J. Electromagnetic Analysis & Applications, 2010, 2, 654-663
doi:10.4236/jemaa.2010.212086 Published Online December 2010 (http://www.SciRP.org/journal/jemaa)
The Confinement of Electromagnetic Radiation of
Nanoemitters in a Multilayered Microsphere with
Left-Handed Layers
Gennadiy Burlak, Alfredo Díaz-de-Anda
CIICAp, Universidad Autónoma del Estado de Morelos, Cuernavaca, Mexico.
Received October 26th, 2010; revised November 23rd, 2010; accepted November 29th, 2010.
ABSTRACT
The electromagnetic rad iation of nanoemitters pla ced into a multila yered microsphere with dispersive left-han ded (LH)
layers included is studie d num erically . It is fo und that in the frequency r ange w here LH lay ers have a n egative ref raction
index the fiel d frequency spectrum c onsists of a series of narrow and w ell separated reso nances. In the band of such peaks,
the great part of the field energy is loca ted in a LH layer and practically does not leave the microsphere.
Keywords: Left-Handed Material, Coated Microsphere, Alternative Multilayered Stack, Numerical Calculations
1. Introduction
The recently emerging fields of metamaterials and trans-
formation optics promise a family of exciting applications
in nanophotonics with the potential for much faster in-
formation processing. The possibility of creating optical
negative-index metamaterials (NIM) using nanostructured
metal--dielectric composites has triggered intense basic
and applied research [1-7]. In very recent experiments [8]
it has been demonstrated that the incorporation of gain
material in the metamaterial makes it possible to fabricate
an extremely low-loss and an active optical NIM that is
not limited by the inherent loss in its metal constituent.
Since in such materials the electric field, the magnetic
field, and the wave vector of a plane wave form a left-
handed system, they are also called left-handed materials
(LHMs). Investigations of the electromagnetic properties
of LHMs open various promising directions in the elec-
trodynamics of materials with simultaneously significant
electric and magnetic properties, including LHMs.
2. Multilayered Microspheres
One of such directions is the use of microcavities and
microspheres that provides a new view of various effects
and interactions in structured and layered media. Nowa-
days, the basic regime of the operation of open (uncoated)
dielectric microspheres is the whispering gallery mode
(WGM) for a microsphere with a radius of the order
100 1m


or less. The extremely high quality factors
(Q-factors) have already been realized [9].
But since fabricating the coated dielectric spheres of the
submicron sizes, the problem arises of studying the optical
oscillations in microspheres beyond the WGM regime for
harmonics with small spherical numbers. The peculiarity
of such a system (nanoemitter + multilayered microsphere)
consists of the following. The ratio of the typical sizes of a
nanoemitter () to the typical sizes of a microsphere
(100 ) is small
8
10 10
9
10 nm
0.0
9
0nm 1

. However, the range of the
wavelengths of a nanoemitter is comparable to
the width of a layer in the coated microsphere; hence the
retardation effects cannot be neglected already.
600 nm
It is well known that in general, a dielectric sphere has a
complex spectrum of the electromagnetic low quality
factor eigenoscillations because of the energy leakage into
the outer space [10]. The case of the compound structure:
the dielectric sphere coated by an alternative stack, is
richer. The -factor of such oscillations strongly de-
pends on the properties of the stack. It has a large value in
the frequency regions of high reflectivity, and beyond
these regions remains small, [11,12]. The combination
of such factors causes a large variety of optical properties
of microspheres with a multilayer stack. In particular,
such a system can serve as a spherical symmetric photonic
band gap (PGC) structure, which possesses strong selec-
tive transmittance properties [13,14], and can arrange the
nanometer-sized photon emitters. These possibilities al-
Q
Q

