## Nonlocal effects in a hybrid plasmonic waveguide for nanoscale confinement |

Optics Express, Vol. 21, Issue 2, pp. 1430-1439 (2013)

http://dx.doi.org/10.1364/OE.21.001430

Acrobat PDF (1250 KB)

### Abstract

The effect of nonlocal optical response is studied for a novel silicon hybrid plasmonic waveguide (HPW). Finite element method is used to implement the hydrodynamic model and the propagation mode is analyzed for a hybrid plasmonic waveguide of arbitrary cross section. The waveguide has an inverted metal nano-rib over a silicon-on-insulator (SOI) structure. An extremely small mode area of~10^{−6}*λ*^{2} is achieved together with several microns long propagation distance at the telecom wavelength of 1.55*μ*m. The figure of merit (FoM) is also improved in the same time, compared to the pervious hybrid plasmonic waveguide. We demonstrate the validity of our method by comparing our simulating results with some analytical results for a metal cylindrical waveguide and a metal slab waveguide in a wide wavelength range. For the HPW, we find that the nonlocal effects can give less loss and better confinement. In particular, we explore the influence of the radius of the rib’s tip on the loss and the confinement. We show that the nonlocal effects give some new fundamental limitation on the confinement, leaving the mode area finite even for geometries with infinitely sharp tips.

© 2013 OSA

## 1. Introduction

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

2. Z. Han, A. Y. Elezzabi, and V. Van, “Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform,” Opt. Lett. **35**(4), 502–504 (2010). [CrossRef] [PubMed]

3. D. F. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. **29**(10), 1069–1071 (2004). [CrossRef] [PubMed]

4. 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]

5. D. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. **87**(6), 061106–061103 (2005). [CrossRef]

8. 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]

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

13. D. X. Dai, Y. C. Shi, S. L. He, L. Wosinski, and L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express **19**(14), 12925–12936 (2011). [CrossRef] [PubMed]

15. J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. **103**(9), 097403 (2009). [CrossRef] [PubMed]

16. J. M. McMahon, S. K. Gray, and G. C. Schatz, “Calculating nonlocal optical properties of structures with arbitrary shape,” Phys. Rev. B **82**(3), 035423 (2010). [CrossRef]

17. G. Toscano, S. Raza, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express **20**(4), 4176–4188 (2012). [CrossRef] [PubMed]

22. C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science **337**(6098), 1072–1074 (2012). [CrossRef] [PubMed]

23. A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. **108**(10), 106802 (2012). [CrossRef] [PubMed]

22. C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science **337**(6098), 1072–1074 (2012). [CrossRef] [PubMed]

24. G. C. Aers, B. V. Paranjape, and A. D. Boardman, “Non-radiative surface plasma-polariton modes of inhomogeneous metal circular cylinders,” J. Phys. F **10**(1), 53–65 (1980). [CrossRef]

25. R. Ruppin, “Effect of non-locality on nanofocusing of surface plasmon field intensity in a conical tip,” Phys. Lett. A **340**(1-4), 299–302 (2005). [CrossRef]

26. R. Ruppin, “Non-local optics of the near field lens,” J. Phys. Condens. Matter **17**(12), 1803–1810 (2005). [CrossRef]

27. F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C **112**(46), 17983–17987 (2008). [CrossRef]

27. F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C **112**(46), 17983–17987 (2008). [CrossRef]

^{−6}

*λ*

^{2}, while still maintaining propagation distances exceeding several microns at telecom wavelengths. We also investigate how nonlocal effects alter the confinement of the hybrid plasmonic waveguide. However, we don’t consider the quantum effect at the metal surface [28

28. D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: Nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett. **12**(3), 1333–1339 (2012). [CrossRef] [PubMed]

