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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9836–9847
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Long-range channel plasmon polaritons in thin metal film V-grooves

Sangjun Lee and Sangin Kim  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9836-9847 (2011)
http://dx.doi.org/10.1364/OE.19.009836


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Abstract

We numerically investigate the propagation characteristics of guided modes in a thin metal film V-groove embedded in a dielectric medium, with a particular emphasis on long-ranging channel plasmon polaritons (LR-CPPs). The LR-CPP shows several orders of magnitude larger propagation length than the previously studied short-range channel plasmon polariton (SR-CPP). Moreover, the LR-CPP possesses a peculiar mode cutoff mechanism when surrounding dielectric media are asymmetric and this makes its propagation characteristics very sensitive to index change of the surrounding dielectric media.

© 2011 OSA

1. Introduction

Surface plasmon-polaritons (SPPs) have attracted a lot of interest and been topics of numerous studies due to their sub-diffraction-limit confinement of light which opens huge opportunity to subwavelength optics [1

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

]. The strong light confinement of the SPP results in field enhancement and hence, enhanced light-matter interaction such as optical nonlinearity. Energy-efficient nonlinear-optic device operations can also be potential applications of the SPP.

2. Guiding principle of thin metal film V-grooves: two types of channel plasmon polariton modes

The thin metal(gold) film V-groove waveguide considered in this work is depicted in Fig. 1(a)
Fig. 1 (a) Geometry of thin metal film V-groove waveguide. Electric field profiles, Ex for (b) Symmetrically-coupled LR-SPPs, (c) Symmetrically-coupled SR-SPPs, (d) Antisymmetrically-coupled LR-SPPs, and (e) Antisymmetrically-coupled SR-SPPs in one-dimensional dielectric/metal/dielectric/metal/dielectric (D/M/D/M/D) structure. Also, (f) and (g) shows the complex effective index as functions of gap distance for (b) and (c) cases, respectively: t = 10nm, n1 = n2 = 2.38, and λ = 1.55μm.
which is similar to the structure considered in ref [21

21. J. Dintinger and O. J. F. Martin, “Channel and wedge plasmon modes of metallic V-grooves with finite metal thickness,” Opt. Express 17(4), 2364–2374 (2009). [CrossRef] [PubMed]

]. All the corners are rounded to avoid field singularities in the same way as in ref [21

21. J. Dintinger and O. J. F. Martin, “Channel and wedge plasmon modes of metallic V-grooves with finite metal thickness,” Opt. Express 17(4), 2364–2374 (2009). [CrossRef] [PubMed]

]: the radius of curvature of the inner corner is 10 nm and that of the outer corner is (t + 10) nm. For the surrounding medium, chalcogenide glass (As2S3, n = 2.38) is selected. Asymmetric surrounding media cases (n1n2) as well as symmetric surrounding medium cases (n1 = n2) are considered. θ and h are the angle and the depth of the groove, respectively. The wave propagates along the z-direction and complex permittivity εAu of gold is extracted directly from Palik’s experimental data in wavelength range of 0.7~2.0 μm [22

22. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

], and for the wavelength longer 2.0 μm, a modified Drude-Lorentz model is used since experimental data are not available [23

23. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef]

].

As reported in ref [21

21. J. Dintinger and O. J. F. Martin, “Channel and wedge plasmon modes of metallic V-grooves with finite metal thickness,” Opt. Express 17(4), 2364–2374 (2009). [CrossRef] [PubMed]

], there is another type of mode in the thin metal V-groove originated from WPP, whose guiding principle is quite different from those of the CPP modes. In this work, however, the WPP mode is not investigated in depth since our focus is the radiative behavior of the LR-CPP at the optical communication wavelengths.

