## Manipulating the directivity of antennas with metamaterial

Optics Express, Vol. 16, Issue 15, pp. 10962-10967 (2008)

http://dx.doi.org/10.1364/OE.16.010962

Acrobat PDF (10182 KB)

### Abstract

In this paper we use spatially variant metamaterial substrate to manipulate the directivity of antennas. We show theoretically that by embedding a dipole at different locations inside this substrate, the emitted rays can be directed to different orientations as required. As a result, spatial multiplexing can be realized by carefully selecting proper parameters of this substrate. It can also be observed that the electric field received in this antenna system is enhanced when it is used for reception. Simulations based on finite element method are used to validate our theoretical analysis, showing a controllable high directive property. In order to simplify the physical realization process, we propose the reduced parameters for practical design and also study it with numerical simulations.

© 2008 Optical Society of America

## 1. Introduction

1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced
nonlinear phenomena,” IEEE Trans. Microwave Theory Technol. **47**, 2075 (1999). [CrossRef]

4. Z. Weng, Y. Jiao, G. Zhao, and F. Zhang, “Design and Experiment of One Dimension and Two Dimension Metamaterial Structures for Directive Emission,” Progress In Electromagnetics Research-PIER **70**, 199 (2007). [CrossRef]

7. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys.
Rev. Lett. **89**, 213902 (2002). [CrossRef] [PubMed]

8. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of
refraction,” Phys. Rev. E **70**, 046608 (2004). [CrossRef]

11. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780 (2006). [CrossRef] [PubMed]

12. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks
and the mirage effect,” Opt. Lett. **32**, 1069 (2007). [CrossRef] [PubMed]

7. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys.
Rev. Lett. **89**, 213902 (2002). [CrossRef] [PubMed]

8. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of
refraction,” Phys. Rev. E **70**, 046608 (2004). [CrossRef]

*ε*

_{eff}=0 and

*μ*

_{eff}=0 can be tuned to the desired specification to produce directional emission. With this metamaterial substrate, any obliquely incident wave will be totally reflected and the electric field will evanesce rapidly after passing the incidence surface. It is easily seen that no matter at which position the source is embedded, the radiated rays will always be confined to a small solid angle around the normal.

## 2. Field distribution in the metamaterial substrate

*ρ*′=

*α*

*x*,

*φ*′=

*y*/

*l*,

*z*′=

*z*, which maps a cylinder to an infinite slab, the relative permittivity and permeability tensors of the transformation medium are designed to be

*α*and

*l*are definite constants. To begin with, we first examine the field distribution in this metamaterial substrate with the thickness of

*d*. For TE polarization (the following idea is also applicable to TM polarization case), we write the electric field vector as

*E*̄=

*z*̂

*E*

_{z}(

*x*,

*y*).

*E*

_{z}(

*x*,

*y*) is obtained

*E*

_{z}=

*B*

_{k,l}(

*α*

*k*

_{0}

*x*)

*e*

^{ik,y}, where

*B*

_{n}(

*ξ*) is the

*n*–th order of Bessel function.

*E*

^{i}

_{z}=

*E*

_{0}

*e*

^{-ikxx+ikyy}(Here

*k*

^{2}

_{x}+

*k*

^{2}

_{y}=

*k*

^{2}

_{0}.) is incident upon the slab at the interface

*x*=

*d*, as shown in Fig. 1. By applying phase matching at the boundary, the reflected field (

*x*>

*d*), the transmitted field (

*x*< 0), and the field inside the slab (0 <

*x*<

*d*) can be obtained respectively:

*J*

_{n}(ζ) and

*N*

_{n}(ζ) representing the Bessel functions of the first, and the second kind respectively. Here

*R*,

*T*,

*A*, and

*B*are all unknown coefficients. By applying the continuities of

*E*

_{z}and

*H*

_{z}at the two interfaces

*x*=

*d*and

*x*= 0, we can get

*a*=

*J*

_{kyy}(

*α*

*k*

_{0}

*d*) and

*b*=

*J*′

_{kyl}(

*α*

*k*

_{0}

*d*)

*α*

*k*

_{0}

*d*/

*k*

_{x}

*l*are both real numbers. It is interesting to see that the total transmitted field is zero and seen at the interface

*x*=

*d*, all the incident waves will travel back to the free space. In addition, only the wave normally (

*k*

_{y}= 0) incident upon the substrate slab can result in non-zero electric field at the left interface (

*E*

^{Slab}

_{z}=

*AE*

_{0}

*J*

_{0}(0) =

*AE*

_{0}at

*x*= 0). Any wave obliquely incident at the right interface of the slab (

*k*

_{y}≠ 0) will be bent back completely before it get to the left interface and cannot reach the left boundary (

