## Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surfaces |

Optics Express, Vol. 19, Issue 13, pp. 12208-12219 (2011)

http://dx.doi.org/10.1364/OE.19.012208

Acrobat PDF (3855 KB)

### Abstract

We present an advanced numerical formulation to calculate the optical properties of 3D nanoparticles (single or coupled) of arbitrary shape and lack of symmetry. The method is based on the (formally exact) surface integral equation formulation, implemented for parametric surfaces describing particles with arbitrary shape through a unified treatment (Gielis’ formula). Extinction, scattering, and absorption spectra of a variety of metal nanoparticles are shown, thus determining rigorously the localised surface-plasmon resonances of nanocubes, nanostars, and nanodimers. Far-field and near-field patterns for such resonances are also calculated, revealing their nature. The flexibility and reliability of the formulation makes it specially suitable for complex scattering problems in Nano-Optics & Plasmonics.

© 2011 OSA

## 1. Introduction

1. X. Lu, M. Rycenga, S. E. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem. **60**, 167–92 (2009). [CrossRef]

3. A. Ono, J. Kato, and S. Kawata, “Subwavelength optical imaging through a metallic nanorod array,” Phys. Rev. Lett. **95**, 267407 (2005). [CrossRef]

4. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science **311**, 189–193 (2006). [CrossRef] [PubMed]

6. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics **5**, 83–90 (2011). [CrossRef]

7. J. A. Sánchez-Gil and J. V. García-Ramos, “Local and average electromagnetic enhancement in surface-enhanced Raman scattering from self-affine fractal metal substrates with nanoscale irregularities,” Chem. Phys. Lett. **367**, 361–366 (2003). [CrossRef]

9. H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface- enhanced Raman scattering,” Phys. Rev. E **62**, 4318–4324 (2000). [CrossRef]

5. V. Giannini, A. Fernandez-Dominguez, Y. Sonnefraud, T. Roschuk, R. Fernandez-García, and S. A. Maier, “Controlling light localization and light–matter interactions with nanoplasmonics,” Small **6**, 2498–2507 (2010). [CrossRef] [PubMed]

10. P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, and B. Hecht, “Resonant optical antennas,” Science **308**, 1607–1609 (2005). [CrossRef] [PubMed]

13. T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. V. Hulst, “*λ*/4 resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. **7**, 28–33 (2007). [CrossRef] [PubMed]

14. C. Bohren and D. Huffman, *Absorption and Scattering of Light by Small Particles* (Wiley, 1998). [CrossRef]

15. W. L. Barnes, “Comparing experiment and theory in plasmonics,” J. Opt. A, Pure Appl. Opt. **11**, 114002 (2009). [CrossRef]

16. K. S. Yee, “Numerical Solution of initial value problems of Maxwells equations,” IEEE Trans. Antenn. Propag. **14**, 302–307 (1966). [CrossRef]

17. R. Clough, “The finite element method after twenty-five years: a personal view,” Comput. Struct. **12**, 361–370 (1980). [CrossRef]

18. C. Girard and A. Dereux, “Near-field optics theories,” Rep. Progr. Phys. **59**, 657 (1996). [CrossRef]

2. T. R. Jensen, G. C. Schatz, and R. P. V. Duyne, “Nanosphere lithography: surface plasmon resonance spectrum of a periodic array of silver nanoparticles by ultraviolet-visible extinction spectroscopy and electrodynamic modeling,” J. Phys. Chem. B **103**, 2394–2401 (1999). [CrossRef]

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

20. M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Tranfer. **111**, 650–658 (2010). [CrossRef]

21. V. Myroshnychenko, E. Carbó-Argibay, I. Pastoriza-Santos, J. Pérez-Juste, L. M. Liz-Marzán, and F. García de Abajo, “Modeling the optical response of highly faceted metal nanoparticles with a fully 3D boundary element method,” Adv. Mater. **20**, 4288–4293 (2008). [CrossRef]

22. A. A. Maradudin, T. R. Michel, A. Mcgurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. **203**, 255–307 (1990). [CrossRef]

23. J. A. Sanchez-Gil and M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am A **8**, 1270 (1991). [CrossRef]

15. W. L. Barnes, “Comparing experiment and theory in plasmonics,” J. Opt. A, Pure Appl. Opt. **11**, 114002 (2009). [CrossRef]

24. S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. **30**, 409–418 (1982). [CrossRef]

25. A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonics and high permittivity nanostructures,” J. Opt. Soc. Am. A **26**, 732–740 (2009). [CrossRef]

