Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial
We propose a simple design of an all-dielectric silicon-based planar metamaterial that manifests an extremely sharp resonant reflection and transmission at the wavelength of about 1550 nm due to both low dissipative losses and the trapped mode operating method particularities. The resonance Q-factor...
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nasplib_isofts_kiev_ua-123456789-1002092025-02-23T17:36:56Z Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial Підсилення люмінесценції квантових точок за допомогою повністю діелектричного метаматеріалу Усиление люминесценции квантовых точек с помощью полностью диэлектрического метаматериала Khardikov, V.V. Prosvirnin, S.L. Радиофизические явления в твердом теле и плазме We propose a simple design of an all-dielectric silicon-based planar metamaterial that manifests an extremely sharp resonant reflection and transmission at the wavelength of about 1550 nm due to both low dissipative losses and the trapped mode operating method particularities. The resonance Q-factor dozens of times exceeds that of resonances in the common infrared plasmonic structures. The designed metamaterial is considered for aggregation with a pumped gain medium to achieve the enhancement of luminescence and produce an all-dielectric analog of a “lasing spaser”. We report that an essential enhancement (above 500 times) of luminescence of a layer containing pumped quantum dots can be achieved by using the designed metamaterial. This value scores of times exceeds the known luminescence enhancement by the plasmonic planar metamaterials. Запропоновано повністю діелектричний метаматеріал на базі кремнію, який підтримує надзвичайно різкий резонанс відбиття та проходження поблизу 1550 нм. Такий характер резонансної взаємодії метаматеріалу зі світлом обумовлений особливостями резонансу на замкненій моді та низькими дисипативними втратами в діелектрику. Добротність резонансу в десятки раз перевищує добротність резонансу у звичайних плазмонних структурах інфрачервоного діапазону. Розглянуто можливість поєднання розроблюваного метаматеріалу з активним середовищем з метою одержання підсилення люмінесценції та створення повністю діелектричного аналога приладу, відомого як "lasing spaser". Показано, що, використовуючи даний метаматеріал, вдається досягти суттєвого підсилення (понад 500 разів) люмінесценції шару з активними квантовими точками. Значення коефіцієнта підсилення люмінесценції багаторазово перевищує відомий коефіцієнт підсилення для планарних плазмонних метаматеріалів. This work was partially supported by the Ministry of Education and Science of Ukraine under the Program “Electrodynamics of layered composites with chiral properties and multifunctional planar systems”, Project No. 0112 U 000561 and the Ukrainian State Foundation for Basic Research, Project F40.2/037. 2013 Article Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial / V.V. Khardikov, S.L. Prosvirnin // Радиофизика и радиоастрономия. — 2013. — Т. 18, № 4. — С. 331-340. — Бібліогр.: 28 назв. — англ. 1027-9636 https://nasplib.isofts.kiev.ua/handle/123456789/100209 en Радиофизика и радиоастрономия application/pdf Радіоастрономічний інститут НАН України |
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Радиофизические явления в твердом теле и плазме Радиофизические явления в твердом теле и плазме |
| spellingShingle |
Радиофизические явления в твердом теле и плазме Радиофизические явления в твердом теле и плазме Khardikov, V.V. Prosvirnin, S.L. Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial Радиофизика и радиоастрономия |
| description |
We propose a simple design of an all-dielectric silicon-based planar metamaterial that manifests an extremely sharp resonant reflection and transmission at the wavelength of about 1550 nm due to both low dissipative losses and the trapped mode operating method particularities. The resonance Q-factor dozens of times exceeds that of resonances in the common infrared plasmonic structures. The designed metamaterial is considered for aggregation with a pumped gain medium to achieve the enhancement of luminescence and produce an all-dielectric analog of a “lasing spaser”. We report that an essential enhancement (above 500 times) of luminescence of a layer containing pumped quantum dots can be achieved by using the designed metamaterial. This value scores of times exceeds the known luminescence enhancement by the plasmonic planar metamaterials. |
| format |
Article |
| author |
Khardikov, V.V. Prosvirnin, S.L. |
| author_facet |
Khardikov, V.V. Prosvirnin, S.L. |
| author_sort |
Khardikov, V.V. |
| title |
