- Email: [email protected]
Statistical analysis of the CuBr laser efficiency improvement
ARTICLE IN PRESS
Optics & Laser Technology 40 (2008) 641–646 www.elsevier.com/locate/optlastec
Statistical analysis of the CuBr laser efficiency improvement I.P. Ilieva, S.G. Gocheva-Ilievab,, D.N. Astadjovc, N.P. Denevc, N.V. Sabotinovc a
Department of Physics, Technical University—Plovdiv, 25 Tz. Djusstabanov St, 4000 Plovdiv, Bulgaria Department of Applied Mathematics and Modelling, University of Plovdiv, 24 Tzar Assen St, 4000 Plovdiv, Bulgaria c Metal Vapour Lasers Department, Georgi Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee, 1784 Sofia, Bulgaria b
Received 8 February 2007; received in revised form 9 September 2007; accepted 21 September 2007 Available online 30 October 2007
Abstract A new approach for improving the efficiency of a copper bromide vapour laser with wavelengths of 510.6 and 578.2 nm is implemented. Multi-factor and regression analyses of a large amount of experimental data have outlined the parameters with highest impact on the laser efficiency. They are electric input power, inside diameter of the rings, distance between the electrodes, inside diameter of the laser tube, hydrogen pressure and electric input power per unit length. The results obtained allow discovery of the internal structure of dependences among parameters, to better account for the physical processes that influence efficiency, to improve the planning of further experiments and laser production technology. r 2007 Elsevier Ltd. All rights reserved. Keywords: Laser efficiency; Metal vapour laser; Factor analysis
1. Introduction A great number of parameters have an effect upon the performance of gaseous lasers (metal vapour lasers included)—laser geometry, electric excitation routine, working gas medium, optical resonator and more. The detailed experimental study of all these variables and exceptionally the strength of their internal interaction are not always practically possible. The alternative is to assess the level of influence these parameters have upon the laser output characteristics as output laser power, efficiency, long-term laser performance, lifetime, laser beam quality, cost issues, etc. The parameters of low impact can be further neglected while the high-impact parameters may be evaluated with regard to the improvement of certain output characteristics. The development of any laser source requires classification of basic input variables. A relation structure between them and between each of them and the other output parameters can be assigned. Such relation structures make Corresponding author. Tel.: +359 32 265 842; fax: +359 32 635 049.
E-mail address: [email protected] (S.G. Gocheva-Ilieva). 0030-3992/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2007.09.009
it possible to explain the physical processes in a laser source; improving physical experiment planning, production technology and the methods for computer simulation of relevant processes. The focus of our study is the copper bromide vapour laser (CuBr laser). It is a low-temperature brand of copper laser (510.6 and 578.2 nm) where a halide (CuBr) is substituted for elemental copper. The CuBr laser operation temperature is lower than that of the elemental copper laser by 800–1000 K. The power and efficiency of these lasers are amongst the highest of the lasers operating in the visible spectrum [1]. Although their behaviour has been systematically examined for decades, due to the vast applications and modernization, the presented study is theoretically and practically justified. The aim of this work is, by means of mathematical statistics, to: (1) identify the input laser parameters which influence efficiency; (2) carry out classification of the parameters by grouping those that correlate strongly between themselves and do not correlate with the rest; (3) determine the level of influence of the grouped variables on efficiency; (4) set up a linear model of this dependency; (5) carry out interpretation of the results and clarify the
ARTICLE IN PRESS 642
I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
physical processes with largest contribution to CuBr laser efficiency. The experimental database of the Laboratory of Metal Vapour Lasers with the Institute of Solid State Physics, Bulgarian Academy of Sciences has been used for the purpose of this statistical analysis [2–10]. The work is a continuation of results reported earlier [11,12]. The statistical package SPSS has been used (see Ref. [13]). 2. Factor analysis In this paper, we examine 12 parameters which determine the CuBr laser functionality. They are: D— inside diameter of the laser tube, dr—inside diameter of the rings, L—length of the active area (electrode separation), Pin—input electrical power, PL—input electrical power per unit length, Prf—pulse repetition frequency, Pne—neon gas pressure, PH2 —hydrogen gas pressure, C—equivalent capacity of the capacitor bank, Pout—output laser power, Tr—temperature of the CuBr reservoirs and Eff—laser efficiency. It must be noted that, in general, the input data and, in particular, the data for hydrogen pressure, include nonoptimized data. Only for Tr the data are optimized in the temperature interval (480–490 1C) and it could be seen later that its influence is negligible. Our main goal is to identify and group the considered variables by the level of correlation with the efficiency— one of the most important laser output parameters. Thus, it is possible to establish the relationship between parts of these variables. Details of the factor analysis of our experimental data can be found in Appendix. Below we discuss the critical aspects of the analysis. Consider the correlation matrix derived from the factor analysis shown in Table A.1 in Appendix. Correlation coefficients of the variables Prf, Pne, C and Tr with the efficiency Eff are less than .3, show there is minor correlation dependency between any of them and the efficiency itself. Also their coefficients with the rest of the variables are much lower than .5 (with only two exceptions), so no correlation or a weak one is observed. In addition, the four variables have unacceptable significance levels. For that reason, the variables Prf, Pne, C and Tr will not be included in the statistical consideration. The output laser power Pout is a dependant output variable, which cannot be considered as an influence on efficiency, so it is also excluded. Finally, only the variables D, dr, L, Pin, PL and PH2 will take part in the factor and regression analyses. They have correlation coefficients with Eff amounted to .604; .761; .836; .700; (.648) and .561, respectively. According to the factor analysis (see Appendix), these six variables were classified in three factors. The first factor F1 groups the variables D, dr, L and Pin that are highly correlated with each other, presumably because they all are influenced by the same underlying dimension (factor). The second factor F2 loaded highly and negative on PL and the
Efficiency, Eff
Tube geometry, D, dr, L
Input power, Pin
Input power per unit length, PL
Hydrogen Pressure, PH 2
Fig. 1. Schematic diagram for basic groups of parameters, having high influence on laser efficiency.
