## Analysis of sub-wavelength light propagation through long double-chain nanowires with funnel feeding

Optics Express, Vol. 15, Issue 7, pp. 4216-4223 (2007)

http://dx.doi.org/10.1364/OE.15.004216

Acrobat PDF (355 KB)

### Abstract

The surface integral equation (SIE) method is utilized to characterize plasmonic waveguide made of two parallel chains of silver nanowires with radius of 25nm fed by a V-shaped funnel at a working wavelength of 600nm. The efficiency of energy transport along the waveguide due to surface plasmonic coupling is investigated for different dimensions and shapes. The opening angle of the V-shaped funnel region for optimum light capturing is included in the investigation as well. A long plasmonic double-chain waveguide of length ∼3.3μm has been analyzed and optimized.

© 2007 Optical Society of America

## 1. Introduction

05. J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B **60**,9061–9068 (1999). [CrossRef]

11. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. **23**,1331–1333 (1998). [CrossRef]

## 2. Surface integral equation method

*ε*(ω) =

_{r}*ε*(ω) +

_{Re}*jε*(ω) with a negative real value and strongly dependent on the frequency [15]. Thus, several simulation techniques which are limited to lossless, non-dispersive materials are not applicable to plasmonic devices. In time-domain methods the dispersion properties of metals are approximated by suitable analytical expressions. In most cases the Drude model is invoked to characterize the frequency dependence of the metallic dielectric function [8

_{Im}08. W. M. Saj, “FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice,” Opt. Express **13**,4818–4827 (2005). [CrossRef] [PubMed]

13. S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain results for silver cylinders,” Phys. Rev. B **68**, 045415 (1–11) (2003). [CrossRef]

16. A. F. Peterson, S. L. Ray, and R. Mittra, *Computational Methods for Electromagnetics* (Wiley-IEEE Press, 1997). [CrossRef]

## 3. Model

13. S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain results for silver cylinders,” Phys. Rev. B **68**, 045415 (1–11) (2003). [CrossRef]

05. J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B **60**,9061–9068 (1999). [CrossRef]

*d*, and

_{fn}*d*, respectively. The two parallel chains are separated by a distance of

*h*(center-center separation of adjacent cylinders). For simplicity, the medium surrounding the waveguide is assumed to be air (

*ε*= 1.0). Throughout the entire paper, the cylindrical silver nanowires have a radius of

_{r}*r*= 25 nm. The light source is located between the first cylinders in the funnel region and in the center plane of the double-chain waveguide, as shown in Fig. 1. The structure is illuminated by a TE

^{Z}mode, which means that all E-field components are transverse to the z-direction [16

16. A. F. Peterson, S. L. Ray, and R. Mittra, *Computational Methods for Electromagnetics* (Wiley-IEEE Press, 1997). [CrossRef]

*ε*(

_{r}*ω*) = -13.98 +

*j*0.95. The computed field intensity is extracted along two planes; one along the horizontal symmetry plane (x-z plane or line AB), that is along the axis of propagation and the other transverse to this plane (z-y plane or line CD) that cuts through the center of the second last cylinder pairs as shown in Fig. 1.

## 4. Results and discussion

*d*, the center-arm,

_{fn}*d*, and between the double-chain,

*h*. The excitation at different wavelengths will also be investigated as this may reveal some extra information on how the wavelength-dependent permittivity affects the optimum waveguide dimensions. Finally the effect of the opening angle in the funnel region will be determined.

*d*= 2.04

_{fn}*r*, 2.2

*r*, 2.6

*r*, 3

*r*and 4

*r*, which correspond to actual values of

*d*= 51, 55, 65, 75 and 100 nm. Fig. 2 illustrates the relative field intensity along the AB and CD planes (defined in Fig. 1). It is evident that for dfn = 2.2

_{fn}*r*, 2.6

*r*, 3

*r*and 4

*r*there is little effect on the field distribution along the double-chain region or transverse to the propagation direction at the output of the waveguide [CD, Fig. 2(b)], 3190 nm away from the last pair of cylinders in the double-chain. This changes, however, when two nanowires in the funnel region become too close (or too far), such as for

*d*= 2.04

_{fn}*r*. Figure 2(b) also illustrates that the light is well confined in the center of the double-chain and decays rapidly away from the symmetry plane in the ±y-direction of the waveguide. From both figures a value of

*d*= 2.6

_{fn}*r*can be considered as optimum center-center distance for two adjacent nanowires to obtain maximum energy transport along the double-chain.

*d*= 2.6r. The center-center separation varies over a range of

_{fn}*d*= 2.04

*r*to 4

*r*. It can be seen from Fig. 3 that, for a spacing of

*d*= 2.04

*r*and

*d*= 4

*r*, the field intensity at the end of the waveguide becomes quite weak with best values obtained for a spacing of

*d*= 2.2

*r*. Judging from the results displayed in Fig. 3 it is evident that the center-center spacing in the double-chain region has a much higher influence on the field distribution than that in the funnel region.

*h*separating the parallel chains. The separation between the AgNW in the funnel region and the double-chain region are kept constant at

*d*= 2.6

_{fn}*r*= 65 nm and

*d*= 2.2

*r*= 55 nm, respectively. Figure 4(a) shows that for values of

*h*= 2.04

*r*and 2.2

*r*, the field intensity decays rapidly along the propagation direction and that the intensity at the end of the double-chain drops to zero for

*h*= 2.04

*r*, which corresponds to 5 nm. For gap widths ranging from

*h*= 2.2

*r*to

*h*= 3

*r*the field intensities change only slightly reaching a maximum for

*h*= 3

*r*= 75nm. This value can be considered the optimum distance between the double-chain to obtain the maximum light energy propagating along the structure.

