EIT based tunable metal composite spherical nanoparticles
Ghassem Rostami a, Mahmoud Shahabadi a, Ali Afzali-Kusha a, Ali Rostami b,*a School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
b School of Engineering-Emerging Technologies, University of Tabriz, Tabriz 51666, Iran
Received 6 August 2010; received in revised form 22 July 2011; accepted 9 August 2011
Available online 16 September 2011
Abstract
Optical polarizability of a composite metal–dielectric and dielectric–metal spherical nanoparticle is investigated in view of
achieving all-optical tunability. In this work, the Electromagnetically Induced Transparency (EIT) implemented in the shell or core
of the nanoparticle is used. For the proposed nanoparticle, we show that EIT phenomenon can be utilized to tune the resonance
frequency in the frequency response of polarizability. We present a quasi-static analysis for determining polarizability of the
nanoparticle. According to our simulation results, all-optical tunability of polarizability can be achieved for reasonable values of
optical pump power. Also, we demonstrate how the extinction quality factor and electric field distribution can be controlled for the
introduced nanoparticle.
# 2011 Elsevier B.V. All rights reserved.
Keywords: EIT; Metal composite nanoparticles; Quasi-static
www.elsevier.com/locate/photonics
Available online at www.sciencedirect.com
Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111
1. Introduction
Composite nanoparticles are key elements for
nanoscale engineering. They are composed of two or
more homogeneous layers. The optical properties of
these particles, especially their nonlinear behavior, can
be engineered using optical and geometrical parameters
[1]. It is well known that for suitable arrangement of
layers desirable optical characteristics can be achieved.
For example these nanoparticles can be used to
implement as basic elements of nanoscale optical
amplifiers in quantum dot semiconductor optical
amplifiers (QD-SOA) [2]. Also, they can be used to
realize array of nanoparticles as optical and quantum
passive elements such as waveguides [3]. The optical
* Corresponding author.
E-mail addresses: [email protected], [email protected]
(A. Rostami).
1569-4410/$ – see front matter # 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.photonics.2011.08.001
properties of composite nanoparticles have been the
subject of many articles on enhancement of nonlinear
optical properties [4], management of different shapes
and distribution of nanoparticles [5], linear optical
properties of gold nanoparticles [6], effect of incident
frequency changes on nanoparticles’ response [7],
properties of hollow gold nanospheres [8], control of
luminescent host materials by doping of metal nano-
particles [9], investigation of optomechanical properties
of composites [10], electromagnetic field distribution and
manipulation of optical parameters of composite
materials [11,12], temperature dependent optical para-
meters of composite materials [13,14], controlling
efficiency of nonlinear optical phenomenon in composite
materials [15], and polarizability management of nano-
shells [16–20]. Metallic nanoshell structure with a
metallic core is one of the most common structures in
composite nanoparticles. Recently, many researchers
have been focused on the investigation of the linear
optical properties of metal nanoshells [20].
G. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111 103
For implementation of tunable complex frequency
responses, a tunable subsystem is necessary. To this end,
one commonly shapes the geometry of a nanoparticle
[5]. For metal composite nanoparticles, tunability of the
plasmon resonance frequency has been achieved
experimentally using variation of geometrical para-
meters [6–8]. However, all-optical tunability of
composite nanoparticle is not reported so far [14–20].
It is the objective of this work to introduce an optically
tunable nano-structure. Here, Electromagnetically
Induced Transparency (EIT) implemented in spherical
nanoshells is used to achieve tunability. To demonstrate
this property, we make use of a quasi-static analysis to
extract the linear polarizability of spherical nanopar-
ticles composed of various layers. We show how with
the help of EIT the electric field intensity inside the
particle as well as the plasmon resonance frequency can
be tuned over a wide range.
EIT phenomenon is one of the useful optical effects
described by quantum optics. In this phenomenon, a
three-level atom (such as Si/SiO2) becomes transparent
at a certain wavelength (the probe wavelength) once a
strong laser field is applied at the wavelength
corresponding to the second transition of the material.
