Periodic structures with spherical alumina particles: Transmission and reflection spectra
Coherent transmission and reflection coefficients of individual monolayers of spherical alumina particles in the wavelength range from 0.3 to 1.0 μm and multilayers consisting of a set of such monolayers are computed. Quasi-crystalline approximation of the theory for multiple scattering of waves...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2013
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irk-123456789-1176852017-05-27T03:06:42Z Periodic structures with spherical alumina particles: Transmission and reflection spectra Miskevich, A.A. Loiko, V.A. Coherent transmission and reflection coefficients of individual monolayers of spherical alumina particles in the wavelength range from 0.3 to 1.0 μm and multilayers consisting of a set of such monolayers are computed. Quasi-crystalline approximation of the theory for multiple scattering of waves and the transfer matrix method are used. Three types of coherent transmittance minima inherent to close-packed regular disperse systems are singled out and described. Spectra of a glass plate with one-sided and double-sided coating with alumina particles are analyzed. Systems for creating antireflection coatings, diffuse light scattering structures, selective and nonselective filters of transmitted and reflected light are considered. 2013 Article Periodic structures with spherical alumina particles: Transmission and reflection spectra / A.A. Miskevich, V.A. Loiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 2. — С. 177-184. — Бібліогр.: 17 назв. — англ. 1560-8034 PACS 42.25.Dd, 42.70.Df, 42.79.Kr http://dspace.nbuv.gov.ua/handle/123456789/117685 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Coherent transmission and reflection coefficients of individual monolayers of
spherical alumina particles in the wavelength range from 0.3 to 1.0 μm and multilayers
consisting of a set of such monolayers are computed. Quasi-crystalline approximation of
the theory for multiple scattering of waves and the transfer matrix method are used.
Three types of coherent transmittance minima inherent to close-packed regular disperse
systems are singled out and described. Spectra of a glass plate with one-sided and
double-sided coating with alumina particles are analyzed. Systems for creating
antireflection coatings, diffuse light scattering structures, selective and nonselective
filters of transmitted and reflected light are considered. |
| format |
Article |
| author |
Miskevich, A.A. Loiko, V.A. |
| spellingShingle |
Miskevich, A.A. Loiko, V.A. Periodic structures with spherical alumina particles: Transmission and reflection spectra Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Miskevich, A.A. Loiko, V.A. |
| author_sort |
Miskevich, A.A. |
| title |
Periodic structures with spherical alumina particles: Transmission and reflection spectra |
| title_short |
Periodic structures with spherical alumina particles: Transmission and reflection spectra |
| title_full |
Periodic structures with spherical alumina particles: Transmission and reflection spectra |
| title_fullStr |
Periodic structures with spherical alumina particles: Transmission and reflection spectra |
| title_full_unstemmed |
Periodic structures with spherical alumina particles: Transmission and reflection spectra |
| title_sort |
periodic structures with spherical alumina particles: transmission and reflection spectra |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/117685 |
| citation_txt |
Periodic structures with spherical alumina particles:
Transmission and reflection spectra / A.A. Miskevich, V.A. Loiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 2. — С. 177-184. — Бібліогр.: 17 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT miskevichaa periodicstructureswithsphericalaluminaparticlestransmissionandreflectionspectra AT loikova periodicstructureswithsphericalaluminaparticlestransmissionandreflectionspectra |
| first_indexed |
2025-07-08T12:37:23Z |
| last_indexed |
2025-07-08T12:37:23Z |
| _version_ |
1837082353363582976 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
177
PACS 42.25.Dd, 42.70.Df, 42.79.Kr
Periodic structures with spherical alumina particles:
Transmission and reflection spectra
A.A. Miskevich, V.A. Loiko
B.I. Stepanov Institute of Physics of NAS of Belarus,
68, Nezalezhnastsi ave., Minsk, 220072, Belarus,
E-mail: miskevic@dragon.bas-net.by, loiko@dragon.bas-net.by
Abstract. Coherent transmission and reflection coefficients of individual monolayers of
spherical alumina particles in the wavelength range from 0.3 to 1.0 μm and multilayers
consisting of a set of such monolayers are computed. Quasi-crystalline approximation of
the theory for multiple scattering of waves and the transfer matrix method are used.
Three types of coherent transmittance minima inherent to close-packed regular disperse
systems are singled out and described. Spectra of a glass plate with one-sided and
double-sided coating with alumina particles are analyzed. Systems for creating
antireflection coatings, diffuse light scattering structures, selective and nonselective
filters of transmitted and reflected light are considered.
Keywords: alumina particle, display, periodic structure, photonic crystal, transmittance
and reflectance.
Manuscript received 12.12.12; revised version received 25.02.13; accepted for
publication 19.03.13; published online 25.06.13.
1. Introduction
Alumina (aluminium oxide, Al2O3) is a promising
material for numerous applications in optics,
optoelectronics, electromechanics, display technologies,
etc. We consider spectra of spatially ordered systems
made of spherical alumina particles.
One of the approximations allowing to describe
light propagation through the layer of the spatially-
ordered particles is the quasi-crystalline approximation
(QCA) [1] of the statistical theory for multiple scattering
of waves. In this approximation, the transmittance and
reflectance are determined by the sum of waves scattered
by particles with accounting of their re-irradiation and
correlation of spatial location. Correlation is taken into
account by the radial distribution function (RDF) [2].
The method of the radial distribution function for
non-ideal planar photonic crystal (PPC) modeling has
been described previously in [3]. Its essence is as
follows. At first, we select the center of any particle as
the coordinate origin and evaluate the radii of
coordination circles of the ideal lattice and the number
of particle centers located in each circle. The dependence
of the number of particle centers on their distance in
lattice is a set of the infinitely narrow peaks at distances
which are equal to the coordination circles radii.
Accordingly, the radial distribution function of an ideal
photonic crystal (PC) has nonzero values at these
distances and equals to zero at the other ones. Actual
PCs typically have non-ideal lattice with “blurred”
coordination circles. It was shown [3] that the expression
for calculating the radial distribution function g(u) of the
real PPC (consisting of spherical monodisperse particles)
can be written as:
i
i
i
i
u
Ru
uR
N
u
)(σ2
exp
)σ(2
1
2
)g(
2
2
1
0 . (1)
Here, σ(u) is the blurring function which
characterizes the “blurring” (broadening) of coordination
circles (into the fuzzy ring-shaped areas) as a function of
the distance u relatively to the coordinate origin. It is
reasonable to use the linear blurring function:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
178
bauu 0)σ( . (2)
The function (2) takes into account asymmetry of
individual peaks of RDF, which are observed in
experiment. It allows to simulate RDFs of PCs in a wide
range of their ordering degree. The RDFs obtained with
using (2) are well adapted to be used in QCA. Moreover,
the calculation of RDF and, consequently, coherent
transmission and reflection coefficients of PPC in QCA,
is fast and requires low amounts of computational
resources. In the equations (1) and (2) u = R/D is the
dimensionless distance expressed in diameters D of
particles (u ≥ 1); R is the distance in a monolayer plane
relatively to the coordinate origin; ρ0 – averaged
numerical particle concentration in the monolayer; Ni –
number of particle centers in the coordination circle with
the radius Ri of an ideal crystal; σ0 – dimensionless
initial dispersion of coordination circle radii expressed in
particle diameters; a and b are the coefficients chosen as
dependent on the filling coefficient (the ratio of the area
of particle projections on the layer plane to the area
where they are distributed) of layer, type of lattice, and
σ0. The ordering degree and scale of the spatial order of
the simulated crystal can be specified by changing of σ0,
and a and b coefficients, respectively.
Note that at σ(u) = const, Eq. (1) is transformed
into the known expression for RDF which describes the
Gaussian blurring of coordination circles in crystal
lattice [4].
