Negative refractive index in coaxial
plasmon waveguides
Ren
́
e de Waele,
1
,
∗
Stanley P. Burgos,
2
Harry A. Atwater,
2
and Albert
Polman
1
1
Center for Nanophotonics, FOM-Institute for Atomic and Molecular Physics (AMOLF),
Science Park 104, 1098 XG Amsterdam, The Netherlands
2
California Institute of Technology, 1200 East California Boulevard, Pasadena, California
91125, USA
*waele@amolf.nl
Abstract:
We theoretically show that coaxial waveguides composed of
a metallic core, surrounded by a dielectric cylinder and clad by a metal
outer layer exhibit negative refractive index modes over a broad spectral
range in the visible. For narrow dielectric gaps (10 nm GaP embedded in
Ag) a figure-of-merit of 18 can be achieved at
λ
0
=
460 nm. For larger
dielectric gaps the negative index spectral range extends well below the
surface plasmon resonance frequency. By fine-tuning the coaxial geometry
the special case of
n
=
−
1 at a figure-of-merit of 5, or
n
=
0 for a decay
length of 500 nm can be achieved.
© 2010 Optical Society of America
OCIS codes:
(240.6680) Optics at surfaces; (160.3918) Metamaterials; (222.0220) Optical de-
sign and fabrication.
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1. Introduction
Controlling the propagation of light at the nanoscale is one of the challenges in photonics.
Surface plasmons, electromagnetic modes that propagate at a metal/dielectric interface provide
a key opportunity to achieve this goal, due to their relatively small evanescent fields [1, 2].
Moreover, as their dispersion can be strongly controlled by geometry, their effective wave-
length can be shrunk well below the free-space wavelength, enabling further miniaturiza-
tion of optical components. Initial experiments on plasmon optics were carried out at planar
metal/dielectric interfaces, demonstrating basic control of plasmons. Plasmonic components
such as mirrors [3, 4] and waveguides [5, 6] were realized, however still of relatively large
size due to the
>
100 nm evanescent tails, and with limited control over dispersion. Subse-
quently, insulator-metal-insulator structures were investigated, and have demonstrated confine-
ment of light to
<
100 nm length scales in taper geometries [7, 8], though at high loss. The
reverse, metal-insulator-metal (MIM) geometries, have demonstrated lower loss, higher disper-
sion [9,10], and recently the attainment of a negative index of refraction [11–13].
A disadvantage of planar MIM structures is that they only confine light in one transverse
direction. Recently, coaxial MIM waveguides, composed of a metal core surrounded by a di-
electric cylinder clad by a metal outer layer have been introduced, that confine light in all trans-
verse directions [14,15]. We have recently reported optical transmission measurements through
single coaxial waveguides, from which the dispersion diagram for these nanoscale waveguides
was determined [16].
Inspired by the earlier work on MIM waveguides, a natural question arises whether coaxial
waveguides would posses a negative refractive index, and if so, for what geometry and over
what spectral range. Since the coaxial waveguides are essentially 3-dimensional objects, the
observation of negative index in individual coaxes, also inspires the design of 3-dimensional
negative-index metamaterials [17–21] composed of arrays of coaxial waveguides [22].
Here, we theoretically study the dispersion of coaxial Ag/Si/Ag plasmon waveguides and
demonstrate that for well-chosen geometries modes with a negative refractive index are ob-
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12771
−2
0
2
4
0
1
2
3
4
k’ (10
8
m
-1
)
k” (10
8
m
-1
)
Arg(Det(M))
π
−π
0
r
z
φ
(b)
(a)
Field
max
min
0
Fig. 1. Coaxial plasmon waveguide geometry and numerical mode solving method. (a)
Schematic cross-section of a coaxial waveguide with the definition of the cylindrical polar
coordinates,
r
,
φ
and
z
. The metallic inner core and outer cladding separate a dielectric
channel. A schematic wave propagating in the waveguide in the direction of positive
z
is
also indicated. (b) Argument
θ
of the determinant, det
[
M
(
k
)]
, plotted in the complex
k
-
plane for a Ag/Si/Ag waveguide with 75 nm inner core diameter and 10-nm-wide dielectric
channel at
ω
=
3
×
10
15
rad/s. By cycling around the closed loop indicated by the dashed
square the net number of discontinuities in
θ
is determined. Zero positions are indicated by
the white circles.
served. These modes are dominant over other waveguide modes for a wide range of frequencies
above the surface plasmon resonance frequency. We discuss the influence of waveguide geom-
etry and material on the mode index and demonstrate that the figure-of-merit (FOM), defined
as the magnitude of the real part of the propagation constant in the waveguide divided by the
imaginary part [13,23], can be as high as 18.
2. Method
The azimuthal dependence of the fields is described by the harmonic function
e
in
φ
of order
n
.
In the remainder we only consider modes with
n
=
1, since these are the lowest order modes
that couple to free-space radiation. The radial dependence of the fields in all three domains
(metal-dielectric-metal) is described by solutions to the 2
nd
order Bessel differential equation.
