Modeling of Output Characteristics of a UV Cu + Ne-CuBr Laser Snezhana

This paper examines experiment data for a Ne-CuBr UV copper ion laser excited by longitudinal pulsed discharge emitting in multiline regime. The flexible multivariate adaptive regression splines MARSs method has been used to develop nonparametric regression models describing the laser output power and service life of the devices. The models have been constructed as explicit functions of 9 basic input laser characteristics. The obtained models account for local nonlinearities of the relationships within the various multivariate subregions. The built best MARS models account for over 98% of data. The models are used to estimate the investigated output laser characteristics of existing UV lasers. The capabilities for using the models in predicting existing and future experiments have been demonstrated. Specific analyses have been presented comparing the models with actual experiments. The obtained results are applicable for guiding and planning the engineering experiment. The modeling methodology can be applied for a wide range of similar lasers and laser devices.


Introduction
During the process of engineering design, together with the purely experimental investigations, various methods for mathematical modeling are also applied.Standard mathematical models usually include differential and integral equations, optimization, or other types of mathematical problems through which the processes occurring in the laser medium are described and the behavior of the system is explored by simulating its various states 1-5 .The disadvantage of these methods is the difficulty in obtaining an explicit estimate of emission range of just a few spectral lines and the high coherence of the beam, it is used for processing, which requires high resolution, such as recording information, fluorescence, high-precision drilling, cutting, cleaning, modification of newly developed materials, and so forth.Technological applications of the UV lasers created at the Laboratory of Metal Vapor Lasers are achieved.Laser-induced modification by the use of UV laser radiation with 248.6 nm wavelength has been performed in a conducting polymer.A significant growth in the electrocrystallization of copper on polymer layers was obtained.The refractivity of polymer layers in ultraviolet spectral regions was determined.Micronic holes of 10, 20, and 40 microns in size were drilled in glass, polymer, and ZnSe 20 .
The schematic of the ultraviolet Ne-CuBr laser tube is shown in Figure 1.Its main technical characteristics are given in Table 1.
The goal of subsequent development of the examined UV lasers is to investigate the possibilities for improving its output characteristics: output power and service life.

Data Description
A total of 9 laser characteristics independent variables, predictors are studied, which are of physical significance and are considered independent.The variables we will examine are as follows: D, mm: inside diameter of the laser tube; DR, mm: inside diameter of the internal rings in the ceramic tube or quartz tube insert; L, cm: length of the active zone distance between the electrodes ; P IN2 , kW: electric power supplied to the active volume, taking into account 50% losses of the total supplied power P IN ; P NE , Torr: neon gas pressure; P RF , kHz: pulse rate frequency; P H2 , Torr: hydrogen gas pressure; C, pF: equivalent capacity of the condensation battery; TR, • C: temperature of the copper bromide reservoir equal to the temperature of the outer side of the quartz tube .
The dependent variables are laser output power laser generation P out , W, and laser service life L Time , hours.This study uses the data from n 238 experiments for 11 different laser devices, reported in 11-20 .The scatter plot of the relationships for each two of the eleven variables 9 independent and 2 dependent is given in Figure 2.
The descriptive properties of the data indicate that applying the standard conventional parametric methods is not recommendable as the basic conditions for their validity are violated 6 .More specifically, the two dependent variables are not normally distributed.There is also certain collinearity between independent variables.For that reason, in this paper, nonparametric statistical techniques are used, choosing in this case the MARS method.

