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

Abstract

An improved theoretical model of the gas temperature profile in the cross-section of an ultraviolet copper ion excited copper bromide laser is developed. The model is based on the solution of the one-dimensional heat conduction equation subject to special nonlinear boundary conditions, describing the heat interaction between the laser tube and its surroundings. It takes into account the nonuniform distribution of the volume power density along with the radius of the laser tube. The problem is reduced to the boundary value problem of the first kind. An explicit solution of this model is obtained. The model is applied for the evaluation of the gas temperature profiles of the laser in the conditions of free and forced air-cooling. Comparison with other simple models assumed constant volume power density is made. In particular, a simple expression for calculating the average gas temperature is found.

1. 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 or to induce some phenomena in different materials, such as material and its surface modification, image recording, and fluorescence. The photon energy is sufficient first, to produce photochemical ablation rather than melting and subsequent vaporization, or second, to initiate the induced processes. The DUV laser ablation produces clean precision cutting and drilling with a minimum of thermal or mechanical damages on the target. DUV beams, in addition to heatless etching, offer the advantage of focusing with less limitation due to diffraction. The good spatial structure of the laser beam is generally important for the good-quality image projected, while narrow linewidth would reduce the chromatic aberrations. Chromatic aberrations are very difficult to correct because of the limited number of optical materials that transmit in the DUV range. The excimer lasers, which are generally used as light sources in the DUV range, can neither reach the spatial structure nor the linewidth required. Their wide application is also impeded by their cost and considerable overheads. Lately, there has been widespread interest in the ion metal vapor lasers, operating below 300 nm in the DUV spectral region. High beam quality and narrow-linewidth are inherent for metal vapor lasers. For the first time laser oscillation on four DUV Cu⁺ lines: 248.6 nm, 252.9 nm, 260.0 nm, and 270.3 nm, in a nanosecond, pulsed longitudinal Ne-CuBr discharge was obtained in [1]. In [2] an active zone diameter scanning of the DUV Cu⁺ Ne-CuBr laser was carried out and the optimal discharge conditions were found for each active zone diameter. A record average output power of 1.3 W was obtained at multiline operation. A record specific average output power was 57 mW/cm³ at an active volume of 23 cm³. The highest peak pulse power and average laser power on the 248.6 nm laser line for the DUV Cu⁺ lasers were 3.25 W and 0.85 W, respectively.

For the metal and metal halide vapor lasers, the thermal mode, and in particular the radial temperature distribution, is of great importance for the stability of the laser operation and for the achievement of high output characteristics as well [8, 9]. In addition, cross-sections and rate constants for heavy particle collisions, which thoroughly determine the inverse population and laser output parameters, such as asymmetric charge transfer, Penning ionization, depend strongly on the gas temperature. That is why the radial temperature distribution in the active zone of the DUV Cu⁺ Ne-CuBr laser is object of a detailed investigation. The gas temperature problem is also topical in radio-frequency discharges, widely encountered in gas lasers, electron and ion technologies, nanotechnology, plasma-chemical technology, and more [8].

It is well known that the experimental techniques for gas temperature measurement, using spectral lines broadening and thermocouple are definitely imprecise. That is why mathematical and computer models are applied to carry out the temperature distribution. These models are used not only to evaluate the temperature but also for complex modeling of plasma kinetics processes [9–14].

The determination of the gas temperature profile in metal vapor lasers has been firstly considered in [9], where self-consistent mathematical models have been developed based on an exact solution of the one-dimensional steady-state heat conduction equation subject to the simple boundary conditions of the first and second kind. Till now, the same method has been used in all other papers in this area, when the buffer gas temperature is calculated, see for instance, [9–16]. In these studies, the expected average temperature and profile of the gas are obtained using the following simplifying assumptions: (1) the average power input is considered uniform in the discharge—no radial or longitudinal variations in power density are considered (𝑞_𝑣=constant); (2) the wall temperature is assumed to be constant.

However, in practice, and in computer modeling and simulations, these assumptions are not fully applicable. That is so because the real values of the volume power density 𝑞_𝑣 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 air-cooling of the laser tube.

2. 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.

Figure 1 

Construction of discharge tube.

