Вибір функції якості в задачах синтезу оптичних покриттів
The article presents general information on the use of optical coatings in various industries and analyzes the main approaches to optimizing optical filter structures. An approach to solving a class of optical coating synthesis problems is proposed, based on the formation of a new optimization model...
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Репозитарії
System research and information technologies| _version_ | 1867334443194122240 |
|---|---|
| author | Mitsa, Oleksandr Stetsyuk, Petro Zhukovskyi, Serhii Levchuk, Oleksandr Petsko, Vasyl Shapochka, Ihor |
| author_facet | Mitsa, Oleksandr Stetsyuk, Petro Zhukovskyi, Serhii Levchuk, Oleksandr Petsko, Vasyl Shapochka, Ihor |
| author_institution_txt_mv | [
{
"author": "Oleksandr Mitsa",
"institution": "Uzhhorod National University, Uzhhorod"
},
{
"author": "Petro Stetsyuk",
"institution": "V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine, Kyiv"
},
{
"author": "Serhii Zhukovskyi",
"institution": "Zhytomyr Ivan Franko State University, Zhytomyr"
},
{
"author": "Oleksandr Levchuk",
"institution": "Uzhhorod National University, Uzhhorod"
},
{
"author": "Vasyl Petsko",
"institution": "Uzhhorod National University, Uzhhorod"
},
{
"author": "Ihor Shapochka",
"institution": "Uzhhorod National University, Uzhhorod"
}
] |
| author_sort | Mitsa, Oleksandr |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2025-11-09T00:01:30Z |
| description | The article presents general information on the use of optical coatings in various industries and analyzes the main approaches to optimizing optical filter structures. An approach to solving a class of optical coating synthesis problems is proposed, based on the formation of a new optimization model. The primary attention is paid to the formalization and analysis of the target function. To determine the quality of the optical coating, the deviation of the spectral characteristics from the required ones was estimated using the least squares, least absolute deviation, and minimum criteria. As a result, both smooth and two non-smooth target functions are proposed and analyzed. The peculiarities of their application in solving optimization problems related to optical coating synthesis are described, and corresponding numerical experiments are presented. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2025.3.04 |
| first_indexed | 2025-11-09T02:11:02Z |
| format | Article |
| fulltext |
O.V. Mitsa, P.I. Stetsyuk, S.S. Zhukovskyi, O.M. Levchuk, V.I. Petsko, I.V. Shapochka, 2025
48 ISSN 1681–6048 System Research & Information Technologies, 2025, № 3
UDC 519.87; 535.345.67
DOI: 10.20535/SRIT.2308-8893.2025.3.04
SELECTION OF TARGET FUNCTION IN OPTICAL COATINGS
SYNTHESIS PROBLEMS
O.V. MITSA, P.I. STETSYUK, S.S. ZHUKOVSKYI, O.M. LEVCHUK,
V.I. PETSKO, I.V. SHAPOCHKA
Abstract. The article presents general information on the use of optical coatings in
various industries and analyzes the main approaches to optimizing optical filter
structures. An approach to solving a class of optical coating synthesis problems is
proposed, based on the formation of a new optimization model. The primary atten-
tion is paid to the formalization and analysis of the target function. To determine the
quality of the optical coating, the deviation of the spectral characteristics from the
required ones was estimated using the least squares, least absolute deviation, and
minimum criteria. As a result, both smooth and two non-smooth target functions are
proposed and analyzed. The peculiarities of their application in solving optimization
problems related to optical coating synthesis are described, and corresponding nu-
merical experiments are presented.
Keywords: optical coatings synthesis, wide bandpass filters, mathematical model-
ing, optimization, r-algorithm.
INTRODUCTION
Optical layered coatings have been used in a vast array of applications across dif-
ferent industries for many decades. They are used to modify the behaviour of
light, enhancing the performance of optical devices in several ways. These coat-
ings are commonly made of thin films of different materials that are deposited
onto a substrate using various techniques, including sputtering, evaporation, and
chemical vapour deposition [1]. One of the most prominent applications of optical
layered coatings is in the field of optics. Optical lenses, filters, and mirrors are
coated with thin layers of materials such as titanium dioxide, silicon dioxide, and
aluminium to modify their refractive index, reflectivity, and transmission proper-
ties. These coatings help to reduce unwanted reflections, increase the light trans-
mission, and improve colour accuracy, resulting in sharper, clearer images [2].