The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 655
low us to essentially expand the operational properties of
microspheres with the engineering of nanometer-sized
photon emitters as attractive artificial light sources for
advanced optical technologies. Equally important, this
system can provide a compact and simple building block
for studying the quantum aspects of light. The attachment
of semiconductor nanoclusters onto a spherical microcavity
has already allowed the observation of the Rabi splitting
[15].
Various properties of the electromagnetic waves in
microspheres have been studied both experimentally and
theoretically by a number of authors [16-20]. Recently
various properties of nanoemitters are discussed (see
[21,22] and reference therein).
The incorporation of nanoemitters in the structures with
LH materials, e.g. microspheres, can open new possibili-
ties in the electrodynamics of such systems. As far as the
author is aware, the radiation of active nanosources, placed
in multilayered microspheres with included LH material,
has inadequately been studied yet, though it is a logical
extension of previous works in the case of bare micro-
spheres. Due to recent synthesization of the microspheres
with radial variation in the refractive index [23,24], it is of
interest to study new electromagnetic phenomenon when
nanosources are incorporated not only onto the surface of
a microsphere, but also inside various layers of the spheri-
cal stack consisting of the LH and conventional materials.
The analysis of the refractive index properties of optical
metamaterials, as a function of real and imaginary parts of
dielectric permittivity and magnetic permeability demon-
strated a specific interplay between the resonant response
of constituents of metamaterials that allows efficient dis-
persion management. The use of one-dimensional plane
structures, including dispersiveless LH layers, allows
considerable widening of the band gap of layered struc-
tures [25]. Moreover, similarly constructed spherical stack
allows extremely narrow frequency resonances in a qua-
siperiodic structure [26] Nevertheless, up to now, the case
of a radiated nanosource placed in a LH dispersive spheri-
cal stack containing both conventional and LH materials
has not been studied.
In this paper, we study this situation of a microsphere
coated by alternating layers with the dispersive LH layers
included. We explored both the frequency and radial de-
pendencies of the field radiation in such a frequency area
to answer the question of whether or not the spherical stack
can confine the electromagnetic fields and form the new
photon states. Our approach is based on the dyadic Green’s
function (GF) technique that provides an advanced ap-
proximation for a multilayered microsphere, in particular
in cases where the field is arrested in a LH layer. Such a
numerical approach has allowed us to study the multi-
layered microspheres with any structure of the superficial
layers and to evaluate the total contribution of various
field states in unified framework.
This paper is organized as follows. In Section 2, our
approach and basic equations are formulated for optical
fields in a dielectric microsphere coated by a multilayered
stack. In Section 3, the properties of permittivity, per-
meability, and the refractive index of LH layers are dis-
cussed. In Section 4, we outline the numerical scheme of
applying the GF technique and also discuss our numerical
results for the cavity field states radiated by nanoemitters
placed into such combined multilayered microsphere. In
the last sections, we discuss and summarize our conclu-
sions.
2.1. Basic Equations the Green Function
The spatial scale of the nanoemitter objects (1100nm

)
is at least one order of magnitude smaller than the spatial
scale of microspheres (). Therefore in the
coated microsphere (Figure 1), we can represent the na-
noemitter structure as a point source placed at and
having a dipole moment0. It is well known that the
solution of the wave equation for the radiated electro-
magnetic field due to a general source is [27]
34
1010 nm
d'r
)

E(Jr






,, ,
0
id
V
 
 


ErrGrrJr (1)
where (, , )


Grr
it
e
is the dyadic GF, which depends on the
type of boundary conditions imposed on and con-
tains all the physical information necessary for describing
the multilayered structure (the time dependence is assumed
to be
()Er

). Equation (1) is complemented by the standard
boundary conditions: limitation of the fields in the center
of the microsphere and continuity of the tangential com-
ponents of the fields at the interfaces of layers. We also use
E
Ø
Figure 1. Geometry of a multilayered microsphere. A stack
of multilayers with LH layers included is deposited on the
surface of the microsphere.

The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 656
ditions, where there is only an
Sommerfeld’s radiation conchnique for multilayered microspheres and introduce our
notations. Following the approach [28], we write down
DGF of such a system as follows:
outgoing wave in the external boundary of the microsphere.
In this case, the electromagnetic field E in the coated
structure consists of the sum of the waves radiating in the
surrounding medium and the multiple wave reflections
due to the interfaces between layers. Substituting the
nanoemitter source in the form i

Jd,


0


dd rr
in (1), we obtain
 
0
,,,, ,



r pGrr
Er (2)
where In such a
itter frequency spectrum


22
000
//c  pd situation, the
nanoem is identical to the dyadic
GF spectrum. Thus, the equation of the field generated by
a nanoemitter assumes the form of the GF


,,


Grr
equation, and is given by [27]


2,, ,
c
 

 


Grrr r (3)
where is the point where the field is observed, while
2



r

r
is the nanoemitter (point source) location.
Let us consider the multilayered spherical structure: a
co
2a Multilayered Microsphere
unction te-



,,,,,,,
fs
Vfs

 

GrrG rrGrr
 (4)
where


,,
V


Grr represents the contribution of the
direct waves from the radiation sources in the unbounded
medium, whereas


() ,, fs


Grr describes the contribu-
tion of the multiple reflection and transmission waves due
to the layer interfaces. The dyadic GF (, , )
V


Grr in (4)
is given by
  
3
,
2
1,0 10
3
ˆˆ
,, ,,
4
n
VV
nmq nm
qnm
ik C
k









rr
GrrrrG rr
(5)
with




0
!
21 (2 ),
1!
nm m
nm
n
Cnnn m




 (6)
where the prime denotes the nanoemitter coordinates
(,,r



), and are spherical and azimuthal quantum
numbers, respectively, while
n m
s
k is the wave number of
the medium where the radiated nanoemitters are located.
It is worth noting that due to the dyad , the
ˆˆ
rr