This work is the theoretical study of coherent
transmittance and reflectance inherent to a layered
system of plane-parallel monolayers (multilayer) of
spatially-ordered monodisperse spherical alumina
particles. The system is illuminated normally to the
planes of monolayers. At first, in QCA [1, 3, 5, 6] the
spectra of coherent transmission and reflection
coefficients of individual monolayers are calculated.
Then, using the obtained coefficients and the transfer
matrix method (TMM) [7, 8] the coherent transmission
and reflection coefficients of a multilayer are evaluated.
Spectra of coherent transmission and reflection
coefficients of the glass plate with one- and double-sided
coating with alumina particles are investigated.
Antireflection coatings, selective and nonselective
reflectors, transmission and reflection filters, diffuse
scattering plates are considered as examples for possible
application of the obtained results.
2. Spectra of coherent transmission
and refraction of monolayer
Consider a plane-parallel monolayer of aluminium oxide
particles. Spectral dependence of the real part of the
refractive index for alumina is taken from the ref. [9].
Pay attention that imaginary part of the refractive index
in the considered spectral range is zero.
The spectra of coherent transmission Tс and
reflection Rс coefficients of regularly-packed monolayer
(PPC) with hexagonal lattice and partially-ordered
monolayer consisting of monodisperse spherical Al2O3
particles are shown in Fig. 1. Note that in this work, the
notation “regularly-packed” designates non-ideal
packing of particles in PPC’s lattice. The monolayer
filling coefficient η = 0.5. Coherent transmission and
reflection coefficients of monolayers were calculated in
QCA [1, 3, 5, 6]. The radial distribution function of PPC
was evaluated by the method described schematically
above with using the blurring function (2) for
coordination circles as follows: σ(u) = σ0u, σ0 = 0.01. To
simulate RDF of the partially-ordered monolayer of
particles, the solution of Ornstein-Zernike integral
equation [10] was used. It was numerically calculated in
the Percus-Yevick approximation [11] for a system of
hard spheres by the iteration method described in [12].
The obtained results demonstrate influence of spatial
arrangement of particles on coherent transmittance and
reflectance of monolayer. In the spectra of partially-
ordered monolayers, the transmission minimum caused
by interference of incident and scattered waves takes
place. This minimum is situated in the vicinity of the
extinction efficiency factor maximum. For a partially-
ordered monolayer of monodisperse spherical Al2O3
particles with the diameter D = 0.3 μm and the filling
coefficient η = 0.5, this minimum occurs at λ ≈ 0.357 μm
(see minimum 1′ in the dashed line in Fig. 1a). Its depth
and position depend on the concentration and optical
properties of particles. Such a minimum also takes place
in the monolayer with a regular packing of particles.
However, its position is shifted relatively to the first one.
For PPC with hexagonal lattice, it occurs at λ ≈
0.339 μm (see the minimum 1 in the solid line in
Fig. 1a). Unlike the spectrum of the partially-ordered
monolayer, in addition, the spectrum of PPC has a sharp
deep minimum caused by the particle location
periodicity. It can have a fine structure. For parameters
indicated in the figure caption, this minimum is located
in the wavelength range 0.52 to 0.53 μm (see the
minimum 2 in the solid line in Fig. 1a). There is also a
sharp maximum of the coherent reflection coefficient
(see the solid line in Fig. 1b) within the same range.
Such sharp maxima observed in light diffraction by
metallic gratings are known as Wood’s anomalies [13].
Thus, at normal illumination of the planar monolayer of
regularly-packed particles (planar photonic crystal),
there is an effect similar to the photonic band gap effect,
which takes place in PCs at light propagation along the
periodicity of the multilayer structure.
The transmission minima (1, 1′) can be very deep.
Their values depend on the concentration and optical
characteristics of particles. For example, the value of the
transmission minimum 1 of PPC with hexagonal lattice
of particles with the diameter D = 0.3 μm is 810cT at
the monolayer filling coefficient η = 0.533 and
wavelength λ ≈ 0.327 μm.
The particles are non-absorbing within the spectral
range under consideration. As a result, the incoherent
(diffuse) component Σinc of light transmitted and
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
179
reflected by a monolayer is equal to ccinc RT 1 .
The results of calculation of the coherent transmittance
and reflectance sum Tc+Rc for PPC аnd partially-ordered
monolayer are displayed in Fig. 1b. For PPC at the
wavelengths λ ≥ 0.52 μm, practically all the energy of
incident light is either transmitted through the regularly-
packed monolayer straight forward or reflected by it
straight backward. Accordingly, the diffuse (incoherent)
component is absent. Pay attention that the minimum 2
is located at the beginning of this spectral range.
Fraction of the incoherent component of radiation
Σinc transmitted and reflected by the partially-ordered
monolayer is nonzero over the entire considered
wavelength range. For the parameters indicated in the
figure caption, the incoherent component Σinc is less than
0.1 at λ ≥ ~0.6 μm and decreases with increasing the
wavelength. With the filling coefficient decreasing, the
value of Σinc decreases in this spectral range.
0.3 0.4 0.5 0.6 0.7 0.8
1E-5
1E-4
1E-3
0.01
0.1
1
1'
2
Tc
m
(a)
1
0,30 0,35 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80
0,0
0,2
0,4
0,6
0,8
1,0
Rc, hex
Tc+Rc, hex
Rc, part.ord.
Tc+Rc, part.ord.Tc,Rc
, m
(b)
Fig. 1. Spectral dependences of the coherent transmission
Tс (a), reflection Rс coefficients and their sum Tс+Rс (b) for the
monolayer of alumina particles: regularly-packed monolayer
(PPC) with the hexagonal lattice (solid lines) and partially-
ordered monolayer (dashed lines). Gray solid line is Tс+Rс of
regularly-packed monolayer, gray dashed line is Tс+Rс of
partially-ordered monolayer (b). The monolayer filling
coefficient η = 0.5. The radial distribution function of PPC is
calculated at σ(u) = σ0u, σ0 = 0.01. The particle diameter D =
0.3 μm for all the monolayers. The numbers 1, 1′, and 2
indicate the types of the transmission minima (see description
in the text).
3. Light propagation in a multilayer
Consider light propagation in a multilayer with taking
into account multiple reflections between monolayers by
using the transfer matrix method [7, 8]. We suppose the
monolayers as interfaces and the spaces between them as
layers.
In Fig. 2, the coherent transmission and reflection
coefficients of multilayer consisting of 12, 30 and 60
monolayers with hexagonal packing of monodisperse
spherical Al2O3 particles in air are displayed. The
numerical calculation results were obtained for the
spectral ranges where the value of incoherent component
for the monolayer ccinc RT 1 was small
(according to the results presented in Fig. 1b, the Σinc-
value is small at λ ≥ 0.6 μm). The diameter of particles
D = 0.3 μm, monolayers filling coefficient η = 0.5,
distances between adjacent monolayers h = 0.3 μm. The
0.6 0.7 0.8 0.9 1.0
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
3
12 monolayers
30 monolayers
60 monolayersT
c
m
(a)
0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
12 monolayers
30 monolayers
60 monolayers
R
c
m
(b)
Fig. 2. Spectra of the coherent transmittance Tс (a) and
reflectance Rс (b) of multilayer consisting of 12, 30, and 60
PPCs with hexagonal packing of Al2O3 particles in air.
Distances between adjacent monolayers are h = 0.3 μm.
The filling coefficient of each monolayer η = 0.5, diameter
of particles D = 0.3 μm. The radial distribution function of
PPC is calculated at σ(u) = σ0u, σ0 = 0.01. The digit 3
indicates the transmission minimum (photonic band gap)
inherent to the layered disperse system (see description in
the text).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
180
calculation for the monolayer (in this case it is a planar
photonic crystal) was implemented into QCA. The
calculation for the multilayer was carried out using
TMM. We name this combination of the methods as
QCA-TMM.
