## Abstract

Zero-index material is a typical metamaterial with an effective zero refractive index, possessing a variety of exotic electromagnetic properties and particular functionalities. We have considered two kinds of zero-index materials with the first one a nearly matched zero index made of magnetic metamaterial and the second one a radially anisotropic zero index. The magnetic metamaterial-based systems are shown to be significant in wavefront engineering and flexibly tunable by an external magnetic field and a temperature field. The radially anisotropic zero-index-based systems can remarkably enhance the omnidirectional isotropic radiation by enclosing a line source and a dielectric particle within a shell configuration. The physical origin lies in that the dielectric particle effectively rescatters the trapped anisotropic higher order modes and converts them into the isotropic 0th order mode radiated outside the system. The case for the system with the loss is then examined and the energy compensation with a gain particle is also demonstrated.

### Keywords

- zero index materials
- magnetic metamaterials
- multiple scattering theory
- omnidirectional isotropic radiation
- Mie theory

## 1. Introduction

Metamaterials are a kind of composite electromagnetic (EM) materials consisting of subwavelength “meta-atoms” with either electric or magnetic response or even both, which possess nearly arbitrary profile of effective permittivity *ε* and permeability *μ* in principle [1–3]. A great deal of novel and unique EM properties, not occurring in natural materials, such as negative refraction [4, 5], cloaking [6, 7], illusion [8], and subwavelength propagation [9, 10] are theoretically predicted and experimentally implemented from the microwave region to the optical region. Zero-index material (ZIM) is a typical metamaterial with effective zero refractive index, including *ε*-near-zero [11–13], *μ*-near-zero [14, 15], or both *ε* and *μ* near zero, a matched ZIM (MZIM) [16–18], and even more generally the ZIM with anisotropy [19–21].

Due to the extraordinary refractive index of the ZIMs, a great variety of bizarre EM behaviors and potential applications based on ZIM have been extensively investigated and reported. It has been demonstrated both theoretically and experimentally that the ZIM can be used to squeeze the electromagnetic wave and make it tunnel through a deep subwavelength channel with arbitrary shape, serving as a highly efficient coupler between incoming and outgoing waveguides [11, 13, 22]. Due to the zero phase delay in the ZIM, the phase pattern of an EM wave can be flexibly engineered, enabling the wavefront shaping [16, 23, 24]. The ZIM can also be used to modify and enhance the directive emission with high efficiency [20, 25, 26]. Compact omnidirectional metamaterial antennas can also be designed based on the anomalous transmittance of *ε*-near-zero ultranarrow radial channels [27]. Interestingly, by incorporating dielectric defects into the MZIM or *ε*-near-zero material the transmission and reflection can be switched by delicately controlling the defects [16, 28, 29]. Later on, Luo et al. [30] have implemented nearly perfect bending waveguides with anisotropic ZIM, while Cheng et al. [19] have used another kind of anisotropic ZIM, a radially anisotropic ZIM (RAZIM), to combine multiple sources and acquire omnidirectional radiation.

The ZIMs are usually composed of the building blocks made of metallic materials or dielectric materials, or sometimes are supposed to homogeneous media with the desired parameters in theoretical research. Here, we first present another kind of metamaterials composed of building blocks made of ferrite materials with intrinsic magnetic response, which are accordingly called magnetic metamaterials. The effective electric permittivity *ε*_{eff} and magnetic permeability *μ*_{eff} can be tuned by an external magnetic field (EMF) or temperature due to the dependence of ferrite materials on EMF and temperature, thus providing us with more degrees of tunability. In addition, the permeability of ferrite material is a second rank tensor with nonzero off-diagonal elements, indicating the time-reversal-symmetry breaking nature in magnetic metamaterials [31]. As a result, nonreciprocal behaviors can be observed in such system, even for the geometrically symmetric ZIM made of magnetic metamaterial. Then, we will consider a composite system constructed by a RAZIM shell enclosing a dielectric rod inside, which is used to implement a strongly enhanced two-dimensional (2D) isotropic radiation with relatively high efficiency. In addition, different from the configuration with the gain particles inserted inside the MZIM [32], our proposal does not alter the structure of the RAZIM shell and keep its homogeneity, which makes it experimentally feasible.

