## Effect of retardation on localized surface plasmon resonances in a metallic nanorod

Optics Express, Vol. 17, Issue 26, pp. 23655-23663 (2009)

http://dx.doi.org/10.1364/OE.17.023655

Acrobat PDF (246 KB)

### Abstract

The localized surface plasmon resonances in a metallic nanorod are determined using the “electrostatic approximation” and by a finite-difference time-domain numerical solution of Maxwell’s equations. The difference between the two methods is related to the effects of re-radiation, or retardation, which is not included in the electrostatic formulation. It is shown that high-order modes in a metallic nanorod can be modeled by both methods, even beyond the point where the electrostatic method is supposed to fail. This suggests that the simple analytical expressions derived from the electrostatic approximation are valid for describing the large range of resonant modes associated with metallic nanoparticles, including dark modes.

© 2009 OSA

## 1. Introduction

1. K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. **14**(3), 302–307 (1966). [CrossRef]

3. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A **11**(9), 2494–2502 (1994). [CrossRef]

5. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A **12**(5), 1077–1086 (1995). [CrossRef]

6. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**(4), 1491–1499 (1994). [CrossRef]

7. F. J. García de Abajo and A. Howie, “Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics,” Phys. Rev. Lett. **80**(23), 5180–5183 (1998). [CrossRef]

8. F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” A., Phys. Rev. B **65**(11), 115418 (2002). [CrossRef]

9. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B **72**(15), 155412 (2005). [CrossRef]

10. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B **79**(15), 155423 (2009). [CrossRef]

11. T. J. Davis, K. C. Vernon, and D. E. Gómez, “A plasmonic “ac Wheatstone bridge” circuit for high-sensitivity phase measurement and single-molecule detection,” J. Appl. Phys. **106**(4), 043502 (2009). [CrossRef]

12. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. **101**(4), 047401 (2008). [CrossRef] [PubMed]

14. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. **8**(9), 758–762 (2009). [CrossRef] [PubMed]

15. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets,” Proc. SPIE **7394**, 739423 (2009). [CrossRef]

## 2. Electrostatic approximation and the resonances of a nanorod

9. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B **72**(15), 155412 (2005). [CrossRef]

## 3. Resonances of the nanorod

### 3.1 Electrostatic resonances

### 3.2 Finite Difference Time Domain model

1. K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. **14**(3), 302–307 (1966). [CrossRef]

10. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B **79**(15), 155423 (2009). [CrossRef]

*x*,

*y*, and

*z*components of the electric field were sampled just below the nanorod, at the end opposite to that of the excitation pulse. Data were sampled as functions of time and then Fourier transformed to obtain the spectra. This process was repeated for four different scales corresponding to nanorod lengths of 90 nm, 230 nm, 460 nm and 690 nm. The resonances were identified by the order of the peak positions in the spectra.

10. T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B **79**(15), 155423 (2009). [CrossRef]

## 4. Discussion

9. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B **72**(15), 155412 (2005). [CrossRef]

*A*is the surface area of the nanoparticle,

*λ*is the wavelength of light in vacuum and

*A*to the square of the resonance wavelength

*λ*in the embedding medium. For a spherical particle, this gives a scaling with the ratio of the square of the diameter

*L*and its radius

*R*so that we should use

## 5. Summary

## References and links

1. | K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. |

2. | A. Taflove, |

3. | E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A |

4. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

5. | M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A |

6. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

7. | F. J. García de Abajo and A. Howie, “Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics,” Phys. Rev. Lett. |

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

9. | I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B |

10. | T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B |

11. | T. J. Davis, K. C. Vernon, and D. E. Gómez, “A plasmonic “ac Wheatstone bridge” circuit for high-sensitivity phase measurement and single-molecule detection,” J. Appl. Phys. |

12. | S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. |

13. | N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. |

14. | N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. |

15. | T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets,” Proc. SPIE |

16. | P. W. Barber, R. K. Chang, and H. Massoudi, “Electrodynamic calculations of the surface-enhanced electric intensities on large Ag spheroids,” Phys. Rev. B |

17. | V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. |

18. | W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B |

19. | A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B |

20. | C. F. Bohren, and D. R. Huffman, |

21. | I. D. Mayergoyz, Z. Zhang, and G. Miano, “Analysis of dynamics of excitation and dephasing of plasmon resonance modes in nanoparticles,” Phys. Rev. Lett. |

22. | T. J. Davis, “Modelling and fabrication of tuned circuits for optical meta-materials,” Proc. SPIE |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: October 23, 2009

Revised Manuscript: November 30, 2009

Manuscript Accepted: December 1, 2009

Published: December 10, 2009

**Citation**

Timothy J. Davis, Kristy C. Vernon, and Daniel E. Gómez, "Effect of retardation on localized surface plasmon resonances in a metallic nanorod," Opt. Express **17**, 23655-23663 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23655

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### References

- K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]
- A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, London, 1995).
- E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11(9), 2494–2502 (1994). [CrossRef]
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]
- M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A 12(5), 1077–1086 (1995). [CrossRef]
- B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]
- F. J. García de Abajo and A. Howie, “Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics,” Phys. Rev. Lett. 80(23), 5180–5183 (1998). [CrossRef]
- F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” A., Phys. Rev. B 65(11), 115418 (2002). [CrossRef]
- I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, “Electrostatic (plasmon) resonances in nanoparticles,” Phys. Rev. B 72(15), 155412 (2005). [CrossRef]
- T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009). [CrossRef]
- T. J. Davis, K. C. Vernon, and D. E. Gómez, “A plasmonic “ac Wheatstone bridge” circuit for high-sensitivity phase measurement and single-molecule detection,” J. Appl. Phys. 106(4), 043502 (2009). [CrossRef]
- S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). [CrossRef] [PubMed]
- N. Verellen, Y. Sonnefraud, H. Sobhani, F. Hao, V. V. Moshchalkov, P. Van Dorpe, P. Nordlander, and S. A. Maier, “Fano resonances in individual coherent plasmonic nanocavities,” Nano Lett. 9(4), 1663–1667 (2009). [CrossRef] [PubMed]
- N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]
- T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems: applications to dark modes in nanoparticle pairs and triplets,” Proc. SPIE 7394, 739423 (2009). [CrossRef]
- P. W. Barber, R. K. Chang, and H. Massoudi, “Electrodynamic calculations of the surface-enhanced electric intensities on large Ag spheroids,” Phys. Rev. B 27(12), 7251–7261 (1983). [CrossRef]
- V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37(9), 1792–1805 (2008). [CrossRef] [PubMed]
- W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004). [CrossRef]
- A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74(3), 033402 (2006). [CrossRef]
- C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 5.
- I. D. Mayergoyz, Z. Zhang, and G. Miano, “Analysis of dynamics of excitation and dephasing of plasmon resonance modes in nanoparticles,” Phys. Rev. Lett. 98(14), 147401 (2007). [CrossRef] [PubMed]
- T. J. Davis, “Modelling and fabrication of tuned circuits for optical meta-materials,” Proc. SPIE 6038, Y380 (2005).

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