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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 16722–16732
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Realization of subwavelength guiding utilizing coupled wedge plasmon polaritons in splitted groove waveguides

Jian Pan, Kaiting He, Zhuo Chen, and Zhenlin Wang  »View Author Affiliations


Optics Express, Vol. 18, Issue 16, pp. 16722-16732 (2010)
http://dx.doi.org/10.1364/OE.18.016722


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Abstract

A detailed numerical study of propagation characteristics of a coupled wedge plasmon polariton (CWPP) in splitted groove waveguide (SGW) formed with two metal wedges is performed by using the finite element method (FEM). It is shown that the SGW structure could confine CWPP modes tightly to the nano-gap region between the wedge tips, operating in a much broad bandwidth. The effect of the glass substrate, wedge roundness, gap width, groove wedge angle, and groove depth are also investigated. Particularly, our SGWs are found to be quite robust against groove depth reduction, which could be beneficial to minimize the waveguide structure dimensions. Feasibility of using such SGWs for the design of efficient subwavelength plasmonic elements is also discussed on the nanoscale whispering gallery resonators as an example.

© 2010 OSA

1. Introduction

Among aforementioned plasmonic waveguides, V-grooves engraved in a thick metal film supporting channel plasmon polartions (CPPs) have been demonstrated to show strong lateral confinement of the SPP fields at the bottom of grooves, simultaneously with low propagation losses at telecommunication wavelengths [13

13. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). [CrossRef] [PubMed]

]. Furthermore, based on such kind of V-groove waveguide (VGW) structure, plasmonic functional devices such as ring resonators, Mach-Zehnder interferometers, and add-drop multiplexers, have been successfully realized [15

15. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]

, 16

16. V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007). [CrossRef] [PubMed]

]. In contrast to the extreme plasmon confinement in narrow insulator films buried inside metal [17

17. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]

], VGW with a finite groove depth always involve electromagnetic fields extending significantly away from the groove bottom upon increasing the operating wavelength, and consequently producing a substantial degree of cross talk between neighboring waveguides, which thus limits the integration level and operation bandwidth of plasmonic circuits [18

18. E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef] [PubMed]

].

In general, when a trapezoid shaped gap instead of a rectangular gap is introduced into a metal film, the formed structure - a splitted groove waveguide (SGW) can support highly confined coupled wedge plasmon polariton (CWPP) modes due to its reduced symmetry, which will be demonstrated later in the text. Actually, the dependencies of the fundamental CWPP modes on wedge angle, nano-gap width, and radius of curvature of the rounded tips have been investigated by Pile et al. in a geometry composed of a twin two-fold symmetric metal wedge with a nano-gap [20

20. M. Haraguchi, D. F. P. Pile, T. Okamoto, M. Fukui, and D. K. Gramotnev, “New Plasmon Waveguides Composed of Twin Metal Wedges with a Nano Gap,” Opt. Rev. 13(4), 228–230 (2006). [CrossRef]

, 21

21. D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. 100(1), 013101 (2006). [CrossRef]

]. Very recently, Manjavacas et al. suggested a structure consisting of parallel metallic nanowires for highly confining plasmon modes to the gap region defined by two neighboring wires, and thus allowing extremely compact plasmonic circuits in three-dimensional spaces [22

22. A. Manjavacas and F. J. García de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. 9(4), 1285–1289 (2009). [CrossRef]

]. As compared with the aforementioned coupled-SPP waveguides [20

20. M. Haraguchi, D. F. P. Pile, T. Okamoto, M. Fukui, and D. K. Gramotnev, “New Plasmon Waveguides Composed of Twin Metal Wedges with a Nano Gap,” Opt. Rev. 13(4), 228–230 (2006). [CrossRef]

22

22. A. Manjavacas and F. J. García de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. 9(4), 1285–1289 (2009). [CrossRef]

], the proposed SGW structure is much easier to fabricate.

