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Mathematical modeling of CO2 TEA laser
Optics & Laser Technology 30 (1998) 451±457
Mathematical modeling of CO2 TEA laser M. Soukieh *, B. Abdul Ghani, M. Hammadi Atomic Energy Commission, P.O. Box 6091, Damascus, Syria Received 4 August 1998; accepted 30 November 1998
Abstract Five and six-temperature models for the CO2±N2±He system are used to describe the process of the dynamic emission in the TEA CO2 laser. All physical constants and relaxation rates related to these models are examined to estimate the output pulse parameters as a function of the input parameters. The two pumping processes implemented; empirical function and dierential equation show a good agreement with the experimental data. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Modeling; Carbon dioxide lasers; Gas laser
1. Introduction Due to their interesting physical and chemical parameters, pulsed CO2 TEA lasers are widely utilized in scienti®c and industrial applications. However, there are many processes determining the dynamic transitions between vibrational energy levels of molecules of gas mixture CO2±N2±He, are still not fully understood [1, 2]. For instance, there are many factors may cause a decrease in the output power, such as nonsystematically electrical discharge and dissociation of CO2 to CO molecules. The degree of dissociation could reach 80±90% according to experimental data, particularly in sealed systems [3]. In addition to the dissociation phenomenons, many reactions in the gas mixture, could produce dierent molecular gases such as NO2, O2, N2O, NO, CO2ÿ , CO2+ , . . ., etc. However, in spite of the unfavorite role of the above mentioned dissociation process, a part of the CO molecules could participate in the laser action [1, 3, 4]. The interesting feature in this type of lasers resulted from the possibility of producing high output power pulses with high eciency. The output pulse par* Corresponding author. Tel.: +963-6111-926/7; fax: +963-6112289.
ameters and their delay time depend on the ratio of gas mixture and pumping rates [5, 6]. In literature, there are several models describing the dynamic emission of TEA CO2 laser [4, 5, 7, 8] i.e. four, ®ve and six-temperature models. In fact, four and ®ve-temperature models are special case of the sixtemperature model which will be considered in this work. Actually, this model is considered because it recognizes almost all the transitions that can take place between dierent vibrational±rotational levels in the gas mixture. A fully and a comprehensive survey of the previous works, related to the above mentioned models, show many discrepancies, such as: 1. Most of the workers neglect the decay processes and energy losses in the resonator. 2. Consider dierent values for the relaxation rates [8]. 3. Huge dierences between the numerical and experimental data in the output laser parameters [5, 8]. 4. Most workers have used the empirical function method only for pumping process [6, 8]. In this paper, a full investigation on ®ve and sixtemperature models and a comparison between the numerical results of these models with the experimental data are sustained.
0030-3992/98/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 7 7 - 2
452
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
During numerical calculations, considering empirical function as a pumping process method, a comparison was suggested using a dierential equation.
2. Mathematical model The following three equations describe the variation in the stored energy density in unit volume (erg/cm3) as a function of time variation in CO2 symmetrical, bending and antisymmetrical modes respectively [8]: dE1 fNe tNCO2 W1 X1 T cW1 DNWUu t=h dt W1 E3 ÿ E3 T, T1 , T2 E1 ÿ E1 T ÿ W3 t3 T, T1 , T2 t10 T E1 ÿ E1 T2 ÿ t12 T2 W1 E5 ÿ E5 T, T1 , T2 : W5 t5 T, T1 , T2 dE2 E1 ÿ E1 T2 fNe tNCO2 W2 X2 T dt t12 T2 W2 E3 ÿ E3 T, T1 , T2 E2 ÿ E2 T ÿ t3 T, T1 , T2 t20 T W3 W2 E5 ÿ E5 T, T1 , T2 : W5 t5 T, T1 , T2
1
2
dE3 fNe tNCO2 W3 X3 T ÿ cW3 DNWUu t=h dt E3 ÿ E3 T, T1 , T2 E4 ÿ E4 T3 t3 T, T1 , T2 t43 T W3 E5 ÿ E5 T, T3 : t53 T, T3 W5
ÿ
3
The time evolution of the stored energy density in unit volume for N2 molecule is given by the following equation [8]: dE4 E4 ÿ E4 T3 Ne tNN2 W4 X4 T ÿ t43 T dt W4 E5 ÿ E5 T, T4 : W5 t54 T, T4
