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.
- zero index materials
- magnetic metamaterials
- multiple scattering theory
- omnidirectional isotropic radiation
- Mie theory
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
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
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
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 . 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
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
with , , , and , where is the resonance frequency with MHz/Oe the gyromagnetic ratio,
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
where is the structural factor that transforms the scattered wave from the
In Eq. (5), and are, respectively, the Bessel function and the Hankel function of the first kind, the superscript ‘′’ represents the derivative with respect to with
The magnetic field can be derived easily from Maxwell’s equations. To calculate the photonic band diagram, we should set
For convenience, we recapitulate the results for the effective-medium theory; more details are referred to the published literature . 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
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 mm and arranged periodically with square lattice with the lattice separation mm. The results are shown in Figure 1(a) and (c), corresponding, respectively, to the magnetic metamaterials under the EMF Oe and Oe. It can be found that there appear no eigenmodes below the first band, suggesting the formation of the photonic band gap. The first band possesses the negative slope, namely, , corresponding to the negative
From the photonic band diagrams and the effective constitutive parameters
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 K, allowing a wide controlling temperature range and thus a better tunability on effective refractive index. The temperature field ranging from 0 (273.15 K) to 100°
To examine the thermal effect on the magnetic metamaterials, we keep Oe unchanged, and tune the temperature
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 . The results are shown in Figure 5, where we can observe that under the temperature K the incident Gaussian beam is split into two separated beams propagating with the refractive angle as shown in panels (a) and (e), equal to the incident angle , implying that the effective index of the triangular prism is , consistent with the results from effective-medium theory given in Figure 4(a). Under the temperature K the effective constitutive parameters and , corresponding nearly to an MZIM, the electric field exhibits an invariant phase inside the prism, resulting in two perpendicularly outgoing beams with the same phase at two lateral interfaces as shown in panels (b) and (f). With further increasing the temperature to K, we obtain the effective constitutive parameters and , corresponding to the effective index , the Gaussian beam experiences a little bit focusing and collimation as shown in panels (c) and (g). When the temperature reaches K, the effective constitutive parameters and , corresponding to the effective index , a strong focusing with the outgoing beam waist radius shrunk nearly to
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 . 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
where , corresponding to the radially anisotropic zero index. The origin of the cylindrical coordinate fixed at the center of the RAZIM shell. A line source of TM polarization is considered to interact with the RAZIM shell. For convenience, we first consider the simple system schematically illustrated in Figure 6(a) to depict the physical picture, based on which the system with further introducing a dielectric particle as shown in Figure 6(b) can be solved by further taking account of the mutual scattering between the dielectric particle and the RAZIM shell.
3.1.1. RAZIM shell enclosing only a single line source
where with the wavenumber in the vacuum, and are, respectively, the th order Bessel functions and Hankel functions of first kind with the order , and the summation
where the coefficients
By matching the boundary conditions, namely, the continuity of the tangential components of the EM field
where the generalized Mie coefficients are given by
Regarding the RAZIM shell considered in our system, , implying that the order of the cylindrical functions and in Eqs. (10), (13), and (16) tends to infinity for . As a result, and , leading to the vanishment of the Mie coefficient for . Therefore, it follows from Eq. (14) that , , and for . This indicates that the permitted propagating EM waves in the RAZIM shell are nearly independent of the azimuthal angle
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
To consider the scattering behavior of the dielectric rod and obtain
The parameters in Eq. (19) are defined as , , and is the distances from the dielectric rod to the line source
which determine the coefficients
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
To optimize the performance of the dielectric rod, we calculate the total power radiating out of the RAZIM shell, which is defined as
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 , we can compensate the energy loss from the RAZIM shell, yielding an enhancement of the output radiating power by a factor of about 7 as indicated by the red solid line in Figure 9. For comparison, the case for the system without the RAZIM shell but with a gain particle is also simulated as indicated by the blue solid line. Neither significant increase nor isotropy in the output radiation is achieved, suggesting once again the crucial role of the RAZIM shell.
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
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.
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.
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