Electric coupling to the magnetic resonance of split ring resonators

Abstract

We study both theoretically and experimentally the transmission properties of a lattice of split ring resonators (SRRs) for different electromagnetic (EM) field polarizations and propagation directions. We find unexpectedly that the incident electric field E couples to the magnetic resonance of the SRR when the EM waves propagate perpendicular to the SRR plane and the incident E is parallel to the gap-bearing sides of the SRR. This is manifested by a dip in the transmission spectrum. A simple analytic model is introduced to explain this interesting behavior. Comment: 4 pages, 4 figures

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arXiv:cond-mat/0407369v1 [cond-mat.mtrl-sci] 14 Jul 2004
Electric coupling to the magnetic resonance of split ring resonators
N. Katsarakis, T. Koschny, and M. Kafesaki
Institute of Electronic Structure and Laser (IESL), Foundation for Research and Technology Hellas (FORTH),
P.O. Box 1527, 71110 Heraklion, Crete, Greece.
E. N. Economou
IESL-FORTH, P.O. Box 1527, 71110 Heraklion, Crete, Greece, and Dept. Physics, University of Crete, Greece
C. M. Soukoulis
IESL-FORTH, and Dept. of Materials Science and Technology, 71110 Heraklion, Crete, Greece
Ames Laboratory and Dept. Physics and Astronomy, Iowa State University†, Ames, Iowa 50011
We study both theoretically and experimentally the transmission properties of a lattice of split ring
resonators (SRRs) for different electromagnetic (EM) field polarizations and propagation directions.
We find unexpectedly that the incident electric field E couples to the magnetic resonance of the
SRR when the EM waves propagate perpendicular to the SRR plane and the incident E is parallel
to the gap-bearing sides of the SRR. This is manifested by a dip in the transmission spectrum. A
simple analytic model is introduced to explain this interesting behavior.
PACS: 41.20.Jb, 42.70.Qs, 73.20.Mf
Metamaterials with a negative index of refraction have
attracted recently great attention due to their fascinat-
ing electromagnetic (EM) properties. It was Veselago
that introduced the term “left-handed substances” in his
seminal work published in 1968 [1]. He suggested that
in a medium for which the permittivity ǫ and perme-
ability µ are simultaneously negative, the phase of the
EM waves would propagate in a direction opposite to
that of the EM energy flow. In this case, the vectors k,
E and H form a left-handed set and therefore Veselago
referred to such materials as “left-handed”. The inter-
est in Veselago’s work was renewed since Pendry et al.
proposed an artificial material consisting of the so-called
split-ring resonators (SRRs) which exhibit a band of neg-
ative µ values in spite of being made of non-magnetic
materials, and wires which provide the negative ǫ be-
havior [2]. Based on Pendry’s suggestion and targeting
the original idea of Veselago, Smith et al. demonstrated
in 2000 the realization of the first left-handed material
(LHM) which consisted of an array of SRRs and wires,
in alternating layers [3]. Since the original microwave
experiment by Smith et al. several composite metamate-
rials (CMMs) were fabricated [4,5] that exhibited a pass
band in which it was assumed that ǫ and µ are both neg-
ative. This assumption was based on transmission mea-
surements of the wires alone, the SRRs alone, and the
CMMs. The occurrence of a CMM transmission peak
within the stop bands of the SRR and wire structures
was taken as evidence for the appearance of LH behav-
ior. Further support to this interpretation was provided
by the demonstration that such CMMs exhibit negative
refraction of EM waves [6,7]. Moreover, there is a sig-
nificant amount of numerical work [8–11] in which the
transmission and reflection data are calculated for a fi-
nite length of metamaterial. A retrieval procedure can
then be applied to obtain the effective metamaterial pa-
rameters ǫ and µ, under the assumption that it can be
treated as homogeneous. This procedure was applied in
Ref. 12 and confirmed that a medium composed of SRRs
and wires could indeed be characterized by effective ǫ
and µ whose real parts were both negative over a finite
frequency band, as was the real part of the refractive
index n. However, it was recently shown [13,14] that
the SRRs exhibit resonant electric response in addition
to their resonant magnetic response. As a result of this
electric response and its interaction with the electric re-
sponse of the wires, the effective plasma frequency, ω′
is much lower than the plasma frequency of the wires,
ωp. An easy to apply criterion was proposed [13] to iden-
tify if an experimental transmission peak is left-handed
(LH) or right-handed (RH): If the closing of the gaps of
the SRRs in a given LH structure removes only a single
peak from the T data (in the low frequency regime), this
is strong evidence that the T peak is indeed LH. This
criterion is valuable in experimental studies, where one
cannot easily obtain the effective ǫ and µ. It was applied
experimentally and it was found that some T peaks that
were thought to be LH turn out to be RH [14]. It seems
that a careful study of the SRR behavior, both electric
and magnetic, is necessary for the design and realization
of LH structures. Marqu´ es et al. considered bianisotropy
in SRR structures and developed an analytical model to
evaluate the magnitude of cross-polarization effects [15].
