Abstract

A new class of bowtie antennas with Sierpiński fractal features is proposed for sensing molecular vibration modes in the near- to mid-infrared. These antennas offer a compact device footprint and an enhanced confinement factor compared to a bowtie antenna. Through extensive simulations, it is shown that these characteristics are related to the ability of this fractal geometry to become polarized. Simulation results demonstrate that these antennas may be tuned between 700nm ≤ λ ≤ 3.4µm and that electric field enhancement by 56 is possible at the center of the antenna gap.

© 2011 OSA

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    [CrossRef]

2010 (1)

2009 (2)

A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. 102(11), 113901 (2009).
[CrossRef] [PubMed]

J. S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of nanoantennas in an optical nanocircuit,” Nano Lett. 9(5), 1897–1902 (2009).
[CrossRef] [PubMed]

2008 (4)

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008).
[CrossRef] [PubMed]

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[CrossRef] [PubMed]

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008).
[CrossRef]

2007 (3)

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007).
[CrossRef] [PubMed]

L. Wang and X. Xu, “High transmission nanoscale bowtie-shaped aperture probe for near-field optical imaging,” Appl. Phys. Lett. 90(26), 261105 (2007).
[CrossRef]

2006 (1)

G. W. Hanson, “On the applicability of the surface impedance integral equation for optical and near infrared copper dipole antennas,” IEEE Trans. Antenn. Propag. 54(12), 3677–3685 (2006).
[CrossRef]

2005 (2)

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

J. Matteo and L. Hesselink, “Fractal extensions of near-field aperture shapes for enhanced transmission and resolution,” Opt. Express 13(2), 636–647 (2005).
[CrossRef] [PubMed]

2004 (2)

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, ““Gap-Dependent Optical Coupling of Single “Bowtie” Nanoantennas Resonant in the Visible,” Nano Lett. 4(5), 957–961 (2004).
[CrossRef]

2003 (1)

D. H. Werner and S. Ganguly, “An overview of fractal antenna engineering research,” IEEE Trans. Antennas Propag. 45, 38–57 (2003).

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Agrawal, A.

A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. 102(11), 113901 (2009).
[CrossRef] [PubMed]

Alù, A.

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008).
[CrossRef]

Aquistapace, F.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Bao, Y.-J.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Biagioni, P.

J. S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of nanoantennas in an optical nanocircuit,” Nano Lett. 9(5), 1897–1902 (2009).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Chusseau, L.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Coutaz, J.-L.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

de la Chapelle, M. L.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Duvillaret, L.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Engheta, N.

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2(5), 307–310 (2008).
[CrossRef]

A. Alù and N. Engheta, “Input impedance, nanocircuit loading, and radiation tuning of optical nanoantennas,” Phys. Rev. Lett. 101(4), 043901 (2008).
[CrossRef] [PubMed]

Feichtner, T.

J. S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of nanoantennas in an optical nanocircuit,” Nano Lett. 9(5), 1897–1902 (2009).
[CrossRef] [PubMed]

Fromm, D. P.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, ““Gap-Dependent Optical Coupling of Single “Bowtie” Nanoantennas Resonant in the Visible,” Nano Lett. 4(5), 957–961 (2004).
[CrossRef]

Ganguly, S.

D. H. Werner and S. Ganguly, “An overview of fractal antenna engineering research,” IEEE Trans. Antennas Propag. 45, 38–57 (2003).

Garet, F.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Gasquet, D.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Gaubert, C.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Giani, A.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Grimault, A.-S.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Hanson, G. W.

G. W. Hanson, “On the applicability of the surface impedance integral equation for optical and near infrared copper dipole antennas,” IEEE Trans. Antenn. Propag. 54(12), 3677–3685 (2006).
[CrossRef]

Hao, X.-P.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Hecht, B.

J. S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of nanoantennas in an optical nanocircuit,” Nano Lett. 9(5), 1897–1902 (2009).
[CrossRef] [PubMed]

Hesselink, L.

Hou, B.

B. Hou, X. Q. Liao, and J. K. S. Poon, “Resonant infrared transmission and effective medium response of subwavelength H-fractal apertures,” Opt. Express 18(4), 3946–3951 (2010).
[CrossRef] [PubMed]

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

Huang, J. S.

