Doubly negative metamaterials in the near
infrared and visible regimes based on thin film
nanocomposites
Vitaliy Lomakin†, Yeshaiahu Fainman†, Yaroslav Urzhumov‡, and Gennady Shvets‡
†
Department of Electrical and Computer Engineering, University of California, San Diego
‡
Department of Physics, University of Texas at Austin
[email protected]
Abstract: An optical metamaterial characterized simultaneously by
negative permittivity and permeability, viz. doubly negative metamaterial
(DNM), that comprises deeply subwavelength unit cells is introduced. The
DNM can operate in the near infrared and visible spectra and can be
manufactured using standard nanofabrication methods with compatible
materials. The DNM’s unit cell comprise a continuous optically thin metal
film sandwiched between two identical optically thin metal strips separated
by a small distance form the film. The incorporation of the middle thin
metal film avoids limitations of metamaterials comprised of arrays of paired
wires/strips/patches to operate for large wavelength / unit cell ratios. A
cavity model, which is a modification of the conventional patch antenna
cavity model, is developed to elucidate the structure’s electromagnetic
properties. A novel procedure for extracting the effective permittivity and
permeability is developed for an arbitrary incident angle and those
parameters were shown to be nearly angle-independent. Extensions of the
presented two dimensional structure to three dimensions by using square
patches are straightforward and will enable more isotropic DNMs.
©2006 Optical Society of America
OCIS codes: (160.4670) Optical Materials; (310.6860) Thin Films, Optical Properties
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1. Introduction
Metamaterials are artificial composite materials that possess electromagnetic properties that
are not found in natural environments. Doubly negative metamaterials (DNM) are
metamaterials that are characterized by permeability, permittivity, and index of refraction
simultaneously having negative real parts [1]. Due to their unique electromagnetic properties,
DNMs have a number of important potential applications including the construction of perfect
lenses, transmission lines, and antennas [2-7].
First practical realizations of DNM were introduced in the microwave and then terahertz
regimes [8-12]. For example, microwave and terahertz DNMs were constructed from periodic
unit cells comprising split ring resonators and straight wires [10]. In these DNMs the split ring
resonators and wires support strong magnetic and electric resonance that result in frequency
bands of negative permittivity and permeability that can be tuned to overlap. Extending the
operational spectrum of DNMs to optical frequencies is an important ongoing task among
physical and engineering communities.
However, realizations of DNMs in the optical and, especially, near-infrared (IR) and
visible spectra are challenging. For instance, it has been demonstrated that scaling of the split
ring resonator based metamaterials to the visible regime fails for realistic metals due to the
saturation of the magnetic resonance frequency and increased loss[13]. Recently, several
structures have been suggested to operate as DNMs in the optical regime [12, 14-16]. These
optical DNMs can be classified into two types. One type incorporates arrays of plasmonic
rods [17] or spheres [18] of subwavelength size to construct two- and three-dimensional
DNMs. The operation of these structures is based on the existence of quasi-static resonances
supported by subwavelength particles when the frequency of operation approaches the plasma
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frequency of the particles in the ambient environment. Unfortunately, due to this property,
such DNMs will not operate in spectral ranges extended to near-IR. Moreover, these designs
may lead to excessively high losses for realistic materials and cannot be easily realized using
standard nanofabrication techniques. The second type of optical DNMs represents several
variations of pairs of patterned thin metal films, including arrays of paired wires, paired strips,
staples, and paired perforated plates [11, 12, 14-16]. The operation of these structures is based
on the existence of plasmonic resonances of magnetic and electric type supported by cavities
formed between the pairs of particles. These structures allow a greater flexibility in tuning
their electromagnetic properties and they can be manufactured using standard nanofabrication
techniques. However, none of these structures were shown to operate in a wide spectral range
from near-IR to visible. Moreover, all these structures comprise unit cells that are only
marginally subwavelength (with the wavelength / unit cell size ratio being around 2.5 in
vacuum) when the frequency of operation is in the near-IR or visible parts of the spectrum.
For instance, it was shown that arrays of paired thin metal strips (and, hence, arrays of paired
wires and patches) cannot support overlapping frequency bands of magnetic and electric
resonances when the wavelength / unit cell size ratio is large [11]. This restriction represents a
major limitation of the use of these structures as true DNMs. Indeed, it is well known that a
structure can be regarded as a homogeneous DNM only when its unit cell is much smaller
than the wavelength of operation scaled to the effective index of refraction. Otherwise, the
diffraction phenomena will dominate the DNM behavior.
