Page 1
All-angle negative refraction and imaging in a
bulk medium made of metallic nanowires in the
visible region
Yongmin Liu1, Guy Bartal1, Xiang Zhang1,2,*
1NSF Nanoscale Science and Engineering Center (NSEC), 5130 Etcheverry Hall,
University of California, Berkeley, CA 94720-1740, USA
2Materials Sciences Division,Lawrence Berkeley National Laboratory,1 Cyclotron Road,
Berkeley, CA 94720, USA
*Corresponding author: xiang@berkeley.edu
Abstract: We theoretically demonstrated that all-angle negative refraction
and imaging can be implemented by metallic nanowires embedded in a
dielectric matrix. When the separation between the nanowires is much
smaller than the incident wavelength, these structures can be characterized
as indefinite media, whose effective permittivities perpendicular and
parallel to the wires are opposite in signs. Under this condition, the
dispersion diagram is hyperbolic for transverse magnetic waves propagating
in the nanowire system, thereby exhibiting all-angle negative refraction.
Such indefinite media can operate over a broad frequency range (visible to
near-infrared) far from any resonances, thus they offer an effective way to
manipulate light propagation in bulk media with low losses, allowing
potential applications in photonic devices.
©2008 Optical Society of America
OCIS codes: (160.4760) Optical properties; (160.3918) Materials: Metamaterials; (240.6680)
Surface Plasmons
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1. Introduction
Negative refraction and superlensing have been extensively studied in negative-index
metamaterials (NIMs) and photonic crystals (PCs) [1-15]. Veselago theoretically predicted
that at the interface of a positive-index material and a negative-index material, the Snell’s law
is reversed [1]. Pendry has further extended this concept, showing that a flat slab of NIMs can
make a perfect lens for sub-diffraction-limited imaging [2]. Such a superlens has been
implemented by a silver film with the properly designed thickness and working wavelength
[3,4]. Both negative refraction and imaging effects have been observed in the microwave
region based on artificial NIMs, where the electric permittivity and magnetic permeability are
simultaneously negative [5-7]. Researchers have also explored similar novel phenomena by
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engineering the dispersion of photonic bands in dielectric and metallic PCs [8-15], proving
that negative refraction can exist without a negative refractive index.
So far, demonstration of negative refraction and imaging in bulk materials at visible
frequencies still remains very challenging [16-19]. As NIMs rely on resonance, they are
usually accompanied by significant losses, especially in the visible region. Moreover, the
feature size of NIMs, as well as the period of PCs should be controlled at nanometer precision
within the entire bulk volume. Very recently, people have investigated negative refraction of
surface plasmon polaritons (SPPs) at visible frequencies, arising from the anomalous
dispersion of SPPs in specific frequency range in planar plasmonic waveguide structures
[20,21]. Negative refraction and imaging at visible frequencies were also proposed in metal-
dielectric multilayers [22-24], whereas an indirect observation of negative refraction in mid-
infrared region was reported in a semiconductor multilayer structure with the total thickness
about one wavelength [25]. Using curved metal-dielectric multilayers, researchers have
demonstrated the far-field magnifying hyperlens [26-29].
In the long-wavelength limit, the metal-dielectric multilayer structure in refs. [22-29] can
be described as indefinite materials, where the transverse and longitudinal effective
permittivities are different in signs as introduced by Smith and Schruig [30]. In this paper we
study and design a bulk indefinite material composed of aligned arrays of metallic nanowires
in a dielectric matrix, to implement all-angle negative refraction over a broad band in the
optical frequency region. Such negative diffraction is not sensitive to the azimuthal angle,
nanowire arrangement or geometry imperfection. More importantly, these structures can
operate far from any resonances. Therefore the material loss is significantly lower than
negative-index materials and plasmonic structures at visible frequencies [18,20,21], allowing
much longer propagation distances. Our analyses, based on the dynamical Maxwell-Garnet
theory, are in good agreements with finite-element simulations considering the actual
nanowire structure and material property.
2. Effective material properties of metallic nanowires
(a)
Fig. 1. (a). Schematic of metallic nanowires embedded in a dielectric matrix. (b) Illustration of
the hexagonal lattice of nanowires fabricated by an anodized alumina template. The rectangular
unit cell is adopted in the finite-element simulations.
