Abstract

The sub-wavelength resonances, known to exist in metamaterial radiators and scatterers of circular cylindrical shape, are investigated with the aim of determining if these resonances also exist for polygonal cylinders and, if so, how they are affected by the shape of the polygon. To this end, a set of polygonal cylinders excited by a nearby electric line current is analyzed numerically and it is shown, through detailed analysis of the near-field distribution and radiation resistance, that these polygonal cylinders do indeed support sub-wavelength resonances similar to those of the circular cylinders. The dispersion and loss, inevitably present in realistic metamaterials, are modeled by the Drude and Lorentz dispersion models to study the bandwidth properties of the resonances.

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References

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  1. G. V. Eleftheriades, and K. G. Balmain, eds., Negative-refraction metamaterials – fundamental principles and applications (John Wiley & Sons, 2005).
  2. C. Caloz, and T. Itoh, eds., Electromagnetic metamaterials – Transmission Line Theory and Microwave Applications (John Wiley & Sons, 2006).
  3. N. Engheta, and R. W. Ziolkowski, eds., Metamaterials – physics and engineering explorations (John Wiley & Sons, 2006).
  4. N. Engheta and R. W. Ziolkowski, “A positive future for double negative materials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
    [CrossRef]
  5. A. Alù, and N. Engheta, “Resonances in sub-wavelength cylindrical structures made of pairs of double-negative and double-positive or epsilon-negative and mu-negative coaxial shells,” in Proceedings of the International Electromagnetics and Advanced Applications Conference, (Turin, Italy, 2003), pp. 435–438.
  6. A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and∕or double-positive metamaterial layers,” J. Appl. Phys. 97(9), 094310 (2005).
    [CrossRef]
  7. R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antenn. Propag. 54(7), 2113–2130 (2006).
    [CrossRef]
  8. H. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
    [CrossRef]
  9. S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation from concentric metamaterial spheres excited by an electric Hertzian dipole,” Radio Sci. 42(6), RS6S16 (2007).
    [CrossRef]
  10. S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation and scattering from concentric metamaterial cylinders excited by an electric line source,” Radio Sci. 42(6), RS6S15 (2007).
    [CrossRef]
  11. H. Wallén, H. Kettunen, and A. Sihvola, “Electrostatic resonances of negative-permittivity interfaces, spheres, and wedges,” in Proceedings of The First Intl. Congress on Advanced Electromagnetic Materials for Microwave and Optics, (Rome, Italy, 2007).
  12. H.-Y. She, L.-W. Li, O. J. F. Martin, and J. R. Mosig, “Surface polaritons of small coated cylinders illuminated by normal incident TM and TE plane waves,” Opt. Express 16(2), 1007–1019 (2008).
    [CrossRef] [PubMed]
  13. S. Arslanagić, N. C. J. Clausen, R. R. Pedersen, and O. Breinbjerg, “Properties of sub-wavelength resonances in metamaterial cylinders,” in Proceedings of NATO Advanced Research Workshop: Metamaterials for Secure Information and Communication Technologies, (Marrakesh, Morocco, 2008).
  14. S. Arslanagić, and O. Breinbjerg, “Sub-wavelength resonances in polygonal metamaterial cylinders,” in Proceedings of IEEE AP-S USNC/URSI National Radio Science Meeting, (San Diego, USA, 2008).
  15. ANSOFT, Version 10.1.3, Copyright (C), 1984–2006 Ansoft Corporation.
  16. C. A. Balanis, Advanced Engineering Electromagnetics (John Wiley & Sons, 1989).
  17. It should be noted that the initial HFSS model utilized radiation boundaries instead of the perfectly matched layers. However, such a model resulted in inconsistent results, in particular with varying side length w, despite the fact that the distance from the perfectly matched layers to the polygonal cylinders and the ELC was larger than λ0/4 as suggested by HFSS, and despite improved discretization along the radiation boundaries. This problem was alleviated by use of perfectly matched layers for which the default discretization options were sufficient to obtain consistent and convergent results.
  18. It is important to note that the delta energy, ∆E, which is the difference in the relative energy error from one adaptive solution to the next, and serves as a stopping criterion for the solution, was set to 0.01 in all cases. This value of ∆E was targeted and obtained in 3 consecutive adaptive solutions for the 48-, 24-, 12-, and 8-sided PCs, and in 2 consecutive adaptive solutions for the 4-sided PCs.
  19. For the 4-sided PC, the MNG shell in the non-dispersive model is described by permeability μ2=−4μ0 and a loss tangent of 0.001 for all frequencies.

