Metamaterial Resonators: Theory and Applications
Metamaterials (MTM) are artificial composite materials engineered to possess extraordinary electromagnetic properties, such as negative index characteristics (ε< 0 and µ < 0). The characteristics of MTMs depend on the properties of the host materials, embedded material, volume of the fraction, operating frequency and the morphology of the composite material such as the dimensions and shapes of the host structure. The array of MTM unit cells provides the effect of new medium when the sizes of unit cells are sub-wavelength, features that are actually smaller than the wavelength of the waves they affect. As depicted in Figure 1, MTMs accommodate the missing quadrant III of the ε– µ domain.
The trend of nature is to favor conventional materials, illustrated in quadrant I of Figure 1, and, to a lesser degree, single negative materials (quadrants II and IV). The sign (+ or -) of permittivity/permeability are not restricted by physical law so long as generalized entropy conditions for dispersive media are satisfied.1 This can be mathematically verified by inserting a plane wave into Maxwell’s curl equations with ε< 0 and µ < 0.
The negative index material n' (n= , ε< 0 and µ < 0), also referred to as left-handed (LH) material, exhibits anti-parallelism between phase and group velocity (v_{p}-||v_{g}). This causes strong localization and enhancement of fields, thereby enhancement in effective group velocity of resonators realized from MTM structures. Increased group velocity yields improvement in the slew rate and quality factor of the resonator, which is advantageous for lower phase noise oscillator circuits. Additionally, several important phenomena of classical physics reverse in negative index or LH media, as illustrated in Figure 2^{1}.
Figure 2 depicts the following: (a) Doppler effect (Dw > 0 for an observer seeing a retreating source), (b) Vavilov – Cerenkov radiation (backward radiation of a fast-wave particle in motion), (c) Snell’s law, (d) Goos – Hänchen effect (lateral shift in total internal reflection), (e) lensing effect (convex/concave LH lenses are diverging/converging) and (f) sub-wavelength focusing of an image by a flat slab (low spatial frequency features are focused by reversed Snell’s law while high spatial frequency feature are enhanced, due to a reverse transfer function associated with surface wave or surface Plasmon excitation^{2-8}).
Metamaterial Characterization
MTMs characterized by double (ε < 0 and µ < 0) or single negative (ε < 0 and µ > 0 or ε > 0 and µ < 0) structure can exhibit independent electric (E) and magnetic (H) responses described by ε and µ.
The E field causes magnetic polarization while the H field induces electrical polarization, known as magneto-electric coupling. Such media are called bi-isotropic. Media that exhibit magneto-electric coupling and are also anisotropicare called bi-anisotropic.
Four material parameters are intrinsic to magneto-electric coupling of bi-isotropic media. These are: ε, µ, κ and Χ, or permittivity, permeability, strength of chirality and the Tellegen parameter, respectively. In this type of media, the material parameters do not vary with changes along a rotated coordinate system of measurements. They are invariant or scalar.
The intrinsic magneto-electric parameters κ and Χ affect the phase of the wave. The effect of the chirality parameter is to split the refractive index. In isotropic media, this results in wave propagation only if ε and µ have the same sign. In bi-isotropic media with κ assumed zero and Χ a non-zero value, different results appear. Both a backward wave and a forward wave can occur. Alternatively, two forward waves or two backward waves can occur, depending on the strength of the chirality parameter.
The realization of negative index material for broadband operation from a set of arbitrary passive structure unit cells arranged in predefined order is challenging.^{9} On the other hand, it is important to understand the limits imposed to negative index material (ε < 0, µ < 0) by phase reversal between phase and group velocity and losses, as required by causality.
The estimation of refractive index calculated by^{1}
where n is the refractive index, z is the wave impedance, k is wave factor, and d is the physical length.
