## Light scattering from 2D arrays of monodispersed Ag-nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling

Optics Express, Vol. 16, Issue 20, pp. 15312-15324 (2008)

http://dx.doi.org/10.1364/OE.16.015312

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### Abstract

Two dimensional arrays of monodispersed Ag-nanoparticles separated by different gaps with sub-10 nm precision are fabricated on anodic alumina substrates with self-organized pores. Light scattering spectra from the arrays evolve with the gaps, revealing plasmonic coupling among the nanoparticles, which can be satisfactorily interpreted by analytical formulae derived from generic dipolar approximation. The general formulism lays down a foundation for predicting the *Q* factor of an array of metallic nano-particles and its geometric characteristics.

© 2008 Optical Society of America

## 1. Introduction

1. M. Faraday, “On the color of colloidal gold,” Phil. Trans. R. Soc. London **147**, 145–181 (1857). [CrossRef]

2. G. Mie, “Beitrage zur optik truber medien speziel kolloidaler metallosungen,” Ann. Phys. **25**, 377–445 (1908). [CrossRef]

3. M. Geissler and Y. Xia, “Patterning: Principles and some new developments,” Adv. Mater. **16**, 1249–1269 (2004). [CrossRef]

4. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

8. H. H. Wang, *et al*., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. **18**, 491–495 (2006). [CrossRef]

9. S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science **275**, 1102–1106 (1997). [CrossRef] [PubMed]

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. **3**, 1087–1090 (2003). [CrossRef]

12. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. **7**, 2080–2088 (2007). [CrossRef]

14. W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. **220**, 137–141 (2003). [CrossRef]

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. **3**, 1087–1090 (2003). [CrossRef]

15. J. P. Kotmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express **8**, 655–663 (2001). [CrossRef]

16. X. -D. Xiang, *et al*., “A Combinatorial Approach to Materials Discovery,” Science **268**, 1738–1740 (1995). [CrossRef] [PubMed]

18. V. A. Markel, “Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure,” J. Mod. Opt. **40**, 2281–2291 (1993). [CrossRef]

8. H. H. Wang, *et al*., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. **18**, 491–495 (2006). [CrossRef]

## 2. Sample preparation and measurements

8. H. H. Wang, *et al*., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. **18**, 491–495 (2006). [CrossRef]

*r*, was adjusted by varying the anodization voltage. A fixed diameter of 25 nm was prepared by subsequent etching of the channel walls after the anodization process. Finally, Ag nanorods were grown inside the channels by AC electro-deposition. In the spectroscopy study, the Ag/AAO sample was illuminated in dark-field mode by unpolarized white light from a halogen lamp in an inverted microscope. In dark-field configuration, the sample was illuminated obliquely and the scattered light was collected along the normal to the sample surface. It was then directed to a monochrometer plus charge-coupled device for spectral detection. The smallest mean interparticle spacing,

*r*̄, achieved is 30 nm, yielding a mean gap of 5 nm [19]. The corresponding histogram of the interparticle spacing, as given in the inset of Fig. 1, shows a full-width at half-maximum, Δ

*r*, of 3 nm. Figure 1 displays Δ

*r*as a function of

*r*̄ for all the samples prepared for the spectroscopic measurement. A typical scattering spectrum exhibits two peaks (not shown), corresponding to the transverse and longitudinal modes [20] of the plasmonic resonance which can be assigned by polarization dependency of these two modes. The rod length of all the samples used in this study was adjusted to be approximately 100 nm such that only the transverse mode is visible in the spectral range of interest.

## 3. Theory

*(*

**E**_{loc}*) for the particle located at*

**r**_{i}*is the sum of the incident field*

**r**_{i}*(*

**E**_{inc}*) and the fields induced by the rest of particles. The effective polarizability of the arrays can be derived accordingly:*

**r**_{i}*α*is the polarizability of a single prolate spheroid of the transverse mode in an electrostatic field,

_{T}*k*=2

*π*/

*λ*,

*θ*

*is the angle between*

_{ij}*and*

**r**_{ij}*(*

**E**_{inc}*), and*

**r**_{i}*=*

**r**_{ij}*-*

**r**_{i}*. Considering long-wavelength approximation, Eq. (2) becomes*

**r**_{j}*r*is the center-to-center distance between two adjacent particles and

*F*is the lattice sum of the hexagonal particle array.