*E*

^{Slab}

_{z}=

*AE*

_{0}

*J*

_{kyl}(0)

*e*

^{ikyy}= 0at

*x*= 0). Based on this interesting phenomenon, we can see that a dipole embedded at the left boundary

*x*= 0 of the substrate can only receive the normally incident wave. As a matter of fact, this can be easily understood by considering the dispersion relation of the metamaterial (

*k*

^{2}

_{x}+

*k*

^{2}

_{y}/

*μ*

_{x}=

*k*

^{2}

_{0}for TE wave). From the right boundary

*x*=

*d*to the left boundary

*x*= 0,

*μ*

_{x}varies spatially from

*μ*

_{x}=

*d*/

*l*to

*μ*

_{x}= 0. Consequently, for all the oblique incidences,

*k*

_{x}should vary from real to imaginary since

*k*

_{y}is kept a non-zero constant. And the switching point depends on the y component of the wavenumber vector, which is determined by the incident angle. As an example, we plot the

*E*

_{z}distributions in a metamaterial slab with TE polarized incidences of three different angles (0°, 8°, and 30°) analytically in Fig. 2. In the case of normal incidence, the wave can reach the left boundary, as shown in Fig. 2(a). As the incident angle increases, the depth that the wave can penetrate the slab decreases, which can be observed in Fig. 2(b), and (c).

*a*=

*J*

_{0}(

*α*

*k*

_{0}

*d*) and

*b*=

*J*′

_{0}(

*α*

*k*

_{0}

*d*)α

*d*/

*l*. Obviously the enhancement can be controlled by the three parameters α,

*l*, and

*d*. Since the thickness

*d*is always fixed in a practical realization, we can adjust α and

*l*to modify the field distributions inside the slab and hence achieve the required properties. Fig. 2(d) depicts the

*E*

_{z}distribution inside the 0.05 m thick metamaterial slab corresponding to three incident angles (0°, 8°, and 30°). Different with traditional homogeneous metamaterial with zero refractive index [7-8

7. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys.
Rev. Lett. **89**, 213902 (2002). [CrossRef] [PubMed]

*x*= 0, as expected. As the incident angle increases to a large enough one (e.g., 30°), the electric field near the incident surface will tend to attenuate, which is more close to the situation in a traditional zero index metamaterial. In Fig. 2(d), we carefully selected the parameters (α and

*l*) so that the

*E*

_{z}peak of one incidence (0°) is approximately at the nodal point of the other (8°), which can be utilized in the wave multiplexing technique. Here if we place two dipoles in different positions (corresponding to the peak locations of two incidences) in the substrate, waves emitted from the two different directions can be well received at the same time.

## 3. Numerical simulation

*E*

_{z}distribution and when the metamaterial substrate is fed by z-polarized line source embedded at the different positions in the slab while Fig. 4(c) shows normalized power densities in the far field region with the corresponding two different locations of sources. As can be observed in the figures, when the source is at the left boundary, a beam with high directivity to the normal is produced. But as we move the line source along the normal, the radiation pattern changes. When the source is 0.025m from the left boundary, two narrow beams in ±8° directions are produced, which also confirms the results in Fig. 3. The black solid line corresponds to the case when the source embedded at the left boundary, showing a half-power beamwidth of about 4.8° with a 40 dB suppression of side-lobes. The red dashed line shows power peaks in two symmetric directions, and the suppression of side-lobes is about 30 dB.

## 4. Simplified approach for realization

*ε*are all spatially varied (so as

*μ*) while the

*y*component is infinite at the origin. This increases the difficulty in fabrication and hence we need to figure out ways of simplifying the parameters to make it possible to realize. When the line source is TE polarized along z direction, only

*ε*

_{z},

*μ*

_{x}and

*μ*

_{y}in Eqs. (1) enter into Maxwell’s equations. Moreover, the dispersion properties and wave trajectory in the slab remain the same, as long as the value

*ε*

_{z}

*μ*

_{x}and

*ε*

_{z}

*μ*

_{y}are kept constant. This gives the ability to choose one of the three constitutive parameters arbitrarily to achieve some favorable condition [14

14. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of
electromagnetic cloaking structures,” Phys. Rev. E **74**, 036621 (2006). [CrossRef]

*E*

^{i}

_{z}=

*E*

_{0}

*e*

^{-ik0x}, the fields are still distributed as standing wave inside the slab, but the transmitted coefficient is non-zero, indicating that the wave can transmit through the slab. When the wave is obliquely incident with electric field

*E*

^{i}

_{z}=

*E*

_{0}

*e*

^{-ikxx+ikyy}, the calculation shows that the transmitted fields are exactly equal to zero and penetration depth decreases as the incident angle increases, which is similar to the ideal case, but the fields distributed in the slab will no longer be enhanced.