In isotropic media, the group velocity is v_{g}
v_{g} is the group velocity, v_{p} is the phase velocity, ω is angular frequency, k is the wave number. From equations 4 and 5,
From equation 6, group velocity is anti-parallel to the phase velocity in MTM media. For MTM media, the group index n_{g} given by
where n = (µε)^{0.5}
From equations 6 and 7, group and phase velocity of negative refractive index material manipulated for amplification of the evanescent mode waves in a resonator cavity improve resonator quality factor.^{4} The determination of n, ε and µ can be evaluated from S-parameters.^{10}
Metamaterial Resonator
Figure 3 shows the typical arrangement of planar split ring resonator (SRR) and complementary split ring resonator (CSRR) structures that exhibit negative permeability and negative permittivity.^{11} They can be characterized as a magnetic and electric dipole excited by the magnetic (H) and electric (E) fields along the ring axis.^{12}
Interestingly, single negative property (–ε or –µ) supported by a SRR or CSRR structure can offer a sharp stopband at the vicinity of resonant frequency. This enables storage of EM energy into the SRR or CSRR structure through an evanescent-mode coupling mechanism, resulting in a high quality factor.
As shown in Figure 4, SRR and CSRR structures act as an LC resonator driven by an external electromotive force. The value of inductance and capacitance of the SRR is described by^{14-16}
where I(k) is the Fourier–Bessel transform of the current function on the ring, is the azimuthal surface current density on the ring, c is the slot width, r_{0} is the average radius of the ring of the SRR, a and b are the geometrical parameters shown in Figure 4b, and β(x) = S_{0}(x)J_{1}(x)-S_{1}(x)J_{0}(x) with S_{n} and J_{n} being nth-order Struve and Bessel functions.^{16}
The behavior of SRRs and CSRRs, and their derived geometries, are strictly dual for perfectly conducting and infinitely thin metallic screens placed in a vacuum. In reality, this is not the case and a shift in resonance frequency occurs from losses, the finite width of the metallization and the presence of a dielectric substrate.
The realization of MTM media using SRRs, as shown in Figure 4, is based on its unit cell size being much smaller than the wavelength of the incident wave. The SRR behaves as a quasi-static LC resonant circuit fed by the external magnetic flux linked by the SRR unit cell. The SRR unit cell, realized by two coupled conducting rings printed on a dielectric slab, requires precise placement of metal patterns at two sides of a dielectric substrate. A recent publication reports the use of spiral resonators (SR), which provide a potential reduction in the electrical size of the MTM unit cell. Moreover, the SR is not bi-anisotropic and uniplanar, making for an easier fabrication process compared to SRR unit cells, if the metamaterial is viewed as a continuous medium for superlens and cloaking purposes.^{17}
Figure 5 shows the typical configuration of the SRR and SR. The SRR is shaped by two coupled conducting rings fabricated on a dielectric slab, whereas spiral resonators SR2 (two turns) and SR3 (three turns) are made by a single strip rolled up to form a spiral. Baena^{17} describes the topology of the SRR and SR and their equivalent circuits, where C_{0} is defined as the edge capacitance between the two rings over the whole circumference, L_{s} is the inductance of a single ring and r_{0} the average radius.
As shown in Figure 5, the total current in the unit cell is the sum of the currents on each ring. For a given value of the angular polar coordinate (Φ), the current lines go from one ring to another across the slot between the rings, in the form of field displacement current lines.^{17} The normalized quasi-static voltage V (Φ) and the electric current intensity I (Φ) along the strips, as a function of the angular polar coordinate (Φ), illustrate the size reduction advantages of SRs as a metamaterial medium. From the equivalent circuits depicted in Figure 5b, there is a reduction of the SR’s resonance frequency with respect to the SRR. The capacitance value for a SRR is C_{0}/4, whereas for an SR it is C_{0} and for an SR3 it is 2C_{0}. Therefore, the resonance frequency ratios are .
Figure 6 shows the comparative size reduction of the SR2 and SR3 with the identical resonant frequency. The size reduction is about 50 percent for the SR2 and 65 percent for the SR3 as compared to the SRR unit cell.18 The SR structures shown in Figures 5 and 6 are also amenable for Möbius “TWIST” (discussed in the first part of this series, published in the November 2014 issue), resulting in high Q-factor Möbius metamaterial strips resonators^{18-19}.
