## Rainbow-like radiation from an omni-directional source placed in a uniaxial metamaterial slab

Optics Express, Vol. 17, Issue 9, pp. 7068-7073 (2009)

http://dx.doi.org/10.1364/OE.17.007068

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### Abstract

In this paper, the radiation of an omni-directional line source placed in a uniaxial metamaterial slab is experimentally presented. The anisotropic slab made of metallic symmetrical rings with dispersive permeability is investigated both theoretically and experimentally. For low value of the permeability, a directive radiation at the broadside of the slab can be obtained. Due to the excitation of the leaky wave mode supported by this structure, the emitted electromagnetic wave transmits at a greater angle from the normal of the slab as the value of permeability increases along with the frequency. Thus a rainbow-like radiation will be formed since waves of different frequencies will deflect into different directions.

© 2009 Optical Society of America

## 1. Introduction

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ” Sov. Phys. Usp . **10**, 509–514 (1968). [CrossRef]

2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity” Phys. Rev. Lett . **84**, 4184–4187 (2000). [CrossRef] [PubMed]

3. J. B. Pendry, “Negative refraction makes a perfect lens” Phys. Rev. Lett . **85**, 3966–3969 (2000). [CrossRef] [PubMed]

4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies” Science . **314**, 977–980 (2006) [CrossRef] [PubMed]

7. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction” Phys. Rev. E . **70**, 046608-1-4 (2004) [CrossRef]

11. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission” Phys. Rev. Lett . **89**, 213902-1-4 (2002). [CrossRef] [PubMed]

## 2. Theoretical analysis

*h*extending itself infinitely in

*y*direction with a line source placed in the center along the

*x*axis, as shown in Fig. 1. Assume the metamaterial is characterized by constitutive parameters

*ε̿*=

*diag*[

*ε*;

_{x}*ε*;

_{y}*ε*] and

_{z}*μ̿*= [

*μ*;

_{x}*μ*;

_{y}*μ*], where

_{z}*μ*is assumed to be dispersive and obey the Lorentz model, while other components are assumed to be unit for simplicity. In such a configuration, the radiation of the line source inside the slab can be regarded as

_{z}*TE*-polarized with the electric field along the

*x*direction and the corresponding dispersion relation of the metamaterial slab is written as

*k*

_{y}^{2}/

*μ*+

_{z}ε_{x}*k*

_{z}^{2}/

*μ*=

_{y}ε_{x}*k*

_{0}

^{2}, where

*k*

_{0}is the wave number in free-space,

*k*and

_{y}*k*are the wave numbers along the

_{z}*y*and

*z*directions, respectively. Since

*μ*is assumed to obey Lorentz model, there is a frequency

_{z}*f*at which

_{0}*μ*=0. It can be seen that when 0≤

_{z}*μ*≪1, the k surface becomes a very flat ellipse with its minor axis along the

_{z}*y*direction, while the k surface of air is an isotropic circle with a unit radius, as shown in the inset Fig. 1. Therefore, after the phase matching at the boundary of the metamaterial slab, the omni-directional radiation from the line source inside the slab will be compressed into a narrow angle of 2

*θ*outside the slab, where

_{c}*θ*is defined by arctan (√

_{c}*μ*/

_{z}*μ*). When

_{y}*μ*approaches to zero,

_{z}*θ*also approaches to zero and therefore the outgoing wave is along the normal of the boundary, i.e., along the

_{c}*z*direction, to obtain a high directivity, which is just the case discussed in [11–13

11. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission” Phys. Rev. Lett . **89**, 213902-1-4 (2002). [CrossRef] [PubMed]

_{0},

*θ*increases with the increase of

_{c}*μ*, implying that the outgoing wave is no longer required to propagate perpendicularly to the surface, and intuitively, any refractive angle of the outgoing wave between -

_{z}*θ*and +

_{c}*θ*is permitted. However, if the so-called leaky wave mode is considered, we will find that the outgoing wave will actually select a specific refractive angle to propagate.

_{c}*μ*is no longer zero, the emitted wave from the line source in the slab is permitted to have a non-zero

_{z}*k*, therefore the wave could be guided in the slab along the y direction, and the outgoing wave can be regarded as a summation of the refractive waves when the guided wave reflects at the both boundaries of the slab again and again, as illustrated in the lower part of Fig. 1. For a given frequency and a thickness

_{y}*h*of the slab, a different

*k*will lead to a different phase difference

_{y}*∆φ*defined by the additional propagation distance 2

*l*between two adjacent reflections, shown in Fig. 1. It is interesting that when there is a specific

*k*denoted by

_{y}*k*, such that 2

_{ys}*l*equals to integral numbers of the propagating wavelength, yielding

*∆φ*=2

*mπ*,

*m*=0,1,2,…, all the reflected waves will have the same phase and we can observe an enhanced outgoing radiation power in free space along a specific direction denoted by the outgoing angle

*θ*=arcsin(

_{s}*k*/

_{ys}*k*

_{0}). For other (

*k*, the adjacent reflected waves are out of phase and will hence cancel out. So, for a frequency a little higher than

_{y}*f*

_{0}, the outgoing wave deflects from the normal while still keeps a high directivity.