29. R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat Commun **3**, 825 (2012). [CrossRef] [PubMed]

## 2. Theoretical formalism

*δ*in the bulk metalNote that, when

_{L}*ω*is higher or

*β*is larger,

*δ*becomes larger. Thus, the nonlocal effect becomes important at high frequencies. The penetration depth is in the order of 0.1nm in noble metals, such as gold or silver [20

_{L}20. A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. **12**(6), 3308–3314 (2012). [CrossRef] [PubMed]

17. G. Toscano, S. Raza, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express **20**(4), 4176–4188 (2012). [CrossRef] [PubMed]

**J**and

**E**have the same propagation constant. We can write Eq. (2) into a weak form resulting in an integral expression [30]where

_{J˜}denotes the vector-valued test function of

**J**. Note that one may obtain a more simplied (but less accurate) integral expression as [21

21. K. R. Hiremath, L. Zschiedrich, and F. Schmidt, “Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using Nédélec finite elements,” J. Comput. Phys. **231**(17), 5890–5896 (2012). [CrossRef]

**E**, and the continuity of the normal component of

**J**at the surface due to the finite values of charge and current densities [31

31. A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B **2**(4), 835–850 (1970). [CrossRef]

## 3. Validation for benchmark problems

24. G. C. Aers, B. V. Paranjape, and A. D. Boardman, “Non-radiative surface plasma-polariton modes of inhomogeneous metal circular cylinders,” J. Phys. F **10**(1), 53–65 (1980). [CrossRef]

25. R. Ruppin, “Effect of non-locality on nanofocusing of surface plasmon field intensity in a conical tip,” Phys. Lett. A **340**(1-4), 299–302 (2005). [CrossRef]

26. R. Ruppin, “Non-local optics of the near field lens,” J. Phys. Condens. Matter **17**(12), 1803–1810 (2005). [CrossRef]

*n*, the propagation distance

_{eff}*L*and the mode profile for the metal cylindrical waveguide and the metal slab waveguide. The propagation distance is given by

_{prop}*L*= 1/(2Im(

_{prop}*n*)

_{eff}*k*

_{0}), where

*k*is the wave number in vacuum. We take parameters for silver, namely, the plasma frequency

_{0}*ηω*= 8.59eV, Drude damping coefficient

_{p}*ηγ*= 0.075eV, and nonlocal parameter

*β =*0.0036

*c*[20

_{0}20. A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. **12**(6), 3308–3314 (2012). [CrossRef] [PubMed]

*β*on the frequency due to transition absorption.

*R*= 2nm with the exact analytical solution, both for local (

*β*= 0) and nonlocal response (

*β*= 0.0036

*c*). For these tiny nanowires, the nonlocal effect can be considerable. To give two examples, the relative difference of

_{0}*L*between the nonlocal and local responses in the figure can be up to 4.6%. The relative difference of

_{prop}*n*between the nonlocal and local responses in the figure can be up to 2.5%. The mode profile in the metal is different due to the fact that the nonlocal parameter

_{eff}*β*will increase the longitudinal mode penetration depth

*δ*. Effectively, the electric field penetration into the metal is increased at the same time.

_{L}*L*and

_{prop}*n*for the nonlocal response are always less than 0.2% and 0.4%, respectively, while for the local response the relative errors are always less than 10

_{eff}^{−5}.

*d*= 2nm. As one can see, the maximum relative differences in

*L*and

_{prop}*n*mode profile between the nonlocal and local models are 1.8% and 0.9%, respectively. This indicates that the infinity along one dimension of the slab waveguide (which has only one-dimensional confinement) reduces the difference between the local and nonlocal models as compared to the cylinder waveguide, which has a two-dimensional confinement. The maximum relative errors in the numerically computed

_{eff}*L*and

_{prop}*n*for the nonlocal response in Fig. 1(b) are 0.16% and 0.07%, respectively, while for the local response it is always smaller than 10

_{eff}^{−5}.