The photonic band gap (PBG) caused by the
multilayer periodic structure arises with increasing the
number of monolayers. It is formed in the wavelength
range ~0.75 to 0.77 μm for the considered parameters of
the system. See the transmission minimum indicated as 3
in Fig. 2а. As seen from the results in Fig. 2, already for
12 identical monolayers the noticeable PBG arises:
Tc ≈ 0.12 at λ ≈ 0.761 μm. For 30 monolayers, the
coherent transmission coefficient Tc ≈ 4101.8 at λ ≈
0.757 μm, and for 60 monolayers Tc ≈ 7107.1 at λ ≈
0.756 μm. In the same spectral range, the coherent
reflectance maximum comparable by its width with the
coherent transmittance minimum 3 is formed.
As it follows from the results of calculations, these
three types of coherent transmittance minima are
inherent to the close-packed regular disperse system (see
Fig. 1a where they are indicated by digits 1, 2, and digit
3 in Fig. 2а). The transmittance minimum 1 is caused by
the concentration and optical properties of particles in a
single monolayer. It takes place both for PPC and
partially-ordered monolayer. The transmittance
minimum 2 is caused by regularity in particle locations
within a single monolayer. It takes place for PPC at sizes
of particles and distances between them compared with
the wavelength of the incident light. The transmittance
minimum 3 is PBG. It takes place in spectra of the
periodic system at wave propagation along periodicity of
the multilayer structure.
The depths, widths, locations, and the number of
PBGs of the multilayer depend on distances between its
monolayers. Calculations for the same monolayer
parameters as in Fig. 2 show: when the distance between
monolayers is changed from 0.3 up to 0.5 μm, the depth
and width of PBG are increased, and PBG location is
shifted to the larger wavelengths. With further increasing
the distance, location of this PBG is shifted to the larger
wavelengths, and PBG depth decreases. Simultaneously,
in the range of short wavelengths the additional more
narrow PBGs appear. The longer is the distance between
monolayers, the higher is the number of appearing
PBGs. The results show that multilayers consisting of
PPCs can be used to develop and manufacture the band-
pass radiation filters in a wide spectral range.
Comparison with the results obtained for the
partially-ordered monolayers has shown that the spectra
of the system of partially-ordered monolayers and the
system of PPCs have both general and different features.
In particular, for both systems (with the same
particle diameter D = 0.3 μm) PBG formation within the
range λ ≈0.73…0.78 μm and an increase in the number
of spectral oscillations with the number of monolayers
takes place. That is caused by periodicity of the
multilayer structure along the direction of the incident
wave and multiple wave reflections in a system. The
absolute values of the minima of transmission in PBG
differ slightly.
There are significant differences in PBG formation
between the systems of planar photonic crystals and
partially-ordered monolayers of particles. PBG of the
PPCs system is more symmetric and has more sharp
edges in comparison with PBG of partially-ordered
monolayers system. The values of the reflection maxima
in PBG are also noticeably different. Coherent
reflectance maximum of the partially-ordered
monolayers system in PBG Rc,max ≈ 0.81 (at the
wavelength λ ≈ 0.738 μm), whereas the respective
maximum of the PPC system Rc,max ≈ 1.
4. Glass plate covered with alumina particles:
transmittance and reflectance spectrum
The coefficient of light reflection by a plane-parallel
glass plate in air is approximately 0.08 (~0.04 from each
glass-air interface). To reduce the reflection of light
from the plate, the antireflection coatings on its surfaces
are used. These are typically the thin dielectric layers of
a certain refractive indices and thicknesses. It is
promising to create such coatings from aluminum oxide
[14, 15], due to their relative simplicity and low cost
production.
Consider a two-layer system consisting of a
homogeneous plane-parallel layer of aluminum oxide on
a plane-parallel glass plate (Fig. 3).
Let us calculate the coherent transmission and
reflection coefficients of this system under TMM. The
technique for calculating the reflection and transmission
of a system consisting of sufficiently thin layers is based
on the use of complex amplitude reflection and
transmission coefficients. When the system includes a
thick layer (e.g., layer L2 in Fig. 3), the energy
transmission and reflection coefficients are used instead
of the amplitude ones [7]. Coherent transmission and
reflection coefficients of this system are shown in Fig. 4.
In this figure, the transmission and reflection spectra of
the plane-parallel glass plate without coating are shown
as well.
Glass
Incident light
Rс
Tс
L0
L1
L2
L3
If1
If2
If3
h1
h2
Al2O3
Fig. 3. Schematic presentation of homogeneous plane-parallel
alumina layer on the surface of a plane-parallel glass plate
(side view).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
181
0.5 0.6 0.7 0.8 0.9 1.0
0.84
0.85
0.86
0.87
0.88
0.89
0.90
0.91
0.92
0.93
T
c
m
(a)
0.05
0.1
0.51.0
Uncoated glass
0.5 0.6 0.7 0.8 0.9 1.0
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
R
c
m
(b)
Uncoated glass
0.05
0.1
0.5
1.0
Fig. 4. Spectra of the coherent transmittance Tс (a) and
reflectance Rс (b) of an uncoated plane-parallel glass plate
(thick line) and system of plane-parallel homogeneous Al2O3
layer on the surface of glass plate (thin lines) at different
thicknesses of Al2O3 layer. The thickness of glass plate is
2 mm. Numbers near lines indicate the thicknesses of Al2O3
layer in micrometers.
The results displayed in Fig. 4 were obtained under
the assumption that there are three interfaces If1…If3 in
the system (see Fig. 3). It was assumed that the
interfaces are flat boundaries of Al2O3 layer L1 (with the
thickness h1) and glass layer L2 (with the thickness h2).
The amplitude reflection and transmission coefficients of
planar interfaces between the layers are calculated by the
Fresnel formulae [16]. The layer L2 of glass has the
thickness h2 = 2 mm, which is much higher than the
wavelengths under consideration. Therefore, L2 is a
“thick” layer, and the energy transmission and reflection
coefficients are used accordingly to [7]. The layers L0
and L3 are semi-infinite layers of the environment (air).
The refractive indexes of the Corning 1737 glass were
used in the calculations [17]. As it follows from the
results shown in the figure, coating of glass with the
homogeneous layer of alumina does not reduce the
coherent reflection coefficient of a system in comparison
with that of the uncoated glass plate. Changes in the
Al2O3 layer thickness lead to a change in the oscillation
frequency of the transmission and reflection as a
function of wavelength.
Glass
Incident light
Rс
Tс
L0
L1
L2
L3
If1
If2
If3
h1
h2
Al2O3
Fig. 5. Schematic presentation of system consisting of
monolayer of monodisperse spherical Al2O3 particles on glass
(side view).
Now consider a system consisting of a glass plate
covered with a monolayer of Al2O3 particles on its upper
surface (Fig. 5).
Let us calculate coherent transmission and
reflection coefficients of this system using the scheme
described above. Find the conditions under which a
coating of alumina particles will serve as an
antireflection layer (at normal illumination). Calculation
of the coherent transmission and reflection coefficients
of the system is fulfilled with the assumption that it
consists of three interfaces: a monolayer of alumina
particles If1 and the two flat interfaces of glass plate If2
and If3 (see Fig. 5). Accordingly, there are two layers
between interfaces: L1 with the thickness h1 of air and L2
with the thickness h2 of glass. The thickness h1 is equal
to the radius of particles. Layers L0 and L3 are semi-
infinite layers of the environment (air). Calculation of
coherent transmission and reflection of interface If1
(planar PC) is carried out in QCA with σ(u) = σ0u,
σ0 = 0.01, and of interfaces If2 and If3 (planar boundaries
between the layers) by using the Fresnel formulae.
