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Theoretical and experimental analyses of the TEA CO2 lasers dynamics by six temperature vibrational-rotational model
Optik 135 (2017) 238–243
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Theoretical and experimental analyses of the TEA CO2 lasers dynamics by six temperature vibrational-rotational model R. Torabi a,∗ , H. Saghafifar a , A.M. Koushki b a b
Malek-Ashtar University of Technology, Physics Department, Shahinshahr, Iran Lasers and Optics Research School, Nuclear Science and Technology Research Institute, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 5 November 2016 Accepted 28 January 2017 Keywords: TEA CO2 laser simulation Dynamic emission simulation Vibrational-rotational six temperature model
a b s t r a c t A six-temperature dynamic model, modified in terms of vibrational-rotational transitions, has been used to describe the dynamic processes in TEA CO2 lasers. Significant improvement in the predicted laser characteristics was obtained, taking into account the CO2 molecules vibrational-rotational transitions. The coupled differential equations were numerically solved by the fourth order Runge–Kutta method. The simulated output characteristics of the laser have been compared with simple vibrational dynamics model and experimental data due to a homemade TEA CO2 laser. It has been found that the best agreement with experimental results regards to the vibrational-rotational dynamic model. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction The CO2 lasers are regarded, for decades, as the most economical coherent radiation source producing high power beams with good quality, suitable for a wide variety of applications. Among them, transversely excited at atmospheric pressure (TEA) CO2 lasers are subjects of particular interest due to their relatively simple construction along with high available intensities and repetition rates. Moreover, distinct rotational-vibrational laser transitions with different frequencies, leading to a discrete output spectrum containing several tens of emission lines, has made the CO2 lasers an exceptional ideal tunable gas molecular laser for experimental [1–3] as well as theoretical studies [3–8]. Many works have been so far done in the field of TEA CO2 lasers dynamic emission simulation, mainly based on population number density equations [9,10] and temperature model such as four, five and six-temperature models [3,6]. Among these, the later (6TM) takes into account almost all the influential vibrational-rotational transitions in active medium molecules and is currently used to describe the kinetic processes of CO2 lasers as a powerful model. However, most of works on CO2 lasers dynamic emission simulation by 6TM employs an empirical electron density distribution equation in discharge plasma column, without essential considerations of the laser electric discharge circuit. In addition, whether 6TM is used or not, the populations of rotational sublevels within each vibrational level are assumed, as a rule, to be continuously in Boltzmann distribution, based on instantaneous rotational relaxation assumption [5,11]. Nevertheless, the rotational processes are always neglected and only the vibrational kinetics is analyzed. The omission of such processes may be the responsible for some well-known contradictions between modeling and experimental results, appeared in literature. This suggestion can be reinforced regarding the reports indicating a rotational relaxation time of R ∼ 0.1 ns for TEA CO2 lasers with pulse durations
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (R. Torabi). http://dx.doi.org/10.1016/j.ijleo.2017.01.091 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
R. Torabi et al. / Optik 135 (2017) 238–243
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Fig. 1. Energy diagram of the effective V-R levels in CO2 lasers.