MARS Method
The relatively new nonparametric MARS technique that performs well with complex data structures has been applied 21, 22, 26 .MARS allows building flexible models capable of describing local relationships in a multidimensional region of data, within which subregions are defined.For each of these multidimensional subregions, the most suitable type of relationship is chosen and a general model is built, which approximates as closely as possible the actual data in accordance with a specific criterion.
Let us denote the independent variables by X x 1 , x 2 , . . ., x p and their dependent one by y y X .We assume that y, x 1 , x 2 , . . ., x p are vectors with dimension n.For our data n 238, p 9. The general form of a MARS model y M of the data for y has the form where b 0 , b j , j 1, 2, . . ., M are the sought constant coefficients of the model, M the number of functions included in the model, and BF j X basis functions of the selected type.
In the linear case, basis functions are presented in the form of one of the following two one-dimensional "mirror" functions: Here c k is a constant, called a knot of the basis function with values in the definition interval of the predictor variable x k .In the linear case, model 4.1 presented as a sum of basis functions of type 4.2 is plotted graphically in the form of a piecewise linear function.When building nonlinear MARS models, basis functions BF j X can also include the products of two or more functions of the form 4.2 without repeating the index k.The maximum number of multipliers is called order of interaction.The linear model has an order of interaction 1.
We have to note that unlike ordinary interpolation splines, in the MARS model 4.1 , the knots of the basis functions, their number M, the selection of functions, and the subintervals of determination are not known in advance.These are determined by different optimization conditions and prerequisites for estimating the proximity of the model to real data on y X .
When building the models, the researcher sets an initial maximum number M 0 of basis functions and the maximum order of interaction.The recommendation is M 0 ≥ 3p 21 .MARS procedures systematically select subintervals and subregions for each predictor, estimate the degree of significance to the model of a given basis function, and exclude those that do not contribute to the improvement of the model.Several different criteria are used to evaluate the model after each step.One criterion is the minimization of the sum of squares of errors:  which "the best" model of M 1 models with m basis functions is the one that minimizes the expression where C m m δ m − 1 /2, δ ∈ 2, 3 , X i x 1i , x 2i , . . ., x pi , y i y x 1i , x 2i , . . ., x pi .As a whole, the implementation of MARS method algorithms allows the automatic definition of those predictors that influence the examined dependent variable, as well as the degree of this influence, removing the statistically insignificant predictors.The final model usually requires that the residuals of the model are normally distributed.
An advantage of the method is that the resulting models are simpler in form and can easily be interpreted during their practical application.
Within this study only the best MARS models are presented.All nine variables are initially introduced as predictors.We use the notation M 0 , r for a model with M 0 initial maximum number of allowed basis functions BFs and up to rth order of interactions, r ≥ 1.

Building MARS Models of the Laser Generation of an UV Laser
First, we will present the best MARS models of laser generation P out .
The basic statistic figures of the constructed models are given in Table 2.We will describe in more detail two of these models: the best piecewise linear model with no interactions and the best model with first-order interactions.

Linear MARS Models of P out
Out of the linear models, we will present model 40;1 .The figures of other linear models are given in the upper part of Table 2.

5.1
The obtained regression model for laser output power P out with these functions is

5.2
The model 5.1 -5.2 can easily be used to estimate or predict laser generation.For example, let us take case i 128, with output laser power P out 128 900 mW 20 at the following values of the independent variables: We calculate each of the functions in 5.1 consecutively.We have BF1 1.4, BF3 1500/2 − 525 225, BF4 BF5 BF8 BF9 0, and so forth.We substitute in model 5.2 obtaining the estimate P out 128 840 mW.

5.4
The absolute error in case i 128 is about 60 mW, and the relative one is 7%.The overall relative error for the model is 8-9%, which is a satisfactory figure.
Model 5.1 -5.2 is the best MARS model of a given type, which is selected so as to allow no overfitting of the model, as well as by using the measures of data fit SSE and GCV.The obtained basic statistics are given in Table 2.The model is significant at level 0.000.
The relative influence of individual predictors in model 5.2 is given in Table 3, column 2. It is apparent that the most significant variable is P H2 , whose influence is measured as 100%, and the influence of the others is calculated against it.
The local behavior of the main predictors and their contribution to model 5.1 -5.2 are shown in Figures 3 a -3 h in pure ordinal units.

Nonlinear MARS Models
Of the second-order models we present the best model with up to 40 BF, which is denoted as model 40;

5.5
The corresponding regression model for laser output power P out includes all of these functions with 8 independent variables.It has the following form:

5.6
Likewise, model 5.5 -5.6 can be used to estimate and predict output power.For example, in the same case i 128, with the data 5.3 , we obtain the estimate P out 128 856.4 mW.

5.7
The absolute error in case i 128 is about 44 mW, and the relative one is 5%.
A part of the partial contributions of pairs of predictors in model 5.5 -5.6 to the value of P out are presented in Figures 3 a -3 d .The biggest contribution of more than 6000 units is made by the interaction between P NE and P H2 , which reaches the highest value in a large 2D region.Predictors C and DR provide almost the same contribution.The other two interactions also have given effects.

