Mathematical modeling is performed in order to analyze the efects of technological errors originated from partial inhomogeneity, variations of the refractive index and the diferences 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.
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
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areas [
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 diferent step-functions having linear, quadratic, logarithmic, and exponential distributions. 2.
When the refractive index , geometric thickness of a layer and wavelength are
selected as parameters, the characteristic matrix of one layer can be written in the
following way [
⃦ ) = ⃦⃦⃦− cos (, , sin (, , ) ) − sin (, , cos (, , ⃦ ) ) ⃦⃦⃦ , (, ) = ︁( 1 ∑︁ =1 2(, , ( ))︁) 12 , where is the number of points of the net of spectral interval from 1 to 2, ( ) 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 (, , ) = 2 .
4 (2.1) (2.2) (2.3) where (, ,
( ) =
While identifying the characteristic matrix of partially inhomogeneous structure
we will use the theoretical considerations described in [
Using the characteristic matrix of the whole layered structure the transmittance coeficient dependent on wavelength
can be found by following equation: 2 + 0 121( ) + 0 222( ) + 0 122( ) + 01 221( ) where 0, – refractive indexes of the external medium and substrate, respectively.
The function of quality can be established as follows [
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 , interval boundaries (, ) from which parameter of the layer assumes a random The second step is to define that the first layer is considered and that the
The fourth step is to identify parameters of a layer though the obtained random The fifth step is to calculate characteristics of the coating and compute the value of objective function ( ) in the experiment No .
The sixth step is to use calculated ( ) value for formation of sums of the The seventh step is to check the condition of conducting of the planned number of experiment. If this condition is met, then the numerical characteristics of distribution of the objective functions can be calculated: ︁∑ , ︁∑
2. 1 ∑︁
=1 =
( ), ⎯ ⎸ = ⎷⎸
1 (︁ ∑︁ − 1 =1 2( ) − ︁) 2 , where value.
and are mathematical expectation and dispersion of a random ( )
Let us to consider the influence of partially inhomogeneous areas on spectral characteristics 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 max value, while transmission half-width Δ 0.5 and bandwidth Δ 0.1 decrease.
Except exponential distribution of refractive index with increasing the average value of refractive index the transmission coeficient, max, is increasing in comparison with the ideal case. For exponential distribution of refractive index, the value of transmission coeficient max 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 max value from ideal case for all types of refractive index distribution. It was found that the transmission coeficient max 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 Δ 0.5 and band-width Δ 0.1. 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 − . . . 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 (4 + 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 (2 + 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 lowrefracting layer is much lower than to errors induced by refractive index changes. The exceptions are 2 and (2 + 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 diferent distributions of refractive index are not ranked (unlike for a cutting filter) in order of magnitude of average refractive index value of spectral range.
Main characteristics of wide-band filters are bandwidth Δ 0.5, bandwidth Δ 0.1 and the middle of transmittance range . The middle of transmittance range is given by the formula: = 0.5 + 0.5 , 2 where 0.5, 0.5 are left and right boundaries of the transmittance range with the transmittance value of 0.5, respectively.
The middle of the transmittance range of most filters with diferent distributions of refractive index, , 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 0 = 480 . Except exponential distribution of refractive index, the bandwidths of Δ 0.1 and Δ 0.5 at working wavelengths ( 0) of 480, 630, 750 and 1000 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 wavelength 0 = 3000 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 and bandwidth Δ 0.5) is increasing with increasing of number of layers. The bandwidth Δ 0.1 for 17 layers is higher than for 25 layers. This is due to more rapid decreasing of transmittance coeficient at the boundaries of the transmittance range.
The role of small changes of layers’ parameters with diferent distributions of refractive index on stability of spectral characteristic of the −2 2 · · · 2 2 type wideband filter were studies using the Monte Carlo method.
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 of a wide-band filter are more stable to errors in layers’ parameters than spectral characteristics of a narrow-band filter.
The detailed analysis of results indicate that the stability to diferent technological 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 and values decreases very slowly (inversely proportional to square root of number of experiments). The second problem relate with the retrieving information on the law of distribution of errors in defining layers’ parameters. This particularly applies to refractive index of layers, which value depends on numerous technological parameters including substrate and evaporation temperatures, evaporation and condensation rates, residual gas in coating chamber, presence or absence of oxidizing medium, cleanness of raw materials and other parameters.