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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23712–23723
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Giant omnidirectional radiation enhancement via radially anisotropic zero-index metamaterial

Neng Wang, Huajin Chen, Wanli Lu, Shiyang Liu, and Zhifang Lin  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23712-23723 (2013)
http://dx.doi.org/10.1364/OE.21.023712


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Abstract

We demonstrate a remarkable enhancement of isotropic radiation via radially anisotropic zero-index metamaterial (RAZIM). The radiation power can be enhanced by an order of magnitude when a line source and a dielectric particle is enclosed by a RAZIM shell. Based on the extended Mie theory, we illustrate that the basic physics of this isotropic radiation enhancement lies in the confinement of higher order anisotropic modes by the RAZIM shell. The confinement results in some high field regions within the RAZIM shell and thus enables strong scattering from the dielectric particle therein, giving rise to a giant amplification of isotropic radiation out of the system. The influence of the loss inherent in the RAZIM shell is also examined. It is found that the attenuation of omnidirectional power enhancement due to the loss in the RAZIM can be compensated by gain particles.

© 2013 OSA

1. Introduction

Generating omnidirectional radiation is of great importance for radio communication and broadcasting, mobile devices, as well as base stations for resource dispatch [1

1. J. D. Kraus and R. J. Marhefka, Antennas: For All Applications (McGraw Hill, 2002).

]. As is known, the three-dimensional isotropic coherent radiation is forbidden due to the requirement of Maxwell’s equations [2

2. C. J. Boukamp and H. B. G. Casimir, “On multipole expansions in the theory of electromagnetic radiation,” Physica 20, 539–554 (1954). [CrossRef]

4

4. Y. Yuan, N. Wang, and J. H. Lim, “On the omnidirectional radiation via radially anisotropic zero-index metamaterials,” Europhys. Lett. 100, 34005 (2012). [CrossRef]

]. Fortunately, omnidirectional radiation exhibiting uniform power distribution in one plane is permitted and realizable with different types of antennas such as discone antenna, coaxial collinear antenna [5

5. T. J. Judasz and B. B. Balsley, “Improved theoretical and experimental models for the coaxial colinear antenna,” IEEE Trans. Antennas and Propagat. 37, 289–296 (1989). [CrossRef]

], microstrip antenna [6

6. R. Bancroft, “Design parameters of an omnidirectional planar microstrip antenna,” Microw. Opt. Technol. Lett. 47, 414–418 (2005). [CrossRef]

], and metamaterial antenna [7

7. H. X. Xu, G. M. Wang, M. Q. Qi, and Z. M. Xu, “A metamaterial antenna with frequency-scanning omnidirectional radiation patterns,” Appl. Phys. Lett. 101, 173501 (2012). [CrossRef]

]. Recently, near-three-dimensional omnidirectional radiations are also reported [8

8. J. Ahn, H. Jang, H. Moon, J. W. Lee, and B. Lee, “Inductively coupled compact RFID tag antenna at 910 MHz with near-isotropic radar cross-section (RCS) patterns,” IEEE Antennas Wirel. Propag. Lett. 6, 518–520 (2007). [CrossRef]

,9

9. S. L. Chen, K. H. Lin, and R. Mittra, “Miniature and near-3D omnidirectional radiation pattern RFID tag antenna design,” Electron. Lett. 45, 923–924 (2009). [CrossRef]

]. To realize long range transmittance of electromagnetic (EM) radiation with good quality, a strong radiating power is necessary. Spatial power combination is an effective engineering technique to combine multiple solid-state components to implement high power sources in microwave region [10

10. R. A. York and R. C. Compton, “Quasi-optical power combining using mutually synchronized oscillator arrays,” IEEE Trans. Microwave Theory Tech. 39, 1000–1009 (1991). [CrossRef]

12

12. M. P. DeLisio and R. A. York, “Quasi-optical and spatial power combining,” IEEE Trans. Microwave Theory Tech. 50, 929–936 (2002). [CrossRef]

]. Nonetheless, at terahertz or even higher frequency this traditional technique can not ensure the required efficiency.

Recently, metamaterials [13

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

15

15. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

], a kind of artificial composite materials consisting of resonant “meta-atoms”, which possess arbitrary effective permittivity ε and permeability μ, are employed to realize spatial power combination [16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

]. In their work, a radially anisotropic zero-index metamaterial (RAZIM) is shown to be capable of combining multiple sources and obtaining omnidirectional radiation. Zero-index material (ZIM) is a typical metamaterial, which corresponds to ε-near-zero (ENZ) [17

17. N. Garcia, E. V. Ponizovskaya, and John Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80, 1120–1122 (2002). [CrossRef]

21

21. R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008). [CrossRef] [PubMed]

], μ-near-zero (MNZ) [22

22. M. G. Silveirinha and P. A. Belov, “Spatial dispersion in lattices of split ring resonators with permeability near zero,” Phys. Rev. B 77, 233104 (2008). [CrossRef]

