## A simple model for the resonance shift of localized plasmons due to dielectric particle adhesion |

Optics Express, Vol. 20, Issue 1, pp. 524-533 (2012)

http://dx.doi.org/10.1364/OE.20.000524

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### Abstract

Ultrasensitive detectors based on localized surface plasmon resonance refractive index sensing are capable of detecting very low numbers of molecules for biochemical analysis. It is well known that the sensitivity of such sensors crucially depends on the spatial distribution of the electromagnetic field around the metal surface. However, the precise connection between local field enhancement and resonance shift is seldom discussed. Using a quasistatic approximation, we developed a model that relates the sensitivity of a nanoplasmonic resonator to the local field in which the analyte is placed. The model, corroborated by finite-difference time-domain simulations, may be used to estimate the magnitude of the shift as a function of the properties of the sensed object – permittivity and volume – and its location on the surface of the resonator. It requires only a computation of the resonant field induced by the metal structure and is therefore suitable for numerical optimization of nanoplasmonic sensors.

© 2011 OSA

## 1. Introduction

1. P. Englebienne, “Use of colloidal gold surface plasmon resonance peak shift to infer affinity constants from the interactions between protein antigens and antibodies specific for single or miltiple epitopes,” Analyst **123**, 1599–1603 (1998). [CrossRef] [PubMed]

7. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. **7**, 442–453 (2008). [CrossRef] [PubMed]

8. J. Homola, *Surface plasmon resonance based sensors*, Springer Series on Chemical Sensors and Biosensors (Springer-Verlag, Berlin-Heidelberg-New York, 2006). [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]

10. H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, “Spectroscopy of single hemoglobin molecules by surface enhanced raman scattering,” Phys. Rev. Lett. **83**, 4357–4360 (1999). [CrossRef]

13. D.-K. Lim, K.-S. Jeon, H. M. Kim, J.-M. Nam, and Y. D. Suh, “Nanogap-engineerable Raman-active nan-odumbbells for single-molecule detection,” Nat. Mater. **9**, 60–67 (2010). [CrossRef]

14. A. Unger, U. Rietzler, R. Berger, and M. Kreiter, “Sensitivity of crescent-shaped metal nanoparticles to attachment of dielectric colloids,” Nano Lett. **9**, 2311–2315 (2009). [CrossRef] [PubMed]

15. T. Sannomiya, C. Hafner, and J. Voros, “In situ sensing of single binding events by localized surface plasmon resonance,” Nano Lett. **8**, 3450–2455 (2008). [CrossRef] [PubMed]

*et al.*, who studied the coupling between a plasmonic nanoparticle and a small dielectric sphere in the quasistatic limit [16

16. T. Davis, D. Gómez, and K. Vernon, “Interaction of molecules with localized surface plasmons in metallic nanoparticles,” Phys. Rev. B **81**, 045423 (2010). [CrossRef]

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

## 2. Coupled dipole approximation

*P*=

_{i}*α*(

_{i}*E*

_{i}_{0}– Σ

_{j}_{≠}

*) dependent on the local electric field coming from the source*

_{i}A_{ij}P_{j}*E*

_{i}_{0}and other dipoles

*A*and its polarizability

_{ij}P_{j}*α*. To make our point we drop all vector and matrix indices and look at the following set of equations for a dipole

_{i}*P*

_{1}coupled to a dipole

*P*

_{2}[18]: where

*E*

_{0}is the external field strength (assumed to be the same for both dipoles) and

*A*

_{12}is the dipole coupling factor. We are interested in the new mode frequencies of the second particle, assumed to be metallic, due to its interaction with the first being a dielectric. Substituting Eq. (1a) into Eq. (1b) we obtain the self-consistent polarizability of the metal object This two dipole coupling approach is very general and the result was used

*e.g.*to calculate the anisotropic polarizability tensor of a nanosphere dimer near a planar substrate [19