3. Characteristics of long-range channel plasmon polaritons

W(r)=12(d(ε(r)w)dw|E(r)|2+μ0|H(r)|2).
(2)

4. Radiative behavior of long-range channel plasmon polaritons in asymmetric environment

In this section, the mode cutoff behavior of the LR-CPP is investigated in comparison to that of the SR-CPP. In order to investigate the cutoff behaviors, dispersion curves are calculated for the thin metal (gold) V-groove of h = 5 μm, θ = 30°, and n 1 = n 2 = 2.38. Figure 4(a)
Fig. 4 Dispersion curves for LR-CPP, SR-CPP, and WPP modes with (a) t = 10nm and (b) t = 20nm in a uniform medium of n1 = n2 = 2.38. The dashed line represents the light line of n = 2.38. (c) The propagation length of the modes for t = 10nm. (θ = 30°, h = 5μm in all calculations.)
and 4(b) show the calculated dispersion curves for t = 10 nm and t = 20 nm, respectively. For reference, the dispersion curve of the WPP is also included. As seen in Fig. 4(a) and 4(b), all the modes in V-grooves with uniform dielectric medium show cutoff due to the finite groove size. In our calculation, the mode cutoff is defined as an occasion that the real part of N eff becomes equal to the highest value of the surrounding dielectric media. For the wavelengths longer than the cutoff wavelengths, mode becomes radiative since the effective index of the bound mode cannot be smaller than the index of the surrounding dielectric medium. For t = 10 nm, cutoff energies (wavelengths) for the LR-CPP, the SR-CPP, and the WPP are approximately 0.667 eV (1.86 μm), 0.105 eV (11.84 μm), and 0.077 eV (16.06 μm), respectively. For t = 20 nm, cutoff energies (wavelengths) for the LR-CPP, the SR-CPP, and WPP are approximately 0.506 eV (2.45 μm), 0.137 eV (9.05 μm), and 0.097 eV (12.75 μm), respectively. The LR-CPPs show much shorter cutoff wavelengths (higher cutoff energies) than the SR-CPP and the WPP and most sensitive change to the film thickness change because of the weakest mode confinement. As the metal film gets thinner, the dispersion curve of the LR-CPP becomes closer to the light line and its cutoff energy increases since the confinement of the LR-CPP gets weaker. In contrast, the SR-CPP is more strongly confined for the thinner metal film, and thus, its dispersion curve and cutoff energy show opposite tendencies. These opposite tendencies of the LR-CPP and the SR-CPP can be understood from the same behaviors of the LR-SPP and the SR-SPP in the flat metal film for the film thickness change. Somehow, the WPP shows similar tendency to the SR-CPP. Figure 4(c) shows the calculated propagation lengths for t = 10 nm. The propagation lengths monotonically increase as photon energy gets closer to the cutoff energy.

In order to investigate the effect of index asymmetry, the dispersion curves for the CPP modes are calculated for the V-groove of h = 5 μm, θ = 30°, and t = 10 nm with various index asymmetries. Figure 5(a)
Fig. 5 Dispersion curves for two CPP modes with various index asymmetries; (a) LR-CPP and (b) SR-CPP with n1>n2 = 2.38, (c) LR-CPP and (d) SR-CPP with n2>n1 = 2.38. The dashed line represents the light line of n = 2.38. The number in each legend denotes the index in the higher-index region. Dispersion curves for symmetric environment are also plotted for reference. (t = 10nm, θ = 30°, h = 5μm in all calculations.)
and 5(b) show the calculated dispersion curves near the cutoff energy for n1 = 2.384 and 2.39 with n2 = 2.38. The cutoff energy (wavelength) of the LR-CPP varies to 0.827 eV (1.5 μm) and 0.984 eV (1.26 μm) for n1 = 2.384 and 2.39, respectively. For the SR-CPP shown in Fig. 5(b), the cutoff energy (wavelength) varies to 0.1055 eV (11.75 μm) and 0.106 eV (11.7 μm) for n1 = 2.384 and 2.39, respectively. Another type of index asymmetry with a higher-index lower clad (n2 > n1 = 2.38) is also considered, and the calculated dispersion curves are plotted in Fig. 5(c) and 5(d). The cutoff energy (wavelength) of the LR-CPP varies to 0.713 eV (1.74 μm) and 0.821 eV (1.51 μm) for n2 = 2.384 and 2.39, respectively. For the SR-CPP shown in Fig. 5(d), the cutoff energy (wavelength) varies to 0.107 eV (11.6 μm) and 0.112 eV (11.1 μm) for n2 = 2.384 and 2.39, respectively.

