Towards a systematic design of isotropic bulk magnetic metamaterials using the cubic point
groups of symmetry
J. D. Baena,
*
L. Jelinek,
†
and R. Marqués
‡
Departamento de Electrónica y Eletromagnetismo, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain
共Received 8 May 2007; revised manuscript received 27 September 2007; published 17 December 2007
兲
In this paper, a systematic approach to the design of bulk isotropic magnetic metamaterials is presented. The
roles of the symmetries of both the constitutive element and the lattice are analyzed. For this purpose, it is
assumed that the metamaterial is composed of cubic split ring resonators 共SRRs兲 arranged in a cubic lattice.
The minimum symmetries needed to ensure an isotropic behavior are analyzed, and some particular configu-
rations are proposed. Besides, an equivalent circuit model is proposed for the considered cubic SRRs. Experi-
ments are carried out in order to validate the proposed theory. We hope that this analysis will pave the way to
the design of bulk metamaterials with strong isotropic magnetic response, including negative permeability and
left-handed metamaterials.
DOI: 10.1103/PhysRevB.76.245115 PACS number共s兲: 42.70.Qs, 41.20.Jb, 42.25.Bs
I. INTRODUCTION
Metamaterials are artificial media exhibiting exotic elec-
tromagnetic properties not previously found in nature.
Among them, media showing simultaneously negative elec-
tric permittivity and magnetic permeability in some fre-
quency range, or “left-handed” metamaterials, are of particu-
lar interest. The striking properties of left-handed
metamaterials, including backward-wave propagation, nega-
tive refraction, and inverse Cerenkov and Doppler effects
were first reported by Veselago
1
in 1968. However, the real-
istic implementations of left-handed metamaterials came
several decades later, as a combination of split ring resona-
tors 共SRRs兲 and metallic wires.
2
SRRs are small planar reso-
nators exhibiting a strong magnetic response, which were
proposed in 1999 by Pendry et al.
3
as suitable “atoms” for
the development of negative magnetic permeability metama-
terials. One year later, Smith et al. demonstrated the possi-
bility of making up a left-handed medium by periodically
combining metallic wires—which provide an effective nega-
tive permittivity at microwaves
4
—and SRRs.
2
In subsequent
works, other SRR designs were proposed,
5–8
in order to re-
duce electrical size and/or cancel the bianisotropic behavior
of the original Pendry’s design. However, all the aforemen-
tioned implementations of negative permeability and left-
handed metamaterials are highly anisotropic—or even
bianisotropic
5
—providing only a uniaxial resonant magneti-
zation, while isotropy is needed for many interesting appli-
cations of metamaterials, as, for instance, the “perfect lens”
proposed by Pendry.
9
The aforementioned implementations are, in fact, a com-
bination of two separate systems, one providing the negative
magnetic permeability 共the SRR system兲 and the other pro-
viding the negative electric permittivity 共the wire system兲.
How both subsystems can be combined in order to obtain a
new system, whose electromagnetic properties were mainly
the superposition of the magnetic and the electric properties
of each subsystem, is an interesting and controversial
issue
10,11
that is, however, beyond the scope of this paper. In
what follows, we will assume that it is possible to find some
combination of two isotropic subsystems, one made of me-
tallic wires 共or other elements providing a negative electric
permittivity兲 and the other made of SRRs, whose superposi-
tion gives a left-handed metamaterial, and will focus our
attention on the design of isotropic systems of SRRs. Actu-
ally, since isotropic media with negative magnetic permeabil-
ity are not found in nature, an isotropic system of SRRs
providing such property in some frequency range will be an
interesting metamaterial by itself. These metamaterials could
provide the dual of negative electric permittivity media, with
similar applications 共in imaging,
9
for instance兲. They would
be also of interest for magnetic shielding and other practical
applications.
A first attempt to design an isotropic magnetic metamate-
rial was carried out by Gay-Balmaz and Martin,
12
who de-
signed a spherical magnetic resonator—formed by two SRRs
crossed in right angle—which is isotropic in two dimensions.
