Complex modes and effective refractive index in 3D periodic arrays of plasmonic nanospheres |
Optics Express, Vol. 19, Issue 27, pp. 26027-26043 (2011)
http://dx.doi.org/10.1364/OE.19.026027
Acrobat PDF (1384 KB)
Abstract
We characterize the modes with complex wavenumber for both longitudinal and transverse polarization states (with respect to the mode traveling direction) in three dimensional (3D) periodic arrays of plasmonic nanospheres, including metal losses. The Ewald representation of the required dyadic periodic Green’s function to represent the field in 3D periodic arrays is derived from the scalar case, which can be analytically continued into the complex wavenumber space. We observe the presence of one longitudinal mode and two transverse modes, one forward and one backward. Despite the presence of two modes for transverse polarization, we notice that the forward one is “dominant” (i.e., it contributes most to the field in the array). Therefore, in case of transverse polarization, we describe the composite material in terms of a homogenized effective refractive index, comparing results from (i) modal analysis, (ii) Maxwell Garnett theory, (iii) Nicolson-Ross-Weir retrieval method from scattering parameters for finite thickness structures (considering different thicknesses, showing consistency of results), and (iv) the fitting of the fields obtained through HFSS simulations. The agreement among the different methods justifies the performed homogenization procedure in case of transverse polarization.
© 2011 OSA
1. Introduction
2. Simulation model
3. Dispersion diagrams for 3D periodic arrays of plasmonic nanospheres
3.1 Transverse polarization (T-pol)
3.2 Longitudinal polarization (L-pol)
4. Parametric analysis: comparison of modal dispersion diagrams
4.1 Transverse polarization (T-pol)
4.2 Longitudinal polarization (L-pol)
4.3 Figure of merit
5. 3D lattice with finite thickness along z: transmission, reflection and absorption
6. Homogenization: effective refractive index
6.1 NRW retrieval method and comparison with theoretical results
6.2 Field fitting retrieval method
7. Conclusion
Appendix A: Ewald representation for the dyadic GF for 3D periodic arrays
Acknowledgment
References and links
1. | A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B 74(20), 205436 (2006). |
2. | R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42(6), RS6S21 (2007). |
3. | R. Sainidou and G. F. de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express 16(7), 4499–4506 (2008). |
4. | D. Van Orden, Y. Fainman, and V. Lomakin, “Optical waves on nanoparticle chains coupled with surfaces,” Opt. Lett. 34(4), 422–424 (2009). |
5. | R. A. Shore and A. D. Yaghjian, Complex Waves on 1D, 2D, and 3D Periodic Arrays of Lossy and Lossless Magnetodielectric Spheres (Air Force Research Laboratory, Hanscom, AFB, MA, 2010; revised, 2011). |
6. | S. Campione, S. Steshenko, and F. Capolino, “Complex bound and leaky modes in chains of plasmonic nanospheres,” Opt. Express 19(19), 18345–18363 (2011). |
7. | A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B 28(6), 1446–1458 (2011). |
8. | S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: Fabrication, optical properties and applications,” in Selected Topics in Photonic Crystals and Metamaterials, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 143–196. |
9. | A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B 75(2), 024304 (2007). |
10. | C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). |
11. | S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 8.1. |
12. | P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett. 8(3), 124–126 (1998). |
13. | I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett. 49(6), 1353–1357 (2007). |
14. | G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech. 56(9), 2069–2075 (2008). |
15. | A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express 19(3), 2754–2772 (2011). |
16. | J. D. Jackson, Classical Electrodynamics (Wiley, 1998). |
17. | V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70(5), 054202 (2004). |
18. | D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11(11), 2851–2861 (1994). |
19. | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965). |
20. | A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). |
21. | S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express 1(6), 1077–1089 (2011). |
22. | A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006). |
23. | I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000). |
24. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). |
25. | M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76(24), 245117 (2007). |
26. | A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011). |
27. | S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1 Pt 2), 016607 (2011). |
28. | A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970). |
29. | W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62(1), 33–36 (1974). |
30. | A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech. 45(1), 52–57 (1997). |
31. | D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). |
32. | C. R. Simovski, “On the extraction of local material parameters of metamaterials from experimental or simulated data,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 11.1. |
33. | S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009). |
34. | A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999). |
35. | A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 9.1. |
36. | S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Characterization of the optical modes in 3D-periodic arrays of metallic nanospheres,” in URSI General Assembly and Scientific Symposium (Istanbul, Turkey, 2011). |
37. | X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1 Pt 2), 016608 (2004). |
38. | P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin) 369(3), 253–287 (1921). |
39. | K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys. 63(1), 222–235 (1986). |
40. | A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett. 10(5), 168–170 (2000). |
OCIS Codes
(160.1245) Materials : Artificially engineered materials
(260.2065) Physical optics : Effective medium theory
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics
ToC Category:
Metamaterials
History
Original Manuscript: September 13, 2011
Revised Manuscript: November 16, 2011
Manuscript Accepted: November 16, 2011
Published: December 7, 2011
Citation
Salvatore Campione, Sergiy Steshenko, Matteo Albani, and Filippo Capolino, "Complex modes and effective refractive index in 3D periodic arrays of plasmonic nanospheres," Opt. Express 19, 26027-26043 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26027
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References
- A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B74(20), 205436 (2006).
