Abstract

A numerical and theoretical study is presented on the exhibition of tunable narrow band coupled-induced transparency phenomenon in a hybrid waveguide-plasmon system consisting of gold twin nanowires array embedded in a slab waveguide. We show that, at slightly non-normal incidence, a properly designed splitting of transmission with narrow transparency peaks may occur at a given wavelength, depending on the angle of incidence. This leads to the wavelength-selective high quality coupled-induced transparency resonance at optical frequencies. By adjusting the gap distance of the pair gratings, the coupled-induced transparency band can be switched between on-state and off-state, which provides us possibilities to develop controllable plasmonic functional devices employing plasmonic nanostructures.

©2013 Optical Society of America

1. Introduction

The merge of photonic resonance with plasmonic electronic resonance has invoked considerable interest for its exotic and fascinating optical property involves periodic subwavelength metallic nanostructures [1, 2]. The excitation and manipulation of surface plasmon polaritons (SPPs), the collective oscillations of conduction electrons trapped at metal-dielectric interface, is of vital significance for various of plasmonic nanophotonics applications, such as extraordinary optical transmission (EOT) [3], electromagnetic-induced-transparency (EIT) [47], enhanced Raman scattering [8], plasmonic sensor [9], imaging [1012], negative index metamaterials (NIM) [1315].

Among these rich phenomena, the coupling and interaction of hybrid plasmonic system is of particular interest, which provides feasible and efficient ways to manipulate the photons and tailor the optical response of system. All-optical EIT-like effect in artificial designed plasmonic system has been observed and investigated in numerous of researches [47, 1620]. However, due to the intrinsic loss of plasmon resonance, the linewidth (quality factor) of many plasmonic systems is much limited. Recent theoretical and experimental results have shown that by employing the coherent coupling between photonics mode and plasmonic mode, the performance of the hybrid system and the spectra profile of EIT-effect can be finely engineered [6, 7, 19].

In this paper, we report that properly designed coupled hybrid plasmonic system can exhibit controllable coupled induced transparency band at slightly non-normal incidence. The incident light can be coupled to both direction propagating waveguide mode through the diffraction Bloch modes of the grating; it is the interference between the localized plasmonic resonance and the delocalized Bloch modes that leads to the narrow coupled induced transparency band in the broad symmetric plasmon resonance. The important structural modification allows a distinct observable coupled induced transparency bands in the transmission spectra. The gap value between the pair metallic nanowires plays the role of a switch that enables the control of on-state and off-state of the coupled-induced transparency peak.

2. Structures and method

The proposed system consists of periodic sandwiched metal-dielectric-metal (MDM) nanowires embedded in the high-index dielectric slab waveguide layer on top of a lower dielectric index substrate. In our analysis, the nanowires made of gold with a cross-section of 100 × 15 nm2 are used. In this modeling, the dielectric constants of Au is fitted by Drude-Lorentz model [21] and the refractive index of the substrate and the waveguide layer are set as ns = 1.52 and nwg = 2.1, respectively.

The scheme of the periodic MDM array system with gold nanowires thickness t, width w, dielectric spacer layer height h, slab waveguide layer thickness T, and periodicity p is shown in Fig. 1. The incident light with the electric field polarization along the x direction illuminates the system with an incident angle θ. The rigorous coupled wave analyses (RCWA) method is employed to model the optical properties of the structure [6, 19].

 figure: Fig. 1

Fig. 1 Schematic view of the hybrid slab metallic pair nanowires array structure. The dielectric waveguide layer of thickness T on top of quartz substrate with embedded double metallic grating of gap distance h is investigated.

Download Full Size | PPT Slide | PDF

3. Transmission properties of the system for varied geometry

The importance of the waveguide layer in the hybrid system, which contributes to the waveguide-plasmon coupled-induced transparency arises from the coupling between the localized plasmon resonance and the waveguide mode, has been investigated and demonstrated in various previous studies [6, 16, 17]. The result exhibiting narrowing of the transmission band in dependence on the waveguide layer thickness is depicted in Fig. 2(a). The cut-off wavelength of this hybrid transmission transparency band is related to the thickness of the waveguide layer. The thinner waveguide layer thickness will leads to a shorter cut-off wavelength. Below the cutoff of the waveguide mode, only a kink with abrupt transmission due to the Wood Anomaly (WA) is observed. The transmission window becomes increasingly sharp as the waveguide approaches its cutoff. The sharp transmission peak in the broad plasmonic resonance with a very high quality factor indicates extremely slowed down group velocity of light traversing the system, which is difficult to achieve in purely plasmonic systems with intrinsic metal damping [4, 7, 18]. The corresponding group index ng can be approximately evaluated byng=λ2/4TΔλ [19, 22], whereΔλis the FWHM of the transparent window.