Application of the Models for P out
In 5.4 and 5.7 , it was demonstrated how the predicted values of the model P out are found.
The properties of approximation of the first-and second-order models can be easily examined using graphs.For example, for the piecewise linear model 5.1 -5.2 , the graphs presented in Figure 3 show the local relationships between individual predictors and the dependent variable P out . Figure 3 a indicates that the pressure of the buffer gas neon P NE , which makes the biggest contribution to the model, should be taken within the interval 16, 21 Torr.For the inner diameter of the laser tube, the high values are achieved at 4 < D < 10 mm Figure 3 b , which has been established experimentally in 16, 17 .The behavior of the pressure of hydrogen admixtures P H2 also fits quite well for all experiments, exhibiting maximum local influence in 0.02, 0.04 Torr Figure 3 e .In addition to the mutual influence of predictors, it is also possible to plot slices in order to examine the local behavior of each of the two variables with one fixed at a value chosen by the researcher and the other changing within the whole interval of definition.
As an example, Figure 5 a shows a slice from Figure 4 b for the modeled behavior of the variable P NE neon gas pressure for a fixed value of the hydrogen pressure P H2 0.03 Torr.In this case, there is a clearly identifiable maximum in the interval 11, 26 Torr.The slice in Figure 5 b shows the distribution of the influence of the pair of predictors {P NE , P IN2 } for a fixed P NE 19.5 Torr, depending on the increase of P IN2 , as an element of Figure 4 c , which shows the maximum importance.
By carefully studying the resulting estimates of P out from a given model and the defined influences of the significant variables, as well as their interactions, it is possible to guide the experiment in order to improve the output characteristics.
In addition to the plots, the models can be used to examine the local influence of the main laser input quantities on output ones, as well as to predict new experiments.In order to demonstrate this, we will consider the local behavior of P out for the laser from 18 when varying the pressure of the neon buffer gas P NE in the interval 15.5, 18.5 Torr.For P IN 1300 W we choose experiment data with fixed D DR 7.1 mm, L 86.5 cm, P H2 0.03 Torr, P RF 19.5 kHz, and C 735 pF.In Figure 6

MARS Models for Laser Service Life
This type of research has not been performed for metal vapor lasers so far.The models utilize as predictors all of the 9 input laser characteristics from Section 3 with response L Time -laser service life, measured in hours.Again, our objective is to find a model with a sufficiently high  coefficient of determination over 95% and the smallest possible mean-square error under 5% comparable with the experimental error.The statistics of the obtained models and basic statistics are given in Table 5.The first best MARS model with 98% coefficient of determination is model 40;2 .

Linear MARS Models of L Time
Of the linear-type models, we will present model 30;1 .The figures of other linear models are given in the upper part of Table 5.
6.2 Model 6.1 -6.2 can be used to estimate and predict the service life of the examined lasers as shown by the models of P out .The model is significant at level 0.000.
The relative influence of individual predictors in model 6.2 is given in Table 6, column 2. It is apparent that the most significant variables are TR, C, P NE , and P IN2 in descending order according to their influence .The behavior of the 4 main predictors in their intervals of determination and contribution in model 6.1 -6.2 are shown in Figures 7 a -

6.4
The absolute error of model 6.3 -6.4 is about 0.006%, and the relative one is 4%.The importance of the two main predictors is shown in Figures a -8 b .

Application of the Obtained MARS Models of L Time
Model 6.3 -6.4 analogically allows for estimation and prediction of the service life of the considered laser device depending on a given set of values of participating predictors.What is essential here is the relative influence of these predictors, which is to be taken into consideration by the design engineer when planning the experiment, in our case maintaining strictly the temperature of the reservoirs TR equal to the temperature maintained at the outer wall of the laser tube quartz or ceramic , and keeping the ratio between the pressure of the neon buffer gas P NE and the bank of condenser within the established limits.

Comparison and Diagnosis of the Obtained MARS Models
Model comparison can be performed using the general statistical indices from Tables 2 and 5.It has to be noted that in the last column the predictors for the respective model are presented in descending order according to their relative importance.The lowest values of R 2 and GCV R 2 are in linear models r 1 both for P out and for L Time .With the increase of the number of basis functions BFs the parameters of the models increase slightly.However, the investigation of the residuals of the linear models shows that they are not sufficiently adequate and that the relative error is comparatively high within 10-15%.This means that the relationship of the input independent variables with P out or L Time is not linear.
For output power P out , Table 2 shows that second-order models from 40;2 exhibit a 98% coefficient of determination and that the participation of the predictors is stable.Since for 40;2 the GCV R 2 estimate is over 95%, this model is quite good.The next models up to 60;2 demonstrate practically the same fitting properties.
Of all models of L Time , model 30;2 exhibits the best qualities.Third-order models are comparable with the respective second-order models and demonstrate almost the same or slightly lower indices than second-order ones.Since these models are more complex in form, we conclude that the examined data demonstrate second degree local nonlinearities and it is best to describe these using second-order models.
In order to examine the diagnostics we have to note that all presented models are statistically significant at level P 0.00000.The coefficients of the models are significant with P ≤ 0.005.The standardized error of the estimate is small.Model residuals are normally distributed.
Figure 9 shows the comparison of experimental data for laser output power P out with those calculated by model 40;2 .It is observed that the model fits data quite well.For the coefficient of determination we have R 2 0.98, which means that the model accounts for 98% of all data.The corresponding GCV R 2 0.95 see Table 2 .The residuals of this model are normally distributed with N 0, 0.998 .This way, the two main criteria and statistical indices show very good predictive properties and goodness of fit of the constructed MARS model 40;2 .
The fitting property of model 30;2 for L Time against the experiment is given in Figure 10.

