## Multiple-multipole simulation of optical near-fields in discrete metal nanosphere assemblies

Optics Express, Vol. 16, Issue 3, pp. 1820-1835 (2008)

http://dx.doi.org/10.1364/OE.16.001820

Acrobat PDF (1076 KB)

### Abstract

We applied a multiple-multipole method to calculate the field enhancement of discrete metal nanosphere assemblies due to plasma resonance, thus performing the first full electromagnetic simulation of a variety of nanoparticle assemblies for efficient field focusing, including the self-similar geometric series of spheres first proposed by Li, Stockman and Bergman. Our study captures electromagnetic resonance effects important for optimizing nanoparticle assemblies to achieve maximum electric field focusing. We predict optical frequency electric fields can be enhanced in gold nanoparticle assemblies in aqueous solution by the order of ~450, within a factor of 2 of that achievable in silver nanostructures. We find that both absorption and far-field scattering resonances of nanoparticle assemblies must be carefully interpreted when inferring near-field focusing properties.

© 2008 Optical Society of America

## 1. Introduction

2. K. Kneipp, M. Moskovits, and H. Kneipp, *Surface-Enhanced Raman Scattering: Physics and Applications* (Springer, Berlin, 2006). [CrossRef]

*ν*is the scattered photon frequency.

_{s}3. L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B **71**, 235408 (2005). [CrossRef]

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

5. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404 (2004). [CrossRef] [PubMed]

6. J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, “Controlling the surface enhanced Raman effect via the nanoshell geometry,” Appl. Phys. Lett. **82**, 257–259 (2003). [CrossRef]

*et al.*, [7

7. B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, “Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles,” Nano Lett. **5**, 2246–2252 (2005). [CrossRef] [PubMed]

8. F. Aldaye and H. F. Sleiman, “Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching,” J. Am. Chem. Soc. **129**, 4130–4131 (2007). [CrossRef] [PubMed]

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. **34**, 4573–4588 (1995). [CrossRef] [PubMed]

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. **127**, 285–298 (1996). [CrossRef]

*et al.*, in which metal bispheres and clusters of small spheres dressing a large sphere have been analyzed. Thus far, there has been no full electromagnetic analysis of the focusing properties of sphere assemblies explicitly designed for efficient field focusing, such as the self-similar structure. The quasi-static approximation or dipole approximation are often made in studies of light scattering by assemblies of spherical nanoparticles; Khlebtsov et al. have conclusively shown that a multipole approach is needed for accurate prediction of far-field light scattering even for simple sub-wavelength bispheres [17

17. B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology **17**, 1437–1445 (2006). [CrossRef]

## 2. Numerical technique

### 2.1 Multiple-multipoles

*N*small spheres, each characterized by a normalized radius

_{s}*x*=

^{j}*ka*=2

^{j}*πm*/

_{medium}a^{j}*λ*, a normalized position

_{vacuum}**R**

^{j}=(

*X*,

^{j}*Y*,

^{j}*Z*) and a relative refractive index

^{j}*iωt*) for all field quantities. Mie theory accurately describes the scattering of uncoupled spheres, but generalization of Mie theory to multiple-multipoles is necessary to account for successive light scattering amongst nanospheres at sub-wavelength separations.

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. **34**, 4573–4588 (1995). [CrossRef] [PubMed]

**N**

^{(1)},

**M**

^{(1)},

**N**

^{(3)},

**M**

^{(3)}[9

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. **34**, 4573–4588 (1995). [CrossRef] [PubMed]

*η*=

*η*

_{0}/

*m*is the wave impedance of the ambient medium,

_{medium}*η*=

^{j}*η*/

*m*is the wave impedance in metal sphere

^{j}*j*, and

**N**

^{(1)},

**M**

^{(1)}indicate vector spherical harmonics centered on the origin of sphere

*j*. We assume here that the vector harmonics are defined at each sphere with respect to a common set of direction axes

*x*,

*y*and

*z*. The complete scattered electric field is given by a sum over all

*N*spheres,

_{s}*j*with respect to the system origin. The prefactor

*E*=|

_{mn}*E*

_{0}|

*i*(2

^{n}*n*+1)(

*n*-

*m*)!/(

*n*+

*m*)!, where |

*E*

_{0}| is the magnitude of the incident wave, is introduced as a normalization constant to maintain numerical stability [9

**34**, 4573–4588 (1995). [CrossRef] [PubMed]

*m*=±1. In the above, the order

*m*and the degree

*n*are integers such that -

*n*≤

*m*≤

*n*, and describes the multipole (i.e. dipole, quadrupole, etc.) nature of the electromagnetic field dressing each sphere. In numerical work, the harmonic degree is truncated to a finite number 1≤

*n*≤

*N*.

