The Field Confinement, Narrow Transmission Resonances, and Green Function of a Multilayered Microsphere with Metamaterial Defects

Abstract

We numerically investigate the optical transmission through a compound spherical stack with conventional and metamaterial (MM) layers and also embedded MM defect. A formation of extremely narrow resonant peak with nearly complete transmittance in area of a band gap is found. We demonstrate that photon fields of certain frequencies can be strongly confined by a left-handed (LH) defect. The influence of a random deviation in the width of compound spherical layers as well the transit to the whispering gallery mode (WGM) is also discussed.

Full text

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Hindawi Publishing Corporation
Journal of Atomic, Molecular, and Optical Physics
Volume 2011, Article ID 217020, 13 pages
doi:10.1155/2011/217020
Research Article
TheFieldConfinement,NarrowTransmissionResonances,
andGreenFunctionof a MultilayeredMicrospherewith
Metamaterial Defects
GennadiyBurlakand A.D´ ıaz-de-Anda
Centro de Investigaci´ on en Ingenier´ ıa y Ciencias Aplicadas, Universidad Aut´ onoma del Estado de Morelos,
62209 Cuernavaca, MOR, Mexico
Correspondence should be addressed to Gennadiy Burlak, gburlak@uaem.mx
Received 30 June 2011; Revised 3 August 2011; Accepted 6 August 2011
Academic Editor: Alan Migdall
Copyright © 2011 G. Burlak and A. D´ ıaz-de-Anda. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We numerically investigate the optical transmission through a compound spherical stack with conventional and metamaterial
(MM) layers and also embedded MM defect. A formation of extremely narrow resonant peak with nearly complete transmittance
in area of a band gap is found. We demonstrate that photon fields of certain frequencies can be strongly confined by a left-handed
(LH) defect. The influence of a random deviation in the width of compound spherical layers as well the transit to the whispering
gallery mode (WGM) is also discussed.
1.Introduction
Controlling the electromagnetic properties of materials,
going beyond the limit that is attainable with naturally
existing substances, has become a reality with the advent
of metamaterials [1, 2]. The range of various structured
artificial structures has promised a vast variety of otherwise
unexpected physical phenomena [3–5] among which the
experimental realization of a negative refractive index has
been one of the main advantage.
A typical metamaterial combines continuous metal films
(about 50nm) with nanostrip magnetic resonators that
finally yield a negative-index material. The film can be
formed with a mixture of dielectric (e.g., silica) and metal,
such as silver or gold, forming a semicontinuous metal
film. Such structures are fabricated using techniques as
the evaporation of a metal onto a dielectric substrate (see
more details in review [5]). Preparation of metamaterial
structure in the optical range still is an advanced task. In
recent experiment [6] it has been demonstrated that the
incorporation of the gain material (rhodamine 800) in the
metamaterial makes it possible to fabricate an extremely
low-loss and an active optical negative-index metamaterial
(NIM) that is not limited by the inherent loss in its metal
constituent. In this experiment the optical (NIM) structure
(silverlayers+rhodamine800)isthefishnetwithperiodicity
about 300nm. Epoxy doped with rhodamine 800 (Rh800)
and dye is used as a gain medium. The loss compensation
mechanism in the sample is straightforward to understand.
When Rh800 is excited by a pump pulse with sufficiently
high power, a population inversion is formed inside the
dye molecules. This provides amplification for a properly
delayed probe pulse whose wavelength is coincident with the
stimulated emission wavelength of the dye molecules (see
details in [6]).