We apply a Bessel function of the first kind,
J
n
, to the metal core, as that function remains finite
at the waveguide axis. A Hankel function of the first kind,
H
(
1
)
n
, is applied to the metal cladding.
Inside the dielectric channel the radial field is described by two linearly independent cylinder
functions. The arguments of the cylinder functions in each of the three domains is
κ
i
r
, where
κ
i
is the radial wave number in medium
i
, defined via
κ
2
i
≡
ε
i
ω
2
c
2
−
k
2
(1)
where
ε
i
is the complex dielectric constant in domain
i
. To satisfy the condition that fields decay
to zero at radial infinity we take the square root of Eq. (1) such that the radial wave number has
a positive imaginary part.
On each domain boundary we formulate four continuity conditions for the tangential com-
ponents of the electric and magnetic fields. The optical eigenmodes of the coaxial waveguide
are found when the determinant of the resulting homogeneous system of eight equations with
eight unknown coefficients vanishes,
det
[
M
(
k
)]=
0
(2)
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12772
ω
(10
15
rad/s)
FOM, |k’/k”|
k” (m
-1
)
10
6
10
7
10
8
10
9
Energy (eV)
−2
0-1
1
2 3
1
2
3
4
5
3
2
1
k’ (10
8
m
-1
)
ω
SP
max
0
min
H
y
y
x
H
y
y
x
10
-3
10
-1
10
1
10
3
(a)
(b)
(c)
Fig. 2. Dispersion relations of the three lowest-order modes of a coaxial waveguide with 75-
nm-diameter Ag core, 25-nm wide Si channel, and infinite outer Ag cladding. Radial fre-
quency is plotted versus propagation constant
k
(a), attenuation constant
k
(b), and figure-
of-merit
k
/
k
(c). The Ag/Si surface plasmon resonance frequency
ω
SP
=
3
.
15
×
10
15
rad/s
(
λ
0
=
598 nm) is indicated by the horizontal line. Panel (a) shows two modes with positive
index (blue dashed curve and green dotted curve) and one mode with a negative index be-
low a frequency of
∼
3
.
8
×
10
15
rad/s (red drawn curve). The insets in (a) show the
H
y
field
distribution in the transverse plane of the waveguide at 2
.
8
×
10
15
rad/s for the positive-
index mode (blue dashed dispersion curve) and at 3
.
6
×
10
15
rad/s for the negative-index
mode.
where
M
is the matrix of the system of equations. We have used two independent methods
for determining the optical modes,
k
(
ω
)
, of the structure. One involved a numerical procedure
developed to detect local minima of the determinant of the system in the complex
k
-plane. The
other method relies on the fact that the argument,
θ
, given by
det
[
M
(
k
)]=
|
det
[
M
(
k
)]
|
e
i
θ
(3)
is undefined when det
[
M
(
k
)]=
0. This can be visualized in a plot of
θ
in the complex
k
-plane.
An example is shown in Fig. 1(b) where
θ
is plotted for a coaxial waveguide with a Ag core and
cladding and a 10 nm silicon spacer layer. We used empirically determined optical constants
for the metal [24] and dielectric [25]. Contour lines in the figure appear to close in on each
other at each of the zeros, which are indicated by the white dots in the figure. By counting
each discontinuity
−
π
→
π
and
π
→−
π
about a closed loop in the figure [for an example,
see the dashed square loop in Fig. 1(b)] we are able to determine the number of zeros in the
enclosed area. In case we find that one or more zeros reside in the area, we split the area up
in smaller pieces and repeat the procedure until the location of the zero(s) is determined with
double computer precision.
Using this method, solutions for
k
were found for real frequency, so that dispersion rela-
tions,
ω
(
k
)
, could be constructed. Calculations were performed in the optical angular frequency
regime 1
×
10
15
rad/s
<
ω
<
5
×
10
15
rad/s (free-space wavelength,
λ
0
=
377–1884 nm). We
only consider modes with positive energy velocity,
ν
e
, or equivalently, positive attenuation con-
stant
k
[13]. Therefore, to achieve antiparallel energy and phase velocity, which is the unique
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12773
ω
(10
15
rad/s)
−4
−3
−2
−1
0
1
1
2
3
4
5
k’ (10
8
m
-1
)
ω
SP
w=10nm
30nm
70nm
k” (m
-1
)
10
6
10
7
10
8
10
9
w=70nm
30nm
10nm
1
2
3
Energy (eV)
(b)
(a)
Fig. 3. Dispersion relations for negative-index coaxial waveguides with Ag core and
cladding and Si dielectric channel, (a):
ω
(
k
)
; (b):
ω
(
k
)
. The inner core diameter is fixed
at 75 nm and the Si-channel thickness
w
is 10 nm, 30 nm, and 70 nm. Positive-index modes
[as shown in Fig. 2(a)] are not shown in the figure. The bold sections of the dispersion
curves indicate the spectral range over which the negative-index mode is dominant, i.e. has
lower loss than the positive index modes. The frequency where the red and green dispersion
curves cross
k
=
0 is indicated by the star-symbols.
requirement for a negative mode index, the propagation constant
k
needs to be negative.
