swiss roll ensemble homogenization by full-wave simulations

implementing the filter. However, beyond f0, the third transmis-

sion zero position and the filter upper skirts are well matched

with theory. The insertion loss is less than 0.5 dB up to 0.58

GHz and less than 2 dB in most of the upper passband. The

measured return loss is better than 29 (14.7) dB in the lower

(upper) passbands.

4. CONCLUSIONS

The new wideband bandstop filter configuration of Ref. 1 with

four transmission zeros is reanalyzed to show that this configu-

ration also produces three transmission zeros in the stopband.

The generalized conditions for transmission zeros are derived

explicitly. With the tree transmission zeros, an equal-ripple stop-

band is always achieved as compared with the four transmission

zeros, where an additional optimization is required. Further, the

three zeros filter has the same bandwidths and cut-off rates of

the four zeros filter in Ref. 1. A prototype filter with coupled

line impedance ratio same as [1] with 91% 20 dB FBW and bet-

ter than 30 dB return loss in the passband is designed at 2 GHz

for validation purposes.

REFERENCES

1. M.A. Sanchez-Soriano, G. Torregrosa-Penalva, and E. Bronchalo,

Compact wideband bandstop filter with four transmission zeros,

IEEE Microwave Wireless Compon Lett 20 (2010), 313–315.

2. D.M. Pozar, Microwave engineering, 2nd ed., Wiley, New York,

New York, 1998.

VC 2011 Wiley Periodicals, Inc.

SWISS ROLL ENSEMBLEHOMOGENIZATION BY FULL-WAVESIMULATIONS

Bart Michiels, Ignace Bogaert, Jan Fostier,and Daniel De ZutterDepartment of Information Technology (INTEC), Ghent University,Sint-Pietersnieuwstraat 41, Ghent B-9000, Belgium; Correspondingauthor: [email protected]

Received 12 January 2011

ABSTRACT: This paper investigates a magnetic metamaterial, built

from so-called Swiss rolls, by means of full-wave simulations. Afterdetermining the resonance frequencies of a single Swiss roll, the

macroscopic material parameters of an ensemble of Swiss rolls aredetermined by S-parameter retrieval, using a bianisotropic model, thatdoes not assume reciprocity a priori. As a result, the macroscopic

permeability, permittivity, and magnetoelectric coupling coefficients areobtained as a function of frequency. VC 2011 Wiley Periodicals, Inc.

Microwave Opt Technol Lett 53:2268–2274, 2011; View this article

online at wileyonlinelibrary.com. DOI 10.1002/mop.26257

Key words: Swiss roll; metamaterial; homogenization; S-parameterretrieval; bianisotropy

1. INTRODUCTION

Metamaterials have attracted considerable attention in recent

years. In general, metamaterials are ensembles of microscopic

(i.e., much smaller than the wavelength) structures that can be ho-

mogenized into a macroscopic medium with effective material pa-

rameters. The microscopic structure can be designed to allow the

construction of metamaterials with remarkable material parame-

ters, for example, chiral, negative permittivity, negative permeabil-

ity, and even negative refractive index materials. However, the re-

trieval of meaningful material parameters from these microscopic

metamaterial structures remains a challenging issue and a topic of

high interest in the metamaterial research community [1–4].

In this paper, a metamaterial structure built from so-called

Swiss rolls will be homogenized by means of full-wave simula-

tions. Swiss rolls are rolled-up perfectly electrically conducting

(PEC) plates that, when arranged into a periodic lattice, form a

two-dimensional (2D) magnetic metamaterial. This metamaterial

was first proposed in Refs. 5–7 and continues to attract much in-

terest [8–10]. An incident transverse-electric (TE)-polarized

plane wave induces a current along the surface of the Swiss roll

and the magnetic field in the center exhibits resonant behavior

as a function of frequency, giving rise to a negative permeability

in certain frequency ranges.