EIT has been observed in atoms [21], rare-earth-ion
doped crystals [22], semiconductor quantum wells [23],
quantum dots [24] and Bose–Einstein condensates
[25,26]. Basic principles and characteristics of EITwere
introduced in [27]. Some interesting applications of EIT
in optoelectronic systems were introduced in [28,29].
Applications of EIT for design of tunable optical
waveguides were introduced in [30]. In this paper,
exploiting EIT, we propose a nano-structure with
optically tunable characteristics. For this purpose, the
paper is organized as follows.
In Section 2, calculation of polarizability using a
quasi-static approximate analysis is presented. Numer-
ical results generated using this analysis are presented
and described in Section 3. Finally, we bring the paper
to an end with some concluding remarks.
Fig. 1. (a) Structure of the composite spherical nanoparticle and (b) compo
2. Polarizability formulations
In this section, an analysis for calculation of electric
polarizability of a three-layer nanoparticle is presented.
We also take into account EIT properties of the layers
composing this nanoparticle.
2.1. Single spherical metal–dielectric composite
nanoparticles
In this section, electromagnetic properties of a
metal–dielectric and dielectric–metal spherical compo-
site nanoparticle illustrated in Fig. 1(a) will be analyzed
with the help of the method described in [12], and
Chapters 14 [31] and 2 [32]. The particle is exposed to a
linearly polarized incident electric field described by~Eð~r; tÞ ¼ ~Eo expðið~k �~r � vtÞÞ where ~k denotes its
wave-vector. Now, a quasi-static approximation is used.
As shown in Fig. 1, the nano-structure under
investigation is placed in a host material and has a
spherical core of radius a and a spherical shell of radius
b. Following the notations assumed by Neeves and
Birnboim in [12], we denote the complex absolute
permittivity of the core, the shell, and the host material
by eC, eS, and eH, respectively.
Since the outer radius of the nanoparticle under
investigation is much smaller than the wavelength of the
incident plane wave, we can approximately determine
the electric field inside the nanoparticle with the help of
an electrostatic analysis. In this analysis, the nanopar-
ticle is assumed to be in a uniform electric field as
shown in Fig. 1(b). Under this assumption, the electric
field everywhere inside the nanoparticle can be
approximately expressed in terms of the amplitude of
the electric field and the relative permittivity of the
different layers of the nanoparticle. This has been
carried out in [12]. According to electromagnetic
theory, for obtaining response of the material to incident
electric field, at first electric potential and then electric
field must be calculated [12]. Then electric field outside
site spherical nanoparticle exposed to a uniform applied electric field.
G. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111104
the composite spherical nanoparticle resembles the
electric field generated by a dipole whose effective
dipole moment ~p based on classical electromagnetic
theory is given by [12]:
~p ¼ a~EoP; (1-1)
where a denotes the polarizability of the composite
spherical nanoparticles and is expressed by [12]:
a ¼ 4peHb3 eSea � eHeb
eSea þ 2eHeb; (1-2)
where
ea , eC � ð3 � 2 f Þ þ 2 f � eS; (1-3)
eb , eC � f þ eS � ð3 � f Þ; (1-4)
with
f ¼ 1 � a
b
� �3
; (1-5)
in which f can be interpreted as the ratio of the volume
of the shell 4p(b3 � a3)/3 to the volume of composite
particle 4pb3/3 [12]. Using the effective dipole moment
given by (1-1) and (1-2), one can calculate the radiation
intensity of the equivalent dipole Idipole according to the
calculations presented in [31]. The result is
Idipole ¼1
hH
Ed � E�d
¼ nH
n0
� �jaj2ðkÞ6½ðjcj2 þ j’j2Þ � j’j2 sin2ðuÞ
þ 2Reðc � ’Þ cos2ðuÞ� � jE0j2;(1-6)
where k = 2p/l
cðkrÞ ¼ 1
krþ i
ðkrÞ2� 1
ðkrÞ3
!; (1-7a)
fðkrÞ ¼ 1
kr� 3i
ðkrÞ2þ 3
ðkrÞ3
!: (1-7b)
The composite nanoparticle cross sections for
scattering, absorption, and extinction (Csca, Cabs, and
Cext) as well as their quality factors can be obtained
from the relations presented in Chapter 2 of [32].