24

24. Y. Jin and S. L. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18, 16587–16593 (2010). [CrossRef] [PubMed]

], or both ε and μ near zero, a matched ZIM (MZIM) [25

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

27

27. X. Q. Huang, Y. Lai, Z. H. Hang, H. H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011). [CrossRef] [PubMed]

], and even the ZIM with anisotropy [16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

, 28

28. Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008). [CrossRef]

32

32. W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Application of zero-index metamaterials for surface plasmon guiding,” Appl. Phys. Lett. 102, 011910 (2013). [CrossRef]

]. A great deal of interesting phenomena and potential applications based on ZIM have been reported, which, among others, include squeezing and tunneling EM wave through subwavelength channel [19

19. B. Edwards, A. Alù, M. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008). [CrossRef] [PubMed]

,21

21. R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008). [CrossRef] [PubMed]

,33

33. Q. Cheng, R. P. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91, 2341052007.

,34

34. M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76, 245109 (2007). [CrossRef]

], modifying and enhancing directive emission [28

28. Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008). [CrossRef]

,29

29. Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94, 044107 (2009). [CrossRef]

,31

31. Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett. 99, 131913 (2011). [CrossRef]

,35

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

], shaping wavefront [27

27. X. Q. Huang, Y. Lai, Z. H. Hang, H. H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011). [CrossRef] [PubMed]

, 36

36. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007). [CrossRef]

, 37

37. S. M. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108, 193904 (2012). [CrossRef]

], constructing reflectionless sharp bends [38

38. B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105, 044905 (2009). [CrossRef]

, 39

39. J. Luo, P. Xu, H. Y. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012). [CrossRef]

], and switching transmission and reflection by controlling the embedded defect in ZIM [27

27. X. Q. Huang, Y. Lai, Z. H. Hang, H. H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011). [CrossRef] [PubMed]

, 40

40. J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96, 101109 (2010). [CrossRef]

42

42. Y. Xu and H. Chen, “Total reflection and transmission by epsilon-near-zero metamaterials with defects,” Appl. Phys. Lett. 98, 113501 (2011). [CrossRef]

]. However, the efficiency of the spatial power combination by use of RAZIM shell is seriously limited since only the isotropic 0-th order mode of the EM wave can be radiated out [16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

]. When the line sources are positioned far from the shell center, the higher order anisotropic modes trapped in the shell become evident while the magnitude of the isotropic mode decreases, resulting in the low efficiency of the system. Adding gain medium can be an alternative approach to enhance the radiation through energy compensation [43

43. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010). [CrossRef]

45

45. Z. Huang, T. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012). [CrossRef] [PubMed]

]. Very recently, Zhu and coworkers also achieve the amplification of the isotropic radiation by inserting gain particle into circular matched ZIM [46

46. W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Light amplification in zero-index metamaterial with gain inserts,” Appl. Phys. Lett. 101, 031907 (2012). [CrossRef]

]. Nonetheless, since the line source is located inside the matched ZIM, the EM field inside is nearly homogenous [25

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

], leading again to the low efficiency [45

45. Z. Huang, T. Koschny, and C. M. Soukoulis, “Theory of pump-probe experiments of metallic metamaterials coupled to a gain medium,” Phys. Rev. Lett. 108, 187402 (2012). [CrossRef] [PubMed]

]. As a result, a single gain particle can not achieve a remarkable effect. One has to resort to multiple gain particles for significant enhancement of radiated power [46

46. W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Light amplification in zero-index metamaterial with gain inserts,” Appl. Phys. Lett. 101, 031907 (2012). [CrossRef]

]. Besides, since ZIM is constructed based on effective medium theory, inserting gain particles inside the metamaterial may destroy its homogeneity and result in the change of the effective permittivity and permeability. Accordingly, their proposal might meet serious difficulty in the experiments and practical applications.

In this work, we propose an efficient and experimentally realizable design to implement a remarkably enhanced two-dimensional (2D) isotropic radiation by enclosing a line source together with a conventional dielectric rod with a RAZIM shell, which is physically permitted so that a perfect 2D EM mode can be realizable. In addition, this proposal does not destroy the homogeneity of the RAZIM shell, which makes it feasible in the experiment. The physics of the 2D isotropic radiation power enhancement lies in that the RAZIM shell traps the anisotropic higher order modes, although it is transparent for isotropic 0-th order EM modes. It thus yields a strong inhomogeneity of EM field in the space enclosed by the RAZIM shell. In the strong EM field region, a single dielectric particle can efficiently re-scatter the anisotropic higher order EM modes into isotropic 0-th order modes, inducing a great enhancement of omnidirectional radiation out of the system. Later on, we will present an exact theoretical approach, and then provide the numerical calculation and simulation results, which demonstrate a 10 times amplification of the radiating power over that of a line source in free space, while keeping the radiation omnidirectional. Finally, the intrinsic loss of the RAZIM shell are also examined and the gain-particle-compensation-based power amplification are demonstrated as well.

