## Nonlocal propagation and tunnelling of surface plasmons in metallic hourglass waveguides |

Optics Express, Vol. 21, Issue 22, pp. 27509-27518 (2013)

http://dx.doi.org/10.1364/OE.21.027509

Acrobat PDF (7012 KB)

### Abstract

The nanofocusing performance of hourglass plasmonic waveguides is studied analytically and numerically. Nonlocal effects in the linearly tapered metal-air-metal stack that makes up the device are taken into account within a hydrodynamical approach. Using this hourglass waveguide as a model structure, we show that spatial dispersion drastically modifies the propagation of surface plasmons in metal voids, such as those generated between touching particles. Specifically, we investigate how nonlocal corrections limit the enormous field enhancements predicted by local electromagnetic treatments of geometric singularities. Finally, our results also indicate the emergence of nonlocality assisted tunnelling of plasmonic modes across hourglass contacts as thick as 0.5 nm.

© 2013 OSA

## 1. Introduction

1. K. Nerkararyan, “Superfocusing of a surface polariton in a wedge-like structure,” Phys. Lett. **237**, 103–105 (1997). [CrossRef]

2. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404 (2004). [CrossRef] [PubMed]

3. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon. **1**, 641–648 (2007). [CrossRef]

4. H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon. **6**, 838–844 (2012). [CrossRef]

5. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**, 053002 (2006). [CrossRef] [PubMed]

6. T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett. **110**, 263901 (2013). [CrossRef] [PubMed]

7. K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London) **491**, 574–577 (2012). [CrossRef]

9. A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano **7**, 6287–6296 (2013). [CrossRef] [PubMed]

11. 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**, 3308–3314 (2012). [CrossRef] [PubMed]

11. 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**, 3308–3314 (2012). [CrossRef] [PubMed]

15. 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 **84**, 121412 (2011). [CrossRef]

11. 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**, 3308–3314 (2012). [CrossRef] [PubMed]

16. 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**, 106802 (2012). [CrossRef] [PubMed]

## 2. Methods

2. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404 (2004). [CrossRef] [PubMed]

**12**, 3308–3314 (2012). [CrossRef] [PubMed]

*L′*and angle

*α*(measured wall to wall). The gap thickness is reduced (increased) linearly from

*D*to

*W*(from

*W*to

*D*) in the region −

*L′*<

*x*< 0 (0 <

*x*<

*L′*). The length of the two waveguide arms along the

*x*-direction,

*L*=

*L′*sec(

*α*/2), is chosen to ensure that the SPP optical path is of identical length for all taper angles

*α*. The resultant

*x*-dependent modulation of the gap thickness

*d*(

*x*) can be expressed as

*d*(

*x*) = 2|

*x*|tan(

*α*/2).

2. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404 (2004). [CrossRef] [PubMed]

*A*(

*x*) is a slowly varying amplitude factor ensuring power flow conservation. Within the WKB approximation, the propagation wavevector

*k*(

*x′*) experiences only a small variation over a distance

*dx′*(i.e., |

*dk*(

*x′*)

^{−1}/

*dx′*| ≤ 1) [17

17. D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys. **104**, 034311 (2008). [CrossRef]

*x*,

*k*(

*x*) can be obtained from the dispersion relation of an infinite, spatially dispersive, plasmonic MIM geometry with gap thickness

*d*(

*x*). Treating nonlocality within the hydrodynamical Drude model we can calculate the dispersion relation of the lowest SPP mode supported by MIM waveguides, obtaining [18

18. S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B **88**, 115401 (2013). [CrossRef]

*q*(

_{i}*x*) (

*i*= 1, 2, 3) are the various normal wavevectors associated to the SPP mode:

*q*

_{1}(

*x*) = [

*k*(

*x*)

^{2}−

*ε*

_{1}(

*ω/c*)

^{2}]

^{1/2}and

*q*

_{2}(

*x*) = [

*k*(

*x*)

^{2}−

*ε*

_{2}(

*ω/c*)

^{2}]

^{1/2}correspond to the transverse wavevector components into free space and metal, respectively, and

*ε*

_{1}= 1 and

*ε*

_{2}=

*ε*

_{D}(

*ω*), where

*ω*

_{P}and damping constant

*γ*.

*c*is the velocity of light. Note that in the limit

*β*= 0, Eq. (2) recovers the local dispersion relation [19]

*α*, which are consistent with the geometric assumptions presented above.