2.2. Resonance condition
From (1-2), it is obvious that the polarizability of the
composite nanoparticle can be maximized if the
denominator of (1-2) meets the resonance condition,
i.e., if the real part of the denominator vanishes. This
condition, which is also used in [12], results in
ReðeSea þ 2eHebÞ ¼ 0: (1-8)
In terms of real and imaginary part of the permittivity
functions, i.e., eC ¼ e0C þ ie00C; eS ¼ e0S þ ie00S , and
eH ¼ e0H þ ie00H , one can exress (1-8) as
ð3 � 2 f Þðe00Ce0S þ e0Ce00SÞ þ 2 f ð2e0Se00S þ e00Ce
0H þ e0Ce
00HÞ
þ 2ð3 � f Þðe0He00S þ e00He0SÞ ¼ 0: (1-9)
The above condition determines the resonance
frequencies for various compositions of the nanopar-
ticle as described below.
2.2.1. First case
In this case, we assume that the core is made of a
metal whose complex permittivity can be expressed as
eC ¼ 1 �v2
p
vðv þ igÞ ; (1-10)
where vp, and g are plasma frquency and free electron
relaxation frequency, respectively. In other words, we
model the core using a Drude model. The above ex-
pression is used in (1-9) to determine the resonance
frequency. For this purpose, a numerical computation is
required; however, in the special case of loss-less shell
and host medium, i.e., e00S ¼ e00H ¼ 0, the resonance
frequency for the system, that is equivalent to zero
imaginary part, is obtained as follows.
vr ¼ v p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3 � 2 f Þ � e0S þ 2 f � e0H
ð3 � 2 f Þ � e0S þ 2 f � e0H þ 2 f � e0S2
þ2ð3 � f Þ � e0He0S
� g2
v2p
vuuut :
(1-11)
2.2.2. Second case
For metallic shell, the resonance frequencies can be
obtained as follows.
v6 þ b1v4 þ b2v
2 � b3 ¼ 0: (1-12)
In this equation coefficients b1, b2, and b3 defined as
b1 ,
2g2 � ð2 f þ P2Þ � P1 � v2p � 4v2
p � f
P2 þ 2 f
" #;
(1-13a)
;
– Fundamentals and Applications 10 (2012) 102–111 105
b2 ,
g4 � ð2 f þ P2Þ � P1 � v2p � g2 þ 2 f � v4
p
�4 f � v2p � g
P2 þ 2 f
26664
37775;(1-13b)
b3 , �2 f � v4
p � g2
P2 þ 2 f
" #; (1-13c)
P1 , ð3 � 2 f Þ � e0C þ 2ð3 � f Þ � e0H ; (1-14a)
P2 , ð3 � 2 f Þ � e0C þ 2ð3 � f Þ � e0H þ 2 f � e0C � e0H :
(1-14b)
According to Eqs. (1-11) and (1-12) the resonance
frequency can be tuned by applying the pump field.
2.3. Electromagnetically Induced Transparency
(EIT)
In this section, we utilize the EIT phenomenon as an
effective approach for realization of all-optical tuning of
the composite nanoparticles. Atomic systems with three
or more levels which are usually realized by quantum
dots can exhibit EIT. A simple atomic structure for
realization of EIT is illustrated in Fig. 2. In this figure,
we consider Si nanocrystal in SiO2 host to achieve EIT
at frequencies desirable for plasmonic systems. Type
and size of the nanoparticles are determined depending
G. Rostami et al. / Photonics and Nanostructures
Fig. 3. (a) Real and (b) imaginary parts of the susceptibility of the Si/SiO2 EIT
field (Dac = 0). For these figures, the Rabi frequency is set at 0.05 � 101
frequencies are equivalent to an applied field of 3.3 � 103 and 5.28 � 105
Fig. 2. Schematic of EIT atomic system (Si nanocrystal inside SiO2
vac = 5.6019 � 1015 rad/s).
on the plasmonic frequency range of the metal used in
the composite nanoparticles.