2. Geometry and formulations

The geometry of the system is schematically illustrated in Fig. 1, where the shadowed blue region is the RAZIM shell with a and b the inner and outer shell radii, and the positions of the dielectric rod and the line source are denoted byD and S, respectively. In the cylindrical coordinate, the permittivity and the permeability tensors of the RAZIM are characterized by [4

4. Y. Yuan, N. Wang, and J. H. Lim, “On the omnidirectional radiation via radially anisotropic zero-index metamaterials,” Europhys. Lett. 100, 34005 (2012). [CrossRef]

, 16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

, 47

47. Y. X. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homegeneous cylindrically anisotropic cylinders,” Plamonics 5, 251–258 (2010). [CrossRef]

]
ε¯=ε0(r^r^εr+ϕ^ϕ^εϕ+z^z^εz),μ¯=μ0(r^r^μr+ϕ^ϕ^μϕ+z^z^μz),
(1)
where μr → 0. The origin of the cylindrical coordinate is at the center of the RAZIM shell. In our work, radiation behavior of a transverse magnetic (TM) line source with the electric field polarized along z direction is considered. For convenience in illustrating physics, we first consider a simple system schematically illustrated in Fig. 1(a), based on which we can then solve the system when a dielectric particle is introduced as shown in Fig. 1(b) by taking account of the scattering effect between the particle and the RAZIM shell.

Fig. 1 A schematic illustration showing the radiation enhancement system consisting of a RAZIM shell covering (a) a line source only and (b) both a line source and a dielectric rod. The shell center, the line source position, and the dielectric rod position are denoted, respectively, by O, S and D. The medium inside and outside the shell is vacuum. The inner and outer shell radii are a and b, respectively. The radius of the dielectric rod is rd, while |OD| = d and |OS| = s.

2.1. The system with a single line source

In the framework of the generalized Lorenz-Mie theory, the EM field in the RAZIM region can be expanded into the linear combination of the eigenmodes [16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

, 47

47. Y. X. Ni, L. Gao, and C. W. Qiu, “Achieving invisibility of homegeneous cylindrically anisotropic cylinders,” Plamonics 5, 251–258 (2010). [CrossRef]

]
Ez=m[BmJν(ksr)+CmHν(ksr)]eimϕ,arb,
(2)
where ks2=k2μϕεz with k the wavenumber in the vacuum, Jν and Hν are, respectively, the ν-th order Bessel functions and Hankel functions of first kind, with ν=|m|/μϕ/μr, and the summation m runs from −∞ to ∞. The corresponding magnetic field in the transverse xoy plane can be calculated by
iωμrμ0Hr=1rEzϕ,iωμϕμ0Hϕ=Ezr,
(3)
for the TM waves. The electric field radiated by a line source can also be expanded around the RAZIM shell center [48

48. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graph, and Mathematical Tables(Dover, 1964).

, 49

49. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

]
Ez=H0(k|rls|)=mJm(ks)Hm(kr)eimϕ,r>s,Ez=H0(k|rls|)=mHm(ks)Jm(kr)eimϕ,r<s,
(4)
where r is the position vector, and ls denotes the position of the line source, with s = ls = |ls| denoting the separation between the line source and the RAZIM shell center. For convenience and without loss of generality, the line source is supposed to be located at (xs, ys) with ys = 0. With these expansions, we can write the total electric field in different regions according to
Ez=m[AmJm(kr)+Jm(kd)Hm(kr)]eimϕ,s<ra,Ez=mDmHm(kr)eimϕ,rb,
(5)
where coefficients Am characterize the reflection of EM wave from the RAZIM shell, and Dm describes the EM wave radiating out of the shell.

Boundary conditions requires that the tangential components of the EM field Ez and Hϕ should be continuous at the interface, based on which we can work out the partial wave expansion coefficients for the electric fields in different regions,
Bm=qmCm,Dm=pmCm,
(6a)
Am=qmJm(ks),Cm=pmJm(ks),
(6b)
where the generalized Mie coefficients are given by
pm=ksHν(ksb)Jν(ksb)ksHν(ksb)Jν(ksb)ksHm(kb)Jν(ksb)kμϕHm(kb)Jβ(ksb),
(7a)
qm=kμϕHν(ksb)Hm(kb)ksHν(ksb)Hm(kb)ksHm(kb)Jν(ksb)kμϕHm(kb)Jν(ksb),
(7b)
pm=kμϕHm(ka)Jm(ka)kμϕHm(ka)Jm(ka)kμϕ[Hν(ksa)+qmJν(ksa)]Jm(ka)ks[Hν(ksa)+qmJν(ksa)]Jm(ka),
(7c)
qm=ksHm(ka)[Hν(ksa)+qmJν(ksa)]kμϕHm(ka)[Hν(ksa)+qmJν(ksa)]kμϕ[Hν(ksa)+qmJν(ksa)]Jm(ka)ks[Hν(ksa)+qmJν(ksa)]Jm(ka).
(7d)

