymmetric transmission [24

24. V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, V. V. Khardikov, and S. L. Prosvirnin, “Asymmetric transmission of light and enantiomerically sensitive plasmon resonance in planar chiral nanostructures,” Nano Lett. 7(7), 1996–1999 (2007). [CrossRef]

]. Those chiral phenomena have possible applications on circular polarizer and waveplate [19

19. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Extrinsic electromagnetic chirality in metamaterials,” J. Opt. A, Pure Appl. Opt. 11(7), 074009 (2009). [CrossRef]

,25

25. 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]

]. We have found that the reflections of SRRs at 45° incidence are very different for RCP and LCP EM radiation, a phenomenon caused by extrinsic chirality.

2. Simulation structures and methods

The simulated structure was an array of SRRs [Fig. 1(a)
Fig. 1 (a) Simulated SRR array. (b) Unit cell of the SRR pattern. The blue (ki) and red (kr) arrows are the wave vectors of incidence and reflection; the green arrow (n) is the normal vector. The incident plane is perpendicular to the arms of the SRR.
]. Each SRR was made from gold. The geometrical parameters were the period (p), base length (b), arm length (a), wire thickness (t), and wire width (w). The total length of the SRR (L) was equal to 2a + b. L was set to be much longer than t and w because our discussion was based on the wire geometry assumption. In all of the simulations, the wire thickness (t) and the wire width (w) were both 50 nm and the total length (L) was 1100 nm. The choice of the structure dimension was based on the capability of currently nano-fabrication technology and the resulting resonance range from visible to short-wavelength infrared regions. The refractive index of the environment was 1. The SRRs were freestanding structures so that the analysis would focus on the plasmon resonances of the gold structures while neglecting the effects of a substrate. At oblique incidence, the incident plane was perpendicular to the arms of the SRR. The incidence angles (θ) in the simulations were 0 and 45°. Four polarization states were simulated: perpendicular polarization, parallel polarization, RCP, and LCP. The perpendicular and parallel polarizations were defined as incident electric fields perpendicular and parallel to the incident plane, respectively. Any other polarization states can be linear combinations of the above states. A rigorous coupled-wave analysis (RCWA) algorithm [26

26. L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997). [CrossRef]

,27

27. C. H. Lin, H. L. Chen, W. C. Chao, C. I. Hsieh, and W. H. Chang, “Optical characterization of two-dimensional photonic crystals based on spectroscopic ellipsometry with rigorous coupled-wave analysis,” Microelectron. Eng. 83(4-9), 1798–1804 (2006). [CrossRef]

] was the numerical method used in this study. RCWA has been used widely to analyze the optical diffraction of periodic structures. The advantage of using RCWA is that the reflected and transmitted fields from the periodic structures are expressed in terms of Rayleigh expansions, which can be directly employed to calculate the diffraction efficiencies. Its computing time is significantly faster and the required memory size is much smaller as compared with the finite-difference time domain (FDTD) method. The refractive indexes of gold were obtained through interpolation of tabulated values provided in the literature [28

28. D. W. Lynch, and W. R. Hunter, “Gold (Au),” in Handbook of optical constants of solids, E. D. Palik, ed. (Academic Press, Inc., 1985), pp. 286–287.

].

3. Results and discussions

3.1. Plasmon resonances at normal incidence

Figure 2
Fig. 2 Reflections of SRRs at normal incidence. The reflection peaks are labeled with the plasmon mode, with subscripts indicating the polarization states of the incident waves.
displays the reflection spectra of SRRs at normal incidence. The base length (b) was 350 nm, the arm length (a) of the SRR was 375 nm. The period (d) was 600 nm. The multiple reflection peaks correspond to the plasmon resonance modes [10

10. C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express 14(19), 8827–8836 (2006). [CrossRef] [PubMed]

,11

11. A. K. Sheridan, A. W. Clark, A. Glidle, J. M. Cooper, and D. R. S. Cumming, “Multiple plasmon resonances from gold nanostructures,” Appl. Phys. Lett. 90(14), 143105 (2007). [CrossRef]

]. The incident EM wave drives the conducting electrons in SRRs and, thus, induces plasmon currents. The electrons reradiate forward and backward. The forward radiation tends to cancel the incident wave and the backward radiation appears as the reflection [29

29. E. Hecht, Optics, 4 ed. (Person Education, Inc., 2002), pp. 333–334.

]. Therefore, the reflection peaks indicate the spectral positions of strong plasmon currents.