Natural extension of such directions is the analysis of
a compacted spherical multilayered system with embedded
metamaterial layers. It is well known that except for the
whispering gallery mode (WGM) regime [7–10], a bare
dielectric sphere has a complex spectrum of the electromag-
netic low-quality (Q factor) eigenoscillations because of the
energy leakage into the outer space [11]. The case of the
compound structure, when the dielectric sphere is coated by
an alternative stack, is much richer. The Q-factor of optical

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2 Journal of Atomic, Molecular, and Optical Physics
oscillations has a large value in the frequency regions of weak
transmittance and, beyond these regions Q, remains small
[12–14]. This gives rise to a large variety of optical properties
of microspheres coated with a multilayer stack. Such a
system can serve as a spherical symmetric photonic band
gap structure, which possesses strong selective transmittance
properties [15, 16], with the incorporated nanometer-sized
photon emitters. These possibilities essentially allow the
expansion of the operational properties of microspheres
with attractive artificial light sources for advanced optical
technologies. The energy confinement into small volumes
has applications in many fields such as photonics and
quantum electrodynamics. In this sense, the microstructures
that have the greatest ability to store energy for long periods
of time are the dielectric spherical resonators. Recently, it has
been feasible to construct such a microsphere accurately, and
the parameters may be precisely controlled and measured
[17]. New materials are used to extend the optical properties
of layered microspheres [18].
In studying such an effect in multilayered microspheres,
the following question emerges: how does such an effect will
be changed at a deviation from strict periodicity of a stack?
Generally such a case can be interpreted as a translation
symmetry breaking due to placing of defect layers in the
periodic stack. The question rises whether it is possible
to obtain narrow transmittance peaks in an alternating
spherical stack with inserting a defect layer?
A translation symmetry breaking can be achieved in
a quasiperiodic system. In [19–21] optical properties of
layered microsphere with a dielectric stack, in which optical
layersareconstructedfollowingtheFibonaccisequence,were
investigated. Such a quasiperiodic structure serves as an
intermediate case between a strictly periodic and cleanly
casual sequence of alternative layers in a stack. It was found
that when the number of layers (Fibonacci order) increases,
the structure of the spectrum acquires a fractal form. If
the metamaterial layers (with negative refraction index) are
included in such a stack then the width of the resonant peaks
in the frequency spectrum becomes extremely narrow [22].
It is found the formation and coupling of defect modes in
two-dimensional photonic crystal (PC) band gaps associated
with degenerate edges [23]. In very recent article [24] it has
been shown that by varying the radii and permittivity of
the defects, total transmission or reflection of the impinging
electromagnetic wave can be achieved.
Inthispaper,westudythephotontransmissions through
the compound spherical stack containing conventional and
metamaterial (MM) layers and also an embedded MM
defect that are transparent and can bend light in opposite
directions. A narrow well-separated resonant peak with
nearly complete transmittance in a bandgap arises when
such a MM defect layer is embedded to the center of the
spherical stack. We demonstrate that photon fields of certain
frequencies can be strongly confined by a left-handed (LH)
defect. The influence of a random deviation in the width of
compound spherical layers as well the transit to whispering-
gallery mode (WGM) is also discussed.
2.Basic Equation
The geometry of the multilayered microsphere with the
embedded defect is depicted in Figure 1.
Despite that the peculiarities of the propagation of opti-
cal waves in 1D plane systems with defects are investigated
in a number of articles the results of such studies cannot
be applied directly to a spherical stack for the following
reasons. (i) In a spherical case, the transfer matrices Mj
have the determinant depending on number j of a layer
as det(Mj)=(rj+1/rj)2[14] (is not unimodular), where ri+1
and rj are radii of the external and internal boundaries
of the layer, respectively. Such a structure of the transfer
matrix is physically caused by preservation of the energy
flux in a radial direction in a solid spherical angle. (ii) In
a spherical stack, the transfer matrix Mjis quite involved
since it is written through the complex Hankel functions.
Physically,itisduetothepreferentialroleofthepole(center)
of a microsphere that breaks the translational symmetry
in such a system. Furthermore, in a spherical geometry,
the transmittance coefficient T depends on the spherical
quantum number m (angular momentum), for great m ? 1
that leads to a whispering gallery mode (WGM) regime with
practically zero transmittance in a frequency range.
In order to study the optical properties of a spherical
stack with embedded defect layer, let us first formulate the
transfer matrix method exploited in spherical multilayered
geometry. Various approaches were proposed [14, 25–28]
(see [29, 30] and references therein). Here we follow [25].