Overall, the LR-CPP shows more sensitive cutoff energy changes for the index asymmetries than the SR-CPP. For the SR-CPP, cutoff occurs due to the finite size of the groove, and the index asymmetry does not considerably affect dispersion characteristic and mainly changes the cutoff conditions. (Note that the mode cutoff is defined as an occasion that the real part of N eff becomes equal to the highest value of the surrounding dielectric media.) Consequently, the index asymmetry causes the cutoff energy change. In contrast to the SR-CPP, the cutoff behavior of the LR-CPP is greatly affected by the index asymmetry via two mechanisms. For the case of n1 > n2, the cutoff mechanism induced by the index asymmetry in the LR-CPP can be explained by the one-dimensional model used for the guiding principle of the CPP modes in section 2. By using the semi-analytic method [16

16. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

], the guided modes are calculated for two coupled flat metal (gold) films of t = 10 nm where the index of the core (n1) is different from that of the clad (n2 = 2.38). For comparison, both symmetrically-coupled LR-SPP and SR-SPP are considered. Figure 6
Fig. 6 Dependence of complex effective indices on gap distance in two coupled metal films with different core indices, n1: (a) real and (b) imaginary part of effective index for symmetrically-coupled LR-SPP, (c) real (d) imaginary part of effective index for symmetrically-coupled SR-SPP. The insets show corresponding mode shapes. The operating wavelengths are chosen close to the cutoff wavelengths of the LR-CPP and the SR-CPP: λ = 1.55μm in (a) and (b), and λ = 11.8 μm in (c) and (d). (t = 10nm, and n2 = 2.38 in all calculations.)
shows the calculated complex effective indices as functions of the gap distance. The operating wavelengths close to the cutoff wavelengths of the LR-CPP and the SR-CPP are chosen, which are 1.55 μm and 11.8 μm, respectively. One can see that for the symmetrically-coupled LR-SPP mode, the trend of the effective index changes completely with a small index difference between the core and the clad. The effective index is almost a constant in the case n1 = 2.39 and it increases as d increases in the case of n1 = 2.40. This implies that if the index of the core is larger than 2.39, the symmetrically-coupled LR-SPP mode cannot be confined inside the groove so that the LR-CPP mode becomes radiative even in the infinite V-groove. On the other hand, for the symmetrically-coupled SR-SPP mode, the effective index shows the same decreasing trend even for n1 = 3.38 with increasing d. Therefore, even with introduction of a large index asymmetry, radiative behavior is not expected for the SR-CPP mode in the infinite groove. Although this one-dimensional model directly applies only to an infinite V-groove, it reveals that besides the finite size effect there is another cutoff mechanism for the LR-CPP induced by the index asymmetry of the higher-index core (n1 > n2). It is surmised that this is why the LR-CPP shows much larger cutoff wavelength change for such index asymmetry.

For the large enough index asymmetry of the higher-index clad (n1 < n2), the cutoff of the LR-CPP is caused by the cutoff of the LR-SPP in each side of the V-groove. (Note that in the flat metal film, the index asymmetry induces only the cutoff of the LR-SPP [18

18. I. G. Breukelaar, “Surface plasmon-polaritons in thin metal strips and slabs: waveguiding and mode cutoff,” Master Thesis, University of Ottawa (2004).

].) This also makes the cutoff characteristic of the LR-CPP more sensitive to the index asymmetry (n1 < n2) than the SR-CPP.

Those two cutoff mechanisms for the LR-CPP in a finite V-groove are discussed in detail later in this section.

For t = 20 nm case, which is not show here, the cutoff energy of the LR-CPP is less sensitive to the index asymmetry than t = 10 nm case, but still much more sensitive than the SR-CPP.