22. G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE **7757**(2010). [CrossRef]

23. J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. **9**, 887–891 (2009). [CrossRef] [PubMed]

25. 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**, 1333–1339 (2012). [CrossRef] [PubMed]

7. K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London) **491**, 574–577 (2012). [CrossRef]

26. J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett. **13**, 564–569 (2013). [CrossRef]

## 3. Results and discussion

*d*schematically depicted in Figure 2(b).

*ω*(

*k*) only for the lowest SPP modes is shown. Note that the modal frequency and wavevector are normalized to the surface plasmon frequency, defined as Re(

*ε*

_{D}(

*ω*

_{SPP})) = −1. Results for various gap thicknesses ranging from 1 nm to 50 nm are rendered. SPP bands obtained within the local (

*β*= 0) and the nonlocal (

*β*= 0.0036

*c*) descriptions are plotted in black solid and red dashed lines, respectively. In all calculations, the local Drude constants were taken from the fitting to Ag experimental data [27] used in [11

**12**, 3308–3314 (2012). [CrossRef] [PubMed]

*β*for silver was considered in the nonlocal treatment [28

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

*d*can be interpreted as an increase of the mode binding to the structure surfaces [29

29. A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics **14**, 1515–1521 (2008). [CrossRef]

*d*very suitable for light localisation purposes. Note that the insets of Figure 2 demonstrate the long propagation lengths (Λ = (2Im{

*k*})

^{−1}) of these plasmonic modes over a wide spectral window, which reach many SPP wavelengths even for very small waveguide dimensions.

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

13. R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Commun. **190**, 205–209 (2001). [CrossRef]

*ω*(

*k*) due to spatial dispersion is more pronounced for smaller

*d*, where the thickness of the gap approaches the longitudinal plasmon decay length (

*δ*

_{LP}≈ 0.1 nm for realistic Ag parameters).

*z*-component of the electric field,

*E*

_{z}/E_{0}, along the surface of the waveguide defined in Figure 1(b), with

*L*= 1000 nm,

*D*= 60 nm,

*W*= 0 nm, and

*α*= 3.44°. Note that these parameters imply a geometrical singularity at the point (

*x*,

*y*) = (0, 0). The fields have been normalized to the incident amplitude

*E*

_{0}. Both local (black) and nonlocal (red) results are rendered at 0.7

*ω*

_{SPP}, where the modes depicted in Figure 2(a) exhibit a convenient balance between confinement and propagation length. This balance arises because, whilst the binding of modes to the metal surfaces is tighter at high frequencies, the propagation length increases towards lower mode energies due to the reduced absorption damping (see Figure 2(a)). Excellent agreement between WKB analytical (lines) and FEM numerical (dots) results is obtained. Remarkably, we observe that in the nonlocal description SPPs are able to tunnel through the geometrical singularity located at

*x*= 0 nm, penetrating into the non-illuminated side of the waveguide in the region

*x*> 0. This is in contrast to the local limit, where SPPs are stopped adiabatically as they approach this singularity. Similar effects have been already reported theoretically in other singular MIM geometries that support EM surface modes. For example, as we have demonstrated in [16

16. 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**, 106802 (2012). [CrossRef] [PubMed]

**12**, 3308–3314 (2012). [CrossRef] [PubMed]

*x*= 0 nm, and vanish in regions where the metal surfaces are far apart.

*z*-field amplitude |

*E*

_{z}/E_{0}| at the waist of the hourglass waveguide as a function of frequency, for waist diameters ranging from 0.1 nm to 5 nm (

*α*= 3.44°). Note that we plot the component in the direction perpendicular to the metal surfaces (

*z*-direction) as it is the one that undergoes the highest amplification. Local (black) and nonlocal (red) results are rendered, and good agreement between WKB analytical (lines) and FEM numerical (dots) results is observed. Both descriptions yield a reduction of the field amplification at the hourglass waist at larger frequencies, which comes as a result of the increased metal absorption damping. The difference between local and nonlocal results is most pronounced for

*W*= 0.1 nm, the narrowest hourglass waist thickness considered in Figure 4. In such diminutive junctions, the real part of the nonlocal wavevector in the SPP propagation direction decreases significantly due to spatial dispersion, which results in lower field enhancements due to the weaker SPP binding to the metal surfaces.

