Techniques for Determining the Relative Weights of Spatial Harmonics Induced in Truncated Periodic Structures such as Metamaterials Rodney Gomez1, Katherine Lugo1, Nader Farahat1, Raj Mittra2 and Lakshmy Iyer1 1
Polytechnic University of Puerto Rico P.O. box 192017 San Juan, PR 00919 E-mail:
[email protected] 2
The Pennsylvania State University, University Park, PA 16802
Abstract In this paper we determine the weights of the spatial harmonics in a periodic structure, with quasi-periodicity along the propagation direction, by detailed analysis of the fields inside the unit-cells. We use both the Prony’s Method and Fourier transform to extract the spectrum of the fields inside the unit cells along the propagation direction and compare the results obtained by each method. Two examples, one-dimensional dielectric layers and split-ring resonators with wires, are considered. The results show that the Fourier transform approach is more accurate and consistent in comparison to Prony’s Method.
1. Introduction Periodic structures have been used in many engineering applications [1-2]. In recent years the interest in periodic structures has been mainly focused in the construction of the artificial material, known as metamaterials, with unusual characteristics such as left-handedness, negative refraction and extraordinary transmission [3-5]. The dispersion diagram of an ideal, infinite structure, can be computed from its frequency response and, hence, the presence of harmonics (Floquet modes) can be identified by analyzing the field distribution in such a structure. However, in a realistic and practical design, we need to deal with a finite number of unit-cells, and the procedure for identifying the excited harmonic modes and their relative levels is more involved. In order to address the above question the fields inside the unit-cells must be closely analyzed. In this paper we use two methods, based on Prony and Fourier techniques, to analyze the fields and determine the levels of excited spatial harmonics. Two configurations are investigated to illustrate the procedure: (i) dielectric material layered in one dimension; and (ii) split-ring resonator with wires.
2. Numerical Results We begin with the case of a planar dielectric structure whose dispersion diagram is shown in Fig. 1. We model this structure using a 1-D FDTD Method [6] using 64 unit-cells. The magnitude of the electric field inside 47 cells along the structure is shown in Fig. 2 at 2.5 GHz. First we use the Prony method to fit this distribution to a set of decaying exponentials. We also use the Fourier transform of this distribution as a function of the distance. Both techniques yield the information we seek about the dominant spatial harmonics. The numerical results for the modal weights are compared with those of the analytical ones, and are shown in Table 1. Both methods show reasonably close agreement for the two most significant modes, viz., β0 and β-1, where βn = β0 + 2Лn/d. Next we turn to the split-ring and wire resonator [5] shown in Figs. 3 and 4. We use the FDTD Method along with the periodic boundary conditions [7] imposed on the transverse walls, and analyze a structure comprising of 10 unitcells along the propagation direction that is illuminated by a uniform plane wave with its electric field polarized along the wires (see Fig 3). There are two transmission peaks, at 10 and 16 GHz, in the obtained S-parameters, shown in Fig. 5. We show the phase of the co-polar electric fields along the cells in the middle of the unit-cell in Figs. 6 and 7. The connected dots show the locations of the unit-cells. Although the behavior inside each cell is quite complex, we observe, by following the dots, that there are positive and negative phase progressions at 10 and 16 GHz, respectively. At 10 GHz, the field behavior shows that the group velocity is positive, but that phase velocity is negative, implying backward wave propagation. This phenomenon is also seen to be present in the dispersion diagram of the SRRs, shown in Fig. 8a, which displays it for several periods. Note that the lowest spatial harmonic β0 is a backward mode as opposed to that excited in dielectric layers in Fig. 1 that supports forward mode instead. To
determine the weights of spatial harmonics for this structure, once again we analyze the field distributions in Fig. 6 by using both the methods. Since, in contrast to the first example, the SRR is a three-dimensional structure, we are at liberty to sample the fields at different locations in the unit-cells. Through many experiments we have found that the Fourier transform approach is relatively independent to the choice of the locations where the field is sampled. Figs. 8 b, c and d show the spectra of the fields (obtained by Fourier transform) at three locations (see Fig. 3), namely the wire side; the center; and, the loop side of the unit-cell respectively. The level of the spectrum out of the plotted range is insignificant. A close examination of the maxima in all three plots shows that β0, β-1 and β-2 harmonics located at the left side of the diagram have positive group velocities (indicated by the direction of the arrows) that are initially excited by the plane wave impinging from the left side of the structure. On the right side of the diagram, however, β0, β1 and β2 correspond to the reflection from the end of the truncated structure and these have negative group velocities. Furthermore we note that the maxima in all three samples are in the backward section (shown by “b”) of the dispersion diagram at 10 GHz.
