## Dispersion relation, propagation length and mode conversion of surface plasmon polaritons in silver double-nanowire systems |

Optics Express, Vol. 21, Issue 12, pp. 14591-14605 (2013)

http://dx.doi.org/10.1364/OE.21.014591

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### Abstract

We study the surface plasmon modes in a silver double-nanowire system by employing the eigenmode analysis approach based on the finite element method. Calculated dispersion relations, surface charge distributions, field patterns and propagation lengths of ten lowest energy plasmon modes in the system are presented. These ten modes are categorized into three groups because they are found to originate from the monopole-monopole, dipole-dipole and quadrupole-quadrupole hybridizations between the two wires, respectively. Interestingly, in addition to the well studied gap mode (mode 1), the other mode from group 1 which is a symmetrically coupled charge mode (mode 2) is found to have a larger group velocity and a longer propagation length than mode 1, suggesting mode 2 to be another potential signal transporter for plasmonic circuits. Scenarios to efficiently excite (inject) group 1 modes in the two-wire system and also to convert mode 2 (mode 1) to mode 1 (mode 2) are demonstrated by numerical simulations.

© 2013 OSA

## 1. Introduction

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## 2. Computational method

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*D*is 200 nm and the wire-wire distance

*w*will be varied to study the gap size effects. For simplicity, the surrounding medium is chosen to be air. The permittivity of silver is described by

34. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

*ω*is determined by an iterative calculation, and the obtained

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*m*= 0, 1, and 2 SPP modes in the single silver nanowire are shown as the solid lines in Figs. 2(a) and 2(b), respectively. Here

*m*denotes the angular quantum number and the mode fields in a cylindrical system have the form of ~

*ϕ*is the azimuthal angle. The electric field patterns and the surface charge distributions calculated based on the Gauss law, are also displayed in the insets in Fig. 2. Clearly, the

*m*= 0, 1, 2 modes are, respectively, the monopole, dipole and quadrupole modes of the single nanowire. At a fixed frequency, the propagation length increases with the

*m*value (Fig. 2). Furthermore, the real part

*k*

_{z}of the parallel momentum decreases with the increasing

*m*. This indicates that the SPP mode with a larger

*m*is less localized in the vicinity of the nanowire and thus has a smaller Ohmic loss, giving rise to an increased propagation length. Interestingly, the propagation length of the

*m*= 2 mode becomes extremely large at ~960 THz. Note that the dispersion relation in this region comes very close to the light line and the SPP modes becomes very delocalized. Therefore, the mode can travel a very long distance, almost like a real propagating wave. Similar behaviors can be found for the other modes.

*ψ*at the wire center

*ψ*must be the Hankel function of the first kind since it approaches the propagating wave form of ~

*r*tends to infinite. The Helmholtz equation can then be written aswhere the wave functions inside and outside the cylinder being, respectively,Here

*m*= 0, 1, 2 modes from Eq. (4) [36

36. J. C. Ashley and L. C. Emerson, “Dispersion relations for non-radiative surface plasmons on cylinders,” Surf. Sci. **41**(2), 615–618 (1974). [CrossRef]

## 3. Results and discussion

**9**(4), 1285–1289 (2009). [CrossRef] [PubMed]

21. V. Myroshnychenko, A. Stefanski, A. Manjavacas, M. Kafesaki, R. I. Merino, V. M. Orera, D. A. Pawlak, and F. J. García de Abajo, “Interacting plasmon and phonon polaritons in aligned nano- and microwires,” Opt. Express **20**(10), 10879–10887 (2012). [CrossRef] [PubMed]

**9**(4), 1285–1289 (2009). [CrossRef] [PubMed]

**20**(10), 10879–10887 (2012). [CrossRef] [PubMed]

*m*= 0, 1, 2 modes in the single nanowire are displayed. Figure 3(a) shows that in the double-wire system, these three SPP bands now split into ten distinguished ones, indicating rather strong hybridizations between the

*m*= 0, 1, 2 modes residing on the two different nanowires. Previously, the attention was mainly focused on the gap SPP mode (mode 1) [18

