IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 12, NO. 6, NOVEMBER/DECEMBER 2006 1097
Photonic Metamaterials: Magnetism at
Optical Frequencies
Stefan Linden, Christian Enkrich, Gunnar Dolling, Matthias W. Klein, Jiangfeng Zhou, Thomas Koschny,
Costas M. Soukoulis, Sven Burger, Frank Schmidt, and Martin Wegener
(Invited Paper)
Abstract—Photonic metamaterials are man-made materials
with “lattice constants” smaller than the wavelength of light. Tailor-
ing the properties of their functional building blocks (atoms) allows
one to go beyond the possibilities of usual materials. For example,
magnetic dipole moments at optical frequencies (µ =1)become
possible. This aspect substantially enriches the possibilities of op-
tics and photonics and forms the basis for the so-called negative-
index metamaterials. Here, we describe the underlying physics and
review the recent progress in this rapidly emerging field.
Index Terms—Metamaterial, negative permeability, split-ring
resonator (SRR).
I. INTRODUCTION
I
N a usual crystal, the atoms are arranged in a periodic fashion
with lattice constants of the order of half a nanometer. This
is orders of magnitude smaller than the wavelength of light.
For example, green light has a wavelength of about 500 nm.
Thus, for a given direction of propagation, the light field expe-
riences an effective homogeneous medium, that is, it does not
“see” the underlying periodicity but only the basic symmetries
Manuscript received December 8, 2005; revised June 12, 2006. This work
was supported by the Deutsche Forschungsgemeinschaft (DFG) and the State
of Baden-W
¨
urttemberg through the DFG-Center for Functional Nanostruc-
tures under Subproject A1.4 and Subproject A1.5. The work of M. Wegener
was supported by Project DFG-We 1497/9-1. The work of C. M. Soukoulis
was supported by the Alexander von Humboldt Senior-Scientist Award 2002,
by the Ames Laboratory under Contract W-7405-Eng-82, by EU FET Project
DALHM, and by DARPA under Contract HR0011-05-C-0068.
S. Linden is with the Institut f
¨
ur Nanotechnologie, Forschungszentrum
Karlsruhe in der Helmholtz-Gemeinschaft, D-76021 Karlsruhe, Germany.
C. Enkrich, G. Dolling, and M. W. Klein are with the Institut f
¨
ur Ange-
wandte Physik, Universit
¨
at Karlsruhe (TH), D-76131 Karlsruhe, Germany
(e-mail: christian.enkrich@physik.uni-karlsruhe.de).
J. Zhou is with the Department of Electrical and Computer Engineering and
the Microelectronics Research Center, Iowa State University, Ames, IA 50011
USA.
T. Koschny is with the Ames Laboratory and the Department of Physics
and Astronomy, Iowa State University, Ames, IA 50011 USA, and also with
the Institute of Electronic Structure and Laser, Foundation for Research and
Technology Hellas, Unversity of Crete, 71110 Heraklion, Crete, Greece.
C. M. Soukoulis is with the Department of Materials Science and Technology,
University of Crete, 71110 Heraklion, Crete, Greece, and also with the Institute
of Electronic Structure and Laser, Foundation for Research and Technology
Hellas, 71110 Heraklion, Crete, Greece, and also with the Ames Laboratory
and the Department of Physics and Astronomy, Iowa State University, Ames,
IA 50011 USA.
S. Burger and F. Schmidt are with the Zuse Institute Berlin, D-14195 Berlin,
Germany, and also with the DFG Forschungszentrum Matheon, D-10623 Berlin,
Germany.
M. Wegener is with the Institut f
¨
ur Nanotechnologie, Forschungszentrum
Karlsruhe in der Helmholtz-Gemeinschaft, D-76021 Karlsruhe, Germany, and
also with the Institut f
¨
ur Angewandte Physik, Universit
¨
at Karlsruhe (TH),
D-76131 Karlsruhe, Germany.