2. The fundamental properties of CWPP modes in SGWs

Throughout this paper, we use the commercial FEM software package COMSOL Multiphysics [24

24. COMSOL Multiphysics 3.3a.

] to evaluate the performances of the proposed waveguide structures. The complex effective index Neff is obtained from the mode analysis solver of COMSOL. The propagation wave vector β and propagation length Lprop are thus defined as follows: β=Re(Neff)k0 and Lprop=12Im(Neff)k0, where k0 is the wave vector in free space. In the calculations, the metal that constitutes the waveguides is gold with its dielectric constants described by a Drude-Lorentz model [25

25. A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005). [CrossRef]

]. To avoid any nonphysical situations, all the sharp corners of the considered waveguide structures in this paper are rounded, and the maximum length of the triangular meshes is set to be less than 5 nm along rounded corners or at the metal-air interfaces.

Figure 1(a)
Fig. 1 Schematic cross sections of a SGW structure (a) and a conventional VGW structure (b).
schematically shows the lateral cross section (perpendicular to the direction of mode guiding) of the considered SGW structure, where d is the groove depth (metal film thickness), g is the gap width, and θ 1, θ 2 are two wedge angles. Obviously, the aforementioned rectangular gap MIM structure is a special case of the considered SGW structure with θ 1 = θ 2 = 90°. The SGW structure is assumed to be surrounded by air (later we will demonstrate the effect of a higher refractive index substrate). Such a kind of freestanding SGW structure is possible to be fabricated experimentally by first milling a trapezoid shaped gap in a thin metal film supported on a silica substrate, and then using hydrofluoric to etch away a small part of silica substrate around the gap.

In order to check the validity of the code used in this paper, we first calculate the dispersion relation, propagation length, and field distributions for a conventional VGW with the same geometrical parameters as those given in ref [18

18. E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef] [PubMed]

], i.e., d = 1.172 μm, θ 1 = θ 2 = 77.5°, and the corners are rounded with a 10 nm-radius curvature. The numerical results for the conventional VGW are shown in Fig. 2
Fig. 2 (Color online) Dispersion relations (a) and propagation lengths (b) for the fundamental guiding modes of a conventional VGW (orange) and SGWs having three different gaps g = 50 nm (red), 100 nm (green) and 200 nm (blue). For both waveguide structures, the groove depth d = 1.172 μm and the wedge angles θ 1 = θ 2 = 77.5°. The dash-dotted line in (a) indicates the SPP dispersion curve on a flat gold/air interface.
(orange line). Here we focus on the fundamental CPP mode. It is seen from Fig. 2(a) that as the wavelength increases the CPP mode line gradually approaches the SPP dispersion curve, and at λ = 1.44 μm (cut-off wavelength) it crosses the dispersion line of SPP mode on a flat gold/air surface (SPPgold-air), which is found to agree very well with the numerical results reported in Ref [18

18. E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef] [PubMed]

].

Geometrically, the present SGW structure can be viewed as a truncated VGW with a finite groove depth [Fig. 1(b)], where the planar part below the groove is cut away, followed by separating two groove side walls with a small gap. For the purpose of a meaningful comparison the geometrical parameters of SGWs are set to be the same as those in the conventional VGW. The dispersion characteristics of SGWs with different gap widths g = 50 nm, 100 nm, and 200 nm are also shown in Fig. 2(a). Similar to the behavior of the fundamental CPP mode line of the VGW, the dispersion line of fundamental CWPP mode supported on SGWs also gradually approaches the SPPgold-air dispersion line as the wavelength increases. It is worth noting that similar to the rectangular gap structures the proposed SGW structures do not exhibit cutoff for fundamental guiding modes [26

26. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006). [CrossRef] [PubMed]

], which means that in the proposed SGW structures a much broader operation bandwidth can be achieved than in conventional VGW.

