Abstract

In this paper, we study the role of nanoparticle shape and aperiodic arrangement in the scattering and spatial localization properties of plasmonic modes in deterministic-aperiodic (DA) arrays of metal nanoparticles. By using an efficient coupled-dipole model for the study of the electromagnetic response of large arrays excited by an external field, we demonstrate that DA structures provide enhanced spatial localization of plasmonic modes and a higher density of enhanced field states with respect to their periodic counterparts. Finally, we introduce and discuss specific design rules for the engineering and optimization of field enhancement and localization in DA arrays. Our results, which we fully validated by rigorous Generalized Mie Theory (GMT) and transition matrix (T-matrix) theory, demonstrate that DA arrays provide a robust platform for the design of a variety of novel optical devices with enhanced and controllable plasmonic fields.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, and G. C. Schatz, "Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
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    [CrossRef] [PubMed]

2009

L. Dal Negro, C. Forestiere, G. Miano, and G. Rubinacci, "Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays," Phys. Rev. B 79, 85404 (2009).
[CrossRef]

A. Gopinath, S. Boriskina, B. Reinhard, and L. Dal Negro, "Deterministic aperiodic arrays of metal nanoparticels for surface-enhanced Raman scattering," Opt. Express 17, 3741- 3753 (2009).
[CrossRef] [PubMed]

2008

A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, "Photonic-plasmonic scattering resonances in determinsitic aperiodic structures," Nano. Lett. 8, 2423-2431 (2008).
[CrossRef] [PubMed]

L. Dal Negro, N. N. Feng, and A. Gopinath, "Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays," J. Opt. A, Pure Appl. Opt. 10, 064013 (2008).
[CrossRef]

2007

L. Dal Negro and N. Feng, "Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles," Opt. Express 22, 14396-14403 (2007).
[CrossRef]

2006

E. Macia, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397-441 (2006).
[CrossRef]

D. W. Brandl, N. A. Mirin, and P. Nordlander, "Plasmon modes of nanosphere trimers and quadrumers," J. Phys. Chem. B 110, 12302-12310 (2006).
[CrossRef] [PubMed]

S. Zou and G. C. Schatz, "Theoretical studies of plasmon resonances in one dimensional nanoparticles chains: narrow lineshapes with tunable widths," Nanotech. 17, 2813-2820 (2006)
[CrossRef]

2005

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides: etimation of waveguide loss," Phys. Rev. B 67, 205402 (2005).
[CrossRef]

2004

P. Nodlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, "Plasmon hybridization in nanoparticle dimers," Nano. Lett. 4, 899-903 (2004)
[CrossRef]

2003

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Comm. 220, 137-141 (2003)
[CrossRef]

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]

K. L. Kelly, E. Coronado, L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

L. Zhao, K. L. Kelly, and G. C. Schatz, "The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonance wavelength and width," J. Phys. Chem. B 107, 7343-7350 (2003).
[CrossRef]

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, and G. C. Schatz, "Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
[CrossRef]

2002

L. Kroon, E. Lennholm, and R. Riklund, "Localization-delocalization in aperiodic systems," Phys. Rev. B 66, 094204 (2002).
[CrossRef]

1998

A. Rudinger and F. Piechon, "On the multifractal spectrum of the Fibonacci chain," J. Phys. A: Math. Gen. 31, 155-164 (1998).
[CrossRef]

T. Wriedt, "A review of elastic light scattering theories," Part. Part. Syst. Charact. 15, 67-74 (1998).
[CrossRef]

1995

1992

M. Dulea, M. Johansson, and R. Riklund, "Localization of electrons and electromagnetic waves in a deterministic aperiodic system," Phys. Rev. B 45, 105-114 (1992).
[CrossRef]

1989

J. M. Luck, "Cantor spectra and scaling of gap widths in deterministic aperiodic systems," Phys. Rev. B 39, 5834-5849 (1989).
[CrossRef]

1988

B. T. Draine, "The discrete dipole approximation and its application to interstellar graphite dust," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1973

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides: etimation of waveguide loss," Phys. Rev. B 67, 205402 (2005).
[CrossRef]

Aussenegg, F. R.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Comm. 220, 137-141 (2003)
[CrossRef]

Bergman, D. J.

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]

Boriskina, S.

Boriskina, S. V.

A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, "Photonic-plasmonic scattering resonances in determinsitic aperiodic structures," Nano. Lett. 8, 2423-2431 (2008).
[CrossRef] [PubMed]

Brandl, D. W.

