## 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]. As is known, the three-dimensional isotropic coherent radiation is forbidden due to the requirement of Maxwell’s equations [2–4]. 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], microstrip antenna [6], and metamaterial antenna [7]. Recently, near-three-dimensional omnidirectional radiations are also reported [8,9]. 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 [10–12]. Nonetheless, at terahertz or even higher frequency this traditional technique can not ensure the required efficiency.

Recently, metamaterials [13–15], 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]. 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) [17–21], *μ*-near-zero (MNZ) [22–24], or both *ε* and *μ* near zero, a matched ZIM (MZIM) [25–27], and even the ZIM with anisotropy [16, 28–32]. 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 [19,21,33,34], modifying and enhancing directive emission [28,29,31,35], shaping wavefront [27, 36, 37], constructing reflectionless sharp bends [38, 39], and switching transmission and reflection by controlling the embedded defect in ZIM [27, 40–42]. 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]. 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 [43–45]. Very recently, Zhu and coworkers also achieve the amplification of the isotropic radiation by inserting gain particle into circular matched ZIM [46]. Nonetheless, since the line source is located inside the matched ZIM, the EM field inside is nearly homogenous [25], leading again to the low efficiency [45]. 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]. 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 by*D* and *S*, respectively. In the cylindrical coordinate, the permittivity and the permeability tensors of the RAZIM are characterized by [4, 16, 47]

*μ*→ 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

_{r}*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.

#### 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, 47]

*k*the wavenumber in the vacuum,

*J*and

_{ν}*H*are, respectively, the

_{ν}*ν*-th order Bessel functions and Hankel functions of first kind, with $\nu =\left|m\right|/\sqrt{{\mu}_{\varphi}/{\mu}_{r}}$, and the summation

*m*runs from −∞ to ∞. The corresponding magnetic field in the transverse

*xoy*plane can be calculated by

**is the position vector, and**

*r**denotes the position of the line source, with*

**l**_{s}*s*=

*l*= |

_{s}*| 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 (*

**l**_{s}*x*,

_{s}*y*) with

_{s}*y*= 0. With these expansions, we can write the total electric field in different regions according to

_{s}*A*characterize the reflection of EM wave from the RAZIM shell, and

_{m}*D*describes the EM wave radiating out of the shell.

_{m}Boundary conditions requires that the tangential components of the EM field *E _{z}* and

*H*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,

_{ϕ}For the RAZIM considered in our system, *μ _{r}* → 0, suggesting that the order of the cylindrical functions

*J*and

_{ν}*H*in Eqs. (2), (5), and (7)

_{ν}*ν*→ ∞ for

*m*≠ 0. Therefore, |

*H*| → ∞ and |

_{ν}*J*| → 0, resulting in the Mie coefficient

_{ν}*p′*→ 0 for

_{m}*m*≠ 0. It follows from (6) that

*B*→ 0,

_{m}*C*→ 0 and

_{m}*D*→ 0 for

_{m}*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

*ε*=

_{z}*μ*= 1, the Mie coefficients

_{ϕ}*p*

_{0}=

*p′*

_{0}= 1,

*q*

_{0}=

*q′*

_{0}= 0, and

*D*

_{0}=

*J*

_{0}(

*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]. 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], 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] and even realizable in terahertz region [50].

#### 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 *A _{m}*,

*B*,

_{m}*C*and

_{m}*D*read

_{m}*E*are the expansion coefficients of the scattered EM wave by the dielectric rod

_{m}*D*. Since the RZAIM shell is intact when the dielectric rod

*D*is inserted inside, the coefficients

*p*,

_{m}*q*,

_{m}*p′*, and

_{m}*q′*that characterize the scattering property of the RAZIM shell remain the same in the new system.

_{m}To obtain *E _{m}*, 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 ${E}_{z}^{i}$ and scattered by the rod ${E}_{z}^{s}$ are, respectively,

*denotes the position of the dielectric rod, with*

**l**_{d}*d*=

*l*= |

_{d}*| denoting the separation between the dielectric rod center and the RAZIM shell center, and*

**l**_{d}*r*is the radius of the dielectric rod

_{d}*D*. The partial wave expansion coefficients are given by

*I*and

_{m}*R*corresponding to the contribution from the line source and the EM wave scattered inside by the RAZIM shell, In Eq. (11),

_{m}*ϕ*= ∠

_{c}*AOS*,

*ϕ′*= ∠

*ASO*,

*l*

^{2}=

*d*

^{2}+

*s*

^{2}−2

*ds*cos

*ϕ*is the distances from the dielectric rod to the line source

_{c}*S*, with

*l*/sin

*ϕ*=

_{c}*d*/ sin

*ϕ′. a*and

_{m}*b*are the Mie coefficients of the dielectric rod, which can easily obtained from the Mie theory [51]

_{m}*ε*and

_{d}*μ*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

_{d}*R*= 0. Combining Eqs. (8), (10), and (11), we can arrive at

_{m}*E*, with the Mie coefficients

_{n}*a*and

_{m}*q′*given by (12b) and (7d), respectively, whereas the partial wave expansion coefficient

_{m}*I*is given by (11a).