in ns order [12] and also, determinative role of rotational coupling effects in gain-switched regime when the laser pulse duration is greater than 100 R [13]. In our previous work [3] on a pin-array UV pre-ionized TEA CO2 laser, temporal behavior of the discharge current pulse was simulated considering electrical properties of the discharge circuit as well as the pre-ionization geometry, together with 6TM for simulation of the laser dynamic emission process. The main discharge region was modeled by a non-linear distributed lumped RLC circuit, and the associated electron densities were calculated by the transmission line method (TLM), resulting in a good agreement with experimental data on laser pulse shape and energy with, of course, some deviations in pulse durations. This model is improved in the present work taking into account the determinative role of rotational dynamic processes. 2. Materials and methods 2.1. Kinetic model Vibrational–Rotational (V-R) energy levels diagram of the CO2 –N2 –CO system, related to typical CO2 lasers with CO2 :N2 :He gas mixture is shown in Fig. 1. The mathematical model of CO2 –N2 –He–CO system for tunable TEA CO2 lasers is established based on two old assumptions [3,9]: 1. All of the vibrational-rotational transitions are homogeneously pressure broadened due to high total pressure of the gas mixture. 2. The couple of (1,0,0) and (0,2,0) lower laser levels acts as a single energy level due to their Fermi resonance. And a new additional assumption [5,11]: 3. Rotational energy is distributed among the rotational levels via relaxations with related time constants. Regarding different exciting and de-exciting energy transfer processes between translational and V-R energy levels of CO2 , N2 and CO molecules (including electron collisions), the following equations can be obtained to describe the variations in the stored energy density (ergs/cm3 ) as a function of time, known as Teller-Landau equations [3]: E˙ 1 = Ne (t)NCO2 W1 fX1 (T ) + WıJ W1 I (t) + (
E˙ 2 = Ne (t)NCO2 W2 fX2 (T ) + (
W1 E3 − E3 (T, T1 , T2 ) W1 E5 − E5 (T, T1 , T2 ) E1 − E1 (T2 ) E1 − E1 (T ) ) ) − − +( W3 W5 3 (T, T1 , T2 ) 10 (T ) 12 (T2 ) 5 (T, T1 , T2 ) (1)
W2 E3 − E3 (T, T1 , T2 ) W2 E5 − E5 (T, T1 , T2 ) E2 − E2 (T ) E1 − E1 (T2 ) ) ) − + +( W3 W5 3 (T, T1 , T2 ) 20 (T ) 12 (T2 ) 5 (T, T1 , T2 )
(2)
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E˙ 3 = Ne (t) NCO2 W3 fX3 (T ) − WıJ W3 I (t) − E˙ 4 = Ne (t) NN2 W4 X4 (T ) −
E4 − E4 (T3 ) + 43 (T )
E˙ 5 = Ne (t)(1 − f )NCO2 W5 X5 (T ) −
E3 − E3 (T, T1 , T2 ) E4 − E4 (T3 ) + + 3 (T, T1 , T2 ) 43 (T )
W E − E 5 5 (T, T4 ) 4 W5
W E − E 5 5 (T, T3 ) 3 W5
54 (T, T4 )
E5 − E5 (T, T1 , T2 ) E5 − E5 (T, T3 ) E5 − E5 (T, T4 ) − − 5 (T, T1 , T2 ) 53 (T, T3 ) 54 (T, T4 )
53 (T, T3 )
(3) (4) (5)
Eqs. (1)–(3) describe the time evolution of the energy stored in v1 , v2 and v3 vibrational modes of CO2 molecules while, Eqs. (4) and (5) represent the time evolution of energy stored in the vibrational modes of N2 and the CO molecules, respectively [3,5]. In these equations, Ne (t) is the electron density function, NCO2 , NN2 and NHe are the number densities of CO2 , N2 and He in the gas mixture, respectively; f is the non-dissociation ratio of the CO2 molecules; Wi = hi (i = 1, 2, 3, 4, 5) gives the respective energy levels; Xi (i = 1, 2, 3, 4, 5) represents the electron-molecule excitation rates; W is the stimulated emission rate; T is the ambient temperature; T1 , T2 and T3 are respectively the equivalent vibrational temperature of v1 , v2 and v3 vibrational modes of CO2 molecules; T4 and T5 are the equivalent vibrational temperature of N2 and CO molecules, respectively; E, E1 , E2 , E3 , E4 and E5 are stored energies per unit volume (defined by T, T1 , T2 , T3 , T4 and T5 ) and E5 (T, T3 ), E5 (T, T4 ), E3 (T, T1 , T2 ) and E5 (T, T1 , T2 ) are respectively the equilibrium energy for: CO-CO2 (001) collisions, CO-N2 collisions, transferring from v3 to v1 , v2 modes of CO2 molecule and transferring from CO levels to v1 , v2 modes of CO2 molecule [14]. The relaxation rates relevant to different vibrational i and j levels, ij , can be found in [5]. After analysis by synthesis on stimulating radiation, spontaneous emission and losses in the laser cavity, the time evolution of cavity-field intensity I (erg/cm2 s) can be expressed as: d I (I ) = − + ch c dt
ıJ WI + N001 P(J)S h
(6)
Where, c is the photon life time within the resonator [3,5]; ıJ is the population inversion density between rotational sublevels of the upper and lower laser vibrational levels; N001 is the population of the first excited level of v3 mode of CO2 molecules; c is the light velocity; h is the Plank constant; is the laser frequency; P(J) is the ratio of the Jth rotational level population to of the J + 1 level and S = 22 / A sp (d/) is also the spontaneous emission constant in which, sp is the spontaneous emission life time; A is the smallest reflecting area in the laser resonator, is the collisional broadening of the vibrational-rotational sublevels and d is the line width of collisional emission [3,5]. The stimulated emission rate W is defined as: W=
