Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 2730, 2012 131

Determination of Effective Constitutive Parameters, MaterialBoundaries and Properties of SRR-rod and FishnetMetamaterials by Drude/Lorentz Dispersion Models

Feng-Ju Hsieh1, Cheng-Ling Chang1, and Wei-Chih Wang1, 2, 3

1Department of Mechanical Engineering, University of Washington, Seattle, Washington, USA2Department of Electrical Engineering, University of Washington, Seattle, Washington, USA

3Medical Device Innovation Center, National Cheng-Kung University, Tainan, Taiwan

Abstract This paper presents two effective extraction methods to fully retrieve effectiverefractive indices (n), impedances (z), and material properties, such as dielectric permittivity() and permeability () of metamaterials (MMs). The first full extraction method based onthe material continuity allows missing results in the strong resonant band where the imaginarypart of permittivity or permeability is negative to be retrieved. The second method describes agenetic algorithm and optimization of properly defined goal functions to retrieve the parametersfrom Drude and Lorentz dispersion models for and , respectively. Proper initial values of theparameters of models and partial retrieved and are also derived for the second method inorder to reduce the computation time and prevent the optimization solution from falling intolocal minimum. Finally, the refractive index, impedance, permittivity, and permeability of twoslabs composed of SRR-based and fishnet MMs are retrieved and compared using the proposedmethods.

1. INTRODUCTION

Metamaterials (MMs) [1, 2] operated in quasi-optical or Terahertz (THz) frequency range havebeen attracting attention over the past several years due to their unique phenomena, such as thenegative refractive index and strong resonance in THz range, lacking in naturally occurring media.In order to utilize these phenomena of MMs in applications for bio-chemical sensing, spectroscopy,bio-imaging, security and communications,the effective boundaries of MMs must be determinedfirst, before the constitutive parameters and frequency-dependent material properties of MMs arecharacterized. To meet this demand, we propose two optimization models to determine effectiveboundaries of MMs. The optimization models are defined based on the assumption of effectivemedium theory and characteristic of homogeneous media. The results are then compared with thetraditional definition proposed by Kong [3].

Several methods [47] have been reported based on the assumption of effective medium the-ory, which means that the lattice size is much smaller than the operating wavelengths inside themedium.The common approach is to calculate effective refractive index (n) and impedance (z)from reflection (S11) and transmission (S21) coefficients of scattering parameters (S-parameters),and derive the effective constitute parameters and from n and z. There are known issues tothis process that may fail the extraction, such as when either the magnitude of S11 or S21 is smallor when the effective slab boundaries are not well estimated. The proposed methods will helpto clarify the determination of effective boundaries of materials and simplify the determination ofrefractive index from the multiple-branch problem caused by the inverse of logarithm or arccosineoperator.

Another major issue revealed by [3] indicates missing results in real part of n near resonantband where the imaginary part of or is negative. As a result, no constitutive parameters can bedetermined in the resonance band. However, there should be a value of and at any frequencyto describe material properties, where transmission and reflection coefficients can also be derivedor measured. Despite the controversy in considering whether negative and should exist or not,we propose two extraction methods for both sides to retrieve material properties of MMs. Thefirst method allows the retrieval of constitutive parameters with negative imaginary part of or at resonant band, and the second method based on partial retrieved results and passive materialdispersive Drude and Lorentz model allows to retrieve constitutive parameters with all positiveimaginary parts of and .

132 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 2730, 2012

2. SIMULATION MODELS

Two different types of MMs split ring resonator with a rod (SRR-Rod) and fishnet MMs as shownin Fig. 1, are simulated and used to calculate constitutive parameters and material properties. Thedimensions of SRR-Rod unit cell are the lattice size a = 50m, ring spacing b = 5m, outer SRRheight h = 30m, strip width w = 2.5m, and gap width g = 5m. The dielectric material with2.5m thickness between SRR and rod, are characterized by r = 3.84 and tan = 0.018. The slabthickness Lm is determined by the numbers of unit cell in x direction. A planar EM wave is incidentfrom the side of the unit cell and propagating in z direction with simulation ports set originallyL1o = L2o = 75m away from the slab/air interfaces. For the fishnet MM, dielectric is sandwichedbetween the two fishnet-shape conductors, and a planar EM wave propagates in z direction withL1o = L2o = 150m and impinges normal to the unit cell surface with the square lattice sizec = 150m, square conductive pad length d = 104m, and e = 10m. The slab thickness is thethickness of dielectric layer ts plus the thickness of two conductive layers 2tc.

3. EXTRACTION METHODS

A partial extraction method with modifications in determining effective boundaries and real partof refractive index (n) from the well-known method [3] is shown in Fig. 2. Before n and z canbe calculated from S11 and S21 using Equations (1)(2), effective boundaries of a MM slab mustbe determined firstby proposed optimization models defined in (3)(4). Since MMs are resonantin frequencies where n is negative, the characteristic of a homogenous material which means itsimpedance is independent of its slab thickness is adopted in defining the goal functions. By inspect-ing the impedance difference of two different slabs with different numbers of the unit cell layered

(a) (b) (c)Figure 1: (a) Geometry of a two-cell SRR-Rod MM slab with effective boundaries set at new referenceplanes. For a single cell slab, the slab thickness Lm is a. For the two cells slab that has two unit cells stackedup together in z direction, the slab thickness Lm = 2a. (b) Single-layered fishnet MM array with dielectricsandwiched between the top and bottom fishnet-shaped conductors. (c) Geometry of a fishnet MM unit cell.