The research content of the present chapter consists of two main parts. In the first part, we present the design of ZIM with magnetic metamaterials by optimizing the parameters of configuration. Then, by calculating the photonic band diagrams and retrieving the effective constitutive parameters we can confirm the implementation of the ZIM. After that, the field patterns are simulated to demonstrate the zero-phase delay of the ZIM and wavefront modulation by sculpturing typical outgoing interfaces. Finally, the effective refractive index is shown to be thermally controlled due to the dependence of the saturation magnetization of ferrite materials on the ambient temperature. In the second part, we demonstrate a remarkable enhancement of omnidirectional radiation with the RAZIM shell by surrounding the line source together with a particular particle. A rigorous theoretical approach is presented to analyze the phenomenon and optimize the related parameters. Then, the influence of the loss is examined, which can be compensated by introducing a gain particle. Finally, all the results are summarized in the conclusion part.

## 2. ZIM made of magnetic metamaterials

Magnetic metamaterials concerned in present work consist of an array of ferrite rods arranged periodically in air with either square or triangular lattice. Actually, configurations with some randomness introduced to the system are similar in the long wavelength limit, provided that the filling ratio of the ferrite materials is the same. Single-crystal yttrium-iron-garnet (YIG) is a good candidate for designing magnetic metamaterials due to its extremely low loss. In particular, the magnetic permeability of ferrite materials is dependent on an EMF, allowing for the realization of magnetically manipulable negative-index materials [33]. Combining the time-reversal symmetry breaking nature under an EMF, the one-way waveguiding tunable by an external magnetic field has been realized as well [34–36]. Bi and coworkers have shown that the thermally tunable negative index can also be realized with the magnetic metamaterials around the Curie temperature *T*_{c} based on the ferrimagnetic-paramagnetic transition of ferrite material. However, due to the narrow temperature range for the Mn-Zn ferrite material used in their work, the tunability is highly limited. Differently, for the single-crystal YIG employed in present design an even wider temperature range is permitted so that the effective refractive index can be adjusted from negative to zero and then to positive [37], signifying a great possibility to realize the gradient negative-zero-positive index material (NZPIM) [38, 39] when an appropriate gradient temperature field is applied to the system.

### 2.1. Photonic band diagrams and effective-medium theory

To examine the eigenmodes of the magnetic metamaterials, we employ the multiple-scattering theory to calculate the photonic band diagrams, which is proved to be powerful for the systems consisting of nonoverlap spheres or circular cylinders [40–45]. As for the effective electric permittivity *ε*_{eff} and magnetic permeability *μ*_{eff}, we will present simply a coherent potential approximation based effective-medium theory to retrieve these two constitutive parameters [46]. First thing first, for definiteness we should give the magnetic permeability of the single-crystal YIG ferrite rods fully magnetized along the *z* direction, parallel to the rod axes, which is a second rank tensor given by [47]

with *H*_{0} is the sum of the EMF applied in the *z* direction and the shape anisotropy field [47], *α* = 0 [31] and *ε*_{s }= 25 to obtain the eigenmodes. For the 2D system, the transverse electric (TE) mode and the transverse magnetic (TM) mode are decoupled and we consider only the TM mode with the electric field polarized along the rod axis. In this case, the magnetic field of the incident wave is perpendicular to the EMF so that the magnetic field will interact with the precessing magnetic dipoles of the ferrite, thus leading to the control of an EMF on the magnetic permeability.

To calculate the eigenmodes and simulate the electric field patterns, we illustrate here how the multiple scattering theory is used to serve the purpose. For an incident TM wave, the electric field impinging to the *i*th ferrite rod can be expanded in terms of the vector cylindrical wave functions

where *E*_{0} is the amplitude of the electric field, *m*th order cylindrical Bessel function, *k*_{0} is the wavenumber in the vacuum, **r**_{i} is the position vector of the polar angle *ϕ*_{i} in the coordinate system with the origin at the *i*th ferrite rod, **e**_{z} is the unit vector along rod axis, *p*_{m} is the expansion coefficient for the *m*th order partial wave of an incident field. The total scattering electric field can also be obtained by summarizing the scattering electric field from all the ferrite rods

where *m*_{c} is the critical angular momentum in the simulation to ensure the numerical convergence, *N* is the number of the ferrite rods, *m*th order Hankel function of the first kind, *m*th order scattering coefficient for the *i*th ferrite rod, which can be obtained according to

where *j*th ferrite rod into the incident wave on the *i*th ferrite rod and *i*th ferrite rod, which was obtained exactly in literature [48].

In Eq. (5), *r*_{s} the radius of the ferrite rod,

The magnetic field can be derived easily from Maxwell’s equations. To calculate the photonic band diagram, we should set *p* = 0 to solve the stationary-state equations so that the eigenfrequencies corresponding to the wavevectors in reduced Brillouin zone can be obtained.

