## Subwavelength plasmonic kinks in arrays of metallic nanoparticles |

Optics Express, Vol. 20, Issue 3, pp. 2733-2739 (2012)

http://dx.doi.org/10.1364/OE.20.002733

Acrobat PDF (1627 KB)

### Abstract

We analyze nonlinear effects in optically driven arrays of nonlinear metallic nanoparticles. We demonstrate that such plasmonic systems are characterized by a bistable response, and they can support the propagation of dissipative switching waves (or plasmonic kinks) connecting the states with different polarization. We study numerically the properties of such plasmonic kinks which are characterized by a subwavelength extent and a tunable velocity.

© 2011 OSA

## 1. Introduction

*plasmons*[1–3

3. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics **4**, 83–91 (2010). [CrossRef]

4. J. Takahara, S. Yamagishi, H. Taki, A. Moromoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. **22**, 475–477 (1997). [CrossRef] [PubMed]

5. K. Li, M. Stockman, and D. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. **91**, 227402 (2003). [CrossRef] [PubMed]

6. N. C. Panoiu and R. M. Osgood, “Subwavelength nonlinear plasmonic nanowire,” Nano Lett. **4**, 2427–2430 (2004). [CrossRef]

7. W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. **6**, 1027–1030 (2006). [CrossRef]

9. G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. **97**, 057402 (2006). [CrossRef] [PubMed]

10. A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. **106**, 073104 (2009). [CrossRef]

11. R. E. Noskov, A. A. Zharov, and M. V. Tsarev, “Generation of widely tunable continuous-wave terahertz radiation using a two-dimensional lattice of nonlinear metallic nanodimers,” Phys. Rev. B **82**, 073404 (2010). [CrossRef]

12. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. **99**, 153901 (2007). [CrossRef] [PubMed]

14. A. Marini, A. V. Gorbach, and D. V. Skryabin, “Coupled-mode approach to surface plasmon polaritons in nonlinear periodic structures,” Opt. Lett. **35**, 3532 (2010). [CrossRef] [PubMed]

15. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. **104**, 106802 (2010). [CrossRef] [PubMed]

12. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. **99**, 153901 (2007). [CrossRef] [PubMed]

15. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. **104**, 106802 (2010). [CrossRef] [PubMed]

16. K. M. Leung, “Optical bistability in the scattering and absorption of light from nonlinear microparticles,” Phys. Rev. A **33**, 2461 (1986). [CrossRef] [PubMed]

## 2. Model and basic equations

_{2}host medium with permittivity

*ɛ*, and is exited by the optical field

_{h}**E**

^{(ex)}(see Fig. 1). We assume that the particle radii and the center-to-center distance are

*a*= 10 nm and

*d*= 30 nm, respectively. Ratio

*a*/

*d*satisfies the condition

*a*/

*d*≤ 1/3, so that we can employ the point dipole approximation [17

17. S. Yong and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B **69**, 125418 (2004). [CrossRef]

*ɛ*

_{∞}= 4.96,

*h̄ω*= 9.54 eV,

_{p}*h̄ν*= 0.055 eV [18

18. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **9**, 4370 (1972). [CrossRef]

*iωt*) time dependence); whereas dispersion of SiO

_{2}can be neglected since

*ɛ*≃ 2.15 for wavelengths 350 – 450 nm [19]. Nonlinear dielectric constant of silver is

_{h}*n*-th particle. We keep only cubic susceptibility due to spherical symmetry of particles. According to the model suggested in Ref. [20

20. V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, “Size dependent *χ*^{(3)} for conduction electrons in Ag nanoparticles,” Nano Lett. **4**, 1535 (2004). [CrossRef]

*χ*

^{(3)}≃ 3 ×10

^{−9}esu, in comparing to which the cubic nonlinearity of SiO

_{2}is negligibly weak (∼ 10

^{−15}esu [21]).