Depicted in Fig. 6 are the coherent transmission
and reflection spectra of the uncoated glass plate and the
plate coated with a monolayer possessing the hexagonal
lattice of monodisperse spherical alumina particles. As it
follows from the presented results, when the size of
Al2O3 particles is much smaller than wavelengths, one
can obtain an almost uniform spectrum with the
reflectance smaller than the reflectance, and the
transmittance larger than the transmittance of the
uncoated glass plate. In the visible spectral range, when
the coating monolayer contains Al2O3 particles with the
diameter D = 0.1 μm, the reflection coefficient decreases
down to ~0.047. With increasing the Al2O3 particle
diameter up to 0.5 μm, the transmittance minimum and
reflectance maximum in the spectrum within the range
of wavelengths 0.85 to 0.88 μm appear. They are caused
by regularity in particle locations inside the monolayer.
For sufficiently large particles, in some spectral ranges
the values of the coherent reflectance are lower than the
reflectance of the bare interface (for the glass-air
boundary Rc ≈ 0.04), and the coherent transmission
coefficients are small. It implies that the major part of
the radiation is incoherent.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
182
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.90
0.92
0.94
0.96
T
c
m
(a)
Uncoated glass
0.05
0.1
0.51.0
0.5 0.6 0.7 0.8 0.9 1.0
1E-3
0.01
0.1
1
0.5
R
c
m
(b)
0.05
0.1
Uncoated
glass
1.0
Fig. 6. Spectra of the coherent transmittance Tс (a) and
reflectance Rс (b) of the uncoated plane-parallel glass plate
(thick line) and system of monolayer with the hexagonal
packing of monodisperse spherical Al2O3 particles on this plate
for various particle sizes. The thickness of plate is 2 mm,
filling coefficient of monolayer η = 0.5. Ciphers near the lines
indicate diameters of Al2O3 particles in micrometers.
Glass
Incident light
Rс
Tс
L0
L1
L2
L3
If1
If2
If3
h1
h2
If4
L4
h3
Al2O3
Fig. 7. Schematic presentation of multilayer system with two
monolayers of monodisperse spherical Al2O3 particles on top
and bottom surfaces of the glass plate.
To simultaneously reduce the coherent reflectance
and increase the coherent transmittance, we should
“exclude” reflections not only from the top, but also
from the bottom interface of the plate. To solve this
problem, consider the double-sided coating (Fig. 7).
Using QCA-TMM, calculate the spectra of this system
when monolayers of aluminum oxide particles are
regularly-packed and partially-ordered. Fig. 8 shows the
spectral dependences of coherent transmittance and
reflectance when the top and bottom monolayers of
Al2O3 particles are identical. Thin solid lines depict the
spectra of the system with the regularly-packed
monolayers (PPCs with hexagonal lattice), the dashed
lines show the spectra of systems with partially-ordered
monolayers of Al2O3 particles. Pay attention that the
displayed results have a good accuracy in the spectral
range where the value of Tc+Rc ≈ 1. As it follows from
the calculation results, the coatings with monolayers of
Al2O3 particles on both surfaces of the glass plate can
significantly reduce the coherent reflection and increase
the coherent transmission coefficients of the system as
compared with those for the one-sided coated system. At
a double-sided coating with identical monolayers of
Al2O3 particles of diameter D = 0.1 μm, the calculated
coherent reflection coefficient Rc < 0.025 in the visible
spectral range with the minimum lower than 0.01. If the
size of Al2O3 particles significantly smaller than the
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.80
0.85
0.90
0.95
1.00
T
c
m
(a)
0.05
0.1
0.5
1.0
Uncoated glass
0.5 0.6 0.7 0.8 0.9 1.0
1E-3
0.01
0.1
1
R
c
m
(b)
0.05
0.1
0.5
1.0
Uncoated
glass
Fig. 8. Spectra of the coherent transmittance Tс (a) and
reflectance Rс (b) of the uncoated plane-parallel glass plate
(thick line) and system of identical partially-ordered (dotted
lines) and regularly-packed monolayers with hexagonal
packing of Al2O3 particles (thin solid lines, σ(u) = σ0u,
σ0 = 0.01) on top and bottom surfaces of the glass plate for
various particle sizes. The thickness of glass plate is 2 mm,
filling coefficient of monolayers η = 0.5. Diameters (in
micrometers) of particles are indicated near the lines.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
183
incident light wavelength, the differences in coherent
transmittance and reflectance spectra of the systems
coated with PPCs and partially-ordered monolayers are
negligible (see lines for D = 0.05 μm and D = 0.1 μm in
Fig. 8). If the particles size is comparable with the
incident light wavelength, the differences in the spectra
of systems with PPCs and partially-ordered monolayers
become significant. They are more pronounced in the
spectral range of the transmission minimum and
reflection maximum (see solid lines for D = 0.5 μm
within the range of wavelengths
~0.85 μm < λ < ~0.88 μm in Fig. 8) caused by the
regularity of particle packing in a planar PC. It follows
that if the creation of antireflection coatings from the
Al2O3 particles does not required their high ordering,
then for the construction of selective reflectors and
transmission filters the ordering of particles in
monolayers plays a crucial role.
0.5 0.6 0.7 0.8 0.9 1.0
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.00
0.1, 0.12
0.12, 0.12
T
c
m
(a)
Uncoated glass
0.1, 0.1
0.5 0.6 0.7 0.8 0.9 1.0
1E-3
0.01
0.1
0.1, 0.12
0.12, 0.12R
c
m
(b)
0.1, 0.1
Uncoated glass
Fig. 9. Spectra of the coherent transmittance Tс (a) and
reflectance Rс (b) of the uncoated plane-parallel glass plate
(thick line) and system of highly-ordered monolayers with the
hexagonal packing of Al2O3 particles (thin solid lines,
σ(u) = σ0u, σ0 = 0.01) on top and bottom surfaces of the glass
plate. The thickness of the glass plate is 2 mm, filling
coefficient of monolayers η = 0.5. Ciphers near the lines before
and after comma indicate diameters of Al2O3 particles (in
micrometers) of top and bottom monolayer, respectively.
Consider the multilayer system (Fig. 7) in which
the top and bottom surfaces of the glass plate are coated
with monolayers containing spherical particles of
different diameters (which are monodisperse in each
monolayer). Fig. 9 shows the transmittance and
reflectance spectra of a multilayer system when the
surfaces of the glass plate are coated with monolayers
with a hexagonal lattice of particles. As it follows from
the results of calculations presented in the figure, if the
top and bottom interfaces of the plate have different
coatings, the spectral dependences of coherent
transmittance and reflectance are more homogeneous as
compared with the case of identical coatings. For the
considered system, the coherent reflection coefficient in
the visible spectral range is less than 0.02 with the
minimum value less than 0.003.
5. Conclusion
The spectra of the coherent transmission and reflection
coefficients of monolayers of Al2O3 particles have been
calculated in the wavelength range from 0.3 up to
1.0 μm.
Three types of coherent transmittance minima
inherent for regular disperse system have been observed
and described.
Coherent transmission and reflection coefficients of
the glass plate coated with spherical Al2O3 particles have
been calculated. As it follows from the results, creation
of antireflection coatings do not need a high ordering of
particles. Inversely, to create selective reflectors and
transmission filters the arrangement of particles in
monolayer has a crucial role. It is shown that the
reflection coefficient can be reduced significantly by
using the two-sided coating of glass plate with
monolayers of alumina particles.
The obtained results can be used in the design and
manufacture of the antireflection coatings, diffuse light
scattering structures, transmission and reflection filters
(for example, in display and solar cells applications), the
lattice quality control in photonic crystals, etc.
Acknowledgement
This work was supported in part by the state scientific
and technical program of Belarus “Electronics and
photonics”.
References
1. M. Lax, Multiple scattering of waves II. The
effective field for dense systems // Phys. Rev. 85,
p. 621-629 (1952).
2. J.M. Ziman, Models of Disorder. Cambridge,
Cambridge University, 1979.