- R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci.42(6), RS6S21 (2007).
- R. Sainidou and G. F. de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express16(7), 4499–4506 (2008).
- D. Van Orden, Y. Fainman, and V. Lomakin, “Optical waves on nanoparticle chains coupled with surfaces,” Opt. Lett.34(4), 422–424 (2009).
- R. A. Shore and A. D. Yaghjian, Complex Waves on 1D, 2D, and 3D Periodic Arrays of Lossy and Lossless Magnetodielectric Spheres (Air Force Research Laboratory, Hanscom, AFB, MA, 2010; revised, 2011).
- S. Campione, S. Steshenko, and F. Capolino, “Complex bound and leaky modes in chains of plasmonic nanospheres,” Opt. Express19(19), 18345–18363 (2011).
- A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B28(6), 1446–1458 (2011).
- S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: Fabrication, optical properties and applications,” in Selected Topics in Photonic Crystals and Metamaterials, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 143–196.
- A. Alù and N. Engheta, “Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials,” Phys. Rev. B75(2), 024304 (2007).
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 8.1.
- P. Myun-Joo, P. Jongkuk, and N. Sangwook, “Efficient calculation of the Green's function for the rectangular cavity,” IEEE Microw. Guid. Wave Lett.8(3), 124–126 (1998).
- I. Stevanoviæ and J. R. Mosig, “Periodic Green's function for skewed 3-D lattices using the Ewald transformation,” Microw. Opt. Technol. Lett.49(6), 1353–1357 (2007).
- G. Lovat, P. Burghignoli, and R. Araneo, “Efficient evaluation of the 3-D periodic Green's function through the Ewald method,” IEEE Trans. Microw. Theory Tech.56(9), 2069–2075 (2008).
- A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express19(3), 2754–2772 (2011).
- J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
- V. A. Markel, V. N. Pustovit, S. V. Karpov, A. V. Obuschenko, V. S. Gerasimov, and I. L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B70(5), 054202 (2004).
- D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A11(11), 2851–2861 (1994).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).
- A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt.37(22), 5271–5283 (1998).
- S. Campione, M. Albani, and F. Capolino, “Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited],” Opt. Mater. Express1(6), 1077–1089 (2011).
- A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express14(4), 1557–1567 (2006).
- I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B62(23), 15299–15302 (2000).
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
- M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B76(24), 245117 (2007).
- A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
- S. Steshenko, F. Capolino, P. Alitalo, and S. Tretyakov, “Effective model and investigation of the near-field enhancement and subwavelength imaging properties of multilayer arrays of plasmonic nanospheres,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(1 Pt 2), 016607 (2011).
- A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas.19(4), 377–382 (1970).
- W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE62(1), 33–36 (1974).
- A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech.45(1), 52–57 (1997).
- D. R. Smith, S. Schultz, P. Markoscaron, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002).
- C. R. Simovski, “On the extraction of local material parameters of metamaterials from experimental or simulated data,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 11.1.
- S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009).
- A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).
- A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), 9.1.
- S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Characterization of the optical modes in 3D-periodic arrays of metallic nanospheres,” in URSI General Assembly and Scientific Symposium (Istanbul, Turkey, 2011).
- X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1 Pt 2), 016608 (2004).
- P. P. Ewald, “Die Berechnung optischer und elektrostatischer Gitterpotentiale,” Ann. Phys. (Berlin)369(3), 253–287 (1921).
- K. E. Jordan, G. R. Richter, and P. Sheng, “An efficient numerical evaluation of the Green's function for the Helmholtz operator on periodic structures,” J. Comput. Phys.63(1), 222–235 (1986).
- A. Kustepeli and A. Q. Martin, “On the splitting parameter in the Ewald method,” IEEE Microw. Guid. Wave Lett.10(5), 168–170 (2000).
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