Modeling of the Radial Heat Flow and Cooling Processes in a Deep Ultraviolet Cu Ne-CuBr Laser

1 Department of Physics, Technical University of Plovdiv, 25 Tzanko Djusstabanov Street, 4000 Plovdiv, Bulgaria 2 Department of Applied Mathematics and Modeling, Faculty of Mathematics and Informatics, Paisii Hilendarski University of Plovdiv, 24 Tsar Assen Street, 4000 Plovdiv, Bulgaria 3 Metal Vapour Lasers Department, Georgi Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tsarigradsko Shaussee Boulevard, 1784 Sofia, Bulgaria


Introduction
Deep ultraviolet DUV gas laser sources have been objects of great interest in the recent years, because of a wide variety of applications, such as high-precision processing of different materials, high-resolution laser lithography in microelectronics, high-density optical recording of information, laser-induced modification in various materials newly developed, as well as laser-induced fluorescence in plasma and wide-gap semiconductors 1-7 .These applications require a DUV laser source in order to achieve the necessarily high resolution are not uniformly distributed and vary strongly along the tube radius taking its maximum in the central axis of the discharge.In addition, the temperature of the outer surface of the laser tube under insulation is unknown and will change with variation of the laser geometry, input electric power, and other laser parameters including temperature of the surroundings.In 17, 18 , the temperature profile in the case of a copper bromide vapor laser with wavelength 510.6 nm and 578.2 nm was determined by using a new approach.It is based on solving the heat conduction equation with q v constant at nonlinear boundary value conditions for a given temperature of the surroundings.
In this study, for the first time, the analytical investigation of the temperature profile in the cross-section of the laser tube is performed with the assumption of a specified qualitative distribution of q v , dependent on the tube radius, namely, q v q v r .Using the approach similar to this in 17, 18 , in the case of UV Cu CuBr lasers, a new improved analytical model consisting of the one-dimensional heat conduction equation, subject to nonlinear boundary conditions of the third and fourth kind is derived.At a given air temperature of the surroundings, due to the heat convection and heat radiation, the proposed model allows to take into account the heat exchange processes between the outer surface of the laser tube and its surroundings.The gas temperature profile in the tube and the wall temperature are expressed by an explicit solution of the obtained problem and are directly dependent on the basic input laser parameters.The model is applied for evaluating the natural and forced aircooling of the laser tube.

Experimental Setup
The construction of the gas-discharge laser tube described by the model is presented in Figure 1.The basic tube with an 18 mm inside diameter and 24.5 mm outside diameter is made of fused quartz.The active length is 86 cm.The CuBr powder is placed in five quartz side-arm reservoirs.A ceramic tube insert with an inside diameter of 5.2 mm is sleeved in the basic tube.In the ceramic insert, five holes are made over each reservoir for the CuBr vapor diffusion into the active zone.The optic cavity of the laser studied is formed by two dielectric-coated mirrors.Mirror separation is 1.8 m.
The laser is excited by a pulsed electrical scheme with Interacting Circuits IC .The IC excitation of CuBr lasers, operating on self-terminating copper atom transitions, was described in detail in 19 .