3. A.A. Miskevich, V.A. Loiko, Coherent
transmission and reflection of a two-dimensional
planar photonic crystal // J. Experiment. Theor.
Phys. 113(1), p. 1-13 (2011).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
184
4. A.F. Skryshevskiy, Structural Analysis of Liquids
and Amorphous Solids. Moscow, Vysshaya Shkola,
1980 (in Russian).
5. V.A. Loiko, A.A. Miskevich, Light propagation
through a monolayer of discrete scatterers: analysis
of coherent transmission and reflection coefficients
// Appl. Opt. 44(18), p. 3759-3768 (2005).
6. K.M. Hong, Multiple scattering of electromagnetic
waves by a crowded monolayer of spheres:
Application to migration imaging films // J. Opt.
Soc. Am. 70, p. 821-826 (1980).
7. C.C. Katsidis, D.I. Siapkas, General transfer-matrix
method for optical multilayer systems with
coherent, partially coherent, and incoherent
interference // Appl. Opt, 41, p. 3978-3987 (2002).
8. E. Centurioni, Generalized matrix method for
calculation of internal light energy flux in mixed
coherent and incoherent multilayers // Appl. Opt.
44(35), p. 7532-7539 (2005),
9. E.D. Palik, editor, Handbook of Optical Constants
of Solids. San Diego, Academic, 1985.
10. L.S. Ornstein, F. Zernike, Accidental deviations of
density and opalescence at the critical point of a
single substance // Proc. Acad. Sci. 17, p. 793-806
(1914).
11. J.K. Percus, G.J. Yevick, Analysis of classical
statistical mechanics by means of collective
coordinates // Phys. Rev. 110(1), p. 1-13 (1958).
12. A.P. Ivanov, V.A. Loiko, V.P. Dick, Propagation
of Light in Close-packed Disperse Media. Minsk,
Nauka i Tekhnika, 1988 (in Russian).
13. R.W. Wood, Anomalus diffracting gratings // Phys.
Rev. 48, p. 928-937 (1935).
14. N.I. Mukhurov, Aluminium-oxide Micro-
nanostructures for Microelectromechanical
systems. Minsk, Bestprint, 2004 (in Russian).
15. A.A. Lutich, M.B. Danailov, S. Volchek,
V.A. Yakovtseva, V.A. Sokol, S.V. Gaponenko,
Birefringence of nanoporous alumina: dependence
on structure parameters // Appl. Phys. B, 84, p. 327-
331 (2006).
16. M. Born, E. Wolf, Principles of Optics. Cambridge
University Press, 2002.
17. Corning site http://www.corning.com
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 177-184.
PACS 42.25.Dd, 42.70.Df, 42.79.Kr
Periodic structures with spherical alumina particles:
Transmission and reflection spectra
A.A. Miskevich, V.A. Loiko
B.I. Stepanov Institute of Physics of NAS of Belarus,
68, Nezalezhnastsi ave., Minsk, 220072, Belarus,
E-mail: miskevic@dragon.bas-net.by, loiko@dragon.bas-net.by
Abstract. Coherent transmission and reflection coefficients of individual monolayers of spherical alumina particles in the wavelength range from 0.3 to 1.0 μm and multilayers consisting of a set of such monolayers are computed. Quasi-crystalline approximation of the theory for multiple scattering of waves and the transfer matrix method are used. Three types of coherent transmittance minima inherent to close-packed regular disperse systems are singled out and described. Spectra of a glass plate with one-sided and double-sided coating with alumina particles are analyzed. Systems for creating antireflection coatings, diffuse light scattering structures, selective and nonselective filters of transmitted and reflected light are considered.
Keywords: alumina particle, display, periodic structure, photonic crystal, transmittance and reflectance.
Manuscript received 12.12.12; revised version received 25.02.13; accepted for publication 19.03.13; published online 25.06.13.
1. Introduction
Alumina (aluminium oxide, Al2O3) is a promising material for numerous applications in optics, optoelectronics, electromechanics, display technologies, etc. We consider spectra of spatially ordered systems made of spherical alumina particles.
One of the approximations allowing to describe light propagation through the layer of the spatially-ordered particles is the quasi-crystalline approximation (QCA) [1] of the statistical theory for multiple scattering of waves. In this approximation, the transmittance and reflectance are determined by the sum of waves scattered by particles with accounting of their re-irradiation and correlation of spatial location. Correlation is taken into account by the radial distribution function (RDF) [2].
The method of the radial distribution function for non-ideal planar photonic crystal (PPC) modeling has been described previously in [3]. Its essence is as follows. At first, we select the center of any particle as the coordinate origin and evaluate the radii of coordination circles of the ideal lattice and the number of particle centers located in each circle. The dependence of the number of particle centers on their distance in lattice is a set of the infinitely narrow peaks at distances which are equal to the coordination circles radii. Accordingly, the radial distribution function of an ideal photonic crystal (PC) has nonzero values at these distances and equals to zero at the other ones. Actual PCs typically have non-ideal lattice with “blurred” coordination circles. It was shown [3] that the expression for calculating the radial distribution function g(u) of the real PPC (consisting of spherical monodisperse particles) can be written as:
(
)
å
÷
÷
ø
ö
ç
ç
è
æ
-
-
p
p
r
=
-
i
i
i
i
u
R
u
u
R
N
u
)
(
σ
2
exp
)
σ(
2
1
2
)
g(
2
2
1
0
.
(1)
Here, σ(u) is the blurring function which characterizes the “blurring” (broadening) of coordination circles (into the fuzzy ring-shaped areas) as a function of the distance u relatively to the coordinate origin. It is reasonable to use the linear blurring function:
(
)
b
au
u
+
s
=
0
)
σ(
.
(2)
The function (2) takes into account asymmetry of individual peaks of RDF, which are observed in experiment. It allows to simulate RDFs of PCs in a wide range of their ordering degree. The RDFs obtained with using (2) are well adapted to be used in QCA. Moreover, the calculation of RDF and, consequently, coherent transmission and reflection coefficients of PPC in QCA, is fast and requires low amounts of computational resources. In the equations (1) and (2) u = R/D is the dimensionless distance expressed in diameters D of particles (u ≥ 1); R is the distance in a monolayer plane relatively to the coordinate origin; ρ0 – averaged numerical particle concentration in the monolayer; Ni – number of particle centers in the coordination circle with the radius Ri of an ideal crystal; σ0 – dimensionless initial dispersion of coordination circle radii expressed in particle diameters; a and b are the coefficients chosen as dependent on the filling coefficient (the ratio of the area of particle projections on the layer plane to the area where they are distributed) of layer, type of lattice, and σ0. The ordering degree and scale of the spatial order of the simulated crystal can be specified by changing of σ0, and a and b coefficients, respectively.
Note that at σ(u) = const, Eq. (1) is transformed into the known expression for RDF which describes the Gaussian blurring of coordination circles in crystal lattice [4].
This work is the theoretical study of coherent transmittance and reflectance inherent to a layered system of plane-parallel monolayers (multilayer) of spatially-ordered monodisperse spherical alumina particles. The system is illuminated normally to the planes of monolayers. At first, in QCA [1, 3, 5, 6] the spectra of coherent transmission and reflection coefficients of individual monolayers are calculated. Then, using the obtained coefficients and the transfer matrix method (TMM) [7, 8] the coherent transmission and reflection coefficients of a multilayer are evaluated.
Spectra of coherent transmission and reflection coefficients of the glass plate with one- and double-sided coating with alumina particles are investigated. Antireflection coatings, selective and nonselective reflectors, transmission and reflection filters, diffuse scattering plates are considered as examples for possible application of the obtained results.
2. Spectra of coherent transmission
and refraction of monolayer
Consider a plane-parallel monolayer of aluminium oxide particles. Spectral dependence of the real part of the refractive index for alumina is taken from the ref. [9]. Pay attention that imaginary part of the refractive index in the considered spectral range is zero.