 figure: Fig. 2

Fig. 2 (a) Transmission spectra as a function of the waveguide layer thickness (p = 550 nm, h = 40 nm). The left black dashed line shows the guide for the eye. The right black dashed line shows the localized plasmon resonance. The curves have been plotted with subsequent vertical offsets of 1 for clarity. (b) The calculated transmission (straight black line) and phase (dotted blue line) for the result in (a) with a waveguide layer thickness T = 200 nm.

Download Full Size | PPT Slide | PDF

In order to gain deeper understanding and detailed demonstration on the EIT effect in the system, we plot the phase change and the calculated transmission for this configuration. The result is shown in Fig. 2(b). The transmission peaks with a FWHM of Δλ=25nmwithin the broadband low transmission plasmonic resonance zone together with a dramatic phase change of nearlyπ/2can be seen, which indicates a good quality factor Q = 35 EIT resonance with slow group velocity of photon of ng = 38.4.

With the waveguide layer of thickness T = 200 nm, the influence of the nanowires period on the optical property of the system is investigated and analyzed. Excitation of plasmonic resonances and waveguide mode leads to strong scattering and absorption with low transmission [6, 7, 19]. The result plotted in Fig. 3 shows the clear transmission spectra with narrow coupling-induced transparency effect. It is also worth noting that clear anti-crossing occurs when the plasmonic resonance crosses the Wood Anomaly band.

 figure: Fig. 3

Fig. 3 Calculated spectra of transmission in dependence on the nanowires period p for T = 200 nm, h = 40 nm.

Download Full Size | PPT Slide | PDF

The coupling of the waveguide mode with the plasmonic resonance is sensitive to the thickness of the waveguide layer (shown in Fig. 2(a)) and the period (see Fig. 3) of the gold pair gratings. As the period of the nanowires increases, the interaction of the localized plasmonic resonance and the diffractive wave leads to a suppressed transmission window. The coupling-induced transparency cancellation of the high energy branch localized plasmonic resonance opens a narrow high-transmission band in the broad low-transmission zone.

To gain a better understanding and picture of the two branches (high energy and low energy) plasmonic resonance, we depict the corresponding spatial electrical field distribution of the modes corresponding to the transparency peak and the dips as shown in Fig. 2(b). The results are shown in Figs. 4(a)-4(d). The symmetric mode at λ = 844 nm with its electric field in the two gold wires oriented in the same direction behaves as the electric resonance, which shows electric dipole character. At resonance, the transmission has a minimum, where strong localized resonance with tight field confined on the surface of gold nanowires can be seen clearly. The other anti-symmetric mode locates at λ = 1200 nm (see right dashed line in Fig. 2 (a) and red line in Fig. 3) with its electric field in the gold wires points in opposite directions can be regarded as the magnetic resonance mode [23, 24]. The enhanced magnetic field in proximity of the pair gold nanowires can be seen from Fig. 4(e). The formation of these two different coupled plasmon resonances is schematically shown in Fig. 5(a).

 figure: Fig. 4

Fig. 4 Calculated electrical field vector distribution for the symmetric plasmon mode (a-b) and the antisymmetric or magnetic plasmon mode (c). (d) Calculated spatial field distribution of the electric fields component Ez at maxima of the transmission in Fig. 2(b). White lines denote the cross-section of the proposed structure. (e) Calculated spatial field distribution of the magnetic fields component Hy of the magnetic plasmon resonance mode.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5

Fig. 5 (a) An energy-level diagram describing the plasmon hybridization in pair metal nanowires resulting from the interaction between two neighboring nanowires. The two coupled plasmons are an anti-symmetrically coupled (magnetic) plasmon mode and a symmetrically (electric) plasmon mode. (b) Transmission spectra as a function of the nanowires spacer distance h (p = 550 nm, T = 200 nm).