F2 0 sp
(7)
42 h
Where, F = l/L is the filling factor (L and l are the resonator and active medium length respectively). The temporal variation of ıJ is given by [5,11]: d J ıJ − P(J) (ı ) = −2ıJ WI − R dt
(8)
Where, = N001 − N100 is the population inversion density between the upper and lower vibrational laser levels. For a gas mixture at pressure P with RCO2 : RHe : RN2 ratio, the rotational relaxation time is also given by: R −1 = (RCO2 · KCO2 −CO2 + RHe · KCO2 −He + RN2 · KCO2 −N2 ) · P (in s−1
(9) Torr−1 ) can be found in Ref. [15]. Eqs. (1)–(6) and (8) form
For which, rate constants KCO2 −CO2 , KCO2 −He and KCO2 −N2 a system of coupled differential equations that should be numerically solved to obtain laser output characteristics. Besides, as it was mentioned, in contrast to other works in which an empirical Ne (t) equation for electron density distribution is used, here the Ne (t) is calculated based on transmission line method (TLM) (details can be found in our previous report [3]). 2.2. Experimental setup
The model was executed using constructional data of a homemade pin-array UV pre-ionized TEA CO2 laser and the results were also compared with the experimental data obtained via direct measurements on the laser device. The electrical configuration of the laser is schematically shown in Fig. 2. The required high voltage is supplied by the 3-stage Marx generator, triggered by the main spark gap switch. This pulsed voltage is simultaneously fed to the preionization and main discharge capacitors after switching the main spark gap. The laser head consists of two 40 cm long Ernst profiled electrodes with 3.2 cm gap located inside a 80 cm long optical resonator including a total reflectance Cu mirror with 20 m radius of curvature and a %65 reflectance AR coated ZnSe output coupler with 13 cm2 effective area. The laser produces 3cm × 3 cm nearly square shaped pulses at 10P(20) line when is driven with %10CO2 : %10N2 : %80He gas mixture at 1 atm total pressure and 45 kV optimum discharge voltage, generated
R. Torabi et al. / Optik 135 (2017) 238–243
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Fig. 2. Schematic diagram of the TEA CO2 laser electrical configuration.
Fig. 3. The measured waveforms of discharge current (a) and voltage (c) together with the simulated ones (b) and (d), respectively.
by a homemade 3-stage Marx generator. Lower and higher discharge voltages were respectively involved with energy and glow discharge instabilities. 3. Results and discussion A MATLAB computer program based on Runge-Kutta method has been employed to solve the system of coupled differential Eqs. (1)–(6) and (8) for the five energy densities and light intensity. In order to evaluate the time evolution of the rotational population variation, the Eq. (8) must be solved simultaneously to obtain the population inversion density between rotational sublevels. The initial values chosen for the mentioned differential equations are: Ti (t = 0) = 300K, i = 1,2,3,4,5 I (t = 0) = 30erg/cm2 · s At first, the measured and simulated discharge current and voltage waveforms are typically shown in Fig. 3. The first spike in Fig. 3-a represents the prior pre-ionization discharge [16]. In overall, a good agreement between simulated and measured pulse shapes can be found from the qualitative as well as quantitative points of view.
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R. Torabi et al. / Optik 135 (2017) 238–243
Fig. 4. The measured laser pulse shapes for 45 (a) and 35 kV (c) discharge voltages respectively together with the simulated ones (b) and (d) based on different models. The discharge is assumed to be initiated at t = 0.
Table 1 Various characteristic values of the simulated and measured pulse shapes at different discharge voltages. Data Source
Vd (kV)
tb (s)
Ep (J)
p (ns)
Pmax (MW)
Es /Ep a
VRM with Empirical Ne (t) VRM with Empirical Ne (t) VM with Calculated Ne (t) VM with Calculated Ne (t) VRM with Calculated Ne (t) VRM with Calculated Ne (t) Experimental Experimental
45 35 45 35 45 35 45 35
0.9 1.2 0.7 1 0.7 1 0.75 1
6.26 2.1 5.3 1.8 5.17 1.7 5 1.7
56.2 79.2 39 61 47.5 70 110 75
38.14 14 57 15 45 12.7 26 12
0.43 0.70 0.50 0.63 0.54 0.53 0.56 0.46
a b
Calculated by measuring the areas under the pulse spike and tail sections in the simulated and measured pulse shapes [17]. Defined as the time interval between discharge initiation (t = 0 at Fig. 4) and laser pulse starting.