*, the corresponding width, Γ, and the peak scattering intensity at Ω*

_{T}*,*

_{T}*I*(Ω

*), are expressed as:*

_{T}*ε*is the dielectric constant of the surrounding medium,

_{m}*S*

^{3}=

*R*

^{2}

*h*/3

*L*,

*C*=(1-

*L*)/

*L*,

*R*and 2

*h*are the radius and the length of the spheroid, respectively,

*L*is the depolarization factor, and

*N*(

*r*) is the surface particle density. In the derivation, Drude model is assumed for the dielectric function of Ag with plasma frequency,

*ω*, and relaxation time,

_{p}*τ*, as parameters. The detailed derivation of Eqs. (4)-(6) is presented in Ref. 19. In the case of arrays of spherical particles in vacuum, the resultant resonance frequency agrees with the derived result by Persson and Liebsch [17] and exhibits a dependence on

*R*/

*r*which was also obtained by many research groups [10

10. K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. **3**, 1087–1090 (2003). [CrossRef]

12. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. **7**, 2080–2088 (2007). [CrossRef]

**3**, 1087–1090 (2003). [CrossRef]

12. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. **7**, 2080–2088 (2007). [CrossRef]

*U*and therefore extra contributions to the imaginary part of

*α*, as shown in Eq. (1). This additional quadrature component in

_{eff}*α*causes effective absorption that reflects the energy flow from a specific dipole to its surrounding dipoles and therefore gives rise to extra broadening in Γ. For narrower interparticle spacings, Ω

_{eff}*exhibits red shifting and*

_{T}*B*becomes larger, thus leading to an increase in Γ. Finally, Eq. (6) presents the fact that the peak intensity is proportional to the square of the

*Q*factor of the system, which is equal to Ω

*/Γ and directly reflects the field strength at the resonance frequency in this plasmonic system [21].*

_{T}## 4. Results and discussion

*r*

^{3}

*and 1/*

_{ij}*r*terms represent the near- and far-zone contributions, respectively, while the 1/

_{ij}*r*

^{2}

*term dominates in the intermediate zone and has a 90° phase shift with respect to the other terms. The far-zone contribution becomes prominent only when*

_{ij}*r*is comparable to the wavelength and therefore is insignificant in the discussion of the coupling induced peak shifting and broadening. Because the near-zone term is much larger than the intermediate-zone term under the long-wave approximation, it dominates the real part of

_{ij}*U*in the spatial range of this study, and acts as the single source in the coupling-induced peak shifting. In contrast, both the near- and intermediate-zone terms contribute to the spectral width broadening [18

18. V. A. Markel, “Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure,” J. Mod. Opt. **40**, 2281–2291 (1993). [CrossRef]

**3**, 1087–1090 (2003). [CrossRef]

**7**, 2080–2088 (2007). [CrossRef]

*r*̄=30, 35, 40, 45 and 50 nm are shown in Fig. 2. Clearly, the transverse mode is red-shifted and broadened as

*r*̄ decreases. This is consistent with previous experimental findings [10

**3**, 1087–1090 (2003). [CrossRef]

**7**, 2080–2088 (2007). [CrossRef]

*and width Γ*

_{T}*. The dependence of Ω*

_{V}*on*

_{T}*r*̄ is shown in Fig. 3 and was fitted with Eq. (4). The

*ω*obtained from the fitting of Ω

_{p}*, as shown in the third column of Table 1, agrees well with that extracted from fitting the optical constant of Ag to the Drude model [24] (in the second column of Table 1).*

_{T}*, ΔΩ*

_{T}*, on the variation in*

_{T}*r*, Δ

*r*, can be derived [19]. This formula was used to calculate the inhomogeneous contribution, which has a Gaussian distribution, to the observed spectral width based on the result in Fig. 1 and the extracted parameters from the spectral peak-shift analysis. On the basis of the relationship between the extracted Voigt width, Γ