In this manuscript we introduce a novel DNM structure that modifies the structures based
on arrays of paired thin metal strips or rods [11, 14] by adding a thin metal film in the middle
plane between the strips. We show that this simple modification entirely avoids the limitations
of the original paired strip structure (paired wire and paired patch structures) [11]. The
introduced structure provides the following unique properties and features: (i) tunable
operation in a wide optical spectral range from near-IR to visible, (ii) true metamaterial design
consisting of a periodic unit cells of size much smaller than the effective wavelength of
operation, and (iii) design compatible with standard nanofabrication techniques. The
introduced DNM structure is modeled analytically and numerically to elucidate the physics
behind its operation as well as to provide simple means to tune its effective permittivity,
permeability, and index of refraction.
2. Layout of the unit cell
Consider an inhomogeneous nanocomposite DNM consisting of a periodic array of unit cells
as shown in Fig. 1. The unit cells are arranged periodically in the x and z directions with
periods Lx and Lz , respectively. The structure comprises a finite number of ml layers in the
z direction and an infinite number of unit cells in the x direction. Every layer comprises an
infinite metal film of thickness d f and an infinite array of metal strips of width w and
thickness d s (Fig. 1). In the z dimension, the strips are arranged in pairs symmetrically with
respect to the unit cell’s symmetry plane ( z = 0 ). The distance between the bottom face of the
top strip and the top face of the bottom strip is 2h . The structure is uniform in the y
dimension. The strips and the film are assumed to be made of an identical metal characterized
by a relative permittivity ε m with Re{ε m } < 0 in the optical frequency regime (e.g. silver or
gold). It is assumed that d s , d f , h , Lx and Lz are much smaller than the (free-space)
wavelength of illumination λ to assure that the unit cells’ geometry is much smaller than the
effective wavelength of the incident radiation. In addition, d s , d f are assumed to be smaller
than w such that possible charge and current distribution variations in the strips and film
occur primarily in the horizontal ( x ) dimension. The whole metallic structure is embedded
into a dielectric material with permittivity ε d of a total thickness H = ml Lz . A harmonic time
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dependence e j 2π f t of the optical field is assumed and suppressed in what follows. Here,
f = c λ is the frequency of illumination and c is the speed of light in vacuum.
z
ds
df
h
x
−w 2
ds
−h
w2
Lz
Lx
Fig. 1. Structure’s unit cell
As is shown below, the presence of the middle metal film avoids the limitations of the
double-strip (double wire or patch) structure in achieving simultaneously negative effective
permeability and permittivity in deeply subwavelength optical regime [11]. Moreover, it will
be shown that for a TM polarization (magnetic field being along the y axis) and for special
combinations of the structures’ parameters and frequency of illumination, the structure in Fig.
1 is equivalent to a slab made of a DNM characterized simultaneously by negative real parts
of the effective permittivity, permeability, and index of refraction. It is noted that the
introduced array of strips is a 2D counterpart of a 3D structure comprising doubly periodic
arrays of rectangular/square patches and therefore the results presented here are directly
extendable to more general 3D configurations leading to e DNMs with properties nearly
independent of light polarization and plane of incidence.
3. Optical properties of DNM structures
3.1 Cavity model
The structure in Fig. 1 can be viewed as a periodic array of cavities formed in the volumes
between the strips that support resonances, viz. source-free fields. Understanding the behavior
of these resonances is essential for unraveling the structure’s optical properties. To describe
the cavity resonances, the cavity model [19] successfully used in the analysis of patch
antennas can be modified to take into account the penetration of the fields through the thin
metal films. This model is introduced in two steps considering a single unit cell. In the first
step, anticipating that the resonance fields are concentrated primarily between the strips, the
region | x |< w 2 is closed by (virtual) vertical perfect magnetically conducting walls (Fig. 2).
In the second step, the optically thin top and bottom metal strips and the central metal film are
replaced by thin (inductive) admittance sheets characterized by normalized surface
admittances Ysstrip = jk0 (ε m − 1)d s at z = ± h and Ysfilm = jk0 (ε m − 1)d f at z = 0 , respectively
[20]; k0 = 2π λ is the free space wavenumber. Due to its symmetry around z = 0 , the
resulting simplified cavity supports resonances for which magnetic field has either even or
odd parity with respect to the z = 0 plane.