Typical laser pulse shapes for discharge voltages of 45 and 35 kV and CO2 :N2 :He 1:1:8 gas mixture, traced by THORLABS PM100D photon drag and Tektronix TDS 2022B digital oscilloscope are shown in Fig. 4. The associated simulated pulse shapes based on different models such as vibrational model (VM), vibrational-rotational model (VRM) and VRM with empirical electron density equation are also depicted in Fig. 4. Different characteristic values of the measured and simulated (by the above mentioned methods) laser pulses are given in Table 1. This table contains some overall parameters such as onset delay time t and total pulse energy Ep and also some sectional characteristic values related to different parts of the pulse structure such as spike duration p , peak power Pmax and energies ratio of the pulse spike Es to of the total pulse Ep . Since it regards the overall characteristics of the laser pulses, the best agreements between measured and simulated data for both discharge voltages are respectively due to VRM and VM models with calculated Ne (t). On other hand, the worst coincidences are seen between the measured and all simulated pulse durations for 45 kV discharge voltage that are somewhat reflected in the associated peak powers. This disparity that is known as a common problem in such simulation results [4–7] is significantly abated at 35 kV voltage in which, again, the VRM and VM models with calculated Ne (t) respectively show the most reasonable coincidences with experimental data. These results clearly indicate the advantage of taking account the V-R transitions in the simulation procedure.
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4. Conclusion The process of dynamics in TEA CO2 lasers has been simulated by a six-temperature vibrational-rotational model (6TVRM) together with transmission line method (TLM) for electron density distribution calculation. At first, it has been shown that remarkable corrections in the simulated data are available using calculated electron density instead of the prevalent empirical function. Significant improvements have also been obtained in prediction of overall as well as sectional characteristics of the laser pulse shapes at different discharge voltages via taking account the molecular vibrational-rotational transitions in the laser gas mixture. Acknowledgements The authors wish to thank Mr. K. Silakhori from Laser and Optics Research School for his useful discussion and Mr. B. Kazemi and Mr. A. Pezh from Malek-Ashtar University of Technology for their helpful technical support. References [1] N. Menyuk, P.F. Moulton, Development of a high-repetition rate mini-TEA CO2 laser, Rev. Sci. Instrum. 51 (1980) 216–220. [2] I. Chis, A. Ciura, V. Draganescu, D. Dragulinescu, D. Grigorescu, C. Grigoriu, M.L. Pascu, V.G. Velculescu, M. Abrudean, D. Axente, A. Baldea, M. Gligan, Design and performance of a high repetition rate TEA CO2 laser, J. Phys. E: Sci. Instrum. 21 (1988) 393–396. [3] R. Torabi, H. Saghafifar, A.M. Koushki, A.A. Ganjovi, Simulation and initial experiments of a high power pulsed TEA CO2 laser, Phys. Scr. 91 (2016) 015501. [4] K.R. Manes, H.J. Seguin, Analysis of the CO2 TEA laser, J. Appl. Phys. 43 (1972) 5073. [5] K. Smith, R.M. Thomson, Computer Modeling of Gas Lasers, Plenum Press, New York, 1978. [6] B. Abdul Ghani, TEA CO2 laser simulator: a software tool to predict the output pulse characteristics of TEA CO2 laser, Comput. Phys. Commun. 171 (2005) 93–106. [7] A.M. Koushki, S. Jelvani, K. Silakhori, H. Saeedi, Kinetic modeling of a pulsed CO2 laser, Lasers Eng. 21 (2011) 265–280. [8] A.M. Koushki, K. Silakhori, S. Jelvani, Kinetic modeling of a slow flow CW CO2 laser, Opt. Quantum Electron. 43 (2012) 23–33. [9] S. Al-Hawat, K. Al-Mutaib, Numerical modeling of a fast-axial-flow CW-CO2 laser, Opt. Laser Technol. 39 (2007) 610–615. [10] Manoj Kumar, Jai Khare, A.K. Nath, Numerical solution of Boltzmann tranport equation for TEA CO2 laser having nitrogen-lean gas mixtures to predict laser characteristics and gas lifetime, Opt. Laser Technol. 