The spectra of coherent transmission Tс and reflection Rс coefficients of regularly-packed monolayer (PPC) with hexagonal lattice and partially-ordered monolayer consisting of monodisperse spherical Al2O3 particles are shown in Fig. 1. Note that in this work, the notation “regularly-packed” designates non-ideal packing of particles in PPC’s lattice. The monolayer filling coefficient η = 0.5. Coherent transmission and reflection coefficients of monolayers were calculated in QCA [1, 3, 5, 6]. The radial distribution function of PPC was evaluated by the method described schematically above with using the blurring function (2) for coordination circles as follows: σ(u) = σ0u, σ0 = 0.01. To simulate RDF of the partially-ordered monolayer of particles, the solution of Ornstein-Zernike integral equation [10] was used. It was numerically calculated in the Percus-Yevick approximation [11] for a system of hard spheres by the iteration method described in [12]. The obtained results demonstrate influence of spatial arrangement of particles on coherent transmittance and reflectance of monolayer. In the spectra of partially-ordered monolayers, the transmission minimum caused by interference of incident and scattered waves takes place. This minimum is situated in the vicinity of the extinction efficiency factor maximum. For a partially-ordered monolayer of monodisperse spherical Al2O3 particles with the diameter D = 0.3 μm and the filling coefficient η = 0.5, this minimum occurs at λ ≈ 0.357 μm (see minimum 1′ in the dashed line in Fig. 1a). Its depth and position depend on the concentration and optical properties of particles. Such a minimum also takes place in the monolayer with a regular packing of particles. However, its position is shifted relatively to the first one. For PPC with hexagonal lattice, it occurs at λ ≈ 0.339 μm (see the minimum 1 in the solid line in Fig. 1a). Unlike the spectrum of the partially-ordered monolayer, in addition, the spectrum of PPC has a sharp deep minimum caused by the particle location periodicity. It can have a fine structure. For parameters indicated in the figure caption, this minimum is located in the wavelength range 0.52 to 0.53 μm (see the minimum 2 in the solid line in Fig. 1a). There is also a sharp maximum of the coherent reflection coefficient (see the solid line in Fig. 1b) within the same range. Such sharp maxima observed in light diffraction by metallic gratings are known as Wood’s anomalies [13]. Thus, at normal illumination of the planar monolayer of regularly-packed particles (planar photonic crystal), there is an effect similar to the photonic band gap effect, which takes place in PCs at light propagation along the periodicity of the multilayer structure.
The transmission minima (1, 1′) can be very deep. Their values depend on the concentration and optical characteristics of particles. For example, the value of the transmission minimum 1 of PPC with hexagonal lattice of particles with the diameter D = 0.3 μm is
8
10
-
»
c
T
at the monolayer filling coefficient η = 0.533 and wavelength λ ≈ 0.327 μm.
The particles are non-absorbing within the spectral range under consideration. As a result, the incoherent (diffuse) component Σinc of light transmitted and reflected by a monolayer is equal to
(
)
c
c
inc
R
T
+
-
=
S
1
. The results of calculation of the coherent transmittance and reflectance sum Tc+Rc for PPC аnd partially-ordered monolayer are displayed in Fig. 1b. For PPC at the wavelengths λ ≥ 0.52 μm, practically all the energy of incident light is either transmitted through the regularly-packed monolayer straight forward or reflected by it straight backward. Accordingly, the diffuse (incoherent) component is absent. Pay attention that the minimum 2 is located at the beginning of this spectral range.
Fraction of the incoherent component of radiation Σinc transmitted and reflected by the partially-ordered monolayer is nonzero over the entire considered wavelength range. For the parameters indicated in the figure caption, the incoherent component Σinc is less than 0.1 at λ ≥ ~0.6 μm and decreases with increasing the wavelength. With the filling coefficient decreasing, the value of Σinc decreases in this spectral range.
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.2
0.4
0.6
0.80
0.85
0.90
0.95
1.00
T
c
l,m
m
(a)
0.05
0.1
0.5
1.0
Uncoated glass
0.3
0.4
0.5
0.6
0.7
0.8
1E-5
1E-4
1E-3
0.01
0.1
1
1'
2
T
c
l,m
m
(a)
1
0,300,350,400,450,500,550,600,650,700,750,80
0,0
0,2
0,4
0,6
0,8
1,0
Rc, hex
Tc+Rc, hex
Rc, part.ord.
Tc+Rc, part.ord.
T
c
,R
c
, m
(b)
Fig. 1. Spectral dependences of the coherent transmission Tс (a), reflection Rс coefficients and their sum Tс+Rс (b) for the monolayer of alumina particles: regularly-packed monolayer (PPC) with the hexagonal lattice (solid lines) and partially-ordered monolayer (dashed lines). Gray solid line is Tс+Rс of regularly-packed monolayer, gray dashed line is Tс+Rс of partially-ordered monolayer (b). The monolayer filling coefficient η = 0.5. The radial distribution function of PPC is calculated at σ(u) = σ0u, σ0 = 0.01. The particle diameter D = 0.3 μm for all the monolayers. The numbers 1, 1′, and 2 indicate the types of the transmission minima (see description in the text).
3. Light propagation in a multilayer
Consider light propagation in a multilayer with taking into account multiple reflections between monolayers by using the transfer matrix method [7, 8]. We suppose the monolayers as interfaces and the spaces between them as layers.
In Fig. 2, the coherent transmission and reflection coefficients of multilayer consisting of 12, 30 and 60 monolayers with hexagonal packing of monodisperse spherical Al2O3 particles in air are displayed. The numerical calculation results were obtained for the spectral ranges where the value of incoherent component for the monolayer
(
)
c
c
inc
R
T
+
-
=
S
1
was small (according to the results presented in Fig. 1b, the Σinc-value is small at λ ≥ 0.6 μm). The diameter of particles D = 0.3 μm, monolayers filling coefficient η = 0.5, distances between adjacent monolayers h = 0.3 μm. The calculation for the monolayer (in this case it is a planar photonic crystal) was implemented into QCA. The calculation for the multilayer was carried out using TMM. We name this combination of the methods as QCA-TMM.
The photonic band gap (PBG) caused by the multilayer periodic structure arises with increasing the number of monolayers. It is formed in the wavelength range ~0.75 to 0.77 μm for the considered parameters of the system. See the transmission minimum indicated as 3 in Fig. 2а. As seen from the results in Fig. 2, already for 12 identical monolayers the noticeable PBG arises: Tc ≈ 0.12 at λ ≈ 0.761 μm. For 30 monolayers, the coherent transmission coefficient Tc ≈
4
10
1
.
8
-
×
at λ ≈ 0.757 μm, and for 60 monolayers Tc ≈
7
10
7
.
1
-
×
at λ ≈ 0.756 μm. In the same spectral range, the coherent reflectance maximum comparable by its width with the coherent transmittance minimum 3 is formed.
As it follows from the results of calculations, these three types of coherent transmittance minima are inherent to the close-packed regular disperse system (see Fig. 1a where they are indicated by digits 1, 2, and digit 3 in Fig. 2а). The transmittance minimum 1 is caused by the concentration and optical properties of particles in a single monolayer. It takes place both for PPC and partially-ordered monolayer. The transmittance minimum 2 is caused by regularity in particle locations within a single monolayer. It takes place for PPC at sizes of particles and distances between them compared with the wavelength of the incident light. The transmittance minimum 3 is PBG. It takes place in spectra of the periodic system at wave propagation along periodicity of the multilayer structure.