dE5 Ne t 1 ÿ f NCO2 W5 X5 T dt ÿ
E5 ÿ E5 T, T3 E5 ÿ E5 T, T4 ÿ t53 T, T3 t54 T, T4
ÿ
E5 ÿ E5 T, T1 , T2 , t5 T, T1 , T2
5
where f is the nondissociated fraction of CO2 molecules. Note: For simplicity we consider f as a constant (in general f is a function of time, electrical ®eld intensity and electron number density). By taking the sum of Eqs. (1)±(5) in the steady state (i.e. Ne(t) = 0), the following equation which describe the time evolution of the stored energy density in gas mixture CO2±N2±He±CO will be obtained: dE E1 ÿ E1 T E2 ÿ E2 T dt t10 T t20 T W1 W2 E3 ÿ E3 T, T1 , T2 ÿ 1ÿ W3 W3 t3 T, T1 , T2 W4 E5 ÿ E5 T, T4 1ÿ W5 t54 T, T4 W3 E5 ÿ E5 T, T3 1ÿ W5 t53 t, T3 W1 W2 E5 ÿ E5 T, T1 , T2 , ÿ 1ÿ W5 W5 t5 T, T1 , T2
6
where ÿ E 52 1 ÿ f NCO2 52 NN2 32 NHe 52 fNCO kT: Eqs. (1)±(6) are Landau±Teller equations which describe the time evolution of the energy stored in the three dierent CO2, N2, He, CO vibrational modes. The time evolution of the volume density of ®eld intensity in the cavity is given by the following equation: dUu Uu DNWUu S ÿ N001 P J : 7 chu c dt tc h Then, the intensity of laser beam will be given by the following relation:
4
The time evolution of the stored energy density in unit volume for CO molecules is given by the following equation [8]:
Iu cUu : The population dierences between upper and lower laser levels is [7, 8] DN N001 P J 001 ÿ
2J 1 N100 P J 1 2J 3
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
where the population number density of upper and lower levels are N001 fNCO2 exp ÿW3 =kT3 Z,
The laser output and the energy loss could be determined by the following formulae: Eout ÿ
N100 fNCO2 exp ÿW1 =kT1 Z, Z 1 ÿ exp ÿW1 =kT1 1 ÿ exp ÿW2 =kT2 1
The rotational distribution function is given by 2hcBCO2 hcBCO2 J J 1 2J 1exp ÿ : P J 001 kT kT where BCO2 is the rotational constant of molecule CO2, c the light velocity, h the Plank constant, k the Boltzman constant, J (001) the rotational quantum number, W = Fl2/4puDutSP (cm2 s), F = l/L the ®lling factor, l and n the wave length and the frequency of laser emission, l the cavity length, L the resonator length, tc = (ÿ2L)(1 ÿ Rout)/(c ln Rout)(1 ÿ Rout ÿ Kloss) the photon life time in the cavity, Rout the re¯ection coecient of output mirror, Kloss the loss coecient, tSP the spontaneous emission rate, Dn the line width of collision emission, S = (2l2/pAtSP)(du/Du) (s ÿ 1), dn the line width of laser emission, A the re¯ecting area of the smallest cavity mirror, Ne(t) the electron density in unit volume, Ni the molecule number (i = CO2, CO, N2, He) in unit volume, Wi = hni the energies of dierent levels, Xi the electron±molecule excitation rates, T the gas temperature, Ti the simultaneous mode temperature, (1 ÿ f) the dissociation factor of CO2 molecules. The relaxation rates tij among dierent vibrational states (i, j) in the gas mixture were found to have dierent values in most references. The most suitable experimentally relaxation rates and the formulae of dierent energy levels are stated in [8]. The ampli®cation coecient Kn can be calculated as follow: hcBCO2 Jl2 2J 1 N100 N001 ÿ Ku 2 2J ÿ 1 2p DutSP kT hcBCO2 J J ÿ 1 , exp ÿ kT where the laser transition line width is 1=2 X Nt Qt 8kT 1 1 : Du p p MCO2 Mt t Here MCO2 , Mt masses of molecules, Nt the number of molecules in unit volume, Qt collision cross section of molecules.
A lnRout 1 2
ÿ Kloss
2
ÿ exp ÿW3 =kT3 :
453
1 ÿ Rout ÿ Kloss Rout Kloss 1 ÿ Rout 1 ÿ Kloss
t 0 0
Iu tdt
Eloss Eout Kloss = 1 ÿ Rout ÿ Kloss Kloss Rout :
3. Pumping processes Usually the pumping process of gas mixture is simulated by the following empirical function [6, 8]: Ne t 7 1013 1 ÿ exp ÿt exp ÿ2t:
8
12 cm ÿ 3. This suggests that the value of with Nmax e 110
Table 1 Parameter
Numerical value
W1 W2 W3 W4 W5 X1 X2 X3 X4 X5 h c l MCO2 MN2 du/Du MHe MCO k BCO2 QCO2 QN2 QHe QCO Ntot Rgas N0 e tSP J Kloss
1388 cm ÿ 1 667 cm ÿ 1 2349 cm ÿ 1 2330 cm ÿ 1 2150 cm ÿ 1 510 ÿ 9 cm3/s 310 ÿ 9 cm3/s 810 ÿ 9 cm3/s 2.310 ÿ 8 cm3/s 310 ÿ 8 cm3/s 6.62510 ÿ 27 erg/s 2.9981010 cm/s 10.6 mm 7.310 ÿ 23 g 4.610 ÿ 23 g 210 ÿ 3 6.710 ÿ 24 g 4.610 ÿ 23 g 1.3810 ÿ 16 erg/K 0.4 cm ÿ 1 1.310 ÿ 14 cm2 1.1410 ÿ 14 cm2 3.710 ÿ 15 cm2 1.1410 ÿ 14 cm2 3.221016 Ptot cm ÿ 3 8.31107 erg/Kmol 6.0251023 mol ÿ 1 1.610 ÿ 19 C 0.2 s 18 10 ÿ 3 cm ÿ 1
The values of the equation parameters [6, 8].