The full-wave method, used throughout this paper to perform the

simulations, is a method of moments (MoM) solver [11] using the

Electric Field Integral Equation accelerated with the MultiLevel Fast

Multipole Algorithm (MLFMA) [12]. Such solvers typically require

much less unknowns and have a higher accuracy compared with, for

example, Finite Difference Time Domain and Finite Element solvers,

at the cost of being more mathematically involved. To solve the low-

frequency breakdown of the MLFMA, the Normalized Plane Wave

Method [13] is invoked. All these methods and algorithms are imple-

mented in Nero2d, an open source, full-wave solver for 2D scattering

problems [14]. A validation of the algorithms for complex structures

is discussed in Ref. 15. Performing simulations at frequencies close

to the resonance frequencies of the Swiss rolls is a real challenge for

MoM-MLFMA solvers, mainly because of the high condition num-

ber of the MoM-matrix. However, the fact that the structures do not

have to be physically built is a considerable advantage when com-

pared with measurements.

The material parameters are calculated using S-parameter re-

trieval, assuming that the Swiss roll metamaterial behaves as a

bianisotropic material. This approach is similar to Refs. 16 and

17, where split-ring resonators are analyzed. However, in con-

trast with Refs. 16 and 17, reciprocity is not assumed a priori by

our model. Of course the reciprocity property will be checked to

further validate the results of our model, as Swiss rolls are PEC

objects and therefore reciproque.

Other methods to retrieve the material parameters also exist,

such as the field-averaging method [18].

A waveguide set-up allows the material parameters to be

retrieved while simulating only a single row of Swiss rolls

stacked inside a waveguide. According to image theory, this is

equivalent to a grid of Swiss rolls extending to infinity in the

direction perpendicular to the waveguide, but it requires signifi-

cantly less computing capacity.

The outline of the paper is as follows. First, in Section 2 a

single Swiss roll is analyzed. Based on a frequency sweep, the

resonance frequencies are determined, along with a sufficiently

accurate discretization of the Swiss roll. This discretized Swiss

roll is subsequently used to determine the homogenized material

parameters. Next, in Section 3, a waveguide set-up is introduced

to reduce the required amount of computing capacity. Then, in

Section 4, the material parameters as a function of the S-param-

eters are calculated and discussed. Finally, Section 5 presents

the obtained macroscopic permeability, permittivity, and magne-

toelectric coupling coefficients as a function of the frequency.

Section 6 contains some concluding remarks.

2. SINGLE SWISS ROLL

In this paper, by way of example, we consider Swiss rolls with

W ¼ 6 windings, a maximal radius Rmax ¼ 1 m, a distance

between the plates of D1Rmax ¼ 5 cm (D1 ¼ 0.05) and a plate

2268 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 10, October 2011 DOI 10.1002/mop

thickness of D2Rmax ¼ 5 cm (D2 ¼ 0.05). The parametric equa-

tion representing the Swiss roll is

½x; y� ¼ Rð/Þ½cos/; sin/� (1a)

Routerð/Þ ¼ Rmax 1� ðD1 þ D2Þ /2p

� �; / ¼ ½0:::2pW� (1b)

Rinnerð/Þ ¼ Rmax 1� D2 � ðD1 þ D2Þ /2p

� �; / ¼ ½2pW:::0�

(1c)

with Router and Rinner the outer and inner radius of the rolled-up

plate, respectively.

To find the resonance frequencies of a single Swiss roll, the

Swiss roll is illuminated by a TE-polarized plane wave, propa-

gating along the positive x-axis (Hinz ¼ H0 e

�jk0x).

To discretize the Swiss roll, the parameter u in Eq. (1) is

discretized with a stepsize of Du. If Du ¼ 2pW/P, the total

number of segments is 2P þ 2. The discretization of a Swiss

roll for Du ¼ 0.1 is shown in Figure 1. The resonance frequen-

cies, that is, the frequencies for which the absolute value of the

magnetic field in the center of the Swiss roll reaches a maxi-

mum, are now found by performing frequency sweeps (with fre-

quency steps Df).