*, and*

_{V}*r*̄ [19], the Lorentzian width, Γ

*, was then obtained according to the empirical relation among the Voigt, Gaussian and Lorentzian width [25*

_{L}25. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt linewidth: A brief review” J. Quant. Spectrosc. Radiat. Transfer **17**, 233–236 (1977). [CrossRef]

*r*̄ is displayed in Fig. 3. Γ

*was then fitted with Eq. (5) to obtain the parameters listed in the fourth column of Table 1. Similar to the spectral peak-shifting analysis, both*

_{L}*ω*and

_{p}*τ*obtained from the fitting are in good agreement with that extracted from fitting the optical constant of Ag to the Drude model.

*per nanorod,*

_{T}*I**(Ω

*), as a function of*

_{T}*r*̄, as well as the fitting of Eq. (6) to the data. Again, both

*ω*and

_{p}*τ*obtained from the fitting, as listed in the fifth column of Table 1, are in agreement with the ones derived from the spectral peak shifting and line-width broadening data. As shown in Eq. (6),

*I**(Ω

*) is proportional to (Ω*

_{T}*/Γ)*

_{T}^{2}and therefore to

*Q*

^{2}. Because the transverse mode exhibits red shifting and its line width shows broadening as

*r*̄ decreases,

*I**(Ω

*) is expected to decreases monotonically with the decrease in*

_{T}*r*̄, which is consistent with the experimental data shown in Fig. 4. It indicates that the electric field energy stored in the system dissipates faster for smaller interparticle spacings. This

*r*̄-dependent energy dissipation behavior can be attributed to the fact that the coupling between dipoles facilitates the energy flow between them, increasing energy loss channels due to electron-phonon scattering, inherent in

*τ*. In addition, as pointed out previously, the

*Q*factor reflects the effective electric field enhancement factor of the system at the resonance frequency. For the applications of using these nanorod arrays in surface-enhanced Raman scattering (SERS) [8

*et al*., “Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps,” Adv. Mater. **18**, 491–495 (2006). [CrossRef]

*Q*on

*r*̄ provides direct indication how the electromagnetic enhancement factor varies with the interparticle spacing. Since the resonance frequency is varied with the interparticle spacing simultaneously, the actual enhancement factor also depends on the laser wavelength used in Raman scattering. The comprehensive model developed here can thus provide a design tool for SERS-substrates based on plasmonic coupling in nanoparticle arrays.

*term) to account for energy conservation [26*

*k*^{3}26. B. T. Draine, “The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains,” Astrophys. J. **333**, 848–872 (1988). [CrossRef]

*k*

^{2}correction term was considered by many groups [27

27. M. J. Collinge and B. T. Draine, “Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry,” J. Opt. Soc. Am. **A 21**, 2023–2028 (2004). [CrossRef]

*k*term) indicates that the decrease of the

^{3}*Q*factor with the decrease in

*r*is caused mainly by the ohmic loss due to the electron-phonon scattering instead of the radiative loss. Another issue of concern is, for a gap as small as 5 nm (

*r*̄=30 nm), whether the dipole approximation is still valid and therefore other multipolar effects should be included in the analysis. In a recent study of two coupled silver nanospheres, Khlebtsov and coworkers compared the difference in the spectral peak shifting between the calculated results with an electrodynamic dipole approximation as well as a more complete electrodynamic multipole solution [29]. Their results indicate that large deviation in the spectral peak shifting only occurs for

*r*̄/

*R*< 2.5, which is consistent with our observation. The simple analytical formulae, Eqs. (4)-(6), are therefore valid in the variation range of the interparticle spacing used in this study.

## 5. Conclusions

## Appendix

## B. Derivation of the formula

*R*and 2

*h*are the radius and the length of the spheroid.

*e*

^{2}=1-(

*R*/

*h*)

^{2}.