third factor F3 loaded on PH2 . The factors are rated in accordance with their partial percentage in the total variance explained. It should be noticed here, that according to the coefficients in Table A.1, the tube geometry, input power and input power per unit length all have a high level of correlation with one another. Therefore, the influence of F1 and F2 on efficiency cannot be treated separately from the physical point of view. As to F3 (hydrogen gas pressure), it has its own significance—its straight impact on the laser efficiency. Fig. 1 is an illustration of the established structural dependency. The place of the rest of parameters within the structure given cannot be rated using the factor analysis because they have weak correlation with efficiency as well as with each other and the other observed variables. We consider the resulting structural dependency as partial but even so, it includes the dominant physical variables, describing the laser functionality. 3. Regression analysis The aim of the regression analysis is the establishment of the direct dependency between the efficiency from the chosen factors. Three models of multi-regression analysis were carried out: linear, stepwise and backward. For the linear regression model with the obtained orthogonal factors F1, F2 and F3 we got the results in Table A.4 in Appendix. The other methods of regression produce the same results. The found linear dependencies for the efficiency Eff and its standardized variable Eff , respectively, are: Eff :475F 1 þ :308F 2 þ :316F 3 þ 1:333,
(1)
Eff :623F 1 þ :404F 2 þ :414F 3 .
(2)
The derived regression coefficients show level of influence of the three factors (respectively of the combination of the physical variables, grouped in each of them) on laser efficiency. From these equations, we can see that the parameters with high loadings on F1—Pin, dr, L and D have the largest influence. After that are F3 (hydrogen gas pressure) and F2—input power per unit length, having almost equal contribution to the laser efficiency.
ARTICLE IN PRESS I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
The obtained ANOVA at Sig. F ¼ .000, and model summary with R ¼ :850, Rsquare ¼ :722 and standard error of the estimate ¼ .405 confirmed the correctness of the performed multi-regression analysis. The technique of regression analysis allows calculation, with the help of regression Eqs. (1) and (2), of approximate values for Eff at given values of the independent variables. As a check to the obtained model, for the case D=58 mm, dr=58 mm, L=200 cm, Pin=2.5 kW, PL=.63 kW/cm and PH2 ¼ :2 Torr the measured value of efficiency Eff=2.60%, and the foreseen by formula (1) value EffE2.485% with a standard error .067%. 4. Discussion Here we shall give a concise interpretation of statistical analysis outcome of our CuBr laser experimental data set. We shall determine the possible physical processes beyond the structural relationship of underlying physical parameters enlisted into the statistical analysis. Factor F1 explains 67.8% of the data; it is the most important factor, unifying four geometrical and power characteristics. The presence of the rings increases the inside surface of laser tube. They attribute to the cooling and heat balance of laser tube. Rings confine the tube active volume and form a buffer volume between them and the tube wall. The high-current electrical pulse concentrates the discharge in the axial area of tube. The buffer volume gives rise to faster and more efficient plasma relaxation, i.e. diffusion of charged and neutral particles and their consequent recombination in the volume or upon the walls. In the buffer volume, a process of great significance takes place as well—the intense CuBr restoration from fragmentation. The increase of L (the laser tube length) within the limits examined has an effect on the amplification (eaL). Together with the tube diameter D, it affects the volume density of the electric power deposition, and consequently the gas temperature profile of the laser tube. In particular, an increase in tube length reduces the longitudinal electric field and therefore the electron temperature during the excitation phase. The inside diameter D of the tube could have similar effect. As D increases so does the ambient heat-exchange coefficient of the tube (Grashof criteria: GrD3). This improves the heat balance of the laser tube and reduces the thermal population of the lower laser levels. Higher input electrical power Pin could lead to higher electron energy and better excitation of the upper laser levels. It is the parameter with the biggest contribution to the efficiency improvement. The two other factors give together lower percentage (25%) in the total variance explained, but have their physical importance. In the second factor F2, the input electrical power per unit length PL is dominating and figures as separable variable. It is of an energetic character and the only negative loading equal to .909 (see
643
Table A.3). The latter features the inverse proportionality that appears between PL and Eff. Physically it reflects the fact that reduction of the gas temperature radial gradient leads to a higher efficiency since the radial component of the gas temperature gradient is proportional to PL. Any (e.g. radial) gradients of the gas temperature create inhomogeneities in the active medium which are detrimental to laser operation. Reduced PL reduces ionization of the medium as well. Residual ionization and gas temperature are important factors that exacerbate inhomogeneity of the active medium. Moreover, residual ionization also affects the efficiency by redistributing energy coupling as within the active medium so within the circuit formed by the laser tube and the electric pulse. In the third factor F3, the hydrogen gas pressure PH2 has the highest loading. Hydrogen effect on output laser power and efficiency has been established long time ago [2,3]. Fig. 1 is an illustration of the idea that changes in the geometrical dimensions and the input electrical power should go together. Thus, the optimal volume density of electric power and the optimal profile of gas temperature (via minimum linear density of electric power) are necessary to be maintained. As for one of the basic problems—the experiment planning, we can make the following conclusions. An increase of laser efficiency can be realized via simultaneous increase of the geometrical size and the input electric power while keeping the tendency to diminish the linear density of electric power. The hydrogen gas pressure has a strong influence on efficiency. Since small deviation from the optimum pressure can considerably affect efficiency, more experimental data are necessary in order to determine the optimum hydrogen pressure at new geometrical and energetic conditions. Factors of low impact are pulse repetition frequency, neon gas pressure, capacitor bank equivalent capacity and CuBr reservoir temperature. So, in future experiments their influence may not be examined if they are kept within wellknown optima. The results obtained are quite reasonable and can be qualified as valid and consistent with experimental data [2–11]. They can be associated with real physical processes contributing to laser efficiency. 