454
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
Fig. 1. The output power of CO2 pulse laser as a function of gas temperature T based upon ®ve and six-temperature models: (I) six T. model; (II) ®ve T. model.
Nmax did not lead to the mentioned value in [6] with e temperature increment of DT 0 40 K. In this work, another equation was used for describing the pumping process as follows: dNe t a ÿ gue Ne t ÿ bN 2e t ÿ dNe tN, dt
9
where a, b, g, d are the rates of the electron impact ionization, electron±positive ion recombination, electron neutral attachment, negative-ion detachment processes respectively, and ue is the electron drift velocity [8±10]; a (cm ÿ 1) = A1Ptotexp(DE/Ptot), D = 0.265 (Torr/V), A1 = 3.3 10 ÿ 7 (cm Torr) ÿ 1, b (cm3/s) = 1.107, g (cm ÿ 1)1 0.01, d 13 10 ÿ 14 cm3/s), ue = 0.94 1016(E/N + 3.92)(1 ÿ exp(ÿ1.96 1016E/N))106 (cm/s) [7], N = 3.22 1016Ptot (cm ÿ 3).
4. Numerical solution of rate equations The rate Eqs. (1)±(7) represent a system of sti ordinary nonlinear dierential equations. A FORTRAN computer program, based on Rung±Kutta method, was used to solve these equations. The integration of the equations was achieved using an error criterion eR10 ÿ 4. The equations' constants are given in Table 1 and the geometrical dimensions of the laser cavity are: L = 120 cm, F = 0.6, Rout = 0. 65, A = 0.7 cm2 [5]. The following concepts were respected in this work: 1. The excitation process was in the range of harmonic oscillation. 2. Veri®cation of temperature balance of dierent modes. 3. Maximum value of laser pulse intensity In is corresponded to the value when the ampli®cation coe-
Fig. 2. CO2 laser pulse shape as a function of dissociation coecient and pumping function: (I) empirical function; (II) dierential equation.
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
cient Kn is equal to the value of loss coecient Kloss. The initial values of the rate equations were chosen as follows: 1 Ei t 0 Wi Ni exp Wi =kT ÿ 1 Uu t 0 10ÿ9 erg=cm3 T t 0 300 K PCO2 :PN2 :PHe 1:1:8:
455
5. Results and discussions Fig. 1 shows that, using the ®ve-temperature model, the output pulse parameters were improved by increasing the gas temperature and this contradicts with the experimental fact that by increasing temperature the output power will be reduced (in both cases empirical and dierential equation). While the six-temperature model is in good agreement with the experimental analysis. The previous contradiction, in the ®ve-temperature model, could be resulted from the negligence of the dissociation process and many relaxation rates (such as t5i, t3i, . . .).
Fig. 3. CO2 laser pulse shape as a function of pumping methods and pressure ratios of PCO2 :PN2 :PHe for (1), (2) and (3), respectively, 1:1:3, 2:1:6 and 1:1:8.