This result was later generalized in Ref. 13, where a fully
isotropic spherical magnetic resonator was proposed. How-
ever, from a practical standpoint, it is usually easier to work
with cubic designs. A first attempt on such direction was
made by Simovski and co-workers in Refs. 14–16, where
cubic arrangements of planar SRRs and omega particles were
proposed 关see Figs. 1共a兲 and 1共b兲兴. If only the magnetic
and/or electric dipole representations of the SRRs and/or
omega particles are considered, these arrangements are in-
variant under cubic symmetries. However, it has been
shown
13,17
that this invariance is not enough to guarantee an
isotropic behavior since couplings between the planar reso-
nators forming the cubic arrangement can give rise to an
anisotropic behavior, even if its dipole representations sug-
gest an isotropic design. The first isotropic metamaterial de-
sign fully invariant under the whole group of symmetry of
the cube was proposed and simulated in Ref. 18. It is formed
by volumetric square SRRs with four gaps, in order to pro-
vide 90° rotation symmetries about any of the cube axes.
However, this design is unfortunately very difficult to imple-
ment in practice because it cannot be manufactured by using
standard photoetching techniques, as previous SRR
designs,
2,3,5–8,13–17
and the gaps of the SRR have to be filled
with a high relative permittivity dielectric 共about 100兲. The
idea of using spatial symmetries to design isotropic metama-
PHYSICAL REVIEW B 76, 245115 共2007兲
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terials was further developed in Refs. 13, 17, and 19 leading
to the structures depicted in Figs. 1共c兲 and 1共d兲.
A second group of attempts to design isotropic metama-
terials is developed in Ref. 20 and 21. In these works, lattices
of dielectric and/or paramagnetic spheres with very high re-
fractive index are proposed. If the refractive index of the
spheres is high enough, the internal wavelength becomes
small with regard to the macroscopic wavelength, and Mie
resonances of the spheres can be used to produce the nega-
tive effective permittivity and/or permeability. Since the
metamaterial “atoms” are spheres, the isotropy is ensured by
simply placing them in a cubic lattice. However, practical
difficulties to implement such proposals are not easy to over-
come. First of all, lossless media with the very high refrac-
tive index needed for the spheres are difficult to obtain. Sec-
ondly, the system has a very narrow band.
21
All the previously reported proposals for isotropic mag-
netic metamaterial design use a “crystal-like” approach. That
is, they are based on the homogenization of a system of
magnetic resonators which, according to causality laws, ex-
hibit a strong diamagnetic response above resonance. There
is, however, another approach widely used in the microwave
community which is based on the transmission line analogy
to effective media. Initially proposed for two-dimensional
metamaterial design,
22
it was recently generalized to three-
dimensional isotropic structures.
23–26
The main advantage of
this approach is its broadband operation, since no resonators
are necessary for the design. However, it also presents dis-
advantages with regard to crystal-like approaches. The trans-
mission line approach to metamaterials does not seem to be
applicable beyond the microwave range, whereas a signifi-
cant magnetic response of the SRR has been shown in the
terahertz range and beyond.
27,28
In addition, the coupling to
free space of the reported transmission line metamaterials
seems to be difficult and sometimes needs an additional spe-
cific interface 共e.g., an antenna array
25
兲, whereas this cou-
pling is direct in crystal-like metamaterials.
Finally, regarding isotropic left-handed metamaterial de-
sign, it should be mentioned that some recent proposals
based on random arrangements of chiral particles
29,30
have
the advantage of providing simultaneously both electric and
magnetic negative polarizabilities. This approach can be
straightforwardly extended to the design of SRR magnetic
metamaterials, by simply considering random arrangements
of such elements. There is, however, a major difficulty with
this approach: the constitutive elements in a random compos-
ite have to be very small in comparison with the macroscopic
wavelength to show a true statistical behavior, but it is not
easy to design a SRR much smaller than one-tenth of the
wavelength. Due to this fact, periodic arrangements will be
considered in what follows.