The first column of Table 1 contains the total number of seg-

ments and the corresponding resonance frequency, obtained

from the frequency sweep, is shown in the third column. The

second column displays the ratio of the largest segment (Dlmax)

over the wavelength k0 that corresponds with the resonance fre-

quency of the Swiss roll simulation with finest discretization,

that is, for 2P þ 2 ¼ 1602. One of the first observations is that

the numerically obtained resonance frequencies depend on the

chosen discretization, which is intuitively clear: simulations with

a finer discretization give more accurate results. However, they

also require more simulation time and memory. Therefore, it is

advantageous to make a trade-off between accuracy and simula-

tion time. From this point on, a discretization into 756 segments

will be used, for which the relative error on the first resonance

frequency is about 0.1%. For this discretization, the results of a

frequency sweep between 0.1 and 10 MHz and a close up near

the first resonance frequency are given in Figure 2. Table 2

shows the first 10 resonance frequencies for a discretization into

756 segments.

Figure 1 A Swiss roll (W ¼ 6, Rmax ¼ 1 m, D1 ¼ D2 ¼ 0.05) discre-

tized by Du ¼ 0.1 (756 segments). The endpoints of the segments are

denoted by the black dots

TABLE 1 First Resonance Frequency as a Function of theDiscretization

2P þ 2 Dlmax/k0 f0 (MHz)

202 3.1e�3 4.8288

402 1.5e�3 2.3737

602 1.0e�3 2.4460

756 8.2e�4 2.4453

802 7.7e�4 2.4451

1002 6.1e�4 2.4444

1202 5.1e�4 2.4439

1402 4.4e�4 2.4436

1602 3.8e�4 2.4434

Figure 2 Frequency sweeps for 2P þ 2 ¼ 756. The magnetic field in

the center of the Swiss roll is normalized to the amplitude of the inci-

dent magnetic field. (a) f < 1 MHz: Df ¼ 9 � 103 Hz, f > 1 MHz: Df¼ 9 � 104 Hz. (b) Df ¼ 102 Hz. [Color figure can be viewed in the

online issue, which is available at wileyonlinelibrary.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 10, October 2011 2269

For the chosen geometry, with the maximal radius equal to 1

m, the first resonance frequency occurs at a frequency of the

order of 1 MHz. It is worthwhile to point out that the absolute

size of this structure is not essential here. Indeed, when the

structure is scaled down by a factor a, the resonance frequency

is increased by the same factor and all physical phenomena

remain the same.

In Ref. 9, the explanation for the appearance of the higher

order (n > 0) resonance frequencies is given: these frequencies

activate the modes between the conducting plates of the Swiss

roll, which acts as a spiral waveguide.

3. WAVEGUIDE SET-UP

A possible way to homogenize an ensemble of Swiss rolls is to

simulate large grids of such rolls in free space. The field distri-

bution in and outside the grid can be fitted to a field distribution

caused by scattering at a homogeneous medium, and as a result

one could obtain the equivalent macroscopic material parame-

ters. A disadvantage of this method is the large amount of com-

puting capacity that is required to perform the simulations.

A computationally more efficient method to obtain the mac-

roscopic material parameters is to consider the situation, where

a single row of Swiss rolls is inserted into a parallel-plate wave-

guide, as shown at the top of Figure 3. This configuration can

be converted into the equivalent layered media problem, shown

at the bottom of Figure 3, using image theory.

The lattice constant d of the row of Swiss rolls is chosen

equal to 3 m, and the centers of the Swiss rolls are located at x¼ (2n � 1)d/2 for n ranging from 1 to N, with N the total num-

ber of Swiss rolls in the waveguide. Because of symmetry

reasons, the boundaries of the homogeneous medium, with

unknown material parameters e1, l1, n1, and f1, are located at x¼ 0 and x ¼ d1 ¼ Nd, as depicted in Figure 3. Indeed, dividing

a row of 2N Swiss rolls with a lattice constant d into two equal

parts of N Swiss rolls each, the boundary created between the

two equivalent homogeneous media must be located at an equal

distance of d/2 from the centers of the most nearby Swiss roll.

Hence, it seems logical and consistent to model N Swiss rolls as

an equivalent medium with a total thickness of d/2 þ (N � 1)dþ d/2 ¼ Nd.