The film industry also relies heavily on optical coatings to improve the perform-
ance of cameras and lenses. Antireflective coatings applied to camera lenses re-
duce lens flare and ghosting, leading to crisper, higher-quality images. Similarly,
polarizing filters are used to eliminate reflections and glare, resulting in better
contrast and richer colours in the final footage. Optical layered coatings are also
crucial in the medical field [3]. They are used to improve the performance of
various medical devices, such as endoscopes, surgical lasers, and imaging sys-
tems. These coatings help to increase light transmission, reduce unwanted reflec-
tions, and improve the resolution and contrast of medical images, resulting in
more accurate diagnoses and better treatment outcomes. In the field of electronics,
optical layered coatings are used in the production of various displays, including
LCDs and OLEDs [4]. These coatings help to increase the brightness and contrast
of displays, reduce glare and reflections, and improve colour accuracy. They are
Selection of target function in optical coatings synthesis problems
Системні дослідження та інформаційні технології, 2025, № 3 49
also used in the production of solar panels to increase the efficiency of light ab-
sorption and conversion into electricity [5].
There are various approaches to optimizing the structures of optical layered
coatings [6]. The trial-and-error method [7] involves manually adjusting the
thickness and refractive index of each coating layer until the desired optical per-
formance is achieved. However, this method can be time-consuming and does not
always lead to an optimal coating design. Analytical methods use mathematical
equations to calculate the thickness and refractive index of each coating layer.
Some common analytical methods [8] are based on quarter-wave structures or
structures that use bandwidth matching. These methods are relatively easy to use
but do not always result in the optimal coating structure.
Numerical methods use computer algorithms to model the behavior of light
waves within the coating and optimize the structure based on predefined criteria.
Some common numerical methods include the transfer matrix method and the
reverse wave analysis (RWA) method. The transfer matrix method [9] does not
provide a natural way to model these optical properties, making it insufficient for
synthesizing optical coatings. This method also assumes linear transformations,
which do not account for light dispersion as it passes through materials. In optical
coatings, materials are typically used where dispersion is a significant factor and
must be considered in the design. The RWA method [10] can be very sensitive to
initial conditions or input data. Even minor errors or inaccuracies in measure-
ments or models can lead to incorrect results. However, these methods can be
highly accurate and consider a wide range of structural criteria, but they are com-
putationally complex [11].
Genetic algorithms [12] can be effective for the synthesis of optical coatings,
but they may require a significant amount of computational resources and can be
quite slow. The method of microstructured surfaces [13] uses structured micro-
elements on the surface to create the desired optical properties. However, their
production can be complex and require high-precision processing. Optical coat-
ings created using the phase mask method [14] can be sensitive to changes in
temperature, humidity, and mechanical stresses, leading to changes in their optical
properties.
When using numerical methods, the choice of the objective function plays an
important role. This work proposes several objective functions that can be used to
optimize the parameters of optical coatings. One smooth and two non-smooth ob-
jective functions are presented. The effectiveness of their use is demonstrated
with an example of a non-smooth objective function.
PROBLEM STATEMENT AND MATHEMATICAL MODEL
Multilayer optical coatings represent a structure consisting of N layers. The j-th
layer is characterized by two parameters: the refractive index ( jn ) and the geo-
metric thickness ( jd ) (Fig. 1). There are two main tasks associated with them.
The first task, known as the direct or analysis task, involves determining the spec-
tral characteristics (transmission, reflection, and absorption coefficients) of a
known multilayer thin-film system based on the known characteristics of the coat-
ing. The task of calculating the characteristics of an interference coating is based
on solving the stationary wave equation in the plane wave approximation. To
date, a large number of computational schemes have been developed for calculat-
O.V. Mitsa, P.I. Stetsyuk, S.S. Zhukovskyi, O.M. Levchuk, V.I. Petsko, I.V. Shapochka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 3 50
ing optical coatings. Perhaps the most common approach is based on calculating
the tangential components of the electric and magnetic field vectors sequentially
at all layer boundaries that form the coating. Introducing the matrix form of re-
cording equations that connect the field amplitudes at adjacent boundaries al-
lowed for a compact and consistent consideration of interference effects in lay-
ered structures of all types.
The second task, known as the inverse or synthesis task, involves determin-
ing the parameters of the multilayer optical structure that would optimally repro-
duce its predetermined spectral characteristics. In other words, the synthesis problem
is to find such parameters of multilayered optical coating — refractive indices
),,,( 21 Nnnnn
, and geometric thicknesses of layers ),,,( 21 Ndddd
(N —
number of layers), — under which, function, chosen to estimate transmittance
factor quality, will be minimal in a given wavelength range ],[ 21 :
,),(min),(
,
*** dnFdnFF
dn
(1)
subject to
, , ,2 ,1 , maxmin Ninnn iii (2)
, , ,2 ,1 ,maxmin Niddd iii (3)
where *F — minimum value of a coating target function.