The main aim of this paper is to present a systematic
approach to the design of metamaterial structures based on
periodic arrangements of SRRs. The first section is focused
on the spatial symmetries which are necessary to ensure an
isotropic behavior in the metamaterial. Cubic arrangements
of SRRs placed on cubic lattices are considered, and the
minimum symmetry requirements for both the individual
resonators and the lattices are investigated. The second sec-
tion is devoted to a deeper analysis of the isotropic cubic
SRRs forming the basis of the crystal structure. In the third
section, an equivalent circuit model for such cubic SRRs is
developed and applied to some specific examples. The fourth
section is focused on the experimental verification of the
analysis developed in the previous ones. Finally, the main
conclusions of the work are presented.
II. ROLE OF CUBIC SYMMETRIES
Let us assume that constitutive elements and the unit cell
of the material are much smaller than the operating wave-
length. In such a case, the interaction of electromagnetic field
with the material is described by means of constitutive rela-
tions. Besides, the material is supposed to be linear, so the
most general way to express those relations between electro-
magnetic intensities and electromagnetic flux densities is
31
D =
· E +
· H,
B =
· E +
· H, 共1兲
where
,
are second rank constitutive tensors and
,
are
second rank constitutive pseudotensors. In order to get a
macroscopic isotropic behavior, all constitutive tensors and
pseudotensors
,
,
, and
must become scalars or pseu-
doscalars.
Let us now address the problem of forcing the tensors 共or
pseudotensors兲 in Eq. 共1兲 to be scalars 共or pseudoscalars兲 for
the specific case of a periodic structure. It is well known
32,33
that there are 32 symmetry point groups for periodic crystals
which can be classified in 7 crystallographic systems. It is
also known that the cubic system is the only one that forces
any second rank tensor 共or pseudotensor兲 to be a scalar 共or a
pseudoscalar兲.
33
Since any material satisfying the linear con-
stitutive relations 关Eq. 共1兲兴 and being invariant under the cu-
FIG. 1. Cubic constitutive elements for isotropic metamaterial
design. Cubes 共a兲 and 共b兲 were studied in Refs. 14–16. Their hidden
faces are arranged in such a way that the cube satisfies the central
symmetry to avoid magnetoelectric coupling. Cubes 共c兲 and 共d兲
were proposed in Ref. 13 as truly three-dimensional 共3D兲 isotropic
cubic resonators.
BAENA, JELINEK, AND MARQUÉS PHYSICAL REVIEW B 76, 245115 共2007兲
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bic symmetries exhibits an isotropic macroscopic behavior,
this section will be focused on the analysis of such cubic
symmetries. It is clear that any structure invariant under all
the symmetry transformations of the cube must be isotropic,
as already proposed by Koschny et al.
18
Furthermore, the full
symmetry group of the cube contains four different sub-
groups also belonging to the cubic system and, thus, provid-
ing an isotropic macroscopic behavior. Since a less symmet-
ric design is subjected to less structural constraints, it may be
guessed that using these subgroups—instead of the whole
symmetry group of the cube—may have practical advan-
tages. Keeping this in mind, we will first give a short over-
view on the five cubic point groups. Next, we shall connect
these point groups with some real structures made of planar
resonators commonly used in metamaterials. This will be
done in two parts: the study of the symmetries of the consti-
tutive element, or the basis, and the analysis of the suitable
periodic arrangements, or the lattice. At the end of the sec-
tion some practical isotropic structures will be specifically
analyzed.
A. Cubic point groups
The five cubic point groups are schematically represented
in Fig. 2. Following Schöenflies’ notation and ordering by
degree of symmetry, these groups and their generators are as
follows:
共1兲 T=具兵1,4
x
·4
y
,4
y
·4
x
其典=proper rotations of the regular
tetrahedron 共12 operations兲;
共2兲 T
h
=具兵1,−1 ,4
x
·4
y
,4
y
·4
x
其典=T expanded by the inver-
sion 共24 operations兲;
共3兲 T
d
=具兵1,−2
x
,4
x
·4
y
,4
y
·4
x
其典=proper and improper rota-
tions of the regular tetrahedron 共24 operations兲;
共4兲 O=具兵1 ,4
x
,4
y
其典=proper rotations of the cube 共24 op-
erations兲;
共5兲 O
h
=具兵1,−1 ,4
x
,4
y
其典=full symmetry group of the cube
共48 operations兲.