In all the simulations we have chosen to close the waveguide

at x ¼ d2 ¼ (N þ 15)d.The width of the waveguide is chosen equal to d, such that

one obtains a square lattice of Swiss rolls after applying image

theory.

The reduction of computer capacity of the waveguide set-up

with respect to large grids is considerable: the number of

unknowns was 5055 and 12,512 for N ¼ 2 and N ¼ 10 Swiss

rolls, respectively. Moreover, the linear dependence of the num-

ber of unknowns as a function of the size of the homogeneous

medium, that is, Nd, is an important advantage of the waveguide

set-up in comparison with large grids.

4. BIANISOTROPIC MODEL

In the previous section, the ensemble of Swiss rolls is con-

verted to a homogeneous medium with effective material pa-

rameters. However, this homogenization procedure has proven

to be a difficult step. Often the extracted material parameters

exhibit antiresonant behavior, violating the conditions of pas-

sivity and/or causality [1]. On a microscopical level, the local

material parameters vary spatially across the unit cell and the

transition from homogenized medium to free space is not

defined in a clear-cut way. Hence, in Refs. 2 and 3, a distinc-

tion between local and nonlocal material parameters has been

drawn, where these latter determine the transfer matrix of a

lattice unit cell of the homogenized medium. However, in this

paper, the spatial dispersion of the Swiss roll metamaterial is

neglected.

To model the homogenized ensemble of Swiss rolls, the rele-

vant material parameters should first be identified. In this step,

the symmetries of the material play a crucial role. For example

in Ref. 19, a medium of randomly placed and oriented PEC spi-

rals is considered and because of the randomness of the me-

dium, one expects the material parameters to be those of a bi-

isotropic medium. Here, the symmetries of the Swiss roll ensem-

ble will also be used to eliminate a large number of parameters.

We start from the most general linear and local constitutive

equations [20, 21]:

D ¼ �¼ �Eþ n

¼�H (2a)

B ¼ f¼�Eþ l

¼ �H (2b)

In the waveguide set-up of Section 3, the homogenized me-

dium of Swiss rolls exhibits two symmetry planes: the xy-plane,as the Swiss roll is a 2D structure, and the xz-plane, due to the

mirroring of the PEC walls of the waveguide. As e and l are ten-

sors and n and f pseudotensors [22], these symmetries lead to

Figure 3 Swiss rolls stacked into a waveguide. Using image theory,

this problem can be converted into a layered media problem

TABLE 2 Resonance Frequencies for a Discretization into756 Segments

n f (MHz)

0 2.4453

1 8.1853

2 14.719

3 21.520

4 28.418

5 35.362

6 42.331

7 49.314

8 56.306

9 63.304


�¼ ¼ �0

�xx 0 0

0 �yy 0

0 0 �zz

24

35 l

¼ ¼ l0

lxx 0 0

0 lyy 0

0 0 lzz

24

35 (3a)

n¼¼ 1

c

0 0 0

0 0 nyz0 nzy 0

24

35 f

¼¼ 1

c

0 0 0

0 0 fyz0 fzy 0

24

35 (3b)

with e0, l0, and c the permittivity, permeability, and speed of

light in free space, respectively.

Using Eq. (3), the propagation constants can be calculated

[20, pp. 66]. For a TE-polarized plane wave propagating along

the x-axis, only the coefficients eyy, lzz, nyz, and fzy come into

play.

As the magnetoelectric coupling tensors are pseudotensors,

the wavenumber and impedance can be different for both direc-

tions: k6x and Z6 for the propagation along the positive and neg-

ative x-axis, respectively. The relationship between the wave-

numbers kþx and k�x and impedances Zþ and Z� and eyy, lzz, nyz,and fzy is:

kþx ¼ k02

nyz þ fzy6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnyz � fzyÞ2 þ 4�yylzz

q� �(4a)

k�x ¼ k02

�nyz � fzy6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnyz � fzyÞ2 þ 4�yylzz

q� �(4b)

Zþ ¼ Z0l

kþx =k0 � fzy(4c)