Download Full Size | PPT Slide | PDF

The hybrid mode in the waveguide layer at the maximum of transmission at 876 nm is observed clearly (see Fig. 4(d)). Significant portion of the mode energy stored in the waveguide with partially decay to the substrate leads to the high transmission. High transmission above 90% can be achieved as the geometry varies. The electric field avoids the region of the metal nanowires and the cancellation between the symmetric plasmon resonance and the quasiguided mode therefore results in suppressed localized electric resonance and a narrow transparency window. The hybrid transverse magnetic (TM) quasiguided mode reaches its cut-off for increased periods of the nanowires (NWs) array, where narrower transparency peak is predicted.

Furthermore, it is noted that the gap distance h between the pair gratings can be tuned by tailoring the coupling strength of the coupled electric resonance mode, which influences the optical EIT-like transmission response of the proposed system. It can be seen from Fig. 5(b) that the distances between the upper and lower metallic strips provides an effective way to switch the EIT effect between on-state and off-state. For larger distance between the metallic strips, the coupled electric resonance is weaker and the EIT transmission peak is decreased. When the gap distance h is increased from 20 to 90 nm, the coupled resonance induced optical response changes from a clear transmission peak at about 876 nm to a clear dip at 860 nm in the transmission spectrum. The parameters of the virtual devices are given in the caption of Fig. 5.

4. Angle response of the proposed hybrid plasmonic system

In view of practical application, the evolution and angular dependence of the transmission spectra are investigated, which is helpful to identify and distinguish the different light coupling channels and regimes. In Fig. 3(a), at normal incidence, only one transmission peak centered in the broad symmetric plasmonic resonance region is observed, which is related to excitation of theWAsub+1mode; at non-normal incidence, however, the grating-coupled hybrid waveguide modes of different orders occurs, leading to splitting transmission transparency branches. The result is shown in Fig. 6(a). One peak with higher transmission energy blue shifts, and mixes itself within the bright transmission zone. The transparency peak of this higher energy branch can only exist when angle-of-incidence below about 10°. The other peak redshifts to survive in the band gap between theWAair-1and theWAsub-1with a large angle tolerance up to 60°. This low energy branch opens up a clearly observable induced transparency peak, which is attributed to the destructive interferences between the hybrid waveguide mode and the electric resonance plasmon mode. It is observed that the full width at half maximum (FWHM) of the transparency peak varies with the angle-of-incidence. The FWHM and the group index versus the incident angle are shown in Fig. 6(b).

 figure: Fig. 6

Fig. 6 (a) Calculated transmission spectra of the system (p = 400 nm, T = 200 nm, h = 40 nm) in dependence of the incidence angle. The superimposed white solid and dashed lines are the calculated dispersion curves of WAs. “sub” and “air” indicate the substrate and upper interfaces. The curves represent the dispersion relation of the layered waveguide mode for the unperturbed three layer slab waveguide are denoted with prefix “WG”. (b) FWHM and group index versus the incident angle. (c) Calculated transmission as a function of the frequency and in-plane wave vector kx. The superimposed white solid and dashed lines are the dispersion curves of air and substrate light line. The red dashed-dotted line represents the folded dispersion relation of the hybrid TM mode. The green curve represents the folded dispersion relation for the unperturbed three layer slab waveguide.

Download Full Size | PPT Slide | PDF

To gain deeper understanding of the energy channels and the coupling mechanisms, the TM waveguide mode (mode effective index neff) of the bare slab dielectric waveguide (air/waveguide/substrate) without metal nanowires is calculated. The confined waveguide mode can be excited by free space light when phase-matching condition is satisfied. The periodic arranged nanowires array enables light to be coupled to waveguide mode in the waveguide layer through the gating condition of the (m)th-order propagating mode [3, 25]:

k0neff=|k0sinθ+mGx|,
where k0 is the momentum of free space light, θ is the incident light angle in the x direction. |Gx|=2π/pis the reciprocal lattice unit vectors in the periodic directions.

It is noted that the effective index of the unperturbed TM waveguide mode and the dispersion of the mode agrees well with the transparency peak for lower incident angle, for large incidence, the quasiguided hybrid TM mode of the system with metallic gating is distinguished from the unperturbed mode, which should be treated rigorously [17, 25]. The hybrid modes can be understood as the characteristic features of the photonic crystal slab (PCS), and be interpreted by using the empty lattice approximation [25, 26]. Without the grating, the waveguide modes in the plane slab are bound modes, in the sense that they are confined in the slab and decay exponentially in to the vacuum and substrate. The introduction of the grating enables the guided modes a finite lifetime and becomes quasiguided modes. The energy dispersion of this quasiguided TM mode is found from the self-consistency transcendent equation [26]:

tan(β¯t)=Neff2β¯(βs+ns2β)ns2β¯2Neff2βsβ,
where β¯=Neff2k02k2,βs=k2ns2k02,β=k2k02. k is the in plane wave vector, and k0 is the free space wave vector.