Digital Object Identifier 10.1109/JSTQE.2006.880600
of the crystal. In such materials, the phase velocity of light c
may depend on the propagation direction and is generally dif-
ferent from the vacuum speed of light c
0
by a factor called the
refractive index n = c
0
/c (the slowness). The physical origin
are microscopic electric dipoles that are excited by the electric
field component of the incoming light and that radiate with a
certain retardation. Hence, the electric permittivity is different
from unity, i.e., =1. In contrast to this, magnetic dipoles play
no role at optical frequencies in natural substances, i.e., the
magnetic permeability is unity, µ =1.
Electromagnetic metamaterials are artificial structures with
inter-“atomic” distances (or “lattice constants”) that are still
smaller than the wavelength of light. Similarly, the light field
“sees” an effective homogeneous material for any given propa-
gation direction (quite unlike in a photonic crystal). The building
blocks (atoms), however, are not real atoms but are rather made
of many actual atoms, often metallic ones. It is this design aspect
that allows us to tailor the electromagnetic material properties,
in particular the corresponding dispersion relation, to a previ-
ously unprecedented degree. For example, it becomes possible
to achieve magnetic dipole moments at optical frequencies, i.e.,
magnetism at optical frequencies (µ =1). It turns out that, for
<0 and µ<0, the refractive index becomes negative with
n = −
√
µ < 0 (rather than n =+
√
µ > 0). This aspect was
pointed out by Veselago many years ago [1], but remained an
obscurity until rather recently.
In this paper, we first describe the physics of “magnetic
atoms” [e.g., the so-called split-ring resonators (SRRs)], which
can be best viewed as the magnetic counterpart of the famous
Lorentz oscillator model for electric dipoles in optical materi-
als. By simple size scaling, these concepts have recently been
brought toward the optical regime. We also discuss the limits
of size scaling. Alternative “magnetic atom” designs can push
the limits somewhat further and can also ease nanofabrication
of these metamaterials.
II. P
HYSICS OF SRRSAS“MAGNETIC ATOMS”
It is well known from basic magnetostatics that a magnetic
dipole moment can be realized by the circulating ring current
of a microscopic coil, which leads to an individual magnetic
moment given by the product of the current and the area of
the coil. This dipole moment vector is directed perpendicular
to the plane of the coil. If such a coil is combined with a plate
capacitor, one expects an increased current at a finite-frequency
resonance, hence, an increased magnetic dipole moment. Thus,
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1098 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 12, NO. 6, NOVEMBER/DECEMBER 2006
Fig. 1. Illustration of the analogy. (a) A usual LC circuit. (b) SRR. (c) Electron
micrograph showing an actually fabricated structure, a gold SRR (t =20nm)
on a glass substrate. Taken from [6].
a popular design for the magnetic “atoms” is to mimic a usual
LC circuit, consisting of a plate capacitor with capacitance C
and a magnetic coil with inductance L, on a s cale much smaller
than the relevant wavelength of light.
Fig. 1 shows the analogy of a conventional LC circuit and a
metallic SRR on a dielectric surface. The right-hand side (RHS)
shows an electron micrograph of a single gold SRR fabricated
by standard electron-beam lithography. The name “split-ring
resonator” goes back to the work of Hardy and Whitehead in
1981 [2] and that of Pendry et al. in 1999 [3]. This name shall
be employed later. However, the SRR has also previously been
discussed under the names “slotted-tube resonator” in 1977 [4]
in the context of nuclear magnetic resonance (NMR) and “loop-
gap resonator” in 1996 [5].