In a multilayered microsphere, the set of the Maxwell
equation is
∇ ×? H = iωε0ε(ω)? E,
where? E and? B are electric and magnetic fields and ε(ω)
is a dielectric permittivity of a layer. We use the complex
exponential multiplier in the form exp(iωt). Equation (1)
in the spherical coordinate frame (r,θ,ϕ) usually reduces
to the Helmholtz equation for a scalar function called the
Debye potential Π(r,θ,ϕ) [31]. Since our calculations did
not register significant difference between TM and TE wave
cases further we will concentrate mainly in the TM case. The
equation for the radial part of the Debye-potential Π = Π(r)
in a layer can be readily obtained from (1) and is given by
?
where k0 = ω/c. Equation (2) is easily solved in terms of
spherical Hankel functions. In each layer of the stack, we use
the next matrix presentation for the fields
⎡
Eθ
b
∇ ×? E = −iω? B,
(1)
d2Π
dr2+
ε(ω)k2
0−l(l +1)
r2
?
Π = 0,
(2)
? u =
⎣Hφ
⎤
⎦= D ·
⎡
⎣a
⎤
⎦= D · ? q,
? q =
⎡
⎣a
b
⎤
⎦, (3)
where a and b are arbitrary constants, and matrix D = D(y)
is given by
⎡
G(2)
m
D =
⎣inP(2)
m
?y?eiyinP(1)
m
?y?e−iy
?y?eiy
G(1)
m
?y?e−iy
⎤
⎦.
(4)

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Journal of Atomic, Molecular, and Optical Physics3
SiO2
SiO2
LH
SiO2
LH
SiO2
LH
LH
LH
X
Media 0
Layer 1
TE-case
Hφ
Er
Eθ
Hθ
Hr
Eφ
TM-case
Media N
Layer k
Layer N −1
rN
rk
Figure 1: Geometry of the system. Multilayered microsphere (LH + SiO2) with a defect LH layer (indicated as X) in a spherical stack.
Here, n
a particular layer (that can be positive or negative for
metamaterial layers); P(1,2)
m
(y) is the rational part of Hankel
spherical functions h(1,2)
rational part of derivative of Hankel spherical functions
(∂/∂y)h(1,2)
harmonic, y =ωn(ω)r/c. The recursive relations for calcula-
tions of P(1,2)
m
(y) and G(1,2)
provide the matrix D, as well as vectors ? q and ? u by indices
according to the number of layers in the stack. As the vector
? q in each layer is constant, for any two points r1and r2of a
k-layer we obtain from (3) the following:
= n(ω)
=
?ε(ω) is the refractive index of
m (y)=P(1,2)
m
(y)e±iy; G(1,2)
m (y) is the
m (y)=G(1,2)
m (y)e±iy; m is the number of a spherical
m (y) are given in [26]. Below, we
? qk= D−1
k(r1) ·? uk(r1) = D−1
k(r2) ·? uk(r2).
(5)
To start the calculation, we assume that the points r1and r2
belong to the boundaries of layers; r2 = r1+ d1, d1is the
thickness of the layer. At the boundary between layers, k and
k + 1, the continuity of fields gives ? uk(r2) = ? uk+1(r2). With
the latter (5) can be rewritten as
? uk(r1) = Dk(r1) ·D−1
k(r2) ·? uk+1(r2) ≡ Mk·? uk+1(r2).
(6)
This relation can be extended as follows. Let us start from the
bottom layer of the stack with the number k = 1. Then from
(6) we have
? u1 ≡ ? u1(r1) = M1·? u2= M1·M2·? u3
= M1·M2···· MN−1·? uN≡M ·? uN= M ·? uN(rN),
(7)
where
M =
N−1
?
k=1
Mk
(8)
is the transfer matrix between inner and outer layers in the
spherical stack.