19. A. Pinchuk and G. Schatz, “Anisotropic polarizability tensor of a dimer of nanospheres in the vicinity of a plane substrate,” Nanotechnology **16**, 2209–2217 (2005). [CrossRef] [PubMed]

20. B. Rolly, B. Stout, and N. Bonod, “Metallic dimers: When bonding transverse modes shine light,” Phys. Rev. B **84**, 125420 (2011). [CrossRef]

*α*

_{1}

*α*

_{2}

*A*

_{12}

*A*

_{21}= 0 allows us to calculate the mode frequencies of the interacting system. We assume the polarizability to be

*i*= 1,2,

*r*is the radius of sphere

_{i}*i*, the permittivities

*ɛ*are in units of

_{i}*ɛ*

_{0}which is the permittivity of free space. We get an expression for the permittivity of the resonator where

*A*

_{12}=

*A*

_{21}=

*A*), their radii and permittivity

*ɛ*

_{1}of the dielectric. Now for

*Q*= 0 we obtain the well-known resonance condition

*ɛ*

_{2}(

*ω*

_{0}) = −2 for a single metal sphere. In the presence of coupling the resonance condition is adjusted to a new frequency

*ω*and we expand

_{Q}*ɛ*

_{2}(

*ω*) into a Taylor series: to find the shifted resonance frequency. This yields which, assuming that

_{Q}*ɛ*

_{2}is described by a Drude model

*Q*≪ 1. We see that the resonance peak shift is directly proportional to the coupling strength

*Q*. In the near-field we can use

*A*∼ 1/(4

*πɛ*

_{0}

*d*

^{3}), where

*d*is the distance between the dipoles. Thus, for a small analyte (

*r*

_{1}) placed on the surface of a metal sphere (

*r*

_{2}) we have

*d*≈

*r*

_{2}≫

*r*

_{1}

*Q*∼

*A*

^{2}), we see that the magnitude of the shift is proportional to the local electric field enhancement. Moreover, in the full case

*A*is a tensor, so the dipole orientation with respect to the external field also influences the observed shift. Comparing this schematic result to the one obtained by Davis

*et al.*[16

16. T. Davis, D. Gómez, and K. Vernon, “Interaction of molecules with localized surface plasmons in metallic nanoparticles,” Phys. Rev. B **81**, 045423 (2010). [CrossRef]

*ɛ*

_{1}. Next, we will describe a way of calculating the resonance shift based on the local field intensities in the constituents of the resonant system, which is not limited to spherical geometries.

## 3. Quasistatic model

*ω*. The extinction spectrum and resonance frequency

*ω*

_{0}depend on its size, shape, and material properties as well as those of the surrounding medium. Binding of an additional object of permittivity different from the surrounding medium to its surface changes the resonance conditions, what can be observed by a slight shift of the spectrum. In the quasistatic limit, letting

*E⃗*denote the induced field due to an external field

*E⃗*

_{0}impinging on the system, we let

*E⃗*= −∇

*⃗*Ψ. We divide a large volume

*V*into subvolumes

*V*along material boundaries. Then, for each electric field

_{i}*E*in

_{i}*V*we define

_{i}*I*≡ ∫

_{i}_{Vi}|

*E*|

_{i}^{2}d

*V*and in the quasistatic limit for the local fields we can write [21

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

*D⃗*), ∇

*⃗*·

*D⃗*= 0, and the volume

*V*is enclosed by a surface

*S*and we use Gauss’ theorem. For the above equation to hold the induced local field generated by a metal resonator needs to decay to zero at the surface

*S*far away from the particle and this is fulfilled for local modes in the quasistatic approximation. Thus, Eq. (7) demands that for a nontrivial solution (

*i.e.*nonvanishing fields everywhere) to exist at least two materials of opposite signs of permittivity must be present. Hence, in the subwavelength scale, pure dielectric structures do not have localized resonances. In our case the considered subvolumes are the metal

*m*, dielectric analyte

*d*and surrounding medium

*s*, as shown in Fig. 1. Without an analyte Eq. (7) becomes which determines the resonance frequency