Z� ¼ Z0l

k�x =k0 þ fzy(4d)

with k0 the wavenumber in free space. For a passive medium,

the wavenumbers satisfy the conditions of Im(kþx ) � 0 and

Im(k�x ) � 0 [23], which determines the correct choice of the

(6)-sign in Eqs. (4a) and (4b). The relations can be inverted to

yield:

�yy ¼ kþx þ k�xk0ðZþ þ Z�Þ (5a)

lzz ¼ZþZ�ðkþx þ k�x Þk0ðZþ þ Z�Þ (5b)

nyz ¼kþx Z

� � k�x Zþ

k0ðZþ þ Z�Þ (5c)

fzy ¼kþx Z

þ � k�x Z�

k0ðZþ þ Z�Þ (5d)

When the waveguide set-up in Section 3 is illuminated by a

TE-polarized plane wave propagating along the x-axis, one can

write the fields of the equivalent layered media problem as:

x < 0 : Hz ¼ Ae�jk0x þ Beþjk0x (6a)

Ey ¼ Z0ðAe�jk0x � Beþjk0xÞ (6b)

0 < x < d1 : Hz ¼ Ce�jkþ1x þ Deþjk�

1x (6c)

Ey ¼ Zþ1 Ce

�jkþ1x � Z�

1 Deþjk�

1x (6d)

d1 < x < d2 : Hz ¼ Fe�jk0x þ Geþjk0x (6e)

Ey ¼ Z0ðFe�jk0x � Geþjk0xÞ (6f)

As k02 � p2/d2 only strongly evanescent higher order modes

can appear, and therefore, only the zeroth order mode has to be

considered in Eq. (6).

First, the complex amplitudes A, B, F, and G in Eq. (6) are deter-

mined by fitting the magnetic field, obtained by the simulation, for x< 0 and d1 < x< d2, to Eqs. (6a) and (6e), respectively.

Second, the unknown wavenumbers k61 and impedances Z61are determined. Applying the boundary conditions, that is, conti-

nuity of the tangential fields at the boundaries x ¼ 0 and x ¼ d1would lead to only four equations for six unknowns: kþ1 , k

�1 , Z

þ1 ,

Z�1 , C, and D. Therefore, we also consider a second simulation,

with all Swiss rolls rotated over an angle p in the xy-plane. Thisoperation results in the transformations nyz ! �nyz and fzy !�fzy, as n and f are pseudotensors. Changing the signs of the

magnetoelectric coupling coefficients corresponds with the trans-

formation (kþ1 , k�1 , Z

þ1 , Z

�1 ) ! (k�1 , k

þ1 , Z

�1 , Z

þ1 ), as can be seen

from Eq. (5). Applying the boundary conditions for both config-

urations, one obtains

Að1Þ þ Bð1Þ ¼ Cð1Þ þ Dð1Þ (7a)

Z0ðAð1Þ � Bð1ÞÞ ¼ Zþ1 C

ð1Þ � Z�1 D

ð1Þ (7b)

Cð1Þe�jkþ1d1 þ Dð1Þeþjk�

1d1 ¼ Fð1Þe�jk0d1 þ Gð1Þeþjk0d1 (7c)

Zþ1 C

ð1Þe�jkþ1d1 � Z�

1 Dð1Þeþjk�

1d1

¼ Z0ðFð1Þe�jk0d1 � Gð1Þeþjk0d1Þ(7d)

Að2Þ þ Bð2Þ ¼ Cð2Þ þ Dð2Þ (7e)

Z0ðAð2Þ � Bð2ÞÞ ¼ Z�1 C

ð2Þ � Zþ1 D

ð2Þ (7f)

Cð2Þe�jk�1d1 þ Dð2Þeþjkþ

1d1 ¼ Fð2Þe�jk0d1 þ Gð2Þeþjk0d1 (7g)

Z�1 C

ð2Þe�jk�1d1 � Zþ

1 Dð2Þeþjkþ

1d1

¼ Z0ðFð2Þe�jk0d1 � Gð2Þeþjk0d1Þ(7h)

with the superscript ‘‘(1)’’ and ‘‘(2)’’ denoting the first and the

second simulation, respectively. By combining the two configu-

rations, we obtain eight equations for eight unknowns: kþ1 , k�1 ,

Zþ1 , Z�1 , C

(1), D(1), C(2), and D(2).