Constrains (2), (3) have been imposed on the following parameters of multi-
layered optical coating — refractive indices and optical thicknesses. The refrac-
tive indices have been selected from the available coating-forming materials. Dif-
ferent sets of them can be created based on the spectral ranges of materials.
For visible and infrared rages, as a rule, the refractive index does not exceed
2.6. For the ultraviolet rage, materials with a higher refractive index can be used.
Constraints (3) have been imposed on the geometric thickness of coating. The
lower limit is tied to the application process, the upper limit, in the process of
making multilayered optical coatings, as a rule, does not exceed the operating
wavelength 0 .
The value of the energy transmittance index for the electromagnetic wave-
length λ through the multilayer optical structure should light fall on the surface at
dN dj
θN
d1
θj
θs
θ0 θ1
nN nj n1 n0
ns
Fig. 1. Scheme of a light transmission through a multilayer optical structure
Selection of target function in optical coatings synthesis problems
Системні дослідження та інформаційні технології, 2025, № 3 51
an angle 0 (Fig.1) has been calculated through the coefficients of the character-
istic matrix )λ,,( dnM
as follows:
),,,( 0dnT
,
),,,(
1
),,,(),,,(),,,(2
4
0
2
21
0
0
2
1200
2
22
0
0
2
11
0
dnM
pp
dnMppdnM
p
p
dnM
p
p
s
s
s
s
where 000 cos np and sss np cos — for TE wave ( s -polarization);
0
0
0 cos
n
p and
s
s
s
n
p
cos
— for TE wave (р -polarization); 0 — angle of
incidence; s — angle of reflection; snn ,0 — refractive indices of an environ-
ment and a substrate, accordingly.
The characteristic matrix of the N-layer structure is equal to the product of
the matrices of each of the layers [15]:
),λ,,( 0dnM
,),λ,,(),λ,,(),λ,,(),λ,,( 111222111 dnMdnMdnMdnM NNNNNN
where the characteristic matrix of the layer equals
,),λ,,( cos),λ,,( sin
,),λ,,( sin),λ,,( cos),λ,,(
dndnni
dn
n
i
dndnM
λ
cos2
),λ,,(
nd
dn — phase thickness of the layer; —angle of incidence.
Angles of incidence for each layer follow the Snell’s law and can be easily
calculated according to the ratio:
00 sinn = . sin sin sin sin sin 2211 ssNNjj nnnnn
If 00 , then the value of transmittance factor for the N-layer optical struc-
ture can be calculated using the following formula
),,( dnT
,
),,(
1
),,(),,(),,(2
4
2
21
0
2
120
2
22
0
2
11
0
dnM
nn
dnMnndnM
n
n
dnM
n
n
s
s
s
s
where the characteristic matrix of the N-layer structure is written as
,)λ,,()λ,,()λ,,()λ,,()λ,,( 112211 dnMdnMdnMdnMdnM NNNN
and characteristic matrix of one layer is given by
.
λ
2
cos
λ
2
sin
λ
2
sin
λ
2
cos
)λ,,(
ndnd
ni
nd
n
ind
dnM
It should be noted, that characteristic matrix of the multilayered optical structure
meets following condition
O.V. Mitsa, P.I. Stetsyuk, S.S. Zhukovskyi, O.M. Levchuk, V.I. Petsko, I.V. Shapochka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 3 52
.1))λ,,(( dnMdet
(4)
This follows from the fact that the characteristic matrix of each layer has the
same property
1))λ,,(( ii dnMdet , i = 1, 2, …, N.
Property (4) has a simple physical meaning. If an electromagnetic wave
propagates in N media that do not absorb its energy, then an arbitrarily combined
(of these N media) medium will not absorb the energy of the electromagnetic
wave.
OPTICAL COATING TARGET FUNCTIONS AND THEIR USE
The following coating target functions can be chosen to solve the synthesis prob-
lem (1)–(3):
,))λ()λ,,((
1
),( 2
1
1 iidealii
L
i
TdnTw
L
dnF
(5)
,)λ()λ,,(
1
),(
1
2 iidealii
L
i
TdnTw
L
dnF
(6)
,)λ()λ,,(max),(
L,,1 i
3 iidealii TdnTwdnF
(7)
where wi — weighting coefficients, which determine the input on the objective
function at wavelength i ; L — the number of grid points on the spectral interval
between 1 and 2; )λ,,( idnT
— the value of the transmission index for parame-
ters ),( dn
and at wavelength i ; )( iidealT — the value of the transmission in-
dex at wavelength i .