3. Results
Figure 2 shows the dispersion relation,
ω
(
k
)
, for the three lowest-order modes in a coaxial
waveguide consisting of a 75-nm-diameter Ag core surrounded by a 25-nm-thick Si layer and
infinite Ag cladding. In (a) the angular frequency is plotted against
k
, while (b) shows the
frequency as function of
k
, which determines the propagation length of light in the waveguide
via
L
=
1
2
k
”
.
(4)
The surface plasmon resonance frequency
ω
SP
=
3
.
15
×
10
15
rad/s (
λ
0
=
598 nm) is indicated
by the horizontal line.
Figure 2(a) shows two coaxial modes (blue dashed line and green dotted line) with positive
propagation constants over the entire spectral range. Both dispersion curves closely resemble
the dispersion of a surface plasmon polariton propagating along a planar Si/Ag interface. How-
ever, the corresponding propagation constants [Fig. 2(a)], are nearly three times as large as for
the planar single interface plasmon. This is due to the fact that confinement of the plasmon in
the coaxial waveguide geometry leads to larger mode overlap with the metal. Figure 2(a) also
shows the existence of a third mode (red drawn curve) that has a negative propagation constant
k
for frequencies below 3
.
8
×
10
15
rad/s (
λ
0
=
496 nm). The effective index
n
=
ck
/
ω
ranges
from
−
9
<
n
<
5 in the frequency range of Fig. 2. The insets in (a) show the
H
y
field in the trans-
verse plane for the negative-index mode (calculated at
ω
=
3
.
6
×
10
15
rad/s) as well as for the
most dispersive positive mode (blue dashed curve in Fig. 2(a),
ω
=
2
.
8
×
10
15
rad/s). From the
images it is clear that the mode with positive effective index has a symmetric field distribution
with respect to the two centers of the dielectric channel on the x-axis. The
H
y
field is primarily
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12774
−4
−2
0
2
4
1
2
3
4
5
ω
(10
15
rad/s)
k’ (10
8
m
-1
)
k” (m
-1
)
10
6
10
7
10
8
10
9
ω
SP
d
core
=45nm
100nm
75nm
d
core
=100nm
45nm
75nm
1
2
3
Energy (eV)
(b)
(a)
Fig. 4. Dispersion relations for coaxial waveguides with Ag core and cladding and 70-nm-
wide Si dielectric channel, (a):
ω
(
k
)
; (b):
ω
(
k
)
. The inner core diameter,
d
core
,is45nm
(blue curves), 75 nm (green curves) and 100 nm (red curves). Only modes with negative
index are plotted. Bold lines indicate the spectral range where the mode is dominant over
the positive-index mode.
concentrated at the boundary between the metal core and dielectric channel. The negative-index
mode in contrast, has its field primarily concentrated at the outermost channel boundary and
has an
H
y
-field distribution that is anti-symmetric about the center of the dielectric channel,
similar to modes with negative index in planar metal-insulator-metal waveguides [13].
Figure 2(b) shows that for frequencies below the surface plasmon resonance frequency
ω
SP
the lowest-order positive-index mode (blue dashed line) has lowest loss and will therefore be
dominant over other modes. Interestingly, above
ω
SP
the negative-index mode (red curve) be-
comes the dominant mode, as its losses are significantly lower than those for the positive-index
modes. Figure 2(c) shows the figure-of-merit (FOM),
k
/
k
, of the modes. As can be seen, the
negative-index mode has a FOM that approaches 10 for a narrow frequency interval around
3
.
4
×
10
15
rad/s (
λ
0
=
554 nm). The data in Fig. 2 clearly demonstrate that dominant modes of
negative index indeed exist in coaxial plasmon waveguides.
Next, we investigate the conditions that are required to achieve a negative index by vary-
ing the geometry and materials of the waveguide. Figure 3 shows the effect of changing the
dielectric layer thickness on the dispersion of the negative index mode. Calculations were per-
formed for a Ag/Si/Ag coaxial waveguide with a core diameter of 75 nm for a dielectric layer
thickness of 10 nm, 30 nm, and 70 nm. Figure 3(a) shows that the variation in dielectric layer
thickness has a very dramatic effect on the dispersion of the negative-index mode. First of all,
the largest negative index is observed for the thinnest dielectric. Second, while for the 10-nm
and 30-nm dielectric gaps the frequency of the resonance associated with the negative index
mode appears close to the surface plasmon resonance at 3
.
2
×
10
15
rad/s, for the 70-nm gap
this resonance is significantly red-shifted to 2
.
5
×
10
15
rad/s. The spectral range over which the
mode is dominant, indicated by the bold curves in Fig. 3(a), also extends to lower frequencies
when increasing the channel width. For the 70-nm gaps a narrow frequency range is found near
2
.
4
×
10
15
rad/s (
λ
0
=
785 nm) where the index is negative and the figure-of-merit is 5.
As the dispersion branches cross the
k
=
0 line the effective refractive index of the mode van-
ishes [26]. Coaxial waveguides with a narrow dielectric gap suffer high loss at this frequency.
The green star in Fig. 3(b) indicates the frequency at which the dispersion curve crosses the
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OSA
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12
/ OPTICS
EXPRESS
12775