456
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
Table 2 Empirical function and dierential equation for f = 0.2, Nmax = 1.04 1013 (cm 0
ÿ3
)
Pressure ratio (PCO2 :PN2 :PHe)
tdel (ms)
Imax (erg/cm2 s) u
Kmax (cm ÿ 1) u
Emax out (mJ)
Empirical function 1:1:3 2:1:6 1:1:8
0.90 0.83 1.35
24.51013 20.21013 8.771013
3.0610 ÿ 3 3.1310 ÿ 3 2.1710 ÿ 3
358 311 205
Dierential equation 1:1:3 2:1:6 1:1:8
0.90 0.83 1.48
18.91013 15.51013 7.111013
2.8010 ÿ 3 2.7710 ÿ 3 2.0010 ÿ 3
399 320 183
From Figs. 2 and 3 and Table 2, according to the six-temperature model, there is no big dierences in the laser output parameters when both methods for pumping process were used. The increase of gas mixture temperature, during the ®rst 4 ms and the maximum vibrational temperature (estimated in Kelvin) using empirical function and dierential equation methods for pumping process, have been found to be as is shown in Table 3. The laser output pulse width was 129±194 ns for f = 0.2 and the delay time is 0.9 and 1.35 ms for pressure ratios 1:1:3 and 1:1:8 respectively (see Fig. 3). Table 2 also shows that the ampli®cation coecient reaches maximum values of 2.1710 ÿ 3 and 210 ÿ 3 cm ÿ 1, respectively, for the empirical function and the dierential equation at pressure ratio 1:1:8. A relatively high concentration of He provides the eective cooling for the discharge gas and reduces the relaxation time of the symmetric stretch mode of CO2 to ground state. The reported experimental results are 200 mJ as maximum energy with a peak power of 1 MW (see [5], Fig. 9), while the theoretical results in [5] show 160 mJ with a peak power of 1.2 MW using a simple four-rate equations model. In this work, applying the six-temperature model with the same input parameters were used as in [5], the numerical results was found 205 mJ as a maximum energy with a peak power of 1.32 MW for empirical function and 183 mJ and 1.07 MW in case of dierential equation (see Fig. 4). The dissociation coecient was set to 0.2, and the ratio of electric ®eld to the total molecular concentration is E/ N = 6.54 10 ÿ 16 V cm2. This corresponds to a ®eld intensity E = 16 kV/cm. All calculations in this paper Table 3
Empirical function Dierential equation
T1
T2
T3
T4
T5
Tgas
615 588
600 573
1775 1680
1870 1775
2270 2518
18.4 13.8
were carried out under these conditions. The agreement with the calculations is surprisingly good (see Fig. 4). The calculations show that both empirical function and dierential equation give similar results with the following dierences: 1. The maximum laser intensity (Imax u ) for empirical function is equal to 1.23 of the maximum laser intensity obtained from dierential equation. 2. The large energetic tail of the pulse for empirical function is equal to 0.634 of the energetic tail for dierential equation. The size and duration of the tail following the initial of pulse depends partly on proportion of N2 to CO2 in the laser. A large proportion of N2 to CO2 is known to produce a large energetic tail. 3. The temperature of the thermal equilibrium (T 1 Ti) for empirical function equal to 381 K, while it is equal 430 K for dierential equation.
6. Conclusion In this work, the ®ve and the six-temperature models, for the dynamics emission of TEA CO2 laser, have been adapted in order to predict the parameters of output pulses. The FORTRAN computer program written, in this work, allows to study the eect of laser input parameters on the output pulse. The program does not take into account the contribution by NO2, N2O, O2, NO, CO2ÿ , CO2+ molecules . . ., etc, in the laser gas mixture. A simulation of electrical pumping was carried out using two approaches; empirical function (see Eq. (8)) and dierential equation (see Eq. (9)). The laser output pulse parameters such as output power Pout, ampli®cation coecient Kn, population inversion DN, laser intensity In, gas temperature Tgas, . . . , etc, can be estimated as functions of time and input parameters (pressure ratio, dissociation coecient, . . .).
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
457
Fig. 4. Theoretical and experimental pulse power output of CO2 ±N2±He laser: (I) empirical function; (II) dierential equation.
Acknowledgements The authors would like to express their thanks to the Director General of AECS Professor I. Othman for his continuous encouragement, guidance and support. They also thank Dr. B. Masarani, Dr. M. Hamo-Leila, and Dr. A. Al-Mohamad for their useful discussion and revision. References [1] Evans JD, Thompson BJ. Selected papers on CO2 lasers, SPIE, MS 22 (1990). [2] Patel CKN. Selective excitation through vibration energy transfer and optical maser action in N2±CO2. Phys. Rev. Lett. 1994;13:617±9.
[3] Gupta NM. Gas recovery system for CO2 laser, Workshop in Gas Laser. Bombay: BARC, 1988. [4] Tyte DC. Carbon dioxide laser. The Advances in Quantum Electronics 1970;1:129±98. [5] Gilbert J et al. Dynamics of the CO2 atmospheric pressure laser with transverse pulse excitation. Can. J. Phys. 1972;50:2523±35. [6] Manes KR, Seguin HJ. Physics of CO2 TEA laser. Appl. Phys. 1972;43(1):5073±8. [7] Pellegrini C. Developments in high power laser and their applications, 1981. [8] Smith K, Thomson RM. Computer modeling of gas laser. NY: Plenum Press, 1978. [9] Nundy U, Chatterjee UK. Modeling of pulse TEA CO2 laser discharge, Workshop in gas laser. BARC, Bombay, 1988, pp. 331±341. [10] Velikhov EP. Molecular gas lasers physics and application, 1981.
