# The Subwavelength Optical Field Confinement in  a Multilayered Microsphere with Quasiperiodic Spherical Stack

Authors: Gennadiy N. Burlak; A. Diaz-de-Anda
Journal: Advances in OptoElectronics
 (2008)

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## Abstract

We study the frequency spectrum of nanoemitters
placed in a microsphere with a quasiperiodic subwavelength
spherical stack. The spectral evolution of transmittancy at the
change of thickness of two-layer blocks, constructed following the
Fibonacci sequence, is investigated. When the number of layers
(Fibonacci order) increases, the structure of spectrum acquires a
fractal form. Our calculations show the radiation confinement and
gigantic field enhancement, when the ratio of layers’ widths in
twolayer blocks of the stack is close to the golden mean
value.


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## Body

## 1. Introduction

The use of microcavities and microspheres in advanced
optoelectronics has provided a new  view of various effects and interactions in
highly integrated, functional photonic devices. A main fundamental question in
this area is how to drastically increase the spectral optical field strength,
using artificially produced alternating layers on a surface of microsphere.
Nowadays the basic regime of the operation for bare (uncoated) dielectric
microspheres is the whispering gallery mode (WGM). The extremely high-quality
factors (Q-factors) ∼108÷109 have been realized [1–3]. But since fabricating the
coated dielectric spheres of the submicron sizes, the problem arises to study
the optical oscillations in microspheres beyond the WGM regime for harmonics
with small spherical numbers.It is well known that a bare dielectric sphere has a
complex spectrum of the electromagnetic low-quality (Q factor) eigenoscillations because of the
energy leakage into the outer space [4]. The case of the compound structure, when the
dielectric sphere is coated by an alternative stack, is much richer. The Q-factor of optical oscillations has a large
value in the frequency regions of weak transmittancy, and beyond these regions Q remains small [5–8]. 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 [9, 10], with incorporating the nanometer-sized photon
emitters. These possibilities allow to expand essentially the operational properties
of microspheres with attractive artificial light sources for advanced optical
technologies. It is now feasible to construct such a microsphere accurately,
and the parameters may be precisely controlled and measured (see [11] and reference therein).
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 already has allowed the observation
of the vacuum Rabi splitting [12].Among the various quasicrystals, the Fibonacci 1D and Penrose 2D structures have been the subject of an
extensive theoretical and experimental efforts [13–19]. However, though the plane Fibonacci layers are
studied well enough, the properties of quasiperiodic spherical structures were
not studied sufficiently yet.The important optical property of a periodic
alternating spherical stack is a possibility to confine the optical radiation.
However, is periodicity necessary for such resonant optical effects?In order to answer this question, we have studied the
optical radiation of a nanosource (nanometer-sized light source), placed into a
microsphere coated by a quasiperiodic multilayered structure (stack) constructed
following the Fibonacci sequence. In such a system the 1D (radial depending) theory is strictly valid.
Such structures are called quasiperiodic, and are lying outside the constraints
of periodicity. One of the main properties of such a stack is re-reflection of
the electromagnetic waves from the interfaces of the layers, that results in
the collective wave contributions. The collective optical effects in a
quasiperiodic spherical stacks are appreciated only if number of layers in the
stack is large enough. In this case, various approximations based on the
decomposition of field states in the partial spherical modes have insufficient
accuracy, so the deeper insight requires more advanced approaches. Our approach
is based on dyadic Green's function technique [20], that provides an advanced
approximation for a multilayered microsphere with nanoemitters [21]. We have applied this
approach to a quasiperiodic spherical stack and found the substantially
enhanced optical resonances (Green function strength), when the ratio of
layers' widths in two-layer blocks in stack (quasiperiodicity parameter) is
close to the golden mean value. As far as the authors are aware, the optical fields
of nanoemitters placed in a microsphere with quasiperiodic spherical stack
still have poorly been considered, though it is a logical extension of previous
works in this area.This paper is organized as follows. In Section2, we
formulate our approach and basic equations for optical fields in a dielectric
multilayered microsphere. We outline the numerical scheme of applying the
dyadic Green function (DGF) technique to evaluate the spectrum of a nanoemitter
placed in such a quasiperiodic system. In Section 3, we present our numerical
results on structure of the cavity field states and resonances in a microsphere
with quasiperiodic spherical stack dependently of the quasiperiodicity
parameter. We found the enhanced field peaks if such a parameter is close to
the golden mean value. In the last section, we summarize our results.