2008

2007

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation from concentric metamaterial spheres excited by an electric Hertzian dipole,” Radio Sci. 42(6), RS6S16 (2007).
[CrossRef]

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation and scattering from concentric metamaterial cylinders excited by an electric line source,” Radio Sci. 42(6), RS6S15 (2007).
[CrossRef]

2006

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antenn. Propag. 54(7), 2113–2130 (2006).
[CrossRef]

H. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
[CrossRef]

2005

N. Engheta and R. W. Ziolkowski, “A positive future for double negative materials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
[CrossRef]

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and∕or double-positive metamaterial layers,” J. Appl. Phys. 97(9), 094310 (2005).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and∕or double-positive metamaterial layers,” J. Appl. Phys. 97(9), 094310 (2005).
[CrossRef]

Arslanagic, S.

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation from concentric metamaterial spheres excited by an electric Hertzian dipole,” Radio Sci. 42(6), RS6S16 (2007).
[CrossRef]

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation and scattering from concentric metamaterial cylinders excited by an electric line source,” Radio Sci. 42(6), RS6S15 (2007).
[CrossRef]

Breinbjerg, O.

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation from concentric metamaterial spheres excited by an electric Hertzian dipole,” Radio Sci. 42(6), RS6S16 (2007).
[CrossRef]

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation and scattering from concentric metamaterial cylinders excited by an electric line source,” Radio Sci. 42(6), RS6S15 (2007).
[CrossRef]

Engheta, N.

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and∕or double-positive metamaterial layers,” J. Appl. Phys. 97(9), 094310 (2005).
[CrossRef]

N. Engheta and R. W. Ziolkowski, “A positive future for double negative materials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
[CrossRef]

Erentok, A.

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antenn. Propag. 54(7), 2113–2130 (2006).
[CrossRef]

Li, L.-W.

Martin, O. J. F.

Mosig, J. R.

Pidwerbetsky, A.

H. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
[CrossRef]

She, H.-Y.

Stuart, H.

H. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
[CrossRef]

Ziolkowski, R. W.

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation from concentric metamaterial spheres excited by an electric Hertzian dipole,” Radio Sci. 42(6), RS6S16 (2007).
[CrossRef]

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation and scattering from concentric metamaterial cylinders excited by an electric line source,” Radio Sci. 42(6), RS6S15 (2007).
[CrossRef]

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antenn. Propag. 54(7), 2113–2130 (2006).
[CrossRef]

N. Engheta and R. W. Ziolkowski, “A positive future for double negative materials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
[CrossRef]

IEEE Trans. Antenn. Propag.

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antenn. Propag. 54(7), 2113–2130 (2006).
[CrossRef]

H. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. 54(6), 1644–1653 (2006).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

N. Engheta and R. W. Ziolkowski, “A positive future for double negative materials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
[CrossRef]

J. Appl. Phys.

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and∕or double-positive metamaterial layers,” J. Appl. Phys. 97(9), 094310 (2005).
[CrossRef]

Opt. Express

Radio Sci.

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation from concentric metamaterial spheres excited by an electric Hertzian dipole,” Radio Sci. 42(6), RS6S16 (2007).
[CrossRef]

S. Arslanagić, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation and scattering from concentric metamaterial cylinders excited by an electric line source,” Radio Sci. 42(6), RS6S15 (2007).
[CrossRef]

Other

H. Wallén, H. Kettunen, and A. Sihvola, “Electrostatic resonances of negative-permittivity interfaces, spheres, and wedges,” in Proceedings of The First Intl. Congress on Advanced Electromagnetic Materials for Microwave and Optics, (Rome, Italy, 2007).

A. Alù, and N. Engheta, “Resonances in sub-wavelength cylindrical structures made of pairs of double-negative and double-positive or epsilon-negative and mu-negative coaxial shells,” in Proceedings of the International Electromagnetics and Advanced Applications Conference, (Turin, Italy, 2003), pp. 435–438.

G. V. Eleftheriades, and K. G. Balmain, eds., Negative-refraction metamaterials – fundamental principles and applications (John Wiley & Sons, 2005).

C. Caloz, and T. Itoh, eds., Electromagnetic metamaterials – Transmission Line Theory and Microwave Applications (John Wiley & Sons, 2006).

N. Engheta, and R. W. Ziolkowski, eds., Metamaterials – physics and engineering explorations (John Wiley & Sons, 2006).

S. Arslanagić, N. C. J. Clausen, R. R. Pedersen, and O. Breinbjerg, “Properties of sub-wavelength resonances in metamaterial cylinders,” in Proceedings of NATO Advanced Research Workshop: Metamaterials for Secure Information and Communication Technologies, (Marrakesh, Morocco, 2008).

S. Arslanagić, and O. Breinbjerg, “Sub-wavelength resonances in polygonal metamaterial cylinders,” in Proceedings of IEEE AP-S USNC/URSI National Radio Science Meeting, (San Diego, USA, 2008).

ANSOFT, Version 10.1.3, Copyright (C), 1984–2006 Ansoft Corporation.

