We consider a chain of N Ag spheres, of radius R = 10 nm and spaced the distance d = R. We vary N and expose them to the y- and x-polarized plane waves described in the previous section. This gives a spectrum for each value of N and for each polarization, and the top panel in Fig. 4 shows the resonance wavelengths as function N.
We observe a clear redshift and a slight blueshift in the y- and x-polarized cases, respectively. For N = 5, Fig. 1 shows shows the LSPs in the z = 0-plane in the two cases. The induced charges of opposite signs in the gaps in the former case (top panel) give rise to strong field enhancements between the spheres, yielding a redshift as more particles are added. In contrast, for x-polarization (bottom panel) the field is essentially located on the individual particles, with no interaction across the gaps.
In the y-polarized case, an exponential-like con-vergence to an asymptotic resonance wavelength is observed for increasing N, suggesting that ∆λ0Res ∼ exp(−ηN/(N − 1)). The bottom panel of Fig. 4 shows ln ∆λ0Res
as function of the inverse number of periods in the chain, 1/(N − 1). The agreement with the linear fit is not excellent, but acceptable, givingηN=1.9. This suggests an interaction length of approximately two pe-riods in the chain, i.e., each particle interacts with its two nearest neighbors. A similar conclusion is reached in [11].
0 5 10 15 20
0.36 0.38 0.4
Number of Scatterers, N
λRes 0[µm
y-pol.
Inverse Number of Chain Periods, 1/(N− 1)
ln¡ ∆λRes 0¢
Linear Fit
FIGURE 4. Chain of N Ag spheres, of radius R = 10 nm, and spaced the distance d = R.Top panel: Resonance wavelengths as function of N.Bottom panel: Logarithm of peak shift ratio as function of inverse number of chain periods, 1/(N − 1).
In conclusion, we have outlined a scheme based on the Lippmann-Schwinger equation and the electromag-netic Green’s tensor for simulating, in 3D, scattering of electromagnetic waves on N spheres. The method can be used for calculations of the Green’s tensor and the LDOS as well as Purcell factors and cavity modes in optical microstructures (including photonic crystals) and plas-monic nanostructures. We presented two example calcu-lations of the latter, where the resonance wavelengths for chains of Ag nanoparticles were analyzed. We found a strong dependence on the polarization of the incoming field, and a finite interaction length along the chain.
REFERENCES
1. L. Novotny, and N. van Hulst, Nat. Photon.5, 83–90 (2011).
2. D. K. Gramotnev, and S. I. Bozhevolnyi, Nat. Photon.4, 83–91 (2010).
3. H. A. Atwater, and A. Polman, Nat. Mater.9, 205–213 (2010).
4. S. Y. Park, and D. Stroud, Phys. Rev. B69, 125418 (2004).
5. P. T. Kristensen, C. V. Vlack, and S. Hughes, Opt. Lett.
37, 1649–1651 (2012).
6. L. Novotny, and B. Hecht, Principles of Nano-Optics, Cambridge University Press, 2006, first edn.
7. P. T. Kristensen, P. Lodahl, and J. Mørk, J. Opt. Soc. Am.
B27, 228–237 (2010).
8. J. D. Jackson, Classical Electrodynamics, Wiley, 1998, third edn.
9. A. F. Koenderink, Opt. Lett.35, 4208–4210 (2010).
10. P. K. Jain, W. Huang, and M. A. El-Sayed, Nano Lett.7, 2080–2088 (2007).
11. N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, J.
Phys. Chem. C113, 2784–2791 (2009).
Bibliography
[1] D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,”
Nat. Photon. 4, 83–91 (2010).
[2] P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nat.
Photon. 6, 16–24 (2012).
[3] M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat.
Photon. 5, 349–356 (2011).
[4] L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photon. 5, 83–90 (2011).
[5] W. Zhang, L. Huang, C. Santschi, and O. J. F. Martin, “Trapping and sensing 10 nm metal nanoparticles using plasmonic dipole antennas,” Nano Lett. 10, 1006–1011 (2010).
[6] K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–
21800 (2008).
[7] H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat.
Mater. 9, 205–213 (2010).
[8] K. Nakayama, K. Tanabe, and H. A. Atwater, “Plasmonic nanoparticle enhanced light absorption in GaAs solar cells,” Appl. Phys. Lett. 93, 121904 –121904–3 (2008).