5. Conclusion Statistical methods are employed as a new approach to the solution of laser efficiency problem. With a quite high accuracy, it produces partial grouping of non-systematic physical variables on the basis of factor analysis, and thus exhibiting the basic variables of high significance with regard to laser efficiency. Multiple linear regression was also performed and the regression equations for laser efficiency were established. The usefulness of statistical techniques can be examined in the context of the results of this paper, as well as in perspective. The strength of the direct dependencies
ARTICLE IN PRESS 644
I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
between the efficiency and any of the independent variables is determined by the correlation matrix. However, the received common dependencies—the linear regression Eqs. (1) and (2), which describe the rate of influence of the factors (of the parameters in them, respectively) and the efficiency, cannot be derived using other methods. By these equations, the efficiency of each separate experiment can be approximated, as well as predicted for a new experiment. The analysis could be further improved by examining the influence of the optimized and nonoptimized data for a given parameter through a so-called cluster analysis [15]. The presented statistical techniques could be valuable for metal vapour lasers in general, if there is a full enough database for them, which satisfy the conditions for adequacy of the methods. The addition of new data is allowed, as long as that data include new independent variables as well as new dependent variables such as output power, laser lifetime, price etc. In the same time, methods do not set requirements to include data from optimized as well as non-optimized laser operating conditions. In general, the statistical outcome could be helpful for planning in experiment practice and further development of CuBr lasers with improved efficiency. Acknowledgement This study is supported by the Scientific National Fund of Bulgarian Ministry of Education and Science: project number VU-MI-205/2006. Appendix. A Factor analysis is a statistical method designed to transfer a set of intercorrelated variables into a new set of uncorrelated components or factors, which account for the most of variance in the original variables. This technique is used in data reduction to explain the pattern of intercorrelations among variables. One may use factor analysis to generate hypotheses regarding causal mechanisms or to screen variables for subsequent statistical analysis (for example, to perform a linear regression analysis). It should be mentioned that factor analysis is generally applied in social sciences, but it also finds various applications in data processing in engineering, physics, chemistry, etc. [14]. In factor analysis, the original observed variables are modelled as linear combinations of the factors, plus some ‘‘error’’ terms in the form X ¼ FL þ E,
(3)
where X 2 Rnm is the matrix of the original intercorrelated data values, F 2 Rnk is the matrix of factor scores, calculated for every respondent, L 2 Rkm is the matrix of the factor loadings, and E 2 Rnm is the error matrix.
It is often more convenient to work with variables that have all been standardized to common mean 0 and common standard deviation 1. The first stage of this study was the data randomization, which was carried out by the appropriate procedure of SPSS. For over 300 initial data on 12 variables from Refs. [2–11], a 50% random sample was obtained. The resulting sample included 157 rows, which was considered as original data. It can be mentioned that, in principle, the data must be chosen in a random way. To this end, any random sample from the general population of data could be used (for example, 30%, 50% or 100%) if the sample is representative, i.e. it satisfies special statistical tests for validity of factor analysis. These tests are Kaiser–Meyer– Olkin measure of sampling adequacy and Bartlett’s test of sphericity. For our sample, we obtained KMO test=.6614.5 and Bartlett’s test df=66 at significance level .000, respectively. Therefore, the factor analysis is adequate. The next stage was the calculation of the correlation matrix of the randomized data set, given in Table A.1, where the upper part represents the correlation coefficients and the lower part contains the relevant levels of significance. In principle if there are any variables that are not correlated with the other variables, one might as well delete them prior to the factor analysis. So some variables were removed from further considerations as it was explained in the above Section 2. Finally m ¼ 6 variables were used in our factor analysis: D, dr, L, Pin, PL and PH2 . In the next stage, the method of principal component analysis was carried out to determine the number of factors to retain. The first component explains only 67.8% of total variance, which is unacceptable. The second and the third components have almost equal percentages of variance, and in common the three components take 93% of total data variance (see Table A.2). Because of this and considering the strength of the relationship among the physical processes, we retained three factors. Then the extraction of the factors (regarding Eq. (3)) was carried out by the method of principal component analysis with consequent Varimax rotation. The obtained rotated solution is given in Table A.3. It is seen that the first factor F1 loaded high and positive on variables Pin, dr, L and D. The second factor F2 shows high and negative loading on PL and the third factor F3 loaded high and positive on PH2 . It should be added that the rotation of factors was carried out by means of the all seven available methods in SPSS. The obtained rotated matrices were very similar and do not differ significantly to each other. The statistical error analyses by means of the reproduced correlations, anti-images and other statistics were carried out that confirmed the validity of performed factor analysis (Table A.4). Finally, the factor scores were derived and their orthogonality was obtained.