*μ*increases at the same time, yielding a

_{z}*k*surface curve of ellipse with longer minor axis on

*y*direction. In such a circumstance,

*k*has to increase to keep

_{ys}*∆φ*still to be integral numbers of 2π, resulting in a larger

*θ*of the outgoing wave.

_{s}14. G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of Directive Radiation From a Line Source in a Metamaterial Slab With Low Permittivity” IEEE Trans. Antennas Propag . **54**, 1017–1030 (2006). [CrossRef]

*h*≠0, the principal mode is

*m*=1, where the outgoing angle increases monotonously as a function of the radiation frequency. Higher order modes for

*m*= 2,3,4… appear in the case radiation frequency is higher enough so that

*θ*has multiple values, leading to multiple refractive angles of the outgoing waves. However, we will at least find in a certain frequency band with only the principal mode of a rainbow-like radiation, in which the omni-directional radiation inside the slab will be turned into directive radiations in free space, and each direction corresponds to one “color” of the wave.

_{s}## 3. Simulation and experiment

19. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena” IEEE. Trans. Microwave Theory Tech . **47**, 2075–2084 (1999). [CrossRef]

*x*direction with a periodicity of 7 mm and 15 unit cells in the

*y*direction with a periodicity of 5 mm, respectively. Eight pieces of such sample are aligned along the

*z*direction with an interval of 8 mm to finally obtain a slab-like sample. In such a slab, for a proper EM wave incidence, a magnetic resonance can be induced by the resonant currents flowing along the metallic rings, yielding an effective negative permeability along the z direction (

*μ*) [19

_{z}19. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena” IEEE. Trans. Microwave Theory Tech . **47**, 2075–2084 (1999). [CrossRef]

*x*and

*y*directions, there are no magnetic resonances, therefore the value of

*μ*and

_{x}*μ*can be regarded as unities. Thus we get a tensor

_{y}*μ̿*= [

*μ*] which fits well with our previous assumptions. Utilizing the homogenization approach proposed in [20

_{x};μ_{y};μ_{z}20. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients” Phys. Rev. B . **65**, 195104-1-4 (2002). [CrossRef]

21. X. D. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials” Phys. Rev. E . **70**, 016608-1-4 (2004). [CrossRef]

*μ*is retrieved from simulation data, shown in Fig. 2(b). It is seen that

_{z}*μ*behaves negative over a frequency range from 9 GHz to 9.53 GHz and is equal to zero at 9.53 GHz, corresponding to the

_{z}*f*

_{0}discussed before. Starting from 9.53 GHz,

*μ*increases along with the frequency but will never be larger than unit, showing a Lorentz model like dispersive property.

_{z}*x*direction placed in the center of the slab. The simulation result at the frequency of 10.6 GHz is shown in Fig. 3(a), in which the propagation direction of the electric field obviously splits into two waves outside the metamaterial slab, and the corresponding far field radiation pattern has two peaks symmetrical to the normal of the surface. At different frequency of 11 GHz, the radiation patterns show different propagation direction as expected.

*μ*and the great loss existing in this band. At the frequency of 9.625 GHz, there is a strong radiation with high directivity along the normal direction, corresponding to the frequency of

_{z}*f*

_{0}. Afterwards, while still keeping a high directivity, the radiation splits into two beams with larger and symmetric angles as frequency increases until 11 GHz, where a higher order leaky wave mode begins to appear. So from 9.625 GHz to 11 GHz, there is only one principal mode exists, and we will see a pattern just like a rainbow in the image plane. In Fig. 4(c), four radiation patterns from Fig. 4(b) for four selected frequencies, i.e., 9.625 GHz, 10.225 GHz, 10.825 GHz, 11.425 GHz, are shown, which clearly show the spectrum spread effect. At higher frequencies of 11 GHz and upwards, although relatively weaker compared with the principal mode, a higher order mode with smaller radiation angle appears as shown in Fig. 4(b), which is also in accordance with our expectation.

## 4. Conclusion

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ” Sov. Phys. Usp . |

2. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity” Phys. Rev. Lett . |

3. | J. B. Pendry, “Negative refraction makes a perfect lens” Phys. Rev. Lett . |

4. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies” Science . |

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

6. | 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 . |

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

8. | F. L. Zhang, S. Potet, and J. Caobonell, “Negative-Zero-Positive Refractive Index in a Prism-Like Omega-Type Metamaterial” IEEE Trans. Microwave Theory Tech . |

9. | A. Alu, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of epsilon-near-zero-filled narrow channels” Phys. Rev. E . |

10. | B Edwards, A. Alu, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-nearzero metamaterial coupling and energy squeezing using a microwave waveguide” Phys. Rev. Lett . |

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

12. | Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, and J. T. Huangfu, “Directive emission based on anisotropic metamaterials” Phys. Rev. A . |