**p**

*, we employ the model based on the dispersion relation method. This technique was firstly applied to investigation of light scattering by nonlinear metal nanodimers [10*

_{n}10. A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. **106**, 073104 (2009). [CrossRef]

11. R. E. Noskov, A. A. Zharov, and M. V. Tsarev, “Generation of widely tunable continuous-wave terahertz radiation using a two-dimensional lattice of nonlinear metallic nanodimers,” Phys. Rev. B **82**, 073404 (2010). [CrossRef]

22. I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Y. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. **4**, 69 (2006). [CrossRef]

23. N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Y. S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. **93**, 743 (2011). [CrossRef]

*γ*=

*ν*/(2

*ω*

_{0}) + (

*k*

_{0}

*a*)

^{3}

*ɛ*/(

_{h}*ɛ*

_{∞}+ 2

*ɛ*) describes both thermal and radiation losses of particles,

_{h}*ω*–

*ω*

_{0})/

*ω*

_{0}and

*τ*=

*ω*

_{0}

*t*. Equations (1) describe temporal nonlinear dynamics of an array of metal nanoparticles driven by arbitrary external optical field with the frequency

*ω*∼

*ω*

_{0}. We stress that model takes into account all particle interactions through the dipole fields, and it can be applied both to finite and infinite arrays.

## 3. Bistable response and plasmonic kinks

*n*–

*m*| =

*j*and taking into account symmetry structure of the series. When the polarization

*υ*. That is why a step-like time dependency of

*υ*overall bistability zone.

*υ*= 0. Generally speaking, this is a common property of nonlinear discrete systems [24] where the effective periodic potential created by the lattice requires a finite value of the applied external force to start kink’s motion. The existence of the kinks with zero velocity suggest a straightforward way to construct spatially localized modes by taking two standing kink of the opposite polarity. The resulting localized modes form the continuous families, and they resemble the similar localized modes discussed earlier for metal-dielectric multilayers [12

12. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. **99**, 153901 (2007). [CrossRef] [PubMed]

14. A. Marini, A. V. Gorbach, and D. V. Skryabin, “Coupled-mode approach to surface plasmon polaritons in nonlinear periodic structures,” Opt. Lett. **35**, 3532 (2010). [CrossRef] [PubMed]

15. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. **104**, 106802 (2010). [CrossRef] [PubMed]

*υ*[see Fig. 5(a)]. We find that the kinks start moving slowly towards each other, then stop and form a stable dissipative plasmon soliton, as shown in Fig. 5(b). The generated plasmonic mode is localized only on 6 particles, and its width is about 150 nm. Because Ω = −0.1 corresponds to the radiation wavelength

*λ*= 440 nm, this plasmon soliton is truly subwavelength because it extends for 0.34

*λ*. It remains open if the kink’s motion may demonstrate bistability, as was recently revealed for a similar nonlinear system with local coupling [23

**104**, 106802 (2010). [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgments

## References and links

1. | M. I. Brongersma and P. G. Krik, eds. |

2. | S. Maier, |

3. | D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics |

4. | J. Takahara, S. Yamagishi, H. Taki, A. Moromoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. |

5. | K. Li, M. Stockman, and D. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. |

6. | N. C. Panoiu and R. M. Osgood, “Subwavelength nonlinear plasmonic nanowire,” Nano Lett. |

7. | W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett. |

8. | J. A. H. van Nieuwstadt, M. Sandke, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett. |

9. | G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett. |

10. | A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. |

11. | R. E. Noskov, A. A. Zharov, and M. V. Tsarev, “Generation of widely tunable continuous-wave terahertz radiation using a two-dimensional lattice of nonlinear metallic nanodimers,” Phys. Rev. B |

12. | Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. |

13. | A. Marini, D. V. Skryabin, and B. Malomed, “Stable spatial plasmon solitons in a dielectric-metal-dielectric geometry with gain and loss,” Opt. Express |

14. | A. Marini, A. V. Gorbach, and D. V. Skryabin, “Coupled-mode approach to surface plasmon polaritons in nonlinear periodic structures,” Opt. Lett. |