ARTICLE IN PRESS I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
645
Table A.1 Correlation matrixa D
dr
L
Pin
PL
PH2
Prf
Pne
C
Pout
Tr
Eff
.847 1.000 .899 .847 .549 .352 .131 .189 .333 .861 .177 .761
.693 .899 1.000 .861 .716 .503 .165 .128 .212 .895 .076 .836
.636 .847 .861 1.000 .343 .359 .140 .097 .295 .934 .070 .700
.573 .549 .716 .343 1.000 .452 .139 .314 .186 .444 .004 .648
.279 .352 .503 .359 .452 1.000 .148 .059 .078 .429 .267 .561
.055 .131 .165 .140 .139 .148 1.000 .491 .083 .184 .061 .245
.238 .189 .128 .097 .314 .059 .491 1.000 .315 .137 .023 .273
.386 .333 .212 .295 .186 .078 .083 .315 1.000 .235 .224 .168
.638 .861 .895 .934 .444 .429 .184 .137 .235 1.000 .031 .829
.080 .177 .076 .070 .004 .267 .061 .023 .224 .031 1.000 .132
.604 .761 .836 .700 .648 .561 .245 .273 .168 .829 .132 1.000
.000
.000 .000
.000 .000 .000
.000 .000 .000 .000
.000 .000 .000 .000 .000
.248 .051 .019 .040 .041 .035
.001 .009 .055 .115 .000 .233 .000
.000 .000 .004 .000 .010 .171 .155 .000
.000 .000 .000 .000 .000 .000 .012 .047 .002
.160 .013 .174 .191 .481 .000 .228 .389 .003 .352
.000 .000 .000 .000 .000 .000 .001 .000 .020 .000 .054
Correlation
D dr L Pin PL PH2 Prf Pne C Pout Tr Eff
1.000 .847 .693 .636 .573 .279 .055 .238 .386 .638 .080 .604
Sig. (one-tailed)
D dr L Pin PL PH2 Prf Pne C Pout Tr D
.000 .000 .000 .000 .000 .248 .001 .000 .000 .160 .000
a
.000 .000 .000 .000 .051 .009 .000 .000 .013 .000
.000 .000 .000 .019 .055 .004 .000 .174 .000
.000 .000 .040 .115 .000 .000 .191 .000
.000 .041 .000 .010 .000 .481 .000
.000 .155 .012 .228 .001
.000 .047 .389 .000
.002 .003 .020
.352 .000
.054
Determinant ¼ 2.48E006.
Table A.2 Total variance explaineda Component
1 2 3 a
.035 .233 .171 .000 .000 .000
Table A.4 Coefficientsa
Initial eigenvalues Total
% of variance
Cumulative %
4.068 .888 .619
67.793 14.796 10.325
67.793 82.589 92.914
Constant REGR factor score 1 for analysis 1 REGR factor score 2 for analysis 1 REGR factor score 3 for analysis 1
Extraction method: principal component analysis.
Table A.3 Rotated component matrixa
a
Component 1 Pin Dr L D PL PH2
2
3
.946 .902 .790 .728 .909
Rotation method: Varimax with Kaiser normalization. Rotation converged in five iterations. a Extraction method: principal component analysis.
.944
t
Sig.
.623
41.141 14.619
.000 .000
.032
.404
9.473
.000
.032
.414
9.721
.000
Unstandardized coefficients
Standardized coefficients
B
S.E.
Beta
1.333 .475
.032 .032
.308
.316
Dependent variable: Eff.
References [1] Sabotinov NV. Copper bromide lasers. In: Little CE, Sabotinov NV, editors. Pulsed metal vapour lasers, NATO ASI series. Disarmament Technologies-5, Dordrecht: Kluwer Academic Publishers; 1996. p. 113–24. [2] Astadjov DN, Sabotinov NV, Vuchkov NK. Effect of hydrogen on CuBr laser power and efficiency. Opt Commun 1985;56(4):279–82. [3] Astadjov DN, Vuchkov NK, Sabotinov NV. Parametric study of the CuBr laser with hydrogen additives. IEEE J Quantum Electron 1988;24(9):1927–35.
ARTICLE IN PRESS 646
I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
[4] Astadjov DN, Dimitrov KD, Little CE, Sabotinov NV. A CuBr laser with 1.4 W/cm3 average output power. IEEE J Quantum Electron 1994;30(6):1358–60. [5] Stoilov VM, Astadjov DN, Vuchkov N K, Sabotinov NV. High spatial intensity 10 W–CuBr laser with hydrogen additives. Opt Quantum Electron 2000;32:1209–17. [6] NATO contract SfP, 97 2685 (50 W copper bromide laser); 2000. [7] Vuchkov NK, Astadjov DN, Sabotinov NV. A new circuit for CuBr laser excitation. Opt Quantum Electron 1991;23:S549–53. [8] Astadjov DN, Dimitrov KD, Jones DR, Kirkov VL, Little CE, Little N, et al. Influence on operating characteristics of scaling sealed-off CuBr lasers in active length. Opt Commun 1997;135:289–94. [9] Dimitrov KD, Sabotinov NV. High-power and high-efficiency copper bromide vapor laser. SPIE 1996;3052:126–30. [10] Astadjov DN, Dimitrov KD, Jones DR, Kirkov VK, Little CE, Sabotinov NV, et al. Copper bromide laser of 120-W average output power. IEEE J Quantum Electron 1997;33(5):705–9.
[11] Denev NP, Astadjov DN, Sabotinov NV. Analysis of the copper bromide laser efficiency. In: Proceedings of fourth international symposium on laser technologies and lasers ’2005, Plovdiv, Bulgaria; 2006. p. 153–6. [12] Iliev IP, Gocheva-Ilieva SG, Denev NP, Sabotinov NV. Statistical study of the copper bromide laser efficiency. In: Proceedings of sixth international conference of the Balkan Physical Union, 22–26 August 2006, Istanbul, Turkey; Cetin SA, Hikmet I, editors, AIP conference proceedings, vol. 899, American Institute of Physics, Melville, New York; 2007. p. 680. [13] Landau S, Everitt BS. Handbook of statistical analyses using SPSS. CRC Pr Llc; 2003. [14] Hatcher L. Step-by-step approach to using the Sas system for factor analysis and structural equation modeling. SAS Publishing; 1994. [15] Everitt B, Landau S, Leese M. Cluster analysis. 4th ed. Edward Arnold Publishers Ltd.; 2001.
