Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function
The analysis of transmission spectrums of periodical multilayer interference thin-films systems is carried out. Spectral dependences of pointer function from wavelength for interference bandpass filters are obtained. These dependences allow to prognostic of regions of high transmission or reflect...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2008
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| Цитувати: | Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function / Ya. Yaremchuk, V.M. Fitio, Ya.V. Bobitski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 1. — С. 50-53. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1186542017-05-31T03:04:27Z Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function Yaremchuk, Ya. Fitio, V.M. Bobitski, Ya.V. The analysis of transmission spectrums of periodical multilayer interference thin-films systems is carried out. Spectral dependences of pointer function from wavelength for interference bandpass filters are obtained. These dependences allow to prognostic of regions of high transmission or reflectance for different periodical multilayer systems. 2008 Article Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function / Ya. Yaremchuk, V.M. Fitio, Ya.V. Bobitski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 1. — С. 50-53. — Бібліогр.: 11 назв. — англ. 1560-8034 PACS 42.25.Bs, 42.25.Hz, 42.79.Ci http://dspace.nbuv.gov.ua/handle/123456789/118654 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The analysis of transmission spectrums of periodical multilayer interference
thin-films systems is carried out. Spectral dependences of pointer function from
wavelength for interference bandpass filters are obtained. These dependences allow to
prognostic of regions of high transmission or reflectance for different periodical
multilayer systems. |
| format |
Article |
| author |
Yaremchuk, Ya. Fitio, V.M. Bobitski, Ya.V. |
| spellingShingle |
Yaremchuk, Ya. Fitio, V.M. Bobitski, Ya.V. Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Yaremchuk, Ya. Fitio, V.M. Bobitski, Ya.V. |
| author_sort |
Yaremchuk, Ya. |
| title |
Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function |
| title_short |
Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function |
| title_full |
Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function |
| title_fullStr |
Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function |
| title_full_unstemmed |
Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function |
| title_sort |
prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118654 |
| citation_txt |
Prediction of a region with high transmission (reflectance) for bandpass interferential filters by using the method of pointer function / Ya. Yaremchuk, V.M. Fitio, Ya.V. Bobitski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 1. — С. 50-53. — Бібліогр.: 11 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T14:23:45Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 50-53.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
50
PACS 42.25.Bs, 42.25.Hz, 42.79.Ci
Prediction of a region with high transmission (reflectance)
for bandpass interferential filters
by using the method of pointer function
Ya. Yaremchuk, V.M. Fitio, Ya.V. Bobitski
1Institute of Telecommunications and Radio Engineering, Department of Photonics
Lviv Polytechnic National University, 12, Bandera str., 79013 Lviv, Ukraine
Phone: 8-032-2582581, e-mail: yaremchuk@polynet.lviv.ua
2Institute of Technology Rzeszow University
16b, T. Rejtana str., 35959 Rzeszow, Poland
Abstract. The analysis of transmission spectrums of periodical multilayer interference
thin-films systems is carried out. Spectral dependences of pointer function from
wavelength for interference bandpass filters are obtained. These dependences allow to
prognostic of regions of high transmission or reflectance for different periodical
multilayer systems.
Keywords: interference filter, pointer function.
Manuscript received 26.12.07; accepted for publication 07.02.08; published online 31.03.08.
1. Introduction
A lot of interesting and useful phenomena arise at
propagation of electromagnetic radiation through
periodical structures. They are used in different optical
devices such as: diffraction gratings [1, 2], photonic
crystals [3], lasers [4] and filters [5, 6]. Actually, thin
film optics can be constructed in the best way by using
the electromagnetic theory. This theory provides rela-
tively full and consecutive consideration of interference
and polarization effects in all types of film multilayer
systems. Multilayer thin-film coatings for visible and
infrared spectral regions are the object of researches for
plenty of researchers [7-11], so far as they are important
elements for electronics, photonics, laser technique and
telecommunications. It should be noted that production
of filters with given band width of transmission is
enough difficult work, for example, filters used in
multiplexing or demultiplexing consist of more than one
hundred individual layers with rigorous tolerances. In
most cases, these interferential systems have structures
that include a large number of periods.
