Broadband Negative Refractive Index in the Visible Spectrum
M. Keshavarz, S. Khosravi, A. Rostami, G. Rostami and M. Dolatyari
OIC Research Group School of Engineering-Emerging Technologies, University of Tabriz, Tabriz 5166614761, Iran
Keywords: Metamaterials, Broad Band Metamaterials, Core-shell Nanoparticles.
Abstract: In this paper, a composite medium based metamaterial with random distribution of nanoparticles in vacuum
host to achieve negative effective refractive index in the visible wavelength range is suggested for
invisibility purposes. Our calculations show that structures including single metal (dielectric) spheres and
core-shell structures with metallic core and dielectric shell, which consists two-layer particles with uniform
sizes cannot support negative effective refractive index. For this purpose the structures consist of two-layer
nanospheres with different sizes and fill fraction has been proposed. Since, band width of negative refractive
index is narrow, the three layer nanospheres has been studied and investigated. We show that in this case
with increasing the refractive index of middle and outer layers, negative value of effective refractive index
can be increased. Also, we show that using different sizes of nanomaterials in host medium, band width is
increased. Finally, superposition of three layer spherical nanoparticles with different outer radius and
applied single doped semiconductor spheres, has been proposed. We show that Band width with negative
permittivity and permeability can be optimized.
1 INTRODUCTION
Nowadays the metamaterials has very interesting
applications such as applications in super lenses and
invisibility and also hot topic for researchers
recently (Cai, Genov and Shalaev, 2005, Cai and
Shalaev, 2010). Although, metals such as silver,
gold and copper can produce the negative
permittivity in the optical range but finding natural
elements with negative permeability is limited to the
hundred gigahertz frequencies (Cai and Shalaev,
2010). In this way different media consists of
nanorods and split ring resonators (SRRs) in order to
achieve negative effective parameters have been
proposed in different wavelength ranges (Cai and
Shalaev, 2010, Zhang, Fan, Panoiu, Malloy, Osgood
and Brueck, 2005, Smith, Padilla, Vier, Nemat-
Nasser and Schultz, 2000). The complex fabrication
methods to produce these media and limitations of
SRRs related to the saturation magnetic response in
the optical wavelength ranges (Zhang, Fan, Panoiu,
Malloy, Osgood and Brueck, 2005) lead to designing
of the media with random distribution of
nanoparticles (Zhou, Koschny, Kafesaki, Economou,
Pendry and Soukoulis, 2005, Dominguez, Tejeiara,
Marques and Gil, 2011) . The random distribution of
nanoparticles also can produce broad band negative
permittivity that it was not possible with the
previous structures. In this paper, we propose single,
two and three layer spherical nanoparticles with
random distribution to exhibit negative effective
refractive index in visible wavelength range. In this
regard first single nanospheres consist of metal (Ag,
Au, Cu, Al…) and dielectric with high relative
refractive index and two layer spherical
nanoparticles which possess of metal core (Ag) and
dielectric shells (Si) with different sizes and fill
fractions will investigate. After that to produce
negative refractive index, we will consider three
layer spherical nanoparticles (such as proposed in
(Dajian, Shumin, Ying and Xiaojun, 2011)). then,
the effect of electrical permittivity of middle layer
and the refractive index of the outer layer on
increasing of the wide of wavelength range with
negative optical parameters has been studied and at
the end structure consist of three layer nanoparticles
with different size and the same filling fraction has
been proposed to broaden the wavelength range with
negative effective permeability. Since, the
wavelength range with negative effective
permittivity is narrow, finally in order to broaden of
this range, we have applied the semiconductor doped
spherical nanoparticles with a proper plasma
frequency and electrical permittivity (obtained by
Drude model) in the structure. To calculate the
effective parameters Clausius-Mossotti relations
have been used.
113
Keshavarz M., Khosravi S., Rostami A., Rostami G. and Dolatyari M..
Broadband Negative Refractive Index in the Visible Spectrum.
