=Paper=
{{Paper
|id=Vol-2650/paper25
|storemode=property
|title=Investigation of the Influence of Errors on the Parameters of the Layers of Optical Filters on the Stability of Their Spectral Characteristics
|pdfUrl=https://ceur-ws.org/Vol-2650/paper25.pdf
|volume=Vol-2650
|authors=Oleksandr Mitsa,József Holovács,Roman Holomb,Oleksandr Levchuk
|dblpUrl=https://dblp.org/rec/conf/icai3/MitsaHHL20
}}
==Investigation of the Influence of Errors on the Parameters of the Layers of Optical Filters on the Stability of Their Spectral Characteristics==
https://ceur-ws.org/Vol-2650/paper25.pdf
Proceedings of the 11th International Conference on Applied Informatics
Eger, Hungary, January 29–31, 2020, published at http://ceur-ws.org
Investigation of the Influence of Errors on
the Parameters of the Layers of Optical
Filters on the Stability of Their Spectral
Characteristics
Oleksandr Mitsaa , Jozsef Holovácsb , Roman Holomba ,
Oleksandr Levchuka
a
Uzhhorod National University, Ukraine
alex.mitsa@uzhnu.edu.ua,holomb@gmail.com,alex.levchuk@uzhnu.edu.ua
b
Transcarpathian Hungarian College of Higher Education, Ukraine
holovacs@kmf.uz.ua
Abstract
Mathematical modeling is performed in order to analyze the effects of
technological errors originated from partial inhomogeneity, variations of the
refractive index and the differences in geometric thickness of layers with high
refractive index on spectral characteristics of narrow- and wide-band filters.
Using a Monte Carlo method the influence of technological errors on optical
parameters of the layers and their influences on spectral characteristics of
interference filters are investigated and discusses in details.
Keywords: Monte Carlo method, slightly inhomogeneous films, light trans-
mission, multilayer interference coating, the narrowband interference filters,
the wideband interference filters.
MSC: 8208; 82B80
1. Introduction
Investigation of physical properties of some new film-formation materials used for
interference and high power optics indicate that the formation of optical coating
on various substrates can create spartially inhomogeneous films with transitional
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
242
areas [1-3]. Therefore, the influence of this effect on spectral characteristics of in-
terference filters should be considered. Apart from partial inhomogeneity of layers,
spectral characteristics are also influenced by technological errors. In this article
the influence of technological errors occurred in layers with high refractive on op-
tical parameters of filters is investigated using the Monte Carlo method [4]. This
investigation builds on the assumption that error has a normal distribution. The
purpose of this paper is to develop an appropriate mathematical model and to
investigate the effects of different technological errors.
The objective of this article is to develop an appropriate mathematical model
and to investigate the influence of error originated from partial inhomogeneity of
layers with high refractive index and technological errors in layers’ parameters on
spectral characteristics of narrow-band and wide-band filters. The refraction index
of inhomogeneous areas was selected with different step-functions having linear,
quadratic, logarithmic, and exponential distributions.
2. Mathematical modeling
When the refractive index 𝑛, geometric thickness of a layer 𝑑 and wavelength 𝜆 are
selected as parameters, the characteristic matrix of one layer can be written in the
following way [5]:
⃦ ⃦
⃦ cos 𝛿(𝑛, 𝑑, 𝜆) − 𝑛𝑖 sin 𝛿(𝑛, 𝑑, 𝜆)⃦
𝑀𝑠 (𝑛, 𝑑, 𝜆) = ⃦
⃦−𝑖𝑛 sin 𝛿(𝑛, 𝑑, 𝜆)
⃦, (2.1)
cos 𝛿(𝑛, 𝑑, 𝜆) ⃦
where 𝛿(𝑛, 𝑑, 𝜆) = 2𝜋𝑛𝑑
𝜆 .
While identifying the characteristic matrix of partially inhomogeneous structure
we will use the theoretical considerations described in [3].
Using the characteristic matrix of the whole layered structure the transmittance
coefficient dependent on wavelength 𝜆 can be found by following equation:
4
𝑇 (𝜆) = , (2.2)
2 + 𝑛𝑛0𝑠 𝑀11
2 (𝜆) + 𝑛𝑠 𝑀 2 (𝜆) + 𝑛 𝑛 𝑀 2 (𝜆) + 1 𝑀 2 (𝜆)
𝑛0 22 0 𝑠 12 𝑛0 𝑛𝑠 21
where 𝑛0 , 𝑛𝑠 – refractive indexes of the external medium and substrate, respectively.
The function of quality can be established as follows [6]:
(︁ 1 ∑︁
𝐿 )︁ 21
𝐹 (𝑛, 𝑑) = 𝑇 2 (𝑛, 𝑑, 𝜆(𝑖) ) , (2.3)
𝐿 𝑖=1
where 𝐿 is the number of points of the net of spectral interval from 𝜆1 to 𝜆2 , 𝜆(𝑖)
are the values of wavelengths on the given net.