## 4. Novel silicon hybrid plasmonic waveguide

^{−6}

*λ*

^{2}, while the propagation distance is still several microns simultaneously at telecom wavelengths. The cross section of the present silicon hybrid plasmonic waveguide is shown in Fig. 2(a) . It consists of a SOI rib with an inverted metal nano-rib on the top. There is a low-index material (e.g., air) between the SOI rib and the metal structure. This novel hybrid plasmonic waveguide combines the advantage of the silicon hybrid plasmonic waveguide structure (which has low propagation loss and strong field confinement in the vertical direction) and the advantage of the WPP waveguide (low propagation loss and strong field confinement in the horizontal direction). When the metal rib height is zero (i.e., a metal slab) in Fig. 2(c), or the SOI rib is absent in Fig. 2(d), the fundamental mode field is not well confined in any of the two directions and the mode area is about 10

^{−4}

*λ*

^{2}. However, when there is an inverted metal nano-rib, the field distribution of the guide mode is significantly modified and the field is strongly confined at the tip of the metal rib (see Fig. 2(b)) without much decrease in the propagation distance.

*w*

_{co}= 22nm,

*h*

_{metal}= 10nm,

*h*

_{rib}= 10nm,

*h*

_{Si}= 50nm,

*θ*= 10°,

*g*= 0.5nm and

*R*= 0.5nm. The frequency-dependent permittivity of the metal, SiO

_{2}and Si are the same as those in Ref [32

32. D. X. Dai, Y. C. Shi, S. L. He, L. Wosinski, and L. Thylen, “Silicon hybrid plasmonic submicron-donut resonator with pure dielectric access waveguides,” Opt. Express **19**(24), 23671–23682 (2011). [CrossRef] [PubMed]

*β*on the frequency due to transition absorption. Panel (a) shows the dispersion relation of the fundamental mode using the Drude method (line) or the nonlocal method (line). The mode has no cutoff wavelength. Panel (b) shows the propagation distance

*L*and the mode area

_{prop}*A*

_{m}_{.}The mode area

*A*is defined as the ratio of the total mode energy to the peak energy density [9

_{m}9. R. F. Oulton, V. J. Sorger, D. A. Genov, D. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics **2**(8), 496–500 (2008). [CrossRef]

*W*is the time averaged electromagnetic energy density. In the Drude model, the average energy density in the metal can be expressed as follows [33

33. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A **299**(2-3), 309–312 (2002). [CrossRef]

34. F. Forstmann and H. Stenschke, “Electrodynamics at metal boundaries with inclusion of plasma waves,” Phys. Rev. Lett. **38**(23), 1365–1368 (1977). [CrossRef]

*β*= 0 in the Drude model, Eq. (9) can be converted to Eq. (7) [33

33. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A **299**(2-3), 309–312 (2002). [CrossRef]

*p*is the ratio of the total dissipation energy to the total mode energy. When

*λ*= 1.55

*μ*m, we have

*p*= 2.17% in the nonlocal model and

*p*= 2.18% in the Drude model for this HPW. Here we can see that a relatively lower loss is obtained in the nonlocal model. This agrees with the simulation result that the propagation distance is longer in the nonlocal model than that in the Drude model. An amazing point is that in the meantime the mode area is smaller in the nonlocal model than that in the Drude model because the continuity of the normal component of

**E**in the nonlocal model enables a peak energy density located in the metal side at the interface between the metal and dielectric. When the propagation distance is 2.5

*μ*m, the mode area is 4.4 × 10

^{−6}

*λ*

^{2}in the traditional Drude model at 1.55

*μ*m. The mode area is 2.8 × 10

^{−6}

*λ*

^{2}and the propagation distance is 2.6

*μ*m in the hydrodynamic model at 1.55

*μ*m. To the best of our knowledge, this is the best result that has ever been reported about the mode area (note that the energy density distribution given in Ref [35

35. R. Hao, E. Li, and X. Wei, “Two-dimensional light confinement in cross-index-modulation plasmonic waveguides,” Opt. Lett. **37**(14), 2934–2936 (2012). [CrossRef] [PubMed]