39 (2007) 86–93. [11] Yanchen Qu, Deming Ren, Liii Zhang, Xiaoyong Hu, Fengmei Liu, Five temperature mathematical modeling of TEA CO2 laser, Proc. SPIE 6263 (2006), 62630J-1-7. [12] (a) P.K. Cheo, R.L. Abrams, Appl. Phys. Lett. 14 (1969) 47; (b) R.L. Abrams, P.K. Cheo, Appl. Phys. Lett. 15 (1969) 177. [13] E.A. Ballik, B.K. Garside, J. Reid, Appl. Phys. Lett. 26 (1975) 380. [14] Tie-Jun Wang, Jin-Yue Gao, Qiong-Yi He, Tao Ma, Yun Jiang, Zhi-Hui Kang, Analysis of the dynamics of a mechanical Q-switched CO2 laser: six-temperature model, J. Appl. Phys. 98 (2005) 073102. [15] Ralph R. Jacobs, Kenneth J. Petti piece, Scott J. Thomas, Rotational relaxation rate constants for CO2 , Appl. Phys. Lett. 24 (1974) 375. [16] W.J. Witteman, The CO2 Laser, in: K. Shimoda (Ed.), Springer-Verlag, 1987. [17] K. Silakhori, S. Jelvani, F. Ghanavati, B. Sajad, M. Talebi, M.R. Sadr, A small size 1–3 atm pulsed CO2 laser with series-connected spark-gaps ultraviolet pre-ionization, Rev. Sci. Instrum. 85 (2014) 013109.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Theoretical and experimental analyses of the TEA CO2 lasers dynamics by six temperature vibrational-rotational model R. Torabi a,∗ , H. Saghafifar a , A.M. Koushki b a b
Malek-Ashtar University of Technology, Physics Department, Shahinshahr, Iran Lasers and Optics Research School, Nuclear Science and Technology Research Institute, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 5 November 2016 Accepted 28 January 2017 Keywords: TEA CO2 laser simulation Dynamic emission simulation Vibrational-rotational six temperature model
a b s t r a c t A six-temperature dynamic model, modified in terms of vibrational-rotational transitions, has been used to describe the dynamic processes in TEA CO2 lasers. Significant improvement in the predicted laser characteristics was obtained, taking into account the CO2 molecules vibrational-rotational transitions. The coupled differential equations were numerically solved by the fourth order Runge–Kutta method. The simulated output characteristics of the laser have been compared with simple vibrational dynamics model and experimental data due to a homemade TEA CO2 laser. It has been found that the best agreement with experimental results regards to the vibrational-rotational dynamic model. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction The CO2 lasers are regarded, for decades, as the most economical coherent radiation source producing high power beams with good quality, suitable for a wide variety of applications. Among them, transversely excited at atmospheric pressure (TEA) CO2 lasers are subjects of particular interest due to their relatively simple construction along with high available intensities and repetition rates. Moreover, distinct rotational-vibrational laser transitions with different frequencies, leading to a discrete output spectrum containing several tens of emission lines, has made the CO2 lasers an exceptional ideal tunable gas molecular laser for experimental [1–3] as well as theoretical studies [3–8]. Many works have been so far done in the field of TEA CO2 lasers dynamic emission simulation, mainly based on population number density equations [9,10] and temperature model such as four, five and six-temperature models [3,6]. Among these, the later (6TM) takes into account almost all the influential vibrational-rotational transitions in active medium molecules and is currently used to describe the kinetic processes of CO2 lasers as a powerful model. However, most of works on CO2 lasers dynamic emission simulation by 6TM employs an empirical electron density distribution equation in discharge plasma column, without essential considerations of the laser electric discharge circuit. In addition, whether 6TM is used or not, the populations of rotational sublevels within each vibrational level are assumed, as a rule, to be continuously in Boltzmann distribution, based on instantaneous rotational relaxation assumption [5,11]. Nevertheless, the rotational processes are always neglected and only the vibrational kinetics is analyzed. The omission of such processes may be the responsible for some well-known contradictions between modeling and experimental results, appeared in literature. This suggestion can be reinforced regarding the reports indicating a rotational relaxation time of R ∼ 0.1 ns for TEA CO2 lasers with pulse durations
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (R. Torabi). http://dx.doi.org/10.1016/j.ijleo.2017.01.091 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
R. Torabi et al. / Optik 135 (2017) 238–243
239
Fig. 1. Energy diagram of the effective V-R levels in CO2 lasers.