[9] S. Xiao, E. Stassen, and N. A. Mortensen, “Ultrathin silicon solar cells with enhanced photocurrents assisted by plasmonic nanostructures,” J. Nanophoton. 6, 061503 (2012).
[10] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005), 3rd ed.
[11] J. Reddy, An Introduction to the Finite Element Method (McGraw-Hill Sci-ence/Engineering/Math, 2005), 3rd ed.
[12] G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
[13] Y. Akimov, K. Ostrikov, and E. Li, “Surface plasmon enhancement of optical absorption in thin-film silicon solar cells,” Plasmonics 4, 107–113 (2009).
[14] S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007), 1st ed.
[15] S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation,” Phys. Rev. B 69, 125418 (2004).
113
[16] Y. lin Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[17] F. J. Garc´ıa de Abajo, “Multiple scattering of radiation in clusters of dielectrics,”
Phys. Rev. B 60, 6086–6102 (1999).
[18] P. T. Kristensen, “Light-matter interaction in nanostructured materials,” Ph.D.
thesis, Technical University of Denmark, Department of Photonics Engineering (2009).
[19] P. T. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc.
Am. B 27, 228–237 (2010).
[20] L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[21] K. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[22] J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. G´erard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photon. 4, 839–846 (2010).
[23] E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
[24] L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), 1st ed.
[25] J. D. Jackson, Classical Electrodynamics (Wiley, 1998), 3rd ed.
[26] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light (Pergamon Pr, 1981), 6th ed.
[27] A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[28] O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable back-grounds,” Phys. Rev. E 58, 3909–3915 (1998).
[29] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), 2nd ed.
[30] A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997), 1st ed.
[31] A. A. Kokhanovsky, Aerosol Optics: Light Absorption and Scattering by Particles in the Atmosphere (Springer, 2008), 1st ed.
[32] A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett.
35, 4208–4210 (2010).
[33] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE Press, 1998), 1st ed.
Bibliography 115
[34] P. A. Martin, Multiple Scattering. Interaction of Time-Harmonic Waves with N Obstacles. (Cambridge University Press, 2006), 1st ed.
[35] H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics (Oxford University Press, 2004), 1st ed.
[36] P. T. Kristensen, C. V. Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
[37] M. R. Spiegel, S. Lipschutz, and J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables (McGraw-Hill, 2008), 3rd ed.
[38] F. Capolino, Theory and Phenomena of Metamaterials (CRC Press, 2009), 1st ed.
[39] N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009).
[40] C. M¨atzler, “MATLAB functions for Mie scattering and absorption, version 2,”
MATLAB code, Institut f¨ur angewandte Physik, Universit¨at Bern (2002).
[41] P. A. Martin, “Multiple scattering and the Rehr-Albers-Fritzsche formula for the propagator matrix,” J. Phys. A 31, 8923 (1998).
[42] 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).
[43] M. Chen, Y.-F. Chau, and D. Tsai, “Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres,” Plasmonics 3, 157–164 (2008).
[44] M. Lermer, N. Gregersen, F. Dunzer, S. Reitzenstein, S. H¨ofling, J. Mørk, L. Worschech, M. Kamp, and A. Forchel, “Bloch-wave engineering of quantum dot micropillars for cavity quantum electrodynamics experiments,” Phys. Rev. Lett.
108, 057402 (2012).
[45] B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calcula-tions,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[46] D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[47] S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[48] N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[49] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals – Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.
[50] J. Jung, T. Søndergaard, T. G. Pedersen, K. Pedersen, A. N. Larsen, and B. B.
Nielsen, “Dyadic Green’s functions of thin films: Applications within plasmonic solar cells,” Phys. Rev. B 83, 085419 (2011).
[51] M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[52] J. Kellendonk and S. Richard, “Weber-Schafheitlin-type integrals with exponent 1,”
Integr. Transf. Spec. F. 20, 147–153 (2009).
[53] S. Richard (2012). Personal communication with author of Ref. [52].
[54] B. Li and C. Liu, “Multi-scattering of electromagnetic waves by nanoshell aggregates,”
J. Nanopart. Res. 14, 1–11 (2012).