## 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, 2], 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], and cloaks of invisibility that can bend rays around the object [4–6]. Recently, metamaterials with “near-zero” permittivity and/or permeability have also attracted a lot of interest [7–10]. 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]. 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*[*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].

However, if f>f_{0}, *θ _{c}*
increases with the increase of

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

*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.

As the frequency continues to increase, obeying the Lorentz model, *μ _{z}* increases at the same time, yielding a

*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}Such phenomenon has also been studied theoretically and numerically in [14–18] 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 *θ _{s}* 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

Next we use a practical metamaterial slab to experimentally verify aforementioned analysis. Since currently, metamaterials have not behaved better performance in optical band, the simulation and experiment are carried out in microwave region without losing universality. (cancel) We utilize the unit cell shown in Fig. 2(a) to construct the metamaterial slab. This kind of metallic pattern is first proposed in [19] for fabricating metamaterial showing effective negative permeability. In our realization, the metallic patterns are printed in alignment on an 1-mm-thick FR4 substrate with a relative permittivity of 4.6 to obtain one piece of the metamaterial sample, containing 40 unit cells in the *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 _{z}*) [19]. However, along the

*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 [20,21], the dispersion property of

_{x};μ_{y};μ_{z}*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}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.

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 *μ _{z}* 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.

## 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

This work is sponsored by the NSFC (No. 60531020, 60671003, 60701007, and 60801005), 863 Project (No. 2009AA01Z227), the NCET-07-0750, the ZJNSF (No. R1080320 and Y1080715) the Ph.D. Programs Foundation of MEC (No. 20070335120 and 200803351025), and the National Key Laboratory Foundation (No. 9140C5304020704).

## 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]