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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 9 — Apr. 27, 2009
  • pp: 7068–7073
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Rainbow-like radiation from an omni-directional source placed in a uniaxial metamaterial slab

Tao Jiang, Yu Luo, Zhiyu Wang, Liang Peng, Jiangtao Huangfu, Wanzhao Cui, Wei Ma, Hongsheng Chen, and Lixin Ran  »View Author Affiliations


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

One of the most important experiments on light spectrum was made by Sir Isaac Newton. Over 300 years ago, he passed a beam of white light through a triangular prism made of a piece of glass, which allows light to spread out into a spectrum of six colors: red, orange, yellow, green, blue and violet. A most common example of a spectrum is the rainbow created in nature. In this work, by utilizing a kind of artificial material called metamaterial, we can also realize a rainbow-like radiation from an isotropic radiation of an ordinary line source.

Metamaterial is a sort of artificial structural composite possessing extraordinary electromagnetic (EM) properties, such as negative effective permittivity and/or permeability [1

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

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]

], which have excited the imaginations of physicists and engineers in the past few years. Most attractive applications of metamaterials are probably perfect lenses that are not limited to the usual half-wavelength diffraction limit [3

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

], and cloaks of invisibility that can bend rays around the object [4–6

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]

]. Recently, metamaterials with “near-zero” permittivity and/or permeability have also attracted a lot of interest [7–10

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]

]. With the near-zero constitutive parameters and therefore the near-zero refractive index of the metamaterial, the rays emitted by the source inside the metamaterial will be refracted at the air-metamaterial interface into a small angular range centered along the normal of the air-metamaterial surface. Therefore, an enhancement of the directivity compared with the original radiation can be expected [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]

]. In this paper, we show by theoretical analysis, simulation and experiments that when the radiating frequency is leaving the frequency of near-zero refractive index to the higher frequencies, a rainbow-like radiation will be obtained in a limited frequency band, in which an omni-directional radiated EM wave consisting of different “colors” will be refracted into different outgoing angles.

2. Theoretical analysis

Consider a metamaterial slab of thickness 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[εx;εy;εz] and μ̿ = [μx;μy;μz], where μz 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 TE-polarized with the electric field along the x direction and the corresponding dispersion relation of the metamaterial slab is written as ky 2/μzεx+kz 2/μyεx = k 0 2, where k 0 is the wave number in free-space, ky and kz are the wave numbers along the y and z directions, respectively. Since μz is assumed to obey Lorentz model, there is a frequency f0 at which μz=0. It can be seen that when 0≤μz≪1, the k surface becomes a very flat ellipse with its minor axis along the 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 2θc outside the slab, where θc is defined by arctan (√μz/μy). When μz approaches to zero, θc also approaches to zero and therefore the outgoing wave is along the normal of the boundary, i.e., along the 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]

].

However, if f>f0, θc increases with the increase of μz, 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 - θc and + θc 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.

When μz is no longer zero, the emitted wave from the line source in the slab is permitted to have a non-zero ky, 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 h of the slab, a different ky will lead to a different phase difference ∆φ defined by the additional propagation distance 2l between two adjacent reflections, shown in Fig. 1. It is interesting that when there is a specific ky denoted by kys, such that 2l equals to integral numbers of the propagating wavelength, yielding ∆φ=2, 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 θs=arcsin(kys/k 0). For other (ky, the adjacent reflected waves are out of phase and will hence cancel out. So, for a frequency a little higher than f 0, the outgoing wave deflects from the normal while still keeps a high directivity.

As the frequency continues to increase, obeying the Lorentz model, μz increases at the same time, yielding a k surface curve of ellipse with longer minor axis on y direction. In such a circumstance, kys has to increase to keep ∆φ still to be integral numbers of 2π, resulting in a larger θs of the outgoing wave.

Fig. 1. Illustration of a rainbow-like radiation from a line source placed in a metamaterial slab. The inset shows phase matching k surfaces of metamaterial and air with red and black line.

Such phenomenon has also been studied theoretically and numerically in [14–18

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]

] referred to as “leaky wave modes” in different circumstances. For a practical metamaterial slab with 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 θs 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.

3. Simulation and experiment

Fig. 2. (a). Unit cell of the metamaterial sample. (b) The retrieved real and imaginary parts of the permeability μz.

Before the experiment, we perform finite-element method (FEM) simulation to verify the performance of the metamaterial slab. In the simulation, the dimensions and parameters are just the same as that described previously, with an additional current line source along the 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.

Fig. 3. (a). Simulation results of the near electric field distribution and far field radiation pattern from the metamaterial slab at the frequency of 10.6 GHz. (b) Simulation results of the near electric field distribution and far field radiation pattern from the metamaterial slab at the frequency of 11 GHz.

The experiment is carried out in a microwave anechoic chamber. The metamaterial slab is put on a rotary table in the quiet zone of the chamber with a small monopole antenna localized in the center to serve as the line source. Figure 4(a) shows the photo of the metamaterial slab consisting of the dielectric substrates with the printed metallic rings, as well as the monopole antenna shown at the right bottom inset. During the experiment, a wide frequency band signal from 7 GHz to 13 GHz is fed into the monopole while the rotary table rotates from -90 degree to +90 degree (the direction of the normal of the metamaterial slab is defined to be zero degree), and a wide band receiving horn antenna is placed at the other side of the chamber to receive the far field radiation from the slab, as shown at the left bottom inset in Fig. 4(b). The recorded data is shown in Fig. 4(b). We see that for the frequencies lower than 9.625 GHz, there is no obvious radiation being detected because of the negative μz 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 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.

Fig. 4. (a). The metamaterial slab and the monopole antenna used in the experiment. (b) The measured far field radiation pattern from 7 GHz to 13 GHz, the bottom-left inset shows the experimental setup. (c) Radiation patterns for four selected frequencies.

4. Conclusion

In conclusion, we demonstrate experimentally the spectrum spread from a planar surface of an anisotropic dispersive metamaterial slab, in which an omni-directional EM radiation consisting of different “colors” are refracted into different outgoing angles, yielding a rainbow-like radiation in a specific frequency range. The spectral property of the leaky mode allows for a beam scanning result through the planar surface of the metamaterial slab by varying the operation frequency, other than by varying the phase in traditional phased array antennas. This can be used to design a new kind of dispersive antenna which radiates EM waves with different frequencies to different directions, and can be also used in many important potential applications, such as wave division modulation (WDM), spatial light splitters and filters.

Acknowledgments

References and links

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]

5.

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]

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 . 74, 036621-1-5 (2006). [CrossRef]

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]

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 . 56, 2566–2573 (2008). [CrossRef]

9.

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]

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 . 100, 033903-1-4 (2008). [CrossRef] [PubMed]

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]

12.

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]

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 51, 295–328 (2005). [CrossRef]

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]

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 . 53, 32–44 (2005). [CrossRef]

16.

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]

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 . 56, 1329–1339 (2008). [CrossRef]

18.

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]

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]

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]

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

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?" 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]
  5. 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]
  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. 74, 036621-1-5 (2006). [CrossRef]
  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]
  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. 56, 2566-2573 (2008). [CrossRef]
  9. 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]
  10. 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]
  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]
  12. 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]
  13. 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]
  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]
  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. 53, 32-44 (2005). [CrossRef]
  16. 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]
  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. 56, 1329-1339 (2008). [CrossRef]
  18. 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]
  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]
  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]

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