Figure 1(a) schematically illustrates the structure of metallic nanowires embedded in a
dielectric host. This structure can be fabricated by electrochemically growing metallic
nanowires in a porous alumina template, which is prepared by the anodization method in a
self-organized way [31]. Such a method has proved to be a low-cost and high-yield technique
for fabricating different kinds of nanostructures including nanowires, nanodots and nanotubes
[32].
When the geometric parameters, i.e., the wire radius (r) and the distance between two
neighboring wires (d), are much smaller than the free space wavelength (
electromagnetic wave, the underlying system can be considered as an effective uniaxial
0 λ ) of the incident
y
x
z
(b)
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medium [33, 34]. We utilize the dynamical Maxwell-Garnett theory to determine the
permittivity parallel to wires (
//
ε ) and perpendicular to wires (
following equations [34, 35]:
p
εε
//
=
⊥
ε ), which are given by the
dm
p
ε
) 1 (
−+
(1a)
effdmd
d
ε
m
ε
d
d
qp
p
) )( 1 (
)(
ε
ε
−
εε
−
εε
+
d ε are the permittivity of metal and dielectric,
−
+=
⊥
(1b)
Here p is the filling ratio of metal,
respectively. In Eq. (1b), the effective depolarization factor perpendicular to nanowires is
defined as
2
(
32
λ
m
ε and
irrqeff
⋅⋅−⋅−=
3
0
2
0
)
2
λ
(
9
2
)
11
ππ
(2)
The first term on the right-handed site of Eq. (2) is the Lorentz depolarizing factor
perpendicular to the nanowire at the long wavelength limit. The second term is the dynamical
depolarization when the nanowire radius is not so small comparing to the incident wavelength.
The third term accounts for damping of the induced dipole due to radiation emission. In the
Cartesian coordinate shown in Fig. 1(a), one can see that
In this paper, we focus on silver nanowires as the material loss in the visible region is
smallest among the noble metals. The permittivity of silver is described by the Drude model
2
p
m
iγωω+
frequency
s rad
p
/ 105 . 1
×=ω
, and the collision frequency
obtained by fitting the model to the experimental data from the literature [36]. Alumina
(Al2O3) is the dielectric matrix with permittivity
our interest [35].
Figure 2 depicts the effective permittivities of metallic nanowires for the filling ratios p =
0.227 and p = 0.325 at the long wavelength limit
ε continuously changes from negative to positive as the wavelength decreases. In contrast,
ε has a resonant behavior (due to localized SPPs) accompanied by a strong dispersion and a
very large imaginary part [37]. When increasing the filling ratio, the critical frequency at
which
0) Re(
ε
shift to shorter wavelengths, while the resonance for
longer wavelengths. In general, there is a very broad spectral region (from visible to infrared)
satisfying
0
//
<⋅
⊥
εε
, which is critical for negative refraction and imaging as discussed in
the following text.
⊥
==εεε
yx
and
//
εε =
z
.
)(
)(
c
ω
εωε−=
∞
, where the high-frequency bulk permittivity
6
=
∞
ε
, the bulk plasmon
16
s rad
c
/ 1073 . 7
13
×=γ
are
4 . 2
=
d ε
for the entire frequency region of
) 2 / 1
=
(
eff
q
. It can be seen that the real part
of
//
⊥
//=
⊥
ε has a red shift to
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Fig. 2. Effective permittivities for silver nanowires embedded in an alumina matrix with two
different filling ratios. (a) and (b) plot the real part of the permittivity parallel and
perpendicular to the nanowire, respectively. (c) and (d) are the corresponding imaginary parts.
3. General theory of all-angle negative refraction by indefinite media
Now let us consider the electromagnetic (EM) wave propagation in a nonmagnetic uniaxial
material with the electric permittivity tensor given by
⎛
=
εε
0
?
⎟
⎠
⎟
⎟
⎞
⎜
⎝
⎜
⎜
z
y
x
ε
ε
0
ε
0
0
0
00
. (3)
The general format for the electric field and magnetic field of a plane wave with frequency ω
and wave vector k?
can be written as
0
EE
=
??
)(
tr
?
k
?
ie
ω−⋅
, (4)
)(
0
tr
?
k
?
ieH
?
H
?
ω−⋅
=
. (5)
In the principle axis, suppose the wave vector k?