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Z sstrip
z
h
−w 2
w2
x
PMC
Z
film
s
PMC
−h
Z sstrip
Fig. 2. Equivalent modified cavity used to model the cavity in Fig. 1.
3.2 Magnetic resonances
First, consider the resonances with even magnetic field symmetry. For this symmetry, no
current flows in the film admittance sheet ( z = 0 ) and hence this sheet has no effect on the
modal field structure (Fig. 2). In contrast, the currents in the top and bottom strip admittance
sheets ( z = ± h ) are strong and they flow in opposite directions thus resulting in an effective
magnetic dipole response; this resonance type will be referred to as magnetic. The magnetic
field of the magnetic resonances behaves as H = yˆ A( f magn , z ) sin (π q( x − w 2) w) , where q is
an integer counting the number of the field oscillations in the x direction within w ,
A( f magn , z ) is an even function separately defined inside and outside the cavity, and f magn are
the magnetic resonance frequencies satisfying the following dispersion relation obtained by
matching the fields outside and inside the cavity:
Ysstrip + Yz (1 − j cot k z h) = 0 ,
(1)
where k z = (2π f magn c)2 ε d − (π q w)2 and Yz = 2π f magn ε d (ck z ) . It is evident from Eq. 1
that when | Ysstrip | is large, occurring when the product | (ε m − 1)d s | is large, then | cot k z h |
(
)
should be large and as a result magnetic resonance frequencies, f magn ≈ cq 2 w ε d , are
approximately frequencies of simple patch antenna resonances. However, when the product
| (ε m − 1)d s | is finite/small, f magn can be made much smaller than the simple path antenna
resonances, which is the crucial point in achieving a subwavelength DNM operation. To
obtain an approximate expression for f magn in this regime, it is assumed that | k z h | 1 ,
2π f magn ε d c
π q w , and ε m ( f )
model ε m ( f ) ≈ − f
assumptions,
2
p
2
f , where
is given by an approximate lossless Drude
f p is the metal plasma frequency. Based on these
ds h
.
(2)
ε d w2
This expression shows that the structure in Fig. 1 supports magnetic resonances even when
the cavity has a subwavelength size. Moreover, Eq. (2) shows that the resonant frequency no
longer depends solely on the length of the structure (as it is the case with simple antenna
resonances) but rather it is determined by the shape and the material properties [21] of the
composites. The physical reason for this is that the fields inside the cavity are essentially
quasistatic and can be described to the zero order approximation by either electrostatic
potential φi ( E = −∇φi ) or by the stream function ψ i ( E = e y × ∇ψ i ) [17], which are
f magn ≈ f p π q
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proportional to the magnetic field strength A used above. Therefore, properties of such subwavelength structures are essentially scale-invariant.
3.3 Electric resonances
Now, consider the resonances with odd magnetic field symmetry. Here, the currents flow not
only in the strip admittance sheets ( z = ± h ) but also in the infinite thin film sheet ( z = 0 ) and
hence the latter affects the modal field significantly (Fig. 2). The currents in the top and
bottom strip admittance sheets flow in the same direction thus resulting in an effective electric
dipole response; these resonances will be referred to as electric resonance. The magnetic field
is described by H = yˆ B( felect , z ) sin (π q( x − w 2) w) , where B (ωelect , z ) is an odd function
around z = 0 and felect are electric resonance frequencies satisfying
Ysstrip + Yz + Yz
Ysfilm + j 2Yz tan k z h
=0.
2Yz + jYsfilm tan k z h
(3)
(i)
Analytical and numerical investigation of Eq. (3) shows that it has two solutions felec
( i = 1, 2 ) that are frequencies of the electric resonances for each value of the integer q . By
(1)
(2)
(1)
(2)
setting Re{ f elect
} < Re{ f elect
} , it is found that Re{ felectr
} < Re{ f magn } < Re{ f electr
} . Moreover,
(2)
}
with | Ysfilm | 1 , the following inequality Re{ f elect
Re{ f magn } holds. As | Ysfilm | increases,
(2)
(2)
Re{ f elect
} decreases towards Re{ f magn } , and in the limit felect
→ f magn as | Ysfilm |→ ∞ . Such
(2)
allows the electric and magnetic resonances to overlap in the same
behavior of felect
frequency range by simply modifying d f .