Using the well-known quantum mechanical density
matrix method [28–30] and assuming a three-level
atomic system like the Si/SiO2 nanoparticles proposed
above, we can obtain the optical susceptibility of the
EIT medium as [28]:
x ¼ x0 þ ix00; (1-15a)
x0 ¼ NajPabj2De0�hZ
g3ðg1 þ g3Þ þ D2 � g1g3 �V2
4
� �� �;
(1-15b)
x00 ¼ NajPabj2
e0�hZD2ðg1 þ g3Þ � g3 D2 � g1g3 �
V2
4
� �� �(1-15c)
Z ¼ D2ðg1 þ g3Þ2 þ D2 � g1g3 �V2
4
� �2
; (1-15d)
where D = vab = vS is the incident signal detuning.
Moreover, we assumed vac = vC. Here, g1 and g3 are
the decay rates for off-diagonal density matrix elements
related to the transitions a–b and a–c, respectively; Na is
the density of nanoparticles; and Pab and V ¼ PabE=�hare the dipole matrix element between a and b and the
Rabi frequency, respectively. In this relation, E is the
amplitude of the pump field. Diagrams of Fig. 3 which
are calculated using (1-15) show the real and imaginary
medium versus probe frequency (vS) (rad/s) in the case of tuned pump0 rad/s (without pump) and 8 � 1010 (with pump) rad/s. These Rabi
V/m, respectively.
host, Eab = 3.74 eV and Eac = 3.7 eV) (vab = 5.6628 � 1015 rad/s,
G. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111106
Fig. 4. (a) Refractive index (refractive index of background medium n0 is 1.48). (b) Absorption coefficient of EIT medium versus pump field (pump
detuning is �10�6vS).
parts of the complex susceptibility. From the complex
susceptibility x, one can compute the refractive index
n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffin2
0 þ xp
, where n0 is the background effective
refractive index. The computation results are given in
Fig. 4. Calculations of Fig. 4 were performed for a
background refractive index of n0 = 1.48, Na = 1023/m3,
Pab = e � 10�10 cm, y1 = 102 GHz, g3 = 100 MHz.
According to Fig. 4, we can tune the medium loss
and the value of the resonance frequency by the ampli-
tude of the pump signal. Note that for the proposed Si/
SiO2 nanocrystal and the parameters assumed above,
the refractive index can be varied from 1.425 to 1.525
for a pump electric field of 2.2 V/m. In the proposed
composite nanostructure, we can employ this 3-level
atomic system in the shell in order to control the
refractive index and absorption coefficient of this layer
as a function of the applied pump signal.
3. Results and discussion
In this section, the computed characteristics of the
proposed composite nanostructure are presented. We
will investigate effect applied pump field on polariz-
ability, resonance frequency, extinction quality factor,
and radiation properties of the composite nanoparticles.
First, we assume that the core of the composite
nanoparticles is made of gold with a radius of 40 nm and
that its shell which is capable of showing EIT has a
thickness of 10 nm which means that the total radius of
the composite nanoparticles amounts to 50 nm. Note
that we have assumed vp = 1.3 � 1016 rad/s and
g = 1.0753 � 1014 rad/s for the core material. In
Fig. 5, the absolute value and phase of the polarizability
of these composite nanoparticles are depicted as a
function of the applied pump field. For these diagrams,
the frequency of operation is set at 5.662797 � 1015 rad/
s. The diagrams are shown for various values of the host
refractive index.
In Fig. 6, we demonstrate the impact of the pump
field on the frequency response of polarizability. The
maximum value of polarizability corresponds to zero
detuning of the incident signal. For increasing pump
field, the refractive index of the shell is also decreased
so the equivalent optical path in the shell is reduced.
This results in a blue shift of the polarizability as can be
seen in Fig. 6(a). Further increase of the pump field will
increase the shell refractive index as a result of which a
red shift of the polarizability is observed. Also the
polarizability phase is influenced by the pump field as
can be seen in Fig. 6(b).
Fig. 7(a) illustrates the extinction quality factor of the
proposed composite nanostructure as a function of both
the host refractive index and applied pump field whereas
Fig. 7(b) is the frequency response of the extinction
quality factor for different values of the pump field.
According to Fig. 7, the extinction quality factor behaves
similar to the polarizability. Also, because of small
resonance frequency and large incident frequency the
input energy cannot absorbed by the cavity made by
nanoparticle and then extinction quality factor is large
enough.