22. A. A. Maradudin, T. R. Michel, A. Mcgurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. **203**, 255–307 (1990). [CrossRef]

22. A. A. Maradudin, T. R. Michel, A. Mcgurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. **203**, 255–307 (1990). [CrossRef]

23. J. A. Sanchez-Gil and M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am A **8**, 1270 (1991). [CrossRef]

26. P. Tran and A. Maradudin, “The scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Opt. Commun. **110**, 269–273 (1994). [CrossRef]

29. I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of Electromagnetic Waves from Two-Dimensional Randomly Rough Penetrable Surfaces,” Phys. Rev. Lett. **104**, 223,904 (2010). [CrossRef] [PubMed]

30. C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two dimensional nanoparticles,” J. Opt. Soc. Am. B **20**, 2150–2161 (2003). [CrossRef]

31. V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A **24**, 2822 (2007). [CrossRef]

32. U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: a boundary integral method approach,” Phys. Rev. B **72**, 1–9 (2005). [CrossRef]

33. J. Jung and T. Sodergaard, “Greens function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B **77**, 245310 (2008). [CrossRef]

34. P. I. Geshev, U. Fischer, and H. Fuchs, “Calculation of tip enhanced Raman scattering caused by nanoparticle plasmons acting on a molecule placed near a metallic film,” Phys. Rev. B **81**, 125,441 (2010). [CrossRef]

35. J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. **90**, 333–338 (2003). [CrossRef] [PubMed]

## 2. General description of the method

### 2.1. Scattering equations for a single scatterer

*ε*. A monochromatic plane wave of frequency

_{out}*ω*(incident field), impinges on the scatterers with module |

**E**

*| at an arbitrary direction of propagation with propagation constant*

_{i}*e*

^{−}

*. The electromagnetic fields outside (*

^{iωt}**E**

^{>},

**H**

^{>}), and inside (

**E**

^{<},

**H**

^{<}) of a scatterer with volume

*V*and arbitrary shape Ω generated by the interaction between the scatterer and the incident field are described by the Stratton-Chu equations [26

26. P. Tran and A. Maradudin, “The scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Opt. Commun. **110**, 269–273 (1994). [CrossRef]

27. K. Pak, L. Tsang, and J. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough surfaces with the sparse-matrix canonical grid method,” J. Opt. Soc. Am. A **14**, 1515 (1997). [CrossRef]

36. J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. **56**, 99–107 (1939). [CrossRef]

*ε*, and the vector

_{in/out}**n**is the normal vector to the surface pointing outwards. In Eqs. (1) , we assume

*μ*=

*μ*=

_{in}*μ*= 1.

_{out}### 2.2. System of integral equations: Surface EM fields

26. P. Tran and A. Maradudin, “The scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Opt. Commun. **110**, 269–273 (1994). [CrossRef]

30. C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two dimensional nanoparticles,” J. Opt. Soc. Am. B **20**, 2150–2161 (2003). [CrossRef]

**r**tend to Ω from either outside or inside, as needed), and connect EM fields outside and inside through the continuity conditions across the scatterer surface, namely: Finally, we choose Eqs (1a,d) (projected to the surface normal) and Eqs (1b,d) (projected to the surface tangents) as the six scalar equations that constitute our system of surface integral equations.

**r**

*and*

_{t}**r**

*are the derivative of the vector position with respect to the parameters*

_{s}*t*and

*s*, (see Section 3 below). In this manner, we are explicitly stating the parametric equations as the descriptor of the surfaces, since it is the more natural way to obtain the vectorial basis.

### 2.3. Numerical implementation: Quadrature scheme

*is an arbitrary element of the mesh and*

_{i}*r*denotes the position vector of the point at the centre of the interval. Making use of the quadrature scheme introduced above all over the integrals, and projecting Eqs. (1) , as mentioned above, onto the vectors of the natural basis of the surface, Eq. (4), we obtain the system of linear equations. In this regard, we have to take the divergences of the Green function into account. If

_{i}*r*∼

*r*′, from the definition of the Green function, it follows that

*G*(

*r,r*′) → ∞ and ∇

*G*(

*r,r*′) → ∞. In order to deal with these difficulties, it is needed to perform an analytical integration of the divergences of Eqs. (1) with the prescriptions in Refs. [22

