Study of the Metamaterial Homogenization

Lu Guizhen, Yin Hongcheng, Yang Li, Zeng Dongdong Information School, Communication University of China，Beijing 100024，China

Address 1luguizhen@cuc.edu.cn

3onion@cuc.edu.cn 4dongdong@cuc.edu.cn

Abstract — The current-driven homogenization theory

combined with FEM is studied in this paper. The

extracted effective parameters are used to compute the

scattering parameters of homogenization medium. The

computed results are compared with related periodic

structure. Two results agree with very well.

Key Words—Metamaterial, Homogenization, FEM, EM parameters, Current-driven method

I. INTRODUCTION

In the recent decades, the study of the metamaterial is a hot research topic in electromagnetic field and microwave technology area. There are several potential applications which include negative index of refraction, fabricating sub diffraction cavity, waveguide and many others. In many works, it is necessary to understand the permittivity and permeability before using the metamaterials. There are many research works that perform the calculation of effective electromagnetic parameters. The classic calculation method of the effective electromagnetic parameters is Clausius-Mossiously (CM) formula. The problem of the CM method is that the volume fraction of inclusion is small which is not satisfied by the most metamaterial. A simple and practical approach to metamaterial homogenization is provided by retrieval techniques [5], which determine the effective bulk parameters based on predetermined macroscopic models, with the objective of matching the measured scattering features of (usually planar) metamaterial arrays excited by plane waves. It turns out that the obtained parameters are often non-physical, and they usually depend on the form of excitation and the array

boundaries. One of the common issues of the retrieval method consists in properly defining the boundary of the metamaterial sample, and the use of transition layers at each interface with free-space has been suggested to partially overcome some of these issues. In addition, retrieval methods implicitly assume a material model that is usually too simplistic to describe the complex physics of metamaterial arrays. Finally, small errors in numerical or experimental evaluation of the scattering parameters may be amplified by retrieval approaches and give rise to large non-physical artifacts in the retrieved bulk parameters.

In [4], a general and rigorous first-principle definition of effective metamaterial parameters has been used, showing some of the misconceptions and artifacts associated with simpler homogenization schemes. In the paper, these concepts to the rigorous homogenization of a dense array of magnetodielectric spheres, a typical configuration that supports negative index of refraction is applied. In paper [1], a new self-consistent rigorous approach to homogenize non-magnetic periodic metamaterials is developed. The proposed method is general and can be used to calculate the effective parameters of arbitrary periodic dielectric/metallic metamaterials, taking into account both spatial and frequency dispersion, even in frequency bands where the propagation of electromagnetic waves is not allowed (band-gaps). The method is not based on the solution of an eigensystem and does not involve band structure calculations. Instead, the homogenization problem can be formulated as a source driven problem. The idea is to

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excite the periodic material with a suitable source distribution. In paper [3], by the homogenization approach [1], the FDFD is used to solve the homogenization problem and illustrate its application to several metamaterials formed by dielectric inclusions.

In this paper, the current-driven method combined with FEM is used to solve the homogenization problem. To validate the extracted EM parameters, the scattered parameters are computed to the metamaterial structure and homogeneous structure. The computed results show two results have a good agreement.

II． THE FEM FORMULA FOR THE

HOMOGENIZATION In order to compute the unknown dielectric function,

the composite material is excited with a Floquet-periodic external distribution of electric current Je of the form

, exp( )e e avJ J jk r= − ⋅r r

(1)

Where ,e avJ is a constant vector. Hence, the induced

“microscopic” electric and induction fields (E , B ) have also the Floquet property. The micro field satisfies the vector wave equation as (2)

e

e

JjkZEkE

JjEErrr

rrr

−=−×∇×∇

−=−×∇×∇2

002 ωμεμω

(2)

By using Galerkin method, there is the FEM functional formula as (3).

><−=−×∇×∇ EJjkZEEkEE er

rrrrrrδδεδ ,,, 2 (3)

For periodic structure, the Bloch-Floquet periodic boundary conditions can be represented as

( , ) ( , )x yjk a jk bx a y b e x y− −Φ + + = Φ (4)

Where ( , )x yΦ represents the ( , )xE x y or

( , )yE x y , xk and yk are the wave vector

components along the x and y direction. After the micro fields are solved, the average macro field and effective parameters can be found by using formula (5).

aveaveeff

rave

ave

DE

dAEA

D

dAEA

E

rr

rr

rr

=⋅

⋅=

⋅=

∫∫

∫∫

ε

εε 01

1

(5)

In the next section, a example is given to verify above theory.

III. RESULTS AND DISCUSSIONS In Fig.1, a periodic structure is considered to

illustrate the above homogenization method. The effective electromagnetic parameters of

metamaterials, which have been of recent interest, pose a significant challenge to retrieval methods. This particular structure is composed of a strip that has the permittivity 3. The background is free space with permittivity 1. In the FEM, using the vector finite element, the microstructure field is computed and the effective permittivity are computed as Fig.2. Fig.1 Periodic structure cell with a strip medium of permittivity 3.0

Fig.2 Effective permittivity of xxε and. yyε

x

y

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

ka

effe

ctiv

e pe

rmiti

vity

Epsxx

Epsyy

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Fig.3 Scattering parameter of periodic structure and homogenious

medium (effective permittivity 2.0xxε = )

Fig.4 Scattering parameter of periodic structure and homogenious

medium (effective permittivity 1.5yyε = )

Among the results, xxε is computed by using x

direction applied current and yyε is computed by

using y direction applied current. From the Fig.2, it is seen that effective permittivity has the approximate

value xxε 2.0 and yyε 1.5. In order to verify the

homogenization results, the scattering parameters are computed for both periodic structure and homogenization structure. The scattering parameters

for the homogenization xxε and related periodic

structure are shown as Fig.3. The scattering parameters

for the homogenization yyε and related periodic

structure are shown as Fig.4. It can be seen that the both results agree with very well. The results state the

homoginazation method can be used to extract the effective parameters for metamaterial, which is very important to the design and application of metamaterials.

REFERENCES [1] Mario G. Silverinha, Metamaterial homogenization approach

with application to the chanracterization of microstructured

composites with negative parameters, Physical Review B 75

115104, 2007

[2] C. Fietz, G. Shvets, Current-driven metamaterial homogenization,

Physical Review B, 2010

[3] J. T Costa，Computation of the Effective Parameters of

Metamaterials Using a Finite- Difference Frequency-Domain

Method, http://www.av.it.pt/conftele2009/Papers/49.pdf

[4] Xing-Xiang Liu, Andrea Alù, First-Principle Homogenization of

Magnetodielectric Metamaterial Arrays, IEEE AP-S/URSI

pp1522,2011

[5] D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis,

“Determination of effective permittivity and permeability of

metamaterials from reflection and transmission coefficients,”

Phys. Rev.B, Vol. 65, p.195104, Apr. 19, 2002.

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