**34**, 4573–4588 (1995). [CrossRef] [PubMed]

10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A **433**, 599–614 (1991). [CrossRef]

*α*and

^{j}_{n}*β*are the single sphere Lorenz-Mie coefficients and

^{j}_{n}**34**, 4573–4588 (1995). [CrossRef] [PubMed]

10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A **433**, 599–614 (1991). [CrossRef]

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. **127**, 285–298 (1996). [CrossRef]

**R**

^{jl}=

**R**

^{l}-

**R**

^{j}.

**T**represents an interaction operator for the vector harmonic coefficients associated with each sphere,

**T**

^{ji}being composed of elements of the form

**a**=(

**1-T**+

**T**

^{2}-

**T**

^{3}+…)

**p**. The strong interaction between metal nanospheres in close proximity, illuminated close to plasmon-polariton resonance, prohibits the use of such an expansion. In our work, the linear system of Eq. (7) was solved using a biconjugate gradient method with commercially available software.

### 2.2 Numerical evaluation of coefficients

*j*depend on the normalized radius

*x*and relative refractive index

^{j}*m*as,

^{j}*ψ*(

_{n}*z*)=

*zj*(

_{n}*z*) and

*ξ*(

_{n}*z*)=

*zh*

^{(1)}

_{n}(

*z*), with

*j*and

_{n}*h*the spherical Bessel and spherical Hankel functions of first kind.

^{(1)}_{n}**34**, 4573–4588 (1995). [CrossRef] [PubMed]

10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A **433**, 599–614 (1991). [CrossRef]

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. **127**, 285–298 (1996). [CrossRef]

21. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A **11**, 2851–2861 (1994). [CrossRef]

**N**

^{(3)}

*(j)*,

**M**

^{(3)}

*(j)*about an origin at

**R**

^{j}into harmonics

**N**

^{(1)}

*(l)*,

**M**

^{(1)}

*(l)*about the origin

**R**

^{l},

*α*=acos(

*X*/|

^{jl}**R**

^{jl}|)/sin

*β*,

*β*=acos(

*Z*/|

^{jl}**R**

^{jl}|) and

*γ*=0 for rotating the angular momentum z-axis into alignment with the inter-origin vector

**R**

^{jl}=

**R**

^{l}-

**R**

^{j}. The rotation matrix elements are,

**R**

^{jl}| were calculated using the following expressions [12

**127**, 285–298 (1996). [CrossRef]

*a*=

_{p}*a*(

*m*′,

*n*,-

*m*′,

*ν*,

*p*),

*p*=

*n*+

*ν*,

*n*+

*ν*-2,… |

*n*-

*ν*|, were calculated using Bruning and Lo’s recursive technique [11

11. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres Part I – Multipole Expansion and Ray-Optical Solutions,” IEEE Tran. Antennas Propag.AP-19, 378–390 (1971). [CrossRef]

*q*-1)!!=(2

*q*-1)(2

*q*-3)…3·1;(-1)!!≡1. The simplicity of our chosen strategy arises from the fact that axial translations are diagonal in harmonic order

*m*=

*µ*, while the rotations are diagonal in harmonic degree

*η*=

*ν*.

**E**

_{sca}+

**E**

_{inc}|/|

**E**

_{inc}|, can be evaluated. The time average Poynting vector is also directly evaluated <

*S*>=1/2Re{

**E**.

**H***}. Finally, the normalized far-field scattering, extinction and absorption cross-sections (also known as the efficiencies) were calculated [9

**34**, 4573–4588 (1995). [CrossRef] [PubMed]

**433**, 599–614 (1991). [CrossRef]

**R**

^{l}) that is chosen as an origin,

*h*replaced by spherical Bessel functions

_{n}^{(1)}*j*in Eqs. (17) and (18), as appropriate to translation of outward propagating vector harmonics

_{n}**N**

^{(3)},

**M**

^{(3)}[12

**127**, 285–298 (1996). [CrossRef]

## 3. Results and analysis

### 3.1 Geometries setup and convergence

*m*=1.3342). The dielectric properties of the silver and gold nanoparticles were assumed to be that of bulk material; measured bulk dielectric values reported by Johnson and Christy [23

_{medium}23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*z*-direction and polarized along the

*x*-direction. The structures are always oriented in such a way that the maximum charge polarization, along

*x*, is obtained. The seed structure in the center of our systems, where the “hot-spot” is located, consists of a bisphere whose spheres have 5nm radii and a separation of 0.5nm – close to the limit of what DNA based self-assembly might achieve. Field enhancement is taken as the ratio of the local electric field at the “hot-spot”, the center of the seed bisphere, to the electric field strength in the illuminating field.