## 2. Basic Equations

A 1D quasiperiodic (QP) spherical stack, where a
Fibonacci sequence is considered, can be constructed following a simple
procedure. Let us consider two neighbor2-layer segments, long and short, denoted,
respectively, by L and S.
If we place them one by one onto surface of a microsphere, we obtain a
sequence: LS.
In order to obtain a QP sequence, these elements are transformed according to
Fibonacci rules as follows: L is replaced by LS, S is replaced by L: L→LS, S→L.
As a result, we obtain a new sequence: LSL.
Iteratively applying this rule, we obtain, in the next iteration, a sequence
with a five-element stack LSLLS,
and so on. One can control the properties of such a QP stack by the use of some
control parameter (see factor γ later in Section 3). For the stack with N-elements, where N≫1,
the ratio of numbers of long to short elements is the golden mean value, Γ0=(1+5)/2≈1.618.Generally in spherical geometry the wave field depends
on position of a source and it is formed on a distance scale of the order
magnitude of the radius of microsphere or thickness of nanolayers of a stack.
For analysis of such a spectrum it is necessary to use more advanced approach:
the Green function method. The base of the latter is representation of optical
field, radiated by a nanosource in a coated microsphere, as a weighed
superposition (sum) of forward and backward waves (reflected from the layers
interfaces). We consider a situation when radiating point source (nanoemitter)
is placed into microsphere coated by a quasiperiodic stack. In this case, the
frequency spectrum is not described longer by a spectrum of bare microsphere
slightly perturbed by the external stack. If the number of layers is large
enough, we have to study the photons field taking into account the spectral
contributions both bottom microsphere and a quasiperiodic stack. In order to
calculate the properties of such a field, we apply the Green function
technique. In this case, a nanosource corresponds to a nanorod or quantum dot
that recently were employed in experiments with microspheres (see [12, 22] and references therein).The spatial scale of the nanoemitter objects (∼1–100 nm) is in at least of one order of magnitude
smaller than the spatial scale of microspheres (∼103–104 nm). Therefore, in the coated microsphere
(Figure 1), we can represent the nanoemitter structure as a point source placed
at r′ and having a dipole moment d0.
It is well known that the solution of the wave equation for the radiated
electromagnetic field E due to a general source J(r′) reads (μ=1)  [23, 24](1)E(r)=iωμ0∫Vdr′G(r,r′,ω)⋅J(r′),where G(r,r′,ω) is the dyadic Green function (DGF), which
depends on the type of the boundary conditions imposed on E(r) and contains all physical information
necessary for describing the multilayered structure (the time dependence is
assumed to be as eiωt). Equation (1) is complemented by the
standard boundary conditions: limitation of fields in the center of