Description of the Mathematical Model
The aforementioned multiline copper-bromide laser operates in the UV-region 1-5 .The total input electric power is 1300 W. Taking into account the losses in power supply, the laser tube is fed with power Q 1 1000 W. The output multiline laser power is 500 mW.The geometric design of the cross-section of the laser tube in the active zone is shown in Figure 2. The laser source is manufactured from quartz 3.2 , in which a ceramic tube of Al 2 O 3 3.1 is inserted along with the active laser volume, and the quartz tube is covered from the outside active volume with extra heat-insulating wadding 3.3 made of felt-glass, mineral material, or zircon oxide.
The model is developed with the following assumptions: i the temperature profile is determined in a quasi-stationary regime; ii the gas temperature does not change substantially in the interimpulse period; iii the total input electric power Q 1 1000 W in the active volume is transformed into heat, the power transferred to the walls as a result of the discharge radiation and the deactivation of the excited and charged particles is not taken into account; iv the thermal radiation of the heated gas in the active volume is ignored.
The temperature distribution T g in the cross-section of the laser tube is governed by the following quasi steady-state two-dimensional heat conduction equation: div λ g grad T g q v 0, 3.1 where λ g is the thermal conductivity of the gas here neon and q v is the volume density of the discharge.Due to the radial symmetry, T g depends only on the variable r along with the radius of the tube.Consequently, in cylindrical configuration, 3.1 is reduced to the form Usually, as it was mentioned earlier, equation 3.1 resp., 3.2 is solved in publications under the boundary conditions: where T w is the measured temperature of the outer wall of the tube under insulation 9-13 .
Commonly, λ g is in the form λ g λ 0 T m g .In the case q v constant, 3.2 possesses an exact solution 9 , written in the form For solving 3.2 , we need to obtain the correct boundary conditions corresponding to 3.3 for r R 1 see Figure 2 : For that purpose, we will apply the distribution of the radial heat flow through the composite laser tube.We consider the following mixed boundary conditions of the third and fourth kind in cylindrical configuration 20-22 : where T j denotes the temperatures at the boundaries of the tubes, j 1, . . ., 4, respectively see Figure 2 .
Boundary conditions 3.6 express the equation of the continuity of the heat flow at the borders of the three mediums of the composed tube.Here q l is the power per unit length, q l Q 1 /l a ; l a is the active length 2, 5, 6 ; λ j , j 1, 2, 3 are the thermal conductivities of the Al 2 O 3 tube, quartz tube and the thermal insulation, respectively; d j , j 1, . . ., 4 are the diameters of the composite tubes see Figure 2 and Table 1 .
The boundary condition 3.7 shows the heat exchange between the outer surface of the laser tube and the surroundings.The first term on the right-hand side of 3.7 evolves from Newton-Riemann's law for heat exchange by convection.The second term represents the Stefan-Boltzmann law for heat exchange by radiation.The value of Q 1 is equal to the electric power of 1000 W, in accordance with assumption iii , as it was stated earlier, α is the heat transfer coefficient, F 4 is the outside area of the insulation, ε is the integral emissivity of the material, c is the black body radiation coefficient, and T air is the temperature of the air.
The two unknown values α and T 4 in boundary condition 3.7 have to be determined.The values of the constants used in this study are given in Table 1.
In this way we obtain the temperature model described by 3.2 and boundary conditions 3.6 -3.7 , equivalent to 3.2 , 3.5 .Our aim is to find an analytical formula for the solution of this model at q v q v r .

Determination of the Gas Temperature T g r at Radial Distribution of the Volume Power Density q v r
In this section, we will obtain an explicit solution for the gas temperature T g r satisfying the proposed theoretical model 3.2 , 3.6 , 3.7 and will discuss its application.

Determining the Variable Radial Distribution
Due to the lack of experimental data for q v q v r , we will derive it as a qualitative theoretical dependency.From q v jE and j ≈ σE, we have q v ≈ σE 2 , where E E r is the electric field intensity and σ is the electrical conductivity of the medium.In 24 the distribution of the field intensity in the cross-section of the tube is represented by the expression E r E 0 J 0 2.4/R 1 r , where J 0 2.4/R 1 r is a Bessel function of the first kind, 0 ≤ r ≤ R 1 .In this way, we have q v r Q 0 J 0 2.4/R 1 r 2 , where Q 0 is a constant, which is found below.The Bessel function J 0 is well known and usually represented in 23, a table in what follows.In this form, it is not suitable for direct engineering-physics calculations.For this reason, we will approximate the term J 0 2.4/R 1 r 2 by a polynomial of the third-degree J 0 x 2 ≈ a 1 a 2 x a 3 x 2 a 4 x 3 , where x 2.4/R 1 r.Based on tabular data from 23 and using the least squares method, we find a 1 1.0044, a 2 −0.01768, a 3 −0.5657, a 4 0.1668.

4.1
For q v r , we obtain:

4.3
The constant Q 0 can be found by using the equality of areas, bounded between the graphics of each of the functions q 0 constant and q v q v r , and the abscissa r see Figure 3 : After integrating in 4.4 and substituting the values of the constants, we find In Figure 3, the distribution of the volume power densities q 0 constant and q v q v r , according to 4.2 and 4.5 , are illustrated in relative units, assuming here, in order to simplify that q 0 constant 1.From Figure 3 and 4.5 , it can be observed that in the center of the discharge the local electrical volume power density for 4.2 is over two times larger than q v constant.This suggests a difference in the distribution of T g r in the two cases being examined: q v constant and q v q v r .

Determining the Gas Temperature T g r
The solution to 3.2 at mixed boundary conditions 3.6 -3.7 and radial distribution q v r of type 4.2 has the following form: where the constants B, C, D were introduced in 4.3 .Detailed determination of 4.6 is given in the appendix.