In the simplest case of the spherical mode m = 1, the
elements of matrix Mkin (8) are
?y2− y1+ y1y2
y2
?−1+ y2
y2
M11=
2
?cos(Φk)
?sin(Φk)
1y2
+
2− y1y2
1y2
,

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4 Journal of Atomic, Molecular, and Optical Physics
M12= nki?y2− y1
M21=−i?1+ y2y1
?cos(Φk)
y12
??−y2+ y1
y2y3
−nki?1 + y2y1
?cosΦ1
?sin(Φk)
y12
,
1nk
,
−i
?−y2+ y1+ y2
?1 + y1y2− y2
?1 − y2
2+ y2y1− y2
1+ y2
1y2nk
?cos(Φk)
?sin(Φk)
1y2
2
?sin(Φk)
y3
,
M22=
1y2
y3
1
+
1
y3
1
,
(9)
where y1 ≡ yk,k, y2 ≡ yk,k+1, yk,l = k0nkrl, y2− y1 =
Φk = k0nkdk; dkis the thickness of the kth layer. From (9),
one can see that in the spherical case, the transition matrix
Mkdepends explicitly on the distance to the center of the
microsphere.
Note that in the case where rk ? dk or y1 ∼ y2 ? 1
in (8), one gets the well-known plane case expression for Mk
[32]:
⎡
⎣
Only if such an approximation is fulfilled, the matrix
elements in (10) are independent of the absolute values of
the coordinates and dependent only on the thicknesses of
layers dk through value y2− y1 = Φk. This means that in
general, the local properties of electromagnetic oscillations
depend on the place of a layer with respect to the center of
the spherical cavity.
Equation (7) connects the values of the fields on both
external and internal boundaries of the spherical stack.
Therefore, solving the Maxwell equations is completely
equivalenttocalculatingproductsofthetransfermatricesM.
We have used the Sommerfeld radiation conditions, where
there is only an outgoing wave at the external boundary that
gives ? qNin (3) bN = 0. As a result, the amplitudes of waves
in the internal substrate become
⎡
R
Mk=
⎢
cosΦk
−i1
−inksinΦk
cosΦk
nksinΦk
⎤
⎦.
⎥
(10)
? u0= a0·
⎣1
⎤
⎦= D−1
0 ·M ·DN· ? qN= a0σ ·Q ·
⎡
⎣T
0
⎤
⎦,
(11)
where the matrix Q = D−1
coefficient; T = aN/σa0is a transmittance coefficient; σ =
(n0/nN)1/2. After solving (11), one can easily obtain R and T
in the next form:
0 ·M ·DN, R = b0/a0is a reflection
R =Q21(ω)
Q11(ω),
T =
1
σQ11(ω).
(12)
In (12), two equations interrelate three variables: R, T, and
frequency ω. Defining ω, one can calculate the frequency
dependence of R(ω), T(ω) for the stack.
We can use (3)–(12) to calculate the reflectance, trans-
mittance, eigenfrequencies, and eigenfields in an arbitrary
spherical stack. Since no specific properties of a spherical
stack were used, one can conclude that (3)–(11) are valid for
any structure of the spherical stack. Now, we exploit such
a technique for studying a combined alternating spherical
stack with conventional and metamaterial layers.
Further we consider metamaterial layers characterized by
a (relative) permittivity ε(r,ω) and a (relative) permeability
μ(r,ω), both of which are spatially varying, complex func-
tions of frequency satisfying the relations [33, 34]
n(r,ω) =
???ε(r,ω)μ(r,ω)??ei[φε(r,ω)+φμ(r,ω)]/2.
In the following, we refer to the material of a layer as being
left handed (LH or metamaterial) if the real part of its
refractive index is negative. In order to allow a dependence
on the frequency of the refractive index, let us restrict our
attention to a single-resonance permittivity
(13)
ε(ω) = 1+
ω2
Pe
ω2
Te−ω2−iωγe
(14)
and a single-resonance permeability
μ(ω) = 1+
ω2
Pm
ω2
Tm−ω2−iωγm,
(15)
where ωPe, ωPm are the coupling strengths, ωTe, ωTm are
the transverse resonance frequencies, and γe, γm are the
absorption parameters. In our case we used the following
typical values: ωTm/2π = fTm= 159.2THz, γm/2π = fγ =
0.001592THz, fTe = 163.9THz, fPe = 119.4THz, fPm =
68.44THz.