Metamaterial Resonator Dynamics
Figure 7 shows the electromagnetic (EM) wave propagation dynamics.^{1} The direction of the Poynting vector is parallel with the direction of phase velocity or wave vector in right-handed (RH) material, but these two directions are anti-parallel in left-handed (LH) material. CAD simulation exhibits a backward wave into the host transmission line, establishing a standing wave when coupled in-phase with the forward EM wave. The standing wave supports strong EM coupling between the host transmission line and the negative index resonator (MTM resonator), causing strong localization and enhancement of evanescent mode-coupled energy. The manipulation of MMT properties of evanescent mode-coupled resonators for tunable oscillator circuit applications has been reported.^{21}
The propagation constant k_{z} for the electromagnetic wave propagation in + ve z-axis is given by^{9}
which describes the evanescent mode propagation waves (ε > 0, µ < 0), and
which describes the forward mode propagation waves (ε > 0, µ > 0).
From equation 11, the evanescent mode propagation waves will have exponential decay along the z-axis.
The transmission and reflection coefficients, T' and R', can be derived by matching the EM fields at the interface of medium 1 and medium 2:
The transmission and reflection coefficients T' and R' of the transition from inside medium 2 to medium 1 are given by^{9}
The expression of the wave transmission coefficient (T) through both interfaces is derived by adding the multiple scattering dynamics, described by^{9}
For metamaterial (ε < 0 and µ < 0), ε= -1 and µ = -1
The overall reflection coefficient can be given by (R_{r})^{9}
From equation 23, the reflection coefficient R_{r} is zero for double-negative MTM. The transmission coefficient can be represented by e^{(jkzd)} at the interface, provided both permittivity and permeability are equal to –1. From equations 11 and 17, the transmission coefficient will increase exponentially with increasing distance traveled inside the MTM slab. Hence, MTM exhibits amplification of the evanescent wave, opening new degrees of freedom in designing high Q-factor resonators for compact, high frequency oscillators. The Q multiplier effect does not violate energy conservation because the evanescent mode only stores the energy and does not transport the energy.
The evanescent mode-coupled MTM based resonator has the potential to make a dramatic impact on the design of compact, tunable oscillators that otherwise cannot be achieved with conventional printed transmission-line resonators. MTM resonators present several advantages compared with conventional planar resonators, such as high Q-factor, improved selectivity, easy integration in MMICs, multi-band multi-mode characteristics and insensitivity to EMI/EMC.
Figure 8 shows the simulated S_{21} of MTM split ring, Möbius strips and MTM Möbius strips resonators on a 12 mil thick substrate with ε_{r} = 2.2.9 As shown in the figure, metamaterial Möbius strips (MMS) exhibit superior S_{21} characteristics, with higher Q-factor and suppression of the spurious resonance modes. This enables stable broadband operation compared to MTM split rings and Möbius strips resonators, which exhibit undesired second-order modes.
Mode jumping is analyzed by solving the boundary conditions of the printed MTM resonator. The Möbius metamaterial resonator (MMR) shown in Figure 8 conserves the quantity that gives invariance of solutions under a 2π rotation with a definite handedness (discussed in the first article of this series, published in the November 2014 issue). The MMS has the unique characteristic of self phase-injection properties along the mutually-coupled surface of the strips, which enables a higher quality factor for a given coupling coefficient β_{j}. The coupling coefficient β_{j} depends on the geometry of the perturbation, given by^{9,22}
where E_{a} and H_{a} are the respective E and H fields produced by the MMS and E_{b} and H_{b} are the corresponding fields due to the perturbation (d ≠ 0) or nearby adjacent resonator.^{9}
In equation 24, the first term represents the coupling due to the interaction between the E fields of the resonators, and the second term represents the H coupling between the resonators.
Depending on the strength of interaction, multi-mode dynamics exist related to E, H, and hybrid coupling. Figure 9 shows the unloaded Q of MTM and multi-coupled planar resonators over a range of operating frequencies from 2 to 16 GHz. The Q is estimated by experimental measurement of S-parameters using a vector network analyzer.^{22}
Conclusion
Artificial negative index materials (metamaterials) have been newsworthy for a number of years. This article investigates how the implementation of MTM microwave resonators can achieve advantageous properties, showing that the metamaterial Möbius resonator can achieve high Q-factors.
The third article, which will be published in the January 2015 issue, will discuss several metamaterial Möbius resonator tunable oscillator circuits,^{22} showing that oscillators using these resonators offer promising performance.
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