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced
nonlinear phenomena,” IEEE Trans. Microwave Theory Technol. |

2. | D. R. Smith, Willie J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with
Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. |

3. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

4. | Z. Weng, Y. Jiao, G. Zhao, and F. Zhang, “Design and Experiment of One Dimension and Two Dimension Metamaterial Structures for Directive Emission,” Progress In Electromagnetics Research-PIER |

5. | N. Engheta and R. W. Ziolkowski, “A positive future for double negative metamaterials,” IEEE Microwave
Theory Tech. |

6. | A. Yu, F. Yang, and A. Z. Elsherbeni,, “A Dual Band Circularly Polarized Ring Antenna Based on Composite
Right and Left Handed Metamaterials,” Progress In Electromagnetics Research-PIER |

7. | S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys.
Rev. Lett. |

8. | R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of
refraction,” Phys. Rev. E |

9. | J. Zhang, Y. Luo, S. Xi, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate
transformation,” Progress in Electromagnetics Research-PIER |

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

11. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

12. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks
and the mirage effect,” Opt. Lett. |

13. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation
media,” Opt. Express |

14. | S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of
electromagnetic cloaking structures,” Phys. Rev. E |

15. | H. Chen, B-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,”
Phys. Rev. Lett. |

16. | R. Weder, “A rigorous analysis of high-order electromagnetic invisibility cloaks,” J. Phys. A: Math. Theor. |

17. | Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express |

18. | A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. |

19. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Designs for optical cloaking with high-order transformations,” Opt. Express |

20. | R. Weder, “The Boundary Conditions for Electromagnetic Invisibility Cloaks,” arXiv 0801.3611, (2008). |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.4010) Other areas of optics : Microwaves

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 27, 2008

Revised Manuscript: June 25, 2008

Manuscript Accepted: June 26, 2008

Published: July 8, 2008

**Citation**

Jingjing Zhang, Yu Luo, Hongsheng Chen, and Bae-Ian Wu, "Manipulating the directivity of antennas with
metamaterial," Opt. Express **16**, 10962-10967 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-15-10962

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### References

- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Technol. 47, 2075 (1999). [CrossRef]
- D. R. Smith, WillieJ. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite Medium with Simultaneously Negative Permeability and Permittivity," Phys. Rev. Lett. 84, 4184 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001). [CrossRef] [PubMed]
- Z. Weng, Y. Jiao, G. Zhao, and F. Zhang, "Design and Experiment of One Dimension and Two Dimension Metamaterial Structures for Directive Emission," Progress In Electromagnetics Research-PIER 70, 199 (2007). [CrossRef]
- N. Engheta and R. W. Ziolkowski, "A positive future for double negative metamaterials," IEEE Microwave Theory Tech. 53, 1535 (2005). [CrossRef]
- A. Yu, F. Yang, and A. Z. Elsherbeni, "A Dual Band Circularly Polarized Ring Antenna Based on Composite Right and Left Handed Metamaterials," Progress In Electromagnetics Research-PIER 77, 285(2007).
- S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002). [CrossRef] [PubMed]
- R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608 (2004). [CrossRef]
- J. Zhang, Y. Luo, S. Xi, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, "Directive emission obtained by coordinate transformation," Progress in Electromagnetics Research-PIER 81, 437 (2008). [CrossRef]
- R. W. Ziolkowski and A. Erentok, "Metamaterial-based efficient electrically small antennas," IEEE Trans. Antennas Propag. 54, 2113, July 2006. [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780 (2006). [CrossRef] [PubMed]
- F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069 (2007). [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006). [CrossRef] [PubMed]
- S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006). [CrossRef]
- H. Chen, B-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic Wave Interactions with a Metamaterial Cloak," Phys. Rev. Lett. 99, 063903 (2007). [CrossRef] [PubMed]
- R. Weder, "A rigorous analysis of high-order electromagnetic invisibility cloaks," J. Phys. A: Math. Theor. 41, 065207 (2008). [CrossRef]
- Y. Huang, Y. Feng, and T. Jiang, "Electromagnetic cloaking by layered structure of homogeneous isotropic materials," Opt. Express 15, 11133 (2007). [CrossRef] [PubMed]
- A. V. Kildishev and E. E. Narimanov, "Impedance-matched hyperlens," Opt. Lett. 32, 3432 (2007). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express 16, 5444 (2008). [CrossRef] [PubMed]
- R. Weder, "The Boundary Conditions for Electromagnetic Invisibility Cloaks," arXiv 0801.3611 (2008).

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