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial |
| title_short |
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial |
| title_full |
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial |
| title_fullStr |
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial |
| title_full_unstemmed |
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial |
| title_sort |
enhancement of the quantum dot luminescence in all-dielectric metamaterial |
| publisher |
Радіоастрономічний інститут НАН України |
| publishDate |
2013 |
| topic_facet |
Радиофизические явления в твердом теле и плазме |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/100209 |
| citation_txt |
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial / V.V. Khardikov, S.L. Prosvirnin // Радиофизика и радиоастрономия. — 2013. — Т. 18, № 4. — С. 331-340. — Бібліогр.: 28 назв. — англ. |
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Радиофизика и радиоастрономия |
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2025-11-24T04:03:11Z |
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2025-11-24T04:03:11Z |
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| fulltext |
ISSN 1027-9636. . . 18, 4, 2013 331
. 2013, . 18, 4, c. 331–340
© V. V. Khardikov and S. L. Prosvirnin, 2013
ÐÀÄÈÎÔÈÇÈ×ÅÑÊÈÅ ßÂËÅÍÈß
 ÒÂÅÐÄÎÌ ÒÅËÅ È ÏËÀÇÌÅ
V. V. KHARDIKOV 1,2 AND S. L. PROSVIRNIN 1,2
1 Institute of Radio Astronomy, National Academy of Sciences of Ukraine,
4, Chervonopraporna St., Kharkiv, 61002, Ukraine
2 V. Karazin National University of Kharkiv,
4, Svoboda Sq., Kharkiv, 61022, Ukraine
E-mail: khav77@gmail.com
ENHANCEMENT OF THE QUANTUM DOT LUMINESCENCE
IN ALL-DIELECTRIC METAMATERIAL
We propose a simple design of an all-dielectric silicon-based planar metamaterial that manifests an extremely sharp resonant
reflection and transmission at the wavelength of about 1550 nm due to both low dissipative losses and the trapped mode
operating method particularities. The resonance Q-factor dozens of times exceeds that of resonances in the common infrared
plasmonic structures. The designed metamaterial is considered for aggregation with a pumped gain medium to achieve the
enhancement of luminescence and produce an all-dielectric analog of a “lasing spaser”. We report that an essential enhancement
(above 500 times) of luminescence of a layer containing pumped quantum dots can be achieved by using the designed metamaterial.
This value scores of times exceeds the known luminescence enhancement by the plasmonic planar metamaterials.
Key words: all-dielectric metamaterials, trapped mode resonance, quantum dot, luminescence
1. Introduction
Modern nanotechnologies enable to design optically
thin layers of materials with a periodic subwavelength
pattern in order to produce planar metamaterials.
These latter are impressive objects-in-research, due
to their some certain fascinating facilities, and have
become mostly well-known over the last decade and
are quite novel as well. Recent works report the
results of aggregating laser materials with planar
metamaterials to design the parametric gain systems
[1–3] and develop the gaining or lasing devices, such
as the spaser [4–6]. Some essential part of this de-
velopment is studying the luminescence of a gain
material hybridized with a planar metamaterial that
can support a coherent high-Q electromagnetic os-
cillation.
Typically, a planar metamaterial for visible and
near-infrared wavelengths is a plasmonic structure
designed on the basis of either a periodic array of
complex-shaped resonant metallic nanowires or slits
in a metal slab. The main factor responsible for the
spectacular properties of these metafilms is some
resonant interaction of light with the patterned layer.
Moreover, numerous prospective applications of the
planar metamaterials incorporating a pumped laser
medium do require the high Q-factor resonance and
strong confinement of electromagnetic fields.
The main factor extremely limiting the prospective
applications of the infrared plasmonic metamaterials
is high losses inherent to metals. Thus, typically, the
Q-factor of the resonances excited in plasmonic struc-
tures is small because of radiation losses and a huge
energy dissipation inherent to metals in the visible
and near-infrared wavelength ranges.
Fortunately, there is some way to significantly
decrease the radiation losses in the planar plasmonic
structures. This way lies in using the so-called “trapped
mode” resonances in the planar structures with a
broken symmetry of periodic elements due to exci-
ting of anti-phased currents in the metallic elements
of a subwavelength periodic cell in microwave [7, 8],
terahertz [9, 10] and in some optical bands [11–13].
