Resonant Properties of Dielectric Metamaterials
*E. Semouchkina1, G. Semouchkin1, R. Mittra2, and M. Lanagan1
1Materials Research Institute, 2Department of Electrical Engineering,
The Pennsylvania State University, University Park, PA, 16802, USA
Electromagnetic metamaterials, artificially created by embedding periodic inhomogeneties in a
host matrix, exhibit many interesting properties, which, in turn, open the way to novel
applications. It has been previously demonstrated that an artificial medium comprising of a
combination of arrays of microstrip split-ring resonators and wires has a negative refractive index
in a certain frequency range [1, 2]. Another well-known class of engineered materials, viz.,
photonic bandgap (PBG) crystals, perform due to the dispersion effects of wave propagation in
periodic structures [3, 4].
In this paper we report the results of a study of dielectric structures consisting of higher-
permittivity posts embedded periodically in a lower-permittivity host matrix. Such structures can
be fabricated by using low temperature co-fired ceramic (LTCC) technology, which allows us to
co-process different dielectrics in a single module . We simulate the transmission and reflection
characteristics, as well as the field distribution in these structures, and show that they exhibit
unusual electromagnetic properties because of interaction between the neighboring dielectric
resonators at their respective resonant frequencies.
2. Results and discussion
Cylindrical DRs are known to support different resonant modes (TE, TM and hybrid (HEM) types
), depending on the frequency, resonator dimensions, permittivity value and the excitation type.
The electromagnetic field patterns of the DR’s resonant modes resemble oscillating magnetic or
electric moments, which can be represented by equivalent magnetic or electric dipoles . We
have simulated a single cylindrical dielectric resonator (DR) made from high permittivity material
surrounded by a low permittivity substrate and excited by using an open-ended microstrip line, as
well as periodically arranged high-permittivity cylinders forming a square lattice in a low-
permittivity host material, with horn-shaped microstrip feedlines on top of the substrate.
We have employed the finite difference time domain (FDTD) method to calculate the amplitude
and phase distributions of the electromagnetic fields, as well as the transmission and reflection
characteristics, of the individual elements and that of the entire periodic structure. The perfectly
matched layer (PML) type of boundary condition is applied at all boundaries of the computational
domain, except for the bottom boundary, which is perfectly conducting ground plane. A Gaussian
type of source modulated by a sinusoidal wave is employed for the excitation. The Fast Fourier
Transformation procedure is used to convert the results of the time-domain simulations into the
The electromagnetic field configuration of the first resonant modes of a cylindrical DR excited by
a TEM-type of incident electromagnetic wave is shown in Fig. 1. This mode can be associated
with the HEM11? type . As seen from the figure, the field patterns of this mode are similar to
those of a magnetic dipole located horizontally on the ground plane. This equivalent magnetic
dipole is oriented perpendicularly to the direction of the incident wave propagation, and in view of
the symmetry of the cylindrical DR, the dipole can be rotated in the horizontal plane if the
direction of incident EM wave is changed. The field patterns of the next higher frequency resonant
mode (TM01?) (Fig. 2) are similar to the fields of a vertical electric dipole.
Individual resonators have been assembled in an array forming a periodic structure. Fig. 3a depicts
a sample fabricated by using the LTCC process, in which Bi-Zn-Ta-O ceramics with the dielectric
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constant of 62 was co-processed with glass-bonded alumina matrix that has a dielectric constant of
7.8 . Cylindrical rods of equal height and radius of 1.53 mm were embedded in the substrate
matrix with the thickness of 1.53 mm forming a square or triangular lattice with the lattice
constant of 5.6 mm. To measure the transmission and reflection characteristics of the structures,
horn-shaped microstrips - with their geometries optimized to provide a maximum input power -
were designed and this microstrip pattern was printed on the substrate using silver ink. The
simulated and measured characteristics of the samples were found to agree well.
For frequencies far from the resonance of individual DRs, the transmission characteristics of the
periodic structures were similar to those of the uniform dielectric media with an effective
permittivity that is intermediate between the high permittivity cylinders and low permittivity
matrix. However, in the vicinity of the resonant frequencies of the individual DRs, the character of
wave propagation characteristics shows drastic change. Strong minima appear in the reflection
characteristics, and there are corresponding peaks in the transmission characteristics centered at
the resonant frequencies of the individual DRs (Fig. 3b).