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].

3. 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 𝑄₁=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 Al₂O₃ (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.

Figure 2 

Principle geometrical design of the cross-section of the laser source: (1) ceramic (Al2O3) tube; (2) quartz tube; (3) thermal insulation. The diameters are, respectively: 𝑑1=5.2 mm, 𝑑2=18 mm, 𝑑3=24.5 mm, and 𝑑4=32.5 mm.

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 𝑄₁=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_𝑔grad𝑇_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 1𝑟𝑑𝑑𝑟(𝑟𝜆_𝑔𝑑𝑇_𝑔(𝑟)𝑑𝑟)+𝑞_𝑣=0,0≤𝑟<𝑅₁,𝑅₁=𝑑₁2. (3.2)

Usually, as it was mentioned earlier, equation (3.1) (resp., (3.2)) is solved in publications under the boundary conditions: 𝑇_𝑔(𝑅₁)=𝑇_𝑤,𝑑𝑇_𝑔(𝑟)|𝑑𝑟_𝑟=0=0, (3.3) 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=𝜆₀𝑇^𝑚_𝑔.

In the case 𝑞_𝑣=constant, (3.2) possesses an exact solution [9], written in the form 𝑇_𝑔(𝑟)=[𝑇_𝑤^𝑚+1+𝑞_𝑣(𝑚+1)4𝜆₀(𝑅²−𝑟²)]^1/(𝑚+1). (3.4)

For solving (3.2), we need to obtain the correct boundary conditions corresponding to (3.3) for 𝑟=𝑅₁ (see Figure 2): 𝑇_𝑔(𝑅₁)=𝑇₁,𝑑𝑇_𝑔(𝑟)|𝑑𝑟_𝑟=0=0. (3.5)

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]: 𝑇₁=𝑇₂+𝑞_𝑙ln(𝑑₂/𝑑₁)2𝜋𝜆₁,𝑇₂=𝑇₃+𝑞_𝑙ln(𝑑₃/𝑑₂)2𝜋𝜆₂,𝑇₃=𝑇₄+𝑞_𝑙ln(𝑑₄/𝑑₃)2𝜋𝜆₃,𝑄₁=𝛼𝐹₄(𝑇₄−𝑇_air)+𝐹₄𝑇𝜀𝑐[(⁴)100⁴𝑇−(^air)100⁴], (3.6)(3.7) 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=𝑄₁/𝑙_𝑎; 𝑙_𝑎 is the active length [2, 5, 6]; 𝜆_𝑗, 𝑗=1,2,3 are the thermal conductivities of the Al₂O₃ 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).

Table 1 

Related parameters of the theoretical calculation.

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 𝑄₁ 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, 𝐹₄ 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 𝑇₄ 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).

4. 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.

4.1. Determining the Variable Radial Distribution 𝐪_𝐯=𝐪_𝐯(𝐫)

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≈𝜎𝐸², 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)=𝐸₀𝐽₀((2.4/𝑅₁)𝑟), where 𝐽₀((2.4/𝑅₁)𝑟) is a Bessel function of the first kind, 0≤𝑟≤𝑅₁. In this way, we have 𝑞_𝑣(𝑟)=𝑄₀[𝐽₀((2.4/𝑅₁)𝑟)]², where 𝑄₀ is a constant, which is found below. The Bessel function 𝐽₀ 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 [𝐽₀((2.4/𝑅₁)𝑟)]² by a polynomial of the third-degree [𝐽₀(𝑥)]²≈𝑎₁+𝑎₂𝑥+𝑎₃𝑥²+𝑎₄𝑥³, where 𝑥=(2.4/𝑅₁)𝑟. Based on tabular data from [23] and using the least squares method, we find 𝑎₁=1.0044,𝑎₂=−0.01768,𝑎₃=−0.5657,𝑎₄=0.1668. (4.1)