*ε*is the dielectric function of the surrounding medium and

_{m}*ω*and

_{p}*τ*are the plasma frequency and relaxation time, respectively. For silver in the visible wavelength range (400 to 800 nm),

*ω*~10

_{p}^{16}rad/sec and

*τ*~10

^{-14}sec [30]. Since

*ω*≫1/

*τ*, Eq. (1) can be approximated as

*S*

^{3}=

*R*

^{2}

*h*/3

*L*and

*C*=(1-

*L*)/

*L*. It indicates that each particle behaves as a harmonic oscillator (Lorentz model) with a resonance frequency

*τ*. In the case of arrays of particles, the local field

*(*

**E**_{loc}*) on each particle is the sum of the incident field*

**r**_{i}*(*

**E**_{inc}*) and the fields induced by the rest of particles and can be written as [31]*

**r**_{i}*is the difference between the position vectors of*

**r**_{ij}*i*-th and

*j*-th particles. Assuming that the local field at each particle is the same,

*i.e*.

*(*

**E**_{loc}*)=*

**r**_{i}*(*

**E**_{loc}*), and considering*

**r**_{j}*=*

**P**_{j}*α*(

_{T}**E**_{loc}*), the effective polarizability of the arrays can be derived accordingly:*

**r**_{j}*θ*is the angle between

_{ij}*and*

**r**_{ij}*(*

**E**_{inc}*). Notice that*

**r**_{i}*U*depends on the interparticle spacing and the packing configuration of the array. In this study, the Ag nanorod array has a hexagonal packing pattern. With the substitution of Eq. (3), Eq. (5) becomes

*A*and

*B*are defined as

*U*=

*A*+

*iB*. The scattering intensity is then the sum of the scattered radiation from each particle with this effective polarizability. This equation exhibits a quasi-Lorentzian form. The resonance peak in a scattering spectrum emerges as the real part of the denominator vanishes and, therefore, the full-width at half-maximum can be solved accordingly. It also reveals that

*A*is responsible for plasmon resonance peak shift, while

*B*plays a role in the linewidth broadening and is also a function of the interparticle spacing of the array,

*r*. Considering long-wavelength approximation,

*i.e*. exp(

*ikr*)≈1, Eq. (6) becomes

*F*is the lattice sum of the hexagonal particle array. On the other hand, under the condition of neglecting the far-field term,

*U*for the wavelength of 550 nm is smaller than 0.5% for the interparticle spacing range considered in this study. The leading term of

*B*with a dependence on the interparticle spacing is proportional to

*k*

^{5}

*r*

^{2}

*which is zero under the long-wavelength approximation. This approximation is therefore not applied the subsequent derivation of*

_{ij}*B*. From Eq. (7), we can derive the resonance frequency, Ω

*, the corresponding linewidth, Γ, and the scattering intensity at Ω*

_{T}*,*

_{T}*I*(Ω

*). According to Eq. (7), Ω*

_{T}*T*, corresponding to the maximum value of

*α*, can be extracted by solving [(1-

_{eff}*S*

^{3}

*A*)

*ω*

^{2}

*-{(1+*

_{p}*Cε*)+(

_{m}*ε*-1)

_{m}*S*

^{3}

*A*}Ω

^{2}

*]=0 and is given by*

_{T}*α*|

_{eff}^{2}and is given by

*I*(Ω

*) can be expressed as*

_{T}*N*(

*r*) is the surface dipole density. To illustrate the physical contents of the spectral peak and width, we can consider the arrays of spherical particles in air. In this case,

*ε*=1,

_{m}*S*=

*R*and

*C*=2. Equations (9)–(11) then become

## C. Voigt linewidth

*, ΔΩ*

_{T}*, on the variation in*

_{T}*r*, Δ

*r*, can be derived as

*, and*

_{V}*r*̄, is shown in the figure below.

## D. Fitting analysis

*ω*, is just 30% away from the one obtained by fitting the optical constant [24] of Ag to the Drude model within the wavelength range of interest. In the fitting,

_{p}*ε*was chosen to be the dielectric constant of alumina [32

_{m}32. D. W. Thompson, “Optical characterization of porous alumina from vacuum ultraviolet to midinfrared,” J. Appl. Phys. **97**, 113511 (2005). [CrossRef]

*C*shows 5% variation for

*h*varied from 25 to 100 nm,

*C*was fixed at 1.2 based on

*R*=12.5 nm and

*h*=50 nm. The fitted results of the spectral broadening with Eq. (10)