Recently, it has been experimentally demonstrated
that a hybridization of semiconductor quantum dots
(QDs) with a plasmonic metamaterial supporting the
trapped mode resonance leads to a multifold intensity
332 ISSN 1027-9636. . . 18, 4, 2013
V. V. Khardikov and S. L. Prosvirnin
increase and narrowing of their photoluminescence
spectrum [14]. The experiment shows that plasmo-
nic metamaterials interact with QDs like a cavity,
and the luminescence enhancement can be explai-
ned with the cavity quantum electrodynamic Purcell
effect. This observation is an essential step towards
understanding the mechanism of the radiation emis-
sion in plasmonic metamaterials and opens amazing
opportunities for developing artificial metamaterial-
enhanced gain media.
The trapped mode resonances have the typical
peak-and-trough Fano spectral profile [15] and can
be excited, for example, in the planar periodic array
composed of twice asymmetrically-split metal-
lic rings. Their specific character arises from the
destructive interference of the radiation by the anti-
phased current distribution. Theoretically, in a hypo-
thetic lossless structure, the Q-factor of such re-
sonance can be infinitely increased by the asymme-
try degree decreasing, because the radiation loss
tends to zero in this case. This property is the main
difference of the trapped-mode systems from the
nano-antenna structures intended for enhancement
of Purcell effect.
Recently, a novel geometry of metallic strip parti-
cles and their arrangement within a periodic cell of
trapped-mode structures has been proposed in [9, 10].
This suitable choice of the structure geometry results
in an additional interference destruction and, as a
consequence, in a radiation losses decrease.
The influence of a plasmon nanostructure with the
Fano lineshape in absorption spectrum on the spon-
taneous emission spectrum and the lifetime of a sin-
gle molecule of gain material was studied in [16, 17].
The opportunity of controlling the linewidths of ato-
mic spontaneous emissions via the anisotropic Pur-
cell effect was shown. Such interesting phenomena
as the atomic spectral line rapid narrowing and the
nanoscale line width pulsing were observed.
In practice, however, the Q-factor is strongly li-
mited by the energy dissipation inherent to metals in
the visible and near-infrared. Actually, the dissipative
losses increase extremely as currents increase in
metallic elements with decrease of radiation from
nearly symmetry elements of an array. Thus the
trapped mode resonances have a greater Q-factor
than the regular resonance in a symmetric structure,
though still a moderate one [13].
But there is a promising way to achieve a high-Q
factor trapped mode resonance in the planar all-di-
electric arrays with a broken symmetry of practically
non-dissipative elements. It is well-known that the
symmetry break in dielectric photonic band gap struc-
tures results in excitation of a high-Q defective mode
resonance [18–20]. It was shown in [21] that a pla-
nar array with square periodic cell composed of two
different-length semiconductor bars enables to achieve
a trapped mode resonance with Q-factor much greater
than that relating to this kind of resonances in plas-
monic structures. For example, the Q-factor of the
trapped mode resonance of the germanium array [21]
can reach the value of about a thousand in the near-
infrared transparency window.
In the case of this array, the dielectric bars work
as open dielectric resonators, and the pair of the
coupled resonators of each periodic cell can be con-
sidered as a metamolecule. The trapped mode re-
sonance of a planar array is formed by a destruc-
tive interference of the fields scattered by such
metamolecules outside the array. Besides the great
Q-factor, a remarkable property of this all-dielectric
arrays resonance is an essential red shift of their
resonance frequency as against the resonance
frequency of an array whose periodic cell consists
of a single dielectric bar (see [21]). This property
opens the way for dielectric materials with a rela-
tively small refractive index in order to produce this
kind of arrays.
We expect that the spectrum narrowing and in-
tensity increasing of the QDs photoluminescence ob-
served in [14] should rise with the Q-factor increase
of the planar structure resonance. Thus, it is of high
interest to study the features of the photolu-
minescence with a hybridization of QDs with a pe-
riodic all-dielectric array in the trapped-mode re-
sonance regime.
2. Problem Statement
and Solution Method
Now consider a double periodic array composed of
dielectric bars placed on a substrate with thickness
sL (see Fig. 1). The periodic array of thickness aL is
immersed into QD layer (polymer layer in which QDs
are dispersed) with total thickness .QDL The unit cell
of the array includes a pair of dielectric bars of diffe-
rent sizes. In [21], the asymmetry of a double perio-
dic array of dielectric elements was achieved by using
the dielectric bars of identical square cross-sections,
whereas different in length. Though, it is more suit-
ISSN 1027-9636. . . 18, 4, 2013 333
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial
able in practice to use the dielectric bars different
in width along the y-direction. If all the bars are iden-
tical both in length and thickness, then such a doub-
le periodic array is very convenient for manufacture.