In the optical range the performance of all NIM appli-
cations is significantly limited by the inherent and strong
energy dissipation in metals, especially in the near-infrared
and visible wavelength ranges [5]. In our study we have
explored mainly the case where the loss compensation can
be attained. It is worth noting that in general case both
dependencies ε(ω) and μ(ω) may be more complicated than
used in (14) and (15). However, since the metamaterials
are artificial structures there are no strict expressions for
ε(ω) and μ(ω) derived from the first principles up to now.
Some analytical approximations were obtained numerically
in order to fit the experiment data smoothly. In this paper
we use a single (or double) resonance structure for ε(ω)
and μ(ω) similarly that was detected in [6, Figure 4(b)]. On
the other hand we use the relations (14)-(15) that already
were used in a number of articles [33, 35, 36] and are in
agreement with the principle of causality as expressed in the
Kramers-Kronig relations (see recent discussion in [37, 38]).
We believe that such an approach cannot lead to significant
errors. This allows the use in (14)-(15) of the dissipation
parameter γ with small value γ/2π = 0.0016THz.
Figure 2(b) shows the LH refractive index n(r,ω) =
Ren(r,ω)+iImn(r,ω) (ω = 2π f , with the permittivity ε(ω)
and the permeability μ(ω) being, resp., given by (14) and
(15)). In the inset, the details of n(r,ω) are shown in the

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Journal of Atomic, Molecular, and Optical Physics5
180 200220
240
0
2
4
n (ω)
f (THz)
(a)
n (ω)
−60
−40
−20
0
180200 220
240
f (THz)
200
250
−6
−4
−2
0
2
(b)
180 200 220
240
0
0.5
1
f (THz)
T
(c)
180
200220
240
f (THz)
0
0.5
1
T
(d)
Figure2:(Coloronline)ComparisonofthespectralpropertiesofthetransmissioncoefficientsT for15-layeredalternatingstackscontaining
conventional (Si + SiO2) and (LH + SiO2) materials correspondingly (spherical number m = 1): (a) the refractive index n(ω) (f = ω/2π)
for the Si (dotted line) and SiO2(solid line); (b) the real (dotted line) and imaginary part (solid line) of the refraction coefficient n(ω) for
LH metamaterial. The spectrum T(ω) is shown in (c) for conventional stack, while (d) shows T(ω) for a combined stack when Si is replaced
with LH layers. See details in text.
frequency interval from 164THz where Ren(r,ω) < 0. It is
worthnotingthatthenegativerealpartoftherefractiveindex
is typically observed together with strong dispersion, so that
absorption cannot be disregarded in general. However, in a
very recent work [6], it was experimentally demonstrated
that the incorporation of gain material in a metamaterial
makes it possible to fabricate extremely low-loss and active
optical devices. Thus, the original loss-limited negative
refractive index can be drastically improved with the loss
compensation in the visible wavelength range.
It is worth noting that the features of the use of tensorial
or scalar ε(ω) and μ(ω) expressions for metamaterials
in spherical geometry (as well for other types of curved
boundaries) up to now are an open problem. Synthesized
technologies are mainly elaborated for preparation of hybrid
spherical layered structures (see [18] and references therein).
To the best of our knowledge the separation of component
ε(ω) and μ(ω) for spherical interfaces is not studied in
details. However, as far as the ratio of typical sizes of the
metamaterial structures (period of fishnet) and microsphere
is small (about 0.1 an less) and to seek for simplicity we use
the average values ε(ω) and μ(ω) that are scalar in the first
approximation.
3.NumericalResults
Our numerical results are shown in Figures (2)–(10). The
following parameters have been used in our calculations: the
geometry of system is A,{B,C},...,G,...{B,C},D, where