*ω*

_{0}. For a spherical particle

*I*= 2

_{s}*I*and we retrieve the well-known condition

_{m}*ɛ*(

_{m}*ω*

_{0}) + 2

*ɛ*= 0 for a surface plasmon resonance [18]. When we add a dielectric object

_{s}*d*, the resonance is shifted and broadened by Δ

*ω*, and Eq. (8) is modified to where ′ indicates changes compared to Eq. (8). To determine the relative resonance shift

*ɛ*(

_{m}*ω*′) of the metal object into a Taylor series, as in Eq. 4. Subtracting Eq. (8) from Eq. (9) and substituting

*ɛ*(

_{m}*ω*′) into the result we obtain

## 4. Results and discussion

*r*= 30 nm) with a spherical analyte (

_{m}*r*= 1 nm) limiting the number to three components

_{d}*m*,

*s*, and

*d*(see Fig. 1). The considered gold sphere has maximum extinction at 585 THz and at this frequency permittivity

*ɛ*= −3.21+2.91i and its derivative

_{m}22. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*i.e.*the difference between the perturbed and unperturbed intensity distributions

*ɛ*′

*|*

_{i}*E*′

*|*

_{i}^{2}–

*ɛ*|

_{i}*E*|

_{i}^{2}) showing the magnitude and sign of the contributions for a parallel (

*β*= 0) and perpendicular (

*β*=

*π*/2) field alignment in Fig. 2a and b, respectively. When

*β*= 0 the analyte is located at the incident side of the metal sphere on the

*z*-axis, while

*β*=

*π*/2 means that the analyte is on the side where the resonant field is the strongest. The energy redistribution due to analyte adhesion in the considered case is the following: the difference of the product of the permittivity and the integral between the perturbed and unperturbed fields in the metal gives a negative contribution to the total shift, while in the surrounding medium it has a positive sign. In the dielectric particle the sign depends on the permittivity of the analyte – in the case presented for

*ɛ*= 4 the contribution is positive.

_{d}*l*, which then approaches the planar image factor (

*ɛ*–

_{m}*ɛ*)/(

_{s}*ɛ*+

_{m}*ɛ*). This allows us to locally model the gold surface as flat. The analyte is placed in the near-field of the metal resonator and its dipolar response is determined by its polarizability

_{s}*α*∝ (

_{d}*ɛ*–

_{d}*ɛ*)/(

_{s}*ɛ*+ 2

_{d}*ɛ*). The induced dipole, decomposed into parallel and perpendicular components relative to the local metal surface, changes the local electric field which we calculate in the quasistatic case using the method of images. The change to the field outside the resonator is generated by the original dipole and a mirrored one inside the metal, the change inside the metal by the original and an image outside, and inside the dielectric particle by the image in the metal and the presence of the particle itself.

_{s}*ɛ*and the angle

_{d}*β*with black points. The shift is a monotonic function of permittivity, because for an increasing

*ɛ*the dipole moment increases asymptotically due to saturation of the dielectric sphere polarizability (full saturation is not shown here). Figure 3b and c show the total shift predicted by the model

_{d}*β*= 0 and

*π*/2, respectively. We notice, that the model agrees quite well with the full FDTD calculation, although the latter is a fully-retarded method and takes into account the radiative components. For small

*ɛ*the main redshift comes from the surrounding medium

_{d}*ɛ*the analyte volume also decreases the redshift. When the field is perpendicular to the surface instead of parallel, the electric field amplitude, determined by

_{d}*β*, increases giving a larger shift.