We have used a widely used notation for symmetry trans-
formations, 1 being the identity operator, −1 the inversion,
n
p
an n-fold rotation axis about the p axis, and −n
p
the n-fold
axis about the p axis followed by the inversion. For example,
the operator −2
x
is the rotation through 180° about the x axis
followed by the inversion.
B. Cubic basis
In order to simplify the problem, the symmetries can be
separately imposed on the basis and the lattice of the struc-
ture. For the sake of simple fabrication, we will assume that
the basis is formed by six planar resonators placed over the
faces of an inert rigid cube, as in Fig. 1. If the crystal was
diluted enough, then the coupling between two neighboring
cubes would be much weaker than the coupling between the
six SRRs of the same cube and thus each cube could be seen
as a single cubic resonator 共CR兲 electromagnetically coupled
to others. Such consideration implies that the interaction be-
tween the CRs forming the material can be described by
dipole-dipole interactions, higher order multipole interac-
tions being negligible. In such approximation, all the CRs are
properly described by second rank polarizability tensors con-
necting the external field, E
ext
and B
ext
, with the dipolar mo-
ments, p and m, induced in the CRs,
31,34
p =
␣
ee
· E
ext
+
␣
em
· B
ext
,
m =
␣
mm
· B
ext
−
␣
em
t
· E
ext
, 共2兲
where
␣
ee
,
␣
mm
, and
␣
em
are the electric, magnetic, and mag-
netoelectric polarizability tensors, and the superscript t
means transpose operation. The constitutive tensors in Eq.
共1兲 can be derived from these polarizabilities and from the
lattice structure by applying a homogenization technique.
In what follows, different kinds of CRs will be named by
its cubic group symmetry followed by the acronym CR
共group-CR兲. In order to design an isotropic CR, we have to
find suitable planar resonators and place them correctly over
the cube so as to fulfill the necessary symmetries. Obviously,
the planar resonators have to be invariant under certain sym-
metry transformations of the square. To classify all different
possibilities, a list of the symmetry subgroups of the square
is shown in Table I, as well as their geometrical representa-
tions, and some examples of planar resonators commonly
used in metamaterial design and obeying these symmetries.
This table also provides a systematic terminology for planar
resonators by using the symbol of the symmetry group fol-
lowed by the term SRR 共group-SRR兲. In what follows, we
will use the term SRR in a general sense covering any type
of geometry derived from the SRR and the omega particle.
By direct inspection on Fig. 2, it can be seen that any of
the five cubic point groups contains three twofold rotation
axes 共180° rotations兲 parallel to the edges of the cube. Thus,
only resonators belonging to the last five rows of Table I are
appropriate for designing isotropic CRs. At this point, it may
be worth mentioning that Pendry’s SRRs
3
as well as Omega
particles
35
are not appropriate for such purpose because they
correspond to the C
1
-SRR and D
1
-SRR topologies. In sum-
mary, in order to get an isotropic CR, we have to choose six
identical SRRs pertaining to the classes C
2
-, D
2
-, C
4
-, or
D
4
-SRR and arrange them according to one of the cubic
point groups T, T
d
, T
h
, O,orO
h
shown in Fig. 2.
Although all five cubic point groups mentioned above are
equally useful to achieve isotropic CRs, a specific choice
may strongly affect the properties of an isotropic metamate-
rial. For instance, using isotropic CRs of low symmetry may
FIG. 2. Objects with the symmetries of the five cubic point
groups.
