IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007
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A Wideband ADI-FDTD Algorithm for the Design of Double Negative Metamaterial-Based Waveguides and Antenna Substrates Nikolaos V. Kantartzis, Dimitrios L. Sounas, Christos S. Antonopoulos, and Theodoros D. Tsiboukis Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki GR-54124, Greece
An enhanced 3-D alternating-direction implicit finite-difference time-domain method for the systematic analysis of double negative metamaterial-based waveguide and antenna devices is presented in this paper. Being fully frequency-dependent, the new scheme introduces a set of generalized multidirectional operators which incorporate the suitable Lorentz–Drude model and subdue inherent lattice deficiencies for broad spectra. So dispersion errors, as time-steps exceed the Courant limit, are drastically minimized yielding fast and accurate solutions for propagating and evanescent waves. The technique is applied in the design of microwave structures, realized via the prior model or networks of thin wires and split-ring resonators. Numerical results certify its merits, without requiring elongated simulations and excessive overheads. Index Terms—ADI-FDTD method, antennas, double negative (DNG) media, metamaterials, split-ring resonators, waveguides.
I. INTRODUCTION
T
HE recognition of double negative (DNG) metamaterials is attributed to their unique structure that offers promising properties not encountered in nature [1]–[3]. Amid them, one can discern the negative tuning of both constitutive parameters attained by thin rods and split-ring resonators (SRRs). These substances have triggered a serious research for their numerical analysis [4]–[11]. However, as the electrical size is smaller than a wavelength, the study of DNG media leads to lengthy simulations. So, the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method is an instructive tool, as it can skip stability limits [12]–[16]. Extensive works though, revealed that its reflection errors depend on resolution and rise as time-steps become larger. This paper develops an optimized ADI-FDTD method for the modeling of waveguides and antenna substrates loaded by DNG media. To analyze their geometries, the algorithm launches novel operators for the proper Lorentz–Drude framework, in curvilinear coordinates, which mitigate mesh defects. Also, the management of spatial stencils allows the evaluation of backward wavefronts in regions much smaller than the incident wavelength, while the ADI treats refractions at interfaces. Thus, dispersion errors are minimized even when time-steps surpass the Courant limit. The technique is verified by diverse problems like waveguides, junctions, or “smart” antennas. The resulting models display high accuracy and fast convergence for very short simulation times.
Fig. 1. (a) SRR-loaded periodic antenna substrate. (b) Rectangular and a circular SRR paired to a thin metallic wire. (c) DNG-based waveguide.
effective magnetic permeability is described by a Lorentz (L) or Drude (D) model. For example,
with
(1)
is the grid constant, the inner ring radius, the width, the light velocity, the perimeter resistance, the damping frequency, and the adjacent SRR distance. Conversely, thin rods of negative permittivity are modeled by
II. ADI-FDTD SCHEMES FOR 3-D DNG MEDIA The most efficient DNG realization comprises periodic arrays of SRR and metallic wires (Fig. 1). The SRR grid with a negative
Digital Object Identifier 10.1109/TMAG.2006.891007
(2)
where is the static dielectric constant and gous plasma and damping frequencies. 0018-9464/$25.00 © 2007 IEEE
,
the analo-
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007
, are the electric losses, the source, In (8), , are diagonal material tensors. and Next, each time-step is divided into two subiterations. The and the second from to first spans from to . Focusing on during the first subiteration, (8a) gives
(9) For symmetry, terms are temporally averaged, while . After the expansion of , we have
Fig. 2. (a) SRR-based spiral inductor. (b) Periodic spiral SRR substrate.
According to the proposed concept, spatial derivatives are computed by a consistent set of th-order operators as
(3)
(10)
where is a general coordinate system, the spathe multi-directional operator that uses adaptial step, tive stencils for the abruptly-varying evanescent waves at DNG, is vacuum interfaces. For instance, if
where is implicitly calculated by the unknown values at , while is acquired via the updated at . To eliminate , the same idea is used in the -directed in (6) and place the result in part of (8b). Then, we plug is computed from (6), in which (10). The unknown values are obtained by a Crank-Nicolson model of (7), as
(4) (11) Parameters , are real degrees of freedom that augment algorithmic flexibility. Also, time update is conducted by
with each along
. Replacement of (11) in (10) for axis gives the sparse tridiagonal system of
with a correction function which controls the wideband behavior of (3), especially for complex DNG substrates (Fig. 2) (5) (12) To derive the new ADI-FDTD forms, we express magnetic and electric flux densities, as (6) are field intensities and , auxiliary magnetic In (6), and electric polarizations for frequency dependence
Parameters are defined by the spatial stencils, whereas for (12), which has the form of , is recursively solved. Similarly, the second subiterareverses the update of and in the tion for to . Once field quantities are evaluated, interval the algorithm proceeds to the next time-step. This discretization offers a drastic suppression of dispersion errors (up to 10 is beyond the Courant criterion. orders), even when
(7) III. STABILITY AND DISPERSION-ERROR ASPECTS The stability of the proposed schemes is studied via the von Neumann method [14]. Expressing the two subiterations as
Using (3)–(5), in Ampere’s and Faraday’s laws, one gets (8a) (8b)
KANTARTZIS et al.: ADI-FDTD ALGORITHM FOR THE DESIGN OF WAVEGUIDES AND ANTENNA SUBSTRATES
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TABLE I FIRST RESONANCE FREQUENCY FOR THE TWO-PORT DNG WAVEGUIDE
Fig. 3. Side and top view of a hemiellipsoidal cavity-backed antenna with one patch and a metamaterial-based substrate for enhanced beam steering.