Mathematical modeling of CO2 TEA laser M. Soukieh *, B. Abdul Ghani, M. Hammadi Atomic Energy Commission, P.O. Box 6091, Damascus, Syria Received 4 August 1998; accepted 30 November 1998
Abstract Five and six-temperature models for the CO2±N2±He system are used to describe the process of the dynamic emission in the TEA CO2 laser. All physical constants and relaxation rates related to these models are examined to estimate the output pulse parameters as a function of the input parameters. The two pumping processes implemented; empirical function and dierential equation show a good agreement with the experimental data. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Modeling; Carbon dioxide lasers; Gas laser
1. Introduction Due to their interesting physical and chemical parameters, pulsed CO2 TEA lasers are widely utilized in scienti®c and industrial applications. However, there are many processes determining the dynamic transitions between vibrational energy levels of molecules of gas mixture CO2±N2±He, are still not fully understood [1, 2]. For instance, there are many factors may cause a decrease in the output power, such as nonsystematically electrical discharge and dissociation of CO2 to CO molecules. The degree of dissociation could reach 80±90% according to experimental data, particularly in sealed systems [3]. In addition to the dissociation phenomenons, many reactions in the gas mixture, could produce dierent molecular gases such as NO2, O2, N2O, NO, CO2ÿ , CO2+ , . . ., etc. However, in spite of the unfavorite role of the above mentioned dissociation process, a part of the CO molecules could participate in the laser action [1, 3, 4]. The interesting feature in this type of lasers resulted from the possibility of producing high output power pulses with high eciency. The output pulse par* Corresponding author. Tel.: +963-6111-926/7; fax: +963-6112289.
ameters and their delay time depend on the ratio of gas mixture and pumping rates [5, 6]. In literature, there are several models describing the dynamic emission of TEA CO2 laser [4, 5, 7, 8] i.e. four, ®ve and six-temperature models. In fact, four and ®ve-temperature models are special case of the sixtemperature model which will be considered in this work. Actually, this model is considered because it recognizes almost all the transitions that can take place between dierent vibrational±rotational levels in the gas mixture. A fully and a comprehensive survey of the previous works, related to the above mentioned models, show many discrepancies, such as: 1. Most of the workers neglect the decay processes and energy losses in the resonator. 2. Consider dierent values for the relaxation rates [8]. 3. Huge dierences between the numerical and experimental data in the output laser parameters [5, 8]. 4. Most workers have used the empirical function method only for pumping process [6, 8]. In this paper, a full investigation on ®ve and sixtemperature models and a comparison between the numerical results of these models with the experimental data are sustained.
0030-3992/98/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 7 7 - 2
452
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
During numerical calculations, considering empirical function as a pumping process method, a comparison was suggested using a dierential equation.
2. Mathematical model The following three equations describe the variation in the stored energy density in unit volume (erg/cm3) as a function of time variation in CO2 symmetrical, bending and antisymmetrical modes respectively [8]: dE1 fNe tNCO2 W1 X1 T cW1 DNWUu t=h dt W1 E3 ÿ E3 T, T1 , T2 E1 ÿ E1 T ÿ W3 t3 T, T1 , T2 t10 T E1 ÿ E1 T2 ÿ t12 T2 W1 E5 ÿ E5 T, T1 , T2 : W5 t5 T, T1 , T2 dE2 E1 ÿ E1 T2 fNe tNCO2 W2 X2 T dt t12 T2 W2 E3 ÿ E3 T, T1 , T2 E2 ÿ E2 T ÿ t3 T, T1 , T2 t20 T W3 W2 E5 ÿ E5 T, T1 , T2 : W5 t5 T, T1 , T2
1
2
dE3 fNe tNCO2 W3 X3 T ÿ cW3 DNWUu t=h dt E3 ÿ E3 T, T1 , T2 E4 ÿ E4 T3 t3 T, T1 , T2 t43 T W3 E5 ÿ E5 T, T3 : t53 T, T3 W5
ÿ
3
The time evolution of the stored energy density in unit volume for N2 molecule is given by the following equation [8]: dE4 E4 ÿ E4 T3 Ne tNN2 W4 X4 T ÿ t43 T dt W4 E5 ÿ E5 T, T4 : W5 t54 T, T4
dE5 Ne t 1 ÿ f NCO2 W5 X5 T dt ÿ
E5 ÿ E5 T, T3 E5 ÿ E5 T, T4 ÿ t53 T, T3 t54 T, T4