A. LC-Resonance Frequency
The position of the anticipated LC-resonance frequency can
be estimated by the following crude approach: suppose we can
describe the capacitance by the usual textbook formula for a
large capacitor with nearby plates (C ∝plate area/distance) and
the inductance by the formula for a “long” coil with N wind-
ings for N =1(L ∝ coil area/length). Using the nomenclature
of Fig. 1(b), i.e., the width of the metal w, the gap size of the ca-
pacitor d, the metal thickness t, and the width of the coil l, we get
C =
0
C
wt
d
(1)
with the relative permittivity of the material in between the
plates
C
, and
L = µ
0
l
2
t
. (2)
This leads to the eigenfrequency
ω
LC
=
1
√
LC
=
1
l
c
0
√
C
d
w
∝
1
size
(3)
and to the LC-resonance wavelength
λ
LC
=
2πc
0
ω
LC
=2πl
√
C
w
d
∝ size. (4)
Despite its simplicity and the crudeness of our derivation, this
formula contains a lot of correct physics, as confirmed by the
numerical calculations (see later): first, it tells us that the LC-
resonance wavelength is proportional to the linear dimension
of the coil l, provided that the ratio w/d is fixed. This scaling
is valid as long as the metal actually behaves like a metal,
i.e., as long as the LC-resonance frequency is much smaller
than the metal plasma frequency ω
pl
. We will describe this
fundamental limitation in the following paragraphs. Second,
for relevant parameters (
C
≥ 1 and w ≈ d), the prefactor is
typically of the order of ten, i.e.,
λ
LC
≈ 10 × l. (5)
Thus, it is possible to arrange these SRRs in the form of an
array in the xy plane such that the lattice constant a
xy
is much
smaller than the resonance wavelength, i.e., a
xy
λ
LC
.Forex-
ample, for a telecommunication wavelength of λ
LC
=1.5 µm,
the linear dimension of the coil would need to be of the order
of l = 150 nm, implying minimum feature sizes around 50
nm or smaller. Under these conditions, typical values for the
capacitance and the inductance are C ≈ 1aF and L ≈ 1pH,
respectively. Third, the dielectric environment influences the
resonance via
C
, which is, e.g., modified by the presence of a
dielectric substrate. Fourth, if one closes the gap, i.e., in the limit
d → 0 or C →∞, the resonance wavelength goes to infinity,
or equivalently, the resonance frequency ω
LC
becomes zero.
B. Limits of Size Scaling
What are the limits of size scaling according to (3)? This
question has been addressed in [7]–[9]: for an ideal metal, i.e.,
for an infinite electron density n
e
, hence an infinite metal plasma
frequency, a finite current I flowing through the inductance is
connected with zero electron velocity, hence, with a vanishing
electron kinetic energy. In contrast, for a real metal, i.e., for a
finite electron density, the current is inherently connected with a
finite electron velocity v
e
. Thus, one must not only provide the
usual magnetic energy (1/2)LI
2
to support the current I, but ad-
ditionally the total electron kinetic energy N
e
(m
e
/2)v
2
e
, where
N
e
= n
e
V is the number of electrons in the SRR contributing
to the current. To conveniently incorporate this kinetic energy
term into our electromagnetic formulation, we recast it into the
form of an additional magnetic energy. Using n
e
ev
e
= I/wt
and the volume (= cross section times the length) of the SRR
wire V = wt(4(l − w) − d), we obtain
E
kin
= N
e
m
e
2
v
2
e
=
1
2
L
kin
I
2
. (6)
Here, we have introduced the “kinetic inductance”
L
kin
=
m
e
n
e
e
2
4(l − w) − d
wt
∝
1
size
. (7)
While the usual inductance L is proportional to the SRR
size [2], the kinetic inductance (7) scales inversely with size—
provided that all the SRR dimensions are scaled down simul-
taneously. Thus, the kinetic inductance is totally irrelevant for
macroscopic coils but becomes dominant for microscopic in-
ductances, i.e., when approaching the optical frequencies. The
kinetic inductance adds to the usual inductance, L → L + L
kin
in (3), and we immediately obtain the modified scaling for the
magnetic resonance frequency
ω
LC
∝
1
size
2
+ constant
. (8)
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LINDEN et al.: PHOTONIC METAMATERIALS: MAGNETISM AT OPTICAL FREQUENCIES 1099
Obviously, the magnetic resonance frequency is inversely pro-
portional to the size for a large SRR, whereas it approaches a
constant for a small SRR. To evaluate this constant and for the
sake of simplicity, we consider the limits w l, d 4l and
the capacitance C according to (1) in air. Inserting the metal
plasma frequency ω
pl
=
(n
e
e
2
)/(
0
m
e
), we obtain the max-
imum magnetic resonance frequency
ω
max
LC
=
1
L
kin
C
= ω
pl
d
4l
. (9)
This saturation frequency is further reduced by the dielec-
tric environment and by the skin effect [9], which we have
tacitly neglected in our simple reasoning. Furthermore, we re-
mind that our results are implicitly based on the Drude model
of the metal intraband transitions. For real metals in the optical
regime, the interband transitions also often play a significant
role. Our results are meaningful only if ω
max
LC
is smaller than the
onset frequency of the interband transitions. For example, the
interband transitions in aluminum (gold) occur for wavelengths
below 800 nm (550 nm).