-function
in (5) contributes to the radial (longitudinal) part. Due to
the equality


ˆ
0  
ˆ
ˆ

r, such a singularity does not
contribute to the field (2) for the considered case of a
tangential dipole.
ncentric system of spherical layers contacting with the
sphere (concentric stack) deposited onto the surface of the
microsphere with nanoemitters placed in such a structure
(see Figure 1.). The layers are localized at the distance k R
from the center, where 1kkk dRR

 is the width of
k-th layer.
.2. GF of
a
Let us first specify some details of the Green’s fThe partial dyadic GF ,(, , )
V
qnm


Grr in (5) has a form

  
 


(1)
,, ,,
,(1) (1)
,, ,1,
,,,,
,, ,,,,},
qnmsqnmsqnmsqnm s
V
qnm qnmsqnm s qnmqnms
kk kk
kk kk


(1)






 

MrMrNrNr rr
GrrMrMrNrNr rr


(7)
In (7), vector and represen- and-waves, respectively, where s MNt TE TM
   


cossin cos
cos ,
cos sin sin
e
o
m
dPn
m
nn n
nm
m
kj
krP mjkrm
d



 

 

 
 
Mee (8)
    
    
,
1cos
cos sin
cos cossin
1
cos,
sin cos sin
e
o
m
nn r
nm
m
nnm
n
nn
kjkrP m
kr
drj krdP m
mP m
kr drd







 
 


 
  

Ne
ee  
(9)
where and stand for spherical Bessel and Han-
()
n
jx
ction
()
n
hx
kel funs respectively, and ()
m
n
Px is the associated
Legendre function. For the sake plicity, we use in
(8,9) and further on, the following short notation

, and

of sim
e
eo
nm
onm
kkMMr


,
ee
oo
nm nm
kk

MMr.
The superscripin (7-15) indicates that in (8) and (9),
the spherical Bessel function has to be replaced by
the first-type spheric
t (1)
()
n
jx
, )
al Hankel function(1) ()
n
hx.
The scattering GF () (,
fs


Gr
ris written as




,
,
,, ,,
4
fs f s
nmq nm

,0
s
qe n1 0
m
C
n
ik







 GrrG rr
 (10)
where

and
s
r
f
denote the layers erd poi
source point ae located;
whe the fielnt and
f
s

is the Kr symbol and onecker

The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 657







11
,,,
,,MPNP
fs
qnmnfqf MqnmfN
kk 
 Grr
(11)





11
1, ,
MQNQ
nm
fqnmfM qnmfN
kk 

with
,
 
1, ,
PM M
fs fs
MsMqnmsNsMqnms
A
kB k

  (12)
,



1
1, ,
PN N
fs fs
NsNqnmsNsNqnms
A
kBk

  (13)



1
1, ,
MM
fs fs
MsMqnmsNsMqnms
CkBk

 Q

, (14)
(1)
1, ,
() (),
fs fs
N
sM qnm sNsM qnm s
CkDk

 QN N
where
(15)
1
f
sfs

 ,
f
s

is the Kronecker symbol,
() /c
ss
kn

 , ()()()
ss
n



 is the refraction 
index of the
s
-lay
ndent coefficients
er (see
fs
k
A(22) in next Sec.). Frequency depe (),

(
fs
k
B)

,()
fs
k
C

and
()
fs
k
D

in (12-15) are defined from the above-mentioned
boundary conditions and theave
r in the interface of the stack layers.
2.3. The Recursive Calculations of the Scattering
Coefficients
describe details o
behavio
tions
f the w
The use of standard boundary conditions yields the rela-
between (
fs
A),(), ()
fs fs
kkk
BC

 and ()
fs
k
D

coeffi-
cients that are defined by the reflection k
j
f
R and the trans-
mittance k
j
f
T coend can
be written in the following recursive matform:
1, ,
,1, 0,
sf fsf
kkk fskfs



 JIJ I (16)
fficients (in the layer interface
ri
s) a
x
f
where , (in the spherically N-
laye, see nd
,kMN
red medium
1... 1fN
Figure 1.) a
,,
,
,,
1/ /
,
kkk
fs fsFf
fs f
kk
kk ,
/1/
Ff Ff
k k
fs fsPf PfPf
kk RT T
CD k
TRT AB

 


 