Electromagnetic field patterns in metamaterial sample in the vicinity of the two first resonant
frequencies of individual DRs are presented in Figs. 3c and 3d. Fig. 3c demonstrates that, when
the resonant conditions for the mode depicted in Fig. 1 are not yet achieved and the field
magnitudes inside the DRs are small (see the patterns sampled at 9.75 GHz and 10 GHz), the
forming magnetic dipoles are oriented perpendicularly to the direction of the incident wave
propagation. However, at resonance, the DRs become coupled through strong magnetic and
electric fields causing re-orientation of the dipoles (see the pattern sampled at 10.5 GHz). Each of
the 3D resonant modes of the DR exhibits a local symmetry of the electric and magnetic fields
that, in turn, defines the interaction with the neighboring resonators in the lattice. Fig. 3d presents
the distributions of magnitudes and phases of the electromagnetic field components at the second
resonant frequency of the individual DR. This figure serves to demonstrate the formation of the
magnetic meta-domains due to the coupling of the modes corresponding to the vertical electric
dipoles (see Fig. 2). We note that the neighboring stripes in the pattern for the phase distribution of
the Hy component have 1800 phase difference.
Coupling between the individual DRs is also responsible for the resonant mode splitting, when a
resonant band consisting of closely located peaks appears (Fig. 4a) instead of a single resonant
peak for the individual DR. The structure of this band depends on the distance between the DRs in
the lattice. A wide spectrum of coupling states for the dipole community appears, especially for
the modes, which provide for many degrees of freedom for the dipole reorientation due to the
symmetry of the cylindrical DRs. Collective reorientation of the meta-dipoles causes formation of
specific paths for electromagnetic wave propagation in the metamaterial. Simulated far-field
patterns, which are shown in Fig. 5, illustrate the variation in the directivity and the number of
beams radiating by a slab of the metamaterial with a square lattice with changing frequency.
It is worth noting that the observed characteristics of the resonant wave propagation described
above are different from the diffraction processes observed in photonic bandgap structures. Fig. 4
compares the transmission characteristic of the metamaterial sample depicted in Fig. 3a with the
characteristic simulated for the structure with the same parameters, excepts that the cylindrical
dielectric resonators with the height of 1.53 mm are replaced by infinitely long dielectric rods
modeled by using the PML boundary at the top surfaces of the rods. The latter structure is a
typical PBG structure, since there are no resonances in the rods terminated by the PML surface.
We note from Fig. 4 that the transmission characteristics for the two structures are quite different
Also, a comparison of the field distributions in these structures showed that enhanced transmission
occurs at resonance in the first type of structure, while band gaps appear in the second one the due
to diffraction effects.
This work demonstrates that periodic dielectric structures, which incorporate arrays of high
permittivity DRs in low permittivity dielectric matrixes, provide specific paths for electromagnetic
wave propagation in conjunction with the effect of coupling between the 3D resonant modes of the
individual DRs, and that the transmission mechanism in the resonant structures is different from
the one in the PBG crystals. The results presented herein open new avenue for the design of meta-
crystals, in which chemical bonds are substituted by the electromagnetic coupling between the
DRs. A broad group of new architectures useful for novel microwave and mm-wave applications
could be based on this concept. We have also shown that periodic dielectric structures can be
fabricated using the framework of the LTCC process, making it possible in the future to design
devices compatible with integrated modules based on the LTCC platform.
This work was supported by the National Science Foundation under Award DMI-0339535 and as
part of the Center for Dielectric Studies under Grant 012081.
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Fig. 1. Field patterns and schematic of electromagnetic field configuration for the HEM11? mode of a DR.
Fig. 2. Field patterns and schematic of electromagnetic field configuration for the TM01? mode of a DR.
Ex Ey Ez
Hx Hy Hz
Ex Ey Ez
(a) (c) Download full-text
Fig. 3. (a) Top view on metamaterial sample, (b), S11 of the sample with DRs (solid curve) and of a uniform
substrate (dashed curve), (c) Hz component distribution in metamaterial in the vicinity of the first resonant
frequency, and (d) field patterns at the second resonant frequency.
Fig. 4. S21(a) of the sample with the same parameters, as in Fig. 3, and (b) of a sample with similar
parameters, but with infinitely long rods instead of cylindrical DRs.
Fig. 5. Examples of far field patterns from metamaterial sample at different frequencies demonstrating
variation in directivity and number of propagating beams.
6 7 8 9 10 11 12 13 14
Frequency (x10-1, GHz)
Frequency (x10-1, GHz)
Ez Hx Hy
phase amplitude amplitude amplitude phase
10.75 GHz 10.0 GHz 10.25 GHz10.5 GHz9.75 GHz
f=9.95 GHz f=10.18 GHz