For convenience, we recapitulate the results for the effective-medium theory; more details are referred to the published literature [46]. The scenarios of the effective-medium theory are as follows: (1) transform the periodic lattice of the magnetic metamaterials into the effective medium with effective constitutive parameters *ε*_{eff} and *μ*_{eff}; (2) take the unit cell of the magnetic metamaterials as an equal-area coated rod with ferrite rod as the inner core and the background medium as the coated layer with radius *r*_{0}, which is evidently an approximation, applicable only for the lattice with high symmetry. For a square lattice the radius *ε*_{eff} and *μ*_{eff} are determined by the condition that the total scattering of this coated rod in the effective medium vanishes in the long wave limit, namely, *ε*_{eff} and the effective magnetic permeability *μ*_{eff}.

where *f* is the filling fraction with

with

### 2.2. Phase patterns and wavefront engineering

By use of multiple-scattering theory, we calculate the photonic band diagrams for the magnetic metamaterials composed of the single-crystal YIG ferrite rods of the radius *ε*_{eff} and *μ*_{eff} as further corroborated by the effective constitutive parameters shown in Figures 1(b) and (d). In particular, the first, second, and the third bands are degenerated at Γ point; meanwhile, the second band is nearly flat, signifying the characteristic of longitudinal mode. This accidental degeneracy can lead to the appearance of effective zero index with *ε*_{eff} and *μ*_{eff} experience a nearly linear transition from negative to zero and then to positive except that a very narrow magnetic resonance appears, which is the difference of the magnetic metamaterials from the dielectric photonic crystals. This might be significant for investigating the EM features of NZPIM in frequency domain [50, 51]. More importantly, by decreasing the EMF from

From the photonic band diagrams and the effective constitutive parameters *ε*_{eff} and *μ*_{eff}, we have obtained a good MZIM at the working frequency

A particular functionality of the ZIM is to tailor the wavefront of the incident EM wave due to the zero phase delay inside the ZIM. We demonstrate such property by designing four typical outgoing interfaces sculptured from the MZIM, which are used to manipulate the wavefront of an incident Gaussian beam. The results are shown in Figure 3, where we can observe that the convex cylindrical face can transform the plane wavefront into the cylindrical one as shown in panel (a), different from the conventional convex lens that focuses the incident beam. It should be noted that the inhomogeneity of the outgoing beam arises from the anisotropy of the magnetic metamaterials. On the contrary, the concave cylindrical face can be used to focus the incident beam as shown in panel (b), behaving like a conventional convex lens but not a concave lens. The triangular prism can be used to split the incident beam into two separated ones propagating perpendicularly to the outgoing interfaces. More generally, we have shown in panel (d) an ordinary undulated interface that transforms the wavefront into the one identical to the interface. Actually, more imaginable configurations can be designed to engineer the wavefront in practice. In addition, the effective index of the magnetic metamaterials can be controlled flexibly by an EMF, which can be used to transform the functionality of the above systems, for example, from focusing to defocusing or in an opposite manner.

### 2.3. Thermally controllable effective index

Another important property of ferrite materials is its saturation magnetization that is dependent on temperature, which can also be handled to control the EM properties of magnetic metamaterials. Single-crystal YIG bears a high Curie temperature *C* (373.15 K) is considered for the magnetic metamaterials of triangular lattice with the lattice separation *a* = 10 mm and the rod radius

To examine the thermal effect on the magnetic metamaterials, we keep *T*. The effective constitutive parameters *ε*_{eff} and *μ*_{eff} are presented in Figure 4(a)–(c), respectively, under three different temperatures 306, 335, and 362 K, corresponding to the saturation magnetization *T* = 306 K a nearly matched negative-index material with

With the above knowledge, we can examine the performance of the magnetic metamaterials for molding EM wave propagation by simulating the field patterns of a TM Gaussian beam incident normally on a triangular prism with the apex angle *λ* can be observed as shown in panels (d) and (h).

## 3. Omnidirectional isotropic radiation via RAZIM

It has been shown that 2D RAZIM shell can be used to generate 2D isotropic radiation due to the fact that it can trap the anisotropic higher order modes, while it is transparent for the isotropic 0th order EM modes [19]. However, the efficiency is quite low since all the energy from the higher order modes are wasted, especially, when multiple sources are used since the energy of the higher order modes in that case occupies an even larger portion. We present an improved system with the 2D RAZIM shell enclosing a line source together with a conventional dielectric rod, which permits the generation of a perfect 2D EM mode; meanwhile, it is experimentally feasible. The dielectric rod can rescatter the anisotropic higher order EM modes and transform them into the isotropic 0th order modes. By positioning the dielectric rod at the strong field region of the anisotropic higher order modes, it can thus realize a great enhancement of omnidirectional radiation. The intrinsic loss of the RAZIM shell can also be examined and a gain-particle is used to compensate this dissipation.