B
E
∂
satisfies the condition of
k
?
lies in the x-z plane. From Maxwell’s
)
, a transverse magnetic wave, which
equations
t
∂
−=×∇
?
?
and
t
E
?
t
D
∂
H
?
∂
⋅∂
=
∂
=×∇
(
?
?
ε
0
=⋅H
?
, can be expressed explicitly as
)](exp[
ˆ
e
0
tzkxkiHH
?
zxy
ω
−+=
, (6)
)]( exp[)
ˆ
e
ˆ
e
(
0
0
tzkxki
k
ε
k
ε
H
E
?
zxz
z
x
x
x
z
ω
ωε
−+−=
. (7)
(a)
(b)
(c)
(d)
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Here
Consequently, the time-averaged Poynting vector for TM waves is readily obtained
x e ˆ ,
ye ˆ
and
ze ˆ are the unit vectors along the x, y and z axis, respectively.
2
0
0
22
1
H
k
?
H
?
E
?
S
?
zxεε ωε
ε
?
⋅
=× >=<
∗
. (8)
From Maxwell’s equations, the dispersion relation for TM waves is given by
2
2
2
z
2
x
c
k
ε
k
ε
xz
ω
=+
, (9)
where c is the light velocity in vacuum. In particular, we study the case of εx > 0 and εz < 0,
representing an “indefinite material” as introduced by Smith and Schruig [30]. Under this
condition, the equifrequency contour of the medium is hyperbolic in the (kx, kz) plane (see Fig.
3(a)). Namely, any real kx has a real solution for kz, indicating that the wave can propagate
without any cutoff [38].This is to say, such materials support propagation of waves with large
wave vectors (i.e., with kx >2π /λ0), which are initially evanescent in free space [30].
(a)
Fig. 3. (a). The equifrequency contour of an indefinite material with εx = 4.515 and εz = -2.530
(green hyperbola), as well as the equifrequency contour of an isotropic material (gray circle).
The refracted wave vector (solid blue arrow) and Poynting vector (solid red arrow) can be
determined by satisfying the causality theorem and the conservation of the tangential wave
vector. While the other set of refracted wave vector (dotted blue arrow) and Poynting vector
(dotted red arrow) are physically incorrect. (b) Schematic diagram of negative refraction for a
TM wave, which is incident from an isotropic material to an indefinite one with εx >0 and εz <
0. (c) Refracted angles for the wave vector (blue) and Poynting vector (red) at various incident
angles. Note the Poynting vector undergoes negative refraction for all incident angles.
Consider a TM polarized light incident from a normal isotropic material, in which the
wave vector and Poynting vector are parallel, to an indefinite medium with εx > 0 and εz < 0
(Figs. 3(a) and 3(b)). At the interface, there are two solutions (presented by solid and dashed
blue arrows in Fig. 3(a)) satisfying the continuity of the tangential component of k?
is,
(that
x k ). However, according to the causality principle [5-15], the energy must flow away from
anisotropic
isotropic
z
x
y
(b)
(c)
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the interface, i.e.,
only the solution represented by the solid blue arrow in Fig. 3(a) satisfies the condition
0
ˆ
e
>⋅=
S
?
S
zz
. It can be readily shown from Eq. (8), that for positive εx
0
2
ˆ
e
0
2
0
>
⋅
=
ωεε
Hk
?
S
x
z
z
. (10)
The tangential component of S?
is written as
0
2
0
2ωεε
Hk
S
z
x
x=
. (11)
Since εz is negative and kx is continuous, the sign of Sx flips at the interface. This gives rise to
negative refraction at the interface as shown in Fig. 3(b), which was also discussed in
reference [39]. In fact, we find that the x-component of the group velocity is negative, which
results in negative refraction in this nanowire composite (or so-called negative group-index).
We can quantitively determine the refraction angles for the wave vector and Poynting
vector defined as
)( tan
1
,
z
x
kr
k
k
−
=θ
, (12a)
)
/
/
( tan)( tan
11
,
xz
zx
z
x
Sr
k
k
S
S
ε
ε
θ
−−
==
, (12b)
respectively. Figure 3(c) plots the numerical calculation based on Eqs. (9) and (12) when εx =
4.515 and εz = -2.530 (corresponding to real parts of εx and εz for nanowires with the filling
ratio p = 0.227 at 632.8nm wavelength). One can see that the refraction angle of the Poynting
vector is always negative for any incident angle while the phase front always propagates in the
positive direction. Moreover, it should be noted that the angle between the Poynting vector
and wave vector is acute, namely
)
)/()/(
/
( cos)
||||
( cos
222
z
2
x
22
11
xzzx
kkkk
c
S
?
k
?