In the present paper, we report numerical and experi-
mental results for the transmission coefficient of a lattice
of SRRs alone for different orientations of the SRR with
respect to the external electric field, E, and the direction
of propagation. Incidence is always normal to some face
of the orthorhombic unit cell of this metamaterial, which
implies six distinct orientations (Fig. 1). It was consid-
p,
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ered an obvious fact that an incident EM wave excites the
magnetic resonance of the SRR only through its magnetic
field; hence one could conclude that this magnetic reso-
nance appears only if the external magnetic field H is
perpendicular to the SRR plane, which in turn implies a
direction of propagation parallel to the SRR (Figs. 1(a),
1(b)). If H is parallel to the SRR (Figs. 1(c), 1(d)) no
coupling to the magnetic resonance was expected. We
show in this paper that this is not always the case. If
the direction of propagation is perpendicular to the SRR
plane and the incident E is parallel to the gap-bearing
sides of the SRR (Fig. 1(d)), an electric coupling of the
incident EM wave to the magnetic resonance of the SRR
occurs. This means that the electric field excites the res-
onant oscillation of the circular current inside the SRR,
influencing either the behavior of ǫ(ω) only (as in Fig.
1(d)) or ǫ(ω) and µ(ω) (as in Fig. 1(b)). Experiments
as well as numerical results based on the transfer ma-
trix (TMM), and on the finite difference time domain
(FDTD) method reveal that for propagation perpendic-
ular to the SRR plane a dip in the transmission spectrum
close to the magnetic resonance ωmof the SRR appears
whenever the mirror symmetry of the SRR with respect
to the direction of the electric field is broken by the gaps
of its rings (Fig. 1(d)). As we point out below the possi-
bility of such electric coupling to the magnetic resonance
does also affect the conventional orientations (Figs. 1(a),
1(b)), that have the direction of propagation along the
SRR plane. A simple analytic model is given that pro-
vides an explanation for the phenomenon.
For the experimental study, a CMM consisting of SRRs
was fabricated using a conventional printed circuit board
process with 30µm thick copper patterns on one side of a
1.6mm thick FR-4 dielectric substrate. The FR-4 board
has a dielectric constant of 4.8 and a dissipation factor
of 0.017 at 1.5GHz. The design and dimensions of the
SRR, which are the same as those of Ref. 5, are described
in Fig. 1. The CMM was then constructed by stacking
together the SRR structures in a periodic arrangement.
The unit cell contains one SRR and has the dimensions
5mm (parallel to the cut sides), 3.63mm (parallel to the
continuous sides), and 5.6mm (perpendicular to the SRR
plane). The transmission measurements were performed
in free space on a CMM block consisting of 25×25×25
unit cells, using a Hewlett-Packard 8722 ES network an-
alyzer and microwave standard-gain horn antennas.
Additionally, numerical simulations using TMM and
FDTD method where performed to understand the cou-
plings to the SRR. Both methods use a discretized model
of the SRR, similar to the one shown in the inset of
Fig. 3, and periodic boundary conditions perpendicular
to the direction of propagation. The TMM directly com-
putes the complex transmission and reflection amplitudes
and thus allows us to obtain the effective medium ǫ(ω)
and µ(ω) via a retrieval procedure [12]. In addition, the
FDTD allows to visualize the spatial distribution of the
fields and currents inside the system, as a function of
time.
We considered the four non-trivial orientations of the
SRR, which are shown in Fig. 1. Figure 2 presents the
measured transmission spectra, T, of the CMM. The con-
tinuous line (line a) corresponds to the conventional case
shown in Fig. 1(a), with H perpendicular to the SRR
plane and E parallel to the symmetry axis of the SRR.