**203**, 255–307 (1990). [CrossRef]

38. H. Ying Yao and Y. Bing Gan, “Regularization of the combined field integral equation on parametric surface for EM scattering problems,” Electromagnetics **26**, 423–438 (2006). [CrossRef]

^{3}surface elements and a precise integration of the interaction terms involving green function within every interval Δ

*is enough to reach convergence in calculations for isolated particles (e.g. cube) with typical size about 50 nm. The calculation takes nearly 150 minutes on an Intel Xeon Quad-Core X5550 2,67 GHz, without any fine tuning.*

_{i}### 2.4. Near field

### 2.5. Far field

**r**as in the case of the near field. In the far field,

**r**′ ≪

**r**holds, so that

**k**, leading to: where we have used the properties of the mixed product in the second term of Eq. (7).

**R**

*,*

_{t}**R**

*are the tangent vectors to the surface of integration in the far field. From the far-field amplitudes, physical magnitudes such as the scattering cross section (*

_{s}*Q*), the extinction cross section (

_{sca}*Q*), and the absorption cross section (

_{ext}*Q*) can be calculated [37]: where

_{abs}*fw*means the forward direction defined by the direction of incidence.

### 2.6. Multi-scatterers

## 3. Flexible surfaces: 3D supeshape

*supershape*(SS) or

*superformula*[35

35. J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. **90**, 333–338 (2003). [CrossRef] [PubMed]

39. P. Bourke, “SuperShape in 3D,” URL http://local.wasp.uwa.edu.au/~{}pbourke/geometry/supershape3d/.

*r*

_{1},

*r*

_{2}are two different realizations of the parameters

*n*

_{1},

*n*

_{2},

*n*

_{3},

*m*,

*a*,

*b*in Eq. (11).

*n*

_{1},

*n*

_{2}, .... For a catalogue of the different shapes allowed to the spherical supershape and to different supershapes, as the toroidal one, see e.g. Ref. [39

39. P. Bourke, “SuperShape in 3D,” URL http://local.wasp.uwa.edu.au/~{}pbourke/geometry/supershape3d/.

## 4. LSPR on complex nanoparticles

### 4.1. Single nanospheres: Mie comparison

14. C. Bohren and D. Huffman, *Absorption and Scattering of Light by Small Particles* (Wiley, 1998). [CrossRef]

*Q*and

_{sca}*Q*in vacuum for a metal sphere made of silver [41

_{ext}41. P. B. Johnson and R. W. Christie, “Optical constants of nobel metals,” Phys. Rev. B **6**, 4370 (1972). [CrossRef]

15. W. L. Barnes, “Comparing experiment and theory in plasmonics,” J. Opt. A, Pure Appl. Opt. **11**, 114002 (2009). [CrossRef]

*x*axis). The near field amplitude is shown along the white line drawn in Fig. 2(e) for both the Mie and the GTm solutions, in Fig. 2(f), confirming the accuracy of the GTm also for near field calculations.

### 4.2. Single supershape: from sharp/rounded nanocubes to nanostars

*Q*of a silver cube in vacuum on the rounded level of the corners. This is indeed a critical issue upon comparing theoretical calculations with experimental measurements, for real life nanoparticles often differ from perfect mathematical shapes [42

_{sca}42. S. Y. Lee, L. Hung, G. S. Lang, J. E. Cornett, I. D. Mayergoyz, and O. Rabin, “Dispersion in the SERS enhancement with silver nanocube dimers,” ACS Nano **4**, 5763–5772 (2010). [CrossRef] [PubMed]

*n*

_{1}=

*n*

_{2}=

*n*

_{3}≡

*n*> 4,

*a*=

*b*= 1, and

*m*= 4. The parameter

*m*and the degeneracy of the parameters

*n*

_{1},

*n*

_{2},

*n*

_{3}to

*n*defines the system as a cube. Taking different values for

*n*, from

*n*= 5 to

*n*= 20, we change the sharpness of the cube corners, the larger the value of

*n*the sharper corners; for

*n*> 20 the cube sharpness does not significantly change. The results are shown in Fig. 3. It is readily observed that, two changes are produced on the

*Q*with increasing

_{sca}*n*. Firstly, if we make the corners sharper the scattering of the cube becomes larger: the charge is accumulated on the corners due to the fast variation of the curvature and that increases the scattered field. Secondly, the LSPR of the cube redshifts about 20 nm as we increase

*n*. This can be understood upon taking into account that the SS changes from a surface close to a sphere (

*n*= 5), to a cube (

*n*> 20). Therefore the resonance wavelentgh has to evolve from a resonance close to that of a sphere with