*b*is a parameter determining the relative sensitivity to the presence of an analyte placed in a parallel or perpendicular electric field with respect to the surface. If the shift is only proportional to the intensity of the local field |

*E*|

^{2}determined by the position of the analyte, and by the permittivity of gold at the resonance frequency

*ω*

_{0}, then

*b*= 1. To estimate

*b*we fit the following function to the calculated data points. The result of the fitting (

*a*

_{0}= 2.71 × 10

^{−6},

*a*

_{1}= 3.98, and

*b*

_{0}= 0.37), as can be seen in Fig. 3a, is presented using a 3D-surface, while the surface’s color scale shows the residuals of the fit. From the fitting procedure we get

*b*

_{0}≈ 0.37, which means that the sensitivity per field unit is larger for the field along the metal surface than perpendicular to it. This is also confirmed in our FDTD simulations.

*V*/

_{d}*V*of the dielectric analyte and is confirmed in Fig. 4, where the largest considered normalized analyte volume corresponds to a radius of 2 nm. The field decays away from the gold sphere, so the resonance shift increases more slowly than a linear function and will approach asymptotically a maximum value in a similar manner as observed for gold nanoislands [23

_{m}23. O. Kedem, A. B. Tesler, A. Vaskevich, and I. Rubinstein, “Sensitivity and optimization of localized surface plasmon resonance transducers,” ACS Nano **5**, 748–760 (2011). [CrossRef] [PubMed]

*ɛ*= 4 and radius of

_{d}*r*= 1 nm and is arranged sequentially in a lattice on the surface of the disk. We calculate the extinction spectra using 3D-FDTD with nonuniform meshing equal to 0.5 nm for the gold disk and 0.2 nm for the dielectric analyte. Resonance shifts, shown in Fig. 5b, are obtained by fitting the data points and extracting the local maximum of the fitted function. Resonance shift values between the particle placements are obtained by interpolating with cubic splines. For the disk, the accuracy of the fit does not depend on the location of the analyte and is ±0.2 × 10

_{d}^{−3}nm. The electric field distribution for the model is calculated at resonance (534.7 nm) of a lone disk and is used, in a similar scheme as discussed for the sphere, to estimate the peak shifts (Fig. 5c). We see, that the agreement is very good when the analyte is placed far away from the edges of the disk, however, decreases when moving away from the center. The largest mismatch is observed when the analyte is placed in the vicinity of the curved edge – the FDTD simulation yields a 1.5 × 10

^{−2}nm shift, while using the model we estimate a 1.0 × 10

^{−2}. The reason behind this is the calculation scheme for the dipole moments of the analyte and its image – we assume a much larger local curvature of the surface of the metal resonator than the analyte. This condition is not fulfilled when the dielectric particle is near the rounded edge. However, a method of calculating electromagnetic fields created by an electric dipole in the vicinity of a sphere in the quasistatic approximation, presented by Zurita-Sánchez [24], may be modified to correct the model prediction.

*r*= 1 nm,

_{d}*ɛ*= 4) arranged at 13 positions along a line as indicated in Fig. 6. The longitudinal dimension of the rod relative to the resonance wavelength is about four times larger so the quasistatic condition is not fulfilled as strongly as in the previous case. Thus, there is a larger discrepancy of the quantitative results obtained via FDTD (simulations and extinction fitting) and model calculated data. The shifts are schematically indicated by lines following the surface of the rod where the analyte molecules are placed: the blue solid line represents fits to FDTD obtained shifts (maximum resonance shift is 3.7 × 10

_{d}^{−3}nm,

## 5. Conclusions

## Acknowledgments

## References and links

1. | P. Englebienne, “Use of colloidal gold surface plasmon resonance peak shift to infer affinity constants from the interactions between protein antigens and antibodies specific for single or miltiple epitopes,” Analyst |

2. | T. Okamoto, I. Yamaguchi, and T. Kobayashi, “Local plasmon sensor with gold colloid monolayers deposited upon glass substrates,” Opt. Lett. |

3. | M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. Van Duyne, “Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers,” J. Am. Chem. Soc. |

4. | N. Nath and A. Chilkoti, “A colorimetric gold nanoparticle sensor to interrogate biomolecular interactions in real time on a surface,” Anal. Chem. |