Coating target functions (5)–(7) have been described below. Function
),(1 dnF
sets the weighted standard deviation of the transmittance indices from
the required for the selected L values of wavelengths. This function is smooth, so
gradient methods, quasi-Newton methods and zero-order methods (use only the
values of the objective function) can be used to minimize it. Function ),(2 dnF
sets the weighted sum of deviations from the mean with respect to the selected L.
Function ),(3 dnF
specifies deviation under minimax control (Chebyshev crite-
rion). The functions ),(2 dnF
and ),(3 dnF
are non-smooth, so Shore r-
algorithms and zero-order methods can be used to minimize them.
When solving the antireflective coating substrate problem, the values of
)λ,,( iideal dnT
are constand and equal to unity. With regard to afford mentioned,
the objective functions takes the form:
, )1)λ,,((
1
),( 2
1
1
ii
L
i
dnTw
L
dnF
,1)λ,,(
1
),(
1
2
ii
L
i
dnTw
L
dnF
Selection of target function in optical coatings synthesis problems
Системні дослідження та інформаційні технології, 2025, № 3 53
.1)λ,,(max),(
L,,1 i
3
ii dnTwdnF
Given that the value of transmittance factor is less than unity, the function
),(2 dnF
can be expressed in the following form
),λ,,(
1
1
1)λ,,(
1
),(
111
2 ii
L
i
i
L
i
ii
L
i
dnTw
L
w
L
dnTw
L
dnF
and will be smooth, when solving the antireflective coating substrate problem. In
the similar fashion, the function ),(3 dnF
can be expressed in the following form
)), λ,,(1(max1)λ,,(max),(
,,1,,1
3 ii
Li
ii
Li
dnTwdnTwdnF
But in contrast to the function ),(2 dnF
, it`s non-smooth.
If all 1iw , we obtain following objective functions:
, )1)λ,,((
1
),( 2
1
1
i
L
i
dnT
L
dnF
,)λ,,(
1
11)λ,,(
1
),(
11
2 i
L
i
i
L
i
dnT
L
dnT
L
dnF
)).λ,,(1(max1)λ,,(max),(
L,,1 iL,,1 i
3 ii dnTdnTdnF
In a number of studies problems of wide bandpass optical coatings synthesis
have been reviewed as maximization problems for similar deviations, and not for
the maximum transmittance, but for the minimum possible, i.e. zero value of the
transmittance [16]. For weighted standard deviation, there is an alternative, where
the maximization problem can be described as
,)λ,,λ(
1
),(Fmax 2
1,
ii
L
idn
dnT
L
dn
(8)
subject to (2) and (3).
In a similar way, for weighted sum of deviations from the mean this problem
can be described as
,)λ,,)((
1
),(Fmax
1,
ii
L
idn
dnT
L
dn
(9)
subject to (2) and (3).
And for deviation under minimax control (Chebyshev criterion) is as follows
.)λ,,)λ((min),(Fmax
L,,1i,
ii
dn
dnTdn
(10)
subject to (2) and (3).
For these models, which use target functions (8)–(10), it is assumed that
there may be a refractive index dispersion. Accordingly, the value of the refrac-
tive index is a function of wavelength and function is defined using approxima-
tion Zellmeier formula
O.V. Mitsa, P.I. Stetsyuk, S.S. Zhukovskyi, O.M. Levchuk, V.I. Petsko, I.V. Shapochka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 3 54
42
42
λλ
λλ
)λ( ii
ii
ii ED
CB
An ,
where iiiii EDCBA , , , , — parameters for refraction index model in the presence
of dispercion. Optical materials can be described either by the values of the dis-
persion formula coefficients, or directly by the values of the refractive index for
different wavelengths. For many optical materials, this information is available in
databases. Also, during the study, one layer can be considered smooth or partially
inhomogeneous [17].
Problem (1)–(3) is multiextremal. It contains 2N variables, where the first N
variables are the refractive indices of the layers, the second N variables are the
geometric thicknesses of the layers. Bilateral constraints on variables are set by
conditions (2)–(3). The local minima of the problem (1)–(3) often provide the re-
quired approximation accuracy and have implementable coating parameters. Such
solutions are often called quasi-optimal. In this work we decided to follow up on
the suggested term, so by quasi-optimal solutions we will always mean such local
extremums of problem (1)–(3), for which the found coating parameters are practi-
cally feasible.
Problem (1)–(3) can be modeled as unconstrained optimization by using
transition from one variables to another
,sin)( 2minmaxmin
jjjjj zxxxx (11)
.,,1 ,
12
min2max
Nj
z
xzx
x
j
jjj
j
(12)
Thus, a solution for each parameter can be found at infinity. An objective
function has been complicated by this. Formula variables (11) provide a smoother
change of the formed surface and have less abrupted transition in comparison to
another formula (12). On the other hand, the transition to unconstrained optimiza-
tion by formula (11) requires the calculation of the value of )(arcsin x , which is a
rather time-consuming operation. For the approach used in this paper, this applies
to both the values of geometric thicknesses and refractive indices. To do this, the
minimum and maximum refractive indices must be selected.