Fot determination of stability of spectral characteristics of interference filters it
is necessary to consider that the error in refraction index values measured for one
layer is not higher than ±0.05 and the deviation in geometric thickness is in range
of ±2 nm. The type of distribution of error values in the given boundaries was not
243
taken into account in the previous study [3]. In this article the normal distribution
of errors was considered.
The Monte Carlo method was used to solve the problem (2.1)–(2.3) within
several steps. The first step is to input information necessary for calculations
including data on the structure of coating, the planned number of experiments 𝑘,
interval boundaries (𝑎, 𝑏) from which parameter of the layer assumes a random
value.
The second step is to define that the first layer is considered and that the
number of experiment is 1.
The third step is to generate a random number.
The fourth step is to identify parameters of a layer though the obtained random
number.
The fifth step is to calculate characteristics of the coating and compute the
value of objective function 𝐹𝑖 (𝑥) in the experiment No 𝑖.
The sixth step is to use calculated 𝐹𝑖 (𝑥) value for formation of sums of the
following form: ∑︁ ∑︁
𝐹𝑖 , 𝐹𝑖2 .
𝑖 𝑖
The seventh step is to check the condition of conducting of the planned num-
ber of experiment. If this condition is met, then the numerical characteristics of
distribution of the objective functions can be calculated:
𝑘
1 ∑︁
𝑀= 𝐹𝑖 (𝑥),
𝑘 𝑖=1
⎯
⎸
1 (︁ ∑︁ 2 )︁
⎸ 𝑘
𝜎=⎷ 𝐹𝑖 (𝑥) − 𝑘𝑀 2 ,
𝑘 − 1 𝑖=1
where 𝑀 and 𝜎 are mathematical expectation and dispersion of a random 𝐹𝑖 (𝑥)
value.
The eighth step is to move to the next layer.
The ninth step is to check the condition for ending the calculation process.
3. Computing experiment
Let us to consider the influence of partially inhomogeneous areas on spectral char-
acteristics of a narrowband filter (Fig. 1).
As it can seen from Fig. 1, the types of distribution of refractive index influences
the deviation of spectral characteristics of 17-layer narrow-band filteroperating at
wavelength of 630 nm. This deviation from the ideal case is depending on value of
average refractive index of spectral range.
For all types of distributions the increase of average refractive index of the
spectral range lead to widening of waoking spectral region to red-shift of 𝜆max
value, while transmission half-width ∆𝜆0.5 and bandwidth ∆𝜆0.1 decrease.
244
Figure 1: Spectral characteristics of 17-layer structure 𝑆 −
𝐻𝐿𝐻..2𝐻..𝐻𝐿𝐻 (𝜆0 = 630𝑛𝑚) in ideal case and in presence of
near-surface and transitional areas with different refractive index
distributions: 1 - ideal case, 2 - step-function distribution; 3 - linear
distribution; 4 - quadratic distribution; 5 - logarithmic distribution;
6 - exponential distribution.
Except exponential distribution of refractive index with increasing the average
value of refractive index the transmission coefficient, 𝑇max , is increasing in compar-
ison with the ideal case. For exponential distribution of refractive index, the value
of transmission coefficient 𝑇max at working wavelengths of 480, 630, 760 and 1000
nm is lower than those found in ideal case.
The increasing of number of layers leads to decrease the deviation of 𝜆max value
from ideal case for all types of refractive index distribution. It was found that the
transmission coefficient 𝑇max changes less than 10−3 during variation of number
of layers. Also, the increasing of number of layers results in decreasing of both
transmission half-width ∆𝜆0.5 and band-width ∆𝜆0.1 . Once again, the higher is
the average value of refractive index of spectral range, the lower are their values.
Let us model the stability of spectral characteristics of a 𝑆 − 𝐵𝐻𝐵. . . 𝐵𝐻𝐵
type narrow-band filter with regard to possible errors of layers’ parameters using
the Monte Carlo method. For a 9-layer narrow-band filter the sensitivity of spectral
characteristics with regard to variations of refractive index of the second layer is
much higher than other variations of layers’ characteristics (Fig. 2). For a (4𝑘 + 1)-
layer narrow-band filter the spectral characteristics are several times more sensitive
to the variations of refractive index of layers with low refractive index than to the
variations of refractive index of layers with high refractive index (Fig. 2). The
exception is (2𝑘 + 1)- layer which is half-wave one and sensitivity to variations of
which is significant too.
The sensitivity of spectral characteristics to errors of geometric thickness of low-
245
Figure 2: Fig. 2. Diagram of dispersion of the objective function
for a 𝑆 −𝐻𝐿𝐻..2𝐻..𝐻𝐿𝐻 type narrowband filter with step-function
distribution of refractive index of near-surface and transitional areas
at the working wavelength = 630𝑛𝑚 based on the results of Monte
Carlo simulation: a) 9-layer structure; b) 17-layer structure.
refracting layer is much lower than to errors induced by refractive index changes.
The exceptions are 2𝑘 and (2𝑘 + 2) layers which are adjacent to the half-wave layer.