S
?
k
?
εε
ω
φ
++
=
⋅
⋅
=
−−
. (13)
This is in contrast to NIMs, where the Poynting vector and wave vector are anti-parallel.
4. Numerical simulations and discussions
The aforementioned analyses of negative refraction are confirmed by full-wave simulations,
which take into account the actual nanowire structure and material property. Using COMSOL
Multiphysics TM3.4, a commercial electromagnetic solver based on the finite element method
(FEM), we numerically demonstrate negative refraction and imaging of TM waves by metallic
nanowires at visible frequencies. In the simulation, the distance between two neighboring
nanowires is set to 110nm and the nanowire radius is 27.5nm (corresponding to the filling
ratio
0.227)d3 /(2
=⋅⋅=
rp
π
). The rectangular unit cell shown in Fig. 1(b) is adopted in
the simulation. Periodic boundary conditions are applied along the y-axis, and matched
boundary conditions are along the x- and z-axes. A transverse magnetic (magnetic field is
polarized along the y-axis) Gaussian beam with a 2μm width is incident at 30 degrees on
1.5μm-long nanowire arrays.
Figures 4(a) and 4(b) plot the cross-sectional view of the absolute electric field and time-
averaged Poynting vector (energy flow) in the x-z plane, respectively, at the excitation
wavelength of 632.8nm. The results evidently show negative refraction at the interfaces, with
refraction angle of about -22 degrees, in a good agreement with the analytical prediction based
on the dynamical Maxwell-Garnett theory (see Fig. 3(c)). It is noted that the field intensity is
enhanced in the gap between nanowires, arising from the existence of SPPs [37]. We also
22
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perform finite element simulation replacing the nanowire structure with a homogenous slab
with effective permittivities
x
056. 0 515 . 4
+=ε
Eqs. (1) and (2) according to the dynamical Maxwell-Garnett theory. Once again, the
excellent agreement between Figs. 4(a) and 4(c), as well as Figs. 4(b) and 4(d) prove the
validity of the effective media approximation for any beams incident from free space to the
underlying indefinite medium.
i
and
i
z
149 . 0530 . 2
−+=ε
, calculated from
Fig. 4. Finite-element simulations showing negative refraction by metallic nanowire structures
at 632.8nm wavelength. Figs. (a) and (c) plot the absolute value of the electric field, and Figs.
(b) and (d) plot the time-averaged Poynting vector. The 3D full-wave simulations ((a) and (b))
considering the real nanowire structure agree very well with the simulation based on the
effective medium approximation ((c) and (d)). In the simulations of (a) and (c), the nanowires
are arranged in a hexagonal lattice as shown in Fig. 1(b), the radius of nanowires is 27.5nm,
and the separation between adjacent wires is 110nm. Those parameters are kept same for
simulations shown in Figs. 5-6.
One of the most attractive advantages to implement negative refraction by nanowires is
that the loss can be significantly reduced, compared to NIMs [18] or plasmonic structures with
anomalous dispersion [20, 21] at visible frequencies. At frequencies far away from the
resonance of
⊥
ε , the system is intrinsically low-loss. For example, at 632.8nm wavelength
with the filling ratio p = 0.227, the intensity of the refracted light only attenuates to 77% after
propagating one micrometer. This indicates a loss that is a few orders of magnitudes lower
than other structures at visible wavelengths [18, 20, 21]. At longer wavelength, the smaller
penetration depth of the field into metal gives rise to even less Ohmic losses. Consequently,
the transmission efficiency of light in the nanowire metamaterials can be further improved.
Moreover, in contrast to photonic crystals [7-14], the negative refraction is much more
tolerant to the deviation of nanowire geometry and lattice, as long as the effective media
approximation is valid. For simplicity, the simulations in this paper consider an ideal
hexagonal lattice of nanowires. But in reality, the nanowires can be disordered, and still
maintain negative refraction, if the spacing between them is much smaller than the
wavelength.
