Abstract

In this paper, we investigate the plasmonic near-field localization and the far-field scattering properties of non-periodic arrays of Ag nanoparticles generated by prime number sequences in two spatial dimensions. In particular, we demonstrate that the engineering of plasmonic arrays with large spectral flatness and particle density is necessary to achieve a high density of electromagnetic hot spots over a broader frequency range and a larger area compared to strongly coupled periodic and quasi-periodic structures. Finally, we study the far-field scattering properties of prime number arrays illuminated by plane waves and we discuss their angular scattering properties. The study of prime number arrays of metal nanoparticles provides a novel strategy to achieve broadband enhancement and localization of plasmonic fields for the engineering of nanoscale nano-antenna arrays and active plasmonic structures.

© 2009 OSA

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References

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  1. 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(6), 064013 (2008).
    [CrossRef]
  2. A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8(8), 2423–2431 (2008).
    [CrossRef] [PubMed]
  3. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544 (2008).
    [CrossRef] [PubMed]
  4. A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic Aperiodic Arrays of Metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17(5), 3741–3753 (2009).
    [CrossRef] [PubMed]
  5. S. G. Williams, Symbolic dynamics and its applications, (American Mathematical Society, 2004).
  6. P. Prusinkiewicz, and A. Lindenmayer, The algorithmic beauty of plants, (Springer Verlag, 1990).
  7. J. Mishra, and S. N. Mishra, L-System Fractals, (Elsevier, 2007).
  8. M. R. Schroeder, Number theory in science and communication, (Springer Verlag, 1985).
  9. G. H. Hardy, and E. M. Wright, An introduction to the theory of numbers, (Oxford University Press, 2008).
  10. S. J. Miller, and R. Takloo-Bighash, An Invitation to Modern Number Theory, (Princeton University Press, 2006).
  11. M. Queffelec, “Substitution dynamical systems-spectral analysis,” (Springer, 1987).
  12. E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69(2), 397–441 (2006).
    [CrossRef]
  13. M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B Condens. Matter 45(1), 105–114 (1992).
    [CrossRef] [PubMed]
  14. L. Dal Negro and N. N. Feng, “Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles,” Opt. Express 15(22), 14396 (2007).
    [CrossRef] [PubMed]
  15. C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays,” Phys. Rev. B Condens. Matter 79(8), 85404 (2009).
    [CrossRef]
  16. L. Kroon, E. Lennholm, and R. Riklund, “Localization-delocalization in aperiodic systems,” Phys. Rev. B Condens. Matter 66(9), 094204 (2002).
    [CrossRef]
  17. A. Gopinath, N. Lawrence, S. Boriskina, L. Dal Negro, “Enhancement of the 1.54μm Erbium emission in aperiodic plasmonic arrays” in preparation.
  18. C. Forestiere, G. Miano, S. V. Boriskina, and L. Dal Negro, “The role of nanoparticle shapes and deterministic aperiodicity for the design of nanoplasmonic arrays,” Opt. Express 17(12), 9648–9661 (2009).
    [CrossRef] [PubMed]
  19. C. Janot, Quasicrystals: A Primer, (Oxford University Press, 1997).
  20. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B Condens. Matter 6(12), 4370–4379 (1972).
    [CrossRef]
  21. T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007).
    [CrossRef] [PubMed]
  22. M. R. Schroeder, “A simple function and its Fourier transform,” Math. Intelligencer 4(3), 158–161 (1982).
    [CrossRef]
  23. M. L. Stein, S. M. Ulam, and M. B. Wells, “A visual display of some properties of the distribution of primes,” Am. Math. Mon. 71(5), 516 (1964).
    [CrossRef]
  24. J. D. Johnston, “Transform Coding of Audio Signals Using Perceptual Noise Criteria,” IEEE J. Sel. Areas Comm. 6(2), 314 (1988).
    [CrossRef]
  25. L. Novotny, and B. Hecht, Principles of Nano-Optics, (Cambridge University Press, 2006).
  26. C. Forestiere, A. Gopinath, G. Miano, L. Dal Negro, S. Boriskina, “Structural resonances in finite-size periodic plasmonic arrays,” in preparation.
  27. A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantennas,” Phys. Rev. B Condens. Matter 78(19), 195111 (2008).
    [CrossRef]
  28. L. Novotny, “Effective Wavelength Scaling for Optical Antennas,” Phys. Rev. Lett. 98(26), 266802 (2007).
    [CrossRef] [PubMed]
  29. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple scattering of light by particles: radiative transfer and coherent backscattering, (Cambridge University Press, 2006).
  30. J. D. Jackson, Classical Electrodynamics, (Wiley, 1998).