3.4 Model for effective parameters
Recalling that the unit cell size of the structure in Fig. 1 is subwavelength, the structure can be
described by its effective permeability and permittivity [22]. Due to the structural geometry,
the effective parameters are expected to be anisotropic and can be characterized by diagonal
permeability and permittivity tensors μeff and εeff . For the TM excitation considered in the
paper, the relevant tensor components affecting the structure’s electromagnetic properties are
μeff , yy , ε eff , xx , and ε eff , zz . As shown below, by judicious choice of parameters, simultaneously
negative permeability and permittivity can be achieved for finite frequency bandwidths.
Interactions of the magnetic and electric resonances with an external field can be described by
the existence of effective magnetic and electric dipole moments, m( f ) = m0 ( f − f m ) yˆ , and
(i )
p (i ) ( f ) = p0(i ) ( f − f elect
) xˆ , respectively, where m0 and p0( i ) are constants determining the
strength of the excitation. Based on this understanding, the effective parameters can be written
as
μeff , yy = μ0,eff , yy −
f p ,magn, yy
f − f magn
, ε eff , ii = ε 0,eff ,ii −
f p(1),elect,ii
(1)
f − felect
−
2)
f p(,elect,
ii
(2)
f − f elect
,
(4)
where ii = xx or zz . The effective parameters in (4) comprise non-resonant components
μ0,eff , yy and ε 0,eff ,ii and resonant components described by the resonance frequencies f magn
(i )
i)
together with constants f p ,magn, yy and f p(,elect,
and felect
ii determined by the resonance
excitation strengths. The resonance excitation strength may depend on the frequency and the
angle and it may be different for ii = xx and zz (see further discussions in Sec. 4.1).
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It is noted that an alternative expression for ε eff ,ii can be derived rigorously based on quasi-
(
static approximation, ε qs ( f ) = ed 1 − F0 / s −
∑ F /(s − s )) , where s( f ) = (1 − ε ( f ) / ε )
i
i
m
i
−1
d
,
and si are the eigenvalues of the differential equation ∇(θ (r )∇φi (r )) = si ∇ φi (r ) . Here φi are
the zero order scalar potential distributions in the nanostructure, and θ (r ) is the Heaviside
function equal to unity when r is inside the metal and zero when r is outside of the metal.
Contributions of different quasi-static resonances are weighted by their strengths Fi that are
2
determined numerically from the functional form of φi (r ) [23].
From the above model, the structure parameters can be chosen such that Re{μeff , yy } ,
Re{ε eff ,ii } are negative. It is important to note that unlike in all previous studies, the effective
parameters here can be tuned nearly independently in the entire range from near-IR to visible.
Indeed, one can first choose w , d s and h to tune the magnetic resonance frequency to a
required frequency of DNM operation. Then the film thickness d f can be chosen so as to
bring the electric resonance frequency close to the magnetic one. This tuning is possible
because the magnetic resonance is (nearly) independent of d f .
(2)
, is
Finally, it is noted that the higher frequency electric resonance, associated with felectr
similar to the electric resonance supported by paired-strip, paired-patch, and paired-wire
(2)
cannot be tuned to be close to f magn (at least for the
structures. Unfortunately, since felectr
same resonance order q ), these structures cannot operate as NIMs. This is the middle film
(1)
and allows for
that provides the existence of the lower frequency tunable resonance with felectr
the NIM operation. It also should be noted that the location of the continuous film in the
middle plane between the strips is critical for proper NIM operation. Displacing the film to a
different location may corrupt the NIM operation significantly as the even and odd resonances
become combined resonances so that the electric and magnetic responses affect each other.
4. Numerical study
4.1 Extraction of the effective parameters
The effective permeability, permittivity, and index of refraction can be obtained by
calculating/measuring the structure’s zeroth order scattering (reflection and transmission)
coefficients and matching appropriate effective parameters. This procedure was presented by
Smith et al [24] to extract effective parameters assuming normally incident plane waves. Here,
this approach is generalized to take into account oblique incident angles and anisotropic
medium effective parameters.