Fig. 5. DNG-loaded four-port junction with longitudinal and transverse SRRs.
Fig. 4. Return loss and radiation patterns of the hemiellipsoidal antenna.
( and are sparse matrices) we solve the eigenand are functions of value problem and derive [ ] and
(13) for , and the wavenumber. The magnitudes of all eigenvalues are less or equal to unity, an issue that confirms the unconditional stability of our algorithm. This property enables us to drastically minimize the CPU time. The improved dispersion relation is extracted by the associated error and the calculated/exact wavenumbers, as (15) as compared to the common ADI-FDTD relation . Its effectiveness holds for preset frequencies and adequately wideband spectra. Note that, apart from some confined storage, the CPU/memory burden is not noticeably raised. IV. NUMERICAL RESULTS The optimized technique is applied to several DNG designs. The first problem studies the hemi-ellipsoidal antenna of Fig. 3, located on a DNG-loaded substrate [Fig. 1(a)] beyond a curvilinear cavity. Beam steering is controlled by a 4.8 2.3 mm . Its dimensions are 42.6 mm, patch 58 mm, 1.12 mm, and 0.85 mm. Fig. 4 gives the return loss and radiation patterns along with the CPU time. Evidently, usual approaches cannot handle the strenuous curvatures, whereas our algorithm is very close to the reference.
Fig. 6. Calculation of junction.
S -parameters
for the metamaterial-based four-port
Let us now focus on the two-port WR137 waveguide of Fig. 1(c) with two parallel metamaterial planes. The SRR di3.26 mm, 0.92 mm, 0.28 mm, and mensions are 0.24 mm. Table I provides the first resonance frequency 15.78 mm and 3.44 mm, B) for three cases: A) 23.92 mm and 4.54 mm, and C) 30.16 mm and 5.60 mm. The proposed method overwhelms Yee’s one, yielding small dispersion errors for coarse curvilinear lattices. Proceeding to the junction of Fig. 5, its interior is loaded with 3.84 mm, two circular SRR types placed both transversely ( 0.98 mm, 0.32 mm, 0.38 mm, and 0.29 mm) 2.56 mm, 0.72 mm, 0.16 mm, and longitudinally ( 0.24 mm, and 0.19 mm) along the structure’s axis. 5.7 mm, 11.3 mm, and The junction with 2.6 mm is discretized in a 60 154 26 mesh. Homogenized DNG parameters are correctly embedded via unit and CFLN 28 cells. Fig. 6 gives the -parameters for (time: 1.3 h) versus Yee’s scheme with a 91.5% finer grid (time: , and CFLN 1.97. 15.9 h), Also, we analyze a bandpass filter with the spiral inductor-SRR pattern of Fig. 2. Typical dimensions are
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ACKNOWLEDGMENT This work was supported by the Greek General Secretariat of Research and Technology under Grant PENED03/03ED936.
Fig. 7. (a) Characteristic impedance and (b) transmitted power of a bandpass 4 4 and 8 8 filter implemented through a spiral inductor SRR substrate.
2
2
H
Fig. 8. snapshots in a rectangular and circular SRR substrate and patch height versus axial ratio frequency for the hemiellipsoidal DNG-based antenna.
8.6 mm, 9.1 mm, 5.2 mm, 5.9 mm, 0.6 mm, 0.25 mm, 0.23 mm, and 17.4 mm. Fig. 7(a) gives the characteristic impedance of two filters, while Fig. 7(b) presents the effect of SRR on the transmitted power. , Results confirm that our algorithm (CFLN 1.489 s, grid: 70 140 36, and time: 56 min) is superior , to the simple ADI FDTD scheme (CFLN 1.015 ns, grid: 134 268 72, and time: 11.3 h). Finally, a substrate for the antenna of Fig. 3 (without the backing cavity) is designed. Keeping the same dimensions, we study a 5 5 7 circular SRR setup. The role of the rectangular and a 7 snapshots and patch is very important as derived from the the plot of Fig. 8 describing the variation of versus the axial ratio frequency. V. CONCLUSION The optimized design of waveguides and antenna substrates with DNG media is developed in this paper by a 3-D ADI-FDTD technique. Its discretization subdues lattice errors far above the Courant limit. Results prove such benefits and unveil the wideband profile of the method.
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Manuscript received April 24, 2006 (e-mail:
[email protected]).