The peaks in Fig. 2 are labeled with the plasmon mode, with subscripts indicating the polarization states of the incident waves. We observe that the parallel polarized wave excites odd modes, namely mode 1 at 3110 nm and mode 3 at 1120 nm. The perpendicular polarized wave excites even modes, namely mode 2 at 1420 nm and mode 4 at 800 nm. The reflection spectra of the RCP and LCP EM waves overlap because of the symmetric configuration at normal incidence. The circularly polarized waves comprised equal amounts of parallel and perpendicular polarized waves. It is distinct that the circularly polarized waves could excite all of the plasmon modes from 1 to 4, with the strength of the reflection being exactly the average value of the parallel and perpendicular polarized waves.

3.2. Plasmon resonances at 45° incidence

3.3. Parallelism factor

The electric fields of EM waves are represented as
E=y^E0cos(kzωt),
(1a)
E=x^E0cos(kzωt),
(1b)
ER=E02[x^cos(kzωt)+y^sin(kzωt)],
(1c)
EL=E02[x^cos(kzωt)y^sin(kzωt)],
(1d)
where k is the wave vector and ω is the angular frequency. Figure 6
Fig. 6 Configuration of an SRR in Cartesian coordinates. The path of integration is represented by the blue dotted arrows along the structure. The incident wavefront is parallel to the xy plane. The incident angle is θ.
displays the configuration of the SRR in Cartesian coordinates; the incident angle is θ and the xy plane is parallel to the incident wavefront. The induced plasmon currents in SRRs exist in the form of standing waves. Therefore, we can simplify the currents to be represented in the form of sinusoidal waves:
I=^I0sin(mπL)cos(ωt+φ),
(2)
where m is the plasmon mode; the number of the current node is equal to m – 1; and ϕ is the phase difference between the induced plasmon current and the applied electric field. The current would automatically adjust its phase to tend to align with the incident electric field. The descriptor represents the path of integration along the structure. The total length of the SRR is L. Integration of the dot product of the electric field and the current is performed as follows:
pi(ϕ)=πE0I0LEiIdT,
(3)
where pi is a function of ϕ and is normalized to 1; i represents the polarization state. We define the P-factor as the maximum value of the integration

Pi=pi(ϕmax).
(4)

The EI part in Eq. (3) is the power dissipation per length [31

31. U. S. Inan, and A. S. Inan, Electromagnetic Waves (Prentice-Hall, Inc., 2000).

]. Equation (3) can be regarded as the normalized time-averaged power dissipation of the entire SRR. Hence, the P-factor is actually the power transferred from the electric field to the plasmon current, but it is normalized to unity.

The concept of P-factor can be explained by taking the following discussion into consideration. Figure 8
Fig. 8 Schematic representations of the incident electric fields and the induced plasmon currents of mode 1 at oblique incidences. Polarization state of the incident EM wave: (a) RCP (dotted red arrows are the currents; solid arrows are the incident electric fields) and (b) LCP.
presents schematic representations of the incident electric fields of RCP and LCP waves and the induced plasmon currents of mode 1. The wavefront is parallel to the xy plane; the dotted red arrows are the plasmon currents; the solid arrows are the incident electric fields. At oblique incidence, the wavefront initially propagates to the left arm of the SRR, passes the base portion of the SRR, and then reaches the right arm. The phases of the EM wave are different in the arms and in the base. In Fig. 8(a), the incident electric field of the RCP wave in the middle of the SRR base is aligned in the + y direction; its projection to the SRR base is parallel to the direction of the plasmon current in the base. The projection of the electric field on the left arm is in + x direction; the projection on the right arm is in the –x direction. Because these projections are aligned anti-parallel to the direction of the currents, they suppress the currents in the arms. When we integrate Eq. (3), we find that the integration along the arms is negative and cancels out the integration along the base. In Fig. 8(b), the electric field of the LCP wave on the arms enhances the currents. Because the LCP wave is parallel to the current of mode 1, P L is higher than P R, and the mode 1 resonance under LCP wave excitation is stronger than that of RCP wave excitation at oblique incidence.