JI (17)
(18)
The explicit expressions for the refle
10 00
, .
00 01






ction k
j
f
R and the
transmittance k
j
f
T coefficients from (17) are wri in the
A
y
e microsphere and the Sommerfeld's radiation
co
tten
ppendix.
The boundarconditions for the limitation of field in the
center of th
ndition at r yield that to (16) one should add the
requirement at ,1
f
N of the form ,, 0
Ns Ns
kk
AB


and 1, 1,ss
kk
CD. It is worth noting that due to the
spherical symmetry of the system th
,
0
,
e coefficients
,
,,
f
sf
kkk
s fs
A
BC and ,
f
s
k
D are functions of n but not ofm.
2.4. Permittivity, Permeability, and Refractive
LH Lrs Index of aye
um
char permittivity
Let us consider a causal linear magnetodielectric medi
acterized by a (relative)(, )

 r and a
(relative) permeability (, )

 r, both of which are spatially
varying, complex functions of frequency satisfying the
relations.
(,)(, ),(,)(, ).



 rrrr (19)
The relation 2(, )(, )(, )n

 rrr
s for the (complex) refractiv
formally offers
two possibilitiee index (,)n

r
 

,/2
,,, ,ne





 r rrr (20)
,i

  r
where


0, ,/2

.



 



rr (21)
her, we follow [29] that allow us to rewrite (20) as Furt
 
 
,,/2
,,,,ne  rrr (22)
i





rr
In the following, we refer to the material of a layer as
being left-handed (or metamaterial) if the real part
re

of its
fractive 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

2
22
1,
e
ee
i




 
P
T
(23)
and a single-resonance permeability

2
22
1
m
mm
 ,
i




 
P
T
(24)
where e

P
, m

P
are the coupling strengths, e

T
, m

T
are
the trnance frequencies, anansverse resod e

, m

arethe
he absorpti peters. Both the permittivity tper-
meability satisfy the Kramers-Kronig relations. Figure 2
where e
on aramand

P
, m

P
are the coupling strengths, e

T
, m

T
are
the transverse resonance frequencies, and e

, m

are the
absorption parameters. Both the permittivity and the
permeability satisfy the Kramers-Kronig relations.
Figure 2 shows the LH refractive index


,n 

r




Re ,Im ,nin

 rr


2
f

 , with the permittiv-
ity ()

 and the permeability ()

 being res
given by (23) and (24)) equency interval from
155 up to 175 THz . In thset, the details of
(, )n
p ivelyect
in the fr
e in THz

r are shown in the frequency interval from 164 THz
up to 175 THz where

Re ,0n  r. It is worth noting
negative real part of the refractive index is typi-
cally observed together wi dispersion, so that
absorption cannot be disregarded in general. However, in a
very recent [8], it was experimentally demonstrated that
the incorporation of gain material in a metamaterial makes
it possible to fabricate an extremely low-loss and active
optical devices. Thus, the original loss-limited negative
that the
th strong

The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 658
(a)
(b)
Figure 2. (Color online.) Real (a) and imaginary (b) parts of
the refractive index

n

2
f

 
/2fof LH layer in the sta
as functions of freck quency

 , with the permittivity


 and the permeability



 being respectively given
by (23) and (24) ;
Tm THz
 0.43 ;

16
10/ 2

 
P
mTm



0.75 ;
P
eTm

  em 7
10 Tm

  (solid lines)]. The val-
the parameters have be be similar to those
in [29]. Insets show
.
refractive index can be drasticaimpro with loss
compenn the visible wavele

en chosen to
ails of Re
in text.
ues of
where the det e detailsn and Se
Im nin the area
Re 0n
lly ved
sation ingth range.
oefficients
3. Numerical Results
Analytical solutions to (16) for scattering c
,,,
,,
f
sfsfs
kkk
A
BCand ,
f
s
k
D fo
spherical stack were derive
r cases of or layers in a
d in [28]t rresponding
1
. Bu
2
co
tr
equations are rather laborious, and thus are hardly suitable
studythe frequencyspecum for the
cases with more than 2layers in the stack already. How-
ever, namely in such a structure, one can expect physically
interesting phenomena due to the wave re-reflections in the
layers of the stack. Similarly to the plane case, such phe-
nomena are most pronounced when the thicknesses of the
alternating spherical layers are approximately equal to
/4
in practice for ing

(quarter-wave layers) [11]. In a general case of al-
ternating layers (having small losses), the equal-
ity 0/
kk
kndl

, (k d is width, 0/kc

) is considered,
t so tha0
/
kk
dlkn  , where l is integer. In this case, the
optical thicknesses of the conventional and LH material
laye rs are the same112 2 dn dn
3.1. The Spherical Stack
We consider a spherical stack with1/
.
00
/2k


of the structure
equality 0/4
kk
dn corresponds to the quarter-wave
uity of the field
we numeore the details of fre-
iation
ternating quarter-wave
case. (Let us remind that the contins in the
layer interfaces requires the continuity of impedances