15. | F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett. |

16. | K. M. Leung, “Optical bistability in the scattering and absorption of light from nonlinear microparticles,” Phys. Rev. A |

17. | S. Yong and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B |

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

19. | E. D. Palik, |

20. | V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, “Size dependent |

21. | M. J. Weber, |

22. | I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Y. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. |

23. | N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Y. S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. |

24. | O. M. Braun and Yu.S. Kivshar, |

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

(250.5403) Optoelectronics : Plasmonics

(190.6135) Nonlinear optics : Spatial solitons

(250.6715) Optoelectronics : Switching

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 2, 2011

Revised Manuscript: November 25, 2011

Manuscript Accepted: December 2, 2011

Published: January 23, 2012

**Citation**

Roman E. Noskov, Pavel A. Belov, and Yuri S. Kivshar, "Subwavelength plasmonic kinks in arrays of metallic nanoparticles," Opt. Express **20**, 2733-2739 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2733

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### References

- M. I. Brongersma and P. G. Krik, eds. Surface Plasmon Nanophotonics (Spinger, 2007), p. 268.
- S. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), p. 219.
- D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4, 83–91 (2010). [CrossRef]
- J. Takahara, S. Yamagishi, H. Taki, A. Moromoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett.22, 475–477 (1997). [CrossRef] [PubMed]
- K. Li, M. Stockman, and D. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett.91, 227402 (2003). [CrossRef] [PubMed]
- N. C. Panoiu and R. M. Osgood, “Subwavelength nonlinear plasmonic nanowire,” Nano Lett.4, 2427–2430 (2004). [CrossRef]
- W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6, 1027–1030 (2006). [CrossRef]
- J. A. H. van Nieuwstadt, M. Sandke, S. Enoch, and L. Kuipers, “Strong modification of the nonlinear optical response of metallic subwavelength hole arrays,” Phys. Rev. Lett.97, 146102 (2006). [CrossRef] [PubMed]
- G. A. Wurtz, R. Pollard, and A. V. Zayats, “Optical bistability in nonlinear surface-plasmon polaritonic crystals,” Phys. Rev. Lett.97, 057402 (2006). [CrossRef] [PubMed]
- A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys.106, 073104 (2009). [CrossRef]
- R. E. Noskov, A. A. Zharov, and M. V. Tsarev, “Generation of widely tunable continuous-wave terahertz radiation using a two-dimensional lattice of nonlinear metallic nanodimers,” Phys. Rev. B82, 073404 (2010). [CrossRef]
- Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett.99, 153901 (2007). [CrossRef] [PubMed]
- A. Marini, D. V. Skryabin, and B. Malomed, “Stable spatial plasmon solitons in a dielectric-metal-dielectric geometry with gain and loss,” Opt. Express19, 6616 (2011). [CrossRef] [PubMed]
- A. Marini, A. V. Gorbach, and D. V. Skryabin, “Coupled-mode approach to surface plasmon polaritons in nonlinear periodic structures,” Opt. Lett.35, 3532 (2010). [CrossRef] [PubMed]
- F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, “Subwavelength plasmonic lattice solitons in arrays of metallic nanowires,” Phys. Rev. Lett.104, 106802 (2010). [CrossRef] [PubMed]
- K. M. Leung, “Optical bistability in the scattering and absorption of light from nonlinear microparticles,” Phys. Rev. A33, 2461 (1986). [CrossRef] [PubMed]
- S. Yong and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B69, 125418 (2004). [CrossRef]
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B9, 4370 (1972). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
- V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, “Size dependent χ(3) for conduction electrons in Ag nanoparticles,” Nano Lett.4, 1535 (2004). [CrossRef]
- M. J. Weber, Handbook of Optical Materials (CRC Press, 2003).
- I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Y. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl.4, 69 (2006). [CrossRef]
- N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Y. S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett.93, 743 (2011). [CrossRef]
- O. M. Braun and Yu.S. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods, and Applications (Springer-Heidelberg, 2004), p. 498.

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