ÿ
E5 ÿ E5 T, T1 , T2 , t5 T, T1 , T2
5
where f is the nondissociated fraction of CO2 molecules. Note: For simplicity we consider f as a constant (in general f is a function of time, electrical ®eld intensity and electron number density). By taking the sum of Eqs. (1)±(5) in the steady state (i.e. Ne(t) = 0), the following equation which describe the time evolution of the stored energy density in gas mixture CO2±N2±He±CO will be obtained: dE E1 ÿ E1 T E2 ÿ E2 T dt t10 T t20 T W1 W2 E3 ÿ E3 T, T1 , T2 ÿ 1ÿ W3 W3 t3 T, T1 , T2 W4 E5 ÿ E5 T, T4 1ÿ W5 t54 T, T4 W3 E5 ÿ E5 T, T3 1ÿ W5 t53 t, T3 W1 W2 E5 ÿ E5 T, T1 , T2 , ÿ 1ÿ W5 W5 t5 T, T1 , T2
6
where ÿ E 52 1 ÿ f NCO2 52 NN2 32 NHe 52 fNCO kT: Eqs. (1)±(6) are Landau±Teller equations which describe the time evolution of the energy stored in the three dierent CO2, N2, He, CO vibrational modes. The time evolution of the volume density of ®eld intensity in the cavity is given by the following equation: dUu Uu DNWUu S ÿ N001 P J : 7 chu c dt tc h Then, the intensity of laser beam will be given by the following relation:
4
The time evolution of the stored energy density in unit volume for CO molecules is given by the following equation [8]:
Iu cUu : The population dierences between upper and lower laser levels is [7, 8] DN N001 P J 001 ÿ
2J 1 N100 P J 1 2J 3
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
where the population number density of upper and lower levels are N001 fNCO2 exp ÿW3 =kT3 Z,
The laser output and the energy loss could be determined by the following formulae: Eout ÿ
N100 fNCO2 exp ÿW1 =kT1 Z, Z 1 ÿ exp ÿW1 =kT1 1 ÿ exp ÿW2 =kT2 1
The rotational distribution function is given by 2hcBCO2 hcBCO2 J J 1 2J 1exp ÿ : P J 001 kT kT where BCO2 is the rotational constant of molecule CO2, c the light velocity, h the Plank constant, k the Boltzman constant, J (001) the rotational quantum number, W = Fl2/4puDutSP (cm2 s), F = l/L the ®lling factor, l and n the wave length and the frequency of laser emission, l the cavity length, L the resonator length, tc = (ÿ2L)(1 ÿ Rout)/(c ln Rout)(1 ÿ Rout ÿ Kloss) the photon life time in the cavity, Rout the re¯ection coecient of output mirror, Kloss the loss coecient, tSP the spontaneous emission rate, Dn the line width of collision emission, S = (2l2/pAtSP)(du/Du) (s ÿ 1), dn the line width of laser emission, A the re¯ecting area of the smallest cavity mirror, Ne(t) the electron density in unit volume, Ni the molecule number (i = CO2, CO, N2, He) in unit volume, Wi = hni the energies of dierent levels, Xi the electron±molecule excitation rates, T the gas temperature, Ti the simultaneous mode temperature, (1 ÿ f) the dissociation factor of CO2 molecules. The relaxation rates tij among dierent vibrational states (i, j) in the gas mixture were found to have dierent values in most references. The most suitable experimentally relaxation rates and the formulae of dierent energy levels are stated in [8]. The ampli®cation coecient Kn can be calculated as follow: hcBCO2 Jl2 2J 1 N100 N001 ÿ Ku 2 2J ÿ 1 2p DutSP kT hcBCO2 J J ÿ 1 , exp ÿ kT where the laser transition line width is 1=2 X Nt Qt 8kT 1 1 : Du p p MCO2 Mt t Here MCO2 , Mt masses of molecules, Nt the number of molecules in unit volume, Qt collision cross section of molecules.
A lnRout 1 2
ÿ Kloss
2
ÿ exp ÿW3 =kT3 :
453
1 ÿ Rout ÿ Kloss Rout Kloss 1 ÿ Rout 1 ÿ Kloss
t 0 0
Iu tdt
Eloss Eout Kloss = 1 ÿ Rout ÿ Kloss Kloss Rout :
3. Pumping processes Usually the pumping process of gas mixture is simulated by the following empirical function [6, 8]: Ne t 7 1013 1 ÿ exp ÿt exp ÿ2t:
8
12 cm ÿ 3. This suggests that the value of with Nmax e 110
Table 1 Parameter
Numerical value
W1 W2 W3 W4 W5 X1 X2 X3 X4 X5 h c l MCO2 MN2 du/Du MHe MCO k BCO2 QCO2 QN2 QHe QCO Ntot Rgas N0 e tSP J Kloss
1388 cm ÿ 1 667 cm ÿ 1 2349 cm ÿ 1 2330 cm ÿ 1 2150 cm ÿ 1 510 ÿ 9 cm3/s 310 ÿ 9 cm3/s 810 ÿ 9 cm3/s 2.310 ÿ 8 cm3/s 310 ÿ 8 cm3/s 6.62510 ÿ 27 erg/s 2.9981010 cm/s 10.6 mm 7.310 ÿ 23 g 4.610 ÿ 23 g 210 ÿ 3 6.710 ÿ 24 g 4.610 ÿ 23 g 1.3810 ÿ 16 erg/K 0.4 cm ÿ 1 1.310 ÿ 14 cm2 1.1410 ÿ 14 cm2 3.710 ÿ 15 cm2 1.1410 ÿ 14 cm2 3.221016 Ptot cm ÿ 3 8.31107 erg/Kmol 6.0251023 mol ÿ 1 1.610 ÿ 19 C 0.2 s 18 10 ÿ 3 cm ÿ 1
The values of the equation parameters [6, 8].