### 3.1. Theoretical approach

The configuration of the system is schematically illustrated in Figure 6, where the shadowed green region is the RAZIM shell with *a* and *b* the inner and outer shell radii, and the dielectric rod and the line source are positioned inside the shell and denoted by *D* and *S*, respectively. In the cylindrical coordinate, the electric permittivity and magnetic permeability tensors of the RAZIM shell are characterized by [19, 52, 53]

where

#### 3.1.1. RAZIM shell enclosing only a single line source

In the framework of the generalized Lorenz-Mie theory, the EM field propagating in the RAZIM region can be expanded into the linear combination of the corresponding eigenmodes [19, 52]

where *m* runs from −∞ to ∞. The corresponding magnetic field in the transverse *xoy* plane can be obtained from Maxwell equations

for the TM waves. The electric field radiated by a TM line source positioned at **l**_{s} can also be expanded around the RAZIM shell center [54, 55]

where **r** is the position vector and *x* axis. With these expansions, we can write the total electric field in different regions according to

where the coefficients *A*_{m} characterize the reflection of *m*th order partial wave from the RAZIM shell, and *D*_{m} describes the transmission of the *m*th order partial wave radiating out of the shell.

By matching the boundary conditions, namely, the continuity of the tangential components of the EM field *E*_{z} and

where the generalized Mie coefficients are given by

Regarding the RAZIM shell considered in our system, *ϕ*, as demonstrated by Eqs. (10) and (13). In addition, for the case when

#### 3.1.2. RAZIM shell enclosing a single line source together with a dielectric rod

To improve the radiating efficiency for the system shown in Figure 6(a), we have to transform the higher order modes confined within the RAZIM shell into the isotropic lower order mode, and then radiating outside the RAZIM shell. To this end, a dielectric particle *D* can be introduced inside the RAZIM shell as illustrated in Figure 6(b), which can rescatter the EM wave so that a part of the higher order modes can be converted into isotropic 0th order mode, thus enhancing the omnidirectional isotropic radiation from the RZAIM-shell-based system. Therefore, in this part we have to incorporate the contribution of the dielectric rod into theoretical framework. As is certain, the dielectric rod will change the scattering field inside central area surrounded by the shell and that propagating inside the shell. Therefore, the partial wave expansion coefficients *A*_{m} and *C*_{m} in Eq. (14b) should be altered

where *E*_{m} are the partial wave expansion coefficients of the scattered EM field from the dielectric rod *D*. As a result, the partial wave expansion coefficients *B*_{m}, and *D*_{m} are altered accordingly as indicated by Eq. (14a). It should be noted that the coefficients *p*_{m}, *q*_{m}, *D* is introduced.

To consider the scattering behavior of the dielectric rod and obtain *E*_{m}, we should transform the expanding partial waves from the shell center to those from the dielectric rod *D*. The electric field inside the dielectric rod

where **l**_{d} is the position of the dielectric rod with *r*_{d} is the radius of the dielectric rod *D*. The partial wave expansion coefficients *T*_{m} and *S*_{m} are given by

where *a*_{m} and *b*_{m} are the Mie coefficients of the dielectric rod, *I*_{m} and *R*_{m} correspond to the contribution from the line source and that scattered inside by the RAZIM shell

The parameters in Eq. (19) are defined as *S* with *a*_{m} and *b*_{m} of the dielectric rod can be easily obtained from the Mie theory [57]

where *ε*_{d} and *μ*_{d} being the permittivity and permeability of dielectric rod, respectively. For a particular case when the RAZIM shell is removed from the system, the corresponding scattering from the shell disappears, leading to *R*_{m} = 0. Combining Eqs. (16), (18), and (19) and after some mathematical manipulations, we can finally arrive at a set of linear equations

which determine the coefficients *E*_{n} For now, we have solved the scattering problem for the RAZIM systems in both Figure 6(a) and (b), based on which we can gain insight into the physical mechanism of the phenomenon as well as the role of the RAZIM shell and surrounded dielectric rod, meanwhile, optimize the configuration to achieve a better radiation efficiency.