For varied angles of incidence, the first order coupled waveguide mode of the system without embedded grating (bare waveguide mode) with different waveguide layer thicknesses is investigated. It is found that the coupled waveguide mode for decreased waveguide layer thickness agrees well the evolution trend of the transparency peak band for increased incidence angles (see Fig. 6(a)). It can be understood that the introduction of period pair grating changes the effective index of the bare waveguide system, where larger angle of incidence leads to smaller value of waveguide effective index. The energy dispersion of the hybrid waveguide plasmonic mode of the plasmonic crystal structure is calculated and plotted in Fig. 6(c). The red dashed-dotted line shows the folded dispersion band of the hybrid mode in the first Brillouin Zone using Eq. (2) with fixed value of Neff = 2.01<nwg = 2.10. The transparency peak band in the broad plasmonic electric resonance zone is seen clearly, which approaches closely to the unperturbed waveguide mode band at the center of the Brillouin Zone, and to the hybrid mode band for increased in-plane wave vector. This can be understood as the contribution of the combination and hybridization of the mode between hybrid mode and waveguide mode, where the introduction of pair metallic nanowires leads to the channel of energy propagating in the waveguide and stored in the nanowires spacer.

The active control of the wavelength-selective switchable coupled induced transparency effect can be feasibly tuned and designed by tailoring the geometry of the system and by adjusting the angle of incidence. Further improved quality of the coupling-induced transparency peak resonance is possible by introduction of active medium with gain and proper design.

5. Conclusions

In conclusion, we have shown that in hybrid waveguide-plasmon system consisting of one-dimension periodic array of pairs of metallic nanowires, the anti-crossing arises from the coupling between localized magnetic resonance of the twin metallic nanowires and the in-plane diffractive Wood Anomaly, and narrow band coupled-induced transparency induced by the coupling from the localized electric resonance to the hybrid quasiguided mode can be obtained by tuning the geometry of the proposed structure. The huge magnetic field enhancement in proximity of the metallic nanowires pairs at the coupled mode resonance can be designed for potential applications based on enhanced magnetic fields [24, 27, 28]. The EIT-like optical effect is promising for slow light, sensing, optical filtering and enhanced nonlinear applications.

Acknowledgments

This work was supported by the National Basic Research Program of China (973 Program) (No. 2011CBA00608, No. 2012CB619203). The authors acknowledge financial support from the National Nature Science Foundation of China under grant no.61036010, and no. 61177070.

References and links

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef]   [PubMed]  

2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef]   [PubMed]  

3. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998). [CrossRef]  

4. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef]   [PubMed]  

5. N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010). [CrossRef]   [PubMed]  

6. T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009). [CrossRef]  

7. V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Electromagnetically induced transparency and slow light in an array of metallic nanoparticles,” Phys. Rev. B 80(3), 035104 (2009). [CrossRef]  

8. K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002). [CrossRef]  

9. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [CrossRef]   [PubMed]  

10. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef]   [PubMed]  

11. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008). [CrossRef]   [PubMed]  

12. V. Yannopapas, “Subwavelength imaging of light by arrays of metal-coated semiconductor nanoparticles: a theoretical study,” J. Phys. Condens. Matter 20(25), 255201 (2008). [CrossRef]  

13. V. M. Shalaev, W. S. 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]   [PubMed]  

14. V. Yannopapas, “Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices,” Phys. Status Solidi 1(5), 208–210 (2007) (RRL). [CrossRef]  

15. 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]  

16. G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17(12), 1791–1802 (2005). [CrossRef]  

17. V. Yannopapas and N. Stefanou, “Optical excitation of coupled waveguide-particle plasmon modes: a theoretical analysis,” Phys. Rev. B 69(1), 012408 (2004). [CrossRef]  

18. A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011). [CrossRef]   [PubMed]  

19. J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011). [CrossRef]  

20. X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012). [CrossRef]  

21. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

22. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]  