Application of the Mathematical Model
The obtained explicit solution 4.6 can be used when the value T 1 of the temperature of the internal tube is known.There are two cases as follows.
1 The temperature T 3 of the outside surface of the laser tube i.e., quartz tube under the insulation is known see Figure 2 .For existing laser devices, it can be measured, for instance, by a thermocouple.Then T 2 and T 1 can be calculated by means of the corresponding boundary conditions from 3.6 .
2 The temperature T 3 is unknown.This problem can arise in the development of new laser sources or implementation of different computer simulations.In this case, the temperature of the surroundings T air must be specified, usually T air 300 K. To use boundary condition 3.7 we need to find α and T 4 .In the following section, we discuss the procedure for determining the heat transfer coefficient α and obtaining a nonlinear algebraic equation for the temperature T 4 .Then, applying 3.6 , we calculate T 3 , T 2 , and T 1 .

Evaluation of Cooling and Discussion
We will apply the derived temperature model 3.2 , 3.6 -3.7 for determining the gas temperature in the cases of free and forced convection.

Cooling of the Laser Tube by Free Convection
In 18 , a simplified temperature model in the case of free convection at q v constant has been used.Here we will compare the results obtained by our new model for the general case q v q v r with those in 18 .
In the case of free convection, the heat transfer coefficient α in 3.

5.2
In the previous expressions 5.1 -5.2 , the numerical values of g, β air , υ air , and λ air are given in Table 1.The data is correct for air temperature T air 300 K 22 .
However 5.2 is a nonlinear equation with respect to the outside temperature of the laser device insulation-T 4 .Also 5.2 can be easily solved by any computer algebra system, for instance by Mathematica 25 .Once the temperature T 4 is calculated, the values of T 3 , T 2 , and T 1 can be evaluated from 3.6 , and the gas temperature T g r in the internal tube is determined by 4.6 .
In Figure 4, on the same coordinate system, the distributions of the gas temperature T g r in the cross-section along with the radius of the laser tube for q v constant and q v q v r are presented.In Table 2, special characteristic temperatures T 4 , T 3 , T 2 , T 1 , and the maximum temperature T 0 in the center of the laser tube are given see also Figure 1 .
Table 2 shows that temperatures T 4 , T 3 , T 2 , and T 1 are equal.Their values are determined by the total electrical power emitted within the active volume and are independent from its radial distribution.In both cases, this power is the same-1000 W. Table 2 and Figure 4 show that T 0 T g 0 T max when q v q v r is 90 • C higher than the Table 2: Gas temperature at special characteristic points in the case of free convection.: q v q 0 54.6 W cm −3 , •: q v q v r from 4.2 .
corresponding value when q v constant.The results for the gas temperature in the case of q v constant have the same behavior as the calculated values in 9, 11 .where v is the velocity of the moving fluid, l a is the length of the laser tube and υ air is the kinematical viscosity of the air.However 5.4 is valid for 40 < Re < 4000 22 .

Cooling of the Laser Tube by Forced Convection
For horizontal tubes with forced air cooling the following equality holds 22 : Nu 0.615Re 0.466 .

5.5
Table 3: Gas temperature at special characteristic points in the case of forced convection.Gas temperature distribution along the half cross-section of the internal tube in the case of forced convection, v 20 m/s: : q v q 0 54.6 W cm −3 , •: q v q v r from 4.2 .

5.6
Substituting 5.6 in boundary condition 3.7 and representing it with respect to the power per unit length q l , we obtain a nonlinear algebraic equation for T 4 in the form q l 0.615πλ air vd 4 υ air 0.466

5.7
In this way by solving 5.7 , we determine T 4 .Then using 3.6 and 4.6 , we find the required gas temperature profile in the cross-section of the laser tube.The obtained values of some characteristic temperatures are given in Table 3, including the maximum value T 0 along the center of the tube.The results of the calculated values of T g r in the two cases q v q 0 constant and q v q v r are shown in Figure 5 for air flow v 20 m/s.
As it is expected the cooling process causes a decrease of the buffer gas temperature in relation with the case of free convection compare Tables 2 and 3, and Figures 4 and 5 .
It can be noted that although the maximum local electric power at the center of the tube is twice higher for q v q v r see Figure 3 , the difference between the corresponding maximum temperatures is only 95 • C.This result is almost the same as in the case of free convection.The deviation is on average around 6%.We can conclude that in principle, solution 3.4 can successfully be used to analyze temperature conditions of existing laser sources, when the temperature T 1 is known.
As an absolute quantity, the difference of 90/95 • C at the center of the discharge should not be overlooked.The model discussed in this paper can better explain and predict the occurrence of a number of negative phenomena connected with the overheating of the laser medium.The increase of 90/95 • C in the temperature at the center of the discharge can lead to a contraction of the gas discharge, thermal ionizing instability, and thermochemical gas degradation, additional thermal population of lower laser levels.In the end, this leads to decreased laser power and deterioration in mode composition.In some cases, the overheating of the discharge at the center of the tube may lead to a cessation of laser generation along its axis and the appearance of dark spots at the center of laser beam.For this reason, regardless the complexity of the new model, its use is fully advisable.