2009

2008

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantennas,” Phys. Rev. B Condens. Matter 78(19), 195111 (2008).
[CrossRef]

R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544 (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(6), 064013 (2008).
[CrossRef]

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

2007

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007).
[CrossRef] [PubMed]

L. Novotny, “Effective Wavelength Scaling for Optical Antennas,” Phys. Rev. Lett. 98(26), 266802 (2007).
[CrossRef] [PubMed]

L. Dal Negro and N. N. Feng, “Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles,” Opt. Express 15(22), 14396 (2007).
[CrossRef] [PubMed]

2006

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69(2), 397–441 (2006).
[CrossRef]

2002

L. Kroon, E. Lennholm, and R. Riklund, “Localization-delocalization in aperiodic systems,” Phys. Rev. B Condens. Matter 66(9), 094204 (2002).
[CrossRef]

1992

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B Condens. Matter 45(1), 105–114 (1992).
[CrossRef] [PubMed]

1988

J. D. Johnston, “Transform Coding of Audio Signals Using Perceptual Noise Criteria,” IEEE J. Sel. Areas Comm. 6(2), 314 (1988).
[CrossRef]

1982

M. R. Schroeder, “A simple function and its Fourier transform,” Math. Intelligencer 4(3), 158–161 (1982).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B Condens. Matter 6(12), 4370–4379 (1972).
[CrossRef]

1964

M. L. Stein, S. M. Ulam, and M. B. Wells, “A visual display of some properties of the distribution of primes,” Am. Math. Mon. 71(5), 516 (1964).
[CrossRef]

Agrawal, A.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007).
[CrossRef] [PubMed]

Alù, A.

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantennas,” Phys. Rev. B Condens. Matter 78(19), 195111 (2008).
[CrossRef]

Boriskina, S. V.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B Condens. Matter 6(12), 4370–4379 (1972).
[CrossRef]

Dal Negro, L.

C. Forestiere, G. Miano, S. V. Boriskina, and L. Dal Negro, “The role of nanoparticle shapes and deterministic aperiodicity for the design of nanoplasmonic arrays,” Opt. Express 17(12), 9648–9661 (2009).
[CrossRef] [PubMed]

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays,” Phys. Rev. B Condens. Matter 79(8), 85404 (2009).
[CrossRef]

A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic Aperiodic Arrays of Metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17(5), 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 deterministic aperiodic structures,” Nano Lett. 8(8), 2423–2431 (2008).
[CrossRef] [PubMed]

R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544 (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(6), 064013 (2008).
[CrossRef]

L. Dal Negro and N. N. Feng, “Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles,” Opt. Express 15(22), 14396 (2007).
[CrossRef] [PubMed]

Dallapiccola, R.

Dulea, M.

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B Condens. Matter 45(1), 105–114 (1992).
[CrossRef] [PubMed]

Engheta, N.

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantennas,” Phys. Rev. B Condens. Matter 78(19), 195111 (2008).
[CrossRef]

Feng, N. N.

A. Gopinath, S. V. Boriskina, N. N. Feng, B. M. Reinhard, and L. Dal Negro, “Photonic-plasmonic scattering resonances in deterministic aperiodic structures,” Nano Lett. 8(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(6), 064013 (2008).
[CrossRef]

L. Dal Negro and N. N. Feng, “Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles,” Opt. Express 15(22), 14396 (2007).
[CrossRef] [PubMed]

Forestiere, C.

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays,” Phys. Rev. B Condens. Matter 79(8), 85404 (2009).
[CrossRef]

C. Forestiere, G. Miano, S. V. Boriskina, and L. Dal Negro, “The role of nanoparticle shapes and deterministic aperiodicity for the design of nanoplasmonic arrays,” Opt. Express 17(12), 9648–9661 (2009).
[CrossRef] [PubMed]

Gopinath, A.