*r*= 25 nm,

*λ*≃ 360 nm, to the resonance of a sharp cube, at

*λ*≃ 400 nm. These two effects have been reported in Ref. [43

43. A. L. González and C. Noguez, “Optical properties of silver nanoparticles,” Phys. Stat. Solidi C **4**, 4118–4126 (2007). [CrossRef]

**E**|

^{4}evaluated on the surface. For the cubes under study we obtain a value from |

**E**|

^{4}∼ 2 · 10

^{4}, (

*n*= 8) to |

**E**|

^{4}∼ 9 · 10

^{4}, (

*n*= 20). Even though those values of the enhancement factor make single nanocubes suitable SERS substrates, they are not large enough to guarantee in general detectable SERS signals.

*n*

_{1}> 0, 2 >

*n*

_{2}=

*n*

_{3}> 1, 1 =

*b*= 1,

*m ≥*3, where

*m*rules the symmetry of the star. In Fig. 4 the extinction cross section, the surface field distribution, and the far field pattern are shown for a 5-fold symmetry nanostar with only a symmetry axis, with parameters

*n*

_{1}= 0.2,

*n*

_{2}=

*n*

_{3}= 1.85,

*a*=

*b*= 1,

*m*= 5 for both realizations of the parameters of the SS, [

*r*

_{1}=

*r*

_{2}in Eq. (11)]. The incident field propagates along the

*z*-axis from bottom to top, and the polarization is oriented along the

*x*-axis (see Fig. 4). There is only a strong LSPR at

*λ*= 380 nm, which exhibits very asymmetric radiation patterns, shown in the far field distribution, Fig. 4(c). (In [44

44. V. Giannini, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Surface plasmon resonances of metallic nanostars/nanoflowers for surface-enhanced raman scattering,” Plasmonics **5**, 99–104 (2010). [CrossRef]

**E**|

^{4}∼ 10

^{7}, which is a value large enough to SERS applications as pointed out in [44

44. V. Giannini, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Surface plasmon resonances of metallic nanostars/nanoflowers for surface-enhanced raman scattering,” Plasmonics **5**, 99–104 (2010). [CrossRef]

45. P. Senthil Kumar, I. Pastoriza-Santos, B. Rodríguez-González, F. Javier García de Abajo, and L. M. Liz-Marzán, “High-yield synthesis and optical response of gold nanostars,” Nanotechnology **19**, 015606 (2008). [CrossRef] [PubMed]

## 5. Conclusion

*Supershape*. Both together the implementation of the 3D-GTm for an arbitrary number of three dimensional scatterers with different shape and electromagnetic properties, and surfaces without any specific symmetry as the flexible surfaces, become a powerful approach to scattering problems involving different kind of shapes, electromagnetic sources, materials on the scatterers and surrounding media. In addition, our method has three critical advantages. Firstly, the formalism is formally exact, so it takes into account the full non retarded multipolar contributions. Secondly, it needs a unique implementation to achieve a huge sort of different shapes. Thirdly, the numerical implementation scales with the particle surface rather than with its volume. Results are presented only for metal nanoparticles, but the formulation can be directly applied to dielectric scatterers, e.g. to tailor the magnetic response of Si nanoparticles [47

47. A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of Silicon nanoparticles in the infrared,” Opt. Express **19**, 4815–4826 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | X. Lu, M. Rycenga, S. E. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem. |

2. | T. R. Jensen, G. C. Schatz, and R. P. V. Duyne, “Nanosphere lithography: surface plasmon resonance spectrum of a periodic array of silver nanoparticles by ultraviolet-visible extinction spectroscopy and electrodynamic modeling,” J. Phys. Chem. B |

3. | A. Ono, J. Kato, and S. Kawata, “Subwavelength optical imaging through a metallic nanorod array,” Phys. Rev. Lett. |

4. | E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science |

5. | V. Giannini, A. Fernandez-Dominguez, Y. Sonnefraud, T. Roschuk, R. Fernandez-García, and S. A. Maier, “Controlling light localization and light–matter interactions with nanoplasmonics,” Small |

6. | L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics |

7. | J. A. Sánchez-Gil and J. V. García-Ramos, “Local and average electromagnetic enhancement in surface-enhanced Raman scattering from self-affine fractal metal substrates with nanoscale irregularities,” Chem. Phys. Lett. |

8. | E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cd,” J. Phys. C |

9. | H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface- enhanced Raman scattering,” Phys. Rev. E |