C. Magnetic Permeability
One can even obtain an explicit and a simple expression
for the magnetic permeability µ(ω) from our simple circuit
reasoning. We start by considering an excitation configuration
where the electric field component of light cannot couple to the
SRR (see later), and where the magnetic field is normal to the
SRR plane. Under these conditions, the self-induction voltage
of the inductance L plus the voltage drop over the capacitance C
equals the voltage U
ind
induced by the external magnetic field,
i.e., U
L
+ U
C
= U
ind
or
L
˙
I +
1
C
Idt= U
ind
= −
˙
φ. (10)
Again assuming a homogeneous magnetic field in the coil, we
obtain the external magnetic flux φ = l
2
µ
0
H, with the external
magnetic field H = H
0
e
−iωt
+c.c. Taking the time derivative
of (10) and dividing by L yields
¨
I +
1
LC
I =
1
L
˙
U
ind
=+ω
2
µ
0
l
2
L
H
0
e
−iωt
+c.c. (11)
Upon inserting the obvious ansatz I = I
0
e
−iωt
+c.c.,we
obtain the current I, the individual magnetic dipole moment
l
2
I, and the magnetization M =(N
LC
/V )l
2
I. Here, we have
introduced the number of LC circuits N
LC
per volume V .
Suppose the lattice constant in the SRR plane is a
xy
≥ l,
and a
z
≥ t in the direction normal to the SRRs. This leads
to N
LC
/V =1/a
2
xy
a
z
. Finally, using M = χ
m
(ω)H, µ(ω)=
1+χ
m
(ω), and (2) brings us to
µ(ω)=1+
F ω
2
ω
2
LC
− ω
2
. (12)
Apart from the ∝ ω
2
numerator, this represents a Lorentz os-
cillator resonance. Here, we have lumped the various prefactors
into the dimensionless quantity F with
0 ≤F=
l
2
t
a
2
xy
a
z
≤ 1. (13)
F =1corresponds to the nearest-neighbor SRRs touching
each other—obviously the ultimate upper bound for the ac-
cessible SRR density. Thus, we can interpret F as a filling
fraction. Ohmic losses, radiation losses, and other broadening
mechanisms can be lumped into a damping γ
m
of the magnetic
resonance.
The bottom line is that the SRR is the magnetic analog of the
usual (electric) Lorentz oscillator model. The permeability of
the closed ring, i.e., the special case of d → 0 ⇒ C →∞⇒
ω
LC
→ 0 in (12), reduces to
µ(ω)=constant =1−F≥0. (14)
In other words, the split in the ring is essential for obtaining
µ(ω) < 0. For example, for 30% lateral spacing (a
xy
=1.3 × l)
and for a spacing in the vertical direction equal to the SRR thick-
ness (a
z
=2× t), we obtain F =0.30 and µ =0.70 for closed
rings. Note, however, that we have tacitly neglected the inter-
action among the rings in our considerations leading to this
conclusion [10]. The assumption of noninteracting rings is jus-
tified for F1, but becomes questionable for F→1. What
qualitative modifications are expected from the interaction of
rings? The fringing field of any particular ring at the location
of its in-plane neighbors is opposite to its own magnetic dipole
moment, hence parallel to the external magnetic field of light.