*n*≤

*N*. In our work, to achieve convergence in near field enhancement at the level of <5%, we have used

*N*=10. This corresponds to a total of 2

*N*(

*N*+1)=220 vector harmonic expansion components

*per sphere*. An example of the convergence in field enhancement is given in Fig. 2 below. The contribution of high degree multipoles is obviously important for the accurate prediction of resonant wavelengths at which field enhancement is maximum, in agreement with Khlebtsov,

*et al.*, findings that the dipole approximation is inadequate. The physical explanation is simple, electric fields confined to a characteristic size

*d*are decomposed into harmonics extending to up to degree

*N*α

*a*/

*d*, where

*a*is particle radius.

### 3.2 Field enhancement in linear chains

*N*is increased from 2 to 20, the resonant wavelength is red-shifted and the quality factor of the field enhancement resonance increases, as shown in Figs. 4 and 5.

_{s}24. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. **98**, 266802 (2007). [CrossRef] [PubMed]

*Q*=

*d*(

*ωε*’)/

*dω*/2

*ε*” where ε’, ε” are real and imaginary parts of the dielectric constant [25

25. F. Wang and Y. Ron Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. **97**, 206806 (2006). [CrossRef] [PubMed]

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

### 3.3 Field enhancement in self-similar structures with constant number of spheres

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

*a*(taken to be 5nm) and interparticle gap

^{0}*d*(taken to be 0.5nm). Successive spheres are added to the structure in geometric series with radii

^{0}*a*and nearest neighbour gap

^{j}*d*enlarged by a factor 1/κ, such that

^{j}*a*

^{j+1}=

*a*/

^{j}*κ*and

*d*

^{j+1}=

*d*/

^{j}*κ*. Note that in the limit where κ=1, the linear chain is recovered. We compared the response of arrays with a fixed number of spheres

*N*=4 and 6 and varying geometric factor

_{s}*κ*and length

*L*, illustrated to scale in Fig. 6 and 7. Since the complexity of self-assembly processes increases with number of particles per discrete unit, optimizing field enhancement for fixed particle number

*N*is important.

_{s}*Q*of silver compared to gold is apparent in the widths of field enhancement resonances, and also in the clear presence of higher order resonances for silver structures but not gold. Interband absorption in gold above 2eV suppresses field enhancement at

*λ*<600nm. The improved efficiency of the geometric series of spheres over the linear chain is due to the increased metallic volume that expels a greater amount of optical energy, and the reduced number of gaps into which optical energy is focused. Importantly, these simulated field enhancement plots reveal that the geometric scale factor

*κ*(which implicitly defines structure length for a fixed number of spheres

*N*) that maximizes field enhancement is a non-trivial function of both sphere material and number of spheres. Full electromagnetic simulation without resorting to the quasi-static approximation is required to accurately model the resonances of the plasma wave excitation along a self-similar structure.

_{s}### 3.4 Field enhancement in self-similar structures of fixed length

*κ*and sphere number

*N*, illustrated in Figs. 12, were also performed. The optimization of structures of fixed linear extent is important for

_{s}*in-vivo*applications [26

26. I. H. El-Sayed, X. Huang, and M. A. El-Sayed, “Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles,” Cancer Lett. **239**, 129–135 (2006). [CrossRef]

*κ*→1). The red-shift can be interpreted as arising from the increase in hybridization of the plasma resonances associated with individual spheres, and hence a reduction in frequency of the lowest order plasma oscillation distributed over the structure [27

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

*κ*<1. The precise value of

*κ*(which implicitly defines number of spheres

*N*for a fixed length structure) that maximizes field enhancement depends upon both sphere material and physical length of the structure, but the general trend is improved efficiency with smaller

_{s}*κ*(fewer spheres

*N*).