For the RAZIM considered in our system, μr → 0, suggesting that the order of the cylindrical functions Jν and Hν in Eqs. (2), (5), and (7) ν → ∞ for m ≠ 0. Therefore, |Hν| → ∞ and |Jν| → 0, resulting in the Mie coefficient p′m → 0 for m ≠ 0. It follows from (6) that Bm → 0, Cm → 0 and Dm → 0 for m ≠ 0. This reveals that the permitted propagating EM waves in the RAZIM shell is nearly independent on the azimuthal angle ϕ, as can be seen from Eqs. (2) and (5). In particular, for the case when εz = μϕ = 1, the Mie coefficients p0 = p′0 = 1, q0 = q′0 = 0, and D0 = J0(kd). Accordingly, only the 0-th order of the isotropic cylindrical EM wave can be radiated out of the RAZIM shell, ensuring its omnidirectionality (ϕ independent), consistent with the results obtained by Cheng and coworkers [16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

]. However, all higher order modes of the cylindrical waves are confined within the RAZIM shell, the introduction of RAZIM shell leads to the decrease of radiation power and reduces the radiation efficiency [4

4. Y. Yuan, N. Wang, and J. H. Lim, “On the omnidirectional radiation via radially anisotropic zero-index metamaterials,” Europhys. Lett. 100, 34005 (2012). [CrossRef]

], although it may implement the spatial power combination for omnidirectional radiation. An important aspect from the theoretical analysis indicates that the RAZIM shell can be considered as a cylindrical resonator for the higher order modes, which results in the creation of the standing wave with strong inhomogeneity. Compared with the ordinary dielectric resonator, it exhibits remarkable difference in that it can trap all the higher order modes and permit the radiating of the isotropic zero order mode. This particular property arises from the anisotropy of the RAZIM, which is an essential aspect for the realization of radiation pattern, unachievable by the ordinary ENZ, MNZ, or MZIM. Besides, it is shown the RAZIM shell can be constructed in microwave region [16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

] and even realizable in terahertz region [50

50. Z. C. Chen, R. Mohsen, Y. D. Gong, T. W. Chong, and M. H. Hong, “Realization of variable three-dimensional terahertz metamaterial tubes for passive resonance tunability,” Adv. Mater. 24, OP143–OP147 (2012). [CrossRef]

].

2.2. The system with an addition of dielectric rod

The key issue to improve the radiating efficiency of the system as shown in Fig. 1(a) is to exploit the field confined within the RAZIM shell. To achieve this purpose, we introduce a dielectric particle D in space enclosed by the RAZIM shell as illustrated in Fig. 1(b). In this case, the dielectric particle re-scatters the EM wave and transforms the waves into the isotropic modes, which can be radiated through the RZAIM shell. So the radiation out of the RAZIM shell orginate not only the isotropic component from the line source S, but also the isotropic component due to the dielectric particle D. In the presence of dielectric rod, the equations in place of Eq. (6b) to determine the partial wave expansion coefficients Am, Bm, Cm and Dm read
Am=qm[Jm(ks)+Em],Cm=pm[Jm(ks)+Em],
(8)
where Em are the expansion coefficients of the scattered EM wave by the dielectric rod D. Since the RZAIM shell is intact when the dielectric rod D is inserted inside, the coefficients pm, qm, p′m, and q′m that characterize the scattering property of the RAZIM shell remain the same in the new system.

To obtain Em, we transform the expanding terms of the EM wave from the shell center to the center of the dielectric rod D. In this way, the electric field inside the dielectric rod Ezi and scattered by the rod Ezs are, respectively,
Ezi=mTmJm(k|rld|)eimϕ,|rld|<rd,
(9a)
Ezs=mSmHm(k|rld|)eimϕ,|rld|>rd,
(9b)
where ld denotes the position of the dielectric rod, with d = ld = |ld| denoting the separation between the dielectric rod center and the RAZIM shell center, and rd is the radius of the dielectric rod D. The partial wave expansion coefficients are given by
Tm=bm(Rm+Im),Sm=am(Rm+Im),
(10)
with Im and Rm corresponding to the contribution from the line source and the EM wave scattered inside by the RAZIM shell,
Im=Hm(kl)einϕ,
(11a)
Rm=nAm+nJn(kd)einϕc,
(11b)
Sm=nEm+nJn(kd)einϕc,
(11c)
In Eq. (11), ϕc = ∠AOS, ϕ′ = ∠ASO, l2 = d2 +s2 −2dscosϕc is the distances from the dielectric rod to the line source S, with l/sinϕc = d/ sinϕ′. am and bm are the Mie coefficients of the dielectric rod, which can easily obtained from the Mie theory [51

51. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

]
bm=kμdJm(krd)Hm(krd)kμdJm(krd)Hm(krd)kdJm(kdrd)Hm(krd)kμdHm(krd)Jm(kdrd),
(12a)
am=kμdJm(krd)Jm(kdrd)kdJm(krd)Jm(kdrd)kdJm(kdrd)Hm(krd)kμdHm(krd)Jm(kdrd),
(12b)
where kd2=k2εdμd, with εd and μd being the permittivity and permeability of dielectric rod, respectively. Note that when the RAZIM shell is removed from the system, the scattering from the shell vanishes, corresponding to Rm = 0. Combining Eqs. (8), (10), and (11), we can arrive at
n(1amqn)Jnm(kc)ei(nm)ϕcEn=namqnJn(kd)Jnm(kc)ei(nm)ϕc+amIm.
(13)
This is a set of linear equations governing the coefficients En, with the Mie coefficients am and q′m given by (12b) and (7d), respectively, whereas the partial wave expansion coefficient Im is given by (11a).