Figure 7(c) displays the P-factors of mode 3. The value of P L is zero at an incident angle of 27°. The phase ϕ max changes by π at that point; i.e., the incident wave reverses the induced current direction when the incident angle is larger than 27°. This preliminary result obtained from Eq. (4) must be checked in an RCWA simulation. Figure 9(a)
Fig. 9 (a) Near field plots for the mode 3 resonance under LCP wave incident angles of 0 and 45°. (b, c) Schematic representations of the incident electric fields of LCP waves and the induced plasmon currents of mode 3 at (b) normal and (c) 45° incidence.
presents the near field plots in the mode 3 resonance under LCP wave incident angles of 0 and 45°; the blue thick arrows represent the directions of the incident electric fields. The incident electric field was uniform over the entire SRR plane in the case of normal incidence. At 45° incidence, the incident field varied with respect to the horizontal position. The blue arrows reveal the direction of the incident field in the middle of the SRR; the dashed arrows reveal that the currents under 0 and 45° incidences were in opposite directions. The near field plots prove that a phase change occurred for the plasmon current. Figures 9(b) and 9(c) are schematic representations of the incident electric fields and the induced plasmon currents of mode 3 of the LCP waves; they are 3D sketches of the first column in Fig. 9(a). The incident angles were 0 and 45°, respectively; the wavefront was aligned parallels to the xy plane. In Fig. 9(b), in the case of normal incidence, the electric field drove the current in the base. The electric fields on the arms did not enhance or suppress the current. In Fig. 9(c), in the case of 45° incidence, the excitation on the arms was sufficiently strong to reverse the direction of the current in the base, making it oppose the electric field in the base and tend to align parallel to the electric fields on the arms. There must be a phase change point between 0 and 45°, confirming the prediction made using the P-factor. As the incident angle increases from zero, the electric field on the arms increases its component in the x direction. When the incident angle is sufficiently large, the electric fields on the arms are sufficiently strong to reverse the plasmon current.

3.4. P-Factor as a function of geometry

4. Conclusions

From the simulated near field plots, we found that parallelism of the incident electric field and the induced plasmon current is the key point affecting excitation. We propose the use of a P-factor to characterize the ability of the incident fields to excite the plasmon currents. The analytical form of the P-factor can be regarded as a measure of the power transferred from the incident wave to the resonance current. Our results revealed that the reflection of the plasmon modes was in positive proportion to the P-factor. The mechanism of the field and current parallelism can be used to explain the resonance behavior of SRRs in terms of the polarization state, incident angle, and geometry of the SRR. The P-factor can be an effective index to estimate the strength of excited plasmon resonances. The analysis of P-factors should be effective not only for SRRs but also for the other wire-like structures.

Acknowledgment

This study was supported by the National Science Council of Taiwan grants NSC 98-2221-E-006-018- and NSC 98-2218-E-009-001.

References and links

1.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

2.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]

3.

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

4.

B. Kanté, A. de Lustrac, J. M. Lourtioz, and S. N. Burokur, “Infrared cloaking based on the electric response of split ring resonators,” Opt. Express 16(12), 9191–9198 (2008). [CrossRef] [PubMed]

5.

M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Negative refraction, surface modes, and superlensing effect via homogenization near resonances for a finite array of split-ring resonators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(4 Pt 2), 046309 (2009). [CrossRef] [PubMed]

6.

Y. T. Chang, Y. C. Lai, C. T. Li, C. K. Chen, and T. J. Yen, “A multi-functional plasmonic biosensor,” Opt. Express 18(9), 9561–9569 (2010). [CrossRef] [PubMed]

7.

A. W. Clark, A. Glidle, D. R. S. Cumming, and J. M. Cooper, “Nanophotonic split-ring resonators as dichroics for molecular spectroscopy,” Appl. Phys. Lett. 93(2), 023121 (2008). [CrossRef]

8.

S. A. Maier, “Localized surface plasmons,” in Plasmonics: fundamentals and applications (Springer, 2007), pp. 65–88.

9.

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84(15), 2943–2945 (2004). [CrossRef]

10.

C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express 14(19), 8827–8836 (2006). [CrossRef] [PubMed]

11.

A. K. Sheridan, A. W. Clark, A. Glidle, J. M. Cooper, and D. R. S. Cumming, “Multiple plasmon resonances from gold nanostructures,” Appl. Phys. Lett. 90(14), 143105 (2007). [CrossRef]

12.

K. B. Crozier, A. Sundaramurthy, G. S. Kino, and C. F. Quate, “Optical antennas: Resonators for local field enhancement,” J. Appl. Phys. 94(7), 4632–4642 (2003). [CrossRef]

13.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher, A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu, B. Monacelli, and G. Boreman, “Plasmon dispersion relation of Au and Ag nanowires,” Phys. Rev. B 68(15), 155427 (2003). [CrossRef]

14.

G. Laurent, N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, and F. R. Aussenegg, “Evidence of multipolar excitations in surface enhanced Raman scattering,” Phys. Rev. B 71(4), 045430 (2005). [CrossRef]

15.