The dimensions of the square periodic cell are chosen
as 900x yd d d= = = nm. The periodic cell is as-
sumed to be symmetric as respects the line drawn
through the cell center and parallel to the y-axis. The
normal incidence of a linearly x-polarized plane wave
is considered.
As a material for the dielectric bars, we propose
the silicon as the most widespread material in elec-
tronics, with the transparency window within the
wavelengths of 1.2 μm to 14 μm. The dimensions
of the array elements are chosen to provide the
trapped mode resonance of optically non-pumped
structure at 1550 nm, the typical wavelength used
in telecommunications. At 1550 nm, the silicon
refractive index is approximately 3.48 at room
temperature [22]. An extremely thin substrate used
enables to exclude any undesired wave interfe-
rence. The substrate is assumed to be a synthetic
fused silica membrane. Its refractive index is ap-
proximately 1.45sn = in the wavelength range un-
der consideration [23]. The substrate thickness is
50sL = nm. Note that there are no high non-eva-
nescent diffraction orders in the substrate (the ine-
quality 1550sn d < nm is valid).
To study the intensity enhancement of the lumi-
nescence of QDs resulting from their hybridization
with a dielectric array, we propose to use the solu-
tions of plane wave diffraction of the structure with
and without an optical pump. In the proposed me-
thod, the photoluminescence emission intensity eW is
evaluated as a difference between the intensity of
the dissipation (or emission) defined by the expres-
sion
2 2
1 ,W R T= − − R and T are the coefficients
of reflection and transmission, in the structure with-
out optical pump and with it
0 ,eW W W+= − (1)
where 0W and W+ are the intensities of the dissipa-
tive losses (or emission) in the structure without and
with optical pump respectively. Note that the value W
will be positive, if any dissipation is observed in the
structure, and negative, if the total intensity of the re-
flected and transmitted waves exceeds the intensity
of the incident wave. This fact is taken into account
in (1), where the photoluminescence intensity eW has
a positive value.
The proposed method was used for calculating
the enhancement luminescence of QDs layer com-
bined with a plasmonic metamaterial, which was
studied experimentally in [14] (see the Appendix).
We use the experimental data on the luminescence
spectrum presented in [14] to define the parame-
ters for the adequate theoretical description of a
certain actual composite material which is QDs
dispersed in a host polymer layer. In addition, the
comparison between the experimental results of that
work and our theoretical ones demonstrates the
validity of the proposed theoretical approach to study
the photoluminescence. The proposed approxima-
tion provides good coincidence of the numerical
results with the experimental ones, despite ignoring
the quantum phenomenon inherent to the interac-
tion of gain medium molecules and a strong local
field. Besides, for simulation, we use the well-known
model of gold and the parameters of QDs provided
in this paper. Therefore, we may conclude that our
simple approach can be effectively used for estima-
tion of the enhancement of luminescence in the all-
dielectric metamaterials.
Thus, to determine the photoluminescence inten-
sity, it is required to solve the problem of plane
wave diffraction by the considered structure. For this
aim, the mapped pseudospectral time domain (PSTD)
method proposed in [24, 25] is used. For the sake
of simplicity, the dispersion of dielectrics is not
taken into account in this paper. Such approxima-
Fig. 1. A sketch of the unite cell of a double-periodic planar
structure. The all-dielectric array composed of two dielectric
bars per periodic cell is immersed into the QD layer
334 ISSN 1027-9636. . . 18, 4, 2013
V. V. Khardikov and S. L. Prosvirnin
tion is justified because the refractive index of sili-
con in the wavelength range under consideration
(from 1500 nm to 1600 nm) varies within 3.48 to
3.47. It is evident that such an extremely weak
dispersion has no effect on the properties of a
trapped-mode resonance.
To take into account the gain in the optically
pumped QDs, the method of additional differential
equations (ADE) [26] is used along with the model
of a negative frequency-dependent conductivity [27].