Optics & Laser Technology 40 (2008) 641–646 www.elsevier.com/locate/optlastec
Statistical analysis of the CuBr laser efficiency improvement I.P. Ilieva, S.G. Gocheva-Ilievab,, D.N. Astadjovc, N.P. Denevc, N.V. Sabotinovc a
Department of Physics, Technical University—Plovdiv, 25 Tz. Djusstabanov St, 4000 Plovdiv, Bulgaria Department of Applied Mathematics and Modelling, University of Plovdiv, 24 Tzar Assen St, 4000 Plovdiv, Bulgaria c Metal Vapour Lasers Department, Georgi Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee, 1784 Sofia, Bulgaria b
Received 8 February 2007; received in revised form 9 September 2007; accepted 21 September 2007 Available online 30 October 2007
Abstract A new approach for improving the efficiency of a copper bromide vapour laser with wavelengths of 510.6 and 578.2 nm is implemented. Multi-factor and regression analyses of a large amount of experimental data have outlined the parameters with highest impact on the laser efficiency. They are electric input power, inside diameter of the rings, distance between the electrodes, inside diameter of the laser tube, hydrogen pressure and electric input power per unit length. The results obtained allow discovery of the internal structure of dependences among parameters, to better account for the physical processes that influence efficiency, to improve the planning of further experiments and laser production technology. r 2007 Elsevier Ltd. All rights reserved. Keywords: Laser efficiency; Metal vapour laser; Factor analysis
1. Introduction A great number of parameters have an effect upon the performance of gaseous lasers (metal vapour lasers included)—laser geometry, electric excitation routine, working gas medium, optical resonator and more. The detailed experimental study of all these variables and exceptionally the strength of their internal interaction are not always practically possible. The alternative is to assess the level of influence these parameters have upon the laser output characteristics as output laser power, efficiency, long-term laser performance, lifetime, laser beam quality, cost issues, etc. The parameters of low impact can be further neglected while the high-impact parameters may be evaluated with regard to the improvement of certain output characteristics. The development of any laser source requires classification of basic input variables. A relation structure between them and between each of them and the other output parameters can be assigned. Such relation structures make Corresponding author. Tel.: +359 32 265 842; fax: +359 32 635 049.
E-mail address: [email protected] (S.G. Gocheva-Ilieva). 0030-3992/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2007.09.009
it possible to explain the physical processes in a laser source; improving physical experiment planning, production technology and the methods for computer simulation of relevant processes. The focus of our study is the copper bromide vapour laser (CuBr laser). It is a low-temperature brand of copper laser (510.6 and 578.2 nm) where a halide (CuBr) is substituted for elemental copper. The CuBr laser operation temperature is lower than that of the elemental copper laser by 800–1000 K. The power and efficiency of these lasers are amongst the highest of the lasers operating in the visible spectrum [1]. Although their behaviour has been systematically examined for decades, due to the vast applications and modernization, the presented study is theoretically and practically justified. The aim of this work is, by means of mathematical statistics, to: (1) identify the input laser parameters which influence efficiency; (2) carry out classification of the parameters by grouping those that correlate strongly between themselves and do not correlate with the rest; (3) determine the level of influence of the grouped variables on efficiency; (4) set up a linear model of this dependency; (5) carry out interpretation of the results and clarify the
ARTICLE IN PRESS 642
I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
physical processes with largest contribution to CuBr laser efficiency. The experimental database of the Laboratory of Metal Vapour Lasers with the Institute of Solid State Physics, Bulgarian Academy of Sciences has been used for the purpose of this statistical analysis [2–10]. The work is a continuation of results reported earlier [11,12]. The statistical package SPSS has been used (see Ref. [13]). 2. Factor analysis In this paper, we examine 12 parameters which determine the CuBr laser functionality. They are: D— inside diameter of the laser tube, dr—inside diameter of the rings, L—length of the active area (electrode separation), Pin—input electrical power, PL—input electrical power per unit length, Prf—pulse repetition frequency, Pne—neon gas pressure, PH2 —hydrogen gas pressure, C—equivalent capacity of the capacitor bank, Pout—output laser power, Tr—temperature of the CuBr reservoirs and Eff—laser efficiency. It must be noted that, in general, the input data and, in particular, the data for hydrogen pressure, include nonoptimized data. Only for Tr the data are optimized in the temperature interval (480–490 1C) and it could be seen later that its influence is negligible. Our main goal is to identify and group the considered variables by the level of correlation with the efficiency— one of the most important laser output parameters. Thus, it is possible to establish the relationship between parts of these variables. Details of the factor analysis of our experimental data can be found in Appendix. Below we discuss the critical aspects of the analysis. Consider the correlation matrix derived from the factor analysis shown in Table A.1 in Appendix. Correlation coefficients of the variables Prf, Pne, C and Tr with the efficiency Eff are less than .3, show there is minor correlation dependency between any of them and the efficiency itself. Also their coefficients with the rest of the variables are much lower than .5 (with only two exceptions), so no correlation or a weak one is observed. In addition, the four variables have unacceptable significance levels. For that reason, the variables Prf, Pne, C and Tr will not be included in the statistical consideration. The output laser power Pout is a dependant output variable, which cannot be considered as an influence on efficiency, so it is also excluded. Finally, only the variables D, dr, L, Pin, PL and PH2 will take part in the factor and regression analyses. They have correlation coefficients with Eff amounted to .604; .761; .836; .700; (.648) and .561, respectively. According to the factor analysis (see Appendix), these six variables were classified in three factors. The first factor F1 groups the variables D, dr, L and Pin that are highly correlated with each other, presumably because they all are influenced by the same underlying dimension (factor). The second factor F2 loaded highly and negative on PL and the
Efficiency, Eff
Tube geometry, D, dr, L
Input power, Pin
Input power per unit length, PL
Hydrogen Pressure, PH 2
Fig. 1. Schematic diagram for basic groups of parameters, having high influence on laser efficiency.