Average Temperature in the Active Volume
During the analysis of the temperature condition of existing or new laser devices, the average gas temperature in the active volume is a characteristic of great importance.It is defined as where T g r is the radial distribution of temperature in the active volume.
The average value of temperature T g for equal configurations depends only on the electric power, supplied to the active volume, and is independent of the radial distribution of T g r .In our case, the electrical power is Q 1 1000 W. The average temperature should not change for the temperature distribution of the type in 3.4 and 4.6 .
All subsequent calculations have been made using the Mathematica software system 25 .
For the radial distribution of T g r given by formula 3.4 , the result is
For the radial distribution of T g r from 4.6 the quantity 6.1 does not have an exact algebraic solution and the integral in 6.1 is solved numerically.The numerical value is T g,Eq.4.5 1339 K.The approximate calculation leads to an insignificant deviation of values for T g,Eq.3.4 and T g,Eq.4.5 , estimated by a relative error of 0.6%.
The presence of a hypergeometric function of type 2 F 1 a, b; c; z in the solution of 6.2 makes it practically difficult to use.For this reason, we represent the function 2 F 1 a, b; c; z by its Gauss series 25, 26 , limited only to the first two terms of the expansion The result is

6.4
In this way by using 6.2 , we obtain

6.5
The average value of the temperature from 6.5 is T g 1358 K, with a relative error of 1%.This shows that 6.5 can be used with sufficient accuracy to determine the average temperature in the active volume: T g ≈ T g,Eq.3.4 .

Conclusion
A theoretical mathematical model for evaluating the buffer gas temperature T g r in UV Cu Ne-CuBr laser is developed.It takes into account the nonuniformly distributed electrical power along the cross-section of the laser tube.Based on common theoretical dependencies, a suggestion is made for the qualitative distribution of volume power density q v q v r .The model includes a heat conduction equation subject to nonlinear boundary conditions.An explicit solution with these conditions is obtained.The model is applied in the cases of free and forced convection.A simple expression is established for calculating the average gas temperature in the active volume.
An evaluation of the previous existing solution has been presented, describing the distribution of the gas temperature T g r under the assumption q v constant.It has been established that, for such an assumption, the error when determining T g r at the center of the tube is about 6%.
The obtained results at q v constant have been compared with similar calculated results by using simple mathematical gas temperature models 9, 11 .
A comparison has been made between the obtained temperature profiles of T g r at q v constant and q v q v r .It has been established that at the center of the tube the temperature when q v q v r is about 90-95 • C higher, both for free and forced convection.
It has to be noted that the simple model when q v constant cannot be used to evaluate radial buffer gas temperature when the temperature of the wall is unknown.The advantage of the model presented here is that when specific geometric dimensions have been chosen, the temperature of the surroundings and other parameters, the temperature T 3 of the outer wall of the laser tube is calculated, after which the values for the gas temperature within the tube are calculated.Therefore, the new model can be applied not only to precise the existing models but to design new laser devices as well.
Coming back to the variable U according to A.
The substitution of C 2 into A.10 gives A.12 Finally, by using A.2 , we obtain the required solution of 3.2 in the form 4.6 : A.13

Figure 3 :
Figure3: Distribution of volume power density along the half cross-section of the internal tube, in relative units: : q v q 0 1, •: q v q v r .
5.1 in boundary condition 3.7 with consequent representation in relation to the power per unit length q l results in 18 q l 0.46πλ air gβ air d 3 4 T 4 − T air υ

Figure 4 :
Figure 4: Gas temperature distribution along the half cross-section of the internal tube with free convection:: q v q 0 54.6 W cm −3 , •: q v q v r from 4.2 .

Figure 5 :
Figure5: Gas temperature distribution along the half cross-section of the internal tube in the case of forced convection, v 20 m/s: : q v q 0 54.6 W cm −3 , •: q v q v r from 4.2 .

Table 1 :
Related parameters of the theoretical calculation.
5 , we obtain 2 is a constant.Using the boundary condition A.4 , we find C 2