Thus, in-plane interaction tends to effectively increase the value
of F in (14). In contrast, interaction with rings from adjacent
parallel planes tends to suppress F in (14). It is presently un-
clear, whether a particular arrangement of rings could allow for
an increase of F sufficient to obtain µ(ω) < 0 (also see [3]).
Interaction similarly influences the behavior of the split rings.
We note in passing that the description of an isotropic
(meta)material in terms of (ω) and µ(ω) may be valid, but it is
not unique. Indeed, it has already been pointed out in [11] that,
alternatively, one can set ˜µ =1and describe the (meta)material
response in terms of the spatial dispersion, i.e., via a wave-
vector dependence of the electric permittivity ˜(ω, k).Onemust
be aware, however, that the resulting “refractive index” ˜n(ω, k)
looses its usual meaning. A more detailed discussion of this
aspect can be found in [12].
Historically, the first demonstration of the negative-index
metamaterials was in 2001 at a frequency of about 10 GHz or
the wavelength of 3cm[13], a regime in which SRR “magnetic
atoms” can easily be fabricated on electronic circuit boards.
The negative permittivity was achieved by the additional metal
stripes. In 2004 [14], µ(ω) < 0 has been demonstrated at a
frequency of about 1 THz(300 µm wavelength) using standard
microfabrication techniques for the SRR ( [15] reviews this early
work).
III. T
OWARD MAGNETISM AT OPTICAL FREQUENCIES
At this point, our experimental team entered this field—partly
driven by the scepticism that similar materials would not be
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1100 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 12, NO. 6, NOVEMBER/DECEMBER 2006
possible at optical frequencies. In our first set of experiments,
we scaled down the lateral size of the SRR by two more or-
ders of magnitude, leading to the f ollowing parameters: l =
320 nm,w =90nm,t=20nm, and d =70nm. On this ba-
sis, we anticipated a magnetic resonance at about λ
LC
=3µm.
These “magnetic atoms” were arranged on a square lattice with
a
xy
= 450 nm ≈ 7 × λ
LC
(and larger ones) and a total sample
area of 25 µm
2
. For normal incidence conditions, however, the
light has zero magnetic field component perpendicular to the
SRR plane. Thus, excitation via the magnetic field is not pos-
sible. Alternatively, the magnetic resonance can also be excited
via the electric-field component of light if it has a component
normal to the plates of the capacitor, i.e., if the incident light
polarization is horizontal. In contrast, for normal incidence and
vertical incident polarization, neither the electric nor the mag-
netic field can couple to the SRR. This “selection rule” can be
used to unambiguously identify the magnetic resonance.
A. SRR at Infrared Wavelengths
Corresponding measured transmittance and reflectance spec-
tra are shown in Fig. 2. Independent of the lattice constant
a
xy
, two distinct resonances are clearly visible. With increas-
ing a
xy
, these resonances narrow to some extent because of the
reduced interaction between the SRRs, but their spectral posi-
tion remains essentially unchanged as expected for t he electric
and magnetic resonant responses of SRRs. This also clearly
shows that Bragg diffraction plays no major role. The long-
wavelength resonance around 3-µm wavelength is present for
the horizontal incident polarization and absent for the vertical
polarization—as expected from the earlier reasoning. Further-
more, as expected from the previous section, this resonance dis-
appears for the closed rings [Fig. 2(g) and (h)], i.e., for d → 0,
hence ω
LC
→ 0. The additional short-wavelength resonance be-
tween 1- and 2-µm wavelength is due to the particle plasmon or
Mie resonance, mainly exhibiting an electric permittivity, which
follows a Lorentz oscillator form according to
(ω)=1+
Fω
2
pl
ω
2
Mie
− ω
2
− iγω
(15)
with the metal Drude model damping γ. The constant F depends
on the SRR volume filling fraction. We will come back to the
Mie resonance in more detail later.