The expression of the corresponding time-domain
conductivity ( )tσ is chosen in the form to ensure
that this value is real and causal,
0
0( ) cos( )exp( ) ( ).t t t u tσσ = ω − τ
τ
(2)
Here ( )u t is the Heaviside unit step function. The
coefficient 0σ is proportional to the peak value of
the gain set by the pumping level and the resulting
population inversion. Such model assumes that the
optical gain medium is homogeneously broadened
and wherein the atoms of the gain medium are indis-
tinguishable and have the same atomic transition fre-
quency, 0.ω Any variation of the QD sizes will cause
broadening of the exciton line of the QDs. The time
constant τ permits to include the relaxation processes
in a phenomenological manner (decay rate is 1 ),τ
and shows that any phase coherence introduced into
the system of atoms by the action of the electric
field will be lost in the time interval ,τ once the elec-
tric field is turned off.
The Fourier transform of (2) reduces to the fol-
lowing frequency dependent conductivity
0
2 2 2 2
0
(1 )
( ) ,
(1 ) 2
i
i
σ + ωτσ ω =
+ ω τ + ωτ − ω τ
(3)
where the time dependence of the electromagnetic
field is chosen in the form exp( ).i tω It is clear from
(3) that the gain coefficient is governed by a single
Lorentzian profile with a width determined by .τ
The resonance frequency, that is the frequency at which
the response is maximized, is given by expres-
sion 2 2
0 .d
−ω = ω + τ The peak of the gain curve is
0( ) 2dσ ω = σ and the full-width-at-half-maximum
band-width is 2 .δω = τ Using (3) one can obtain a
complex propagation constant of a plane wave and
see that the wave amplification will be observed
only for the case of 0 0.σ < The chosen form of the
frequency-dependent conductivity can be easily in-
cluded in the numerical scheme of time domain
approaches by the use of ADE method. It results
in the two additional first order differential equa-
tions in time in each grid node immersed into the
gain medium [27].
A good agreement between our theoretical and
the known experimental results of the study of the
luminescence of a plasmonic metamaterial combined
with QDs (see the Appendix) evidences of correct-
ness of both the model of active QD medium and the
proposed approach to study the luminescence in a
resonant planar metamaterial.
Now let us apply the proposed method to de-
signing a resonant silicon metamaterial combined
with the semiconductor QDs to enhance the photo-
luminescence intensity with the spectral maximum
at 1550 nm.
For numerical study, a laser medium based
on semiconductor QDs was chosen with the follo-
wing parameters:
15 1
0 1.26 10 s−ω = ⋅ which corre-
sponds to wavelength 0 1550λ = nm,
154.85 10 s,−τ = ⋅
2.19QDε = which corresponds to refractive index
1.48QDn = of non-pumped QD laser medium, and
0 500σ = − S/m corresponding to the emission factor
tg 0.021eδ = − in analogy to the lossy factor of the
media. The small value of τ results in a wide-band
QD spectral line, and it enables to exclude from
consideration the effects caused by displacement
between the metamaterial dissipation peak and the
maximum of exciton emission line of QDs. Note that
the pump level (i.e. the value of the parameter 0 )σ
is one order less than it was needed in the case of
plasmonic metamaterials (see the Appendix), because
of the low losses of the all-dielectric array.
The geometry parameters of the dielectric bars array
immersed in the non-pumped QD layer 0( 0)σ = are
chosen to provide a high Q-factor trapped mode reso-
nance of the structure near the wavelength 1550 nm.
The sizes of the silicon structure corresponding to
this condition are the following: 100aL = nm, QDL =
210 nm, 50sL = nm, the dielectric bars length equals
to 800l = nm, the width of the bigger and smaller
bars are 2 260h = nm and 1 160h = nm, respectively,
and the distance between the two different in width
bars is 240 nm (see Fig. 1).
In order to take into account the energy dissipation
in the bars, a model of constant conductive medium
is used to describe the dielectric with the PSTD
ISSN 1027-9636. . . 18, 4, 2013 335
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial
method. This means that the dielectric of bars is
modeled as a medium with both 2
a anε = and aσ
being some constants. Such an approach results in
the frequency-dependent losses.
3. Analysis of the Luminescence
of a Hybridization of QDs
with a Low-Loss Planar
All-Dielectric Metamaterial
The wavelength dependences of the reflection and
transmission coefficients magnitudes of the designed
structure are shown in Fig. 2. There is a trapped
mode resonance near the wavelength 1550 nm which
has the typical peak-and-trough Fano spectral profile.