**9**(4), 1285–1289 (2009). [CrossRef] [PubMed]

**20**(10), 10879–10887 (2012). [CrossRef] [PubMed]

### 3.1 Origin and symmetry of the SPP modes

*w*of the two nanowires first to 100 nm and then to 200 nm. The results of these calculations are shown in Fig. 5. When the

*w*is increased, the energy bands of the SPP modes get closer and eventually converge to the SPP modes of the single nanowire, because the coupling between the two nanowires becomes weaker. In the

*m*= 0, 1, 2) in the single nanowire to Fig. 5(c), clearly reveals the correlation of the SPP modes between the single and double nanowire systems. The three groups of the SPP modes in the double-nanowire system clearly originate from the

*m*= 0, 1, 2 modes in the single nanowire, respectively.

13. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science **302**(5644), 419–422 (2003). [CrossRef] [PubMed]

37. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: Analysis of optical properties,” Phys. Rev. B **75**(3), 035411 (2007). [CrossRef]

*E*field distributions [37

_{y}37. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: Analysis of optical properties,” Phys. Rev. B **75**(3), 035411 (2007). [CrossRef]

*E*field patterns shown in Fig. 7(d). Clearly, modes 1 and 2 come from the monopole-monopole hybridization. Group 2 (modes 3, 4, 5 and 6) can be further categorized into the collinear and parallel subgroups, depending on the directions of the dipoles on the two nanowires. Obviously the coupling strength of the collinear modes would be stronger because the charges on the two nanowires are closer. Therefore, the splitting of the two collinear modes is larger than the parallel modes, as shown in Fig. 7(b). Since the energy level of the symmetric mode is lower than that of the anti-symmetric one [see Fig. 7(d)], it can be deduced that the energy level would monotonically increase from mode 3 to mode 6. Clearly, all the group 2 SPP modes originate from the dipole-dipole interactions. A similar analysis would lead to the energy level sequence for the group 3 modes (modes 7-10) shown in Fig. 7(c). They are the products of the quadrupole-quadrupole interactions. Finally, since the symmetry of the infinite double-nanowire system belongs to the point group D

_{y}_{2h}whose irreducible representations are all one-dimensional, there is no degeneracy for any eigenmodes in our system [38].

### 3.2 Propagation properties

*m*= 0 mode in the single nanowire [Fig. 5(d)]. Interestingly, mode 2 has a larger propagation length than mode 1, as can be seen in Fig. 3 and Fig. 5, indicating that a mode 2-based waveguide may have advantages over the gap SPP-based one.

*w*values with the wavelength fixed at the telecommunication wavelength

*y*-direction (the

*x*= 0 line) [Figs. 8(c)-8(e)] by adopting the analysis method used in [27

27. 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]

27. 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]

*w*. This is because for mode 1, most of the electromagnetic energy is stored in the gap region and hence the energy density

*w*. In contrast, the normalized energy density of mode 2 is almost immune to the gap width variation. This is consistent with the nearly constant propagation length of mode 2, as shown in Fig. 8(a). For mode 2, the surface charges have the same sign and are thus mutually repulsive. Consequently, the surface charges are mainly located on the outer boundaries of the two nanowires. Therefore, the coupling of the two SPP modes is weaker since they are more than 400 nm apart. Consequently, mode 2 is less sensitive to the change of the gap width. This explains why the two modes show contrasting behaviors in the propagation length with respect to the gap width variation.

*w*. Therefore, mode 2 would lose much less energy and could travel a much longer distance [Fig. 8(a)]. In fact, the propagation length of mode 2 is as large as ~170 μm, being the same order of magnitude with that reported in [27

27. 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]

*ε*of the surrounding material to study the effect of the surrounding material. Figure 8(b) shows that, if the permittivity

*ε*of the surrounding material is increased while the gap width

*A*for all the ten SPP modes where the mode area

_{m}/A_{0}*A*is defined as the ratio of the total mode energy to the peak energy density [27

_{m}**2**(8), 496–500 (2008). [CrossRef]