The equations in (7) are invariant under the transformations

(kþ1 , k�1 , Zþ1 , Z�1 , C(1), D(1), C(2), D(2)) $ (�k�1 , �kþ1 , �Z�1 ,�Zþ1 , D

(1), C(1), D(2), C(2)) and k61 $ k61 þ 2p/d1, resulting in

multiple solutions.

The first ambiguity can be solved by applying passivity, for

which the wavenumbers must satisfy Im(kþ1 ) � 0 and Im(k�1 ) �0 [16, 17]. In the case when the wavenumbers have no imagi-

nary part, both solutions can be valid. The second ambiguity

corresponds with the number of interference fringes inside the

Swiss roll medium. The correct physical (kþ1 , k�1 , Z

þ1 , Z

�1 )-solu-

tion is determined by comparing the number of interference

fringes of the simulation to the analytical solution for 0< x <d1.In the case when the wavenumbers have no imaginary part,

the procedure of comparing the number of interference fringes

solves both the first and the second ambiguity.

Finally, after identifying the physical solution, the macro-

scopic material parameters can be derived from Eq. (5). It is

worthwhile to point out that the first ambiguity, in contrast to

the second ambiguity, leaves Eq. (5) unchanged.

The data, used for the fitting of the simulation to the analyti-

cal expressions, are the field distribution along the dashed line

PP0 in Figure 3.

5. EQUIVALENT MATERIAL PARAMETERS

Applying the theory of the previous section to our Swiss roll

example and restricting the analysis to a frequency range near

the first resonance frequency, the S-parameter retrieval is per-

formed from f ¼ 2.356 to 2.556 MHz in frequency steps of 1


kHz. Figure 4 shows the obtained macroscopic permeability and

permittivity for N ¼ 2, denoted by the small circles, and N ¼10, denoted by the full line.

Comparing the curves for N ¼ 2 and N ¼ 10 Swiss rolls in

Figure 4, one observes that the results do not deviate much.

This indicates that the homogenization also works well for a

small number of Swiss rolls and that the finite thickness of the

transition from material to free space, mentioned in the begin-

ning of Section 4, can be neglected.

For frequencies that are very close to the resonance fre-

quency fres ¼ 2.4453 MHz, the absolute value of the wavelength

becomes comparable to the lattice constant of the grid, and the

grid of Swiss rolls cannot be homogenized. Moreover, for fre-

quencies slightly above the resonance frequency, the strong ex-

ponential decrease of the field can no longer be simulated with

sufficient accuracy. Hence, no trustworthy values for lzz, eyy,nyz, and fzy can be derived very close to the resonance.

The permeability and permittivity have no imaginary part,

whereas the magnetoelectric coupling coefficients are purely

imaginary numbers. The obtained permeability and permittivity

are passive (Im(l,e) � 0) and causal (DRe(l,e)/Dx � 0) [23], as

one can see from Figure 4(a).

From Figure 4(b), one sees that fzy ¼ �nyz, so the results

correctly confirm the reciprocity property [24] of the Swiss roll

metamaterial, as the simulation contained only PECs, which are

reciproque. As stated in Section 1, reciprocity could also have

Figure 4 Material parameters as a function of frequency for N ¼ 2 (circles) and N ¼ 10 (full line) Swiss rolls. (a) Relative permeability (blue) and

permittivity (green). (b) Magnetoelectric coupling coefficients: nyz (blue) and fzy (green). [Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com]


been assumed from the very beginning, as in Refs. 9, 16, and

17. However, the successful a posteriori confirmation of reci-

procity is a valuable validation of our results. From Eq. (5) the

bidirectional property kþx ¼ k�x follows, which has been proven

in Ref. 25 for periodic waveguides in general.

