Experimental Demonstration of Anomalous Field Enhancement in All-Dielectric Transition Magnetic Metamaterials

Anomalous field enhancement accompanied by resonant absorption phenomenon was originally discussed in the context of plasma physics and in applications related to radio-communications between the ground and spacecraft returning to Earth. Indeed, there is a critical period of time when all communications are lost due to the reflection/absorption of electromagnetic waves by the sheath of plasma created by a high speed vehicle re-entering the atmosphere. While detailed experimental studies of these phenomena in space are challenging, the emergence of electromagnetic metamaterials enables researchers exceptional flexibility to study them in the laboratory environment. Here, we experimentally demonstrated the strong localized field enhancement of magnetic field for an electromagnetic wave propagating in Mie-resonance-based inhomogeneous metamaterials with magnetic permeability gradually changing from positive to negative values. Although these experiments were performed in the microwave frequency range, the proposed all-dielectric approach to transition metamaterials can be extended to terahertz, infrared, and visible frequencies. We anticipate that these results, besides most basic science aspects, hold the potential for numerous applications, including low-intensity nonlinear transformation optics, topological photonics, and the broader area of surface and interface science.

properties of metamaterials, such as their dielectric permittivities (ε ), magnetic permeabilities (μ ), and refractive indices (n), can be designed to be positive, negative, or even zero at any selected frequency by properly adjusting the dimensions, periodicity, and other properties of their unit cells, or meta-atoms [12][13][14][15][16][17] . Inhomogeneous metamaterial with the effective parameters, ε , μ , or n, gradually changing from positive to negative values, the so called transition metamaterials [9][10][11] , have attracted significant and growing interest over the last few years. In particular, anomalous field enhancement and resonant absorption phenomena have been predicted for obliquely incident transverse-electric (TE) and/or transverse magnetic (TM) waves at the point where the real part of μ and/or ε vanishes [18][19] . These effects, besides most basic science aspects, hold the potential for numerous applications, including low-intensity nonlinear optics 20,21 , wave concentrators [22][23][24] , and polarization-sensitive devices 25 . Moreover, these phenomena may have important implications for different areas of surface and interface science 26 .
In our recent theoretical studies 27,28 , we predicted a resonant field enhancement that occurs near the zero refractive index point under oblique incidence of the electromagnetic wave on a transition metamaterial with magnetic permeability varying from positive to negative values. The physics of the phenomenon of resonant field enhancement in such graded-index metamaterials can be summarized as follows: for incident, transverse-magnetic, or transverse-electric waves, the thin layer near the zero-index point (transition point) can be considered a very thin capacitor or inductor that accumulates infinitely large electric or magnetic field energy, respectively, if we neglect the effects of dissipation and spatial dispersion. Note that this energy accumulation takes place only in the case of obliquely incident waves, as only in this case the electric (magnetic) field at the oblique incidence has a non-zero component in the direction of propagation. Since electric displacement D (magnetic induction B) must be continuous, the electric field E (magnetic field H) anomalously increases as ε (μ ) tends to zero.
In this study, we designed and fabricated graded magnetic permeability metamaterials using a high-index ceramic cube array with graded lattice constants such that the magnetic permeability gradually changes from a positive to a negative value along the propagation direction ( Fig. 1(a)). As opposed to previously realized graded-index structures, such as electromagnetic cloaks that were based on resonant metallic resonators embedded in a dielectric matrix, we designed all-dielectric transition metamaterials. While this first experimental demonstration reported here was performed at microwave frequencies, where metal losses are negligible, an all-dielectric design will be advantageous at optical frequencies, where metallic losses are significant and, moreover, plasmonic enhancement at metal edges may mask the effects of the anomalous field enhancement discussed above.