## 5. Conclusions

## References and links

01. | H. Raether, |

02. | J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B |

03. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

04. | R. Zhia, J. A. Schuller, A. Chandran, and M. Brongersma, “Plasmonics: the next chip-scale technology,” Materials today |

05. | J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, “Plasmon polaritons of metallic nanowires for controlling submicron propagation of light,” Phys. Rev. B |

06. | S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. |

07. | J. R. Krenn, B. Lamprecht, H. Ditlbacher, G. Schider, M. Salerno, A. Leitner, and F. R. Aussenegg, “Non diffraction limited light transport by gold nanowires,” Europhys. Lett. |

08. | W. M. Saj, “FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice,” Opt. Express |

09. | J. C. Weeber, M. U. González, A. L. Baudrion, and A. Dereux, “Surface plasmon routing along right angle bent metal strips,” App. Phys. Lett. |

10. | P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmon-polariton waveguides,” J. Appl. Phys. |

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

12. | M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B |

13. | S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain results for silver cylinders,” Phys. Rev. B |

14. | S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B |

15. | D. W. Lynch and W. R. Hunter, “Comments on the optical constants of metals and an introduction to the data for several metals,” in |

16. | A. F. Peterson, S. L. Ray, and R. Mittra, |

17. | T. K. Wu and L. L. Tsai, “Scattering by arbitrarily cross-sectioned layered, lossy dielectric cylinders,” IEEE Trans. Antennas Propagat . |

18. | Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propagat. |

19. | H. S. Chu, W. B. Ewe, E.P. Li, H. P. Lee, and R. Thampuran, “Surface integral equation method to characterize the nanoplasmonic waveguides with funneling array,” in Nanometa 2007 Conference Digest, Tirol, Austria,8–11 Jan. 2007. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: February 2, 2007

Revised Manuscript: March 12, 2007

Manuscript Accepted: March 19, 2007

Published: April 2, 2007

**Citation**

Hong-Son Chu, Wei-Bin Ewe, Er-Ping Li, and Rüdiger Vahldieck, "Analysis of sub-wavelength light propagation through long double-chain nanowires with funnel feeding," Opt. Express **15**, 4216-4223 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-7-4216

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### References

- H. Raether, Surface Plasmonson smooth and rough surfaces and on gratings (Springer-Verlag, Berlin, 1988).
- J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, "Plasmon polaritons of metallic nanowires for controlling submicron propagation of light," Phys. Rev. B 60, 9061- 9068 (1999). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824 - 830 (2003). [CrossRef] [PubMed]
- R. Zhia, J. A. Schuller, A. Chandran, and M. Brongersma, "Plasmonics: the next chip-scale technology," Materials Today 9, 20 - 27 (2006). [CrossRef]
- J. C. Weeber, A. Dereux, C. Girard, J. R. Krenn, and J. P. Goudonnet, "Plasmon polaritons of metallic nanowires for controlling submicron propagation of light," Phys. Rev. B 60, 9061- 9068 (1999). [CrossRef]
- S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, "Waveguiding in surface plasmon polariton band gap structures," Phys. Rev. Lett. 86, 3008-3011 (2001). [CrossRef] [PubMed]
- J. R. Krenn, B. Lamprecht, H. Ditlbacher, G. Schider, M. Salerno, A. Leitner, F. R. Aussenegg, "Non diffraction limited light transport by gold nanowires," Europhys. Lett. 60, 663-669 (2002) [CrossRef]
- W. M. Saj, "FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice," Opt. Express 13,4818-4827 (2005). [CrossRef] [PubMed]
- J. C. Weeber, M. U. González, A. L. Baudrion, and A. Dereux, "Surface plasmon routing along right angle bent metal strips," Appl. Phys. Lett. 87, 221101 (2005). [CrossRef]
- P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, "Characterization of long-range surface-plasmon-polariton waveguides," J. Appl. Phys. 98, 043109 (2005). [CrossRef]
- M. Quinten, A. Leitner, J. R. Krenn and F. R. Aussenegg, "Electromagnetic energy transport via linear chains of silver nanoparticles," Opt. Lett. 23, 1331-1333 (1998). [CrossRef]
- M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B 62, R16356-R16359 (2000). [CrossRef]
- S. K. Gray and T. Kupka, "Propagation of light in metallic nanowire arrays: Finite-difference time-domain results for silver cylinders," Phys. Rev. B 68, 045415 (1-11) (2003). [CrossRef]
- S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003). [CrossRef]
- D. W. Lynch and W. R. Hunter, "Comments on the optical constants of metals and an introduction to the data for several metals," in Handbook of Optical Constants of Solids, E.D. Palik, ed., (Academic Press, New York, 1985).
- A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (Wiley-IEEE Press, 1997). [CrossRef]
- T. K. Wu and L. L. Tsai, "Scattering by arbitrarily cross-sectioned layered, lossy dielectric cylinders," IEEE Trans. Antennas Propagat. 25, 518-524 (1977). [CrossRef]
- Y. Chang and R. Harrington, "A surface formulation for characteristic modes of material bodies," IEEE Trans. Antennas Propagat. 25, 789-795 (1977). [CrossRef]
- H. S. Chu, W. B. Ewe, E. P. Li, H. P. Lee, and R. Thampuran, "Surface integral equation method to characterize the nanoplasmonic waveguides with funneling array," in Nanometa 2007 Conference Digest, Tirol, Austria, 8-11 Jan. 2007.

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