Figure 7(a)
Fig. 7 Dispersion curves for the WPP modes with index asymmetries of (a) n1>n2 = 2.38 and (b) n2>n1 = 2.38. (t = 10nm, θ = 30°, h = 5μm) The dashed line represents the light line of n = 2.38. The number in each legend denotes the index in the higher-index region. Dispersion curves for symmetric environment are also plotted for reference. (c) Dominant electric field profiles, Ex for the WPP of n1 = 2.39, n2 = 2.38, t = 10nm θ = 30°, h = 5μm, λ = 14μm (0.089 eV).
and 7(b) shows the calculated dispersion curves of the WPP with index asymmetries. The WPP appears to show considerable cutoff energy changes (from 0.0772 eV to 0.089 eV for n 1 = 2.39) with the index asymmetry of the higher-index core, while very small cutoff energy changes (from 0.0772 eV to 0.0775 eV for n 2 = 2.39) are observed with the index asymmetry of the higher-index clad. Physical origin of this phenomenon is not clear at this moment. As seen in Fig. 7(c), fields are confined completely near the outer surface of the groove and there is almost no field in the core in the WPP mode. Therefore, it is expected that the core index change would not affect the property of the WPP. However, this is quite opposite to our observation. This counter-intuitive behavior of the WPP remains as a topic of future research at this moment.

The radiative behavior of the LR-CPP is also observed in the index asymmetry of higher index clad case (n2 > n1 = 2.38). As seen in Fig. 8(a), for n2 > n1 = 2.38, the effective index appears to reach n2 for n2 = 2.389 and the mode cutoff occurs. As seen in the field distribution calculated in the cutoff regime (n1 = 2.38, n2 = 2.392) illustrated in Fig. 8(f), the field radiates in a downward direction of the groove. This can be explained from the cutoff behavior of the LR-SPP mode in a flat metal film that fields radiate into the higher index dielectric medium if an index asymmetry is large enough [18

18. I. G. Breukelaar, “Surface plasmon-polaritons in thin metal strips and slabs: waveguiding and mode cutoff,” Master Thesis, University of Ottawa (2004).

]. Therefore, it can be said that in the higher index clad case, the mode cutoff of the LR-CPP stems from the cutoff of the LR-SPP bound in both sides of the thin metal film groove. As seen in Fig. 8(b), the propagation length increases as the clad index increases since the mode gets to experience less metal as the asymmetry increases.

5. Discussion and conclusion

Similar control of mode boundness can also be obtained for the SR-CPP and the WPP at the operating wavelength near the cutoff wavelengths. To observe the cutoff behavior of the SR-CPP and the WPP at optical communication wavelengths, the dimension of the V-groove should be much smaller than the case of the LR-CPP, since the cutoff wavelengths of the SR-CPP and the WPP are much longer than that of the LR-CPP. This small waveguide dimension may be advantageous in terms of integration density, but disadvantageous in terms of short propagation lengths and large insertion loss for optical fiber coupling.

In conclusion, we have numerically investigated the propagation characteristics of the LR-CPP mode in the thin metal film V-groove and compared to the previously studied SR-CPP mode. In contrast to the SR-CPP mode, the LR-CPP mode is originated from the LR-SPP so that it shows much longer propagation length and its boundness can be more sensitively controlled by the index asymmetry. This LR-SPP-like feature in conjunction with the unique geometric feature will find useful applications.

Acknowledgment

This work was supported by National Research Foundation of Korea Grants (NRF-2010-0000217 and NRF-2010-0001859).

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.

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]

3.

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

4.

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, “Propagation properties of guided waves in index-guided two-dimensional optical waveguides,” Appl. Phys. Lett. 86(21), 211101 (2005). [CrossRef]

5.

A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and M. Soljacić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95(6), 063901 (2005). [CrossRef] [PubMed]

6.

S. Lee and S. Kim, “Plasmonic mode-gap waveguides using hetero-metal films,” Opt. Express 18(3), 2197–2208 (2010). [CrossRef] [PubMed]

7.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]

8.

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]

9.

A. Boltasseva, V. S. Volkov, R. B. Nielsen, E. Moreno, S. G. Rodrigo, and S. I. Bozhevolnyi, “Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths,” Opt. Express 16(8), 5252–5260 (2008). [CrossRef] [PubMed]

10.

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]

11.

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85(26), 6323–6325 (2004). [CrossRef]

12.

D. K. Gramotnev and K. C. Vernon, “Adiabatic nano-focusing of plasmons by sharp metallic wedges,” Appl. Phys. B 86(1), 7–17 (2006). [CrossRef]

13.

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

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002). [CrossRef]

16.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

17.

F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B Condens. Matter 44(11), 5855–5872 (1991). [CrossRef] [PubMed]

18.