### 3.2. Amplifying radiation with dielectric particle

In the simulations and calculations in this part except otherwise specified, the parameters for the RAZIM shell are *a* = 0.5, *b* = 1,

To optimize the performance of the dielectric rod, we calculate the total power radiating out of the RAZIM shell, which is defined as

where **S** is the Pointing vector, the integral curve *L* is a circle around the shell center *O* with the radius larger than outer radius of the RAZIM shell *b*. Considering the fact that only the 0th order cylindrical wave is radiated out, the radiating power can be approximately evaluated according to *I*_{0}. For the radiating power without the RAZIM shell *x*_{d} as indicated by the blue dashed line in panels (a). Even when the dielectric rod is replaced by a gain particle, *I*_{0} with the irradiance is defined as *I*_{wi}, it can be found that the irradiance is reinforced by over 10 times as indicated by the red solid line, consistent with the result shown in Figure 8(a). In addition, a highly isotropic feature is demonstrated as well by examining the irradiance map. For the convenience of comparison, we also present the irradiance *I*_{wo} for the system without the RAZIM shell as denoted by the blue dash line, which is not isotropic anymore and no evident enhancement is achieved with either the dielectric rod or the gain particle. The efficiency of the dielectric rod can be evaluated by comparing *I*_{wi} with the radiance

### 3.3. Amplifying radiation with gain particle

In practice, the loss should be an inevitable issue due to the finite size of the RAZIM shell and its resonant nature. To illustrate the effect of the loss on the radiation enhancement, we present in Figure 9 the results for the system with the loss taken into account, where we can find that the output radiating power is reduced seriously compared to the results shown in Figure 8. To compensate the energy loss, the active coated nanoparticles might be a good choice. By enclosing a gain particle with

The size of the RAZIM shell also has effect on the enhancement of the output radiation, which is illustrated in Figure 10 for the RAZIM shell with *a* = 1 and *b* = 2. It can be found that the enhancement of the omnidirectional radiation becomes difficult in that only in a quite narrow range of the enhancement can be achieved as shown in Figure 10(a). The underlying physics is as follows. Since the field due to the anisotropic higher order modes is trapped inside the RAZIM shell, the field is highly confined and anisotropic when the shell is small so that there appears the region with strong field easily. Then, a dielectric rod located near the position with strong field can rescatter the anisotropic modes into the isotropic mode, inducing a remarkable increase of the output radiating power. Differently, for large RAZIM shell, the anisotropic higher order modes are usually much less confined. As a result, the introduction of a dielectric rod cannot yield a strong rescattering of the trapped anisotropic modes into the isotropic field, resulting in a smaller enhancement of radiation power. Roughly speaking, the design works when the RAZIM shell is small. Nevertheless, the size of RAZIM shell is not necessarily limited to subwavelength scale. As shown in Figure 10(b) by the red solid line, a strong enhancement can still be achieved by a proper arrangement of the dielectric particle.

## 4. Conclusion

In summary, we have designed a kind of zero-index materials (ZIMs) with magnetic metamaterials based on the multiple scattering theory and effective-medium theory. The zero phase delay inside the ZIMs and the wavefront engineering are demonstrated. It is also shown that the effective index can be flexibly tuned by an external magnetic field and temperature, enabling the manipulation on the functionalities. In addition, the anisotropy of the ZIMs and the nonreciprocal feature of the magnetic metamaterials might induce nonreciprocal Goos-Hänchen and other physical consequences. Then, in the second part we have considered a radially anisotropic ZIM (RAZIM) based system with a RAZIM shell enclosing both a line source and a dielectric rod, which can implement a remarkably enhanced omnidirectional radiation. An exact theoretical approach is developed to solve the system, based on which we can optimize the configuration to achieve the high efficiency, more importantly, discover the underlying physics. Actually, the RAZIM shell allows only the 0-th order isotropic mode to radiate outside the system, ensuring the isotropy of the radiation. Differently, the anisotropic higher order modes are trapped, thus creating a strongly inhomogeneous standing wave. The dielectric rod can rescatter the anisotropic modes into isotropic one, enhancing the omnidirectional radiation remarkably. Besides, the present design is experimentally feasible, and meanwhile provides a highly efficient omnidirectional radiation by spatial power combination.

## Acknowledgments

This work was supported by the China 973 Projects (No. 2013CB632701), the National Natural Science Foundation of China (Nos. 11274277, 11574055, and 11574275), MOE of China (B06011), the Zhejiang Provincial Natural Science Foundation of China (LR16A040001), and the open project of SKLSP (KF2016_3) in Fudan University.