All features of the measured spectra (Fig. 2) are reproduced by
numerical calculations using a three-dimensional (3-D) finite-
difference time-domain approach [6] (not shown here). The cor-
responding calculated field distributions [6] (not shown here)
support the simplistic reasoning on SRRs in the previous sec-
tion. Retrieving [16] the effective permittivity (ω) and magnetic
permeability µ(ω) from the calculated spectra, indeed, reveals
µ<0 associated with the λ
LC
=3µm resonance for appropri-
ate polarization conditions [6].
B. SRR at Near-Infrared Wavelengths
Two questions immediately arise: 1) can the magnetic
resonance frequency be further increased by the miniaturization
of the SRRs and 2) can one also experimentally demonstrate
Fig. 2. Measured transmittance and reflectance spectra (normal incidence).
In each row of this “matrix,” an electron micrograph of the corresponding
sample is shown on the RHS. The two polarization configurations are shown
on the top of the two columns. (a) and (b) Lattice constant a = 450 nm.(c)
and (d) a = 600 nm. (e) and (f) a = 900 nm correspond to the nominally
identical SRRs. (g) and (h) a = 600 nm correspond to the closed rings. The
combination of these spectra unambiguously shows that the resonance at about
3-µm wavelength (gray areas)istheLC resonance of the individual SRRs.
Taken from [6].
coupling to the magnetic (or LC) resonance via the magnetic
field component of light at optical frequencies? Both aspects
have been addressed in our earlier work [17]. Electron micro-
graphs of miniaturized structures are shown in Fig. 3(a). 1)
The corresponding measured spectra for horizontal incident
polarization in Fig. 3(b) reveal the same (but blue-shifted)
resonances as in Fig. 2(a). For vertical incident polarization,
compare Fig. 3(c) and Fig. 2(b). 2) In Fig. 4(a), the electric
component of the incident light cannot couple to the LC circuit
resonance for any angle [in Fig. 4(b) it can]. With increasing
angle, however, the magnetic field acquires a component normal
to the SRR plane. This component can induce a circulating
electric current in the SRR coil via the induction law. This
current again leads to the magnetic dipole moment normal to
the SRR plane, which can counteract the external magnetic
field. The magnitude of this resonance (highlighted by the
gray area around 1.5-µm wavelength) is indeed consistent
with theory [17] (not depicted here), and leads to an effective
negative magnetic permeability for propagation in the SRR
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LINDEN et al.: PHOTONIC METAMATERIALS: MAGNETISM AT OPTICAL FREQUENCIES 1101
Fig. 3. (a) Electron micrograph of a split-ring array with a total area of
100 µm
2
. Lower RHS inset shows the dimensions of an individual split ring.
(b) and (c) Corresponding measured normal-incidence transmittance and re-
flectance spectra for the horizontal and vertical polarization, respectively. Taken
from [17].
plane and for a stack of SRR layers rather than just one layer
as considered here. This aspect has been verified explicitly by
retrieving the effective permittivity and permeability from the
calculated transmittance and reflectance spectra [16], [18].
An unexpected feature of the spectra in Fig. 4(a) is that the
950-nm wavelength Mie resonance at normal incidence splits
into two resonances for oblique incidence. This aspect is repro-
duced by numerical calculations [17], [19]. Intuitively, it can be
understood as follows: for normal incidence and vertical polar-
ization, the two similarly shaped vertical SRR arms contribute.
These arms are coupled via the SRR’s bottom arm (and via
the radiation field). As usual, the coupling of the two degen-
erate modes leads to an avoided crossing with two new effec-
Fig. 4. Measured transmittance spectra taken for oblique incidence for the
configurations shown as insets (where α =60
◦
). (a) Coupling to the funda-
mental magnetic mode at 1.5-µm wavelength is possible only via the magnetic
field component of the incident light. (b) Both electric and magnetic fields can
couple. Note the small but significant feature in (a) for 60
◦
around 1.5-µm
wavelength. The lower gray area in (a) is the transmittance into the linear po-
larization orthogonal to the incident one for α =60
◦
. This can be viewed as a
fingerprint of the magnetic resonances under these conditions. Taken from [17].