As the number of layers increases, more parameters for reduction of the tar-
get functions ),( dnF
value in the optimization problems of optical coatings syn-
thesis, can be obtained. Therefore, it is necessary to clarify the criterion for termi-
nation of the search process for solving optimization problem (1)–(3). This goal
can be archived by looking for solution:
.),( *** FdnF
(13)
In case of minimization problem — it will be inequation
*** ),( FdnF
,
and in the case of maximization problem — ),( *** dnFF
.
The introduction of inequality (13) into the optimization model has been
caused by two factors. First, there are a large number of quasi-optimal solutions
that can have a design implementation. Secondly, it is often impossible to achieve
an exact approximation of predetermined spectral characteristics. The spectral
Selection of target function in optical coatings synthesis problems
Системні дослідження та інформаційні технології, 2025, № 3 55
characteristics of the optical coating are analytical functions and can be differenti-
ated an infinite number of times [11]. Accordingly, if the idealized characteristic
is constant or has gaps, then exact approximation cannot be obtained. From a
practical point of view, the definition of the problem should also include a condi-
tion of limiting the number of layers, which would serve as a criterion for termi-
nation of the search process, and can serve for correction of a sufficiently small
value of ε.
An additional condition associated with the manufacture of optical coating
selects one design from a variety of solutions that meet criterion (13), and accord-
ingly, the second Hadamard condition will be met. This condition must also take
into account the characteristics of the selected materials, their interaction with
each other.
The application of the Monte Carlo method allows choosing the most fault
tolerant design solutions [18]. Therefore, for the chosen optical coating, the con-
dition must be met that a slight change in the input parameters will also satisfy
criterion (8), and, respectively, will satisfy the third Hadamard condition.
СOMPUTATIONAL EXPERIMENT
The developed approach has been applied to improve the behaviour of existing
wide bandpass coatings. For this purpose, we used Shor’s R-algorithm [11; 19]
with coating target function represented as
),λ,,(1min),( )(
1λλλ 21
i
L
i
dnTdnF
where ],[ 21 — wavelength range under study; L — number of points in the
wavelength range from 1 to 2 . In this section, the chosen value of L equals
112 , i.e. in the objective function, each integer-value of the interval was
considered ],[ 21 .
Let us demonstrate application of the proposed optimization approach on a
practical example. For this, we will use three optical coatings known in the industry.
In wavelength range between 450 and 800, value of the first coating target
function 404.1),( dnF
(curve 1 — parameters of the optical coating known
in the industry 3.76n1d1=3.76n2d2=0.455n3d3=n4d4=0.250, n1=2.0, n2=1.37,
n3=2.0, n4=1.37), and for the second — ),( dnF
= 0.838 (curve 2 — parameters,
which have been calculated in this article 6.58n1d1=4.06n2d2=0.441n3d3=0.944
n4d4=0.250, n1=2.1, n2=1.35, n3=1.9, n4=1.35). Accordingly, value of the coating
target function ),(F dn
has been improved by 40% (Fig. 2).
Graph of the coating target function can be easily assessed, if we will fix all
parameters, except two (except geometric thicknesses of third and fourth layers,
have been fixed, for optical coating with parameters 0.153n1d1=0.25n2d2=
=0.250, n1=1.35, n2=1.9, n3=1.35, n4=2.1 in the case of antireflection coating
application with refractive index 52.1sn ). As can be seen in the Fig. 3, even
the part of the graph let us assume that this graph has a ravine-type shape. Let’s
consider the sevenlayer antireflection coating, consisting of alternating layers
O.V. Mitsa, P.I. Stetsyuk, S.S. Zhukovskyi, O.M. Levchuk, V.I. Petsko, I.V. Shapochka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 3 56
(1.35 and 2.1), for which layer optical depths in respect to 0 are as follows —
0.05 : 0.071 : 0.062 : 0.257 : 0.018 : 0.12 : 0.2, for which all derived optimal pa-
rameters, except geometric thicknesses of sixth and seventh layers, has been
fixed. Resulting graph (Fig. 6) clearly shows that graph of the estimated target
function has, indeed, a ravine-type shape. It has fixed all the optimal parameters,
except geometric thicknesses of sixth and seventh layers, have been fixed, for
sevenlayer antireflection coating, consisting of alternating layers with refractive
indices 1.35 and 2.1, layer optical depths of the first five layers with respect to 0
are as follows — 0.05 : 0.071 : 0.062 : 0.257 : 0.018.