*λ*decreases, while the ratio of the mode area to the

*λ*is increased as

^{2}*λ*decreases. Panels 3(c), (d) show the electromagnetic energy density distribution at a wavelength of 1.55

*μ*m for both the nonlocal model and the Drude model. The difference between the energy density distributions of the nonlocal model and the Drude model is associated with the boundary condition and the changes in the induced-charge distribution. In the Drude model, the charge is strictly a surface charge, while in the nonlocal model the charge density is finite and the longitudinal mode penetration depth

*δ*tends to spatially smear out the charge distribution. Effectively, this smearing increases the electric field penetration into the metal, which causes the energy density strong in the thin layer near the metal surface.

_{L}*μ*m. In Figs. 4(a) and 4(b), when the gap height decreases from 7nm to 3nm, the mode area keeps decreasing while the propagation distance does not change notably. When the gap height is reduced further, the mode area decreases to the order of 10

^{−6}

*λ*

^{2}. The results of the nonlocal hydrodynamic model and the traditional Drude model have the same tendency. However, the difference between the mode areas simulated by the two models increases as the gap increases.

*L*) to the effective mode radius (

_{prop}*R*) [36

_{m}36. R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express **15**(19), 12174–12182 (2007). [CrossRef] [PubMed]

*R*is defined as the radius of the effective mode area (

_{m}*A*). The FoM for our novel hybrid plasmonic waveguide can be as high as 2700 (or 4000) in the Drude (or hydrodynamic) model. While the FoMs of previously studied wedge plasmonic waveguide, cylinder plasmonic waveguide, channel plasmonic waveguide and the hybrid plasmonic waveguide in [37

_{m}37. R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. **10**(10), 105018 (2008). [CrossRef]

^{−3}

*λ*

^{2}at the telecom wavelength of 1.55

*μ*m in the Drude model. Obviously, our novel HWP can reduce the mode area and in the mean time improve the FoM.

19. G. Toscano, S. Raza, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Surface-enhanced Raman spectroscopy: nonlocal limitations,” Opt. Lett. **37**(13), 2538–2540 (2012). [CrossRef] [PubMed]

*R*from 1nm down to 0.005nm and see the difference between the nonlocal model and the Drude model. As expected, the difference between the two models increase as the tip radius

*R*approaches the length scale of the longitudinal mode penetration depth

*δ*(~0.1nm). In the Drude model, when

_{L}*R*is approaching zero, the mode area monotonously decreases. When

*r*is 0.005nm, the mode area approaches 10

^{−8}λ and a sharp peak appears in the energy density distribution as shown in Fig. 5(b). However, there is a fundamental saturation of the mode area (

*A*= 3 × 10

_{m}^{−7}

*λ*at

^{2}*r*= 0.005nm) in the nonlocal model, as the nonlocal effect relaxes the sharpness of the tip and the energy density distribution is smeared out as shown in Fig. 5(b).

## 5. Conclusions

^{−6}

*λ*

^{2}size in a wide wavelength range from 1.3

*μ*m to 2

*μ*m, and the propagation distance still remains several microns. We have studied theoretically how the nonlocal mode affects this new hybrid plasmonic waveguide. We have found that the nonlocal effects can reduce the loss and improve the confinement. We have shown that when the radius of the rib’s tip is approaching zero, the mode area gets saturated in the nonlocal model, instead of monotonous decrease in the Drude model.