The sensitivity of spectral characteristics of these layers to errors of the geometric
thickness is higher than to the refraction index errors. This also applies to high
refracting layers, except for the first and the last layers of multilayered structure of
interference filter. For a narrow-band filter, unlike a cutting filter, an average value
of refractive index of a spectral range does not influence the stability of spectral
characteristics. The highest dispersion range has the step-function distribution of
refractive index.
Let us now investigate the influence of partially inhomogeneous areas on spectral
characteristics of a wide-band filter (Fig. 3). As can be seen from Fig. 3, for a
17-layer wide-band filter at working wavelength of 630 nm the left boundaries of
transmittance range for different distributions of refractive index are not ranked
(unlike for a cutting filter) in order of magnitude of average refractive index value
of spectral range.
246
Figure 3: Fig. 3. Spectral characteristics of the 17-layer structure
𝑆 − 2𝐵𝐻2𝐵..2𝐵𝐻2𝐵 (𝜆0 = 630𝑛𝑚) in ideal case and in presence
of near-surface and transitional areas with different distributions of
refractive index: 1 – ideal case; 2 with step-function distribution of
refractive index; 3- with linear distribution of refractive index; 4 –
with quadratic distribution of refractive index; 5 – with logarithmic
distribution of refractive index; 6 – with exponential distribution of
refractive index.
Main characteristics of wide-band filters are bandwidth ∆𝜆0.5 , bandwidth ∆𝜆0.1
and the middle of transmittance range 𝜆𝑚𝑑 . The middle of transmittance range is
given by the formula:
𝜆𝐿 + 𝜆𝑅
𝜆𝑚𝑑 = 0.5 0.5
,
2
where 𝜆𝐿 0.5 , 𝜆0.5 are left and right boundaries of the transmittance range with the
𝑅
transmittance value of 0.5, respectively.
The middle of the transmittance range of most filters with different distribu-
tions of refractive index, 𝜆𝑚𝑑 , shifts to the long waves area in comparison with
ideal case at the value which directly depends on average value of refractive index
of spectral range. The exception is the linear distribution of refractive index at
working wavelength 𝜆0 = 480𝑛𝑚. Except exponential distribution of refractive in-
dex, the bandwidths of ∆𝜆0.1 and ∆𝜆0.5 at working wavelengths (𝜆0 ) of 480, 630,
750 and 1000 𝑛𝑚 are decreasing when the average refractive index of spectral range
increase. The values of main characteristics of filters with exponential distribution
of refractive index are higher in comparison with the ideal case. At working wave-
length 𝜆0 = 3000𝑛𝑚 the deviation from ideal case increase with increasing of the
average value of refractive index of spectral range.
The values of main characteristics (middle of transmittance range 𝜆𝑚𝑑 and
bandwidth ∆𝜆0.5 ) is increasing with increasing of number of layers. The bandwidth
∆𝜆0.1 for 17 layers is higher than for 25 layers. This is due to more rapid decreasing
247
of transmittance coefficient at the boundaries of the transmittance range.
The role of small changes of layers’ parameters with different distributions of
refractive index on stability of spectral characteristic of the 𝑆 −2𝐵𝐻2𝐵 · · · 2𝐵𝐻2𝐵
type wideband filter were studies using the Monte Carlo method.
Figure 4: Fig. 4. Dispersion diagram of objective function for
2𝐵𝐻2𝐵 · · · 2𝐵𝐻2𝐵 type wideband filter with the exponential dis-
tribution of refractive index of near-surface and transitional areas at
the working wavelength 𝜆0 = 630𝑛𝑚 obtained using the results of
Monte Carlo simulation: a) 9-layer structure; b) 17-layer structure.
As can be seen from Fig. 4, the sensitivity of spectral characteristics to errors
of refractive index of even layers for both a wide-band and a narrow-band filters
are much higher than to errors of other parameters.
The influence of error in geometric thickness of odd layers on stability of spectral
characteristics of filters is approximately at the same level as those induced by error
in refractive index but more than error in geometric thickness of even layers.
The dispersion range does not always increase with the increasing of number
of layers. However, in general, the increasing trend is maintained: for the 9-layers
structure the maximal value of dispersion range is 0.052, for the 17-layers – 0.054,
for 25-layers – 0.053 and for 33-layers– 0.089. Generally, the spectral characteristics
248
of a wide-band filter are more stable to errors in layers’ parameters than spectral
characteristics of a narrow-band filter.
4. Conclusion
The detailed analysis of results indicate that the stability to different technolog-
ical errors should be taken into account during development of interference filter
structures. Among few structures of interference filters which set roughly similar
spectral characteristics the more stable to errors should be preferred for practical
implementation. The practical implementation of Monte Carlo method faces two
problems. The first problem regards the necessity to conduct the large number of
experiments since the statistical error of 𝑀 and 𝜎 values decreases very slowly (in-
versely proportional to square root of number of experiments). The second problem
relate with the retrieving information on the law of distribution of errors in defin-
ing layers’ parameters. This particularly applies to refractive index of layers, which
value depends on numerous technological parameters including substrate and evap-
oration temperatures, evaporation and condensation rates, residual gas in coating
chamber, presence or absence of oxidizing medium, cleanness of raw materials and
other parameters.
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