microsphere, continuity of tangential components of fields at the interfaces of
layers. Also, we use Sommerfeld's radiation conditions, there is only outgoing
wave in the external boundary of a microsphere. In this case, the
electromagnetic field E in the coated structure consists of sum of the
waves radiating in the surrounding medium and the multiple wave reflections due
to interfaces between layers. Substituting the nanorod source in the form J=iωd,  d=d0δ(r−r′) in (1), we obtain(2)E(r,r′,w)=−p0G(r,r′,ω),where p0=(d0/ε0)(ω2/c2), r is the point where the field is observed,
while r′ is the nanorod (point nanoemitter) location.
In such a situation, the nanoemitter frequency spectrum is identical to dyadic
Green's function (DGF) spectrum.Figure 1
Geometry of multilayered
microsphere. Stack of quasiperiodic multilayers is deposited on the surface of
microsphere.Let us consider the multilayered spherical structure:
a concentric system of spherical layers contacting with the sphere (a
concentric stack) deposited onto the surface of the microsphere with a
nanoemitter placed in such a structure (see Figure1). The layers are localized
at the distances Rk from the center, where dk=Rk+1−Rk is the width of a kth layer.Let us first specify some details of the Green
function technique for multilayered microspheres and introduce our notations.
Following the approach [20],
we write down DGF of such a system as follows:(3)G(r,r′,ω)=GV(r,r′,ω)δfs+G(fs)(r,r′,ω),where GV(r,r′,ω) represents the contribution of the direct
waves from the radiation sources in the unbounded medium, whereas G(fs)(r,r′,ω) describes the contribution of the multiple
reflection and transmission waves due to the layer interfaces. The dyadic Green
tensor GV(r,r′,ω) in (3) is given by(4)G(fe)(r,r′,ω)=iks4π∑p=e,o∑m=1∞∑l=0m2m+1m(m+1)(m−l)!(m+l)!(2−δ0l)Gpml(f,e)(r,r′,ω),where Gcml(f,e)(r,r′,ω) is the particular Green tensor    [20, 21], m is the spherical and l is the azimuth quantum numbers, ki=ωni/c, ni=εi(ω) is a refraction index.  It is worth to note that in a spherical case,
the configuration of photon field strongly depends on the position of source in
a coated microsphere. Such a field has a structure of single spherical wave
only if a nanosource is placed very close to the center [6]. Otherwise, many forward and
backward (reflected) spherical waves contribute, therefore the complete
description of such a system or the Green function (3),   (4) has quite
complicated structure.As already we have mentioned, we represent a nanorod
by a dipole nanoemitter. Since nanorods are highly polarized objects, we pay
more attention to the case when the dipole orientation of a nanoemitter isd=dφ^
[22], so only the tangential
components of the
Green tensor Gφφ contribute.
For a spherical stack, Gφφ(r,r,ω) is expressed through the complex Hankel
functions [20, 23]; therefore an analytical study of the spectrum is
rather difficult problem. Further we use numerical approach; the numerical
scheme for our study is similar to those used in [21].


