## A numerical investigation of sub-wavelength resonances in polygonal metamaterial cylinders

Optics Express, Vol. 17, Issue 18, pp. 16059-16072 (2009)

http://dx.doi.org/10.1364/OE.17.016059

Acrobat PDF (810 KB)

### 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.

© 2009 OSA

## 1. Introduction

*e.g*., [1–4

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]

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]

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]

*e.g*., large radiated power for constant ELC.

*ω*being the angular frequency and

*t*being the time, is assumed and suppressed.

## 2. Configuration and theoretical background

### 2.1 Configuration

*n*sides and a circumscribed circle of radius

*n*sides and a circumscribed circle of radius

*n*= 8-sided polygonal cylinders, this work also investigates 48-, 24-, 12-, and 4-sided polygonal cylinders. Other non-circular configurations could obviously have been employed in the analysis; however, these polygonal cylinder configurations have been selected since they provide a gradual degradation of the perfect circular cylinder.

*z*- axis coinciding with the common axis of the cylinders. The coordinates of the observation point are

### 2.2 Methods of analysis

*e.g*., [16, Ch. 11]. Below, only the main points of this solution procedure are included while the details and the full analytical solution can be found in [10

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]

*i.e*., the scattered field in the region containing the ELC and the total fields in the remaining regions, are likewise expanded in terms of cylindrical wave functions. The expansions of the incident and unknown fields represent multipole expansions of the fields. The unknown fields contain a set of unknown expansion coefficients:

*n*designates the mode number in the multipole expansion,

*i.e*.,

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]

*a*, current

*I*, and axis at

_{e}*h.*By image theory [16, Ch. 7], these plates model the infinite MTM cylinder and ELC. Between the perfectly conducting plates, uniform perfect matching layers which model free-space radiation, have thickness

*d*, circumscribe a square of side length

*w*, and have their corners and edges joined, are inserted. A graphical illustration of the HFSS model of the PC configuration is presented in Fig. 2 , where an 8-sided PC configuration is shown, clearly illustrating the details of the employed model. Additional details as well as the values of specific parameters for the HFSS model of the PC structures are given in Section 3.

### 2.3 Derived quantities and resonance condition

**42**(6), RS6S15 (2007). [CrossRef]

*n*.The symbol

*i.e.*,

*i.e*., sub-wavelength structures, these coefficients become large and thus exhibit a resonance when the approximate conditionis satisfied [3,4

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]

**42**(6), RS6S15 (2007). [CrossRef]

**53**(4), 1535–1556 (2005). [CrossRef]

**42**(6), RS6S15 (2007). [CrossRef]

*i.e*., DNG and/or MNG materials must be incorporated to obtain a resonant sub-wavelength structure. It is noted that these sub-wavelength resonances are due to the presence of specific natural modes in the structure [10

**42**(6), RS6S15 (2007). [CrossRef]

**53**(4), 1535–1556 (2005). [CrossRef]

## 3. Numerical results

### 3.1 Initial remarks

**42**(6), RS6S15 (2007). [CrossRef]

*e.g*., to large values of radiated power (and by reciprocity also scattering cross-section), occurs for the CC configuration if region 1 is free space, region 2 is a MNG material with

*n*-sided polygonal MNG shell excited by a nearby ELC. The frequency of operation is

*w*.

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

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.

### 3.2 Radiation resistance

*n*. For large

*n*,

*i.e*., for

*n*decreases, and the PCs thus deviates considerably from the CC, the outer radius

*x*-axis,

*i.e*., for

### 3.3 Near-field distribution

*e.g.*, Fig. 5(h). Obviously, for the specific ELC location near-by the inner right corner of the 4-sided PC(1) structure, the field attains very high values in the region around the inner right and left corners of the 4-sided PC(1) configuration. Although not shown inhere, as the bend radius decreases, the field values in the said regions increases. As (some of) the rounded corners in the 4-sided PC configurations are located in the region of the high field values, see Fig. 5(h), and as the structures are electrically small, even small changes of the bend radius may lead to different values of the resonances (since the field amplitudes may thus change drastically) as well as different outer radii at which the resonance is attained for the 4-sided PC configuration. This explains the quantitative difference observed between the two 4-sided PCs for the ELC located near-by a corner of these structures.

### 3.4 Loss and Dispersion

*Γ*, respectively, are the magnetic plasma and collision frequencies. For the Lorentz model, the parameter

*n*-sided PCs,

## 4. Summary and conclusions

## Acknowledgements

## References and links

1. | G. V. Eleftheriades, and K. G. Balmain, eds., |

2. | C. Caloz, and T. Itoh, eds., |

3. | N. Engheta, and R. W. Ziolkowski, eds., |

4. | N. Engheta and R. W. Ziolkowski, “A positive future for double negative materials,” IEEE Trans. Microw. Theory Tech. |

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,” |

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. |

7. | R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antenn. Propag. |

8. | H. Stuart and A. Pidwerbetsky, “Electrically small antenna elements using negative permittivity resonators,” IEEE Trans. Antenn. Propag. |

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. |

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. |

11. | H. Wallén, H. Kettunen, and A. Sihvola, “Electrostatic resonances of negative-permittivity interfaces, spheres, and wedges,” |

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 |

13. | S. Arslanagić, N. C. J. Clausen, R. R. Pedersen, and O. Breinbjerg, “Properties of sub-wavelength resonances in metamaterial cylinders,” |

14. | S. Arslanagić, and O. Breinbjerg, “Sub-wavelength resonances in polygonal metamaterial cylinders,” |

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

16. | C. A. Balanis, |

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 |

18. | It is important to note that the delta energy, ∆ |

19. | For the 4-sided PC, the MNG shell in the non-dispersive model is described by permeability |

**OCIS Codes**

(290.5850) Scattering : Scattering, particles

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: July 6, 2009

Revised Manuscript: August 20, 2009

Manuscript Accepted: August 21, 2009

Published: August 26, 2009

**Citation**

Samel Arslanagić and Olav Breinbjerg, "A numerical investigation of sub-wavelength resonances in polygonal metamaterial cylinders," Opt. Express **17**, 16059-16072 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-16059

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### References

- 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).
- 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, “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.
- 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]
- 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]
- 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]
- 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).
- 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]
- 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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