_{s}### 3.5 Near-field distribution in self-similar structures

**E**

_{sca}+

**E**

_{inc}|/|

**E**

_{inc}| in the vicinity of the self-similar structures is useful in explaining the role of structure in near-field enhancement and the importance of vector multipole calculation. Field enhancements in the x-z and x-y planes (the x-z plane is the plane of incidence) for self-similar gold structures of 167.5nm length are presented on a logarithmic scale in Figs. 17, 18. The internal field of each sphere is not shown. Each field enhancement plot is taken with incident light at the wavelength that maximizes field enhancement at the focal spot of the structure, summarized in Table 1. These plots reveal the “hot-spots” associated with each structure, with those structures that approach the linear chain having a preponderance of “hot-spots”. Structures with small

*κ*are dominated by the large volume spheres, which serve to enhance field in a reduced number of “hot-spots”. The higher dielectric quality factor

*Q*of silver versus gold is evident in the increased field enhancement about the silver structures.

*x*≪2π

*m*/

_{medium}a*λ*, is better satisfied, by smaller spheres (in the linear chains), field retardation is reduced and the distinction between incident side near-field and shadow side near-field is diminished.

*S*>=1/2Re{

**E**×

**H***}. The Poynting vector magnitude is plotted in Fig. 19 for self similar gold structures of 167.5 nm length. Energy is clearly drawn into the structures on the incident side, casting shadow regions for spheres as small as 5nm in radius. It should be noted that although isolated spheres may be well modeled under the quasi-static approximation, the complete structure need not be well modeled using a quasi-static approximation. Shadows in the Poynting vector are indeed visible for each structure in Fig. 19.

*N*=8,

_{s}*κ*=0.672 selfsimilar structure is plotted in Fig. 20 for the three lowest order resonances in near-field enhancement (taken from Fig. 16). The three near-field resonances appear to be distinguished by different hybridizations of dipolar plasma excitations on the individual spheres. The geometric series structure does not lend itself to a simple quantitative description of resonances expected for a linear chain.

### 3.6 Far-field properties of self-similar structures

29. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B **24**, 649–657 (1981). [CrossRef]

_{s}-1 gaps where field enhancement is comparable to that at the central focus (Figs. 17, 18), resulting in significantly increased absorption at the near-field enhancement resonance in comparison to structures with

*κ*<1. Strong interband absorption at wavelengths

*λ*< 600nm is evident for gold particle assemblies with small

*κ*, where absorption is expected to be dominated by the largest spheres.

*κ*→1 closely follow the resonances in field enhancement and absorption efficiency. However, in the limit of small geometric factor

*κ*, the scattering efficiency peaks acquire a significant red-shift with respect to the near-field enhancement. The physical origin of this red-shift is unknown, but has been observed in numerical studies of single sphere scattering [29

29. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B **24**, 649–657 (1981). [CrossRef]

29. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B **24**, 649–657 (1981). [CrossRef]

*κ*(and hence larger spheres in the structure). Elastic light scattering data, such as that collected with dark-field microscopy, will therefore need to be carefully interpreted when inferring near-field enhancement properties of self-similar structures with geometric factor

*κ*≪1.

### 3.7 Field enhancement in other geometries

## 4. Conclusions

7. B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, “Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles,” Nano Lett. **5**, 2246–2252 (2005). [CrossRef] [PubMed]

8. F. Aldaye and H. F. Sleiman, “Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching,” J. Am. Chem. Soc. **129**, 4130–4131 (2007). [CrossRef] [PubMed]

*κ*.

## Acknowledgment

## References and links

1. | L. Novotny and B. Hecht, |

2. | K. Kneipp, M. Moskovits, and H. Kneipp, |

3. | L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B |

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

5. | M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. |

6. | J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, “Controlling the surface enhanced Raman effect via the nanoshell geometry,” Appl. Phys. Lett. |

7. | B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, “Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles,” Nano Lett. |

8. | F. Aldaye and H. F. Sleiman, “Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching,” J. Am. Chem. Soc. |

9. | Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. |

10. | D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A |

11. | J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres Part I – Multipole Expansion and Ray-Optical Solutions,” IEEE Tran. Antennas Propag.AP-19, 378–390 (1971). [CrossRef] |

12. | Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. |

13. | F. J. Garcia de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B |

14. | G. Pellegrini, G. Mattei, V. Bello, and P. Mazzoldi, “Interacting metal nanoparticles: Optical properties from nanoparticle dimers to core-satellite systems,” Mat. Sci. Eng. C |

15. | H. Xu, “Calculation of the near field of aggregates of arbitrary spheres,” J. Opt. Soc. Am. A |

16. | R.-L. Chern, X.-X. Liu, and C.-C. Chang, “Particle plasmons of metal nanospheres: Application of multiple scattering approach,” Phys. Rev. E |