The depths, widths, locations, and the number of PBGs of the multilayer depend on distances between its monolayers. Calculations for the same monolayer parameters as in Fig. 2 show: when the distance between monolayers is changed from 0.3 up to 0.5 μm, the depth and width of PBG are increased, and PBG location is shifted to the larger wavelengths. With further increasing the distance, location of this PBG is shifted to the larger wavelengths, and PBG depth decreases. Simultaneously, in the range of short wavelengths the additional more narrow PBGs appear. The longer is the distance between monolayers, the higher is the number of appearing PBGs. The results show that multilayers consisting of PPCs can be used to develop and manufacture the band-pass radiation filters in a wide spectral range.
Comparison with the results obtained for the partially-ordered monolayers has shown that the spectra of the system of partially-ordered monolayers and the system of PPCs have both general and different features.
In particular, for both systems (with the same particle diameter D = 0.3 μm) PBG formation within the range λ ≈0.73…0.78 μm and an increase in the number of spectral oscillations with the number of monolayers takes place. That is caused by periodicity of the multilayer structure along the direction of the incident wave and multiple wave reflections in a system. The absolute values of the minima of transmission in PBG differ slightly.
There are significant differences in PBG formation between the systems of planar photonic crystals and partially-ordered monolayers of particles. PBG of the PPCs system is more symmetric and has more sharp edges in comparison with PBG of partially-ordered monolayers system. The values of the reflection maxima in PBG are also noticeably different. Coherent reflectance maximum of the partially-ordered monolayers system in PBG Rc,max ≈ 0.81 (at the wavelength λ ≈ 0.738 μm), whereas the respective maximum of the PPC system Rc,max ≈ 1.
4. Glass plate covered with alumina particles: transmittance and reflectance spectrum
The coefficient of light reflection by a plane-parallel glass plate in air is approximately 0.08 (~0.04 from each glass-air interface). To reduce the reflection of light from the plate, the antireflection coatings on its surfaces are used. These are typically the thin dielectric layers of a certain refractive indices and thicknesses. It is promising to create such coatings from aluminum oxide [14, 15], due to their relative simplicity and low cost production.
Consider a two-layer system consisting of a homogeneous plane-parallel layer of aluminum oxide on a plane-parallel glass plate (Fig. 3).
Let us calculate the coherent transmission and reflection coefficients of this system under TMM. The technique for calculating the reflection and transmission of a system consisting of sufficiently thin layers is based on the use of complex amplitude reflection and transmission coefficients. When the system includes a thick layer (e.g., layer L2 in Fig. 3), the energy transmission and reflection coefficients are used instead of the amplitude ones [7]. Coherent transmission and reflection coefficients of this system are shown in Fig. 4. In this figure, the transmission and reflection spectra of the plane-parallel glass plate without coating are shown as well.
Glass
Incident light
R
с
T
с
L
0
L
1
L
2
L
3
If
1
If
2
If
3
h
1
h
2
Al
2
O
3
Fig. 3. Schematic presentation of homogeneous plane-parallel alumina layer on the surface of a plane-parallel glass plate (side view).
0.5
0.6
0.7
0.8
0.9
1.0
0.84
0.85
0.86
0.87
0.88
0.89
0.90
0.91
0.92
0.93
T
c
l,m
m
(a)
0.05
0.1
0.5
1.0
Uncoated glass
0.5
0.6
0.7
0.8
0.9
1.0
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
R
c
l,m
m
(b)
Uncoated glass
0.05
0.1
0.5
1.0
Fig. 4. Spectra of the coherent transmittance Tс (a) and reflectance Rс (b) of an uncoated plane-parallel glass plate (thick line) and system of plane-parallel homogeneous Al2O3 layer on the surface of glass plate (thin lines) at different thicknesses of Al2O3 layer. The thickness of glass plate is 2 mm. Numbers near lines indicate the thicknesses of Al2O3 layer in micrometers.
The results displayed in Fig. 4 were obtained under the assumption that there are three interfaces If1…If3 in the system (see Fig. 3). It was assumed that the interfaces are flat boundaries of Al2O3 layer L1 (with the thickness h1) and glass layer L2 (with the thickness h2). The amplitude reflection and transmission coefficients of planar interfaces between the layers are calculated by the Fresnel formulae [16]. The layer L2 of glass has the thickness h2 = 2 mm, which is much higher than the wavelengths under consideration. Therefore, L2 is a “thick” layer, and the energy transmission and reflection coefficients are used accordingly to [7]. The layers L0 and L3 are semi-infinite layers of the environment (air). The refractive indexes of the Corning 1737 glass were used in the calculations [17]. As it follows from the results shown in the figure, coating of glass with the homogeneous layer of alumina does not reduce the coherent reflection coefficient of a system in comparison with that of the uncoated glass plate. Changes in the Al2O3 layer thickness lead to a change in the oscillation frequency of the transmission and reflection as a function of wavelength.
Glass
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с
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с
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L
1
L
2
L
3
If
1
If
2
If
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h
1
h
2
Al
2
O
3
Fig. 5. Schematic presentation of system consisting of monolayer of monodisperse spherical Al2O3 particles on glass (side view).
Now consider a system consisting of a glass plate covered with a monolayer of Al2O3 particles on its upper surface (Fig. 5).
Let us calculate coherent transmission and reflection coefficients of this system using the scheme described above. Find the conditions under which a coating of alumina particles will serve as an antireflection layer (at normal illumination). Calculation of the coherent transmission and reflection coefficients of the system is fulfilled with the assumption that it consists of three interfaces: a monolayer of alumina particles If1 and the two flat interfaces of glass plate If2 and If3 (see Fig. 5). Accordingly, there are two layers between interfaces: L1 with the thickness h1 of air and L2 with the thickness h2 of glass. The thickness h1 is equal to the radius of particles. Layers L0 and L3 are semi-infinite layers of the environment (air). Calculation of coherent transmission and reflection of interface If1 (planar PC) is carried out in QCA with σ(u) = σ0u, σ0 = 0.01, and of interfaces If2 and If3 (planar boundaries between the layers) by using the Fresnel formulae.
Depicted in Fig. 6 are the coherent transmission and reflection spectra of the uncoated glass plate and the plate coated with a monolayer possessing the hexagonal lattice of monodisperse spherical alumina particles. As it follows from the presented results, when the size of Al2O3 particles is much smaller than wavelengths, one can obtain an almost uniform spectrum with the reflectance smaller than the reflectance, and the transmittance larger than the transmittance of the uncoated glass plate. In the visible spectral range, when the coating monolayer contains Al2O3 particles with the diameter D = 0.1 μm, the reflection coefficient decreases down to ~0.047. With increasing the Al2O3 particle diameter up to 0.5 μm, the transmittance minimum and reflectance maximum in the spectrum within the range of wavelengths 0.85 to 0.88 μm appear. They are caused by regularity in particle locations inside the monolayer. For sufficiently large particles, in some spectral ranges the values of the coherent reflectance are lower than the reflectance of the bare interface (for the glass-air boundary Rc ≈ 0.04), and the coherent transmission coefficients are small. It implies that the major part of the radiation is incoherent.
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Fig. 6. Spectra of the coherent transmittance Tс (a) and reflectance Rс (b) of the uncoated plane-parallel glass plate (thick line) and system of monolayer with the hexagonal packing of monodisperse spherical Al2O3 particles on this plate for various particle sizes. The thickness of plate is 2 mm, filling coefficient of monolayer η = 0.5. Ciphers near the lines indicate diameters of Al2O3 particles in micrometers.
Glass
Incident light
R
с
T
с
L
0
L
1
L
2
L
3
If
1
If
2
If
3
h
1
h
2
If
4
L
4
h
3
Al
2
O
3
Fig. 7. Schematic presentation of multilayer system with two monolayers of monodisperse spherical Al2O3 particles on top and bottom surfaces of the glass plate.