3. Results and discussion

Based on the developed theory, we can now perform the simulations on the field patterns and characterize the radiation. In this way, we can illustrate the role of the RAZIM shell and the dielectric rod, thus understanding the mechanism to realize the enhanced isotropic radiation. Then, we can optimize the system to obtain the best performance. In the simulations and calculations except otherwise specified, the parameters for the RAZIM shell are set as a = 0.5, b = 1, μr = 0.01, μϕ = 1, εz = 1, and for the dielectric rod are rd = 0.15, εd = 2, and μd = 1. The wavelength of the line source is set as unit λ = 1. The present theory can even be used to model system for a gain particle or a lossy particle inserted inside the RAZIM shell.

3.1. Field pattern simulation

First, we simulate the electric field amplitude |Ez| pattern inside RAZIM shell, the result is shown in Fig. 2(a), where a line source is positioned at (0.1, 0) and no dielectric rod is inserted. Thus, the role of the RAZIM shell can be illustrated. We can observe a standing wave characterized by strong inhomogeneity, which is created by the higher order partial waves in Eq. (4) due to the high reflection from the RAZIM shell. Accordingly, the RAZIM shell is operated similarly as a filter in that only 0-th order of the partial wave can be radiated out, ensuring the isotropy of the radiation. Simultaneously, it confines all the higher order partial waves inside the system, facilitating the enhancement of the radiation power by the introduction of a dielectric rod. The EM wave radiating outside the RAZIM shell can be calculated approximately by
EzD0H0(kr)=[J0(ks)+E0]H0(kr).
(14)
Accordingly, the performance of the dielectric rod can be evaluated by calculating the amplitude of |D0| given by (6). In Fig. 2(b), we present the map of |D0| as the function of the dielectric rod position (xd, yd), where we can find the optimal position is near to the area where the electric field amplitude |Ez| is strongest. In addition, |D0| bears a much larger value than that of a free line source in a large area, illustrating the outstanding effect of the dielectric rod on the radiation enhancement. Besides, the introduction of a dielectric rod inside the RAZIM shell doesn’t destroy the homogeneity of the RAZIM. This makes the designed system experimentally feasible.

Fig. 2 (a) The electric field amplitude |Ez| pattern inside the RAZIM shell for the configuration in Fig. 1(a), (b) the 0-th order partial wave amplitude |D0| is plotted as the function of the dielectric rod position (xd, yd). The whiteout region denotes the area that the dielectric rod can not reach. The line source is located at (0.1, 0). The parameters of the system are a = 0.5, b = 1, rd = 0.15, μr = 0.01, μϕ = 1, εz = 1, εd = 2, and μd = 1.

3.2. Amplification of the radiation power

To gain further insight into the performance of the dielectric rod, we calculate the total power radiating out of the RAZIM shell. It is defined as
Ps=LSerdl,withS=12Re[E×H*],
(15)
where S is the Poynting vector, the integral curve L is the circle centered at the origin O with the radius larger than the RAZIM shell radius b. For the system showing in Fig. 1(b), only the 0-th order cylindrical wave is radiated out. The radiating power can be approximately evaluated according to
Pwi2|D0|2ωμ0.
(16)
For comparison, we also calculate the radiating power Pwo when the RAZIM shell is removed from the system, which can be measured by
Pwo=2m|amHm(kl)+Jm(kl)|2ωμ0.
(17)
In Fig. 3, we present the radiating power normalized by the radiating power Ps0 of the line source in free space. Both Pwi/Ps0 and Pwo/Ps0 are plotted as the functions of the dielectric rod position xd while keeping yd = 0. Panels (a) and (b) correspond to the cases when the line source is positioned at (0.1, 0) and (0, 0), respectively. For Pwo/Ps0, its value experiences nearly no change with the change of the dielectric rod position, as shown by the blue dashed line in panels (a) and (b). Our simulation shows that even when the dielectric rod is replaced by a gain particle, the value of Pwo/Ps0 remains close to 1, suggesting that an insertion of particle, either passive or active, has nearly no effect on the radiating behavior of the system in the absence of the RAZIM shell. The reason lies in that in free space the line source does not demonstrate the position with a strong electric field amplitude. This explains why Zhu and coworkers have to use multiple gain particles to obtain a strong radiation [46

46. W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Light amplification in zero-index metamaterial with gain inserts,” Appl. Phys. Lett. 101, 031907 (2012). [CrossRef]

]. While for Pwi/Ps0, the radiating power can be significantly improved, as can be observed from the red solid lines shown in panels (a) and (b), indicating that the RAZIM shell plays a crucial role for the amplification of the radiation. The maximum enhancement appears close to the position with the strongest electric field amplitude. It is also noted that the position of the line source has an obvious effect on the radiating power by comparing panels (a) and (b), which arises from the dependence of the standing wave on the source position. In our system, an enhancement of over 10 times is achieved with the insertion of a dielectric rod.