Physical Interpretation of the Results
The developed models correspond quite well with the experiment and reflect nonlinear local relationships in a multidimensional space of 9 variables.
Laser output power is most significantly influenced by input electric power, neon pressure, and hydrogen pressure.Of the geometric dimensions, output power is most dependent on the diameter of the rings DR, which for latest laser devices is equal to D-the diameter of the inner tube, with its optimal value between 7.1 and 7.3 mm.
The new results are those for the service life of the examined lasers, which is one of the most important issues for all types of devices.For the latest lasers 20 this period is quite acceptable, reaching 700-1000 hours.Furthermore, periodically a refreshment of laser tube is made, which involves shutting down the laser, cleaning, and changing the gas mixture.The obtained models unequivocally show that the main contribution to extending the service life of the laser is that of TR inversely proportional-Figure 7 a , C, and P NE .Therefore, these characteristics need to be considered carefully when designing new lasers of the investigated type and their parameters should be constant or nearly constant in relation to those already established.

Conclusion
The basic results of the performed statistical modeling of laser output characteristics, laser generation and laser service life of a multiline ultraviolet copper ion vapor laser excited in a   longitudinal pulsed discharge, have been obtained using nonparametric MARS models.The models demonstrate very good abilities and goodness of fit when predicting existing and future experiments.
It was determined that the best MARS models are nonlinear and contain secondorder members.Of 9 input independent variables used as predictors, 8 influence laser output power with the most significant ones being the pressure of the applied electric power, the inner diameter of the laser tube diameter of the rings , neon buffer gas pressure, and hydrogen pressure.Service life models are simpler and only 3 laser characteristics exhibit significant influence: temperature of the tube, neon gas pressure, and equivalent capacity of the condensation battery.
It is shown that the models can be used to estimate and predict local properties of laser generation and service life of the devices.
The techniques developed can also be further employed in the process of future industrial application of this type of lasers.

Figure 1 :
Figure 1: Longitudinal principle scheme of the laser tube of a UV copper ion laser.

Figure 2 :
Figure 2: Scatter plot of all investigated data of UV Cu Ne-CuBr vapor lasers.
4 d show local relationships between the main pairs of significant input parameters in two-dimensional regions.The supplied electrical power P IN makes the biggest contribution to the model and shows an increasing behavior by comparing Figures 4 a , 4 c and 4 d .Figures 4 b and 4 c , indicate that the neon pressure P NE should be taken within the interval 15, 25 Torr.The hydrogen pressure gives an optimal contribution for 0.03, 0.04 Torr, according to Figures 4 d and 4 b .

2 (Figure 4 :
Figure 4: Graphs of contribution of the selected pairs of predictor variables in MARS model with 5.5 -5.6 in ordinal units.

Figure 5 :Figure 6 :
Figure 5: View of slices from Figures 4 b and 4 c for the influence of P NE and P IN2 on P out in pure ordinal units.

Figure 7 :
Figure 7: Graphs of the main predictors in model 30;1 with 6.1 -6.2 showing their relationship with the response L Time .

Figure 8 :
Figure 8: Graphs of the local mutual contribution of the main predictors to model 30;2 .

Figure 9 :
Figure 9: Values of the experimental P out against the predicted P out by model 40;2 with 5.5 -5.6 with 5% confidential interval.

Figure 10 :
Figure 10: Values of the experimental L Time against the predicted L Time by model 30;2 with 6.3 -6.4 with 5% confidential interval.

Table 1 :
Technical characteristics of a UV copper ion vapor laser 20 .

Table 2 :
Main statistics of the constructed best MARS models of laser output power P out of UV copper ion lasers.

Table 3 :
Relative variable importance in the considered MARS model 40;1 , described by 5.1 -5.2 , and model 40;2 , described by 5.5 -5.6 , for laser output power P out of UV copper ion laser.

Table 4 :
Calculated values for predicted future experiments using MARS model 5.5 -5.6 .
a continuous line indicates the experiment data, Predicted shows estimates from model 40;2 .Results for a future experiment prediction using model 40; 2 are given in Table4.RF 25 kHz, TR 490 • C, and C 372 pF.As expected, laser generation P out increases when input electric power P IN is increased.In this table, the middle row #3 corresponds to an actual experiment with the measured value of P out 1200 mW, and the one predicted by the model is 1145 mW, which represents a relative error of under 5%.

Table 5 :
Main statistics of the constructed best MARS models of laser service life L Time of UV copper ion lasers.

Table 6 :
Relative variable importance in the considered MARS models 30;1 , described by 6.1 -6.2 , and 30;2 with 6.3 -6.4 , for laser service life L Time of UV copper ion laser.