To simultaneously reduce the coherent reflectance and increase the coherent transmittance, we should “exclude” reflections not only from the top, but also from the bottom interface of the plate. To solve this problem, consider the double-sided coating (Fig. 7). Using QCA-TMM, calculate the spectra of this system when monolayers of aluminum oxide particles are regularly-packed and partially-ordered. Fig. 8 shows the spectral dependences of coherent transmittance and reflectance when the top and bottom monolayers of Al2O3 particles are identical. Thin solid lines depict the spectra of the system with the regularly-packed monolayers (PPCs with hexagonal lattice), the dashed lines show the spectra of systems with partially-ordered monolayers of Al2O3 particles. Pay attention that the displayed results have a good accuracy in the spectral range where the value of Tc+Rc ≈ 1. As it follows from the calculation results, the coatings with monolayers of Al2O3 particles on both surfaces of the glass plate can significantly reduce the coherent reflection and increase the coherent transmission coefficients of the system as compared with those for the one-sided coated system. At a double-sided coating with identical monolayers of Al2O3 particles of diameter D = 0.1 μm, the calculated coherent reflection coefficient Rc < 0.025 in the visible spectral range with the minimum lower than 0.01. If the size of Al2O3 particles significantly smaller than the incident light wavelength, the differences in coherent transmittance and reflectance spectra of the systems coated with PPCs and partially-ordered monolayers are negligible (see lines for D = 0.05 μm and D = 0.1 μm in Fig. 8). If the particles size is comparable with the incident light wavelength, the differences in the spectra of systems with PPCs and partially-ordered monolayers become significant. They are more pronounced in the spectral range of the transmission minimum and reflection maximum (see solid lines for D = 0.5 μm within the range of wavelengths ~0.85 μm < λ < ~0.88 μm in Fig. 8) caused by the regularity of particle packing in a planar PC. It follows that if the creation of antireflection coatings from the Al2O3 particles does not required their high ordering, then for the construction of selective reflectors and transmission filters the ordering of particles in monolayers plays a crucial role.
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Fig. 9. Spectra of the coherent transmittance Tс (a) and reflectance Rс (b) of the uncoated plane-parallel glass plate (thick line) and system of highly-ordered monolayers with the hexagonal packing of Al2O3 particles (thin solid lines, σ(u) = σ0u, σ0 = 0.01) on top and bottom surfaces of the glass plate. The thickness of the glass plate is 2 mm, filling coefficient of monolayers η = 0.5. Ciphers near the lines before and after comma indicate diameters of Al2O3 particles (in micrometers) of top and bottom monolayer, respectively.
Consider the multilayer system (Fig. 7) in which the top and bottom surfaces of the glass plate are coated with monolayers containing spherical particles of different diameters (which are monodisperse in each monolayer). Fig. 9 shows the transmittance and reflectance spectra of a multilayer system when the surfaces of the glass plate are coated with monolayers with a hexagonal lattice of particles. As it follows from the results of calculations presented in the figure, if the top and bottom interfaces of the plate have different coatings, the spectral dependences of coherent transmittance and reflectance are more homogeneous as compared with the case of identical coatings. For the considered system, the coherent reflection coefficient in the visible spectral range is less than 0.02 with the minimum value less than 0.003.
5. Conclusion
The spectra of the coherent transmission and reflection coefficients of monolayers of Al2O3 particles have been calculated in the wavelength range from 0.3 up to 1.0 μm.
Three types of coherent transmittance minima inherent for regular disperse system have been observed and described.
Coherent transmission and reflection coefficients of the glass plate coated with spherical Al2O3 particles have been calculated. As it follows from the results, creation of antireflection coatings do not need a high ordering of particles. Inversely, to create selective reflectors and transmission filters the arrangement of particles in monolayer has a crucial role. It is shown that the reflection coefficient can be reduced significantly by using the two-sided coating of glass plate with monolayers of alumina particles.
The obtained results can be used in the design and manufacture of the antireflection coatings, diffuse light scattering structures, transmission and reflection filters (for example, in display and solar cells applications), the lattice quality control in photonic crystals, etc.
Acknowledgement
This work was supported in part by the state scientific and technical program of Belarus “Electronics and photonics”.
References
1.
M. Lax, Multiple scattering of waves II. The effective field for dense systems // Phys. Rev. 85, p. 621-629 (1952).
2.
J.M. Ziman, Models of Disorder. Cambridge, Cambridge University, 1979.
3.
A.A. Miskevich, V.A. Loiko, Coherent transmission and reflection of a two-dimensional planar photonic crystal // J. Experiment. Theor. Phys. 113(1), p. 1-13 (2011).
4.
A.F. Skryshevskiy, Structural Analysis of Liquids and Amorphous Solids. Moscow, Vysshaya Shkola, 1980 (in Russian).
5.
V.A. Loiko, A.A. Miskevich, Light propagation through a monolayer of discrete scatterers: analysis of coherent transmission and reflection coefficients // Appl. Opt. 44(18), p. 3759-3768 (2005).
6.
K.M. Hong, Multiple scattering of electromagnetic waves by a crowded monolayer of spheres: Application to migration imaging films // J. Opt. Soc. Am. 70, p. 821-826 (1980).
7.
C.C. Katsidis, D.I. Siapkas, General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference // Appl. Opt, 41, p. 3978-3987 (2002).
8.
E. Centurioni, Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers // Appl. Opt. 44(35), p. 7532-7539 (2005),
9.
E.D. Palik, editor, Handbook of Optical Constants of Solids. San Diego, Academic, 1985.
10.
L.S. Ornstein, F. Zernike, Accidental deviations of density and opalescence at the critical point of a single substance // Proc. Acad. Sci. 17, p. 793-806 (1914).
11.
J.K. Percus, G.J. Yevick, Analysis of classical statistical mechanics by means of collective coordinates // Phys. Rev. 110(1), p. 1-13 (1958).
12.
A.P. Ivanov, V.A. Loiko, V.P. Dick, Propagation of Light in Close-packed Disperse Media. Minsk, Nauka i Tekhnika, 1988 (in Russian).
13.
R.W. Wood, Anomalus diffracting gratings // Phys. Rev. 48, p. 928-937 (1935).
14.
N.I. Mukhurov, Aluminium-oxide Micro-nanostructures for Microelectromechanical systems. Minsk, Bestprint, 2004 (in Russian).
15.
A.A. Lutich, M.B. Danailov, S. Volchek, V.A. Yakovtseva, V.A. Sokol, S.V. Gaponenko, Birefringence of nanoporous alumina: dependence on structure parameters // Appl. Phys. B, 84, p. 327-331 (2006).
16.
M. Born, E. Wolf, Principles of Optics. Cambridge University Press, 2002.
17.
Corning site http://www.corning.com
� EMBED Origin50.Graph ���� EMBED Origin50.Graph ���
Fig. 8. Spectra of the coherent transmittance Tс (a) and reflectance Rс (b) of the uncoated plane-parallel glass plate (thick line) and system of identical partially-ordered (dotted lines) and regularly-packed monolayers with hexagonal packing of Al2O3 particles (thin solid lines, σ(u) = σ0u, σ0 = 0.01) on top and bottom surfaces of the glass plate for various particle sizes. The thickness of glass plate is 2 mm, filling coefficient of monolayers η = 0.5. Diameters (in micrometers) of particles are indicated near the lines.
� EMBED Origin50.Graph ���
� EMBED Origin50.Graph ���
Fig. 2. Spectra of the coherent transmittance Tс (a) and reflectance Rс (b) of multilayer consisting of 12, 30, and 60 PPCs with hexagonal packing of Al2O3 particles in air. Distances between adjacent monolayers are h = 0.3 μm. The filling coefficient of each monolayer η = 0.5, diameter of particles D = 0.3 μm. The radial distribution function of PPC is calculated at σ(u) = σ0u, σ0 = 0.01. The digit 3 indicates the transmission minimum (photonic band gap) inherent to the layered disperse system (see description in the text).
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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