In wavelength range between 450 and 750, value of the first coating target
function 665.0),( dnF
(curve 1 — parameters of the optical coating known
in the industry, layer optical depths with respect to 0 are as follows — 0.064 :
0.038 : 0.401 : 0.032 : 0.084 : 0.459 : 0.229), and for the second —
Fig. 3. Graph of the quality function of the four-layer coating
Fig. 2. Wide bandpass filter transmittance curve in the case of antireflection coating ap-
plication with refractive index ns=1.51
T
, нм
Selection of target function in optical coatings synthesis problems
Системні дослідження та інформаційні технології, 2025, № 3 57
324.0),(F dn
(curve 2 — parameters, which have been calculated in this arti-
cle, layer optical depths with respect to 0 are as follows — 0.087 : 0.03 : 0.315 :
0.043 : 0.113 : 0.48 : 0.22). Accordingly, value of the coating target function
),(F dn
has been improved by more than 50% (Fig. 4).
It should be noted, that for gradient methods, the use of this objective function
gives a less effective result. For these methods, one must use the target function (10).
In wavelength range between 450 and 750, value of the first coating target
function 953.0),( dnF
(curve 1 — parameters of the optical coating known in
the industry, layer optical depths with respect to 0 are as follows — 0.06 : 0.02 :
0.35 : 0.02 : 0.07 : 0.42 : 0.21), and for the second — 478.0),(F dn
(2 — pa-
rameters, which have been calculated in this article, layer optical depths with re-
spect to 0 are as follows — 0.05 : 0.071 : 0.062 : 0.257 : 0.018 : 0.12 : 0.2). Ac-
cordingly, value of the coating target function ),( dnF
has been improved by
almost 50% (Fig. 5).
CONCLUSIONS
This paper describes three types of target functions, which can be used for solving
optimization problems of optical coatings synthesis. Their reduction to the prob-
lems of unconstrained minimization of smooth and non-smooth functions has
been described and the peculiarities of the transition to new variables for each of
the proposed models has been investigated. The following computer implementa-
tions can be used to accelerate solving optical coating synthesis problems: tabula-
tion of values of trigonometric functions, fast matrix multiplication and the use of
an efficient method for one-dimensional optimization.
A computational experiment has been performed, in which the target func-
tion in the form of the weighted sum of deviations from the mean was taken and
spectral characteristics of the three available wide bandpass antireflection filters
has been improved by using the r-algorithm for optimization. For one of the wide
Fig. 4. Transmittance curve for sevenlayer antireflection coating, consisting of alternat-
ing layers (1.35 and 2.1) of substrate with refractive index ns=1.52
, нм
T
O.V. Mitsa, P.I. Stetsyuk, S.S. Zhukovskyi, O.M. Levchuk, V.I. Petsko, I.V. Shapochka
ISSN 1681–6048 System Research & Information Technologies, 2025, № 3 58
bandpass antireflective coatings, the target function was improved by 40%, and
for the other two, the target function was improved by 50%.
Acknowledgments. The authors are grateful to colleagues from the
Department of Information Management Systems and Technologies of the
Uzhhorod National University and colleagues from the Department of Nonsmooth
Optimization Methods of V.M. Glushkov Institute of Cybernetics of the National
Academy of Sciences for a productive discussion of the topic and the results of
the work.
REFERENCES
1. P. Baumeister, Optical coating technology. SPIE press, 2004. doi: https://
doi.org/10.1117/3.548071
2. A. Piegari, F. Flory (ed.), Optical thin films and coatings: From materials to
applications. Woodhead Publishing, 2018.
3. B. Mansoor, S. Li, W. Chen, “Highly efficient antifogging/antimicrobial dual-
functional chitosan based coating for optical devices,” Carbohydrate Polymers, 296,
119928, 2022. doi: 10.1016/j.carbpol.2022.119928
4. R. Singh, K.N. Unni, A. Solanki, “Improving the contrast ratio of OLED displays:
An analysis of various techniques,” Optical Materials, 34(4), pp. 716–723, 2012.
doi: 10.1016/j.optmat.2011.10.005
5. M.J. Mendes et al., “Design of optimized wave-optical spheroidal nanostructures for
photonic-enhanced solar cells,” Nano Energy, 26, pp. 286–296, 2016. doi:
10.1016/j.nanoen.2016.05.038
6. N. Nedelcu et al., “Design of highly transparent conductive optical coatings
optimized for oblique angle light incidence,” Applied Physics A, 127(8), article
no. 575, 2021. doi: 10.1007/s00339-021-04726-z
7. J.A. Dobrowolski, F.C. Ho, A. Waldorf, D.F. Mitchell, “Trial-and-error optimization
of optical coatings,” Applied Optics, 32(28), pp. 5481–5490, 1993.