*α*. In Figure 4(b) we plot the

*α*-dependent electric field, evaluated at 0.7

*ω*

_{SPP}, at one of the walls of the hourglass waist for

*W*= 0.1 nm (the smallest waist considered in Figure 4(a)). Local (nonlocal) results are rendered in black (red). As

*α*increases, a trend towards larger field enhancements is observed due to the significantly increased propagation length Λ of modes sustained by straight MIM geometries with large gap thicknesses (see the inset of Figure 2(a)).

*L*, to 1000 nm for all

*α*. Only this way, the effect of the tapering angle on the losses experienced by the SPP modes can be investigated. The comparison between WKB analytical results (lines) and FEM numerical results (dots) reveals good agreement between the two approaches for hourglass angles as large as

*α*∼ 15°. At larger angles, the WKB assumption of a reflection-free SPP propagation towards the hourglass waist breaks down, with significant reflection taking place as the modes accommodate to the rapidly changing geometry.

*α*increases, the back scattering of SPPs caused by the metal walls becomes more pronounced, yielding lower field enhancements at the hourglass waist. Note that the latter effect is not described in our analytical WKB treatment of the system. For the geometric parameters considered in Figure 4(b) the balance of these two effects leads to a maximum field enhancement at an angle

*α*∼ 28° for both local and nonlocal predictions..

*x*> 0). Figs. 5(a)–5(c) show results for open hourglass geometries (

*W*> 0). Panel (a) plots the local (black) and nonlocal (red)

*z*-field enhancement |

*E*

_{z}/E_{0}| on the surface of the hourglass waveguide defined in Figs. 1(a)–1(b), but for hourglass waist thicknesses of

*W*= 0.15 nm and

*W*= 5 nm. The fields are evaluated at 0.7

*ω*

_{SPP}using the FEM numerical approach. Nonlocal electric field maps for

*W*= 0.15 nm and

*W*= 5 nm are reproduced in panels (b) and (c), respectively. The comparison between local and nonlocal results in Figure 5(a) reveals that the two models yield extremely similar results for

*W*= 5 nm. This result can be understood from the dispersion bands in Figure 2(a), which reveal that, for 5 nm layer width, local and nonlocal predictions for both k and Λ differ by only 0.2% at

*ω*= 0.7

*ω*

_{SPP}.

*x*> 0 region of the structure. This energy transfer is less efficient for smaller

*W*, which in turn yield much larger field enhancement values. Figure 5(a) shows these two effects for

*W*= 0.15 nm, where the fields at the point

*x*= 10 nm, located in the non-illuminated side of the waveguide, have dropped to less than half of the amplitude when compared to the

*W*= 5 nm case. Interestingly, the nonlocal field amplitudes are larger than the local ones for most of the

*x*> 0 region. This inversion results from absorption losses, which are increased in the local case due to the significantly larger local SPP wavevectors (shorter effective wavelengths

*λ*

_{SPP}= 2

*π/k*). At

*x*= 0 nm the effective wavelength of local and nonlocal SPPs are 5.3 and = 9.0 nm, respectively, which is in agreement with the periodicity observed in the reflection induced beating of |

*E*

_{z}/E_{0}| apparent in Figure 5(a).

*W*< 0. As

*W*becomes negative, a metallic separation layer of thickness

*g*=

*W*/tan(

*α*/2) emerges in between the two arms of the waveguide, occupying the region −

*g*/2 <

*x*<

*g*/2. Figure 5(d) plots |

*E*

_{z}/E_{0}| at 0.7

*ω*

_{SPP}along the hourglass walls of the same geometry considered in Figure 5(a), but for negative values of

*W*.

*g*= 0.5 nm and 5 nm, are considered. For of

*g*= 5 nm the thick stopping layer is seen to be effective in terminating the SPP propagation within both the local (black lines) and nonlocal (red lines) descriptions. Interestingly, the thin metallic stopping layer (

*g*= 0.5 nm) results in a behaviour which is both quantitatively and qualitatively different. As in the case of the singular geometry studied in Figure 3, we observe here that, whereas local SPPs are adiabatically stopped as they approach the hourglass waist, SPPs treated within the nonlocal model are able to tunnel accross this metallic contact, as evidenced by the non-zero nonlocal field amplitude for

*x*> 0 in Figs. 5(d) and 5(f). This result can be understood by considering the SPP group velocity, which in the local limit always approaches zero at the singularity. Thus, even for metallic stopping layers whose thickness is thinner than the skin depth, SPPs are unable to tunnel accross adjacent metallic contacts, irrespective of their thickness. On the other hand, within the nonlocal description, the electronic charge distribution is smeared out accross the metal boundaries, which increases the effective thickness of the dielectric region and, in turn, leads to a non-zero SPP group velocity even at geometrical singularities. This nonlocal smearing of surface charges is measured by the longitudinal plasmon decay length, which for the parameters considered in our calculations is

*δ*

_{LP}∼ 0.1 nm. This value sets the metal thicknesses

*g*, for which nonlocal tunnelling of SPPs in tapered MIM waveguides takes place.