Results
The enhancement effect in the transition metamaterial can be understood by solving the wave equations inside the transition metamaterial. Assume that a TE ( = , , , = , , ) wave at an incident angle of θ is propagating from air (x < -h) into a transition medium with a width of 2h (− < < ) h x h , where relative dielectric permittivity ε is a constant ε r0 , and relative magnetic permeability μ has a graded profile given by µ x h is changing from positive to negative, with zero point at ξ ξ = , ( = ) = f 0 0 0. Since the medium is homogeneous in the y direction, the electric field component can be written as . Then, using the While in the normal incidence case, θ = 0, no unusual behavior is predicted for both electric and magnetic field components, in the case of oblique angle (θ ≠0), a strong enhancement for the longitudinal component of H x at the position where permeability is changing from positive to negative is predicted. Figure 1(b) shows the results of numerical simulations, confirming this prediction.
In order to demonstrate the predicted field enhancement in the vicinity of the zero-μ transition point, we designed and fabricated all-dielectric metamaterials made of high-refractive-index dielectric cubes. Such cubes have been shown to produce electromagnetically induced electric and magnetic resonances. A magnetic resonance originates from the excitation of a particular electromagnetic mode inside the cube with a circular displacement current of the electric field. In contrast to more conventional, split-ring resonator based metamaterials that possess anisotropic electromagnetic response 6-8 , the properties of the all-dielectric structures demonstrated here are isotropic.
The transition metamaterial was realized using a high-refractive-index cube array with graded lattice constants 17 , so that it possessed a graded magnetic permeability profile, varying from positive to negative values, as shown in Fig. 2. The high-index cubes with a side length of 2.2 mm were made from a dielectric ceramic material, CaTiO 3 , doped with 15 wt% ZrO 2 , which was synthesized by the solid state reaction method under 1400 °C, and the measured permittivity of it was 122 + 0.244i. The lattice constant was gradually changing from 6 mm to 5 mm over the length of 66 mm.
Using commercial, full-wave, finite-element simulation software (Microwave Studio, Computer Simulation Technology), we performed a series of scattering (S) parameter simulations for the dielectric cube, with different unit cells' sizes c covering the X band as shown in Fig. 2(a). The standard retrieval procedure 29,30 was used to obtain the effective material parameters, as shown in Fig. 2(b). In our design, 10.472 GHz was chosen as the operating frequency. Then, the array was designed such that the effective magnetic permeability of an array would change from 0.16 to -0.31 along the x direction.
The transition metamaterial sample comprises an array consisting of 12 rows of ceramic cubes. Every two consecutive rows possess the same unit cell sizes, varying in range from 6 mm to 5 mm, as shown in Fig. 2(c). Therefore, from bottom to top, permeability varies gradually from positive to negative in the frequency range around the operating frequency. In the experiments, these cubes were imbedded in an ABS matrix, which makes the entire structure firm and steady.
We utilized the 2D near-field scanning system to measure the power distribution. The corresponding H-field, for which the enhancement was theoretically predicted, was then calculated based on the measurement results. As shown in Fig. 3(a), the near-field scanning system consists of two parallel metal plates with a gap of 11 mm between them, forming a plane waveguide that supports the mode, with an E field pointing in the z direction in the X band. Absorbers on the lower metal plate restrict the measurement area and the path for feeding the incident wave. Microwaves spanning across the X band (8)(9)(10)(11)(12) GHz) that includes the operating frequency of the designed sample are generated by the vector network analyzer (VNR), Agilent ENA 5071 C. A detecting probe, which is also connected to the VNR, is installed in the center of the upper metal plate to collect the amplitude and phase of the E field. The transition metamaterial sample was placed in the middle of the lower plate with an angle of 30 degrees between its normal and the incident direction. During the measurement, the sample was moved with the bottom plate along the U and V directions with a scanning step of 3 mm, and a full, 2D spatial field map of the microwave scattering pattern was acquired, both inside the transition metamaterial and in the surrounding free-space region.
The measurement results of the power, shown in Fig. 3(b), can be used to calculate the field E z and then transformed into the H field, according to Eq. 2. The calculated results of the longitudinal field component (H x ) perpendicular to the boundary of the metamaterial are shown in Fig. 3(c). At the region around the arrays with zero permeability, H x is greatly enhanced, which agrees well with the prediction in Fig. 1.

Conclusions
We experimentally observed a high enhancement of the H field for an electromagnetic wave propagating in inhomogeneous metamaterials with permeability linearly changing from positive to negative values. Although these preliminary results were obtained in the microwave frequency range, the proposed all-dielectric approach to transition metamaterials can be extended to infrared and optical frequencies.
An important advantage of the all-dielectric approach in this spectral range is that unlike plasmonic metal-dielectric meta-atoms, all-dielectric cubes do not suffer conduction losses at optical frequencies.