23. A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006). [CrossRef]  

24. M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010). [CrossRef]  

25. A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004). [CrossRef]  

26. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002). [CrossRef]  

27. C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011). [CrossRef]  

28. H. Liu, X. Sun, Y. Pei, F. Yao, and Y. Jiang, “Enhanced magnetic response in a gold nanowire pair array through coupling with Bloch surface waves,” Opt. Lett. 36(13), 2414–2416 (2011). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
    [Crossref] [PubMed]
  2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
    [Crossref] [PubMed]
  3. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
    [Crossref]
  4. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
    [Crossref] [PubMed]
  5. N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
    [Crossref] [PubMed]
  6. T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).
    [Crossref]
  7. V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Electromagnetically induced transparency and slow light in an array of metallic nanoparticles,” Phys. Rev. B 80(3), 035104 (2009).
    [Crossref]
  8. K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
    [Crossref]
  9. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
    [Crossref] [PubMed]
  10. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [Crossref] [PubMed]
  11. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
    [Crossref] [PubMed]
  12. V. Yannopapas, “Subwavelength imaging of light by arrays of metal-coated semiconductor nanoparticles: a theoretical study,” J. Phys. Condens. Matter 20(25), 255201 (2008).
    [Crossref]
  13. V. M. Shalaev, W. S. 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] [PubMed]
  14. V. Yannopapas, “Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices,” Phys. Status Solidi 1(5), 208–210 (2007) (RRL).
    [Crossref]
  15. 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]
  16. G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17(12), 1791–1802 (2005).
    [Crossref]
  17. V. Yannopapas and N. Stefanou, “Optical excitation of coupled waveguide-particle plasmon modes: a theoretical analysis,” Phys. Rev. B 69(1), 012408 (2004).
    [Crossref]
  18. A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011).
    [Crossref] [PubMed]
  19. J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
    [Crossref]
  20. X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
    [Crossref]
  21. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
  22. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
    [Crossref]
  23. A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
    [Crossref]
  24. M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010).
    [Crossref]
  25. A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
    [Crossref]
  26. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
    [Crossref]
  27. C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
    [Crossref]
  28. H. Liu, X. Sun, Y. Pei, F. Yao, and Y. Jiang, “Enhanced magnetic response in a gold nanowire pair array through coupling with Bloch surface waves,” Opt. Lett. 36(13), 2414–2416 (2011).
    [Crossref] [PubMed]

2012 (1)

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

2011 (4)

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

H. Liu, X. Sun, Y. Pei, F. Yao, and Y. Jiang, “Enhanced magnetic response in a gold nanowire pair array through coupling with Bloch surface waves,” Opt. Lett. 36(13), 2414–2416 (2011).
[Crossref] [PubMed]

A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011).
[Crossref] [PubMed]

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

2010 (2)

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010).
[Crossref]

2009 (3)

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).
[Crossref]

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Electromagnetically induced transparency and slow light in an array of metallic nanoparticles,” Phys. Rev. B 80(3), 035104 (2009).
[Crossref]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

2008 (5)

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]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[Crossref] [PubMed]

V. Yannopapas, “Subwavelength imaging of light by arrays of metal-coated semiconductor nanoparticles: a theoretical study,” J. Phys. Condens. Matter 20(25), 255201 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

2007 (1)

V. Yannopapas, “Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices,” Phys. Status Solidi 1(5), 208–210 (2007) (RRL).
[Crossref]

2006 (2)

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

2005 (3)

V. M. Shalaev, W. S. 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] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17(12), 1791–1802 (2005).
[Crossref]

2004 (2)

V. Yannopapas and N. Stefanou, “Optical excitation of coupled waveguide-particle plasmon modes: a theoretical analysis,” Phys. Rev. B 69(1), 012408 (2004).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

2002 (2)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

1998 (1)

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
[Crossref]

Altug, H.

A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011).
[Crossref] [PubMed]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Artar, A.

A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011).
[Crossref] [PubMed]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

Bai, W.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Bartal, G.

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]

Cai, L.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

Cai, W. S.

Cao, Z. S.

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

Chen, S.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Chen, Z.

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

Cheng, H.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Chettiar, U. K.

Christ, A.

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

Dasari, R. R.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
[Crossref]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Drachev, V. P.

Duan, X.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
[Crossref]

Eigenthaler, U.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Feld, M. S.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
[Crossref]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Gan, Q.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

Gantzounis, G.