A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic Aperiodic Arrays of Metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17(5), 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 deterministic aperiodic structures,” Nano Lett. 8(8), 2423–2431 (2008).
[CrossRef] [PubMed]

R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544 (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(6), 064013 (2008).
[CrossRef]

Johansson, M.

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B Condens. Matter 45(1), 105–114 (1992).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B Condens. Matter 6(12), 4370–4379 (1972).
[CrossRef]

Johnston, J. D.

J. D. Johnston, “Transform Coding of Audio Signals Using Perceptual Noise Criteria,” IEEE J. Sel. Areas Comm. 6(2), 314 (1988).
[CrossRef]

Kroon, L.

L. Kroon, E. Lennholm, and R. Riklund, “Localization-delocalization in aperiodic systems,” Phys. Rev. B Condens. Matter 66(9), 094204 (2002).
[CrossRef]

Lennholm, E.

L. Kroon, E. Lennholm, and R. Riklund, “Localization-delocalization in aperiodic systems,” Phys. Rev. B Condens. Matter 66(9), 094204 (2002).
[CrossRef]

Maciá, E.

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69(2), 397–441 (2006).
[CrossRef]

Matsui, T.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007).
[CrossRef] [PubMed]

Miano, G.

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays,” Phys. Rev. B Condens. Matter 79(8), 85404 (2009).
[CrossRef]

C. Forestiere, G. Miano, S. V. Boriskina, and L. Dal Negro, “The role of nanoparticle shapes and deterministic aperiodicity for the design of nanoplasmonic arrays,” Opt. Express 17(12), 9648–9661 (2009).
[CrossRef] [PubMed]

Nahata, A.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny, “Effective Wavelength Scaling for Optical Antennas,” Phys. Rev. Lett. 98(26), 266802 (2007).
[CrossRef] [PubMed]

Reinhard, B. M.

A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic Aperiodic Arrays of Metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17(5), 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 deterministic aperiodic structures,” Nano Lett. 8(8), 2423–2431 (2008).
[CrossRef] [PubMed]

Riklund, R.

L. Kroon, E. Lennholm, and R. Riklund, “Localization-delocalization in aperiodic systems,” Phys. Rev. B Condens. Matter 66(9), 094204 (2002).
[CrossRef]

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B Condens. Matter 45(1), 105–114 (1992).
[CrossRef] [PubMed]

Rubinacci, G.

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays,” Phys. Rev. B Condens. Matter 79(8), 85404 (2009).
[CrossRef]

Schroeder, M. R.

M. R. Schroeder, “A simple function and its Fourier transform,” Math. Intelligencer 4(3), 158–161 (1982).
[CrossRef]

Stein, M. L.

M. L. Stein, S. M. Ulam, and M. B. Wells, “A visual display of some properties of the distribution of primes,” Am. Math. Mon. 71(5), 516 (1964).
[CrossRef]

Stellacci, F.

Ulam, S. M.

M. L. Stein, S. M. Ulam, and M. B. Wells, “A visual display of some properties of the distribution of primes,” Am. Math. Mon. 71(5), 516 (1964).
[CrossRef]

Vardeny, Z. V.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007).
[CrossRef] [PubMed]

Wells, M. B.

M. L. Stein, S. M. Ulam, and M. B. Wells, “A visual display of some properties of the distribution of primes,” Am. Math. Mon. 71(5), 516 (1964).
[CrossRef]

Am. Math. Mon.

M. L. Stein, S. M. Ulam, and M. B. Wells, “A visual display of some properties of the distribution of primes,” Am. Math. Mon. 71(5), 516 (1964).
[CrossRef]

IEEE J. Sel. Areas Comm.

J. D. Johnston, “Transform Coding of Audio Signals Using Perceptual Noise Criteria,” IEEE J. Sel. Areas Comm. 6(2), 314 (1988).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

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(6), 064013 (2008).
[CrossRef]

Math. Intelligencer

M. R. Schroeder, “A simple function and its Fourier transform,” Math. Intelligencer 4(3), 158–161 (1982).
[CrossRef]

Nano Lett.