3. Acknowledgement This material is based upon work supported by, or in part by, the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF-06-1-0027.
4. References 1. Robert E. Collin, “Foundations for Microwave Engineering” 2nd ed., IEEE Press, 2001 2. David M. Pozar, “Microwave Engineering” 3rd ed. John Wiley & Sons, Inc., 2005 3. V.S. Veselago, The electrodynamics of substances with simultaneously negative values of ε and µ, Sov Phys Usp 10 1968, 4. R.A. Shelby, D.R. Smith, and S. Schultz, Experimental verification of a negative index of refraction, Science 292 2001 , 77_79. 5. D.R. Smith and N. Kroll, Negative refraction index in left-handed materials, Phys Rev Lett 85 2000 , 2933_2936. 6. K. S. Yee “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media”, IEEE Trans. Antennas and propagation, Vol. 14, 1966, pp. 303-307. 7. P. H. Harms, A. Roden, J. Maloney, M. Kesler, E. Kuster, and S. D. Gedney, “Numerical Analysis of Periodic Structures Using the Split Field Update Algorithm”, The Thirteenth Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, March 17-21,1997
εr=1
εr=4
εr=1
d=3 cm
Fig. 1. Dispersion diagram of planar dielectric structure.
Fig. 2. Field distribution of planar dielectric structure along 47 unit-cells at 2.5 GHz.
Most significant Mode
beta
Max amp other m.
-124.3
0.05
-
-129.5
0.0236
0
-120.8
-
Second Most Significant mode
meth
N Mag
Phase
Alpha
beta
N Mag
Phase
P
1
1.161
0.0127
85.03
0.2984
-2.029
F
1
1.231
-
82.42
0.287
-0.820
A
-
-
0
88.57
-
-
Alpha -0.26
2π ( −1) 2π n = −120.86 β −1 = 88.57 + P: Prony, F: Fourier, A: Analytical β n = β 0 + 0.03 d Table 1. Planar dielectric results comparison at 2.5GHz
Measurement xy-plane z
center
Ez
Loop side
…
y 1
2
3
Wire side
z 10
x
Fig. 3. Illustration of the periodic structure and the measurement points 0.4
wire
wire
loops 0.25
5 1.5 0.25 0.25
0.25
2.5
1 2
2.5
loops
Ty=5
Tx=2.2
Fig. 4. Dimensions of 1 unit cell of the split-ring resonator with wire (all dimensions are in mm)
Fig. 5. S-Parameters of 10 unit cells of split-ring resonator with wire.
Frequency (GHz)
Fig. 6. Phase distribution along 10 unitcells of SRR with wire at 10 GHz
10
b 5
f
b
f
b
b
f
b
f
b
-2.5
-2
-1.5
-1
-0.5 a) 0
0.5
1
1.5
2
2.5
-2.5
-2
-1.5
-1
-0.5 b) 0
0.5
1
1.5
2
2.5
-2.5
-2
-1.5
-1
-0.5 c) 0
0.5
1
1.5
2
2.5
-2.5
-2
-1.5
-1
-0.5 d) 0 0.5 kd/(2pi)
1
1.5
2
2.5
1 Wire Side
Fig. 7. Phase distribution along 10 unitcells of SRR with wire at 16 GHz
0.5 0
Center
1 0.5 0
Ring Side
1 0.5 0
Fig. 8. a) Dispersion diagram for several periods. The arrows show the group velocity direction. The letters “f” and “b” stand for forward and backward waves respectively. Fig. 8. b, c, d ) Field spectrum of the 10 unit-cells split-ring and wire resonator at 10 GHz in three locations; the wire side(b), the center(c) and the loop side(d)of the unitcell.