10. | P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, and B. Hecht, “Resonant optical antennas,” Science |

11. | J. J. Greffet, “Nanoantennas for light emission,” Science |

12. | O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. |

13. | T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. V. Hulst, “ |

14. | C. Bohren and D. Huffman, |

15. | W. L. Barnes, “Comparing experiment and theory in plasmonics,” J. Opt. A, Pure Appl. Opt. |

16. | K. S. Yee, “Numerical Solution of initial value problems of Maxwells equations,” IEEE Trans. Antenn. Propag. |

17. | R. Clough, “The finite element method after twenty-five years: a personal view,” Comput. Struct. |

18. | C. Girard and A. Dereux, “Near-field optics theories,” Rep. Progr. Phys. |

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

20. | M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: a 2007–2009 update,” J. Quant. Spectrosc. Radiat. Tranfer. |

21. | V. Myroshnychenko, E. Carbó-Argibay, I. Pastoriza-Santos, J. Pérez-Juste, L. M. Liz-Marzán, and F. García de Abajo, “Modeling the optical response of highly faceted metal nanoparticles with a fully 3D boundary element method,” Adv. Mater. |

22. | A. A. Maradudin, T. R. Michel, A. Mcgurn, and E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. |

23. | J. A. Sanchez-Gil and M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am A |

24. | S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. |

25. | A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonics and high permittivity nanostructures,” J. Opt. Soc. Am. A |

26. | P. Tran and A. Maradudin, “The scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Opt. Commun. |

27. | K. Pak, L. Tsang, and J. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough surfaces with the sparse-matrix canonical grid method,” J. Opt. Soc. Am. A |

28. | I. Simonsen, A. A. Maradudin, and T. A. Leskova, “The scattering of electromagnetic waves from two-dimensional randomly rough perfectly conducting surfaces: the full angular intensity distribution,” Phys. Rev. A |

29. | I. Simonsen, A. A. Maradudin, and T. A. Leskova, “Scattering of Electromagnetic Waves from Two-Dimensional Randomly Rough Penetrable Surfaces,” Phys. Rev. Lett. |

30. | C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two dimensional nanoparticles,” J. Opt. Soc. Am. B |

31. | V. Giannini and J. A. Sánchez-Gil, “Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of Green’s theorem surface integral equations in parametric form,” J. Opt. Soc. Am. A |

32. | U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: a boundary integral method approach,” Phys. Rev. B |

33. | J. Jung and T. Sodergaard, “Greens function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B |

34. | P. I. Geshev, U. Fischer, and H. Fuchs, “Calculation of tip enhanced Raman scattering caused by nanoparticle plasmons acting on a molecule placed near a metallic film,” Phys. Rev. B |

35. | J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. |

36. | J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. |

37. | M. Born and E. Wolf, |

38. | H. Ying Yao and Y. Bing Gan, “Regularization of the combined field integral equation on parametric surface for EM scattering problems,” Electromagnetics |

39. | P. Bourke, “SuperShape in 3D,” URL http://local.wasp.uwa.edu.au/~{}pbourke/geometry/supershape3d/. |

40. | H. Van De Hulst, |

41. | P. B. Johnson and R. W. Christie, “Optical constants of nobel metals,” Phys. Rev. B |

42. | S. Y. Lee, L. Hung, G. S. Lang, J. E. Cornett, I. D. Mayergoyz, and O. Rabin, “Dispersion in the SERS enhancement with silver nanocube dimers,” ACS Nano |

43. | A. L. González and C. Noguez, “Optical properties of silver nanoparticles,” Phys. Stat. Solidi C |

44. | V. Giannini, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Surface plasmon resonances of metallic nanostars/nanoflowers for surface-enhanced raman scattering,” Plasmonics |

45. | P. Senthil Kumar, I. Pastoriza-Santos, B. Rodríguez-González, F. Javier García de Abajo, and L. M. Liz-Marzán, “High-yield synthesis and optical response of gold nanostars,” Nanotechnology |

46. | E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J Chem. Phys. C |

47. | A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of Silicon nanoparticles in the infrared,” Opt. Express |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(290.5850) Scattering : Scattering, particles

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 29, 2011

Revised Manuscript: May 19, 2011

Manuscript Accepted: May 22, 2011

Published: June 8, 2011

**Citation**

Rogelio Rodríguez-Oliveros and José A. Sánchez-Gil, "Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surfaces," Opt. Express **19**, 12208-12219 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12208

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### References

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