3. The effects of the roundness of the sharp corner and substrate on the fundamental CWPP modes

Now, let us consider a situation that a SGW structure is sustained on a dielectric substrate (glass, n = 1.5). Obviously, in this case of asymmetric dielectric environment the dispersion line of SPP mode on a flat gold/glass surface (SPPgold-glass) should be taken into account. As seen from Fig. 5
Fig. 5 (Colour online) Dispersion relations for SGWs on glass substrate having two different wedge angles: θ 1 = θ 2 = 52.5° (red, square) and 77.5° (blue, circle). The dotted line represents SPP curves on a gold/glass interfaces.
, the dispersion curve of fundamental CWPP mode for the SGW structure (d = 1.172 μm, θ 1 = θ 2 = 77.5°, and g = 50 nm) crosses the SPPgold-glass dispersion line at λ = 0.96 μm (cut-off wavelength). This implies that the SGW structure with a substrate significantly shorten the operation bandwidth of the fundamental CWPP mode. One of the ways to extend the operation range is to decrease the wedge angles. Also shown in Fig. 5 is the fundamental CWPP mode dispersion curve for a SGW (d = 1.172 μm and g = 50 nm) with wedge angles θ 1 = θ 2 = 52.5°. In such smaller wedge angles, the fundamental CWPP mode cut-off wavelength is indeed extended to ~1.3 μm. However, the x-directional size of the SGW structure is much enlarged simultaneously, which is a drawback for minimizing the waveguide lateral size. That is the reason why a freestanding SGW structure should be expected here.

4. Dependence on wedge angles

5. Robustness against groove depth variations

Figure 8(b) shows that the cut-off wavelengths of the second mode in SGWs are also dependent on the groove depth. Upon decreasing the groove depth, the cut-off wavelengths of the second-order modes monotonically decreases, which can be explained by the deeper coupling between the bottom groove wedges and top ones. Therefore, a broad single mode operation bandwidth (λ > 0.9 μm) can be achieved when the groove depth of SGW (θ 1 = θ 2 = 77.5°) is decreased to ~0.5 μm for example. Such kind of feature can be exploited to make the SGWs operate in a single mode in a wide region of wavelengths by decreasing the groove depth.

6. Whispering gallery resonator constructed from SGWs

5. Conclusion

Acknowledgements

References and links

1.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, 1st ed. (Springer-Verlag, Berlin, 1988).

2.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006). [CrossRef]

3.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). [CrossRef]

4.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

5.

T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82(5), 668–670 (2003). [CrossRef]

6.

B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. 79(1), 51–53 (2001). [CrossRef]

7.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). [CrossRef] [PubMed]

8.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]

9.

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]

10.

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004). [CrossRef]

11.

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005). [CrossRef] [PubMed]

12.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009). [CrossRef] [PubMed]

13.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). [CrossRef] [PubMed]

14.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100(2), 023901 (2008). [CrossRef] [PubMed]

15.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]

16.

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007). [CrossRef] [PubMed]

17.

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]

18.

E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef] [PubMed]

19.

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]

20.

M. Haraguchi, D. F. P. Pile, T. Okamoto, M. Fukui, and D. K. Gramotnev, “New Plasmon Waveguides Composed of Twin Metal Wedges with a Nano Gap,” Opt. Rev. 13(4), 228–230 (2006). [CrossRef]

21.

D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. 100(1), 013101 (2006). [CrossRef]

22.

A. Manjavacas and F. J. García de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. 9(4), 1285–1289 (2009). [CrossRef]

23.

E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface--plasmon whispering gallery resonators,” Nano Lett. 9(9), 3147–3150 (2009). [CrossRef] [PubMed]

24.

COMSOL Multiphysics 3.3a.

25.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005). [CrossRef]

26.

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006). [CrossRef] [PubMed]

27.

H. Choi, D. F. P. Pile, S. Nam, G. Bartal, and X. Zhang, “Compressing surface plasmons for nano-scale optical focusing,” Opt. Express 17(9), 7519–7524 (2009). [CrossRef] [PubMed]

28.