8. R. Willey, “Design and optimization of optical coatings and optical systems using
analytical methods,” Thin Solid Films, 287(1-2), pp. 24–31, 1996.
9. S. Larouche, L. Martinu, “OpenFilters: An Open Source Software for the Design and
Optimization of Optical Coatings,” in Optical Interference Coatings, OSA Technical
Digest (CD) (pp. WB6). Optica Publishing Group, 2007. doi: 10.1364/OIC.2007.WB6
10. H.A. Macleod, Thin-film optical filters. CRC Press, 2017.
11. P.I. Stetsyuk, A.V. Mitsa, “Parameter Optimization Problems for Multilayer Optical
Coatings,” Cybernetics and Systems Analysis, 41(4), pp. 564–571, 2005. doi:
https://doi.org/10.1007/s10559-005-0092-x
12. S. Martin, J. Rivory, M. Schoenauer, “Synthesis of optical multilayer systems using
genetic algorithms,” Applied Optics, 34, pp. 2247–2254, 1995. doi: https://doi.org/
10.1364/AO.34.002247
13. K. Haas-Santo, M. Fichtner, K. Schubert, “Preparation of microstructure compatible
porous supports by sol–gel synthesis for catalyst coatings,” Applied Catalysis A:
General, 220(1-2), pp. 79–92, 2001. doi: 10.1016/S0926-860X(01)00714-1
14. V. Boominathan, J.K. Adams, J.T. Robinson, A. Veeraraghavan, “Phlatcam:
Designed phase-mask based thin lensless camera,” IEEE Transactions on Pattern
Analysis and Machine Intelligence, 42(7), pp. 1618–1629, 2020. doi:
10.1109/TPAMI.2020.2987489
15. F. Abeles, “The propagation of electromagnetic waves in stratified media,” Annals of
Physics, 3(4), pp. 504–520, 1948.
16. Y.A. Pervak, A.V. Mitsa, J.I. Holovach, I.V. Fekeshgazi, “Influence of transition
film-substrate layers on optical properties of the multilayer structures,” Selected Pa-
pers from the International Conference on Optoelectronic Information Technologies.
International Society for Optics and Photonics, vol. 4425, pp. 321–325, 2001. doi:
10.1117/12.429744
Selection of target function in optical coatings synthesis problems
Системні дослідження та інформаційні технології, 2025, № 3 59
17. A. Mitsa, V. Mitsa, A. Ugrin, “Mathematical modeling of spectral characteristics of
optical coatings with slightly inhomogeneous chalcogenide films,” Journal of Opto-
electronics and Advanced Materials, 7(2), pp. 955–962, 2005.
18. R.Y. Rubinstein, D.P. Kroese, Simulation and the Monte Carlo method. John Wiley
& Sons, 2016.
19. J.V. Burke, A.S. Lewis, M.L. Overton, “The Speed of Shor’s R-algorithm,” IMA Journal
of Numerical Analysis, 28(4), pp. 711–720, 2008. doi: 10.1093/imanum/drn008
Received 02.04.2024
INFORMATION ON THE ARTICLE
Oleksandr V. Mitsa, ORCID: 0000-0002-6958-0870, Uzhhorod National University,
Ukraine, e-mail: alex.mitsa@uzhnu.edu.ua
Petro I. Stetsyuk, ORCID: 0000-0003-4036-2543, V.M. Glushkov Institute of Cybernet-
ics of the National Academy of Sciences, Ukraine, e-mail: stetsyukp@gmail.com
Serhii S. Zhukovskyi, ORCID: 0000-0001-5826-0751, Zhytomyr Ivan Franko State Uni-
versity, Ukraine, e-mail: zss@zu.edu.ua
Oleksandr M. Levchuk, ORCID: 0000-0001-6344-9356, Uzhhorod National University,
Ukraine, e-mail: alex.levchuk@uzhnu.edu.ua
Vasyl I. Petsko, ORCID: 0009-0009-7679-5288, Uzhhorod National University, Ukraine,
e-mail: vasyl.petsko@uzhnu.edu.ua
Ihor V. Shapochka, ORCID: 0000-0003-0904-7879, Uzhhorod National University,
Ukraine, e-mail: ihor.shapochka@uzhnu.edu.ua
ВИБІР ФУНКЦІЇ ЯКОСТІ В ЗАДАЧАХ СИНТЕЗУ ОПТИЧНИХ ПОКРИТТІВ
/ О.В. Міца, П.І. Стецюк, С.С.Жуковський, О.М. Левчук, В.І. Пецко, І.В. Шапочка
Анотація. Наведено загальні відомості про використання оптичних покриттів
у різних галузях промисловості та проаналізовано основні підходи до оптимі-
зації структур оптичних фільтрів. Запропоновано підхід до вирішення класу
задач синтезу оптичних покриттів, заснований на формуванні нової оптиміза-
ційної моделі. Основну увагу приділено формалізації та аналізу цільової фун-
кції. Для визначення якості оптичного покриття використано оцінку відхилен-
ня спектральних характеристик від необхідних за критеріями найменших
квадратів, найменших абсолютних відхилень і мінімаксу. У результаті запро-
поновано та досліджено як гладку, так і дві негладкі цільові функції. Описано
особливості їх застосування в розв’язуванні оптимізаційних задач синтезу оп-
тичних покриттів та наведено відповідні числові експерименти.