J. S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of nanoantennas in an optical nanocircuit,” Nano Lett. 9(5), 1897–1902 (2009).
[CrossRef] [PubMed]

Jin, J.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Kim, S.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kim, S.-W.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kim, Y.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kim, Y.-J.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kino, G.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, ““Gap-Dependent Optical Coupling of Single “Bowtie” Nanoantennas Resonant in the Visible,” Nano Lett. 4(5), 957–961 (2004).
[CrossRef]

Knap, W.

C. Gaubert, L. Chusseau, A. Giani, D. Gasquet, F. Garet, F. Aquistapace, L. Duvillaret, J.-L. Coutaz, and W. Knap, “THz fractal antennas for electrical and optical semiconductor emitters and receptors,” Phys. Status Solidi 1(6c), 1439–1444 (2004).
[CrossRef]

Li, Z.-F.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Liao, X. Q.

Liu, L.

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

Lu, X.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Macias, D.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Matsui, T.

A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. 102(11), 113901 (2009).
[CrossRef] [PubMed]

Matteo, J.

Ming, N.-B.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Miyamaru, F.

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

Moerner, W. E.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, ““Gap-Dependent Optical Coupling of Single “Bowtie” Nanoantennas Resonant in the Visible,” Nano Lett. 4(5), 957–961 (2004).
[CrossRef]

Nahata, A.

A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. 102(11), 113901 (2009).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98(26), 266802 (2007).
[CrossRef] [PubMed]

Park, I.-Y.

S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Peng, R.-W.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Poon, J. K. S.

Saito, Y.

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

Schuck, P. J.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, ““Gap-Dependent Optical Coupling of Single “Bowtie” Nanoantennas Resonant in the Visible,” Nano Lett. 4(5), 957–961 (2004).
[CrossRef]

Shao, J.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Sheng, P.

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

Si, J.-W.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Sundaramurthy, A.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, ““Gap-Dependent Optical Coupling of Single “Bowtie” Nanoantennas Resonant in the Visible,” Nano Lett. 4(5), 957–961 (2004).
[CrossRef]

Takeda, M. W.

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

Vardeny, Z. V.

A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. 102(11), 113901 (2009).
[CrossRef] [PubMed]

Vial, A.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Wang, L.

L. Wang and X. Xu, “High transmission nanoscale bowtie-shaped aperture probe for near-field optical imaging,” Appl. Phys. Lett. 90(26), 261105 (2007).
[CrossRef]

Wang, M.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Wen, W.

F. Miyamaru, Y. Saito, M. W. Takeda, B. Hou, L. Liu, W. Wen, and P. Sheng, “Teraherz electric response of fractal metamaterial structures,” Phys. Rev. B 77(4), 045124 (2008).
[CrossRef]

Werner, D. H.

D. H. Werner and S. Ganguly, “An overview of fractal antenna engineering research,” IEEE Trans. Antennas Propag. 45, 38–57 (2003).

Wu, Z.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Xu, X.

L. Wang and X. Xu, “High transmission nanoscale bowtie-shaped aperture probe for near-field optical imaging,” Appl. Phys. Lett. 90(26), 261105 (2007).
[CrossRef]

Zhang, B.

Y.-J. Bao, B. Zhang, Z. Wu, J.-W. Si, M. Wang, R.-W. Peng, X. Lu, J. Shao, Z.-F. Li, X.-P. Hao, and N.-B. Ming, “Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array,” Appl. Phys. Lett. 90(25), 251914 (2007).
[CrossRef]

Zhu, W.

A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. 102(11), 113901 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

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Figures (3)

Fig. 1
Fig. 1

Schematic representations of the simulations geometries: (a) bowtie antenna and (b) first-, (c) second- and (d) third-iteration Sierpiński fractal plasmonic antennas. The broadband electric field enhancement factor is shown in (e). The antenna color in (a)-(d) corresponds to the line color in (e).

Fig. 2
Fig. 2

Empirical relationships between antenna length and (a) resonant wavelength and (b) resonant enhancement factor for each antenna geometry under consideration. The color of antennas on the right corresponds to the line colors in (a) and (b).

Fig. 3
Fig. 3

Intensity distributions at resonance for (a) bowtie antenna and (b) first-, (c) second- and (d) third-iteration Sierpiński fractal plasmonic antennas.

Tables (1)

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Table 1 Geometric Properties of Antennas under Investigation

Equations (1)

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D = log ( N ) log ( S ) ,

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