F. Neubrech, T. Kolb, R. Lovrincic, G. Fahsold, A. Pucci, J. Aizpurua, T. W. Cornelius, M. E. Toimil-Molares, R. Neumann, and S. Karim, “Resonances of individual metal nanowires in the infrared,” Appl. Phys. Lett. 89(25), 253104 (2006). [CrossRef]

16.

C. Y. Chen, S. C. Wu, and T. J. Yen, “Experimental verification of standing-wave plasmonic resonances in split-ring resonators,” Appl. Phys. Lett. 93(3), 034110 (2008). [CrossRef]

17.

B. Bai, Y. Svirko, J. Turunen, and T. Vallius, “Optical activity in planar chiral metamaterials: Theoretical study,” Phys. Rev. A 76(2), 023811 (2007). [CrossRef]

18.

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]

19.

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Extrinsic electromagnetic chirality in metamaterials,” J. Opt. A, Pure Appl. Opt. 11(7), 074009 (2009). [CrossRef]

20.

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009). [CrossRef] [PubMed]

21.

M. Kuwata-Gonokami, N. Saito, Y. Ino, M. Kauranen, K. Jefimovs, T. Vallius, J. Turunen, and Y. Svirko, “Giant optical activity in quasi-two-dimensional planar nanostructures,” Phys. Rev. Lett. 95(22), 227401 (2005). [CrossRef] [PubMed]

22.

B. F. Bai, K. Konishi, X. F. Meng, P. Karvinen, A. Lehmuskero, M. Kuwata-Gonokami, Y. Svirko, and J. Turunen, “Mechanism of the large polarization rotation effect in the all-dielectric artificially chiral nanogratings,” Opt. Express 17(2), 688–696 (2009). [CrossRef] [PubMed]

23.

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]

24.

V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, V. V. Khardikov, and S. L. Prosvirnin, “Asymmetric transmission of light and enantiomerically sensitive plasmon resonance in planar chiral nanostructures,” Nano Lett. 7(7), 1996–1999 (2007). [CrossRef]

25.

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]

26.

L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997). [CrossRef]

27.

C. H. Lin, H. L. Chen, W. C. Chao, C. I. Hsieh, and W. H. Chang, “Optical characterization of two-dimensional photonic crystals based on spectroscopic ellipsometry with rigorous coupled-wave analysis,” Microelectron. Eng. 83(4-9), 1798–1804 (2006). [CrossRef]

28.

D. W. Lynch, and W. R. Hunter, “Gold (Au),” in Handbook of optical constants of solids, E. D. Palik, ed. (Academic Press, Inc., 1985), pp. 286–287.

29.

E. Hecht, Optics, 4 ed. (Person Education, Inc., 2002), pp. 333–334.

30.

D. J. Griffiths, Introduction to electrodynamics, 3 ed. (Prentice-Hall International, Inc., 1999), pp. 65–77.

31.

U. S. Inan, and A. S. Inan, Electromagnetic Waves (Prentice-Hall, Inc., 2000).

32.

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84(1-2), 219–227 (2006). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: May 4, 2010
Revised Manuscript: June 11, 2010
Manuscript Accepted: June 13, 2010
Published: June 18, 2010

Citation
Wen-Yu Chen and Chun-Hung Lin, "A standing-wave interpretation of plasmon resonance excitation in split-ring resonators," Opt. Express 18, 14280-14292 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-14280


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References

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  25. 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]
  26. L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997). [CrossRef]
  27. C. H. Lin, H. L. Chen, W. C. Chao, C. I. Hsieh, and W. H. Chang, “Optical characterization of two-dimensional photonic crystals based on spectroscopic ellipsometry with rigorous coupled-wave analysis,” Microelectron. Eng. 83(4-9), 1798–1804 (2006). [CrossRef]
  28. D. W. Lynch, and W. R. Hunter, “Gold (Au),” in Handbook of optical constants of solids, E. D. Palik, ed. (Academic Press, Inc., 1985), pp. 286–287.
  29. E. Hecht, Optics, 4 ed. (Person Education, Inc., 2002), pp. 333–334.
  30. D. J. Griffiths, Introduction to electrodynamics, 3 ed. (Prentice-Hall International, Inc., 1999), pp. 65–77.
  31. U. S. Inan, and A. S. Inan, Electromagnetic Waves (Prentice-Hall, Inc., 2000).
  32. C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, “Resonances of split-ring resonator metamaterials in the near infrared,” Appl. Phys. B 84(1-2), 219–227 (2006). [CrossRef]

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