tive oscillation modes, a symmetric and an antisymmetric one,
which are frequency down-shifted and up-shifted as compared
to the uncoupled resonances, respectively. The antisymmetric
mode cannot be excited at all for normal incidence as it has
zero effective electric dipole moment. The red-shifted symmet-
ric mode can be excited. It even has a larger effective electric
dipole moment than a single arm. Indeed, the Mie resonance for
the vertical polarization i s deeper and spectrally broader than
for the horizontal polarization in Fig. 2, and red-shifted with
respect to it. For finite angles of incidence, the phase fronts of
the electric field are tilted with respect to the SRR plane. Thus,
the vertical SRR arms are excited with a small but finite time
delay, equivalent to a finite phase shift. This shift allows cou-
pling to the antisymmetric mode of the coupled system of the
two vertical arms as well. In one half cycle of light, one gets a
positive charge at the lower left-hand side (LHS) corner of the
SRR and a negative charge at the lower RHS corner, resulting
in a compensating current in the horizontal bottom arm. Char-
acteristic snapshots of the current distributions in the SRR have
been shown schematically in [17].
According to this reasoning for oblique incidence (e.g., 60
◦
),
we expect a circulating current component for wavelengths near
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![Fig. 5. Measured spectra of transmittance (solid lines) and reflectance (dashed lines) for cut-wire pairs with vertical incident polarization (LHS column) and horizontal polarization (RHS column). Parameters varied: (a) l = 500nm; (b) l = 400nm; and (c) l = 300nm. Fixed parameters for (a)–(c): w = 150nm, t = 20nm, d = 80nm, ax = 500nm, and ay = 1050nm. The dashed-dotted curves in (a) are spectra from a nominally identical structure, but without the upmost gold wire. (d) as (b), but d = 60nm rather than 80 nm. The insets in (a)–(d) show corresponding electron micrographs (top view). Taken from [27].](/figures/fig-5-measured-spectra-of-transmittance-solid-lines-and-2tju3buk.webp)
![Fig. 4. Measured transmittance spectra taken for oblique incidence for the configurations shown as insets (where α = 60◦). (a) Coupling to the fundamental magnetic mode at 1.5-µm wavelength is possible only via the magnetic field component of the incident light. (b) Both electric and magnetic fields can couple. Note the small but significant feature in (a) for 60◦ around 1.5-µm wavelength. The lower gray area in (a) is the transmittance into the linear polarization orthogonal to the incident one for α = 60◦. This can be viewed as a fingerprint of the magnetic resonances under these conditions. Taken from [17].](/figures/fig-4-measured-transmittance-spectra-taken-for-oblique-6k6n94eb.webp)
![Fig. 3. (a) Electron micrograph of a split-ring array with a total area of 100µm2. Lower RHS inset shows the dimensions of an individual split ring. (b) and (c) Corresponding measured normal-incidence transmittance and reflectance spectra for the horizontal and vertical polarization, respectively. Taken from [17].](/figures/fig-3-a-electron-micrograph-of-a-split-ring-array-with-a-lljvn9ev.webp)
![Fig. 1. Illustration of the analogy. (a) A usualLC circuit. (b) SRR. (c) Electron micrograph showing an actually fabricated structure, a gold SRR (t = 20nm) on a glass substrate. Taken from [6].](/figures/fig-1-illustration-of-the-analogy-a-a-usuallc-circuit-b-srr-3bz9n7gs.webp)
![Fig. 2. Measured transmittance and reflectance spectra (normal incidence). In each row of this “matrix,” an electron micrograph of the corresponding sample is shown on the RHS. The two polarization configurations are shown on the top of the two columns. (a) and (b) Lattice constant a = 450nm. (c) and (d) a = 600nm. (e) and (f) a = 900nm correspond to the nominally identical SRRs. (g) and (h) a = 600nm correspond to the closed rings. The combination of these spectra unambiguously shows that the resonance at about 3-µm wavelength (gray areas) is the LC resonance of the individual SRRs. Taken from [6].](/figures/fig-2-measured-transmittance-and-reflectance-spectra-normal-go83rpy4.webp)