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

Nature

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. B Condens. Matter

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantennas,” Phys. Rev. B Condens. Matter 78(19), 195111 (2008).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B Condens. Matter 6(12), 4370–4379 (1972).
[CrossRef]

M. Dulea, M. Johansson, and R. Riklund, “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Phys. Rev. B Condens. Matter 45(1), 105–114 (1992).
[CrossRef] [PubMed]

C. Forestiere, G. Miano, G. Rubinacci, and L. Dal Negro, “Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays,” Phys. Rev. B Condens. Matter 79(8), 85404 (2009).
[CrossRef]

L. Kroon, E. Lennholm, and R. Riklund, “Localization-delocalization in aperiodic systems,” Phys. Rev. B Condens. Matter 66(9), 094204 (2002).
[CrossRef]

Phys. Rev. Lett.

L. Novotny, “Effective Wavelength Scaling for Optical Antennas,” Phys. Rev. Lett. 98(26), 266802 (2007).
[CrossRef] [PubMed]

Rep. Prog. Phys.

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69(2), 397–441 (2006).
[CrossRef]

Other

L. Novotny, and B. Hecht, Principles of Nano-Optics, (Cambridge University Press, 2006).

C. Forestiere, A. Gopinath, G. Miano, L. Dal Negro, S. Boriskina, “Structural resonances in finite-size periodic plasmonic arrays,” in preparation.

A. Gopinath, N. Lawrence, S. Boriskina, L. Dal Negro, “Enhancement of the 1.54μm Erbium emission in aperiodic plasmonic arrays” in preparation.

C. Janot, Quasicrystals: A Primer, (Oxford University Press, 1997).

S. G. Williams, Symbolic dynamics and its applications, (American Mathematical Society, 2004).

P. Prusinkiewicz, and A. Lindenmayer, The algorithmic beauty of plants, (Springer Verlag, 1990).

J. Mishra, and S. N. Mishra, L-System Fractals, (Elsevier, 2007).

M. R. Schroeder, Number theory in science and communication, (Springer Verlag, 1985).

G. H. Hardy, and E. M. Wright, An introduction to the theory of numbers, (Oxford University Press, 2008).

S. J. Miller, and R. Takloo-Bighash, An Invitation to Modern Number Theory, (Princeton University Press, 2006).

M. Queffelec, “Substitution dynamical systems-spectral analysis,” (Springer, 1987).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple scattering of light by particles: radiative transfer and coherent backscattering, (Cambridge University Press, 2006).

J. D. Jackson, Classical Electrodynamics, (Wiley, 1998).

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Figures (11)

Fig. 1
Fig. 1

Periodic array and its structure in the reciprocal space (logarithmic scale), where L = 3.75 and ∆ is the center to center inter-particle distance, namely ∆ = dmin+2r. The reciprocal space is obtained by the discrete Fourier transform (DFT). The width of the central peak in the reciprocal space is inversely proportional to the array’s dimension.

Fig. 2
Fig. 2

Aperiodic arrays (a-c) and their corresponding structures in the reciprocal space (logarithmic scale) (d-f): (a) and (d) Coprime La=4.9µm, (b) and (e) Prime Lb=10.5µm, (c) and (f) Ulam Lc=11µm. The dimensions of the reciprocal space are the same as in the periodic case (Fig. 1(b)) since the minimum center to center inter-particle distance (∆) is equal to the periodic case.

Fig. 3
Fig. 3

(a) Maximum field enhancement versus the wavelength for an isolated particle, and for Periodic, Coprime, Prime, and Ulam arrays. The arrays are exited by a circularly polarized plane wave at normal incidence. (b) Values of maximum filed enhancement versus the spectral flatness (SF) – filling fraction (FF) product for the different arrays indicated in the figure.

Fig. 4
Fig. 4

Field enhancement spatial distribution (logarithmic scale) on the array’s plane at the frequencies of the maximum field enhancement: (a) periodic at 580nm (b) Coprime at 462nm (c) Prime at 439 nm (d) Ulam at 450 nm. The arrays are exited by a circularly polarized plane wave at normal incidence.

Fig. 5
Fig. 5

The color-maps show the cumulative distributions of field enhancement (CDFE) (logarithmic scale) versus wavelength (x-axis) and field-enhancement (y-axis) for (a) Periodic (b) Coprime (c) Prime and (d) Ulam arrays. The arrays are excited by a circularly polarized plane wave at normal incidence.