DOI: 10.5220/0005337601130117
In Proceedings of the 3rd International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS-2015), pages 113-117
ISBN: 978-989-758-093-2
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
2 THEORETICAL
BACKGROUND AND
SIMULATIN
First, structures including metal (Ag, Au, Cu, Al …)
and high refractive index dielectric (Si, Ge, GaAs
…) as single spheres to produce electric and
magnetic activity in the desired wavelength range
have been studied. The medium has been shown as
bellow:
Figure 1: Single layered spheres with metal and dielectric
spheres in vacuum host.
Although the metal spheres show negative electric
permittivity (since, electric resonance condition for
metal spheres surrounded in a media with electric
permittivity
m
is (
mres
2
)) the electric
resonance of metal nanoparticles in vacuum doesn't
occur in the visible wavelength range. However, the
dielectric with high refractive index with proper
radius produces the magnetic activity in the desired
range. Therefore, the electric and magnetic
resonance don't occur in the desired wavelength
range simultaneously. The media including two
spherical- layer nanoparticles with metallic core and
dielectric shell with same size are considered and
shown as follows.
Figure 2: Two layered spheres with Ag as core and Si as
shell in vacuum host.
Scattering and extinction efficiencies for these
structures can be obtained as (Craig, Bohren and
Donald, 1983):
0
22
2
)12(
2
l
nnsca
bal
y
Q
0
2
Re)12(
2
l
nnext
bal
y
Q
(1)
Where electric and magnetic scattering
coefficients
n
a and
n
b for two layer nanoparticles
can be obtained from (Craig, Bohren and Donald,
1983). Since, the size of nanoparticles is small
related to the incident wavelength, therefore
only
1
a
and
1
b
can be considered for simulations.
Figure 3: (a) and (b) electric and magnetic scattering
efficiencies for the particles with fixed Rout in terms of
Rin and incident light, (c) and (d) electric and magnetic
scattering in the case of fixed Rin versus different Rout
and incident light respectively.
PHOTOPTICS2015-InternationalConferenceonPhotonics,OpticsandLaserTechnology
114
Figure 3 (a) and (b) show electric and magnetic
scattering efficiencies for the particles with fixed
outer layer radii (R
out
=70nm) in term of inner layer
radii (R
in
) and incident light and (c), (d) indicate
scattering efficiencies corresponded to the case of
specific inner layer radius (R
in
=15nm) versus outer
layer radii (R
out
) and incident light wavelength.
It is clear from the figure that with the uniform
particles, it is impossible to adjust inner and outer
radii to resonance both electric and magnetic dipoles
in a specific wavelength range to produce
negative
eff
n
. To resolve this problem, we can apply
a medium consists nanoparticles with different sizes
(R
in
=20nm, R
out
=70nm for f
1
=0.30 and R
in
=25nm,
R
out
=35nm for f
2
= 0.15).
By using the Clausius-Mossotti equation,
Effective permittivity (
eff
) and magnetic
permeability (
eff
) of a host medium with electrical
permittivity (
h
) and magnetic permeability (
h
)
related to the filling fraction
3
4
3
r
Nf
where N
is the number of particles per unit volume, and
polarizabilities of the nano-particles.
3
42 R
f
E
heff
heff
3
42 R
f
E
heff
heff
(2)
Where
the electric and magnetic polarizabilities,
E
and
M
respectively, are directly proportional
to the scattering coefficients
1
a
and
1
b
(factor
13
)6/(
ki
. As can be seen in Figure
4,
eff
n
is negative for a narrow wavelength range.
Broader wavelength range with negative
eff
n
can be
produced by using three layered spherical
nanoparticles according to (Dajian, Shumin, Ying
and Xiaojun, 2011). This goal can be achieved by
applying the middle layer with the refractive index,
smaller than the outer layer in order to increasing the
Plasmon resonance frequency. In this condition the
electric and magnetic resonance simultaneously
occur in a specific wavelength ranges, too. Effective
optical parameters for this structure have been
obtained as (Dajian, Shumin, Ying and Xiaojun,
2011). Since, the magnetic polarizability of the
particle enhances by increase the refractive index of
Figure 4: Effective parameters for a medium consists of
nanoparticles with different sizes (Rin=20nm, Rout=70nm
for f1=0.30 and Rin=25nm,
out
R
=35nm for f2= 0.15).
the outer layer; the effective negative permeability
of the structure will be broader in the desired
wavelength range. This effect can be seen in Figure5
as is shown increase the refractive index of outer
layer leads to broader negative wavelength range.