It’s known that periodical multilayer systems are
equivalent to the one-dimension photonic crystals that
are the volume structures made of material transparent in
a certain wavelength region. The dielectric permittivity
of them is described by a periodical function. Typical
interferential effects at every boundary with increasing
or decreasing the total reflectance (transmission) for
different wavelengths appear for light passing through
the structure of one-dimensional (1D) photonic crystal
with periodical variations of the certain layer thickness
and different refractive index (n). Interferential mirror
coatings are the most known example of 1D photonic
crystal. At correct choice of refractive indices and
optical thickness of layers, in transmission spectra one
can observe the bands with maximal coefficient of
reflectance for a certain wavelength region − the so-
called photonic bandgaps appear.
In the work [12], the methods to calculate a
bandgap in 1D photonic crystal is proposed. It is reduced
to solving the eigenvalue problem of TX = ρX type and
verification of the absolute value of eigenvalue ρ (if it is
equal to unity). The square matrix T has the dimension
2×2. The frequency ν and component of the wave vector
kx are given; in this method, the component kz is defined
with the eigenvalue ρ. The photonic bandgap can be
constructed when we have the frequency ν and
component kz. The only spur of the matrix should be
known to define the eigenvalue. The eigenvectors are
possible to be determined when eigenvalues ρ are
known. In this case, the electromagnetic field of wave
propagating through the crystal can be calculated. In this
work, the transmissions of interferential mirror have
been calculated using this method. It is necessary to note
that the spectral dependence (T11 + T22) / 2 exactly
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 50-53.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
51
identifies the boundary between allowed and forbidden
frequencies. Thus, this dependence is possible to be
called as the pointer function P(λ).
Taking into account that periodical multilayer
systems are equivalent to a one-dimension photonic
crystal, in this work we investigated the possibility to
predict the region of transmission (reflectance) in
multilayer interferential bandpass filters by using the
pointer function, which was applied to photonic crystals.
The analysis of transmission spectra of periodical
multilayer thin-film interferential systems is conducted
using the matrix method.
2. Periodical multilayer interferential
bandpass filters
There are two moments in designing the thin-film filters:
the first one is connected with the low reflectance
(transmission) for certain wavelength region and the
second – simplicity of the constructed structures, which
facilitates their production. The periodical multilayer
structures with equal-thickness layers are used most
often [13], they consisted of layers with high and low
refractive indices, which are repeated. The resources of
obtaining the filters with a relatively wide transmission
band based on these systems are rather limited. There are
different methods of narrowing or widening the main
transmission band. Shown in the work [14] is that
narrowing the high transmission band and widening the
transmission band in a multilayer system take place at
the modification of the thickness of individual layers in
period and at the invariable total thickness of period. In
[15], we proposed the bandpass filter with (HLH)nS type
structure, when H and L are dielectric layers with high
and low refractive indices, respectively, and with
unequal thicknesses. Taking into account that the system
is periodical and consists of only two components like
interferential mirrors, it would be pertinent to verify the
possibility to predict the transmission region by using
the pointer function. In the two-component three-layered
periodic structure, the pointer function P(λ) =
(t11 + t22) / 2 will be as follows:
,cossinsin)//(5.0
cossinsin)//(5.0
cossinsin)//(5.0
coscoscos)(
132332
2311331
3211221
321
2 ΦΦΦ+−
−ΦΦΦ+−
−ΦΦΦ+−
−ΦΦΦ=λ
nnnn
nnnn
nnnn
P
(1)
where λπ=Φ /2 111 dn , λπ=Φ /2 222 dn ,
λπ=Φ /2 333 dn ; n1, n2, n3 – refractive indices of
dielectric layers; d1, d2, d3 – geometric thicknesses of
dielectric layers.
The base wavelength is equal to 5 µm, n1 =
n3 = 4.02, n2 = 1.44, d1 = d3 = 3.11 µm, d2 = 0.87 µm are
used for calculating this system with four periods was
selected. From the dependence shown in Fig. 1, it is the
possibility to immediately say in which spectral region
the transmittance can be high. The region of trans-
mission corresponds to the range of −1 ≤ P ≤ 1.
When using only two different materials within the
period of a periodical multilayer system (n1 = n3 and
d1 = d3), then the formula (1) will be:
.sin2sin)//(5.0
cos2cos)(
211221
21
ΦΦ+−
−ΦΦ=λ
nnnn
P (2)
If the pointer function tends to zero at the base
wavelength λ = 5 µm, then the simplest solutions will be
as follows:
1. π=Φ m1 and 2/2 π=Φ .
2. 4/2/1 π+π=Φ m and π=Φ2 .
3. 8/2/1 π±π=Φ m and
.))//(/1(arctg 12212 nnnn +=Φ
4. 2/))//(/1(arctg2/ 12211 nnnnm +±π=Φ and
4/2 π=Φ ,
where m is an integer.