The zeroth order TM reflection and transmission coefficients for a field incident at an
angle θ on a slab of thickness H made of isotropic or uniaxially anisotropic material can be
written as
T
⎛
= ⎜ cos(k0 nz ,eff H ) +
⎜
⎝
−1
⎞
j ⎛ Z z ,eff cos θ ⎞
+
⎜
⎟ sin( k 0 nz ,eff H ) ⎟ ,
⎜
⎟
⎟
2 ⎝ cos θ Z z ,eff ⎠
⎠
j ⎛ Z z ,eff cos θ ⎞
+
R= ⎜
⎟ sin( k0 nz ,eff H )T ,
2 ⎜⎝ cos θ Z z ,eff ⎟⎠
(5)
where nz ,eff is effective index for the field propagating in the z direction, and Z z ,eff is the
corresponding normalized impedance defined as the ratio between tangential components of
the electric and magnetic fields in the x − y plane. From Eq. (5), nz ,eff and Z z ,eff are found as
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⎛
⎛ 1 − (R
⎞
−T 2) ⎞
−1
⎟ + 2π l ⎟⎟ ( k0 H ) ,
2T
⎠
⎠
⎝
⎝
nz ,eff = ⎜⎜ cos −1 ⎜
Z z ,eff
2
(1 + R 2 ) − T 2
= cos θ
.
(1 − R 2 ) − T 2
(6)
where l is an integer that is chosen as described by Smith at al [24]. In contrast, assuming
that the effective medium is anisotropic and recalling that the field is TM polarized, i.e. that
the field in the slab is of extraordinary type, the quantities nz ,eff and Z z ,eff also satisfy
nz ,eff = ( μ eff , yy ε eff , xx − sin 2θ ε eff , zz ) ,
12
(7)
Z z ,eff = nz ,eff ε eff , xx ,
Eq. (7) with the expressions of Eq. (6), can be used to find the other effective parameters.
However, it is evident that there are more equations than unknowns so that an additional
relation between μeff , yy , ε eff ,xx , and ε eff ,zz has to be imposed. Different relations would lead to
different results for the effective parameters (but all of them would result in the same
structure’s scattering properties). In this paper, we choose a semi-empirical assumption that
ε eff ,zz is a positive constant that is chosen further as ε d . This assumption is based on the
observation that the metal films/strips all are arranged along ( x − y ) planes so that the field
components along the z axis are weakly affected by the presented resonances so that the
i)
excitation constants f p(,elect,
zz in Eq. (4) are weak. In addition, this assumption is based on the
fact that ε eff ,zz extracted by assuming static fields is nearly constant in the range of interest.
Based on this assumption the effective parameters μeff , yy and ε eff ,xx are found as
ε eff , xx =
nz ,eff
Z z ,eff
,
μeff , yy = nz ,eff Z z ,eff +
sin 2 θ
ε eff , zz
(8)
,
where nz ,eff and Z z ,eff are given by Eq. (6) and ε eff ,zz = ε d . In the remaining part of the
manuscript we use the following notations: μeff = μeff , yy , ε eff = ε eff , xx , neff = ( μeff ε eff )1 2 .
4.2 Numerical simulations
To demonstrate the operation of the structure in Fig. 1 and validate the presented analytic
model we performed a series of numerical simulations using the full wave rigorous coupled
wave analysis [25, 26] and finite elements method [27]. In all simulations we used SiO 2 as
an embedding dielectric with the dielectric constant value of ε d = 2.25 . The metal was
assumed to be gold with ε m given by the Drude model ε m = 1 − f p2 ( f ( f − jΓ) ) , where f p is
the plasma frequency and Γ is the scattering frequency characterizing the dissipation rate in
f p = 1.32 × 104 (2π ) THz
and
the metal. Following Ref. [16], we used
Γ = 1.2 ×102 (2π ) THz .
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electric
1
df =0
df =6.5 nm
0.8
df =8.5 nm
|T0|
0.6
0.4
0.2
electric
magnetic
0
400
500
600
700
λ, [nm]
800
900
1000
(a)
6
40
df = 0
5
30
df = 6.5 nm
df = 8.5 nm
Re{ε eff}, Im{ε eff}
4
Re{μeff}, Im{μeff}
df = 0
df = 6.5 nm
3
2
1
df = 8.5 nm
20
10
0
0
-10
-1
-2
-20
400
500
600
700
λ, [nm]
(b)
800
900
1000
400
500
600
700
λ, [nm]
800
900
1000
(c)
Fig. 3. Dependence of the structures scattering coefficients and effective properties on the
middle film thickness. The structure’s parameters are chosen as Lx = 100 nm , w = 50 nm ,
d s = 15 nm for three values of the strip-symmetry plane separation and film thickness being
h = 7 nm and d f = 0 ( Lz = 44.5 nm ), h = 10.25 nm and d f = 6.5 nm ( Lz = 50.5 nm ), as well
as h = 11.25 nm and d f = 8.5 nm ( Lz = 52.5 nm ). (a): magnitude of the zeroth order
transmission coefficient | T0 | ; (b): effective permeability μeff ; (c) effective permittivity ε eff .