*A*of an SPP mode is a good measure of its degree of confinement. Figure 9(a) shows that for all the gap widths considered, mode 1 has a smaller mode area (i.e., a stronger confinement) than the

_{m}/A_{0}*m*= 0 mode of the single nanowire which, however, has a smaller mode area than mode 2. Figure 9(a) also suggests that the confinement (

*A*) of mode 1 becomes weaker (larger) as the gap width

_{m}/A_{0}*w*increases, while the degree of confinement of mode 2 is quite insensitive to the gap width variation. Interestingly, the

*A*versus

_{m}/A_{0}*w*curves in Fig. 9(a) is very similar to the corresponding propagation length versus

*w*curves in Fig. 8(a), thus indicating a strong correlation between the confinement (mode area) and propagation length. Figure 9(b) shows the degree of confinement of all the ten SPP modes at the frequency of 1200 THz (or

*w*= 50 nm. Clearly, the mode area (i.e., mode delocalization) increases nearly monotonically from mode 1 to mode 10. Since the dispersion relations of the ten SPP modes are all far away from the light line and close to each other at 1200 THz [Fig. 3(a)], all the mode areas are small and similar in size [see Fig. 9(b)].

### 3.3 Coupling length of the two neighboring double-nanowire systems

*L*) is defined as

_{c}*L*[39

_{c}39. L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express **13**(17), 6645–6650 (2005). [CrossRef] [PubMed]

40. G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express **16**(3), 2129–2140 (2008). [CrossRef] [PubMed]

*Therefore, the coupling length is an important parameter that determines the integration density of the plasmonic devices. The calculated coupling lengths of modes 1 and 2 for two neighboring double-nanowire systems as a function of their separation*

_{.}*g*are plotted in Fig. 10. As shown in Fig. 10, the coupling lengths of both mode 1 and 2 increase with the unit separation

*g,*because the coupling between the two waveguides becomes weaker. Since the mode energy of mode 1 is mainly stored in the gap region, the coupling effect is relatively weak, thus leading to a large coupling length (Fig. 10).

*g*, the coupling length

*L*also depends on how the two waveguides are placed with respect to each other. For example, the field distributions of mode 1 (the gap mode) in the two double-nanowire systems in the on-top geometry [see the inset in Fig. 10(b)] could overlap more strongly than in the side-by-side geometry [see the inset in Fig. 10(a)]. This explains why the coupling length of mode 1 in the former case [Fig. 10(b)] is orders f magnitude smaller in the latter case [Fig. 10(a)] in all the considered

_{c}*g*values except at

*g*≈95 nm where an anomalously large

*L*appears. For example,

_{c}*L*at

_{c}*g*= 200 nm is ~4100 μm for the side-by-side case and ~80 μm for the on-top case. In the on-top geometry with

*g*≈95 nm, the propagation constants of the symmetric and anti-symmetric modes would cross each other (i.e.,

41. G. B. Hoffman and R. M. Reano, “Vertical coupling between gap plasmon waveguides,” Opt. Express **16**(17), 12677–12687 (2008). [CrossRef] [PubMed]

*L*[Fig. 10(b)]. The energy exchange between the two double-nanowire systems is still strong, and thus the mechanism for the large

_{c}*L*is very different from the non-coupling limit case.

_{c}*g*= 1000 nm, the coupling length of mode 2 is ~60 μm in both cases, being smaller than its propagation length. Nevertheless, this coupling length is comparable with that of the slot waveguide pair [42

42. D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B **93**(1), 99–106 (2008). [CrossRef]

43. Y. Ma, G. Farrell, Y. Semenova, H. P. Chan, H. Zhang, and Q. Wu, “Novel dielectric-loaded plasmonic waveguide for tight-confined hybrid plasmon mode,” Plasmonics **8**(2), 1259–1263 (2013). [CrossRef]

42. D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B **93**(1), 99–106 (2008). [CrossRef]

### 3.4 Selective injection of the SPP modes

*z*-oriented electric dipole source placed at the left end of the two silver nanowires. The field pattern of the excited mode 2 SPP wave is displayed in Fig. 11(b). The SPP wavelength obtained by measuring the periodicity of the simulated