third factor F3 loaded on PH2 . The factors are rated in accordance with their partial percentage in the total variance explained. It should be noticed here, that according to the coefficients in Table A.1, the tube geometry, input power and input power per unit length all have a high level of correlation with one another. Therefore, the influence of F1 and F2 on efficiency cannot be treated separately from the physical point of view. As to F3 (hydrogen gas pressure), it has its own significance—its straight impact on the laser efficiency. Fig. 1 is an illustration of the established structural dependency. The place of the rest of parameters within the structure given cannot be rated using the factor analysis because they have weak correlation with efficiency as well as with each other and the other observed variables. We consider the resulting structural dependency as partial but even so, it includes the dominant physical variables, describing the laser functionality. 3. Regression analysis The aim of the regression analysis is the establishment of the direct dependency between the efficiency from the chosen factors. Three models of multi-regression analysis were carried out: linear, stepwise and backward. For the linear regression model with the obtained orthogonal factors F1, F2 and F3 we got the results in Table A.4 in Appendix. The other methods of regression produce the same results. The found linear dependencies for the efficiency Eff and its standardized variable Eff , respectively, are: Eff :475F 1 þ :308F 2 þ :316F 3 þ 1:333,
(1)
Eff :623F 1 þ :404F 2 þ :414F 3 .
(2)
The derived regression coefficients show level of influence of the three factors (respectively of the combination of the physical variables, grouped in each of them) on laser efficiency. From these equations, we can see that the parameters with high loadings on F1—Pin, dr, L and D have the largest influence. After that are F3 (hydrogen gas pressure) and F2—input power per unit length, having almost equal contribution to the laser efficiency.
ARTICLE IN PRESS I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
The obtained ANOVA at Sig. F ¼ .000, and model summary with R ¼ :850, Rsquare ¼ :722 and standard error of the estimate ¼ .405 confirmed the correctness of the performed multi-regression analysis. The technique of regression analysis allows calculation, with the help of regression Eqs. (1) and (2), of approximate values for Eff at given values of the independent variables. As a check to the obtained model, for the case D=58 mm, dr=58 mm, L=200 cm, Pin=2.5 kW, PL=.63 kW/cm and PH2 ¼ :2 Torr the measured value of efficiency Eff=2.60%, and the foreseen by formula (1) value EffE2.485% with a standard error .067%. 4. Discussion Here we shall give a concise interpretation of statistical analysis outcome of our CuBr laser experimental data set. We shall determine the possible physical processes beyond the structural relationship of underlying physical parameters enlisted into the statistical analysis. Factor F1 explains 67.8% of the data; it is the most important factor, unifying four geometrical and power characteristics. The presence of the rings increases the inside surface of laser tube. They attribute to the cooling and heat balance of laser tube. Rings confine the tube active volume and form a buffer volume between them and the tube wall. The high-current electrical pulse concentrates the discharge in the axial area of tube. The buffer volume gives rise to faster and more efficient plasma relaxation, i.e. diffusion of charged and neutral particles and their consequent recombination in the volume or upon the walls. In the buffer volume, a process of great significance takes place as well—the intense CuBr restoration from fragmentation. The increase of L (the laser tube length) within the limits examined has an effect on the amplification (eaL). Together with the tube diameter D, it affects the volume density of the electric power deposition, and consequently the gas temperature profile of the laser tube. In particular, an increase in tube length reduces the longitudinal electric field and therefore the electron temperature during the excitation phase. The inside diameter D of the tube could have similar effect. As D increases so does the ambient heat-exchange coefficient of the tube (Grashof criteria: GrD3). This improves the heat balance of the laser tube and reduces the thermal population of the lower laser levels. Higher input electrical power Pin could lead to higher electron energy and better excitation of the upper laser levels. It is the parameter with the biggest contribution to the efficiency improvement. The two other factors give together lower percentage (25%) in the total variance explained, but have their physical importance. In the second factor F2, the input electrical power per unit length PL is dominating and figures as separable variable. It is of an energetic character and the only negative loading equal to .909 (see
643
Table A.3). The latter features the inverse proportionality that appears between PL and Eff. Physically it reflects the fact that reduction of the gas temperature radial gradient leads to a higher efficiency since the radial component of the gas temperature gradient is proportional to PL. Any (e.g. radial) gradients of the gas temperature create inhomogeneities in the active medium which are detrimental to laser operation. Reduced PL reduces ionization of the medium as well. Residual ionization and gas temperature are important factors that exacerbate inhomogeneity of the active medium. Moreover, residual ionization also affects the efficiency by redistributing energy coupling as within the active medium so within the circuit formed by the laser tube and the electric pulse. In the third factor F3, the hydrogen gas pressure PH2 has the highest loading. Hydrogen effect on output laser power and efficiency has been established long time ago [2,3]. Fig. 1 is an illustration of the idea that changes in the geometrical dimensions and the input electrical power should go together. Thus, the optimal volume density of electric power and the optimal profile of gas temperature (via minimum linear density of electric power) are necessary to be maintained. As for one of the basic problems—the experiment planning, we can make the following conclusions. An increase of laser efficiency can be realized via simultaneous increase of the geometrical size and the input electric power while keeping the tendency to diminish the linear density of electric power. The hydrogen gas pressure has a strong influence on efficiency. Since small deviation from the optimum pressure can considerably affect efficiency, more experimental data are necessary in order to determine the optimum hydrogen pressure at new geometrical and energetic conditions. Factors of low impact are pulse repetition frequency, neon gas pressure, capacitor bank equivalent capacity and CuBr reservoir temperature. So, in future experiments their influence may not be examined if they are kept within wellknown optima. The results obtained are quite reasonable and can be qualified as valid and consistent with experimental data [2–11]. They can be associated with real physical processes contributing to laser efficiency. 