Figure 2: Flow chart of the proposed extraction method.

Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 2730, 2012 133

up in the wave propagating direction, we can define effective boundaries of the slabs at which theimpedance difference of two slabs with different thickness is minimized. The original goal functionof the optimization model defined in [3] is set up to find the locations of new reference planes of twoslabs such that the mismatch of impedances of these two slabs at the effective boundaries is mini-mized, however, two new goal functions are set to find the effective boundaries where the mismatchof impedances at the investigated frequencies is minimized and equally weighted in magnitude andphase terms or real and imaginary parts of the impedance. Three goal functions are then used togenerate the density plots of mismatch impedance within the searching range of (L1s, L2s) set from80m to 80m, as shown in Fig. 3. The darker zone represents smaller value of mismatch of theimpedances. As a result, the original goal function gives three darker zones while the proposed goalfunctions give only one darker zone for the locations of effective boundaries at L1s = L2s = 73m.

n =1k0d

(cos1

1 S112 + S2122 S21

+ 2pim)

(1)

z = (1 + S11)2 S212(1 S11)2 S212 (2)

min Nffi=1

||z1 (fi, x)| |z2 (fi, x)||max {|z1 (fi, x)| , |z2 (fi, x)|} +

||z1 (fi, x)| |z2 (fi, x)||max {|z1 (fi, x)| , |z2 (fi, x)|} (3)

min Nffi=1

|z1 (fi, x) z2 (fi, x)|max {|z1 (fi, x)| , |z2 (fi, x)|} +

|z1 (fi, x) z2 (fi, x)|max {|z1 (fi, x)| , |z2 (fi, x)|} (4)

where Nf is the total number of samples of interesting frequencies and zj (fi, x) is the impedance ofslab j measured at frequency fi with effective boundaries set at new reference planes x = (L1s, L2s).Positive signs of (L1s, L2s) denote that the new reference planes are shifted outward from the slabwhile negative signs of (L1s, L2s) denote that the new reference planes are shifted inward towardthe slab from the original locations (L10, L20).

The ambiguity of n exists in the signs and the branch index m, and the first proposed methodis to create table of m candidates at each frequency satisfying |nz| nz. Then mathematicalcontinuity of the n can be applied to partially retrieve n by examining Taylor series expansion ofejk0nd at frequency fi and fi+1 to determine the right branch index m at frequency fi+1 by theknown value of m at frequency fi, instead of solving the binomial equation proposed in [3]. Byscanning the m value from all integers through all investigated frequencies using this algorithm,instead of the table of m candidates, n can be fully retrieved.

The second method based on partial retrieved results of n and z, parameterized dispersiveDrude/Lorentz models (5)(6), can retrieve material properties with all positive or by properdefined optimization models (7)(8). Proper initial search ranges for certain parameters, such as

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: (a), (b) and (c) are 2D density graphs of original and the other two goal functions with x, y axesdenote searching range of L1s and L2s, respectively. (d), (e) and (f) are 1D graphs along diagonal andanti-diagonal direction of 2D density graphs of (a), (b), and (c), respectively.

134 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 2730, 2012

(a) (b)Figure 4: (a), (b) are retrieved n, z, and of SRR-Rod and fishnet MMs, respectively.

s, , 0, s and , are necessary to speed up the calculation for the optimization and preventthe solution of optimal parameters to fall into the local minimum. The proper formulae consideringthe right phase terms of and are necessary to obtain n and z.

eff () = p2

( jc)(5)

eff () = +(s )0202 + j 2 (6)

where is the permittivity at the high frequency limit, p is the radial plasma frequency, c is thecollision frequency, s and is the permeability at the low and high frequency limit, respectively,0 is the resonance frequency, and is the damping frequency.

minG = minfi

Abs (|eff (i)| |ref (i)|) + minfi

Abs (eff (i) ref (i)) (7)

minG = minfi

Abs (|eff (i)| |ref (i)|) + minfi

Abs (eff (i) ref (i)) (8)

where ref () and ref () are calculated from partial retrieved n and z.Retrieval results including n, z, and of two simulation models, SRR-Rod and fishnet MMs,

using different extraction methods are plotted in Fig. 4. Solid, dashed, dotted lines represent thedata retrieved by partial extraction, first full extraction, and second Drude/Lorentz extractionmethods, respectively. The first and second extraction methods can both retrieve the missing datawith slightly difference in resonant band.

4. CONCLUSIONS

This paper presents two extraction methods to fully retrieve effective material properties and consti-tutive parameters of MMs based on continuity and dispersion models. Two types of MMs includingSRR-Rod and fishnet MMs are designed and simulated in THz frequencies. The corresponding Sparameters are investigated by two proposed methods to retrieve n, z, and . The retrievalresultsby two methods show good agreements over the investigated frequencies with a slightlydifference in resonant band.

Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 2730, 2012 135

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