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be quite advantageous since the electrical size of the CRs can
be made smaller. This fact can be justified in terms of the LC
circuit models for the SRRs
5–8
because the effective capaci-
tances of low symmetry SRRs are usually higher than those
of high symmetry SRRs,
8
thus providing a smaller resonance
frequency. Following these considerations, the best choice of
basis would be a T-CR made of six planar resonators of the
C
2
-SRR type. A good candidate among all possibilities is the
cube shown in Fig. 1共c兲 made of six nonbianisotropic SRRs
共NB-SRRs兲,
8,36
a configuration already proposed in Refs. 13
and 17. Furthermore, it was shown in Ref. 13 that this con-
figuration shows a bi-isotropic behavior, due to the lack of
inversion symmetry of the cubic arrangement. However,
sometimes, an effective isotropic medium without bi-
anisotropy 共
,
=0兲 is desired. Since
and
are pseudoten-
sors, the invariance of the CR under inversion is required in
order to avoid such property. In this case, the lowest symme-
try group is the T
h
group. A CR invariant under the last group
of symmetry can be made by using planar resonators of the
D
2
-SRR type as, for instance, the symmetric SRR
37
or the
modified double-slit broadside coupled SSR 共BC-SRR兲
shown in Fig. 1共d兲.
13
However, as will be shown in the fol-
lowing, such symmetry requirements can be relaxed if the
lattice symmetries are properly chosen.
C. Cubic lattices
Above findings give precise instructions for choosing
suitable geometries for isotropic metamaterial constitutive
elements. The next step is to create an isotropic metamaterial
with these elements. The cubic shape of the considered con-
stitutive elements suggests that the best periodical arrange-
ments are the simple cubic 共sc兲, body centered cubic 共bcc兲,
and face centered cubic 共fcc兲 lattices shown in Fig. 3. All
these lattices obey the full symmetry group of the cube, O
h
.
Therefore, the whole metamaterial 共lattice plus basis兲 retains
the cubic point group symmetries and the macroscopic iso-
tropic behavior.
Although all previously mentioned lattices can provide
isotropic metamaterials, it is convenient to look deeply into
the possible structures because some particular choices may
offer interesting advantages. Regarding Fig. 3, a is the edge
size of the CR and b is the edge size of the cubic unit cell. In
order to describe CR interactions as dipole-dipole interac-
tions, b must be chosen much larger than a, so that the
metamaterial properties can be deduced from Eq. 共2兲 and the
appropriate homogenization procedure. However, usually,
we are also interested in a high density of dipoles in order to
get a strong electromagnetic response. Therefore, b should
TABLE I. Classification of SRR types based on the symmetry subgroups of the square. The second
column shows Schöenflies’ notation and the generator of groups. The symbols of transformations are 1
=identity; 4 =90° rotation; 2=180° rotation; −4 =−90° rotation; m
x
, m
y
=line reflections respect to the x and
y axes, respectively; m
x,y
, m
x−y
=line reflections respect to both diagonals of the square. Each group is
schematically represented by the objects in the second column which can be replaced by the planar resonators
shown in the third column.
SRR types Symmetry subgroups of the square
Geometrical
representation
Examples of
resonators
C
1
-SRR
C
1
={1}
D
1
-SRR
D
1x
={1, m
x
}
D
1y
={1, m
y
}
D
1,x,y
={1, m
x,y
}
D
1,x,-y
={1, m
x,-y
}
C
2
-SRR
C
2
={1, 2}
D
2
-SRR
D
2x
= D
2y
={1, m
x
, m
y
, 2}
D
2xy
=
2 xy
D ={1, m
x,y
, m
x,-y
, 2}
C
4
-SRR
C
4
={1, 4, 2,-4}
D
4
-SRR
D
4
={1, 4, 2,-4, m
x
, m
y
, m
x,y
, m
x,-y
}
BAENA, JELINEK, AND MARQUÉS PHYSICAL REVIEW B 76, 245115 共2007兲
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be as small as possible. However, decreasing b may lead to a
failure of the aforementioned homogenization procedure.
However, in any case, the combination of a basis and a lat-
tice with the appropriate symmetries will provide an isotro-
pic metamaterial, regardless of the homogenization proce-
dure. Finally, there are some practical limitations to the
values that a and b can reach as, for instance, the obvious
inequality b 艌 a, derived from the fact that CRs are supposed
to be impenetrable.
Additional limitations appear for each specific structure.