1/2
/Z  which is positive for both the LH and the
conventional layers.) Since the amplitude of a spherical
electromagnetic wave depends on the distance to the center
of thphere, such a 0/4 approximation is only
asymptotically close to the plane wave case. So such a
structure can be optimized yet with respect to the local
properties of the layers in the stack.
It is worth to identify the nanosource position in a mi-
crosphere. If a nanoemitter is placed close to the center of a
microsphere, the system is nearly spherically symmetric;
the
e micros
n,
(the
refore the modes with small spherical quantum numbers
mainly contribute to the sums in (5,10). This case is close
to a rotational invariant geometry where the dipole mo-
ment orientation does not need to be specified. Therefore,
we draw more attention to a case where the nanoemitter is
placed rather far from the center in one of the layers of the
spherical stack. In such a system, the preferred direction
(center-source) arises, therefore larger numbers of spheri-
cal modes contribute to DGF (5,10). As a result the fre-
quency spectrum of DGF becomes richer but more com-
plicated.
3.2. The Numerical Scheme of Applying the
Dyadic Green’s Function
In this sectiorically expl
quency and radial dependencies of nanoemitter rad
dyadic Green’s function) for al
layers deposited on the surface of a microsphere (Figure 1).
We use the following steps: 1) we solve (16) for
N-layered spherical structure; 2) we insert the calculated
matrices ,,, ,,
f
sfsfs
kkk
A
BC and ,
f
s
k
D into (12-15), and fi-
nally, 3) to obtain the Green's tensor

,,


Grr (4) we
calculate the sums in (5) and (10).
The realization of such a program requires quite inten-
sive computations. Let us outline some detor a nu-
merical solution, we represent (16) in ththe ma-
tri
ails. F
e form of
x equation


,,, 0
ij
Fabcd


,1,2ij with respect to
the unknown boundary amplitudes1,,
s
k
aA1, ,
s
k
bB
,,
N
s
k
cC,
N
s
k
dD. Our algorithm has been constructed
in such a wayh the re
1, 1,,,
,, ,
, that witquired coefficients
s
sNsNs
kkk k
A
BCD and the intermematrices , diate
f
s
(which are necessary for computing the fields in the in-
ternal layers) have been calculated. Having these matrices
calculated, we further make use them in (4-15). Finally,
this allows us to calculate the dyadic Green's tensor in any
point of the spherical structure and for any number of
layers in the stack. Since the complete Green tensor is
known, there is, of course, no obstacle to performing the
calculations for an arbitrary position of the nanoemitters in
a coated microsphere. We note that such a stack in general
k
J
, where
is the reference wavelength. The
0


The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 659
may have an arbitrary or even random structure.
For our numerical approach, the infinite summation in (5)
and (10) has to be truncated. The truncation index
max
nNmust be chosen, so that a further increase of n
does not change the results within a given accuracy. T
is w
various inaccur
he
e
ph
contribute
facto
to c
ysical factor, determining the value of spherical num-
bers max
nN, is the number of the field eigenmodes that
to the spectrum of DGF in the frequency rang
of interest. One can see that the contribution of high-order
spherical modesnin (5) and (10) may be weakened due to
nm
C ). Because of the complexity of the field
spatial structure, it is difficult to establish some analytical
criterion for max
N. In the process of our calculations, we
have increased the quantity max N till the sum in (4) ceases
hange essentially. Usually max N did not exceed
max 25 30N. We have found that such an approxima-
tion remains king well up to very high spherical
numbers (WGM regime). It orth noting that the con-
tinuity of DGF in the layer interfaces is very sensitive to
acies in calculations and can thus serve as a
good criterion for estimating the global accuracy of the
results.
Since nanoemitters (e.g. nanorods) are highly polarized
objects, we pay more attention to the case where the dipole
orientation of the nanoemitter is ˆ
d=d
r (6
wor

, so only the tan-
gential components of the Green's tensor G


contribute.
en us
3.3. The Parameters of Calculations
The following parameters have beed in our calcula-
tions: the geometry of a system is ..,
A
BCBBDwhereCBC
the letters ,,,
A
BCD indicate the materials
1.75 m in the system,
0

 2/ ,171.5
000
K
fcf THz

. A bottom
microsphere has a refraction index

4
41.52 10ni


(
A
, glass, radius 1000 nm). The refof the
LH layer is (22) (see Figure 2 for details) (B,
2 width
300 nm) and 11n (D, surrounding
the realistic layers case we added to each i
n a small
imaginary part, which corresponds to the material dissipa-
tion. Wat e such a
open s), there are losses due to leakage of the
field into the surrounding space.
3.4. The Frequency Spectrum of the GF
The above-written approach has allowed us to explore both
the frequency spectrum and the
raction index
3
10  (2,SiO ,C
space). To co
given by
7nm ), width 43 1.46 3ni
nsider
e note thven in a material lossless case in
system (ystem
radial distribution of the
here. The
3-7
com
po
Green’s function in the multilayered microsp
results of our calculations are shown in Figures
Mainly, we study the case of 7-layered systems
(spherical stack with 7 layers deposited on the surface of
the microsphere). In Figure 3(a), we plot the frequency
dependence of the imaginary part for the tangential
nent of GF