For d f = 0 , i.e. in the absence of the middle film, only a single magnetic (longer wavelength)
and electric (lower wavelength) resonances are obtained; the resonances manifest themselves
as minima in the transmission coefficient magnitude dependence. For d ≠ 0 , i.e. in the
presence of the middle film, one magnetic are obtained in the wavelength range between two
electric resonances. Associated with the magnetic resonance and longer wavelength electric
resonance are bands of negative Re{μeff } and Re{ε eff } , respectively. The bands are more
separated for d f = 6.5 nm and overlap for d f = 8.5 nm thus resulting in a DNM.
Figure 3(a) shows the magnitudes of the structure’s normal transmission coefficient T0 for
a single layer ( ml = 1 ) in the absence of the middle slab for h = 7 nm , d f = 0 , Lz = 44.5 nm
and in the presence of the middle slab for h = 10.25 nm , d f = 6.5 nm Lz = 50.5 nm as well as
for h = 11.25 nm , d f = 8.5nm , Lz = 52.5 nm . Other structure’s parameters were chosen as
Lx = 100 nm , w = 50 nm , d s = 15 nm . In the absence of the central film, two non-overlapping
electric and magnetic resonances are obtained for λ = 350 nm and 600 nm , respectively. In
the presence of the middle film, for smaller film thickness ( d f = 6.5 nm ), three separate
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13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11172
resonance dips are observed around λ = 435 nm and λ = 640 nm , and λ = 800 nm
corresponding to electric, magnetic, and electric resonances, respectively. As d f increases the
two longer wavelength (magnetic and electric) resonances approach each other and they
almost merge around λ = 680 nm for d f = 8.5nm . The longer wavelength resonance for
d f = 0 and the middle resonance for d f ≠ 0 in Fig. 3(a) correspond to bands of negative
Re{μeff } in Fig. 3(b). The longest wavelength resonances for d f ≠ 0 in Fig. 3(a) correspond
to bands with negative Re{ε eff } in Fig. 3(c). From the obtained results it is evident that in
agreement with Shvets and Urzhumov [11], no simultaneous bands of negative Re{μeff } and
Re{ε eff } are obtained when the middle slab is absent. However, as predicted by the model
and analysis described above, introducing the middle layer of thin metal film at z=0 causes
simultaneously overlapping bands of negative Re{μeff } and Re{ε eff } as required to construct
a DNM. This is because Re{ε eff } tends to be negative between the two electric resonance
frequencies.
To better understand the nature of the resonances in Fig. 3(a) leading to negative Re{μeff }
and Re{ε eff } in Fig. 3(b), we calculated the field distributions assuming static approximation.
Figures 4(a) and 4(b) show the field distribution corresponding to magnetic and electric
resonances within the cavity with the same parameters as those used in Fig. 3(a) for
d f = 6.5 nm and for λ = 640 nm and λ = 800 nm . It is observed that the fields for the middle
and longer wavelength resonances exhibit even and odd symmetries with respect to the z = 0
plane. These symmetries explain the presence of the effective magnetic and electric dipoles
and hence negative Re{μeff } and Re{ε eff } , respectively. From the results in Figs. 3 and 4 it is
evident that the structure in Fig. 1 indeed can operate as a DNM having a deeply
subwavelength unit cell with a wavelength-to-period ratio of about 7 , and that the cavity
model predictions are valid.
50
50
0.02
0.015
0.01
0.01
0.005
0
0
z, [nm]
z, [nm]
0.005
0
0
-0.005
-0.005
-0.01
-0.01
-0.015
-50
-50
-0.02
0
x, nm
50
-50
-50
0
x, [nm]
50
Fig. 4. Electric field distribution corresponding to (a) magnetic resonance and (b) electric
resonances. The field distribution was obtained assuming static approximation and assuming
that SiO 2 ( ε d = 2.25 ) occupies the entire space.
To verify that the phenomena leading to DNM operation are quasi-static in their physical
nature, we have plotted in Fig. 5 the effective permittivity obtained via two methods:
extracting from scattering coefficient as described in Sec. 4.1, and using a quasi-static
expression given after Eq. (4). It is evident that the quasi-static approximation captures the
behavior of ε eff very well. Note that the position of the resonance extracted from fully
electromagnetic simulations is red shifted form its electrostatic value because of the finite
retardation effects proportional to ( L λ ) 2 [28].