Now, we assume that the core of the proposed
nanostructure is made of a medium capable of showing
EIT whereas the shell of the nanostructure is a metal.
The radius of the core and the thickness of the metallic
shell are 40 nm and 10 nm, respectively. Fig. 8
illustrates the calculated extinction quality factor for
this case. According to this figure, increasing the host
refractive index enhances optical coupling into the core
through metallic shell, consequently the extinction
quality factor increases. Note that the quality factor of
the present configuration of the nanoparticles is smaller
than that of the previous case. Fig. 8(a) also depicts the
G. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111 107
Fig. 5. (a) Absolute value and (b) phase (b) of the polarizability of proposed composite spherical nanoparticles with metal (Au) core (radius = 40 nm
and dielectric shell with radius = 50 nm) (vp = 1.3 � 1016 rad/s, g = 1.0753 � 1014 rad/s) for various values of pump field and host refractive index
for incident frequency v = 5.662797 � 1015 rad/s [33].
impact of the pump field on the extinction quality factor
of the nanoparticles. In Fig. 8(b), the extinction quality
factor is shown as a function of frequency. As can be
seen, the resonance frequency increases by increasing
pump field. Furthermore, we observed that additional
increase of the pump field reduces the resonance
frequency. In comparison with the previous case, the
present configuration of the nanoparticles exhibits a
larger amount of frequency shift for the same pump
field. Based on displacement of the resonance frequency
small extinction quality factor can be described too. In
this case the resonance frequency increased because of
the selected geometry, so, the coupling of incident
frequency that is near to the resonance frequency is
increased too. So, the extinction quality factor must be
decreased that is illustrated in this figure.
G. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111108
Fig. 6. (a) Magnitude and (b) phase of the polarizability as a function of frequency when the core is made of metal. Metal core (Au) (radius = 40 nm
and dielectric shell with radius = 50 nm) (vp = 1.3 � 1016 rad/s, g = 1.0753 � 1014 rad/s), for host refractive index n = 1.44 and incident frequency
v = 5.662797 � 1015 rad/s.
Fig. 7. Extinction quality factor of the proposed composite nanoparticles versus (a) pump field and host refractive index and (b) frequency. Metal
core (Au) (radius = 40 nm and dielectric shell with radius = 50 nm) (vp = 1.3 � 1016 rad/s, g = 1.0753 � 1014 rad/s).
G. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111 109
Fig. 8. Extinction quality factor of the composite nanoparticles with EIT core and metallic shell versus (a) pump field and host refractive index and
(b) frequency dielectric core and metal shell (Au) (core radius = 40 nm and shell radius = 50 nm) (vp = 1.3 � 1016 rad/s, g = 1.0753 � 1014 rad/s).
For the present configuration of the nanoparticles,
Fig. 9(a) and (b) represents the magnitude and phase of
particle polarizability as a function of frequency and
applied pump field. These diagrams can be compared
with their counterparts, i.e., Fig. 6(a) and (b).
Fig. 9. Absolute value (a) and phase (b) of polarizab
In this section, we evaluated the proposed core–shell
structure as well as EIT for management of optical
properties of composite materials implemented by these
nanoparticles. We illustrated that by numerical simula-
tion considering EIT lets us tune optical properties of
ility versus frequency for different pump fields.
G. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 102–111110
medium implemented by these nanoparticles. We
considered two cases including core metal and shell
EIT and inverse. Our simulation show that the first case
lets the applied pump field change effectively the
refractive index of the medium than the second case.
This comes back to inherent properties of metals. In
shell metal pump light absorbed more and cannot
effectively change the refractive index.
4. Conclusion
In this paper, EIT phenomenon has been utilized to
introduce composite metal nanoparticles whose polar-
izability can be controlled using an optical pump signal.
We have shown that EIT can be used to tune optical
properties of the nanoparticles such as its extinction
quality factor. Also the resonance frequency of
polarizability can be controlled by the applied pump
signal. The first configuration investigated had a
metallic core and an EIT shell whereas the second
case has an EIT core and a metallic shell. The presented
computational results show that the first configuration
has a more acceptable performance as far as all-optical
tunability is concerned.
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