I. G. Breukelaar, “Surface plasmon-polaritons in thin metal strips and slabs: waveguiding and mode cutoff,” Master Thesis, University of Ottawa (2004).

19.

S. Park and S. H. Song, “Polymer variable optical attenuator based on long range surface plasmon polaritons,” Electron. Lett. 42(7), 402–404 (2006). [CrossRef]

20.

P. Berini, R. Charbonneau, S. Jette-Charbonneau, N. Lahoud, and G. Mattiussi, “Long-range surface plasmon-polaritin waveguides and devices in lithium niobate,” J. Appl. Phys. 101(11), 113114 (2007). [CrossRef]

21.

J. Dintinger and O. J. F. Martin, “Channel and wedge plasmon modes of metallic V-grooves with finite metal thickness,” Opt. Express 17(4), 2364–2374 (2009). [CrossRef] [PubMed]

22.

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

23.

A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef]

24.

J. Chen, G. A. Smolyakov, S. R. Brueck, and K. J. Malloy, “Surface plasmon modes of finite, planar, metal-insulator-metal plasmonic waveguides,” Opt. Express 16(19), 14902–14909 (2008). [CrossRef] [PubMed]

25.

Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20(3), 463–468 (2002). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.7370) Optical devices : Waveguides
(240.0310) Optics at surfaces : Thin films
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: February 18, 2011
Revised Manuscript: April 4, 2011
Manuscript Accepted: May 1, 2011
Published: May 5, 2011

Citation
Sangjun Lee and Sangin Kim, "Long-range channel plasmon polaritons in thin metal film V-grooves," Opt. Express 19, 9836-9847 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9836


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
  2. 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]
  3. K. Tanaka and M. Tanaka, “Simulation of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]
  4. F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, “Propagation properties of guided waves in index-guided two-dimensional optical waveguides,” Appl. Phys. Lett. 86(21), 211101 (2005). [CrossRef]
  5. A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and M. Soljacić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95(6), 063901 (2005). [CrossRef] [PubMed]
  6. S. Lee and S. Kim, “Plasmonic mode-gap waveguides using hetero-metal films,” Opt. Express 18(3), 2197–2208 (2010). [CrossRef] [PubMed]
  7. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]
  8. 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]
  9. A. Boltasseva, V. S. Volkov, R. B. Nielsen, E. Moreno, S. G. Rodrigo, and S. I. Bozhevolnyi, “Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths,” Opt. Express 16(8), 5252–5260 (2008). [CrossRef] [PubMed]
  10. 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]
  11. D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85(26), 6323–6325 (2004). [CrossRef]
  12. D. K. Gramotnev and K. C. Vernon, “Adiabatic nano-focusing of plasmons by sharp metallic wedges,” Appl. Phys. B 86(1), 7–17 (2006). [CrossRef]
  13. E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [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. I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002). [CrossRef]
  16. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]
  17. F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B Condens. Matter 44(11), 5855–5872 (1991). [CrossRef] [PubMed]
  18. I. G. Breukelaar, “Surface plasmon-polaritons in thin metal strips and slabs: waveguiding and mode cutoff,” Master Thesis, University of Ottawa (2004).
  19. S. Park and S. H. Song, “Polymer variable optical attenuator based on long range surface plasmon polaritons,” Electron. Lett. 42(7), 402–404 (2006). [CrossRef]
  20. P. Berini, R. Charbonneau, S. Jette-Charbonneau, N. Lahoud, and G. Mattiussi, “Long-range surface plasmon-polaritin waveguides and devices in lithium niobate,” J. Appl. Phys. 101(11), 113114 (2007). [CrossRef]
  21. J. Dintinger and O. J. F. Martin, “Channel and wedge plasmon modes of metallic V-grooves with finite metal thickness,” Opt. Express 17(4), 2364–2374 (2009). [CrossRef] [PubMed]
  22. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  23. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef]
  24. J. Chen, G. A. Smolyakov, S. R. Brueck, and K. J. Malloy, “Surface plasmon modes of finite, planar, metal-insulator-metal plasmonic waveguides,” Opt. Express 16(19), 14902–14909 (2008). [CrossRef] [PubMed]
  25. Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20(3), 463–468 (2002). [CrossRef]

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