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13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11173
ε
30
Re(εqs)
Im(εqs)
Re(εeff)
Im(εeff)
eff 25
20
15
10
5
0
−5
−10
−15
−20
500
600
700
800
900
1000
wavelength, nm
Fig.
5.
Comparison
ε qs (ω ) = 1 − f 0 / s −
between
∑ f /(s − s ) (where
i
i
the
quasi-static
dielectric
s (ω ) = (1 − ε m (ω ) / ε d ) and
−1
permittivity
ε m is the dielectric
i
permittivity of gold) and the extracted from fully can electromagnetic simulations ε eff (ω ) as
described in Sec. 4.1.
Figure 6 shows that the structure in Fig. 1 can be tuned to operate as a DNM in the entire
range from near-IR to visible by depicting Re{neff } and Im{neff } for three sets of structure
parameters (i. e., set 1, 2 and 3), resulting in DNM operation in three wavelength ranges
820 nm < λ < 1040 nm , 550 nm < λ < 670 nm , and 500 nm < λ < 560 nm . Additional
simulations confirm that DNM operation can be obtained in the entire range from 450 nm to
1800 nm with wavelength-to-period ratios of about 7 and with sufficiently low loss. Notice,
that the wavelength-to- period ratio can be further increased, but on the expense of increased
losses.
Losses of the proposed DNMs are illustrated by Fig. 7, where the ratios Re{neff } Im{neff }
are plotted. Three DNM structures are considered: one embedded in a passive dielectric and
two embedded in a dielectric with gain. Geometrical parameters of the DNMs are listed in the
caption, and gain coefficients correspond to Im{ε d } = 0.03 and Im{ε d } = 0.06 . The
considered values of Im{ε d } correspond to gain coefficient of 1500 cm −1 and 3000 cm −1 ,
respectively. Such values of the gain coefficient can be achieved by semiconductor polymers
or laser dyes [29, 30]. These parameters were chosen to demonstrate a possibility to improve
the DNM operation by means of active materials. It is seen that the largest ratio
Re{neff } Im{neff } is obtained for λ = 620 nm in all three cases with larger ratios
corresponding to larger gains. From these results we learn that for passive structures the losses
are reasonably low thus allowing practical applications of the suggested DNM. It is also
evident that the loss can be reduced significantly by incorporating active materials with
modest gain. Evidently, the loss is modest even without active medium, and is further reduced
by modest gain.
Figure 8 depicts the extracted Re{neff } for the structure parameters as those in Fig. 3 with
d f = 8.5nm for different number of layers ml to demonstrate that the structure can operate as
a bulky material. It is evident that while Re{neff } slightly changes as the number of layers
increases, it is reliably negative in the range 600 nm < λ < 680 nm for any ml .
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13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11174
6
Re{neff}
4
Im{neff}
Set 3
Re{neff}, Im{neff}
2
0
-2
-4
Set 2
-6
Set 1
-8
-10
500
600
700
800
900
1000
1100
λ, [nm]
for different sets of parameters for a single DNM
Fig. 6. Effective index of refraction neff
layer.
Set 1: Lx = 150 nm , Lz = 52 nm , w = 90 nm , d s = 15 nm , d f = 8 nm , h = 11nm ;
Set 2: Lx = 100 nm , Lz = 52.5 nm , w = 50 nm , d s = 15 nm , d f = 8.5 nm , h = 11.25 nm ; and
Set 3: Lx = 100 nm , Lz = 57 nm , w = 40 nm , d s = 15 nm , d f = 10 nm , h = 13.5 nm . It is
evident that the structure can be tuned to operate as a DNM in the range from near-IR to entire
visible spectra and this is with wavelength / periodicity ratio being around 7 , i.e. in deeply
subwavelength regime.
7
Im{ε d}=0.0
Re{neff} / Im{neff}
6
Im{ε d}=0.03
Im{ε d}=0.06
5
4
3
2
1
0
550
600
λ, [nm]
650
700
Fig. 7. The ratio Re{neff } Im{neff } characterizing the losses in the system as a function of the
gain ( Im{ε d } ) in the dielectric layer for a single DNM layer. The structure parameters are
chosen as Lx = 100 nm , Lz = 51.5 nm , w = 50 nm , d s = 15 nm , d f = 7.5 nm , h = 10.75 nm .