## 3. Numerical Results

In this section, we study the frequency spectrum of a
nanoemitter radiation for quasiperiodic Fibonacci layers (spherical stack)
deposited on the surface of a microsphere (Figure1) for different numbers of
layers in the stack. First, let us remind that the Green function relates to
energy of a fluctuating electromagnetic field strength E(r) (at small dissipation) as [25] 〈E(r)2〉=(ℏω2/c2)coth⁡(ℏω/2T0)Im(G(r,r,ω))→T0→0(ℏω2/c2)Im(G(r,r,ω)),
where G(r,r,ω) is the Green function, that is, Gφφ(r,r,ω) in our case, T0 is temperature. We note that such a field
state E is not a photonic state in general, but a
state of the macroscopic medium, dressed by the electromagnetic field [25, 26].The following parameters have been used in our
calculations: the geometry of system isA{L(B,C)⋯S(B,C)⋯}D, where letters A, B, C, D indicate the materials in the spherical stack,
respectively. The bottom microsphere has refraction index n4=1.5+2⋅10−4i (A,
glass, radius 1000nm). The distinct two-layered blocks (L(B,C) and S(B,C)) are stacked according to the Fibonacci
generation rule. For L and S blocks, we use the notation L=(B,C,1) and S=(B,C,γ),
where γ is the ratio of both thicknesses: γ=(C)S/(C)L≤1.
Refraction indices of the layers in blocks are nB=3.58+9⋅10−4i (Si,
width 122nm), nC=1.46+10−3i (SiO2,
width 300nm) [27], and nD=1 (D,
surrounding space). For L-block layers B and C are constructed as λ/4 layers, while for S-block thickness B is the same as for L-block, but the thickness C is 150nm (γ=0.5). For example, for Fibonacci order F9, the total thickness of the microsphere with 68 layers is 13.4μm (34 two-layer blocks). To consider the realistic
layer case, we have added to each ni a small imaginary part that corresponds to a
material dissipation.In order to study the behavior of field in the
microsphere, we have calculated the frequency spectrum of the transmittance
coefficientT and corresponding spectrum of imaginary parts
of the Green function W≡Im(Gφφ(r,r,ω)),
where r is position of a nanoemitter, ω=2πf.
We have calculated evolution of a spectrum for different values γ,
and also for different number of layers in the spherical stack (Fibonacci order Fn) for a range [300–600] THz or [1000–500] nm (DGF is normalized on radius of the bottom
microsphere). Most intensive optical peak was found for F9 stack, when γ=0.618=1/Γ0 (Γ0≈1.618 is golden mean value). The details of field
spectra are shown in Figure 2 for 68 layers in the stack (34 of 2-layer blocks, order Fibonacci is F9). We observe from Figure 2(a), that such a
spectrum consists of peaks with various amplitudes, however the most intensive
peak with Im(Gφφ)≃87 is located at 436.1THz (details of this peak are shown in the inset).
We also have calculated that such a peak relates to eigenfrequency of the
system f=436.098THz with quite large-quality factor Q=7369.02.
Thus, even though periodicity of the stack is broken, well-defined intensive
peak of field is clearly seen. Small peaks in Figure 2(a) have rather indented
form due to the contribution of several close small resonances corresponding to
different spherical modes for DGF in (3)-(4). The transmittancy spectrum for γ=0.618 is shown in Figure 2(b). We observe that as
opposed to just a periodic λ/4 case [6, 7, 28], the Fibonacci stack has a fractal-like transmittancy
spectrum, see Figure 2(b). Such a spectrum consists of narrow resonances
separated by numerous pseudoband gaps, induced by incoherent re-reflections of
optical waves from quasiperiodic layers interfaces. Resonances correspond to
poles of DGF at complex eigenfrequencies f,
where Im(f) determinates the width of a resonance or Q-factor of oscillations Q=Re(f)/2Im(f).
At small dissipation, peaks have a typical Lorentzian line shape Im(Gφφ)∼ε/(δ2+ε2),
where δ is a detuning from a resonance, and ε is the linewidth.(a) Imaginary parts of
tangential component of the Green functionW=Gφφ(r,r) for r=900nm,
(b) Frequency spectrum of transmittance coefficient T   for spherical quantum number m=9.
Microsphere is coated by quasiperiodic stack with N=68 (34 of 2-layers blocks, order Fibonacci F9) for γ=0.618.
See details in text.
(a)(b)Adding more layers should strongly narrow the resonances. The
spectrumT becomes more complicated and new resonances do
form a well-expressed fractal of transmittancy. Corresponding spectrum of the
Green function becomes richer, since a greater number of eigenmodes contribute
into numerous pseudoband gaps of T.
It is important to note that, as one can see from Figure 2(b), the cavity
modes excited by a dipole inside a coated microsphere are distinctly different