454
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
Fig. 1. The output power of CO2 pulse laser as a function of gas temperature T based upon ®ve and six-temperature models: (I) six T. model; (II) ®ve T. model.
Nmax did not lead to the mentioned value in [6] with e temperature increment of DT 0 40 K. In this work, another equation was used for describing the pumping process as follows: dNe t a ÿ gue Ne t ÿ bN 2e t ÿ dNe tN, dt
9
where a, b, g, d are the rates of the electron impact ionization, electron±positive ion recombination, electron neutral attachment, negative-ion detachment processes respectively, and ue is the electron drift velocity [8±10]; a (cm ÿ 1) = A1Ptotexp(DE/Ptot), D = 0.265 (Torr/V), A1 = 3.3 10 ÿ 7 (cm Torr) ÿ 1, b (cm3/s) = 1.107, g (cm ÿ 1)1 0.01, d 13 10 ÿ 14 cm3/s), ue = 0.94 1016(E/N + 3.92)(1 ÿ exp(ÿ1.96 1016E/N))106 (cm/s) [7], N = 3.22 1016Ptot (cm ÿ 3).
4. Numerical solution of rate equations The rate Eqs. (1)±(7) represent a system of sti ordinary nonlinear dierential equations. A FORTRAN computer program, based on Rung±Kutta method, was used to solve these equations. The integration of the equations was achieved using an error criterion eR10 ÿ 4. The equations' constants are given in Table 1 and the geometrical dimensions of the laser cavity are: L = 120 cm, F = 0.6, Rout = 0. 65, A = 0.7 cm2 [5]. The following concepts were respected in this work: 1. The excitation process was in the range of harmonic oscillation. 2. Veri®cation of temperature balance of dierent modes. 3. Maximum value of laser pulse intensity In is corresponded to the value when the ampli®cation coe-
Fig. 2. CO2 laser pulse shape as a function of dissociation coecient and pumping function: (I) empirical function; (II) dierential equation.
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
cient Kn is equal to the value of loss coecient Kloss. The initial values of the rate equations were chosen as follows: 1 Ei t 0 Wi Ni exp Wi =kT ÿ 1 Uu t 0 10ÿ9 erg=cm3 T t 0 300 K PCO2 :PN2 :PHe 1:1:8:
455
5. Results and discussions Fig. 1 shows that, using the ®ve-temperature model, the output pulse parameters were improved by increasing the gas temperature and this contradicts with the experimental fact that by increasing temperature the output power will be reduced (in both cases empirical and dierential equation). While the six-temperature model is in good agreement with the experimental analysis. The previous contradiction, in the ®ve-temperature model, could be resulted from the negligence of the dissociation process and many relaxation rates (such as t5i, t3i, . . .).
Fig. 3. CO2 laser pulse shape as a function of pumping methods and pressure ratios of PCO2 :PN2 :PHe for (1), (2) and (3), respectively, 1:1:3, 2:1:6 and 1:1:8.