0
0.5
1
1.5
2
X10-3
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
2
X10-3
(a)
(b)
(c)
(d)
z
x
y
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Received 8 May 2008; revised 11 Aug 2008; accepted 15 Aug 2008; published 16 Sep 2008
29 September 2008 / Vol. 16, No. 20 / OPTICS EXPRESS 15446
Page 9
Interestingly, while such indefinite materials exhibit negative refraction (related to the
direction of Poynting vector) the phase propagation (associated with the wave-vector) still
remains positive. The refraction properties of nanowires depend on the orientation of the
optical axis with respect to the surface normal. As such, plane wave propagation in a prism-
shaped indefinite material will not always result in negative refraction, as shown in NIMs or
PCs [5-7, 13]. This is demonstrated in Fig. 5, where TM wave propagation through a prism-
shaped sample is simulated. Clearly, a horizontally incident wave is refracted from the 15-
degree angle facet in a positive way, seemingly contradictive to the results shown in Figs.
4(a)-4(d). However, this could be explained considering the ray optics based on the
equifrequncy contour, which is illustrated in Fig. 5(b). Since the continuity of the tangential
wave vector along the interface and causality principle should be satisfied, positive refraction
takes places at the exit surface of the prism.
Fig. 5. (a). The time-averaged Poynting vector for a TM wave transmitted through a prism-
shaped nanowire slab. The surface normal of the prism has a 15-degree angle with respect to
the z-axis. Positive refraction takes place at the exit surface. (b). Interpretation of the refraction
behavior at the exit surface based on the equifrequency contour.
Finally, we also show that a flat slab made of indefinite materials can be used for imaging,
as presented by the ray diagram in Fig. 6(a). The full-wave simulation of the imaging effect
by nanowire arrays is shown in Fig. 6(b), at the wavelength of 632.8nm for a source located
0.5μm in front of a 2.0μm-thick nanowire slab. An elongated focus point is formed after the
nanowire lens. The aberration of the imaging results from the fact that the effective refraction
index of the indefinite material is angle-dependent [41]. By adjusting the geometry of
nanowires, subwavelength and magnified imaging can be achieved [42, 43].
Fig. 6. (a). Ray optics showing that an indefinite-material slab is able to form partial imaging
inside and outside the slab. The source is placed 500nm in front the slab. The gray region
represents a 2μm-thick indefinite slab with εx = 4.515 and εz = -2.530. (b) The time-averaged
Poynting vector in the 3D finite-element simulation. (c) Same as (b), but the realistic nanowire
structure in (b) is replaced by a homogeneous indefinite material with effective permittivies
calculated from the dynamical Maxwell-Garnett theory.
0
0.5
1
1.5
2
2.5
X10-3
(a)
(b)
0
2
4
6
8
X10-6
X10-6
0
2
4
6
8
10
z (μm)
(a)
(b)
(c)
z (μm) z (μm)
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Received 8 May 2008; revised 11 Aug 2008; accepted 15 Aug 2008; published 16 Sep 2008
29 September 2008 / Vol. 16, No. 20 / OPTICS EXPRESS 15447
Page 10
5. Conclusions
To summarize, we have deigned a bulk indefinite metamaterial using metallic nanowires to
realize all-angle negative refraction and imaging in the visible region. Finite-element
simulations taking into account the actual structure and material property agree very well with
the analyses based on the dynamical Maxwell-Garnet theory. The negative refraction, arising
from the hyperbolic equifrequency contour of the system, is fundamentally different from
what have been observed in negative-index materials and photonic crystals. It is intrinsically
low-loss and broad-band, because the mechanism does not rely on any resonances. Moreover,
our approach is insensitive to the variation of wire diameters and separations between wires,
under the effective media approximation at the long wavelength limit. We expect those
intriguing phenomena to stimulate experimental efforts towards imaging, wave guiding and
light manipulation in three dimensions over a very broad frequency band.
Acknowledgments
This work was supported by the Air Force Office of Scientific Research (AFOSR) MURI
program (FA9550-04-1-0434), and the National Science Foundation (NSF) Nanoscale
Science and Engineering Center (DMI-0327077).
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Received 8 May 2008; revised 11 Aug 2008; accepted 15 Aug 2008; published 16 Sep 2008
29 September 2008 / Vol. 16, No. 20 / OPTICS EXPRESS 15448
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