from whispering gallery modes characterized by m≫1 and T→0.We have calculated evolution of a spectrum for
different valuesγ close to γ=1/Γ0 for fixed number of layers in the spherical
stack (Fibonacci order Fn). The results are displayed in Figure 3 for a
range [430–440] THz,
or [0.70–0.68] μm.
One can see that in this area the spectrum consists of rather narrow
resonances, and a set of satellite peaks appears around the main peak. The
intensity of this peak changes for different γ,
giving rise to another main peak. When γ approach to 1/Γ0=0.618,
the field peaks become sharper, as shown in Figure 3. We observe that the
amplitude of resonances is maximal for case γ=0.618.Figure 3
Spectrum of imaginary part of
the Green functionW=Gφφ(r,r) for r=900nm and 68 layers in stack (34 of 2-layer blocks F9) for different values of parameter
quasiperiodicity γ close to the golden mean value: (1) 0.55,
(2) 0.58,
(3) 0.6,
(4) 0.618,
(5) 0.63,
(6) 0.65,
and (7) 0.68.It is of great interest to compare the spectra in
Figure2 to the case with distinct “quasiperiodic parameter” γ, especially to the case γ=1 case, when QP stack effectively becomes a
periodic λ/4 spherical stack. In result of calculations, we have found that
the dominating peak for γ=0.618 (Figure 2(a)) is already 40 times higher with respect to periodic λ/4 case, that is, a signature of a
quasiperiodicity strength in a coated microsphere cases.In previous figures, the frequency spectrum of the
field (∼Im(Gφφ(r,a,f))) for the quasiperiodic stack was shown. The
fractal complicity of the transmittancy spectrum is defined by the intrinsic
properties of the quasiperiodic spherical stack independently on the
nanoemitter location. However in experiments, it is important to identify the
spatial distribution of the field, radiated by nanosources located in such a
quasiperiodic microsphere. Therefore, it is of interest to consider the spatial
field distribution in a cross-section (r,φ,θ=const) that contains both center of the coated
microsphere and nanoemitter for some resonant frequency. Such a distribution is
shown in Figure 4 for θ=1 and the most intensive resonance at f0=436.098THz (see Figure 2(a)), when the quasiperiodicity
parameter γ=0.618.
We observe from Figure 4 that W(r,φ)=Im(Gφφ(r,a,φ)) has a very sharp peak in the place of the
nanosource location. Such a spatial field structure may be treated as a
confinement of the electromagnetic energy ∼Im(Gφφ(r)) inside the coated microsphere. The leakage of
photons through such a structure into the outer space obviously is small. We
observe from Figure 4 that the field structure inside of quasiperiodic stack is
anisotropic and quite intricate, but the field distribution beyond the coated
microsphere has a periodic character.Figure 4
Spatial structureW(r,φ)=Im(Gφφ(r,a,φ)) in a cross-section 0<r<21μm and 0<φ<2π of the microsphere with quasiperiodic stack
for eigenfrequency f=436.09THz.
A nanoemitter is placed at point a=900nm.
Other parameters are same as in Figure 2. One can observe the confinement of
field in the stack. Outer cycle only indicates the external boundary Rext=13.8μm of the quasiperiodic spherical
stack.


## 4. Conclusion

We have studied the frequency spectrum of nanoemitters
placed in a microsphere with a quasiperiodic   subwavelength spherical stack. We
found that the transmittancy spectrum of such a stack consists of quasi-band
gaps and narrow resonances, induced by re-reflection of optical waves. The
spectral evolution of transmittancy at the change of the thickness of two-layer
blocks, constructed following the Fibonacci sequence, is investigated. When the
number of layers (Fibonacci order) increases, the structure of spectrum
acquires a fractal form. We show that the width of resonant peaks in the
frequency spectrum becomesextremely narrow for a quasiperiodic
spherical stack of a high Fibonacci order. In principle, that allows creating a
narrow-band filter with a transmission state within the forbidden band gap of
nanoemitters, incorporated in such a coated microsphere. We have found the
confinement and the gigantic enhancement of the optical field in quasiperiodic
structure, when the ratio of layers' widths in two-layer blocks of stack is
close to the golden mean value. This allows to confine resonantly the field
energy in the quasiperiodic stack in very narrow frequency range in order to
create very selective stop-band filters. Incorporating nanoemitters into such
structured microspheres can open new opportunities for the active control of
light nanosources.



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