As was shown in [21], this resonance results from the
anti-phased excitation of the dielectric bars behaving
as an open half-wavelength dielectric resonators in
this case. Lines 1, 2, and 3 correspond to 0,aσ =
130.3aσ = S/m, and 1303aσ = S/m, or equivalently,
to tg 0,aδ = 3tg 10 ,a
−δ = and 2tg 10a
−δ = at the wa-
velength 1550 nm. Note that actually the silicon lossy
factor tg Siδ is less than 310− at this wavelength.
We estimate the trapped mode resonance Q-fac-
tor by using the following formula proposed in [21]
1 2 1 2
2 2
0 2 1 2 1
2
,Q λ λ λ λ= =
λ λ − λ λ − λ
(4)
where 1λ and 2λ are the wavelengths of the maxi-
mum and minimum of the reflection or transmission
coefficient of the corresponding Fano spectral pro-
file and 0 1 2( ) 2λ = λ + λ is the average wavelength
of the trapped mode resonance. Using (4), the fol-
lowing Q-factor values are obtained: 268, 232 and 83
corresponding to tg 0,aδ = 310− and
210 ,−
respec-
tively. It is of interest that the wavelengths of the
maximum and minimum of the reflection and trans-
mission coefficients coincide only in case of a nondis-
sipative structure. The reflection and transmission
extremes shift with respect to each other if dissipa-
tion in the dielectric is taken into account. Then, the
Q-factor of the trapped mode resonances is estima-
ted as the average value ( ) 2.R TQ Q Q= + Here RQ
and TQ are calculated from the wavelength de-
pendences of the reflection and transmission coeffi-
cients by (4).
Note that the average trapped mode resonan-
ce wavelength ( )0 0 0( ) ( ) 2 1551R Tλ = λ + λ = nm is
observed in all of the considered cases of dissipative
losses.
The wavelength dependences of the photolumi-
nescence intensity eW calculated by using (1) are
presented in Fig. 3. It shows the photoluminescence
intensity of the homogeneous QD layer with thick-
ness 210QDL = nm placed on 50 nm thick silica
membrane (Fig. 3(a)), i.e. the intensity related to the
layered structure without any array. The wavelength
dependences of the photoluminescence of QD hy-
bridization with non-dissipative and low-dissipative
3(tg 10 )a
−δ = metamaterials are shown in Fig. 3(b).
One can see a great enhancement in photolumines-
cence intensity. In the case of non-dissipative bars,
the photoluminescence intensity enhancement makes
1560 times. If the dissipation of silicon, estimated as
3tg 10 ,Si
−δ = is taken into account, this coefficient of
the intensity enhancement equals to 560. These val-
ues mentioned are much more greater than the known
orders of the photoluminescence enhancement in-
herent to plasmonic metamaterials combined with
QDs (see [14] and the Appendix).
Up to this point, we consider the photolumines-
cence in the pumped all-dielectric metamaterial with
its emission factor (tg 0.021)eδ = − which in abso-
lute value exceeds essentially the lossy factor of the
metamaterial
3(tg 10 ).Si
−δ = Now let us study the
case of some pumped lasing medium with the inten-
sity of energy dissipation in the dielectric array
2(tg 10 ),a
−δ = which is comparable with the gain of
QDs. In Fig. 4, we present the wavelength depen-
Fig. 2. Wavelength dependences of the reflection and transmis-
sion coefficients of the periodic array of silicon bars immersed
in a non-pumped QD layer. Lines 1, 2, and 3 correspond
to tg 0,aδ = 3
10
− and 210 ,− respectively
336 ISSN 1027-9636. . . 18, 4, 2013
V. V. Khardikov and S. L. Prosvirnin
dences of the photoluminescence intensity of a QD
layer placed on the silica substrate (line 1), the power
dissipation in the array of dielectric bars hybridized
with non-pumped QDs (line 2), and the photolumi-
nescence intensity of the optically pumped QDs ag-
gregated with the same array (line 3). In the last
case, the QD photoluminescence enhancement is also
observed (see Fig. 4, line 3). However, in the trapped-
mode resonance wavelength, we observe a decrease
of the emission intensity until complete forbidding of
the photoluminescence (see the region near 1550 nm,
corresponding to 0,eW < in Fig. 4). Such wavelength
dependence of the photoluminescence intensity of
QDs aggregated with the array is explained by the
fact that the QD gain results in an increase of
the field amplitude inside the bars and the relating
increase of energy dissipation in the array. In case
of a low energy dissipation (e.g., the dissipation of
silicon array with 3tg 10 ),Si
−δ = these losses result
merely in decrease of the level of photoluminescence
enhancement.