*vs*. interparticle spacing are listed in Table 1 in the main text. The extracted relaxation time,

*τ*, is about 15% higher and

*ω*is only 10% less than the one obtained by fitting the optical constants of Ag to the Drude model. Finally, the fitted results of the spectral peak intensity with Eq. (11)

_{p}*vs*. interparticle spacing yield

*ω*and

_{p}*τ*which are listed in Table 1 in the main text. Both

*ω*and

_{p}*τ*match closely with the values obtained from the fitting of the optical constants of Ag with Drude model. In summary, these fitted values comparably agree among themselves and with the values extracted from the dielectric constants of Ag, supporting the fitting process and Eqs. (9)-(11) used in the fitting.

## Acknowledgment

## References and links

1. | M. Faraday, “On the color of colloidal gold,” Phil. Trans. R. Soc. London |

2. | G. Mie, “Beitrage zur optik truber medien speziel kolloidaler metallosungen,” Ann. Phys. |

3. | M. Geissler and Y. Xia, “Patterning: Principles and some new developments,” Adv. Mater. |

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

5. | H. Raether, |

6. | M. Moskovits, “Surface enhanced spectroscopy,” Rev. Mod. Phys. |

7. | J. C. Hulteen and R. P. Van Duyne, “Nanosphere lithography: A materials general fabrication process for periodic particle array surfaces,” J. Vac. Sci. Technol. |

8. | H. H. Wang, |

9. | S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering,” Science |

10. | K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, “Interparticle coupling effects on plasmon resonances of nanogold particles,” Nano Lett. |

11. | L. Gunnarsson, T. Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, “Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions,” J. Phys. Chem. |

12. | P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation,” Nano Lett. |

13. | C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, “Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays,” J. Phys. Chem. |

14. | W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. |

15. | J. P. Kotmann and O. J. F. Martin, “Plasmon resonant coupling in metallic nanowires,” Opt. Express |

16. | X. -D. Xiang, |

17. | B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. |

18. | V. A. Markel, “Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure,” J. Mod. Opt. |

19. | see Appendix. |

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

21. | J. D. Jackson, |

22. | J. D. Jackson, |

23. | Y. Liu, J. Lin, G. Huang, Y. Guo, and C. Duan, “Simple empirical analytical approximation to the Voigt profile,” J. Opt. Soc. Am. |

24. | E. D. Palik, |

25. | J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt linewidth: A brief review” J. Quant. Spectrosc. Radiat. Transfer |

26. | B. T. Draine, “The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains,” Astrophys. J. |

27. | M. J. Collinge and B. T. Draine, “Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry,” J. Opt. Soc. Am. |

28. | S. Zou and G. C. Schatz, Response to comment on “Silver nanoparticle array structures that produce remarkable narrow plasmon line shapes,” J. Chem. Phys. |

29. | B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches,” Nanotech. |

30. | C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, “Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles,” J. Phys. Chem. |

31. | J. D. Jackson, |

32. | D. W. Thompson, “Optical characterization of porous alumina from vacuum ultraviolet to midinfrared,” J. Appl. Phys. |

**OCIS Codes**

(290.5850) Scattering : Scattering, particles

(160.4236) Materials : Nanomaterials

**ToC Category:**

Scattering

**History**

Original Manuscript: March 31, 2008

Revised Manuscript: April 28, 2008

Manuscript Accepted: May 15, 2008

Published: September 15, 2008

**Citation**

Sajal Biring, Huai-Hsien Wang, Juen-Kai Wang, and Yuh-Lin Wang, "Light scattering from 2D arrays of monodispersed Ag-nanoparticles separated by tunable nano-gaps: spectral evolution and analytical analysis of plasmonic coupling," Opt. Express **16**, 15312-15324 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15312