Ключові слова: синтез оптичних покриттів, широкосмугові фільтри, матема-
тичне моделювання, оптимізація, r-алгоритм.
|
| id | journaliasakpiua-article-301046 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-11-09T02:11:02Z |
| publishDate | 2025 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/d8/40374c54ebf8a9e62e14adb94f87b0d8.pdf |
| spelling | journaliasakpiua-article-3010462025-11-09T00:01:30Z Selection of target function in optical coatings synthesis problems Вибір функції якості в задачах синтезу оптичних покриттів Mitsa, Oleksandr Stetsyuk, Petro Zhukovskyi, Serhii Levchuk, Oleksandr Petsko, Vasyl Shapochka, Ihor синтез оптичних покриттів широкосмугові фільтри математичне моделювання оптимізація r-алгоритм optical coatings synthesis wide bandpass filters mathematical modeling optimization r-algorithm The article presents general information on the use of optical coatings in various industries and analyzes the main approaches to optimizing optical filter structures. An approach to solving a class of optical coating synthesis problems is proposed, based on the formation of a new optimization model. The primary attention is paid to the formalization and analysis of the target function. To determine the quality of the optical coating, the deviation of the spectral characteristics from the required ones was estimated using the least squares, least absolute deviation, and minimum criteria. As a result, both smooth and two non-smooth target functions are proposed and analyzed. The peculiarities of their application in solving optimization problems related to optical coating synthesis are described, and corresponding numerical experiments are presented. Наведено загальні відомості про використання оптичних покриттів у різних галузях промисловості та проаналізовано основні підходи до оптимізації структур оптичних фільтрів. Запропоновано підхід до вирішення класу задач синтезу оптичних покриттів, заснований на формуванні нової оптимізаційної моделі. Основну увагу приділено формалізації та аналізу цільової функції. Для визначення якості оптичного покриття використано оцінку відхилення спектральних характеристик від необхідних за критеріями найменших квадратів, найменших абсолютних відхилень і мінімаксу. У результаті запропоновано та досліджено як гладку, так і дві негладкі цільові функції. Описано особливості їх застосування в розв’язуванні оптимізаційних задач синтезу оптичних покриттів та наведено відповідні числові експерименти. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2025-09-29 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/301046 10.20535/SRIT.2308-8893.2025.3.04 System research and information technologies; No. 3 (2025); 48-59 Системные исследования и информационные технологии; № 3 (2025); 48-59 Системні дослідження та інформаційні технології; № 3 (2025); 48-59 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/301046/330988 |
| spellingShingle | синтез оптичних покриттів широкосмугові фільтри математичне моделювання оптимізація r-алгоритм Mitsa, Oleksandr Stetsyuk, Petro Zhukovskyi, Serhii Levchuk, Oleksandr Petsko, Vasyl Shapochka, Ihor Вибір функції якості в задачах синтезу оптичних покриттів |
| title | Вибір функції якості в задачах синтезу оптичних покриттів |
| title_alt | Selection of target function in optical coatings synthesis problems |
| title_full | Вибір функції якості в задачах синтезу оптичних покриттів |
| title_fullStr | Вибір функції якості в задачах синтезу оптичних покриттів |
| title_full_unstemmed | Вибір функції якості в задачах синтезу оптичних покриттів |
| title_short | Вибір функції якості в задачах синтезу оптичних покриттів |
| title_sort | вибір функції якості в задачах синтезу оптичних покриттів |
| topic | синтез оптичних покриттів широкосмугові фільтри математичне моделювання оптимізація r-алгоритм |
| topic_facet | синтез оптичних покриттів широкосмугові фільтри математичне моделювання оптимізація r-алгоритм optical coatings synthesis wide bandpass filters mathematical modeling optimization r-algorithm |
| url | https://journal.iasa.kpi.ua/article/view/301046 |
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