Fig. 3 The normalized radiating power Ps/Ps0 is plotted as the function of the dielectric rod position xd. The line source is fixed at (0.1, 0) and (0, 0), respectively, for panels (a) and (b). The blue dashed line and the red solid line visualize the radiating power Pwo and Pwi, corresponding to the case without and with the RAZIM shell, respectively. All the other parameters are the same as those in Fig. 2.

It should be emphasized that the over-the-unity total radiation power does not imply a violation of the energy conservation. Actually, in all our simulations, the line source is driven by the same current, or, equivalently, radiates the same driving electric field, see, Eq. (4), which results in the adjustment of the total radiating power for different systems. In the language of antenna theory [1

1. J. D. Kraus and R. J. Marhefka, Antennas: For All Applications (McGraw Hill, 2002).

], this corresponds to different radiation resistance. This resistance can be evaluated according to the theory proposed in Arslanagic and coworkers’ work [52

52. S. Arslanagic, Y. Liu, R. Malureanu, and R. W. Ziolkowki, “Impact of the excitation source and plasmonic material on cylindrical active coated nano-particles,” Sensors 11, 9109–9120 (2011). [CrossRef] [PubMed]

], which is physically consistent with our Eqs. (16) and (17). In the language of photonics, the giant enhancement of radiation power reflects a dramatic increase of local density of states, as at the frequency near the photonic band gap [53

53. A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, “Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice,” Phys. Rev. Lett. 94, 183903 (2005). [CrossRef] [PubMed]

]. The remarkable difference lies in that the radiation here can be made omnidirectional independent of the position of the line source.

To give a clear picture how the dielectric rod works, we carry our idea a step further. In Fig. 4, we present the normalized irradiance I/I0 by that of the line source in free space I0 where the irradiance is defined as I=limr(Sr). The line source is placed at (xs, ys) = (0.1, 0) and the dielectric rod is located at (xd, yd) = (−0.24, 0) close to the position of the strongest electric field amplitude. The red solid line is 1/4 of the irradiance Iwi for the system with the RAZIM shell, from which we can find that the system can amplify the irradiance by over 10 times, consistent with the result shown in Fig. 3(a). In addition, the radiation demonstrates an obvious isotropic characteristic. For comparison, we also present the result when the RAZIM shell is removed from the system. The corresponding irradiance Iwo is shown by the blue dash line, which is not isotropic and no obvious enhancement can be achieved with the dielectric rod or even a gain particle. The performance of the dielectric rod can be evaluated by comparing Iwi to irradiance IwiN when the dielectric rod is removed from the system. The result is given by the green dash-dot line, from which we can find that only 80 percent EM energy of the line source is radiated out due to the trap of the high order waves by the RAZIM shell. This suggests that an insertion of a dielectric rod leads to a nearly 15 times amplification of the radiation power. When multiple line sources are used, even higher enhancement of the radiating power can be expected for omnidirectional spatial power combination [16

16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

]. In the present simulation, the radial component of the permeability μr = 0.01, not exactly equal to 0, suggesting a finite bandwidth of the operating frequency.

Fig. 4 The polar-plot of the normalized irradiance I/I0. The red solid (blue dash) line corresponds to the result Iwi/4 (Iwo) for the system with (without) the RAZIM shell. The green dash-dot line corresponds to the result IwiN for the system with RAZIM shell but without the rod D. The dielectric rod is placed at (−0.24, 0) and all the other parameters are the same as those in Fig. 3(a).

3.3. Influence of the loss due to the RAZIM shell

Fig. 5 (a) The normalized radiating power Ps/Ps0 is plotted as the function of the position xd of the particle with εd = 2.5−0.5i (solid lines) and εd = 2.5 (dashed lines) for the system with (red lines) and without (blue lines) the RAZIM shell, respectively. The line source is fixed at (0.1, 0). (b) The corresponding normalized iiradiance I/I0 is plotted as the function of the polar angle. The red (blue) solid line corresponds to the result IwiA(IwoA) for the system with (without) the RAZIM shell and the gain particle modeled by εd = 2.5 − 0.5i. The red (blue) dashed line corresponds to the result Iwi (Iwo) for the system with (without) the RAZIM shell and the dielectric particle of εd = 2.5. The green dash-dot line is for the system with the RAZIM shell but without the particle inside. The particle with the radius rd = 0.15 is placed at (−0.2, 0). The other parameters are a = 0.5, b = 1, λ = 1, μr = 0.01 + 0.005i, μϕ = 1 + 0.005i, and εz = 2 + 0.005i.

4. Conclusion

In summary, we have designed a system consisting of a RAZIM shell and a dielectric rod to realize a remarkably enhanced omnidirectional radiation. An exact theoretical approach is developed to solve the system rigorously, based on which we can explore the effects of the RAZIM shell and the dielectric rod accurately. It is shown that both the RAZIM shell and the dielectric rod play crucial roles for the enhancement of radiation. The RAZIM shell allows only the 0-th order partial wave to radiate outside the system, ensuring the isotropy of the radiating EM wave. While the higher order partial waves are confined inside the system and establish a strongly inhomogeneous standing wave. The dielectric rod placed close to the position with the largest field amplitude can re-scatter the anisotropic modes into isotropic wave, enhancing the omnidirectional radiation remarkably. Our numerical results suggest that an amplification in radiating power of nearly 10 times can be achieved with present designed system. In addition, we have also considered the influence of the intrinsic loss of the RAZIM shell on the radiation enhancement, which is shown to be effectively compensated by introducing a gain particle. We expect the present design can be feasible in the experiment, and meanwhile provides a high efficiency of the spatial power combination for omnidirectional radiation.