In the case of a sc lattice with T-, T
d
-, or O-CRs, the lack of
inversion symmetry implies that opposite sides of a CR are
not oriented in the same way. Thus, the constrain b ⬎a is
necessary in order to avoid a mutual short circuit between
the SRRs of neighboring CRs. To allow the minimum dis-
tance b =a, the noncentrosymmetric CRs in the sc lattice
must be replaced by T
h
-orO
h
-CRs, so that the SRRs on
contacting sides of neighboring CRs exactly overlap. In the
case of a bcc lattice, the contact between corners implies that
the inequality b艌a must be fulfilled for any type of CR.
Finally, for the fcc lattice, the contact between edges of
neighboring CRs establishes the harder condition b艌2a.
The particular case of an fcc lattice with the minimum cell
size, b=2a, deserves a specific analysis. When T
h
-or
O
h
-CRs are used as the basis of the fcc lattice, the structure
turns into an sc lattice with the highest possible compactness,
i.e., b =a, because the holes between each eight neighboring
CRs have the same shape as the CRs forming the basis. The
case of an fcc lattice with a T-, T
d
-, or O-CR basis is even
more special and interesting because each hole exactly cor-
responds with the inversion of the CR of the basis. There-
fore, although the basis of the structure is not invariant under
inversion, the fcc structure is brought into coincidence with
itself by inversion centered at the center of a CR, followed
by a translation of length a through any of the cube axes.
Since the wavelength of the signal illuminating the structure
is supposed to be much larger than a, the system can be
considered as macroscopically invariant under inversion and,
therefore, any bi-isotropic behavior must disappear. Thus, we
conclude that a very interesting choice in order to obtain an
isotropic metamaterial is the fcc lattice with b =2a and with a
basis formed by T-CRs 关example in Fig. 1共c兲兴 because of its
high compactness, non-bi-isotropic macroscopic behavior,
and low degree of symmetry. It is worth recalling here that
T-CRs have the lowest symmetry among all the possibilities
shown in Fig. 2, which helps to reduce the electrical size of
the unit cell, as explained above.
III. RESONANCES AND POLARIZABILITIES OF CUBIC
RESONATORS
Until now, only the symmetry of CRs and cubic lattices
useful for isotropic periodic metamaterials were analyzed.
However, in order to have a complete characterization of the
metamaterial, polarizabilities and couplings between indi-
vidual SRRs must be considered. In dilute crystals, the ap-
proach of weak coupling between CRs, but strong coupling
between the SRRs of each CR, is valid. Then, the metama-
terial characterization involves two separate problems: ob-
taining the polarizability tensors in Eq. 共2兲 for a single CR
and applying the appropriate homogenization procedure to
obtain the constitutive parameters for the whole structure.
For dense packages, the aforementioned approach is not
valid since couplings between SRRs of different CRs can be
stronger than SRR couplings inside each individual CR.
However, even in these cases, the analysis of the isolated CR
resonances and polarizabilities still provides useful informa-
tion on the behavior of the metamaterial. For instance, it
allows to elucidate if the coupling between SRRs in a prac-
tical low symmetry CR can be neglected or not. In case they
could be neglected, all the analysis in Sec. II B would be-
come irrelevant because the SRRs could be substituted by its
equivalent dipoles 共as it was assumed in Refs. 14–16兲, with-
out more considerations on the SRR structure. Therefore, the
analysis in this section is necessary in order to justify the
practical relevance of the analysis developed in Sec. II. Fur-
ther, in Sec. IV, an experimental validation of this analysis
will be provided.
Let us assume that the CR size is much smaller than the
operating wavelength. Thus, an RLC circuit model is valid
for describing the behavior of single Pendry’s SRRs,
3
as well
as for any type of modified SRRs
5–8
or omega particles.
38
Furthermore, if the resonators are not too close 共so that the
interaction energies are small with regard to the self-energy
FIG. 3. Cubic Bravais’ lattices. Their top views are also depicted
for the particular case of b =2a. Black and gray small cubes repre-
sent cubic resonators on successive planes.
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