The formulas, used in Ref. 9 to extract the material parame-

ters from the S-parameters, are only valid for real wavenumbers,

as stated in Ref. 16. However, in Ref. 9, the use of these formu-

las is extended to complex wavenumbers. Hence, for these com-

plex wavenumbers, the results remain open for discussion.

For N ¼ 10 Swiss rolls, the curves in Figure 4 show physi-

cally incorrect peaks at f ¼ 2.425 and 2.438 MHz. For these fre-

quencies, the wavenumber inside the Swiss roll medium equals

np/d1, with n ¼ 2 and 3, respectively. In this case, the set of

Eqs. (7) becomes singular and no information about the impe-

dances can be obtained.

Figure 5 displays the logarithm of the amplitude of the mag-

netic field as a function of the place coordinate along the dashed

line PP0 in Figure 3 and shows the comparison between the result

of the full-wave simulations and the analytical solutions for N ¼10 Swiss rolls at a frequency f ¼ 2.468 MHz. The red line in Fig-

ure 5 stands for the logarithm of the absolute error between

the full-wave simulation and the analytical model, that is,

log10|Hz,full-wave � Hz,analytical|. For x < 0 and x > Nd, we see that

the plane wave fit corresponds very well to the simulation. The

high error for �75 � x � �70 can be explained by the reflections

at the entrance of the waveguide, which the transmission line rep-

resentation (6a) is unable to represent correctly. Inside the mate-

rial, for 0< x < Nd, the error remains below 1%.

The good agreement between the full-wave results and the

analytical model, for f ¼ 2.468 MHz and for all frequencies in

general, implies that the used bianisotropic model and the set-

up, explained in Section 3, are valid.

6. CONCLUSIONS

In this paper, the Swiss rolls are investigated by means of full-

wave simulations. First, we accurately determined the first 10

resonance frequencies of a single Swiss roll. Next, the macro-

scopic material parameters of an ensemble of Swiss rolls are

determined by S-parameter retrieval, using a bianisotropic

model. Therefore, simulations of a single row of Swiss rolls

stacked into a parallel-plate waveguide are performed. From

image theory, it follows that the obtained results are the same as

the results of simulations with large grids of Swiss rolls. The

advantage using the parallel-plate set-up over large grids is the

strong reduction of the required computational capacity. The

obtained permeability, permittivity, and magnetoelectric cou-

pling coefficients satisfy the physical conditions of passivity,

causality, and reciprocity, and the homogenized model accu-

rately predicts the behavior of the Swiss roll grids.

Finally, to further check the homogenization model, full-wave

results are compared with the analytical model equivalents.

ACKNOWLEDGMENT

The computational resources and services used in this work were

provided by Ghent University.

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VC 2011 Wiley Periodicals, Inc.

TRIPLE-BAND CPW-FED L-SHAPEDMONOPOLE ANTENNA WITH SMALLGROUND PLANE

Kwok L. Chung1 and Sarawuth Chaimool21 Department of Electronic and Information Engineering, HongKong Polytechnic University, Hong Kong China SAR;Corresponding author: [email protected] King Mongkut’s University of Technology North Bangkok, Thailand

Received 16 December 2010

ABSTRACT: This article presents a CPW-fed planar monopole antennawith a small ground-plane size. The monopole antenna is a triple-banddesign and is meant for mobile handsets with WiFi and WiMAX features.

The planar structure comprises an L-shaped monopole backed by a stub-loaded open-loop resonator. The proposed antenna features three

bandwidths of 2.25–2.62 GHz, 3.25–4.97 GHz, and 5.2–6.30 GHz, whichyielded the maximum gain of around 1 to 2 dBi. Reduction effects ofground-plane on antenna’s performance rectified by the parameters of

open-loop resonator are also investigated. The overall dimensions are 46(W) � 34 (L) � 0.8 (d) mm3. VC 2011 Wiley Periodicals, Inc. Microwave

Opt Technol Lett 53:2274–2277, 2011; View this article online at

wileyonlinelibrary.com. DOI 10.1002/mop.26240

Key words: CPW-fed antenna; ground-plane effect; planar monopoleantenna; WiFi-WiMAX applications