The increase of the local field in the silicon array
is shown in Fig. 5. The distribution of x-component
of the electric field in the plane 50z = − nm corre-
sponding to the half array thickness, is presented in
Fig. 5(a) and 5(b) for the structure without optical
pumping and for the gained QD layer, respectively.
A homogeneous increase of the local field is the result
of using a homogeneous-gain-medium approximation
in our simulation. It is evident that the random distri-
bution of quantum dots in the gained layer will result
in nonhomogeneous enhancement of the local field,
which will obviously reduce the level of photolumi-
nescence enhancement. However, the value of the
luminescence enhancement for the all-dielectric
metamaterials is to be much higher than for the plas-
monic ones.
4. Conclusions
We have proposed a simple design of a planar sili-
con-based all-dielectric metamaterial that demon-
strates its extremely sharp resonant reflection and
transmission at the wavelength of about 1550 nm,
Fig. 3. Wavelength dependences of the luminescence intensity
of a QD layer with thickness 210 nm placed on the silica
membrane (a), and QD layer aggregated with metamaterials (b):
1 – tg 0;aδ = 2 – 3tg 10 .Si
−δ = The units of graphs (a) and (b)
have the same scaling
Fig. 4. Wavelength dependences of the photoluminescence
intensity of a QD layer with thickness 210 nm (line 1),
the energy absorption of the QD without optical pump
aggregated with an array of dielectric bars, 2tg 10 ,a
−δ =
(line 2) and the photoluminescence intensity of metamaterials
aggregated with QD (line 3)
ISSN 1027-9636. . . 18, 4, 2013 337
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial
due to both the low dissipative losses and involve-
ment of the trapped mode operating method. Accu-
rate numerical estimations evidence that the Q-fac-
tor of the resonance of the proposed structure do-
zens of times exceeds the Q-factor of the known plas-
monic structures.
The designed planar metamaterial is meant for
aggregating with a layer of pumped active medium in
order to achieve an enhancement of luminescence
and to produce an all-dielectric analog of the lasing
spaser [5].
We have proposed the approach to study the lumi-
nescence intensity of pumped QDs aggregated with
a planar metamaterial. The approach is based on the
comparison of the results of the two diffraction prob-
lems solutions. They are the problems of a plane
electromagnetic wave diffraction by the structure with
and without optical pump. The validity of the method
has been proven by its application to theoretical re-
production of the known experimental data relating
to QD luminescence in plasmonic metamaterials [14].
We demonstrate that essential enhancement (abo-
ve 500 times) of luminescence intensity of a layer
containing pumped QDs can be achieved by using
the designed resonant all-dielectric metamaterial.
This value scores of times exceeds the known lu-
minescence enhancement by plasmonic planar me-
tamaterials.
Appendix: Validation of the Diffraction
Approach to Study Luminescence
For an evaluation of our approach to choosing the
parameters describing the QDs immersed in a host
polymer and the diffraction approach to study lumi-
nescence as well, let us consider one of the plas-
monic metamaterials treated experimentally in [14].
The sample consists of a gold film periodically pat-
terned by asymmetric split-ring slits and placed on
a glass substrate (see Fig. 6). The film thickness is
50gfL = nm. The square cell of the periodic array is
545D = nm on side. The slits of the periodic cell shown
in Fig. 6 are as these: 470a = nm, 170t = nm, and
65w = nm. The slits are filled with a polymer mate-
rial containing semiconductor QDs and the array is
coated with a layer of the same material. The layer is
180QDL = nm thick. In [14] the lead sulfide (PbS) semi-
conductor QDs from Evident Technologies was used
with the luminescence peak around 1300 nm and the
Fig. 5. Distribution of x-component of the electric field in
the cross section of the array periodic cell ( 50z = − nm) for
the nongained (a) and gained (b) QD layer aggregated
with metamaterial at 1551.5 nm that corresponds to the maxi-
mum luminescence of the studied structure, 3tg 10Si
−δ =
Fig. 6. The geometry of the periodic cell of the aggregated with
QD/PMMA metamaterial studied experimentally in [14]
338 ISSN 1027-9636. . . 18, 4, 2013
V. V. Khardikov and S. L. Prosvirnin
mean core diameter of 4.6 nm. These QDs were dis-
persed in polymethylmethacrylate (PMMA).