## 4. Conclusion

**12**, 3308–3314 (2012). [CrossRef] [PubMed]

16. 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**, 106802 (2012). [CrossRef] [PubMed]

31. G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics **2**, 161–240 (2013). [CrossRef]

32. A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett. **12**, 5946–5953 (2012). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | K. Nerkararyan, “Superfocusing of a surface polariton in a wedge-like structure,” Phys. Lett. |

2. | M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. |

3. | S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon. |

4. | H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon. |

5. | D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. |

6. | T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett. |

7. | K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London) |

8. | J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature (London) |

9. | A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano |

10. | N. Ashcroft and D. Mermin, |

11. | 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. | R. Fuchs and F. Claro, “Multipolar response of small metallic sphere: Nonlocal theory,” Phys. Rev. B |

13. | R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Commun. |

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

15. | 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 |

16. | 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. |

17. | D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys. |

18. | S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B |

19. | S. A. Maier, |

20. | A. D. Boardman, |

21. | C. Ciracì, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem |

22. | G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE |

23. | J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. |

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

25. | 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. |

26. | J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett. |

27. | E. D. Palik, |

28. | C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science |

29. | A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics |

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

31. | G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics |

32. | A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett. |

**OCIS Codes**

(230.7400) Optical devices : Waveguides, slab

(240.7040) Optics at surfaces : Tunneling

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Plasmonics

**History**

Original Manuscript: September 5, 2013

Revised Manuscript: October 1, 2013

Manuscript Accepted: October 4, 2013

Published: November 4, 2013

**Virtual Issues**

Surface Plasmon Photonics (2013) *Optics Express*

**Citation**

Aeneas Wiener, Antonio I. Fernández-Domínguez, J. B. Pendry, Andrew P. Horsfield, and Stefan A. Maier, "Nonlocal propagation and tunnelling of surface plasmons in metallic hourglass waveguides," Opt. Express **21**, 27509-27518 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27509

Sort: Year | Journal | Reset

### References

- K. Nerkararyan, “Superfocusing of a surface polariton in a wedge-like structure,” Phys. Lett.237, 103–105 (1997). [CrossRef]
- M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004). [CrossRef] [PubMed]
- S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon.1, 641–648 (2007). [CrossRef]
- H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012). [CrossRef]
- D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97, 053002 (2006). [CrossRef] [PubMed]
- T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett.110, 263901 (2013). [CrossRef] [PubMed]
- K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012). [CrossRef]
- J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature (London)483, 421–427 (2012). [CrossRef]
- A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013). [CrossRef] [PubMed]
- N. Ashcroft and D. Mermin, Solid State Physics (Holt, Rinehart, and Winston, 1976).
- 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, 3308–3314 (2012). [CrossRef] [PubMed]
- R. Fuchs and F. Claro, “Multipolar response of small metallic sphere: Nonlocal theory,” Phys. Rev. B35, 3722–3727 (1987). [CrossRef]
- R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Commun.190, 205–209 (2001). [CrossRef]
- F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008). [CrossRef]
- S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011). [CrossRef]
- 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, 106802 (2012). [CrossRef] [PubMed]
- D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys.104, 034311 (2008). [CrossRef]
- S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B88, 115401 (2013). [CrossRef]
- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
- A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).
- C. Ciracì, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem14, 1109–1116 (2013). [CrossRef] [PubMed]
- G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010). [CrossRef]
- J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett.9, 887–891 (2009). [CrossRef] [PubMed]
- 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]
- 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, 1333–1339 (2012). [CrossRef] [PubMed]
- J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett.13, 564–569 (2013). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
- C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012). [CrossRef] [PubMed]
- A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008). [CrossRef]
- R. Ruppin, “Non-local optics of the near field lens,” J. Phys.: Condens. Matter17, 1803–1810 (2005). [CrossRef]
- G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013). [CrossRef]
- A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

OSA is a member of CrossRef.