For 𝑞_𝑣(𝑟), we obtain: 𝑞_𝑣(𝑟)=𝑄₀[𝑎₁+𝑎₂2.4𝑅₁𝑟+𝑎₃(2.4𝑅₁)²𝑟²+𝑎₄(2.4𝑅₁)³𝑟³]=𝑄₀(𝑎₁+𝐵𝑟+𝐶𝑟²+𝐷𝑟³), (4.2) where 𝐵=𝑎₂2.4𝑅₁,𝐶=𝑎₃(2.4𝑅₁)²,𝐷=𝑎₄(2.4𝑅₁)³. (4.3) The constant 𝑄₀ can be found by using the equality of areas, bounded between the graphics of each of the functions 𝑞₀=constant and 𝑞_𝑣=𝑞_𝑣(𝑟), and the abscissa 𝑟 (see Figure 3): 2𝑞₀𝑅₁=2𝑄₀^𝑅1₀(𝑎₁+𝐵𝑟+𝐶𝑟²+𝐷𝑟³)𝑑𝑟. (4.4) After integrating in (4.4) and substituting the values of the constants, we find 𝑄₀≈2.131𝑞₀. (4.5)

Figure 3 

Distribution of volume power density along the half cross-section of the internal tube, in relative units: ▵: 𝑞𝑣=𝑞0=1, •: 𝑞𝑣=𝑞𝑣(𝑟).

In Figure 3, the distribution of the volume power densities 𝑞₀=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 𝑞₀=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).

4.2. 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: 𝑇_𝑔(𝑟)={𝑇₁^𝑚+1+(𝑚+1)𝑄₀𝜆₀[𝑎₁4(𝑅²₁−𝑟²𝐵)+9(𝑅³₁−𝑟³𝐶)+16(𝑅⁴₁−𝑟⁴𝐷)+25(𝑅⁵₁−𝑟⁵)]}^1/(𝑚+1), (4.6) where the constants 𝐵,𝐶,𝐷 were introduced in (4.3).

Detailed determination of (4.6) is given in the appendix.

4.3. Application of the Mathematical Model

The obtained explicit solution (4.6) can be used when the value 𝑇₁ of the temperature of the internal tube is known. There are two cases as follows.

(1)The temperature 𝑇₃ 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 𝑇₂ and 𝑇₁ can be calculated by means of the corresponding boundary conditions from (3.6).(2)The temperature 𝑇₃ 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 𝑇₄. In the following section, we discuss the procedure for determining the heat transfer coefficient 𝛼 and obtaining a nonlinear algebraic equation for the temperature 𝑇₄. Then, applying (3.6), we calculate 𝑇₃, 𝑇₂, and 𝑇₁.

5. 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.

5.1. 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 (3.7) was calculated in [18] as 𝛼=0.46𝜆_air[𝑔𝛽_air𝑑³₄(𝑇₄−𝑇_air)𝜐²_air]^0.25/𝑑₄. (5.1)

The substitution of (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_air𝑑³₄(𝑇₄−𝑇_air)𝜐²_air]^0.25(𝑇₄−𝑇_air)+𝜋𝑑₄𝑇𝜀𝑐[(⁴)100⁴𝑇−(^air)100⁴]. (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—𝑇₄. Also (5.2) can be easily solved by any computer algebra system, for instance by Mathematica [25]. Once the temperature 𝑇₄ is calculated, the values of 𝑇₃, 𝑇₂, and 𝑇₁ 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 𝑇₄, 𝑇₃, 𝑇₂, 𝑇₁, and the maximum temperature 𝑇₀ in the center of the laser tube are given (see also Figure 1).

Table 2 

Gas temperature at special characteristic points in the case of free convection.

Figure 4 

Gas temperature distribution along the half cross-section of the internal tube with free convection: ▵: 𝑞𝑣=𝑞0=54.6 W cm-3, •: 𝑞𝑣=𝑞𝑣(𝑟) from (4.2).

Table 2 shows that temperatures 𝑇₄, 𝑇₃, 𝑇₂, and 𝑇₁ 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 𝑇₀=𝑇_𝑔(0)=𝑇_max when 𝑞_𝑣=𝑞_𝑣(𝑟) is 90°C higher than the 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].

5.2. Cooling of the Laser Tube by Forced Convection

For all types of convection, the Nusselt criterion, Nu=𝛼𝐻/𝜆, holds [20–22], from which for 𝐻=𝑑₄ and 𝜆=𝜆_air, we find 𝛼=Nu𝜆_air𝑑₄. (5.3)

In the case of forced convection, the Reynolds criterion is represented by [20, 21] Re=𝑣.𝑙_𝑎𝜐_air, (5.4) 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].