_{m}## 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, *μ _{r}* = 0.01,

*μ*= 1,

_{ϕ}*ε*= 1, and for the dielectric rod are

_{z}*r*= 0.15,

_{d}*ε*= 2, and

_{d}*μ*= 1. The wavelength of the line source is set as unit

_{d}*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 |*E _{z}*| 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

*D*

_{0}| given by (6). In Fig. 2(b), we present the map of |

*D*

_{0}| as the function of the dielectric rod position (

*x*,

_{d}*y*), where we can find the optimal position is near to the area where the electric field amplitude |

_{d}*E*| is strongest. In addition, |

_{z}*D*

_{0}| 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.

#### 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

**is the Poynting vector, the integral curve**

*S**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 For comparison, we also calculate the radiating power

*P*when the RAZIM shell is removed from the system, which can be measured by In Fig. 3, we present the radiating power normalized by the radiating power

_{wo}*P*

_{s}_{0}of the line source in free space. Both

*P*

_{wi}/P_{s}_{0}and

*P*

_{wo}/P_{s}_{0}are plotted as the functions of the dielectric rod position

*x*while keeping

_{d}*y*= 0. Panels (a) and (b) correspond to the cases when the line source is positioned at (0.1, 0) and (0, 0), respectively. For

_{d}*P*

_{wo}/P_{s}_{0}, 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

*P*

_{wo}/P_{s}_{0}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]. While for

*P*

_{wi}/P_{s}_{0}, 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.

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], this corresponds to different radiation resistance. This resistance can be evaluated according to the theory proposed in Arslanagic and coworkers’ work [52], 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]. 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/I*_{0} by that of the line source in free space *I*_{0} where the irradiance is defined as
$I=\underset{\mathit{r}\to \infty}{\text{lim}}\left(\mathit{S}\cdot \mathit{r}\right)$. The line source is placed at (*x _{s}*,

*y*) = (0.1, 0) and the dielectric rod is located at (

_{s}*x*,

_{d}*y*) = (−0.24, 0) close to the position of the strongest electric field amplitude. The red solid line is 1/4 of the irradiance

_{d}*I*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

_{wi}*I*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

_{wo}*I*to irradiance ${I}_{wi}^{N}$ 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]. In the present simulation, the radial component of the permeability

_{wi}*μ*= 0.01, not exactly equal to 0, suggesting a finite bandwidth of the operating frequency.

_{r}#### 3.3. Influence of the loss due to the RAZIM shell

To present a proof-of-principle demonstration of our result, we do not consider the intrinsic loss involved in the RAZIM shell for the previous calculations and simulations. However, due to the finite size of the RAZIM shell and its resonant nature, the loss should be an inevitable issue in the system. To illustrate the influence of the loss on the power enhancement, we have performed the simulations. The results are shown in Fig. 5, where we can find that loss can reduce enhancement of the output radiating power seriously compared with Figs. 3 and 4 for the non-loss case as indicated by the red dashed line. This could be a serious problem for practical application, although in our simulation we still have a radiation enhancement of 2. Recent research on the radiation properties of the active coated nanoparticles (NPs) suggested that gain medium can be a good candidate to compensate the energy loss [52,54–56]. Whereas for the work presented in Refs. [50, 52–54] the enhanced radiating field originated from the dipole eigenmode of the active coated NPs, which results in a strong anisotropy of the radiation pattern. In present work, we take both enhancement and isotropy of radiation into account, yielding an increase of radiation while keeping omnidirectional. We also try to compensate the energy attenuation due to the the intrinsic loss of the RAZIM shell by introducing a gain particle with *ε _{d}* = 2.5 − 0.5

*i*. The results are illustrated by the red solid line in Fig. 5, where we can find that an enhancement of the output radiating power is achieved by a factor of about 7. Meanwhile, the omnidirectionality is ensured as well despite of the position of the line source. For comparison, we also present the result for the system without the RAZIM shell but with a gain particle only, which is indicated by the blue solid line. It can be found that neither significant increase nor isotropy in radiation is achieved, imlying once again that the RAZIM shell plays a crucial role in enhancing radiation power and keeping radiation isotropic.

## 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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