S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14(20), 9467–9476 (2006). [CrossRef] [PubMed]

29.

M. Oxborrow, “Traceable 2-D Finite-Element Simulation of the Whispering-Gallery Modes of Axisymmetric Electromagnetic Resonators,” IEEE Trans. Microw. Theory Tech. 55(6), 1209–1218 (2007). [CrossRef]

OCIS Codes
(230.7380) Optical devices : Waveguides, channeled
(240.6680) Optics at surfaces : Surface plasmons
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 26, 2010
Revised Manuscript: June 26, 2010
Manuscript Accepted: July 8, 2010
Published: July 23, 2010

Citation
Jian Pan, Kaiting He, Zhuo Chen, and Zhenlin Wang, "Realization of subwavelength guiding utilizing coupled wedge plasmon polaritons in splitted groove waveguides," Opt. Express 18, 16722-16732 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16722


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References

  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, 1st ed. (Springer-Verlag, Berlin, 1988).
  2. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7-8), 20–27 (2006). [CrossRef]
  3. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61(5), 44–50 (2008). [CrossRef]
  4. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]
  5. T. Nikolajsen, K. Leosson, I. Salakhutdinov, and S. I. Bozhevolnyi, “Polymer-based surface-plasmon-polariton stripe waveguides at telecommunication wavelengths,” Appl. Phys. Lett. 82(5), 668–670 (2003). [CrossRef]
  6. B. Lamprecht, J. R. Krenn, G. Schider, H. Ditlbacher, M. Salerno, N. Felidj, A. Leitner, F. R. Aussenegg, and J. C. Weeber, “Surface plasmon propagation in microscale metal stripes,” Appl. Phys. Lett. 79(1), 51–53 (2001). [CrossRef]
  7. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). [CrossRef] [PubMed]
  8. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003). [CrossRef] [PubMed]
  9. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]
  10. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004). [CrossRef]
  11. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005). [CrossRef] [PubMed]
  12. T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009). [CrossRef] [PubMed]
  13. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). [CrossRef] [PubMed]
  14. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100(2), 023901 (2008). [CrossRef] [PubMed]
  15. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]
  16. V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007). [CrossRef] [PubMed]
  17. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006). [CrossRef] [PubMed]
  18. E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef] [PubMed]
  19. 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]
  20. M. Haraguchi, D. F. P. Pile, T. Okamoto, M. Fukui, and D. K. Gramotnev, “New Plasmon Waveguides Composed of Twin Metal Wedges with a Nano Gap,” Opt. Rev. 13(4), 228–230 (2006). [CrossRef]
  21. D. F. P. Pile, D. K. Gramotnev, M. Haraguchi, T. Okamoto, and M. Fukui, “Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap,” J. Appl. Phys. 100(1), 013101 (2006). [CrossRef]
  22. A. Manjavacas and F. J. García de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. 9(4), 1285–1289 (2009). [CrossRef]
  23. E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface--plasmon whispering gallery resonators,” Nano Lett. 9(9), 3147–3150 (2009). [CrossRef] [PubMed]
  24. COMSOL Multiphysics 3.3a.
  25. A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005). [CrossRef]
  26. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006). [CrossRef] [PubMed]
  27. H. Choi, D. F. P. Pile, S. Nam, G. Bartal, and X. Zhang, “Compressing surface plasmons for nano-scale optical focusing,” Opt. Express 17(9), 7519–7524 (2009). [CrossRef] [PubMed]
  28. S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14(20), 9467–9476 (2006). [CrossRef] [PubMed]
  29. M. Oxborrow, “Traceable 2-D Finite-Element Simulation of the Whispering-Gallery Modes of Axisymmetric Electromagnetic Resonators,” IEEE Trans. Microw. Theory Tech. 55(6), 1209–1218 (2007). [CrossRef]

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