For horizontal tubes with forced air cooling the following equality holds [22]: Nu=0.615Re^0.466. (5.5)

From (5.3), (5.4), and (5.5), the heat transfer coefficient 𝛼 is 𝜆𝛼=0.615^air𝑑₄(𝑣𝑑₄𝜐_air)^0.466. (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 𝑇₄ in the form 𝑞_𝑙=0.615𝜋𝜆_air(𝑣𝑑₄𝜐_air)^0.466⋅(𝑇₄−𝑇_air)+𝜋𝑑₄𝑇𝜀𝑐[(⁴)100⁴𝑇−(^air)100⁴]. (5.7) In this way by solving (5.7), we determine 𝑇₄. 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 𝑇₀ along the center of the tube. The results of the calculated values of 𝑇_𝑔(𝑟) in the two cases 𝑞_𝑣=𝑞₀=constant and 𝑞_𝑣=𝑞_𝑣(𝑟) are shown in Figure 5 for air flow 𝑣=20 m/s.

Table 3 

Gas temperature at special characteristic points in the case of forced convection.

Figure 5 

Gas temperature distribution along the half cross-section of the internal tube in the case of forced convection, 𝑣=20 m/s: ▵: 𝑞𝑣=𝑞0=54.6 W cm-3, •: 𝑞𝑣=𝑞𝑣(𝑟) from (4.2).

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 𝑇₁ 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.

6. 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 𝑇_𝑔=1𝑅₁^𝑅1₀𝑇_𝑔(𝑟)𝑑𝑟, (6.1) 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 𝑄₁=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 𝑇_{𝑔,Eq.(3.4)}=𝑇₁(1+(𝑚+1)𝑞_𝑣𝑅²₁4𝜆₀𝑇₁^𝑚+1)^1/(𝑚+1)×[₂𝐹₁(121,−;3𝑚+12;(1+4𝑇₁^𝑚+1𝜆₀(𝑚+1)𝑞_𝑣𝑇₁^𝑚+1)⁻¹)]. (6.2) The numerical value of (6.2) is 𝑇_{𝑔,Eq.(3.4)}=1347 K.

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 ₂𝐹₁(𝑎,𝑏;𝑐;𝑧) in the solution of (6.2) makes it practically difficult to use. For this reason, we represent the function ₂𝐹₁(𝑎,𝑏;𝑐;𝑧) by its Gauss series [25, 26], limited only to the first two terms of the expansion ₂𝐹₁(𝑎,𝑏;𝑐;𝑧)≈1+𝑎𝑏𝑧𝑐. (6.3) The result is ₂𝐹₁(121,−;3𝑚+12;(1+4𝑇₁^𝑚+1𝜆₀(𝑚+1)𝑞_𝑣𝑇₁^𝑚+1)⁻¹𝑞)≈1−^𝑣𝑅²₁3[(𝑚+1)𝑞_𝑣𝑅²₁+4𝜆₀𝑇₁^𝑚+1]. (6.4)

In this way by using (6.2), we obtain 𝑇_𝑔≈𝑇₁(1+(𝑚+1)𝑞_𝑣𝑅²₁4𝜆₀𝑇₁^𝑚+1)^1/(𝑚+1)𝑞(1−^𝑣𝑅²₁3[(𝑚+1)𝑞_𝑣𝑅²₁+4𝜆₀𝑇₁^𝑚+1]). (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)}.

7. 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 𝑇₃ 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.

The proposed theoretical model and methodology could be useful for solving similar engineering problems where the calculation of the radial heat flow with an internal heat source is required. It gives the opportunity to carry out further computer simulations in order to optimize laser generation, while changing the geometrical design, tube materials, heat insulation, input electric power, and laser operating conditions.