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M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
Table 2 Empirical function and dierential equation for f = 0.2, Nmax = 1.04 1013 (cm 0
ÿ3
)
Pressure ratio (PCO2 :PN2 :PHe)
tdel (ms)
Imax (erg/cm2 s) u
Kmax (cm ÿ 1) u
Emax out (mJ)
Empirical function 1:1:3 2:1:6 1:1:8
0.90 0.83 1.35
24.51013 20.21013 8.771013
3.0610 ÿ 3 3.1310 ÿ 3 2.1710 ÿ 3
358 311 205
Dierential equation 1:1:3 2:1:6 1:1:8
0.90 0.83 1.48
18.91013 15.51013 7.111013
2.8010 ÿ 3 2.7710 ÿ 3 2.0010 ÿ 3
399 320 183
From Figs. 2 and 3 and Table 2, according to the six-temperature model, there is no big dierences in the laser output parameters when both methods for pumping process were used. The increase of gas mixture temperature, during the ®rst 4 ms and the maximum vibrational temperature (estimated in Kelvin) using empirical function and dierential equation methods for pumping process, have been found to be as is shown in Table 3. The laser output pulse width was 129±194 ns for f = 0.2 and the delay time is 0.9 and 1.35 ms for pressure ratios 1:1:3 and 1:1:8 respectively (see Fig. 3). Table 2 also shows that the ampli®cation coecient reaches maximum values of 2.1710 ÿ 3 and 210 ÿ 3 cm ÿ 1, respectively, for the empirical function and the dierential equation at pressure ratio 1:1:8. A relatively high concentration of He provides the eective cooling for the discharge gas and reduces the relaxation time of the symmetric stretch mode of CO2 to ground state. The reported experimental results are 200 mJ as maximum energy with a peak power of 1 MW (see [5], Fig. 9), while the theoretical results in [5] show 160 mJ with a peak power of 1.2 MW using a simple four-rate equations model. In this work, applying the six-temperature model with the same input parameters were used as in [5], the numerical results was found 205 mJ as a maximum energy with a peak power of 1.32 MW for empirical function and 183 mJ and 1.07 MW in case of dierential equation (see Fig. 4). The dissociation coecient was set to 0.2, and the ratio of electric ®eld to the total molecular concentration is E/ N = 6.54 10 ÿ 16 V cm2. This corresponds to a ®eld intensity E = 16 kV/cm. All calculations in this paper Table 3
Empirical function Dierential equation
T1
T2
T3
T4
T5
Tgas
615 588
600 573
1775 1680
1870 1775
2270 2518
18.4 13.8
were carried out under these conditions. The agreement with the calculations is surprisingly good (see Fig. 4). The calculations show that both empirical function and dierential equation give similar results with the following dierences: 1. The maximum laser intensity (Imax u ) for empirical function is equal to 1.23 of the maximum laser intensity obtained from dierential equation. 2. The large energetic tail of the pulse for empirical function is equal to 0.634 of the energetic tail for dierential equation. The size and duration of the tail following the initial of pulse depends partly on proportion of N2 to CO2 in the laser. A large proportion of N2 to CO2 is known to produce a large energetic tail. 3. The temperature of the thermal equilibrium (T 1 Ti) for empirical function equal to 381 K, while it is equal 430 K for dierential equation.
6. Conclusion In this work, the ®ve and the six-temperature models, for the dynamics emission of TEA CO2 laser, have been adapted in order to predict the parameters of output pulses. The FORTRAN computer program written, in this work, allows to study the eect of laser input parameters on the output pulse. The program does not take into account the contribution by NO2, N2O, O2, NO, CO2ÿ , CO2+ molecules . . ., etc, in the laser gas mixture. A simulation of electrical pumping was carried out using two approaches; empirical function (see Eq. (8)) and dierential equation (see Eq. (9)). The laser output pulse parameters such as output power Pout, ampli®cation coecient Kn, population inversion DN, laser intensity In, gas temperature Tgas, . . . , etc, can be estimated as functions of time and input parameters (pressure ratio, dissociation coecient, . . .).
M. Soukieh et al. / Optics & Laser Technology 30 (1998) 451±457
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Fig. 4. Theoretical and experimental pulse power output of CO2 ±N2±He laser: (I) empirical function; (II) dierential equation.
Acknowledgements The authors would like to express their thanks to the Director General of AECS Professor I. Othman for his continuous encouragement, guidance and support. They also thank Dr. B. Masarani, Dr. M. Hamo-Leila, and Dr. A. Al-Mohamad for their useful discussion and revision. References [1] Evans JD, Thompson BJ. Selected papers on CO2 lasers, SPIE, MS 22 (1990). [2] Patel CKN. Selective excitation through vibration energy transfer and optical maser action in N2±CO2. Phys. Rev. Lett. 1994;13:617±9.
[3] Gupta NM. Gas recovery system for CO2 laser, Workshop in Gas Laser. Bombay: BARC, 1988. [4] Tyte DC. Carbon dioxide laser. The Advances in Quantum Electronics 1970;1:129±98. [5] Gilbert J et al. Dynamics of the CO2 atmospheric pressure laser with transverse pulse excitation. Can. J. Phys. 1972;50:2523±35. [6] Manes KR, Seguin HJ. Physics of CO2 TEA laser. Appl. Phys. 1972;43(1):5073±8. [7] Pellegrini C. Developments in high power laser and their applications, 1981. [8] Smith K, Thomson RM. Computer modeling of gas laser. NY: Plenum Press, 1978. [9] Nundy U, Chatterjee UK. Modeling of pulse TEA CO2 laser discharge, Workshop in gas laser. BARC, Bombay, 1988, pp. 331±341. [10] Velikhov EP. Molecular gas lasers physics and application, 1981.