,,Grrf

in the source point


rr

,
(a)
(b)
Figure 3. (Color online.) Frequency spectrum of the imagi-
nary part of the GF (arbitrary units)
in the area of
 
Im, ,WGrrf




0Re n

(see Figure 2(a)), (a) 7 multilays;
(b) 5 multila
itter is located at
r compari
ectrum of the
er
yers.
where the nanoem1480rnm
 in
2
SiO layer. Foson, we also present in Figure 3(b)
the frequency sp

Im G


ure
for a 5-layer
of the spectrumstack. One can see that the struct is
W
similar in both cases.
e observe from Figure 3 (a) that the spectrum of
Im( ) G


consists of peaks with amplitudes; the
highest peaks are located at 164THz and 168THz . They-
have a rather indented
various
formhtion of due to te contribu
se
cal modes
veral close resonances corresponding to various spheri-
. Such peaks have a typical Lorentzian line shape




22
Im /G


 , where

is a detrom the
resonance and
uning f

is the linewidth.
3.5. The Radial Structure of GF in Spherical
Stack
Some interesting special cases can be considered by ana-
lyzing






,Im ,,WrfGrrconstf


 as a func
of r fferent
tion
for a di
f
. In Figure 4, the radial distribu-
tions


corresponding to th
0nm
,Wrfe frequency reso-
nances at 164 THz , 165 THz, and 168 THz (solid lines,
148r

is the position of a nanoem
en
are shown
ffers considerably onance
he va
itter), and the
refrctive index depdencies of the radial stack (dash
lines) . We observe from Figure 4 that the field
structure difor res correspond-
lues of the dispersive refractive index of the
LH layers Re ()nf . Figure 4(a) shows that for
164
a
ing to t
s
f
THz

the field has the form of a resonant oscilla-
tion in the LH layer. For 168
f
THz (see Figure 4(c)),
the field has shape of a single spatial pulse located in the

The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 660
area of the inte th first LH layer with a 2 SiO layer.
165
rface ofe
The case
f
THz Figure 4(b)) is an intermediate
case. We observe that in all shown cases, the field is con-
fined by an LH layer and practically does not leave the
spherical stack.
It is well due to the fluctuation-dissipation
theorem the correlation function of the photon states in the
absorbing environment for temperature T can be written
down with the he
known that
lp of the macroscopic GF as fows [30] llo


2
2coth() Im,,.
2
rr rr
T c




EE G
 (25)
In particular, from (25) one can see that the case rr


yields the inequalitthat correspo
the energy ofgnetic field
y


Im ,0rr G
a fluctuating electroma
nds to


r at E
small dissipation [30]:
 

r
2Im ,GErr , where
(a)
(b)
(c)
Figure 4. (Color online.) Details of radial depe of the
imaginary part of the GF V (abi-
trary units) in the area ofon
frequencies (a)
ndence


,rrf

Im ,WG



0n, (see Figure 2 (a))r , Re
164 ,
f
THz 165,
f
THz (b), and (c)
168
f
THz (solid line). Radius r is normalized on the
radius of the internal microsrce is
placed at 1 /1.48rr (smw shows location of
nanoemitters) in. Distribution
he stack is shown by a dashed line; the stack
consists of 7 layers. Internal microsphere (glass) occupies the
area 1 /rris L H M w i t h

Re 0n, t hen l a yer
2
SiO with

Re 0n, etc. It i s visi ble that for case (a), there
finement of the electromagnetic field already in the
first LHM layer. The spatial variation of W becomes less
with ose of the emitter (field) fsee case(b).
ase (cgle field pulse with large amplitude is
localized in the first LHM layer. The amplitude of the field
decreases appreciably at the increase the rad r.
phere 1 r. The n
all arro
layer of the stack
requency
ius
anosou
,
the secon
, 1- s t l aye r
y a sin
d

Re nalong t
is a con
n increa
For c) on
1
l


,Grr is the GF, which is (,)G

rr in our case. There-
fore, the field structure shown in Figure 4(a,b,c) may be
treated as a strong correlation of the electromagnetic fields
in the vicinity of the LH layer with

Re 0nf. Figure 5
shows that the field structun in Figure 4 for
1480rnm re (show


) does not change considerably at the shift of
the nanoemitter along the layer to 1700rnm
. The
natural question arises, how much cack confine
the field energy for a stack with various numbers of layers?
3.6. GF for Various Numbers of Layers in the
Stack
In Figure 6, the structure
 