Appendix

Let us consider the problem (3.2), (3.5), assuming that the temperature 𝑇₁ is determined. For 𝜆_𝑔=𝜆₀𝑇^𝑚_𝑔, 0≤𝑟<𝑅₁ and 𝑞_𝑣=𝑞_𝑣(𝑟), (3.2) becomes 1𝑟𝑑𝑑𝑟(𝑟𝜆₀𝑇^𝑚_𝑔𝑑𝑇𝑑𝑟)+𝑞_𝑣=0. (A.1)

By the change of variable 𝑈(𝑟)=𝑇_𝑔^𝑚+1, (A.2) we have 𝑇^𝑚_𝑔(𝑑𝑇_𝑔/𝑑𝑟)=(1/𝑚+1)(𝑑𝑈/𝑑𝑟). After some simplification in (A.1) and (3.3), we obtain the problem 𝑑²𝑈𝑑𝑟²+1𝑟𝑑𝑈+𝑞𝑑𝑟^𝑣(𝑚+1)𝜆₀=0,𝑈(𝑅₁)=𝑇₁^𝑚+1,𝑑𝑈|𝑑𝑟_𝑟=0=0. (A.3)(A.4)

By introducing a new variable 𝜏=𝑑𝑈.𝑑𝑟 (A.5) However (A.3) can be rearranged in the form 𝑑(𝑟𝜏)+𝑚+1𝜆₀𝑞_𝑣𝑟𝑑𝑟=0. (A.6)

The integration of (A.6) and substitution of 𝑞_𝑣 by the expression (4.2) yields 𝜏𝑟+((𝑚+1)𝑄₀/𝜆₀)∫(𝑎₁𝑟+𝐵𝑟²+𝐶𝑟³+𝐷𝑟⁴)𝑑𝑟=𝐶₁, where 𝐶₁ is a constant of integration. We find 𝜏+(𝑚+1)𝑄₀𝜆₀(𝑎₁𝑟2𝑟+𝐵²3𝑟+𝐶³4𝑟+𝐷⁴5𝐶)=¹𝑟. (A.7)

Coming back to the variable 𝑈 according to (A.5), we obtain 𝑑𝑈+𝑑𝑟(𝑚+1)𝑄₀𝜆₀𝑟(𝑎2𝑟+𝐵²3𝑟+𝐶³4𝑟+𝐷⁴5𝐶)=¹𝑟. (A.8)

The second boundary condition in (A.4) requires that 𝐶₁=0, so 𝑑𝑈+(𝑚+1)𝑄₀𝜆₀(𝑎₁𝑟2𝑟+𝐵²3𝑟+𝐶³4𝑟+𝐷⁴5)𝑑𝑟=0. (A.9)

The integration of (A.9) yields 𝑈+(𝑚+1)𝑄₀𝜆₀(𝑎₁𝑟²4𝑟+𝐵³9𝑟+𝐶⁴𝑟16+𝐷⁵25)=𝐶₂, (A.10) where 𝐶₂ is a constant. Using the boundary condition (A.4), we find 𝐶₂𝐶₂=𝑈(𝑅₁)+(𝑚+1)𝑄₀𝜆₀(𝑎₁𝑅²₁4𝑅+𝐵³¹9𝑅+𝐶⁴¹𝑅16+𝐷⁵¹25). (A.11)

The substitution of 𝐶₂ into (A.10) gives 𝑈(𝑟)=𝑈(𝑅₁)+(𝑚+1)𝑄₀𝜆₀[𝑎₁4(𝑅²₁−𝑟²𝐵)+9(𝑅³₁−𝑟³𝐶)+16(𝑅⁴₁−𝑟⁴𝐷)+25(𝑅⁵₁−𝑟⁵)]. (A.12)

Finally, by using (A.2), we obtain the required solution of (3.2) in the form (4.6): 𝑇_𝑔(𝑟)={𝑇₁^𝑚+1+(𝑚+1)𝑄₀𝜆₀[𝑎₁4(𝑅²₁−𝑟²𝐵)+9(𝑅³₁−𝑟³𝐶)+16(𝑅⁴₁−𝑟⁴𝐷)+25(𝑅⁵₁−𝑟⁵)]}^1/(𝑚+1). (A.13)

Acknowledgments

This work is partially supported by the National Scientific Fund of Bulgarian Ministry of Education and Science, Grant no. VU-MI-205/2006.