1. INTRODUCTION

Over the past decade, compact multiband planar antennas have

attracted considerable interest for applications in multifunction and

multimode wireless communication systems [1–5]. User markets

demand high functionality and multiple services in compact mo-

bile handsets. For instance, a cell-phone supports simultaneously

multiple cellular voice standards (e.g., GSM/PCS/WCDMA), as

well as data communication standards (e.g., GPS/WiFi/Bluetooth/

FM/DTV). In recent years, Worldwide Interoperability for Micro-

wave Access, known as WiMAX, has resulted in usage of micro-

wave to enable the last mile delivery for wireless data transfer.

There is a tendency to include WiMAX standard in modern cell-

phones designs [6–9]. Hence, when designing a multiband antenna

for modern mobile handsets, one needs to consider the allocated

WiMAX frequency bands: 2.5 GHz band (2.5–2.69 GHz), 3.5

GHz band (3.4–3.69 GHz), and 5.5 GHz band (5.25–5.85 GHz).

Among the number of techniques reported in the literature for

achieving dual-band or multiband operations, coplanar waveguide

(CPW)-fed planar monopole antennas have received much more in-

terest [7–11] than others because of possibility of miniaturization,

single-layer structure, and low fabrication cost. The ground-plane of

CPW-fed monopole, which is mounted on the same side (coplanar)

of monopole element, is regarded as part of the radiating structure.

This is known as the ground-plane-dependent antenna [12]. The sur-

face current distribution on the ground plane is the dominant factor

in determining the radiating performance of the combined (antenna-

ground plane) structure. Literature review has found that studies on

effects of ground-plane size as well as its compensation on multi-

band planar monopole antennas are scant. Effects of ground-plane

size on the microstrip-line fed dual-band antenna [2] and a detailed

study on multiband designs is also available [12]. However, none

covers the cases when CPWs are used as the feeding mechanism.

On the other side, compact and miniature multiband CPW-fed

antennas are often proposed [7–11], but their ground-plane sizes

and/or overall sizes of the completed antennas are neither investi-

gated nor justified. Such a ground-plane effect certainly gives rise to

practical engineering issues such as the total volume and physical

height of the mobile handset. On these lines, this article presents a

CPW-fed L-shaped monopole antenna backed by a stub-loaded

open-loop resonator, using a small ground plane. The triple-band

antenna used in this study is designed to fulfill the requirements of

WiFi and WiMAX bands. In contrast to the larger ground-plane ver-

sion introduced in [8], our aim here is focused on the compensation

of small ground-plane size of such CPW-fed monopole antenna. In

addition to the examination of effects on reducing ground-plane

length, the compensation on these effects by varying the open-

loop’s positions, lengths, and stub-lengths on the impedance match-

ing of the proposed antenna are also studied.

2. ANTENNA GEOMETRY

The geometry and photograph of the triple-band antenna with

small ground-plane size are illustrated in Figure 1. A coplanar

waveguide (CPW)-fed L-shaped monopole antenna is printed on

one side of an inexpensive FR4 dielectric substrate with a

dielectric constant er ¼ 4.4, and thickness d ¼ 0.8 mm. A stub-

loaded open-loop resonator is parasitically coupled on the back-

side of the CPW-fed monopole. The 50-X feed line has a width

of wf ¼ 1.43 mm with gaps of 0.15 mm printed on the same

side of the two symmetrical ground planes of size (Wg � Lg) 22� 12 mm2, which are only 25.5 and 84.6% of length (Lg) and

width (Wg), respectively, of the one presented in [8].

To compensate for the reduction in ground-plane sizes, the reso-

nant length (RL) and the relative position (VL) of the open-loop res-

onator have been increased. The sizes of L-shaped monopole require

fine adjustments to maintain the triple-band for WiFi and WiMAX

operations. However, the open-stub length has been reduced to 3.6

mm as shown in Figure 1(b). The stub-loaded open-loop resonator is

responsible for the generation of resonant modes at 2.45 and 4.15


swiss roll ensemble homogenization by full-wave simulations

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