It is evident that the loss is not high without any gain and it further improves significantly by
increasing the gain; the required values of gain correspond to practically achievable values.
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13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11175
0.5
Re{neff}, Im{neff}
0
-0.5
ml=4
-1
-1.5
ml=3
-2
-2.5
Re{μeff} m =2
l
Im{μ }
eff
-3
600
650
700
750
λ, [nm]
Fig. 8. The effective index of refraction for different number of layers ml . The structure
parameters are chosen as Lx = 100 nm , Lz = 102.5 nm , w = 50 nm , d s = 15 nm , d f = 8.5 nm ,
h = 11.25 nm . DNM operation with stable negative index in the range 640 nm - 680 nm .
6
1
-1
4
-2
3
Re{ε eff}
Re{μeff}
θ=0
θ = π/6
θ = π/3
0
5
2
1
-3
-4
-5
θ=0
θ = π/6
θ = π/3
0
-1
-2
550
600
λ, [nm]
650
(a)
-6
-7
700
-8
550
600
λ, [nm]
650
700
(b)
Fig. 9. The real part of effective parameters μ eff and ε eff for different angles of incidence for
a single DNM layer. The structure parameters are chosen as Lx = 100 nm , Lx = 52.5 nm ,
w = 50 nm , d s = 15 nm , d f = 8.5 nm , h = 11.25 nm . DNM operation is obtained over a wide
range of incident angles.
Figure 9 shows the wavelength dependence of Re{μeff } and Re{ε eff } for different values
of the incident angle θ ; the structure parameters are given in the figure caption. It is observed
that negative Re{μeff } and Re{ε eff } simultaneously are obtained for all angles with some
angular dependence. While further studies are required to eliminate the obtained angular
dependence, still the obtained results, to the best of authors’ knowledge, are the only results
currently available showing subwavelength unit cell DNMs performance in a wide angular
range in the optical (near-IR/visible) regimes. We note that μeff exhibits a much weaker
(almost negligible) variation with θ than ε eff .
Finally, we note that the value of the scattering frequency Γ in the Drude model in all
simulations above is more than 3 times larger as compared to values assumed for simulations
in some other recent works (e.g. [15,18]). Fortunately, as is evident from the demonstrated
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13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11176
results, the structure in Fig. 1 performs very well even with these parameters. It should be
emphasized that reducing the value of Γ by the factor of 3 results in a drastic improvement of
the DNM performance in terms of decrease of dissipation and increase of achievable values of
μeff even without introducing any gain medium.
5. Summary
A novel realization of a DNM comprising unit cells of deeply subwavelength size was
introduced. The DNM can be tuned to operate over a wide bandwidth of the optical spectrum
from near-IR to visible and can be manufactured using standard nanofabrication methods with
materials compatible with these methods.
The DNM is composed of unit cells each comprising a continuous optically thin metal
film sandwiched between two identical thin metal strips separated by a small distance from
the film. The region between the metal strips operates as a nano-scale subwavelength cavity.
To elucidate the electromagnetic properties of the structure a modified cavity model was
developed. In this model, the perfect magnetically conducting boundary conditions are
imposed on the side walls of the cavity. The thin metal film and strips are approximated by
infinitesimally thin impedance sheets. It was shown that the cavities support both magnetic
and electric resonances that can be tuned to occur in overlapping frequency bends. It was
further shown that the crucial role in the ability to achieve DNM operation is played by the
presence of the middle film that enables tuning the electric resonances independently from the
magnetic ones.
Extensions of the presented 2D structure to 3D by using patches instead of strips are
straightforward and will allow for constructing DNMs with effective parameters independent
on the incident plane and wave polarization. These extensions will be presented in the
forthcoming publications. The aforementioned ideas and model can also be incorporated with
other configurations to allow their DNM operation in the near-IR and visible parts of the
spectrum. The presented structure and ideas are anticipated to allow for a number of important
applications in physics and engineering, such as the development of super lenses and other
subwavelength optical elements.
Acknowledgments
This research was supported by Nanoscale Interdisciplinary Research Teams (NIRT) program,
National Science Foundation (NSF).
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Received 16 August 2006; revised 20 October 2006; accepted 26 October 2006
13 November 2006 / Vol. 14, No. 23 / OPTICS EXPRESS 11177