5. | A. J. Haes and R. P. Van Duyne, “A nanoscale optical biosensor: Sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles,” J. Am. Chem. Soc. |

6. | H. Xu and M. Käll, “Modeling the optical response of nanoparticle-based aurface plasmon resonance sensors,” Sens. Actuators B Chem. |

7. | J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. |

8. | J. Homola, |

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. | H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, “Spectroscopy of single hemoglobin molecules by surface enhanced raman scattering,” Phys. Rev. Lett. |

11. | L. Rodríguez-Lorenzo, R. A. Álvarez-Puebla, I. Pastoriza-Santos, S. Mazzucco, O. Stéphan, M. Kociak, L. M. Liz-Marzán, and F. J. G. de Abajo, “Zeptomol detection through controlled ultrasensitive surface-enhanced Raman scattering,” J. Am. Chem. Soc. |

12. | T. Dadosh, J. Sperling, G. W. Bryant, R. Breslow, T. Shegai, M. Dyshel, G. Haran, and I. Bar-Joseph, “Plasmonic control of the shape of the raman spectrum of a single molecule in a silver nanoparticle dimer,” ACS Nano |

13. | D.-K. Lim, K.-S. Jeon, H. M. Kim, J.-M. Nam, and Y. D. Suh, “Nanogap-engineerable Raman-active nan-odumbbells for single-molecule detection,” Nat. Mater. |

14. | A. Unger, U. Rietzler, R. Berger, and M. Kreiter, “Sensitivity of crescent-shaped metal nanoparticles to attachment of dielectric colloids,” Nano Lett. |

15. | T. Sannomiya, C. Hafner, and J. Voros, “In situ sensing of single binding events by localized surface plasmon resonance,” Nano Lett. |

16. | T. Davis, D. Gómez, and K. Vernon, “Interaction of molecules with localized surface plasmons in metallic nanoparticles,” Phys. Rev. B |

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

18. | J. D. Jackson, |

19. | A. Pinchuk and G. Schatz, “Anisotropic polarizability tensor of a dimer of nanospheres in the vicinity of a plane substrate,” Nanotechnology |

20. | B. Rolly, B. Stout, and N. Bonod, “Metallic dimers: When bonding transverse modes shine light,” Phys. Rev. B |

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

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

23. | O. Kedem, A. B. Tesler, A. Vaskevich, and I. Rubinstein, “Sensitivity and optimization of localized surface plasmon resonance transducers,” ACS Nano |

24. | J. R. Zurita-Sánchez, “Quasi-static electromagnetic fields created by an electric dipole in the vicinity of a dielectric sphere: method of images,” Rev. Mex. Fis. |

**OCIS Codes**

(240.6490) Optics at surfaces : Spectroscopy, surface

(240.6680) Optics at surfaces : Surface plasmons

(280.0280) Remote sensing and sensors : Remote sensing and sensors

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: October 3, 2011

Revised Manuscript: November 4, 2011

Manuscript Accepted: November 9, 2011

Published: December 21, 2011

**Citation**

Tomasz J. Antosiewicz, S. Peter Apell, Virginia Claudio, and Mikael Käll, "A simple model for the resonance shift of localized plasmons due to dielectric particle adhesion," Opt. Express **20**, 524-533 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-524