Let us apply the diffraction approach based on the
formula (1) to estimate the photoluminescence inten-
sity of QD/PMMA hybridized with this plasmonic
metamaterial.
The parameters of the gain medium model must
be determined before luminescence intensity is esti-
mated. As mentioned above, the parameters 0 ,ω ,τ
and 0σ are required to determine the gain medium.
Parameters 0ω and τ of QD/PMMA are deter-
mined from the photoluminescence peak frequen-
cy
2 2
0dω = ω + τ and the full-width-at-half-maxi-
mum bandwidth of QD/PMMA photoluminescence
2 .δω = τ We use the values 15 1
0 1.47 10 s−ω = ⋅ and
159.69 10 s.−τ = ⋅ The refractive index of the non-
pumped QD/PMMA is 1.48n = (see [14]). The value
of 0σ depends on the level of population inversion in
QD/PMMA, and determines the intensity of photolumi-
nescence. Here, we use the value 0 5000σ = − S/m
to describe our optically pumped QD/PMMA. It is a
typical value used to describe a gain of semiconduc-
tor media, for example, in [27].
We consider the metamaterial aggregated with
QD/PMMA placed on a semi-infinite silica substrate
( ).sL → ∞ Such a choice enables to exclude the
interference in the substrate from the study. The model
of gold permittivity proposed in [28] is used in our
simulation of a plasmonic metamaterial.
The results of our numerical simulation of the
wavelength dependences of the reflection, transmis-
sion, and dissipation coefficients of the metamaterial
aggregated with non-pumped QD/PMMA are shown
in Fig. 7. These theoretical results can be compa-
red with the experimental ones presented in Fig. 1
of [14]. One can see a very good agreement between
our numerical and the measured results. Some dif-
ference between the levels of reflection, transmis-
sion and absorption in theoretical and experimental
results can be explained by distinctions between the
actual structure and the perfect periodic one.
To estimate the enhancement of the luminescence
intensity due to using metamaterial structure, we
compare its value with the luminescence intensity
of a homogeneous optically pumped QD/PMMA
layer placed on the same substrate. The thickness
of the layer is chosen the same as the total thickness
of the metallic array and its QD/PMMA coating.
The wavelength dependences of the photolumines-
cence intensity of the homogeneous QD/PMMA
layer of 230 nm thickness (line 1) and the metama-
terials aggregated with 180 nm-thick QD/PMMA
layer (line 2) are shown in Fig. 8. In this case, we
obtain 17 times enhancement of photoluminescence
intensity. This value of the luminescence enhance-
ment only twice exceeds that obtained experimen-
tally in [14], that is quite a good result in the mo-
delling of gain structures.
Thus, we have proven that both our model of gain
QD medium and the diffraction approach for stu-
dying the luminescence intensity demonstrate that
Fig. 7. Results of numerical simulation of wavelength
dependences of the reflection, transmission, and dissipation
coefficients of a metamaterial aggregated with non-pumped
QD/PMMA investigated experimentally in [14]
Fig. 8. Wavelength dependences of the photoluminescen-
ce intensity of homogeneous QD/PMMA layer (line 1)
and plasmonic metamaterial aggregated with optically pumped
QD/PMMA (line 2)
ISSN 1027-9636. . . 18, 4, 2013 339
Enhancement of the Quantum Dot Luminescence in All-Dielectric Metamaterial
the numerical data obtained are in good agreement
with the known experimental ones.
ACKNOWLEDGMENTS
This work was partially supported by the Ministry of
Education and Science of Ukraine under the Program
“Electrodynamics of layered composites with chiral
properties and multifunctional planar systems”, Project
No. 0112 U 000561 and the Ukrainian State Foundation
for Basic Research, Project F40.2/037.
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340 ISSN 1027-9636. . . 18, 4, 2013
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Submitted 17.10.2013
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