The spectral curve of the system with the thickness
of layers calculated for the first variant of the solutions
at m = 5 will agree with the spectral curve shown in
Fig. 1. Interference system with layers (their thicknesses
are obtained from the solution 2), will have the
transmission band within the limits that are defined by
the pointer function. However, the spectral curve has
numerous deep valleys, and their number increase with
increasing the number of periods. Thus, the solution 2
cannot find any practical application. The solutions 3
and 4 provide perfectly useful characteristics. For
example, the spectral curve corresponds to system with
layers, thicknesses of which are calculated in accordance
with the solution 4 (see Fig. 2).
The transmission band can be widened practically
by one and a half, if the repeated period of the multilayer
system is constructed from three different materials, and
if it has ABCBA structure, where A, B and C are layers
of the equal optical thickness with refractive indices nA,
nB, and nC, respectively [16, 17].
Fig. 1. Wavelength dependences of transmission for the
interferential filter of the type (HLH)nS and of the pointer
function.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 50-53.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
52
Fig. 2. Wavelength dependences of transmission for the
interferential filter of the type (HLH)10S and of the pointer
function.
In the work [18], expressions for determination of
the effective refractive index N for ABCBA period are
presented. Obtained with these expressions are the
relations that connected with the refractive indices nA,
nB, and nC. At these relations take place in simultaneous
suppression of the two band high reflectance.
To predict the region of transmission in these
filters, we will use the system that is presented in [16],
where all the layers have the optical thickness equal to
one fourth of the wavelength and with the refractive
indices nA = 1.38, nB = 1.90, and nC = 2.30, respectively.
The pointer function is simply calculated by application
of mathematics software. The expression has rather
cumbersome view, so it is not presented here. Fig. 3
shows that the pointer function exactly indicates on the
region of wavelengths where the high transmittance is
possibly achieved at given parameters of the structure.
The transmission bands can be more widened when
multilayer systems that consist of periods ABCСBA or
ABCDDCBA will be used. Widened transmission bands
of these systems take place when the sideband of high
transmission is suppressed. The number of different
materials in the period defines the maximal number of
high transmission bands that are suppressed, the number
of the layer in the period indicates on spectral location of
these bands [19].
Fig. 3. Wavelength dependences of transmission for the
interferential filter of the type (ABCBA)10S and of the pointer
function.
Fig. 4. Wavelength dependences of transmission for the
interferential filter of the type (ABCDDCBA)5S and of the
pointer function.
In all the cases, refractive indices must strongly
correspond to certain relations [18]. When multilayer
systems with repeated ten-layer periods are used, it is the
possibility to suppress high reflectance bands of 4th, 5th,
and 6th orders, etc. [19]. In Fig. 4, the wavelength
dependences of transmission of interferential filters of
the type (ABCDDCBA)5 and the pointer function are
presented. All layers in the offered system have the
equal optical thickness (one fourth of the wavelength)
with the refractive indices nA = 1.65, nB = 1.95,
nC = 2.30, and nD = 2.70, respectively.
Thus, the pointer function exactly indicates on the
region of high transmission in multilayer repeated
system such as in previous cases. The spectral curve has
two valleys in the region of wavelengths 3 and 5 µm.
The height can be modified by varying the number of
periods. So, the pointer function indicates on the same
regions (P ≅ 1), where certain peculiarity takes place.
3. Conclusions
The term of the pointer function based on the theory of
1D photonic crystal is introduced. The spectral
dependences of the pointer function for bandpass and
wide-bandpass interferential filters are obtained. These
dependences allow to predict the region of high
transmission (reflectance) for periodic multilayer
systems with different structures and to indicate the
regions with certain particularities. The base advantage
of using the pointer function is that with its help one can
easily indicate the region of high transmission
(reflectance) in bandpass and cut-off filters. For
periodical systems consisting only of two different
materials, there is the possibility to simply calculate
necessary thicknesses of individual layers to provide
maximal transmission for a certain range of wave-
lengths. Thus, the pointer function facilitates design of
optical filters that have comparatively complicated
structure. It is known that the modern bandpass thin-film
filters include more hundred layers, and the process of
calculation of transmission or reflectance coefficients
take enough large volume of time.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 1. P. 50-53.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
53
This work was supported by the State Fund for
Fundamental Researches of Ukraine under Grant
F25.2/017.
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