Acknowledgments

This work was supported by the 973 Project (No. 2011CB922004), National Natural Science Foundation of China (Nos. 10904020, 11174059, and 11274277), and the open project of SKLSP in Fudan University (No. KL2011_8). Liu is also supported by a program for innovative research team in Zhejiang Normal University.

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H. F. Mathis, “A short proof that an isotropic antenna is impossible,” Proc. IRE 39, 970 (1951).

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Y. Yuan, N. Wang, and J. H. Lim, “On the omnidirectional radiation via radially anisotropic zero-index metamaterials,” Europhys. Lett. 100, 34005 (2012). [CrossRef]

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T. J. Judasz and B. B. Balsley, “Improved theoretical and experimental models for the coaxial colinear antenna,” IEEE Trans. Antennas and Propagat. 37, 289–296 (1989). [CrossRef]

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R. Bancroft, “Design parameters of an omnidirectional planar microstrip antenna,” Microw. Opt. Technol. Lett. 47, 414–418 (2005). [CrossRef]

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H. X. Xu, G. M. Wang, M. Q. Qi, and Z. M. Xu, “A metamaterial antenna with frequency-scanning omnidirectional radiation patterns,” Appl. Phys. Lett. 101, 173501 (2012). [CrossRef]

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J. Ahn, H. Jang, H. Moon, J. W. Lee, and B. Lee, “Inductively coupled compact RFID tag antenna at 910 MHz with near-isotropic radar cross-section (RCS) patterns,” IEEE Antennas Wirel. Propag. Lett. 6, 518–520 (2007). [CrossRef]

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S. L. Chen, K. H. Lin, and R. Mittra, “Miniature and near-3D omnidirectional radiation pattern RFID tag antenna design,” Electron. Lett. 45, 923–924 (2009). [CrossRef]

10.

R. A. York and R. C. Compton, “Quasi-optical power combining using mutually synchronized oscillator arrays,” IEEE Trans. Microwave Theory Tech. 39, 1000–1009 (1991). [CrossRef]

11.

S. Nogi, J. S. Lin, and T. Itoh, “Mode analysis and stabilization of a spatial power combining array with strongly coupled oscillators,” IEEE Trans. Microwave Theory Tech. 41, 1827–1837 (1993). [CrossRef]

12.

M. P. DeLisio and R. A. York, “Quasi-optical and spatial power combining,” IEEE Trans. Microwave Theory Tech. 50, 929–936 (2002). [CrossRef]

13.

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

14.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

15.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

16.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012). [CrossRef] [PubMed]

17.

N. Garcia, E. V. Ponizovskaya, and John Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett. 80, 1120–1122 (2002). [CrossRef]

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M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006). [CrossRef]

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B. Edwards, A. Alù, M. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008). [CrossRef] [PubMed]

20.

A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Phys. Rev. E 78, 016604 (2008). [CrossRef]

21.

R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008). [CrossRef] [PubMed]

22.

M. G. Silveirinha and P. A. Belov, “Spatial dispersion in lattices of split ring resonators with permeability near zero,” Phys. Rev. B 77, 233104 (2008). [CrossRef]

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Y. Jin, P. Zhang, and S. L. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81, 085117 (2010). [CrossRef]

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Y. Jin and S. L. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18, 16587–16593 (2010). [CrossRef] [PubMed]

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

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M. Silveirinha and Nader Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B, 75, 075119 (2007). [CrossRef]

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X. Q. Huang, Y. Lai, Z. H. Hang, H. H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011). [CrossRef] [PubMed]

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Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008). [CrossRef]

29.

Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett. 94, 044107 (2009). [CrossRef]

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Q. Cheng, W. X. Jiang, and T. J. Cui, “Radiation of planar electromagnetic waves by a line source in anisotropic metamaterials,” J. Phys. D: Appl. Phys. 43, 335406 (2010). [CrossRef]

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Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett. 99, 131913 (2011). [CrossRef]

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W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Application of zero-index metamaterials for surface plasmon guiding,” Appl. Phys. Lett. 102, 011910 (2013). [CrossRef]

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Q. Cheng, R. P. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91, 2341052007.

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52.

S. Arslanagic, Y. Liu, R. Malureanu, and R. W. Ziolkowki, “Impact of the excitation source and plasmonic material on cylindrical active coated nano-particles,” Sensors 11, 9109–9120 (2011). [CrossRef] [PubMed]

53.

A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, “Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice,” Phys. Rev. Lett. 94, 183903 (2005). [CrossRef] [PubMed]

54.

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56.