D. W. Brandl, N. A. Mirin, and P. Nordlander, "Plasmon modes of nanosphere trimers and quadrumers," J. Phys. Chem. B 110, 12302-12310 (2006).
[CrossRef] [PubMed]

Coronado, E.

K. L. Kelly, E. Coronado, L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

Dal Negro, L.

A. Gopinath, S. Boriskina, B. Reinhard, and L. Dal Negro, "Deterministic aperiodic arrays of metal nanoparticels for surface-enhanced Raman scattering," Opt. Express 17, 3741- 3753 (2009).
[CrossRef] [PubMed]

L. Dal Negro, C. Forestiere, G. Miano, and G. Rubinacci, "Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays," Phys. Rev. B 79, 85404 (2009).
[CrossRef]

L. Dal Negro, N. N. Feng, and A. Gopinath, "Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays," J. Opt. A, Pure Appl. Opt. 10, 064013 (2008).
[CrossRef]

A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, "Photonic-plasmonic scattering resonances in determinsitic aperiodic structures," Nano. Lett. 8, 2423-2431 (2008).
[CrossRef] [PubMed]

L. Dal Negro and N. Feng, "Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles," Opt. Express 22, 14396-14403 (2007).
[CrossRef]

Draine, B. T.

B. T. Draine, "The discrete dipole approximation and its application to interstellar graphite dust," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Dulea, M.

M. Dulea, M. Johansson, and R. Riklund, "Localization of electrons and electromagnetic waves in a deterministic aperiodic system," Phys. Rev. B 45, 105-114 (1992).
[CrossRef]

Feng, N.

L. Dal Negro and N. Feng, "Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles," Opt. Express 22, 14396-14403 (2007).
[CrossRef]

Feng, N. N.

L. Dal Negro, N. N. Feng, and A. Gopinath, "Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays," J. Opt. A, Pure Appl. Opt. 10, 064013 (2008).
[CrossRef]

A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, "Photonic-plasmonic scattering resonances in determinsitic aperiodic structures," Nano. Lett. 8, 2423-2431 (2008).
[CrossRef] [PubMed]

Forestiere, C.

L. Dal Negro, C. Forestiere, G. Miano, and G. Rubinacci, "Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays," Phys. Rev. B 79, 85404 (2009).
[CrossRef]

Gopinath, A.

A. Gopinath, S. Boriskina, B. Reinhard, and L. Dal Negro, "Deterministic aperiodic arrays of metal nanoparticels for surface-enhanced Raman scattering," Opt. Express 17, 3741- 3753 (2009).
[CrossRef] [PubMed]

A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, "Photonic-plasmonic scattering resonances in determinsitic aperiodic structures," Nano. Lett. 8, 2423-2431 (2008).
[CrossRef] [PubMed]

L. Dal Negro, N. N. Feng, and A. Gopinath, "Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays," J. Opt. A, Pure Appl. Opt. 10, 064013 (2008).
[CrossRef]

Haynes, C. L.

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, and G. C. Schatz, "Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
[CrossRef]

Hohenau, A.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Comm. 220, 137-141 (2003)
[CrossRef]

Johansson, M.

M. Dulea, M. Johansson, and R. Riklund, "Localization of electrons and electromagnetic waves in a deterministic aperiodic system," Phys. Rev. B 45, 105-114 (1992).
[CrossRef]

Kelly, K. L.

K. L. Kelly, E. Coronado, L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

L. Zhao, K. L. Kelly, and G. C. Schatz, "The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonance wavelength and width," J. Phys. Chem. B 107, 7343-7350 (2003).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides: etimation of waveguide loss," Phys. Rev. B 67, 205402 (2005).
[CrossRef]

Krenn, J. R.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Comm. 220, 137-141 (2003)
[CrossRef]

Kroon, L.

L. Kroon, E. Lennholm, and R. Riklund, "Localization-delocalization in aperiodic systems," Phys. Rev. B 66, 094204 (2002).
[CrossRef]

Lamprecht, B.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Comm. 220, 137-141 (2003)
[CrossRef]

Leitner, A.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Comm. 220, 137-141 (2003)
[CrossRef]

Lennholm, E.

L. Kroon, E. Lennholm, and R. Riklund, "Localization-delocalization in aperiodic systems," Phys. Rev. B 66, 094204 (2002).
[CrossRef]

Li, K.

P. Nodlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, "Plasmon hybridization in nanoparticle dimers," Nano. Lett. 4, 899-903 (2004)
[CrossRef]

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]

Luck, J. M.