Im, ,WGrrf


 for a
n such a sta
numbers, the spectral peak practically does not
considered stack (LH
ayered
er the
system with various layers is shown at frequency168THz .
From Figure 6, we observe that with a change in the
er of lay
variate. This means that for a layers
and conventional layers), the field beyond the first LH
layer is so small that the wave boundary re-reflections do
not affect the structure of the field in the LH layer. This
allows us to conclude that the structure of the spectrum is
defined mainly by the intrinsic properties of the spherical
stack and weakly depends of the nanoemitter location.
3.7. The Structure of the GF in a LH Layer
Furthermore, in some experiments it is important to iden-
tify the spatial distribution of the field (for some reso-
nances) radiated by nanosources located in a multil
microsphere. Therefore, it is of interest to consid
spatial field distribution in a cross-section (,,rconst



)
(a)
(b)
(c)
Figure 5. (Color online.) The same as in Figure 4 but for
position of a nanoemitter farther from the center at
0
1
/1.7rr

. We observe that for such a location of nano-
trength

source, the s ImWG



 is less than for case
8
1
/1.4rr

shown in Figure 4.

The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 661
that contains both the center of the coated microsphere and
the nanoemitter. Taking into account the physical meaning
is of interest to calculate the DGF spectrum
fortwo nanoemitters placed at points r and r in a multi-
osphere for a resona
of (25), it
coated micrnt frncy Figure 7, eque. In
such a distribution is shown for168
f
THz (see Figur e
4(c)).
From (25), one can see that
 
Im, ,Grrf

is propor-
Figure 6. (Color online.) Radial dependencies of imaginary
part of the GF (arbitrary units), in
area of , at frequency


Im, ,WGrrf




0n (see Figure 2(a))Re
168
f
THz
A small

stack. for d if f e re n t n u m be r s of l a shift was added to every
n in the stack.
yers in th e sp h e ri ca l
line to separate the
corresponding close curves. The dashed line shows the radial
dependencies of
Figure 7. (Color online.) Spatial structure W (arbitrary
units), )


,Im(,,WrG rr




4000rnmand 02
in a cross-section
0

 
/4
of a coated microsphere
with

stack for eigenfrequency 168
f
THz
rnm . The othe
o
. A
r pa-
ent of the field in the area of
stack. See details
to thetromagnetic f
r, while
nanoemitter is placed at point 1480
rameters are the same as in Figure 4. One canbserve the
confinem the LH layer of the
in text.
tional energy of the fluctuating elecield
in the position of a nanoemitte

 
, ,Grrf


corresponds to the correlations of the photonic states at r
and r
Im

. We observe from Figure 7 that the spatial distri-
bution




Im, ,Grrf

 has well-defined peaks not only
at rr


, but also at angles i

along the circle where
rr


. This means that a strong correlat
produced by two separated nanoemitters can be arisen at
such resonance. It is worth to note that such a field state is
not a pheral but a state of the macroscopic
me dressed by the electromgnetic field [30]. We also
e from Figure 7 that the field structure inside of a
alternating stack is anisotropic and quite intricate, but the
field amplitudes beyond the coated microsphere are small.
Nevertheless, a spectral detector placed outside the micro-
sphere still enables a noncontact monitoring of the proper-
ties of a nanoemitter located inside a multicoated micro-
sphere.
3.8. Discussion
We observe that the use of the of GF technique allows a
clear description and a self-consistent understanding of the
physics
ion of field states
a
otonic state in gen
dium, a
observ
of the electromagnetic phenomena in the spherical
rs included with respect of the standard
the spherical modes of the system. We
stack with LH laye
decompositions on
have shown that the difference in the location of a nano-
source may lead to the essentially distinct wave pictures.
As a result the frequency spectrum of radiations becomes
rather indented as it is defined by the sum of the various
spherical harmonics with the frequency depending field
amplitudes. Such a spectrum differs sufficiently from the
case of a spherical stack with conventional materials.
Therefore the behavior of the considered compound sys-
tem is instructively to compare with a limit infinite plane
alternating case that has an analytical solution.
The latter stack consists of two different materials with
widths i
a and impedances

1/2
/
ii

 , 1, 2i

, where
both i

and i

are dispersiveless quantities. The first
layer is a conventional dielectric, while the second layer
may be LH material with 0

 and 0

. The dispersion
relation with 2
f



versus the Bloch parameter




for a 1Dcase can be written as (see e.g. [25])






coscos cossin sin,ppqp p    (26)
with




21
1/2//1,q  
1,2

 11 /,pnac




22 1
/,na na1,
1


 where 1

 for the LH ma
and nre the refractive indices. The equati
terial,
(26) is
i a
d for
on
solve

to determine the forbi
negative sign of the refractive index
s of parameter
dden bands. He
is shown explicitly
re the
(by
mean

), so in (26) bo
are all positive
th refractive indices
1, 2. For quarter-wave stack 1n


and
1


(conventional layer) (26) is reduced to the following

The Confinement of Electromagnetic Radiation of Nanoemitters in a Multilayered Microsphere with Left-Handed Layers 662
 