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### References

- P. Englebienne, “Use of colloidal gold surface plasmon resonance peak shift to infer affinity constants from the interactions between protein antigens and antibodies specific for single or miltiple epitopes,” Analyst123, 1599–1603 (1998). [CrossRef] [PubMed]
- T. Okamoto, I. Yamaguchi, and T. Kobayashi, “Local plasmon sensor with gold colloid monolayers deposited upon glass substrates,” Opt. Lett.25, 372–374 (2000). [CrossRef]
- M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. Van Duyne, “Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers,” J. Am. Chem. Soc.123, 1471–1482 (2001). [CrossRef]
- N. Nath and A. Chilkoti, “A colorimetric gold nanoparticle sensor to interrogate biomolecular interactions in real time on a surface,” Anal. Chem.74, 504–509 (2002). [CrossRef] [PubMed]
- A. J. Haes and R. P. Van Duyne, “A nanoscale optical biosensor: Sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles,” J. Am. Chem. Soc.124, 10596–10604 (2002). [CrossRef] [PubMed]
- H. Xu and M. Käll, “Modeling the optical response of nanoparticle-based aurface plasmon resonance sensors,” Sens. Actuators B Chem.87, 244–249 (2002). [CrossRef]
- J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater.7, 442–453 (2008). [CrossRef] [PubMed]
- J. Homola, Surface plasmon resonance based sensors, Springer Series on Chemical Sensors and Biosensors (Springer-Verlag, Berlin-Heidelberg-New York, 2006). [CrossRef]
- H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced raman scattering,” Phys. Rev. E62, 4318–4324 (2000). [CrossRef]
- H. Xu, E. J. Bjerneld, M. Käll, and L. Börjesson, “Spectroscopy of single hemoglobin molecules by surface enhanced raman scattering,” Phys. Rev. Lett.83, 4357–4360 (1999). [CrossRef]
- L. Rodríguez-Lorenzo, R. A. Álvarez-Puebla, I. Pastoriza-Santos, S. Mazzucco, O. Stéphan, M. Kociak, L. M. Liz-Marzán, and F. J. G. de Abajo, “Zeptomol detection through controlled ultrasensitive surface-enhanced Raman scattering,” J. Am. Chem. Soc.131, 4616–4618 (2009). [CrossRef] [PubMed]
- T. Dadosh, J. Sperling, G. W. Bryant, R. Breslow, T. Shegai, M. Dyshel, G. Haran, and I. Bar-Joseph, “Plasmonic control of the shape of the raman spectrum of a single molecule in a silver nanoparticle dimer,” ACS Nano3, 1988–1994 (2009). [CrossRef] [PubMed]
- D.-K. Lim, K.-S. Jeon, H. M. Kim, J.-M. Nam, and Y. D. Suh, “Nanogap-engineerable Raman-active nan-odumbbells for single-molecule detection,” Nat. Mater.9, 60–67 (2010). [CrossRef]
- A. Unger, U. Rietzler, R. Berger, and M. Kreiter, “Sensitivity of crescent-shaped metal nanoparticles to attachment of dielectric colloids,” Nano Lett.9, 2311–2315 (2009). [CrossRef] [PubMed]
- T. Sannomiya, C. Hafner, and J. Voros, “In situ sensing of single binding events by localized surface plasmon resonance,” Nano Lett.8, 3450–2455 (2008). [CrossRef] [PubMed]
- T. Davis, D. Gómez, and K. Vernon, “Interaction of molecules with localized surface plasmons in metallic nanoparticles,” Phys. Rev. B81, 045423 (2010). [CrossRef]
- T. J. Davis, K. C. Vernon, and D. E. Gómez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B79, 155423 (2009). [CrossRef]
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, Inc., New York, 1999).
- A. Pinchuk and G. Schatz, “Anisotropic polarizability tensor of a dimer of nanospheres in the vicinity of a plane substrate,” Nanotechnology16, 2209–2217 (2005). [CrossRef] [PubMed]
- B. Rolly, B. Stout, and N. Bonod, “Metallic dimers: When bonding transverse modes shine light,” Phys. Rev. B84, 125420 (2011). [CrossRef]
- F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006). [CrossRef] [PubMed]
- P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
- O. Kedem, A. B. Tesler, A. Vaskevich, and I. Rubinstein, “Sensitivity and optimization of localized surface plasmon resonance transducers,” ACS Nano5, 748–760 (2011). [CrossRef] [PubMed]
- J. R. Zurita-Sánchez, “Quasi-static electromagnetic fields created by an electric dipole in the vicinity of a dielectric sphere: method of images,” Rev. Mex. Fis.55, 443–449 (2009).

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