J. M. Luck, "Cantor spectra and scaling of gap widths in deterministic aperiodic systems," Phys. Rev. B 39, 5834-5849 (1989).
[CrossRef]

Macia, E.

E. Macia, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397-441 (2006).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides: etimation of waveguide loss," Phys. Rev. B 67, 205402 (2005).
[CrossRef]

McFarland, A. D.

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, and G. C. Schatz, "Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
[CrossRef]

Miano, G.

L. Dal Negro, C. Forestiere, G. Miano, and G. Rubinacci, "Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays," Phys. Rev. B 79, 85404 (2009).
[CrossRef]

Mirin, N. A.

D. W. Brandl, N. A. Mirin, and P. Nordlander, "Plasmon modes of nanosphere trimers and quadrumers," J. Phys. Chem. B 110, 12302-12310 (2006).
[CrossRef] [PubMed]

Nodlander, P.

P. Nodlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, "Plasmon hybridization in nanoparticle dimers," Nano. Lett. 4, 899-903 (2004)
[CrossRef]

Nordlander, P.

D. W. Brandl, N. A. Mirin, and P. Nordlander, "Plasmon modes of nanosphere trimers and quadrumers," J. Phys. Chem. B 110, 12302-12310 (2006).
[CrossRef] [PubMed]

Oubre, C.

P. Nodlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, "Plasmon hybridization in nanoparticle dimers," Nano. Lett. 4, 899-903 (2004)
[CrossRef]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Piechon, F.

A. Rudinger and F. Piechon, "On the multifractal spectrum of the Fibonacci chain," J. Phys. A: Math. Gen. 31, 155-164 (1998).
[CrossRef]

Prodan, E.

P. Nodlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, "Plasmon hybridization in nanoparticle dimers," Nano. Lett. 4, 899-903 (2004)
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Rechberger, W.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, "Optical properties of two interacting gold nanoparticles," Opt. Comm. 220, 137-141 (2003)
[CrossRef]

Reinhard, B.

Reinhard, B. M.

A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, "Photonic-plasmonic scattering resonances in determinsitic aperiodic structures," Nano. Lett. 8, 2423-2431 (2008).
[CrossRef] [PubMed]

Riklund, R.

L. Kroon, E. Lennholm, and R. Riklund, "Localization-delocalization in aperiodic systems," Phys. Rev. B 66, 094204 (2002).
[CrossRef]

M. Dulea, M. Johansson, and R. Riklund, "Localization of electrons and electromagnetic waves in a deterministic aperiodic system," Phys. Rev. B 45, 105-114 (1992).
[CrossRef]

Rubinacci, G.

L. Dal Negro, C. Forestiere, G. Miano, and G. Rubinacci, "Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays," Phys. Rev. B 79, 85404 (2009).
[CrossRef]

Rudinger, A.

A. Rudinger and F. Piechon, "On the multifractal spectrum of the Fibonacci chain," J. Phys. A: Math. Gen. 31, 155-164 (1998).
[CrossRef]

Schatz, G. C.

S. Zou and G. C. Schatz, "Theoretical studies of plasmon resonances in one dimensional nanoparticles chains: narrow lineshapes with tunable widths," Nanotech. 17, 2813-2820 (2006)
[CrossRef]

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, and G. C. Schatz, "Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
[CrossRef]

L. Zhao, K. L. Kelly, and G. C. Schatz, "The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonance wavelength and width," J. Phys. Chem. B 107, 7343-7350 (2003).
[CrossRef]

K. L. Kelly, E. Coronado, L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

Stockman, M. I.

P. Nodlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, "Plasmon hybridization in nanoparticle dimers," Nano. Lett. 4, 899-903 (2004)
[CrossRef]

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]

Van Duyne, R. P.

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, and G. C. Schatz, "Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
[CrossRef]

Wriedt, T.

T. Wriedt, "A review of elastic light scattering theories," Part. Part. Syst. Charact. 15, 67-74 (1998).
[CrossRef]

Xu, Y.-L.

Zhao, L.

L. Zhao, K. L. Kelly, and G. C. Schatz, "The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonance wavelength and width," J. Phys. Chem. B 107, 7343-7350 (2003).
[CrossRef]

K. L. Kelly, E. Coronado, L. Zhao, and G. C. Schatz, "The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment," J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, and G. C. Schatz, "Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays," J. Phys. Chem. B 107, 7337-7342 (2003).
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Figures (9)

Fig. 1.
Fig. 1.