5. Conclusion Statistical methods are employed as a new approach to the solution of laser efficiency problem. With a quite high accuracy, it produces partial grouping of non-systematic physical variables on the basis of factor analysis, and thus exhibiting the basic variables of high significance with regard to laser efficiency. Multiple linear regression was also performed and the regression equations for laser efficiency were established. The usefulness of statistical techniques can be examined in the context of the results of this paper, as well as in perspective. The strength of the direct dependencies
ARTICLE IN PRESS 644
I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
between the efficiency and any of the independent variables is determined by the correlation matrix. However, the received common dependencies—the linear regression Eqs. (1) and (2), which describe the rate of influence of the factors (of the parameters in them, respectively) and the efficiency, cannot be derived using other methods. By these equations, the efficiency of each separate experiment can be approximated, as well as predicted for a new experiment. The analysis could be further improved by examining the influence of the optimized and nonoptimized data for a given parameter through a so-called cluster analysis [15]. The presented statistical techniques could be valuable for metal vapour lasers in general, if there is a full enough database for them, which satisfy the conditions for adequacy of the methods. The addition of new data is allowed, as long as that data include new independent variables as well as new dependent variables such as output power, laser lifetime, price etc. In the same time, methods do not set requirements to include data from optimized as well as non-optimized laser operating conditions. In general, the statistical outcome could be helpful for planning in experiment practice and further development of CuBr lasers with improved efficiency. Acknowledgement This study is supported by the Scientific National Fund of Bulgarian Ministry of Education and Science: project number VU-MI-205/2006. Appendix. A Factor analysis is a statistical method designed to transfer a set of intercorrelated variables into a new set of uncorrelated components or factors, which account for the most of variance in the original variables. This technique is used in data reduction to explain the pattern of intercorrelations among variables. One may use factor analysis to generate hypotheses regarding causal mechanisms or to screen variables for subsequent statistical analysis (for example, to perform a linear regression analysis). It should be mentioned that factor analysis is generally applied in social sciences, but it also finds various applications in data processing in engineering, physics, chemistry, etc. [14]. In factor analysis, the original observed variables are modelled as linear combinations of the factors, plus some ‘‘error’’ terms in the form X ¼ FL þ E,
(3)
where X 2 Rnm is the matrix of the original intercorrelated data values, F 2 Rnk is the matrix of factor scores, calculated for every respondent, L 2 Rkm is the matrix of the factor loadings, and E 2 Rnm is the error matrix.
It is often more convenient to work with variables that have all been standardized to common mean 0 and common standard deviation 1. The first stage of this study was the data randomization, which was carried out by the appropriate procedure of SPSS. For over 300 initial data on 12 variables from Refs. [2–11], a 50% random sample was obtained. The resulting sample included 157 rows, which was considered as original data. It can be mentioned that, in principle, the data must be chosen in a random way. To this end, any random sample from the general population of data could be used (for example, 30%, 50% or 100%) if the sample is representative, i.e. it satisfies special statistical tests for validity of factor analysis. These tests are Kaiser–Meyer– Olkin measure of sampling adequacy and Bartlett’s test of sphericity. For our sample, we obtained KMO test=.6614.5 and Bartlett’s test df=66 at significance level .000, respectively. Therefore, the factor analysis is adequate. The next stage was the calculation of the correlation matrix of the randomized data set, given in Table A.1, where the upper part represents the correlation coefficients and the lower part contains the relevant levels of significance. In principle if there are any variables that are not correlated with the other variables, one might as well delete them prior to the factor analysis. So some variables were removed from further considerations as it was explained in the above Section 2. Finally m ¼ 6 variables were used in our factor analysis: D, dr, L, Pin, PL and PH2 . In the next stage, the method of principal component analysis was carried out to determine the number of factors to retain. The first component explains only 67.8% of total variance, which is unacceptable. The second and the third components have almost equal percentages of variance, and in common the three components take 93% of total data variance (see Table A.2). Because of this and considering the strength of the relationship among the physical processes, we retained three factors. Then the extraction of the factors (regarding Eq. (3)) was carried out by the method of principal component analysis with consequent Varimax rotation. The obtained rotated solution is given in Table A.3. It is seen that the first factor F1 loaded high and positive on variables Pin, dr, L and D. The second factor F2 shows high and negative loading on PL and the third factor F3 loaded high and positive on PH2 . It should be added that the rotation of factors was carried out by means of the all seven available methods in SPSS. The obtained rotated matrices were very similar and do not differ significantly to each other. The statistical error analyses by means of the reproduced correlations, anti-images and other statistics were carried out that confirmed the validity of performed factor analysis (Table A.4). Finally, the factor scores were derived and their orthogonality was obtained.