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### References

- M. Faraday, "On the color of colloidal gold," Phil. Trans. R. Soc. London 147, 145-181 (1857). [CrossRef]
- G. Mie, "Beitrage zur optik truber medien speziel kolloidaler metallosungen," Ann. Phys. 25, 377-445 (1908). [CrossRef]
- M. Geissler and Y. Xia, "Patterning: Principles and some new developments," Adv. Mater. 16, 1249-1269 (2004). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424,824-830 (2003). [CrossRef] [PubMed]
- H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer-Verlag, Berlin, 1980), pp. 116.
- M. Moskovits, "Surface enhanced spectroscopy," Rev. Mod. Phys. 57, 783-826 (1985). [CrossRef]
- J. C. Hulteen and R. P. Van Duyne, "Nanosphere lithography: A materials general fabrication process for periodic particle array surfaces," J. Vac. Sci. Technol. A 13, 1553-1558 (1995).
- H. H. Wang, et al., "Highly Raman Enhancing-Substrates Based on Silver nanoparticle Arrays with Tunable Sub-10 nm Gaps," Adv. Mater. 18, 491-495 (2006). [CrossRef]
- S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102-1106 (1997). [CrossRef] [PubMed]
- K. H. Su, Q. H. Wei, X. Zhang, J. J. Mock, D. R. Smith, and S. Schultz, "Interparticle coupling effects on plasmon resonances of nanogold particles," Nano Lett. 3, 1087-1090 (2003). [CrossRef]
- L. Gunnarsson, T. Rindzevicious, J. Prikulis, B. Kasemo, M. Käll, S. Zou, and G. C. Schatz, "Confined Plasmons in Nanofabricated Single Silver Particle Pairs: Experimental Observations of Strong Interparticle Interactions," J. Phys. Chem. B 109, 1079-1087 (2005).
- P. K. Jain, W. Huang, and M. A. El-Sayed, "On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: A Plasmon Ruler Equation," Nano Lett. 7, 2080-2088 (2007). [CrossRef]
- C. L. Haynes, A. D. McFarland, L. L. Zao, R. P. Von Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, "Nanoparticle Optics: The Importance of Radiative Dipole Coupling in Two-Dimensional Nanoparticle Arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
- W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Commun. 220, 137-141 (2003). [CrossRef]
- J. P. Kotmann and O. J. F. Martin, "Plasmon resonant coupling in metallic nanowires," Opt. Express 8, 655-663 (2001). [CrossRef]
- X. -D. Xiang, et al., "A Combinatorial Approach to Materials Discovery," Science 268, 1738-1740 (1995). [CrossRef] [PubMed]
- B. N. J. Persson and A. Liebsch, "Optical properties of two-dimensional systems of randomly distributed particles," Phys. Rev. B 28, 4247-4254 (1983).
- V. A. Markel, "Coupled Dipole approach to Scattering of Light from a One-Diemnsional Periodic Dipole Structure," J. Mod. Opt. 40, 2281-2291 (1993). [CrossRef]
- see Appendix.
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 130.
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 371.
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 407.
- Y. Liu, J. Lin, G. Huang, Y. Guo, and C. Duan, "Simple empirical analytical approximation to the Voigt profile," J. Opt. Soc. Am. B 18, 666-672 (2001).
- E. D. Palik, Handbook of Optical Constants of Solid (Academic Press, London, 1985), pp. 353.
- J. J. Olivero and R. L. Longbothum, "Empirical fits to the Voigt linewidth: A brief review" J. Quant. Spectrosc. Radiat. Transfer 17, 233-236 (1977). [CrossRef]
- B. T. Draine, "The Discrete-Dipole Approxiamtion and its Application to Interstellar graphite Grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
- M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004). [CrossRef]
- S. Zou and G. C. Schatz, Response to comment on "Silver nanoparticle array structures that produce remarkable narrow plasmon line shapes,"J. Chem. Phys. 102, 122 (2005).
- B. Khlebtsov, A. Melnikov, B. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimmer of metal nanospheres: Comparison of dipole and multipole approaches," Nanotech. 17, 1437-1445 (2006). [CrossRef]
- C. Voisin, N. D. Fatti, D. Christofilos, and F. Vallee, "Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles," J. Phys. Chem. B, 105, 2264-2280 (2001).
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 410.
- D. W. Thompson, "Optical characterization of porous alumina from vacuum ultraviolet to midinfrared," J. Appl. Phys. 97, 113511 (2005). [CrossRef]

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