## Acknowledgment

## References and links

1. | L. Liu, Z. H. Han, and S. L. He, “Novel surface plasmon waveguide for high integration,” Opt. Express |

2. | Z. Han, A. Y. Elezzabi, and V. Van, “Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform,” Opt. Lett. |

3. | D. F. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. |

4. | 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 |

5. | D. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. |

6. | T. Ogawa, D. Pile, T. Okamoto, M. Haraguchi, M. Fukui, and D. K. Gramotnev, “Numerical and experimental investigation of wedge tip radius effect on wedge plasmons,” J. Appl. Phys. |

7. | 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. |

8. | 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 |

9. | R. F. Oulton, V. J. Sorger, D. A. Genov, D. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics |

10. | D. X. Dai and S. L. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express |

11. | D. X. Dai and S. L. He, “Low-loss hybrid plasmonic waveguide with double low-index nano-slots,” Opt. Express |

12. | Y. S. Bian, Z. Zheng, Y. Liu, J. S. Liu, J. S. Zhu, and T. Zhou, “Hybrid wedge plasmon polariton waveguide with good fabrication-error-tolerance for ultra-deep-subwavelength mode confinement,” Opt. Express |

13. | D. X. Dai, Y. C. Shi, S. L. He, L. Wosinski, and L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express |

14. | A. D. Boardman, |

15. | J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. |

16. | J. M. McMahon, S. K. Gray, and G. C. Schatz, “Calculating nonlocal optical properties of structures with arbitrary shape,” Phys. Rev. B |

17. | G. Toscano, S. Raza, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express |

18. | S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B |

19. | G. Toscano, S. Raza, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Surface-enhanced Raman spectroscopy: nonlocal limitations,” Opt. Lett. |

20. | A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. |

21. | K. R. Hiremath, L. Zschiedrich, and F. Schmidt, “Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using Nédélec finite elements,” J. Comput. Phys. |

22. | C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science |

23. | A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. |

24. | G. C. Aers, B. V. Paranjape, and A. D. Boardman, “Non-radiative surface plasma-polariton modes of inhomogeneous metal circular cylinders,” J. Phys. F |

25. | R. Ruppin, “Effect of non-locality on nanofocusing of surface plasmon field intensity in a conical tip,” Phys. Lett. A |

26. | R. Ruppin, “Non-local optics of the near field lens,” J. Phys. Condens. Matter |

27. | F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C |

28. | D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: Nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett. |

29. | R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat Commun |

30. | P. Monk, |

31. | A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B |

32. | D. X. Dai, Y. C. Shi, S. L. He, L. Wosinski, and L. Thylen, “Silicon hybrid plasmonic submicron-donut resonator with pure dielectric access waveguides,” Opt. Express |

33. | R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A |

34. | F. Forstmann and H. Stenschke, “Electrodynamics at metal boundaries with inclusion of plasma waves,” Phys. Rev. Lett. |

35. | R. Hao, E. Li, and X. Wei, “Two-dimensional light confinement in cross-index-modulation plasmonic waveguides,” Opt. Lett. |

36. | R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express |

37. | R. F. Oulton, G. Bartal, D. F. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

(250.5300) Optoelectronics : Photonic integrated circuits

(260.3910) Physical optics : Metal optics

(160.4236) Materials : Nanomaterials

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 1, 2012

Revised Manuscript: December 16, 2012

Manuscript Accepted: January 2, 2013

Published: January 14, 2013

**Citation**

Qiangsheng Huang, Fanglin Bao, and Sailing He, "Nonlocal effects in a hybrid plasmonic waveguide for nanoscale confinement," Opt. Express **21**, 1430-1439 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1430

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### References

- L. Liu, Z. H. Han, and S. L. He, “Novel surface plasmon waveguide for high integration,” Opt. Express13(17), 6645–6650 (2005). [CrossRef] [PubMed]
- Z. Han, A. Y. Elezzabi, and V. Van, “Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform,” Opt. Lett.35(4), 502–504 (2010). [CrossRef] [PubMed]
- D. F. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett.29(10), 1069–1071 (2004). [CrossRef] [PubMed]
- 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,” Nature440(7083), 508–511 (2006). [CrossRef] [PubMed]
- D. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett.87(6), 061106–061103 (2005). [CrossRef]
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