Fig. 6
Fig. 6

Reciprocal lattice (logarithmic scale) (a-d) compared to the scattering maps (logarithmic scale) for the forward scattering hemisphere (e-h) for: (a) and (e) Periodic, (b) and (f) Coprime, (c) and (g) Prime, (d) and (h) Ulam arrays. The arrays are excited by a circularly polarized plane wave at normal incidence and at the wavelength of the maximum of scattering efficiency, i.e. (a) 389.5nm, (b) 462nm, (c) 439nm, (d) 450.5nm. The scattering angles can be calculated from the horizontal and vertical axes values as cos2 θ = max{0,1-x2 -y 2}, φ = tan−1 (y/x).

Fig. 7
Fig. 7

Each scattered point represents the couple (magnitude, phase) of the projections of the dipolar moments along the polarization direction (x-axis) for Periodic (a), Coprime (b), Prime Numbers(c), and Ulam (d) arrays. These structures are excited by a x-polarized plane wave, at normal incidence, propagating at the wavelength of the maximum scattering efficiency, i.e. (a) 580nm, (b) 462nm, (c) 439nm, (d) 450nm. The magnitude is normalized to the maximum in each case (i.e. (a) 2.5·10−32, (b) 8.6·10−32, (c) 1.0·10−31, and (d) 5.6·10−32).

Fig. 8
Fig. 8

Each scattered point represents the couple (magnitude, phase) of the projections of the dipolar moments along the vertical array’s axis (y-axis) for (a) Periodic, (b) Coprime, (c) Prime Numbers, and (d) Ulam arrays. These structures are excited by a horizontal-polarized plane wave, at normal incidence, propagating at the wavelength of the maximum scattering efficiency, i.e. (a) 580nm, (b) 462nm, (c) 439nm, (d) 450nm. The magnitude is normalized to the maximum in each case (i.e. (a) 6.79·10−33, (b) 4.6·10−32, (c) 4.8·10−32, and (d) 1.25·10−32) .

Fig. 9
Fig. 9

Magnitude, in the Fraunhofer zone, of the scattered E-field in a plane orthogonal to the array and including the lattice’s vertical axis, for different geometries, excited by a plane wave, propagating at the wavelength of the scattering efficiency‘s maximum: (a) Periodic at 580nm (b) Coprime at 462nm (c) Prime at 439 nm (d) Ulam at 450 nm. The incoming plane wave is at normal incidence and is circularly polarized. The plots are in dB scale.

Fig. 11
Fig. 11

Magnitude, in the Fraunhofer zone, of the scattered E-field in the array’s plane, for different lattices excited by a plane wave at the frequencies of the scattering efficiency‘s maxima: (a) Periodic at 389.5nm, (b) Coprime at 462nm, (c) Prime at 439nm (d), and Ulam at 450.5nm. The incoming plane wave has a wave vector orthogonal to the array’s plane and is circularly polarized. The plots are in dB scale.

Fig. 10
Fig. 10

Magnitude, in the Fraunhofer zone, of the scattered E-field in a plane orthogonal to the array and including the lattice’s vertical axis, for different geometries, excited by a plane wave, propagating at the wavelength of the scattering efficiency‘s maximum: (a) Periodic at 580nm (b) Coprime at 462nm (c) Prime at 439 nm (d) Ulam at 450 nm. The incoming plane wave is at 45° incidence and is circularly polarized. The plots are in dB scale.

Tables (2)

Tables Icon

Table 1 – Geometric parameters describing the main characteristics of investigated arrays.

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Table 2 – The directional parameter δ and the angular width at half maximum Δθ are reported for Periodic, Coprime, Prime and Ulam spiral when the array is excited by a circular polarized plane wave at normal incidence, propagating at the wavelength of the maximum scattering efficiency.

Equations (5)

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S F = n = 0 N 1 | D F T { s ( n ) } | N ( n = 0 N 1 | D F T { s ( n ) } | N )
ρ 1 L 2 2 n 2 d x ' ln x '
E ( r ) 1 4 π ε 0 k 2 exp ( j k r ) r A i = 1 N exp ( j k r ^ r ^ i ) p i
φ ( x ) : R 3 { ¥ i f x Î { r i i = 1 , 2 , ... N } 0 o t h e r w i s e = i = 1 N δ ( x - r i )
F ( k r ^ ) 2 = c o n s t k 2 | Φ ( k r ^ ) |

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