Notice that T exhibits a stop band at 8.5-10.0GHz, due
to the magnetic resonance. The dashed line (line b) shows
T for the orientation of Fig. 1(b); here E is no longer par-
allel to the symmetry axis of the SRR and thus there is
no longer a mirror symmetry of the combined system of
SRR plus EM field. Notice that now T exhibits a much
wider stop band (at 8-10.5GHz), starting at lower fre-
quency. Very interesting results are obtained by compar-
ing T for the two cases shown in Figs. 1(c) and 1(d), for
which there is no coupling to the magnetic field since H
is parallel to the SRR plane. For the case 1(c), where
E is parallel to the symmetry axis, no structure is ob-
served around the magnetic resonance frequency (line c
in Fig. 2), as expected. However in case 1(d), where the
SRR plus EM field exhibit no mirror symmetry, a strong
stop band in T around ωm is observed (line d), similar
to that of the conventional case 1(a). This strongly sug-
gests that the magnetic resonance can be excited by the
electric field provided that there is no mirror symmetry.
These observations are in good agreement with the nu-
merical results, presented in Fig. 3. For the propagation
perpendicular to the SRR we observe a stop band only
if E is parallel to the cut-bearing sides of the SRR and
“sees” its asymmetry (line d); otherwise we have trans-
parency (line c). At low frequencies, the SRR can basi-
cally be represented only by its outer ring. As is shown
in Fig. 4, the SRR ring will experience different spatial
distributions of the induced polarization, depending on
the relative orientation of E with the SRR gap. If E is
parallel to the no gap sides of the SRR its polarization
will be symmetric and the polarization current is only
flowing up and down the sides of the SRR, as shown in
Fig. 4(a). If the SRR is turned by 90 degree, shown in
Fig. 4(b), the broken symmetry leads to a different con-
figuration of surface charges on both sides of the SRR,
connected with a compensating current flowing between
the sides. This current contributes to the circulating cur-
rent inside the SRR and hence couples to the magnetic
resonance. We directly observed both types of currents
in the FDTD simulations; as an example, the component
of the polarization current parallel to the external elec-
tric field is shown in Fig. 4. The retrieval procedure for ǫ
and µ indicates that the electric coupling leads to a reso-
nant electric response in ǫ near ωm. Also the experimen-
tally observed broadening of the conventional SRR dip
for the turned SRR was found numerically as well (line
b in Fig. 3). The reason is the additional electric cou-
pling which adds an electric resonant response (in ǫ(ω))
directly below the resonant magnetic response. Closing
the gaps of the SRR [13] we observed both in the exper-
iment and in the simulations that the dips disappeared.
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In summary, we have studied experimentally and theo-
retically the propagation of EM waves for different orien-
tations of the SRR. It is found that the incident electric
field couples to the magnetic resonance of the SRR, pro-
vided its direction is such as to break the mirror symme-
try. This unexpected electric coupling to the magnetic
resonance of the SRR is of fundamental importance in
understanding the refraction properties of SRRs in the
low frequency region of the EM spectrum. Also this new
finding is very important for the design of LHMs in higher
dimensions.
Acknowledgments. Financial support of EU IST project
DALHM, NSF (US-Greece Collaboration), and DARPA
are acknowledged. This work was partially supported by
Ames Laboratory.
† Permanent address
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l
l
w
t
d
l = 3mm
w = d = t = 0.33mm
(a)
k
E
H
(b)
k
E
H
(c)
k
E
H
(d)
k
E
H
FIG. 1. Left: The SRR geometry studied. Right: The four
studied orientations of the SRR with respect to the triad k, E,
H of the incident EM field. The two additional orientations,
where the SRR are on the H-k plane, produce no electric or
magnetic response.
ω [GHz]
T [dB]
14
13
12 11
10
9
8
7
0
-20
-40
-60
a
b
c
d
FIG. 2. Measured transmission spectra of a lattice of SRRs
for the four different orientations shown in Fig. 1.
ωa/c
T [dB]
0.09
0.0850.08 0.0750.070.065 0.06
0
-30
-60
-90
-120
a
b
c
d
FIG. 3. Calculated transmission spectra of a lattice of
SRRs for the four different orientations shown in Fig. 1. The
curve c practically coincides with the axis. The discretization
of one particular SRR is shown in the inset.
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+δQ
+δQ
−δQ
−δQ
a)
+δQ′
+δQ
−δQ′
−δQ
−δQ′
+δQ′
Jeff
δQ′≪ δQ
b)
FIG. 4. A simple drawing for the polarization in two dif-
ferent orientations of a single ring SRR. The external electric
field points upward. Only in case of broken symmetry (b) a
circular current will appear which excites the magnetic reso-
nance of the SRR. The interior of the ring shows FDTD data
for the polarization current component J?Eat a fixed time for
normal incidence (Figs. 1(c),(d)) as a gray scale plot.
4