17. | B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology |

18. | H.C. van de Hulst, |

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

20. | G. Arfken and H. J. Weber, |

21. | D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A |

22. | A. R. Edmond, |

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

24. | L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. |

25. | F. Wang and Y. Ron Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. |

26. | I. H. El-Sayed, X. Huang, and M. A. El-Sayed, “Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles,” Cancer Lett. |

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

28. | C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. |

29. | B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B |

**OCIS Codes**

(240.5420) Optics at surfaces : Polaritons

(240.6680) Optics at surfaces : Surface plasmons

(290.4020) Scattering : Mie theory

(160.4236) Materials : Nanomaterials

(250.5403) Optoelectronics : Plasmonics

(240.6695) Optics at surfaces : Surface-enhanced Raman scattering

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: December 12, 2007

Revised Manuscript: January 23, 2008

Manuscript Accepted: January 24, 2008

Published: January 25, 2008

**Virtual Issues**

Vol. 3, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Wei-Yin Chien and Thomas Szkopek, "Multiple-multipole simulation of optical nearfields in discrete metal nanosphere assemblies," Opt. Express **16**, 1820-1835 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1820

Sort: Year | Journal | Reset

### References

- L. Novotny and B. Hecht, Principles of Nano-Optics (University Press, Cambridge, 2006).
- K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications (Springer, Berlin, 2006). [CrossRef]
- L. A. Sweatlock, S. A. Maier, and H. A. Atwater, "Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles," Phys. Rev. B 71, 235408 (2005). [CrossRef]
- K. Li. and M. I. Stockman, and D. J. Bergman, "Self-similar chain of metal nanospheres as an efficient nanolens," Phys. Rev. Lett. 91, 227402 (2003). [CrossRef] [PubMed]
- M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004). [CrossRef] [PubMed]
- J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, "Controlling the surface enhanced Raman effect via the nanoshell geometry," Appl. Phys. Lett. 82, 257-259 (2003). [CrossRef]
- B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005). [CrossRef] [PubMed]
- F. Aldaye and H. F. Sleiman, "Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching," J. Am. Chem. Soc. 129, 4130-4131 (2007). [CrossRef] [PubMed]
- Y. Xu, "Electromagnetic scattering by an aggregate of spheres," Appl. Opt. 34, 4573-4588 (1995). [CrossRef] [PubMed]
- D. W. Mackowski, "Analysis of radiative scattering for multiple sphere configurations," Proc. R. Soc. London Ser. A 433, 599-614 (1991). [CrossRef]
- J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part I - Multipole Expansion and Ray-Optical Solutions," IEEE Tran.Antennas Propag. AP-19, 378-390 (1971). [CrossRef]
- Y. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996). [CrossRef]
- F. J. Garcia de Abajo, "Multiple scattering of radiation in clusters of dielectrics," Phys. Rev. B 60, 6086-6102 (1999). [CrossRef]
- G. Pellegrini, G. Mattei, V. Bello, and P. Mazzoldi, "Interacting metal nanoparticles: Optical properties from nanoparticle dimers to core-satellite systems," Mat. Sci. Eng. C 27, 1347-1350 (2007). [CrossRef]
- H. Xu, "Calculation of the near field of aggregates of arbitrary spheres," J. Opt. Soc. Am. A 21, 804-809 (2004). [CrossRef]
- R.-L. Chern, X.-X. Liu and C.-C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, 016609 (2007). [CrossRef]
- B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches," Nanotechnology 17, 1437-1445 (2006). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
- G. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th. ed., (Academic, Orlando, 2005).
- D. W. Mackowski, "Calculation of total cross sections of multiple-sphere clusters," J. Opt. Soc. Am. A 11, 2851-2861 (1994). [CrossRef]
- A. R. Edmond, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]
- F. Wang and Y. Ron Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806 (2006). [CrossRef] [PubMed]
- I. H. El-Sayed, X. Huang and M. A. El-Sayed, "Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles," Cancer Lett. 239, 129-135 (2006). [CrossRef]
- E. Prodan, C. Radloff, N. J. Halas and P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419 - 422 (2003). [CrossRef] [PubMed]
- C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002). [CrossRef] [PubMed]
- B. J. Messinger, K. U. von Raben, R. K. Chang and P. W. Barber, "Local fields at the surface of noble-metal microspheres," Phys. Rev. B 24, 649-657 (1981). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

« Previous Article | Next Article »

OSA is a member of CrossRef.