Figure 5: The effect of different materials for outer layer
on effective optical parameters.
Figure 6: The effect of changing refractive index middle
layer on the effective parameters.
BroadbandNegativeRefractiveIndexintheVisibleSpectrum
115
Increasing of the electric permittivity of the
middle layer leads to a shift in the electric resonance
to the longer wavelength and broaden the
wavelength band with negative effective permittivity
but the effective permeability isn’t influenced as
shown in Figure 6.
Figure 7 shows the effect of increase of the
refractive index of middle and outer layer that broad
wavelength band with negative refractive index can
be achieved.
Figure 7: effective parameters plotted for a medium
contains three layered spheres with n2=1.7 and n3=nGe=4.
Since, the electric and magnetic resonance
simultaneously in the other hand, the magnetic
resonance is influenced by changes in the outer layer
radius, so in order to broadening the wavelength
range with negative permeability the idea of
superposition can be useful as discussed in the
following section.
2.1 Superposition Effect
In this part, we study the influence of outer layer
radius on the broadening band width for negative
refractive index. We use the particles which have the
middle and outer layer with high refractive index.
The results show the change in the outer layer radius
causes the magnetic resonance shifts to longer
wavelength. In fact a particle can be considered as a
cavity which increase in the size of cavity results in
the resonance wavelength shifts to longer
wavelength ranges. So, by using superposition of
nanoparticles with different outer layer radius the
broader wavelength range with negative
permeability, can be achieved. The structure is
shown in Figure 8 and effective parameters are
shown in Figure 9.
The results indicate that the wavelength range
Figure 8: Three layer nanoparticles with different
dimension in vacuum media.f1 and f are fill factors of the
three layer nano particles with r1, r2, r3 radiuses (orange
spheres) and
1
r
,
2
r
,
3
r
radiuses (blue spheres).
Figure 9: The effective optical parameters for a medium
consist of two layer nanoparticles with different outer size
and doped semiconductor.
with negative permittivity is narrow and it leads to a
narrow band negative refractive index and by
applying spherical doped semiconductor
nanoparticles (which electric permittivity follows the
Drude model and have plasma frequency in the
desired wavelength range) in the three layer
nanoparticles (As shown in Figure 10) the
wavelength range with negative refractive index can
be broaden.
Figure 10: Three layer nanoparticles with different
dimension and a single layer doped semiconductor
nanoparticles in vacuum media. f1, f2 and f3 are fill factor
of the three layer nano particles with r1, r2, r3 radiuses
(orange spheres) and
1
r
,
2
r
,
3
r
radiuses (blue spheres)
and the fill factor of doped semiconductor nanoparticles
with a radius (green sphere).The Materials in core, middle
layer and outer layer are Ag, SF5 and Ge respectively.
PHOTOPTICS2015-InternationalConferenceonPhotonics,OpticsandLaserTechnology
116
The effective parameters for this structure are
illustrated in Figure 11.
Figure 11: The effective optical parameters for a medium
consist of two layer nanoparticles with different outer size
and doped semiconductor.
3 CONCLUSIONS
We proposed single, two and three layer spherical
structures with random distribution in order to
achieve broad band effective refractive index in this
paper. The particles are in vacuum media and sizes
of them are in the small nanometer ranges. The
results show Non-overlapping of the scattering
efficiencies for single layer particles (metal and
dielectric nanosphers) and two layer spherical
nanoparticles consist of metal core (Ag) and
dielectric shell (Si) with uniform size causes no
negative
eff
n
in the visible wavelength range. To
solve the problem we proposed the structure with
different size and fill fraction of two layered
nanoparticles. Since, the wavelength range with
negative
eff
n
is narrow in these conditions; we used a
three layer structure. Increase the electrical
permittivity of middle layer and the refractive index
of the outer layer lead to broadening of the negative
eff
n
range. Using different size of three layered
nanoparticles with the same filling fraction to
broadening of the wavelength range with negative
effective permeability was introduced by
superposition idea. In order to increase the negative
refractive index, we applied the doped spherical
semiconductor layer which has proper plasma
frequency and electrical permittivity confirmed by
Drude model.
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