C. A. Balanis, Advanced Engineering Electromagnetics (John Wiley & Sons, 1989).

It should be noted that the initial HFSS model utilized radiation boundaries instead of the perfectly matched layers. However, such a model resulted in inconsistent results, in particular with varying side length w, despite the fact that the distance from the perfectly matched layers to the polygonal cylinders and the ELC was larger than λ0/4 as suggested by HFSS, and despite improved discretization along the radiation boundaries. This problem was alleviated by use of perfectly matched layers for which the default discretization options were sufficient to obtain consistent and convergent results.

It is important to note that the delta energy, ∆E, which is the difference in the relative energy error from one adaptive solution to the next, and serves as a stopping criterion for the solution, was set to 0.01 in all cases. This value of ∆E was targeted and obtained in 3 consecutive adaptive solutions for the 48-, 24-, 12-, and 8-sided PCs, and in 2 consecutive adaptive solutions for the 4-sided PCs.

For the 4-sided PC, the MNG shell in the non-dispersive model is described by permeability μ2=−4μ0 and a loss tangent of 0.001 for all frequencies.

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Figures (7)

Fig. 1
Fig. 1

The cross-section of the circular (a) and polygonal (b) configurations.

Fig. 2
Fig. 2

The HFSS model of the PC configuration. In the shown case, the PC configuration consists of an 8-sided polygonal cylinder structure. The figure is not to scale.

Fig. 3
Fig. 3

Radiation resistance Rrt [ Ω/mm ] as a function of the outer radius ρ2 for the circular (CC) and n-sided polygonal cylinder configurations (PCs), with n = 48, 24, 12, 8, and 4. For the 4-sided PC, the 4-sided(1) PC has the bend radius of the corner rounding ρf1=1/2 mm , whereas the 4-sided(2) PC has ρf2=ρf1/2=1/(22) mm. In all cases, the ELC is located in region 1 at (ρs,ϕs)=(5.75 mm, 0°) .

Fig. 4
Fig. 4

Radiation resistance Rrt [ Ω/mm ] as a function of the ELC location (ρs,ϕs=0°) for the circular (CCs) and n-sided polygonal configurations (PCs), with n = 48, 24, 12, 8, and 4. For the 4-sided PC, the 4-sided(1) PC has the bend radius of the corner rounding ρf1=1/2 mm , whereas the 4-sided(2) PC has ρf2=ρf1/2=1/(22) .

Fig. 5
Fig. 5

The magnitude of the electric near field for the resonant CC (c) and n-sided PCs, with n = 48 (d), n = 24 (e), n = 12 (f), n = 8 (g) and n = 4 (h). In all cases, the field is depicted in the xy-plane and the linear dynamic range is set to [150-170000] V/m, while the ELC is located at (ρs,ϕs)=(5.75 mm, 0°) , see also Figs. 4(a) and (b) for an indication of the ELC location. For the 4-sided PC, only the results for the 4-sided(1) PC are shown. A left-right arrow, indicating the size scale of the figure, and curves, representing the circular and polygonal surfaces of the MNG shell, are indicated in all cases.

Fig. 6
Fig. 6

The magnitude of the electric near field for the 8-sided resonant PC with the ELC located at (ρs,ϕs)= (5.55 mm, 5.625°) (d), (ρs,ϕs)= (5.32 mm, 11.25°) (e), and (ρs,ϕs)=(5.29 mm, 22.50°) (f), see also Figs. 6(a), (b) and (c) for an indication of the ELC location. The latter ELC location is symmetric with respect to a plane of the structure, whereas the former two are not. The field is depicted in the xy-plane and the linear dynamic range is set to [150-170000] V/m. A left-right arrow, indicating the size scale of the figure, and curves, representing the circular and polygonal surfaces of the MNG shell, are indicated in all cases.

Fig. 7
Fig. 7

Radiation resistance Rrt [ Ω/mm ] as a function of frequency f for the resonant CC and n-sided PCs, where n = 48, 24, 12, 8, and 4 in the cases where the MNG shell is modeled with no dispersion (a), Drude dispersion (b), and Lorentz dispersion (c). For the 4-sided PC, only the results for the 4-sided(1) PC are shown.

Tables (1)

Tables Icon

Table 1 The outer radius, ρ2 , and radiation resistance, Rrt , of the resonant CC and PCs. In all cases, the ELC is located in region 1 at (ρs,ϕs)=(5.75 mm, 0°) .

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Prt=(14η0Ie2)[k04n=0Nmaxτn2(3τn)|αn|2] ,
Pri=(14η0Ie2)[k02] .
ρ1ρ2(μ2+μ1)(μ2+μ0)(μ2μ1)(μ2μ0)2n ,              (n1)
Rrt=2PrtIe2 ,
Rri=2PriIe2 ,
μ2(ω)=μ0(1ωp2ω2jΓωωr2) ,

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