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OCIS Codes
(290.4020) Scattering : Mie theory
(350.5610) Other areas of optics : Radiation
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: August 19, 2013
Revised Manuscript: September 5, 2013
Manuscript Accepted: September 5, 2013
Published: September 27, 2013

Citation
Neng Wang, Huajin Chen, Wanli Lu, Shiyang Liu, and Zhifang Lin, "Giant omnidirectional radiation enhancement via radially anisotropic zero-index metamaterial," Opt. Express 21, 23712-23723 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23712


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References

  1. J. D. Kraus and R. J. Marhefka, Antennas: For All Applications (McGraw Hill, 2002).
  2. C. J. Boukamp and H. B. G. Casimir, “On multipole expansions in the theory of electromagnetic radiation,” Physica20, 539–554 (1954). [CrossRef]
  3. H. F. Mathis, “A short proof that an isotropic antenna is impossible,” Proc. IRE39, 970 (1951).
  4. Y. Yuan, N. Wang, and J. H. Lim, “On the omnidirectional radiation via radially anisotropic zero-index metamaterials,” Europhys. Lett.100, 34005 (2012). [CrossRef]
  5. T. J. Judasz and B. B. Balsley, “Improved theoretical and experimental models for the coaxial colinear antenna,” IEEE Trans. Antennas and Propagat.37, 289–296 (1989). [CrossRef]
  6. R. Bancroft, “Design parameters of an omnidirectional planar microstrip antenna,” Microw. Opt. Technol. Lett.47, 414–418 (2005). [CrossRef]
  7. H. X. Xu, G. M. Wang, M. Q. Qi, and Z. M. Xu, “A metamaterial antenna with frequency-scanning omnidirectional radiation patterns,” Appl. Phys. Lett.101, 173501 (2012). [CrossRef]
  8. J. Ahn, H. Jang, H. Moon, J. W. Lee, and B. Lee, “Inductively coupled compact RFID tag antenna at 910 MHz with near-isotropic radar cross-section (RCS) patterns,” IEEE Antennas Wirel. Propag. Lett.6, 518–520 (2007). [CrossRef]
  9. S. L. Chen, K. H. Lin, and R. Mittra, “Miniature and near-3D omnidirectional radiation pattern RFID tag antenna design,” Electron. Lett.45, 923–924 (2009). [CrossRef]
  10. R. A. York and R. C. Compton, “Quasi-optical power combining using mutually synchronized oscillator arrays,” IEEE Trans. Microwave Theory Tech.39, 1000–1009 (1991). [CrossRef]
  11. S. Nogi, J. S. Lin, and T. Itoh, “Mode analysis and stabilization of a spatial power combining array with strongly coupled oscillators,” IEEE Trans. Microwave Theory Tech.41, 1827–1837 (1993). [CrossRef]
  12. M. P. DeLisio and R. A. York, “Quasi-optical and spatial power combining,” IEEE Trans. Microwave Theory Tech.50, 929–936 (2002). [CrossRef]
  13. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000). [CrossRef] [PubMed]
  14. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001). [CrossRef] [PubMed]
  15. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305, 788–792 (2004). [CrossRef] [PubMed]
  16. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett.108, 213903 (2012). [CrossRef] [PubMed]
  17. N. Garcia, E. V. Ponizovskaya, and John Q. Xiao, “Zero permittivity materials: Band gaps at the visible,” Appl. Phys. Lett.80, 1120–1122 (2002). [CrossRef]
  18. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006). [CrossRef]
  19. B. Edwards, A. Alù, M. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett.100, 033903 (2008). [CrossRef] [PubMed]
  20. A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Phys. Rev. E78, 016604 (2008). [CrossRef]
  21. R. P. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett.100, 023903 (2008). [CrossRef] [PubMed]
  22. M. G. Silveirinha and P. A. Belov, “Spatial dispersion in lattices of split ring resonators with permeability near zero,” Phys. Rev. B77, 233104 (2008). [CrossRef]
  23. Y. Jin, P. Zhang, and S. L. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B81, 085117 (2010). [CrossRef]
  24. Y. Jin and S. L. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express18, 16587–16593 (2010). [CrossRef] [PubMed]
  25. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E70, 046608 (2004). [CrossRef]
  26. M. Silveirinha and Nader Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B,75, 075119 (2007). [CrossRef]
  27. X. Q. Huang, Y. Lai, Z. H. Hang, H. H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10, 582–586 (2011). [CrossRef] [PubMed]
  28. Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A77, 053821 (2008). [CrossRef]
  29. Y. G. Ma, P. Wang, X. Chen, and C. K. Ong, “Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial,” Appl. Phys. Lett.94, 044107 (2009). [CrossRef]
  30. Q. Cheng, W. X. Jiang, and T. J. Cui, “Radiation of planar electromagnetic waves by a line source in anisotropic metamaterials,” J. Phys. D: Appl. Phys.43, 335406 (2010). [CrossRef]
  31. Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett.99, 131913 (2011). [CrossRef]
  32. W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Application of zero-index metamaterials for surface plasmon guiding,” Appl. Phys. Lett.102, 011910 (2013). [CrossRef]
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