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17(12), 1791–1802 (2005).
[Crossref]

Geiselmann, M.

M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010).
[Crossref]

Genov, D. A.

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]

Ghaemi, H. F.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
[Crossref]

Giessen, H.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010).
[Crossref]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

Gippius, N. A.

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Grupp, D. E.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
[Crossref]

Gu, C.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Hirscher, M.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Itzkan, I.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
[Crossref]

Jiang, Y.

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Kildishev, A. V.

Kneipp, H.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
[Crossref]

Kneipp, K.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
[Crossref]

Kuhl, J.

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

Langguth, L.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Lezec, H. J.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
[Crossref]

Li, J.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Lippitz, M.

M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010).
[Crossref]

Liu, H.

Liu, N.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Liu, W.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Liu, Z.

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[Crossref] [PubMed]

Lyandres, O.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Mesch, M.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Oulton, R. F.

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).
[Crossref]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

Pan, J.

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

Paspalakis, E.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Electromagnetically induced transparency and slow light in an array of metallic nanoparticles,” Phys. Rev. B 80(3), 035104 (2009).
[Crossref]

Pei, Y.

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Sarychev, A. K.

Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Shalaev, V. M.

Song, G.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

Sönnichsen, C.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

Stefanou, N.

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17(12), 1791–1802 (2005).
[Crossref]

V. Yannopapas and N. Stefanou, “Optical excitation of coupled waveguide-particle plasmon modes: a theoretical analysis,” Phys. Rev. B 69(1), 012408 (2004).
[Crossref]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Sun, X.

Tang, C. J.

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

Thio, T.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
[Crossref]

Tian, J.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Tikhodeev, S. G.

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Ulin-Avila, E.

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]

Utikal, T.

M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010).
[Crossref]

Valentine, J.

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]

Van Duyne, R. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Vitanov, N. V.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Electromagnetically induced transparency and slow light in an array of metallic nanoparticles,” Phys. Rev. B 80(3), 035104 (2009).
[Crossref]

Wang, Z. L.

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

Weiss, T.

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Xu, Y.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Yang, H.

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

Yanik, A. A.

A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011).
[Crossref] [PubMed]

Yannopapas, V.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Electromagnetically induced transparency and slow light in an array of metallic nanoparticles,” Phys. Rev. B 80(3), 035104 (2009).
[Crossref]

V. Yannopapas, “Subwavelength imaging of light by arrays of metal-coated semiconductor nanoparticles: a theoretical study,” J. Phys. Condens. Matter 20(25), 255201 (2008).
[Crossref]

V. Yannopapas, “Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices,” Phys. Status Solidi 1(5), 208–210 (2007) (RRL).
[Crossref]

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17(12), 1791–1802 (2005).
[Crossref]

V. Yannopapas and N. Stefanou, “Optical excitation of coupled waveguide-particle plasmon modes: a theoretical analysis,” Phys. Rev. B 69(1), 012408 (2004).
[Crossref]

Yao, F.

Yuan, H. K.

Zentgraf, T.

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).
[Crossref]

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]

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

Zhan, P.

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

Zhang, J.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

Zhang, S.

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).
[Crossref]

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]

Zhang, X.

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).
[Crossref]

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]

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Zhao, J.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).
[Crossref]

X. Duan, S. Chen, H. Yang, H. Cheng, J. Li, W. Liu, C. Gu, and J. Tian, “Polarization-insensitive and wide-angle plasmonically induced transparency by planar metamaterials,” Appl. Phys. Lett. 101(14), 143105 (2012).
[Crossref]

J. Phys. Condens. Matter (3)

G. Gantzounis, N. Stefanou, and V. Yannopapas, “Optical properties of a periodic monolayer of metallic nanospheres on a dielectric waveguide,” J. Phys. Condens. Matter 17(12), 1791–1802 (2005).
[Crossref]

V. Yannopapas, “Subwavelength imaging of light by arrays of metal-coated semiconductor nanoparticles: a theoretical study,” J. Phys. Condens. Matter 20(25), 255201 (2008).
[Crossref]

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14(18), R597–R624 (2002).
[Crossref]

Nano Lett. (2)

N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Sönnichsen, and H. Giessen, “Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing,” Nano Lett. 10(4), 1103–1107 (2010).
[Crossref] [PubMed]

A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011).
[Crossref] [PubMed]