Comparison of the field enhancement spectra calculated with CDA and GMT code (a-b) for a) 81-nanospheres periodic array and b) 80-nanospheres Fibonacci array. Comparison of the scattering efficiency calculated with the CDA and the T-matrix method (c-f) for Fibonacci array of 80 oblate nano-spheroids with (c) c/a=0.2, and (d) c/a=0.6, and for Fibonacci array of 80 prolate nanoparticles with (e) c/a=1.5 and (f) c/a=3.

Fig. 2.
Fig. 2.

Scattering efficiency versus wavelength for several values of the particles shape factor c/a and for (a) 1936 nanoparticles periodic array b) 1428 nanoparticles Fibonacci array (c) 2048 nanoparticles Rudin Shapiro array (d) 2016 nanoparticles Thue-Morse array.

Fig. 3.
Fig. 3.

Scattering efficiency versus wavelength for a 1428 nanoparticles Fibonacci structure, horizontal dimer and vertical dimer with oblate (c/a=0.2) (a), spherical (c/a=1) and prolate (c/a) shape. Electrical Field Distribution associated to the (d) blue-shifted (492nm) and (e) red-shifted (540nm) Fibonacci scattering peak of Fig. 3(b). In both cases the incoming field polarization is along the horizontal axis.

Fig. 4.
Fig. 4.

Maximum field enhancement versus wavelength for several values of the ratio c/a and for (a) 1936 nanoparticles periodic array (b) 1428 nanoparticles Fibonacci array (c) 2048 nanoparticles Rudin Shapiro array (d) 2016 nanoparticles Thue-Morse array.

Fig. 5.
Fig. 5.

Semilogarithmic plots of calculated field distribution at the wavelength of maximum field enhancement for 1428 nanoparticle Fibonacci array of (a) oblate ellipsoids (rc=25nm, c/a=0.2 wavelength=822nm) (b) sphere (rc=25nm wavelength=528nm) and (c) prolate spheroids (rc=25nm c/a=3 wavelength=474nm).

Fig. 6.
Fig. 6.

Calculated wavelength splitting of the scattering cross section of DA arrays as a function of the maximum field enhancement for Thue-Morse (red triangles), Rudin-Shapiro (green squares), and Fibonacci (black squares) arrays with different nanoparticles eccentricities.

Fig. 7.
Fig. 7.

Semilogarithmic plots of the CDFE function for arrays of various nanoparticle shapes (radius 25 nm) and different morphologies calculated at the wavelength of maximum field enhancement.

Fig. 8.
Fig. 8.

Participation ratio versus wavelength for several values of the ratio c/a and for (a) 1936 nanoparticles periodic array, (b) 1428 nanoparticles Fibonacci array, (c) 2048 nanoparticles Rudin Shapiro array, and (d) 2016 nanoparticles Thue-Morse array.

Fig. 9.
Fig. 9.

Wavelength positions of the maximum field enhancement (square), maximum extinction efficiency (triangle) and minimum participation ratio (circle) versus the particles shape factor c/a for (a) 1936 nanoparticles periodic array, (b) 1428 nanoparticles Fibonacci array, (c) 2048 nanoparticles Rudin Shapiro array, and (d) 2016 nanoparticles Thue-Morse array.

Equations (16)

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Eh=α01ph
α01=diag(α0,a1,α0,b1,α0,c1)=1V0 (Iεε0+Aε0)
At=abc2 0ds(s+t2)(s+a2)(s+b2)(s+c2) witht=a,b,c.
α1=F α01
fi=1(1)6πk3α0,t114πtk2α0,t1t=a,b,c
Eh,k=14πε0 [k2rh,kCh,kBh,k(1rh,k3ikrh,k2)] eikrh,k pk
Bh,k=3r̂h,kr̂h,kI
Ch,k=r̂h,kr̂h,kI
α1ph14πε0k=1,khN[k2rh,kCh,kBh,k(1rh,k3ikrh,k2)]eikrh,kpk=Eext(rh)
Thk={14πε0{k2rh,kBh,kCh,k(1rh,k3ikrh,k2)}hkα1h=k
k=1NThkpk=Eext(rh)h=1....N
E(i)(r)=n=1NDn[aniMn3(kr)+bniNn3(kr)]
E(s)(r)=n=1NDn[ansMn3(kr)+bnNn3(kr)]
[asbs]=T[aiai]
T=Q11(ks,ki)[Q31(ks,ki)]1 ,
PR=AE(x,y)2 AE(x,y)4 ,

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