13. | B. I. Wu, W. Wang, and J. Pacheco et al, “A study of using metamaterial as antenna substrate to enhance gain” Progress in Electromagnetics Research, PIER |

14. | G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, “Analysis of Directive Radiation From a Line Source in a Metamaterial Slab With Low Permittivity” IEEE Trans. Antennas Propag . |

15. | P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G. Lovat, and S. Paulotto, “Effects of Leaky-Wave Propagation in Metamaterial Grounded Slabs Excited by a Dipole Source” IEEE Trans.
Microwave Theory Tech . |

16. | N. Guérin, S. Enoch, G. Tayeb, P. Sabouroux, P. Vincent, and H. Legay, “A Metallic Fabry-Perot Directive Antenna” IEEE Trans. Antennas Propag . |

17. | P. Burghignoli, G. Lovat, F. Capolino, D. R. Jackson, and D. R. Wilton, “Directive Leaky-Wave Radiation From a Dipole Source in a Wire-Medium Slab” IEEE Trans. Antennas Propag . |

18. | A. Alù, F. Bilotti, N. Engheta, and L. Vegni, “Subwavelength Planar Leaky-Wave Components With Metamaterial Bilayers” IEEE Trans. Antennas Propag . |

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

20. | D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients” Phys. Rev. B . |

21. | X. D. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials” Phys. Rev. E . |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(350.5610) Other areas of optics : Radiation

(160.3918) Materials : Metamaterials

**ToC Category:**

Materials

**History**

Original Manuscript: January 30, 2009

Revised Manuscript: April 10, 2009

Manuscript Accepted: April 11, 2009

Published: April 14, 2009

**Citation**

Tao Jiang, Yu Luo, Zhiyu Wang, Liang Peng, Jiangtao Huangfu, Wanzhao Cui, Wei Ma, Hongsheng Chen, and Lixin Ran, "Rainbow-like radiation from an omni-directional
source placed in a uniaxial metamaterial slab," Opt. Express **17**, 7068-7073 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7068

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### References

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?" Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity" Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
- J. B. Pendry, "Negative refraction makes a perfect lens" Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies" Science. 314, 977-980 (2006) [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media" Opt. Express. 14, 9794-9804 (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, 036621-1-5 (2006). [CrossRef]
- R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction" Phys. Rev. E. 70, 046608-1-4 (2004) [CrossRef]
- F. L. Zhang, S. Potet, and J. Caobonell, "Negative-Zero-Positive Refractive Index in a Prism-Like Omega-Type Metamaterial" IEEE Trans. Microwave Theory Tech. 56, 2566-2573 (2008). [CrossRef]
- A. Alu, M. G. Silveirinha, and N. Engheta, "Transmission-line analysis of epsilon-near-zero-filled narrow channels" Phys. Rev. E. 78, 016604-1-4 (2008) [CrossRef]
- B. Edwards, A. Alu, M. E. Young, M. Silveirinha, N. Engheta, "Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide" Phys. Rev. Lett. 100, 033903-1-4 (2008). [CrossRef] [PubMed]
- S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission" Phys. Rev. Lett. 89, 213902-1-4 (2002). [CrossRef] [PubMed]
- Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, and J. T. Huangfu, "Directive emission based on anisotropic metamaterials" Phys. Rev. A. 77, 053821-1-5 (2008). [CrossRef]
- B. I. Wu, W. Wang, J. Pacheco et al, "A study of using metamaterial as antenna substrate to enhance gain" Progress in Electromagnetics Research, PIER 51, 295-328 (2005). [CrossRef]
- G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, "Analysis of Directive Radiation From a Line Source in a Metamaterial Slab With Low Permittivity" IEEE Trans. Antennas Propag. 54, 1017-1030 (2006). [CrossRef]
- P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G. Lovat, and S. Paulotto, "Effects of Leaky-Wave Propagation in Metamaterial Grounded Slabs Excited by a Dipole Source" IEEE Trans. Microwave Theory Tech. 53, 32-44 (2005). [CrossRef]
- N. Guérin, S. Enoch, G. Tayeb, P. Sabouroux, P. Vincent, and H. Legay, "A Metallic Fabry-Perot Directive Antenna" IEEE Trans. Antennas Propag. 54, 220-224 (2006). [CrossRef]
- P. Burghignoli, G. Lovat, F. Capolino, D. R. Jackson, and D. R. Wilton, "Directive Leaky-Wave Radiation From a Dipole Source in a Wire-Medium Slab" IEEE Trans. Antennas Propag. 56, 1329-1339 (2008). [CrossRef]
- A. Alù, F. Bilotti, N. Engheta and L. Vegni, "Subwavelength Planar Leaky-Wave Components With Metamaterial Bilayers" IEEE Trans. Antennas Propag. 55, 882-891 (2007). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena" IEEE. Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
- D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients" Phys. Rev. B. 65, 195104-1-4 (2002). [CrossRef]
- X. D. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials" Phys. Rev. E. 70, 016608-1-4 (2004). [CrossRef]

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