22
coscossin ,pq p  
where 0 /2pff

,00 1/ is the reference frequency
he alternating stack) that has well-known son to
e the properties of the band gaps for a conventional
f
of tlutio
defin
/4

dielectric plan
ver, for 1
e stack, see e.g. Cha. 6 in [31]. p
Howe

 (LH layer) in contrast to 1


case,
ation the equ



22
coscossin ,pqp   (27)
real solution for has a

only for pl

 or 0 2
l
f
flf


(1,2, ...l) in dently of material imped (pa-
rameter q). Teans that in (27) the transm
bands shrink to
depen
his m
zero [This co
odic sphewhextr
ances
ittance
changed for a
e
owever, for
25].
rical stack,
re
nclusion is
re the e
d [26
quasiperi
sidered
mely narrow
transmittance peaks a generate]. H
con here dispersive (with losses) LH case the ref-
erence stack frequency 0171.5
f
THz such an area l
f
with high transmittance is far from the working frequency
range of 175
f
TGz, where Re n is negative, see Fig-
ure 2. Thus, considered here the frequency range corre-
sponds to a forbidden gap of the infinite plane stack with
LH layers included, so t e nt of the electro-
magnetic field by the LH layer can be expected. Further-
more, the e used hes not only to iden-
tify the layers in which such a confinement occurs, but yet
to explore the spatial structure of the field in such an area,
see Figure 4 and Figure 7.
4. Conclusions
We have studied the frequency spectrum and spatial de-
pendences of the electromagnetic field (the dyadic Green's
function), radiated by a na
hconfineme
GF techniqure allow
noemitters (considered in a
ed into the multilayered microsphe
tive quarter-wave spherical stack with
RENCES
trodynamics of Substances with
es of ε and μ,” Sovi et Ph ysi cs
D. Schurig and D. R. Smith, “Controllin
Elctromagnetic Fields,” Science, Vol. 312, No. 5781, 2006,
pp. 1780-1782
-Free
2, 2005, pp. 352-418.
dipole model) plac
coated by the alterna
re
[1
LH layers included. We found that in the frequency range
where LH layers have negative refraction index the fre-
quency spectrum of the field consists of the series of nar-
row and well separated peaks. In the area of such reso-
nances the LH layers can confine the field of nanoemitters;
as a result, the photon states become located in the coated
microsphere and practically do not leak into a surrounding
space. As the width of resonant peaks is rather small, it
allows creating highly polarized nanoemitters oscillating
with high Q factor to actively control the polarization
state of microcavity photons. Such states can be useful not
only for the operational purposes, but also for quantum
information technologies.
5. Acknowledgements
The work of G.B. is partially supported by CONACYT
grant No. 25564.
REFE
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
  

 
111 11
111
,
fff ffff ff
N
Pf
fff fff
ff ff
kJ HJH
TkJH kJH
 




 (34)

I
tr
  


 
111 11
111
,
fff ffff ff
N
Ff
ffffff
ffff
kJ HJH
TkJ HkJ H
 





detailed repre
tation for coefficients reflection if
R and transmittance if T
(,kMN,,iPF) throw the spherical stack is written
as [28],
 
 
k k (35)
with
111
11
,
fffff
1
f
f
fff
M
ff fff
ff ff
kHH kH
RH kJH





 (28)
Pf
f
H
kJ



,
ilni l
J
jkR (36)


1,
ilnil
H
hkR (37)
 
 
111
111
,
ffffff
ff ff
M
Ff
f
ff fff
ff ff
kJJ kJJ
RkJ HkJ H







1|,
il
n
ilk R
dj
Jd







(29)
(38)
 
 
111
111
,
ffffff
f
fff
M
Pf
fff fff
ff ff
kHH kHH
RkJH kJH





 (30)


1
1|
il
n
ilk R
dh
Hd







 
 
111
111
,
ffffff
ffff
N
Ff
f
ff fff
ff ff
kJJ kJJ
RkJ HkJ H




 
 (31)
 

 
111 11
111
,
and stand for s
Hank ectively,
-thterial, while is th
om) wserve th
g
(39)
Here ()
n
jx
el functio
ma
(28-35
hbors
()
n
hx
ns resp
l
R
e ob
pherical Bessel and
is the wave vector in the
i k
e rad
at
i
Fr
nei
ius of the i-th layer.
if the parameters of
, (32)
fff ffff ff
M
Pf
fff fff
ffff
kJ HJ H
TkJH kJH
 





f
and )(1f

layers are cl
 

 
111 11
111
,
fff ffff ff
M
Ff
ffffff
ffff
kJ HJ H
TkJ HkJ H
 





ose then te reflec-
on of
h
ti such ,(ff1)

0
k
if  interface is smallR
(33)