*E*field is 1481 nm, being in perfect agreement with the wavelength derived from the calculated dispersion relations shown in Fig. 3. The eigen-momentum

_{y}*k*of mode 2 at

^{−1}(Fig. 3), and hence, the SPP wavelength is 1490 nm. If we rotate the dipole away from the

*z*-axis in the

*x-z*plane, mode 2 could still be excited because the symmetries of the charge pattern of mode 2 and the exciting dipole remains partially matched, although with a much reduced injection efficiency. The relative injection efficiency here is defined as the ratio (

*I*/

_{SPP}*I*) of the total energy of the SPP wave flowing towards the right side of the absorbing boundary (

_{0}*I*) for an angle

_{SPP}*θ*to that for

*I*). Figure 11(a) shows clearly that ratio

_{0}*I*/

_{SPP}*I*decreases monotonically as angle

_{0}*θ*increases. For example, the electric field

*z*-axis, though

*I*/

_{SPP}*I*is now reduced by a factor of five. This is because the electric field of the dipole becomes increasingly transverse as angle

_{0}*θ*increases and is completely transverse at

*y*-direction) [see Figs. 11(c) and 11(d)]. The measured SPP wavelength is about 1242 nm based on the simulated

*E*field patterns, being again consistent with the value of ~1243 nm derived from the calculated dispersion relation of mode 1 shown in Fig. 3. Figure 11(c) shows that when the dipole is rotated away from the

_{y}*y*-axis in the

*x*-

*y*plane, mode 2 could still be excited but with a decreased injection efficiency. The injection efficiency at

*y*-axis).

*y-z*plane [see Fig. 11(e)]. Furthermore, the percentages of mode 1 and mode 2 may be controlled by varying angle

*φ*between the dipole and the

*y*-axis. For example, the

*E*field pattern at

_{y}*z*(

*y*) direction could excite a mode 2 (mode 1) SPP wave that is about 3.5 (2.4) times stronger than the corresponding SPP wave excited by one dipole source. Finally, we note that modes 3-6 may also be excited by a pair of dipole sources with appropriate configurations of the dipoles. Nonetheless, a detailed description of this is beyond the scope of this paper.

### 3.5 Mode conversion

44. H. R. Park, J. M. Park, M. S. Kim, and M. H. Lee, “A waveguide-typed plasmonic mode converter,” Opt. Express **20**(17), 18636–18645 (2012). [CrossRef] [PubMed]

45. Y. T. Hung, C. B. Huang, and J. S. Huang, “Plasmonic mode converter for controlling optical impedance and nanoscale light-matter interaction,” Opt. Express **20**(18), 20342–20355 (2012). [CrossRef] [PubMed]

*L*of the upper bending nanowire. At the right side (output), mode 1 is converted to mode 2. Unfortunately, as shown in Fig. 12(a), there is some energy loss at the bending area of the circuit and also some small portion of mode 1 survives in the right side region (output) due to robustness of mode 1. Nevertheless, mode 1 is more or less successfully converted to mode 2 by the converter, as demonstrated by the calculated surface charge distribution shown in Fig. 12(b). Clearly, the unlike surface charges at the input region of the circuit become the like ones at the output region due to the transformation by the mode converter. In Figs. 12(c) and 11(d), an inverse conversion from mode 2 to mode 1 by the same system is illustrated. Our simulations prove the scenario in principle, although it could be further optimized by tuning the geometry or employing some other designs for the mode converter.

## 4. Conclusions

*m*= 0, 1, 2 SPP modes in the single nanowire agree very well with that given by the Mie theory, thereby verifying the reliability of the present approach. Dispersion relations, surface charge distributions, field patterns and propagation lengths of ten lowest energy SPP modes in the silver double-nanowire system are presented. These ten SPP modes are naturally categorized into three groups since they are found to result from the monopole-monopole (2 modes), dipole-dipole (4 modes) and quadrupole-quadrupole (4 modes) interactions, respectively. Group 1 which consists of an antisymmetrically coupled charge mode (i.e., the well studied gap mode) (mode 1) and a symmetrically coupled charge mode (mode 2), are studied in detail. In particular, mode 2 is found to have a larger group velocity and a longer propagation length than the gap mode, and this suggests that mode 2 may be employed as a signal transporter, in addition to mode 1. Scenarios to efficiently excite (inject) these two modes in the two-wire system and also to convert mode 2 (mode 1) to mode 1 (mode 2) are demonstrated by numerical simulations.