Nat. Mater. (3)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7(6), 435–441 (2008).
[Crossref] [PubMed]

Nat. Photonics (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[Crossref]

Nature (2)

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]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (9)

A. Christ, T. Zentgraf, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Controlling the interaction between localized and delocalized surface plasmon modes: Experiment and numerical calculations,” Phys. Rev. B 74(15), 155435 (2006).
[Crossref]

M. Geiselmann, T. Utikal, M. Lippitz, and H. Giessen, “Tailoring the ultrafast dynamics of the magnetic mode in magnetic photonic crystals,” Phys. Rev. B 81(23), 235101 (2010).
[Crossref]

A. Christ, T. Zentgraf, J. Kuhl, S. G. Tikhodeev, N. A. Gippius, and H. Giessen, “Optical properties of planar metallic photonic crystal structures:” Phys. Rev. B 70(12), 125113 (2004).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

C. J. Tang, P. Zhan, Z. S. Cao, J. Pan, Z. Chen, and Z. L. Wang, “Magnetic field enhancement at optical frequencies through diffraction coupling of magnetic plasmon resonances in metamaterials,” Phys. Rev. B 83(4), 041402 (2011).
[Crossref]

V. Yannopapas and N. Stefanou, “Optical excitation of coupled waveguide-particle plasmon modes: a theoretical analysis,” Phys. Rev. B 69(1), 012408 (2004).
[Crossref]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (1998).
[Crossref]

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).
[Crossref]

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Electromagnetically induced transparency and slow light in an array of metallic nanoparticles,” Phys. Rev. B 80(3), 035104 (2009).
[Crossref]

Phys. Status Solidi (1)

V. Yannopapas, “Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices,” Phys. Status Solidi 1(5), 208–210 (2007) (RRL).
[Crossref]

Science (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

Other (1)

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Schematic view of the hybrid slab metallic pair nanowires array structure. The dielectric waveguide layer of thickness T on top of quartz substrate with embedded double metallic grating of gap distance h is investigated.
Fig. 2
Fig. 2 (a) Transmission spectra as a function of the waveguide layer thickness (p = 550 nm, h = 40 nm). The left black dashed line shows the guide for the eye. The right black dashed line shows the localized plasmon resonance. The curves have been plotted with subsequent vertical offsets of 1 for clarity. (b) The calculated transmission (straight black line) and phase (dotted blue line) for the result in (a) with a waveguide layer thickness T = 200 nm.
Fig. 3
Fig. 3 Calculated spectra of transmission in dependence on the nanowires period p for T = 200 nm, h = 40 nm.
Fig. 4
Fig. 4 Calculated electrical field vector distribution for the symmetric plasmon mode (a-b) and the antisymmetric or magnetic plasmon mode (c). (d) Calculated spatial field distribution of the electric fields component Ez at maxima of the transmission in Fig. 2(b). White lines denote the cross-section of the proposed structure. (e) Calculated spatial field distribution of the magnetic fields component Hy of the magnetic plasmon resonance mode.
Fig. 5
Fig. 5 (a) An energy-level diagram describing the plasmon hybridization in pair metal nanowires resulting from the interaction between two neighboring nanowires. The two coupled plasmons are an anti-symmetrically coupled (magnetic) plasmon mode and a symmetrically (electric) plasmon mode. (b) Transmission spectra as a function of the nanowires spacer distance h (p = 550 nm, T = 200 nm).
Fig. 6
Fig. 6 (a) Calculated transmission spectra of the system (p = 400 nm, T = 200 nm, h = 40 nm) in dependence of the incidence angle. The superimposed white solid and dashed lines are the calculated dispersion curves of WAs. “sub” and “air” indicate the substrate and upper interfaces. The curves represent the dispersion relation of the layered waveguide mode for the unperturbed three layer slab waveguide are denoted with prefix “WG”. (b) FWHM and group index versus the incident angle. (c) Calculated transmission as a function of the frequency and in-plane wave vector kx. The superimposed white solid and dashed lines are the dispersion curves of air and substrate light line. The red dashed-dotted line represents the folded dispersion relation of the hybrid TM mode. The green curve represents the folded dispersion relation for the unperturbed three layer slab waveguide.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

k 0 n eff =| k 0 sinθ+m G x |,
tan( β ¯ t)= N eff 2 β ¯ ( β s + n s 2 β) n s 2 β ¯ 2 N eff 2 β s β ,

Metrics