ARTICLE IN PRESS I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
645
Table A.1 Correlation matrixa D
dr
L
Pin
PL
PH2
Prf
Pne
C
Pout
Tr
Eff
.847 1.000 .899 .847 .549 .352 .131 .189 .333 .861 .177 .761
.693 .899 1.000 .861 .716 .503 .165 .128 .212 .895 .076 .836
.636 .847 .861 1.000 .343 .359 .140 .097 .295 .934 .070 .700
.573 .549 .716 .343 1.000 .452 .139 .314 .186 .444 .004 .648
.279 .352 .503 .359 .452 1.000 .148 .059 .078 .429 .267 .561
.055 .131 .165 .140 .139 .148 1.000 .491 .083 .184 .061 .245
.238 .189 .128 .097 .314 .059 .491 1.000 .315 .137 .023 .273
.386 .333 .212 .295 .186 .078 .083 .315 1.000 .235 .224 .168
.638 .861 .895 .934 .444 .429 .184 .137 .235 1.000 .031 .829
.080 .177 .076 .070 .004 .267 .061 .023 .224 .031 1.000 .132
.604 .761 .836 .700 .648 .561 .245 .273 .168 .829 .132 1.000
.000
.000 .000
.000 .000 .000
.000 .000 .000 .000
.000 .000 .000 .000 .000
.248 .051 .019 .040 .041 .035
.001 .009 .055 .115 .000 .233 .000
.000 .000 .004 .000 .010 .171 .155 .000
.000 .000 .000 .000 .000 .000 .012 .047 .002
.160 .013 .174 .191 .481 .000 .228 .389 .003 .352
.000 .000 .000 .000 .000 .000 .001 .000 .020 .000 .054
Correlation
D dr L Pin PL PH2 Prf Pne C Pout Tr Eff
1.000 .847 .693 .636 .573 .279 .055 .238 .386 .638 .080 .604
Sig. (one-tailed)
D dr L Pin PL PH2 Prf Pne C Pout Tr D
.000 .000 .000 .000 .000 .248 .001 .000 .000 .160 .000
a
.000 .000 .000 .000 .051 .009 .000 .000 .013 .000
.000 .000 .000 .019 .055 .004 .000 .174 .000
.000 .000 .040 .115 .000 .000 .191 .000
.000 .041 .000 .010 .000 .481 .000
.000 .155 .012 .228 .001
.000 .047 .389 .000
.002 .003 .020
.352 .000
.054
Determinant ¼ 2.48E006.
Table A.2 Total variance explaineda Component
1 2 3 a
.035 .233 .171 .000 .000 .000
Table A.4 Coefficientsa
Initial eigenvalues Total
% of variance
Cumulative %
4.068 .888 .619
67.793 14.796 10.325
67.793 82.589 92.914
Constant REGR factor score 1 for analysis 1 REGR factor score 2 for analysis 1 REGR factor score 3 for analysis 1
Extraction method: principal component analysis.
Table A.3 Rotated component matrixa
a
Component 1 Pin Dr L D PL PH2
2
3
.946 .902 .790 .728 .909
Rotation method: Varimax with Kaiser normalization. Rotation converged in five iterations. a Extraction method: principal component analysis.
.944
t
Sig.
.623
41.141 14.619
.000 .000
.032
.404
9.473
.000
.032
.414
9.721
.000
Unstandardized coefficients
Standardized coefficients
B
S.E.
Beta
1.333 .475
.032 .032
.308
.316
Dependent variable: Eff.
References [1] Sabotinov NV. Copper bromide lasers. In: Little CE, Sabotinov NV, editors. Pulsed metal vapour lasers, NATO ASI series. Disarmament Technologies-5, Dordrecht: Kluwer Academic Publishers; 1996. p. 113–24. [2] Astadjov DN, Sabotinov NV, Vuchkov NK. Effect of hydrogen on CuBr laser power and efficiency. Opt Commun 1985;56(4):279–82. [3] Astadjov DN, Vuchkov NK, Sabotinov NV. Parametric study of the CuBr laser with hydrogen additives. IEEE J Quantum Electron 1988;24(9):1927–35.
ARTICLE IN PRESS 646
I.P. Iliev et al. / Optics & Laser Technology 40 (2008) 641–646
[4] Astadjov DN, Dimitrov KD, Little CE, Sabotinov NV. A CuBr laser with 1.4 W/cm3 average output power. IEEE J Quantum Electron 1994;30(6):1358–60. [5] Stoilov VM, Astadjov DN, Vuchkov N K, Sabotinov NV. High spatial intensity 10 W–CuBr laser with hydrogen additives. Opt Quantum Electron 2000;32:1209–17. [6] NATO contract SfP, 97 2685 (50 W copper bromide laser); 2000. [7] Vuchkov NK, Astadjov DN, Sabotinov NV. A new circuit for CuBr laser excitation. Opt Quantum Electron 1991;23:S549–53. [8] Astadjov DN, Dimitrov KD, Jones DR, Kirkov VL, Little CE, Little N, et al. Influence on operating characteristics of scaling sealed-off CuBr lasers in active length. Opt Commun 1997;135:289–94. [9] Dimitrov KD, Sabotinov NV. High-power and high-efficiency copper bromide vapor laser. SPIE 1996;3052:126–30. [10] Astadjov DN, Dimitrov KD, Jones DR, Kirkov VK, Little CE, Sabotinov NV, et al. Copper bromide laser of 120-W average output power. IEEE J Quantum Electron 1997;33(5):705–9.
[11] Denev NP, Astadjov DN, Sabotinov NV. Analysis of the copper bromide laser efficiency. In: Proceedings of fourth international symposium on laser technologies and lasers ’2005, Plovdiv, Bulgaria; 2006. p. 153–6. [12] Iliev IP, Gocheva-Ilieva SG, Denev NP, Sabotinov NV. Statistical study of the copper bromide laser efficiency. In: Proceedings of sixth international conference of the Balkan Physical Union, 22–26 August 2006, Istanbul, Turkey; Cetin SA, Hikmet I, editors, AIP conference proceedings, vol. 899, American Institute of Physics, Melville, New York; 2007. p. 680. [13] Landau S, Everitt BS. Handbook of statistical analyses using SPSS. CRC Pr Llc; 2003. [14] Hatcher L. Step-by-step approach to using the Sas system for factor analysis and structural equation modeling. SAS Publishing; 1994. [15] Everitt B, Landau S, Leese M. Cluster analysis. 4th ed. Edward Arnold Publishers Ltd.; 2001.