## Acknowledgments

## References and links

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7. | S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science |

8. | M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. |

9. | S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature |

10. | M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics |

11. | K. Bao, H. Sobhani, and P. Nordlander, “Plasmon hybridization for real metals,” Chin. Sci. Bull. |

12. | V. Klimov and G.-Y. Guo, “Bright and dark plasmon modes in three nanocylinder cluster,” J. Phys. Chem. C |

13. | E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science |

14. | 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. |

15. | M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. |

16. | K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. |

17. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. |

18. | A. Manjavacas and F. J. García de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. |

19. | A. Manjavacas and F. J. García de Abajo, “Coupling of gap plasmons in multi-wire waveguides,” Opt. Express |

20. | W. Cai, L. Wang, X. Zhang, J. Xu, and F. J. Garcia de Abajo, “Controllable excitation of gap plasmons by electron beams in metallic nanowire pairs,” Phys. Rev. B |

21. | V. Myroshnychenko, A. Stefanski, A. Manjavacas, M. Kafesaki, R. I. Merino, V. M. Orera, D. A. Pawlak, and F. J. García de Abajo, “Interacting plasmon and phonon polaritons in aligned nano- and microwires,” Opt. Express |

22. | Z. X. Zhang, M. L. Hu, K. T. Chan, and C. Y. Wang, “Plasmonic waveguiding in a hexagonally ordered metal wire array,” Opt. Lett. |

23. | H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. |

24. | J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. |

25. | M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. |

26. | R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature |

27. | 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 |

28. | J. Tian, Z. Ma, Q. Li, Y. Song, Z. Liu, Q. Yang, C. Zha, J. Åkerman, L. Tong, and M. Qiu, “Nanowaveguides and couplers based on hybrid plasmonic modes,” Appl. Phys. Lett. |

29. | F. J. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B |

30. | M. Schmeits, “Surface-plasmon coupling in cylindrical pores,” Phys. Rev. B Condens. Matter |

31. | J. P. Kottmann and O. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express |

32. | M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express |

33. | C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express |

34. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

35. | A. D. Boardman, |

36. | J. C. Ashley and L. C. Emerson, “Dispersion relations for non-radiative surface plasmons on cylinders,” Surf. Sci. |

37. | Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: Analysis of optical properties,” Phys. Rev. B |

38. | D. M. Bishop, |

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

40. | G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express |

41. | G. B. Hoffman and R. M. Reano, “Vertical coupling between gap plasmon waveguides,” Opt. Express |

42. | D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B |

43. | Y. Ma, G. Farrell, Y. Semenova, H. P. Chan, H. Zhang, and Q. Wu, “Novel dielectric-loaded plasmonic waveguide for tight-confined hybrid plasmon mode,” Plasmonics |

44. | H. R. Park, J. M. Park, M. S. Kim, and M. H. Lee, “A waveguide-typed plasmonic mode converter,” Opt. Express |

45. | Y. T. Hung, C. B. Huang, and J. S. Huang, “Plasmonic mode converter for controlling optical impedance and nanoscale light-matter interaction,” Opt. Express |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(260.2030) Physical optics : Dispersion

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 29, 2013

Revised Manuscript: May 11, 2013

Manuscript Accepted: May 13, 2013

Published: June 12, 2013

**Citation**

Shulin Sun, Hung-Ting Chen, Wei-Jin Zheng, and Guang-Yu Guo, "Dispersion relation, propagation length and mode conversion of surface plasmon polaritons in silver double-nanowire systems," Opt. Express **21**, 14591-14605 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-14591

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### References

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