in ns order [12] and also, determinative role of rotational coupling effects in gain-switched regime when the laser pulse duration is greater than 100 R [13]. In our previous work [3] on a pin-array UV pre-ionized TEA CO2 laser, temporal behavior of the discharge current pulse was simulated considering electrical properties of the discharge circuit as well as the pre-ionization geometry, together with 6TM for simulation of the laser dynamic emission process. The main discharge region was modeled by a non-linear distributed lumped RLC circuit, and the associated electron densities were calculated by the transmission line method (TLM), resulting in a good agreement with experimental data on laser pulse shape and energy with, of course, some deviations in pulse durations. This model is improved in the present work taking into account the determinative role of rotational dynamic processes. 2. Materials and methods 2.1. Kinetic model Vibrational–Rotational (V-R) energy levels diagram of the CO2 –N2 –CO system, related to typical CO2 lasers with CO2 :N2 :He gas mixture is shown in Fig. 1. The mathematical model of CO2 –N2 –He–CO system for tunable TEA CO2 lasers is established based on two old assumptions [3,9]: 1. All of the vibrational-rotational transitions are homogeneously pressure broadened due to high total pressure of the gas mixture. 2. The couple of (1,0,0) and (0,2,0) lower laser levels acts as a single energy level due to their Fermi resonance. And a new additional assumption [5,11]: 3. Rotational energy is distributed among the rotational levels via relaxations with related time constants. Regarding different exciting and de-exciting energy transfer processes between translational and V-R energy levels of CO2 , N2 and CO molecules (including electron collisions), the following equations can be obtained to describe the variations in the stored energy density (ergs/cm3 ) as a function of time, known as Teller-Landau equations [3]: E˙ 1 = Ne (t)NCO2 W1 fX1 (T ) + WıJ W1 I (t) + (
E˙ 2 = Ne (t)NCO2 W2 fX2 (T ) + (
W1 E3 − E3 (T, T1 , T2 ) W1 E5 − E5 (T, T1 , T2 ) E1 − E1 (T2 ) E1 − E1 (T ) ) ) − − +( W3 W5 3 (T, T1 , T2 ) 10 (T ) 12 (T2 ) 5 (T, T1 , T2 ) (1)
W2 E3 − E3 (T, T1 , T2 ) W2 E5 − E5 (T, T1 , T2 ) E2 − E2 (T ) E1 − E1 (T2 ) ) ) − + +( W3 W5 3 (T, T1 , T2 ) 20 (T ) 12 (T2 ) 5 (T, T1 , T2 )
(2)
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R. Torabi et al. / Optik 135 (2017) 238–243
E˙ 3 = Ne (t) NCO2 W3 fX3 (T ) − WıJ W3 I (t) − E˙ 4 = Ne (t) NN2 W4 X4 (T ) −
E4 − E4 (T3 ) + 43 (T )
E˙ 5 = Ne (t)(1 − f )NCO2 W5 X5 (T ) −
E3 − E3 (T, T1 , T2 ) E4 − E4 (T3 ) + + 3 (T, T1 , T2 ) 43 (T )
W E − E 5 5 (T, T4 ) 4 W5
W E − E 5 5 (T, T3 ) 3 W5
54 (T, T4 )
E5 − E5 (T, T1 , T2 ) E5 − E5 (T, T3 ) E5 − E5 (T, T4 ) − − 5 (T, T1 , T2 ) 53 (T, T3 ) 54 (T, T4 )
53 (T, T3 )
(3) (4) (5)
Eqs. (1)–(3) describe the time evolution of the energy stored in v1 , v2 and v3 vibrational modes of CO2 molecules while, Eqs. (4) and (5) represent the time evolution of energy stored in the vibrational modes of N2 and the CO molecules, respectively [3,5]. In these equations, Ne (t) is the electron density function, NCO2 , NN2 and NHe are the number densities of CO2 , N2 and He in the gas mixture, respectively; f is the non-dissociation ratio of the CO2 molecules; Wi = hi (i = 1, 2, 3, 4, 5) gives the respective energy levels; Xi (i = 1, 2, 3, 4, 5) represents the electron-molecule excitation rates; W is the stimulated emission rate; T is the ambient temperature; T1 , T2 and T3 are respectively the equivalent vibrational temperature of v1 , v2 and v3 vibrational modes of CO2 molecules; T4 and T5 are the equivalent vibrational temperature of N2 and CO molecules, respectively; E, E1 , E2 , E3 , E4 and E5 are stored energies per unit volume (defined by T, T1 , T2 , T3 , T4 and T5 ) and E5 (T, T3 ), E5 (T, T4 ), E3 (T, T1 , T2 ) and E5 (T, T1 , T2 ) are respectively the equilibrium energy for: CO-CO2 (001) collisions, CO-N2 collisions, transferring from v3 to v1 , v2 modes of CO2 molecule and transferring from CO levels to v1 , v2 modes of CO2 molecule [14]. The relaxation rates relevant to different vibrational i and j levels, ij , can be found in [5]. After analysis by synthesis on stimulating radiation, spontaneous emission and losses in the laser cavity, the time evolution of cavity-field intensity I (erg/cm2 s) can be expressed as: d I (I ) = − + ch c dt
ıJ WI + N001 P(J)S h
(6)
Where, c is the photon life time within the resonator [3,5]; ıJ is the population inversion density between rotational sublevels of the upper and lower laser vibrational levels; N001 is the population of the first excited level of v3 mode of CO2 molecules; c is the light velocity; h is the Plank constant; is the laser frequency; P(J) is the ratio of the Jth rotational level population to of the J + 1 level and S = 22 / A sp (d/) is also the spontaneous emission constant in which, sp is the spontaneous emission life time; A is the smallest reflecting area in the laser resonator, is the collisional broadening of the vibrational-rotational sublevels and d is the line width of collisional emission [3,5]. The stimulated emission rate W is defined as: W=
F2 0 sp
(7)
42 h
Where, F = l/L is the filling factor (L and l are the resonator and active medium length respectively). The temporal variation of ıJ is given by [5,11]: d J ıJ − P(J) (ı ) = −2ıJ WI − R dt
(8)
Where, = N001 − N100 is the population inversion density between the upper and lower vibrational laser levels. For a gas mixture at pressure P with RCO2 : RHe : RN2 ratio, the rotational relaxation time is also given by: R −1 = (RCO2 · KCO2 −CO2 + RHe · KCO2 −He + RN2 · KCO2 −N2 ) · P (in s−1
(9) Torr−1 ) can be found in Ref. [15]. Eqs. (1)–(6) and (8) form
For which, rate constants KCO2 −CO2 , KCO2 −He and KCO2 −N2 a system of coupled differential equations that should be numerically solved to obtain laser output characteristics. Besides, as it was mentioned, in contrast to other works in which an empirical Ne (t) equation for electron density distribution is used, here the Ne (t) is calculated based on transmission line method (TLM) (details can be found in our previous report [3]). 2.2. Experimental setup
The model was executed using constructional data of a homemade pin-array UV pre-ionized TEA CO2 laser and the results were also compared with the experimental data obtained via direct measurements on the laser device. The electrical configuration of the laser is schematically shown in Fig. 2. The required high voltage is supplied by the 3-stage Marx generator, triggered by the main spark gap switch. This pulsed voltage is simultaneously fed to the preionization and main discharge capacitors after switching the main spark gap. The laser head consists of two 40 cm long Ernst profiled electrodes with 3.2 cm gap located inside a 80 cm long optical resonator including a total reflectance Cu mirror with 20 m radius of curvature and a %65 reflectance AR coated ZnSe output coupler with 13 cm2 effective area. The laser produces 3cm × 3 cm nearly square shaped pulses at 10P(20) line when is driven with %10CO2 : %10N2 : %80He gas mixture at 1 atm total pressure and 45 kV optimum discharge voltage, generated
R. Torabi et al. / Optik 135 (2017) 238–243
241
Fig. 2. Schematic diagram of the TEA CO2 laser electrical configuration.
Fig. 3. The measured waveforms of discharge current (a) and voltage (c) together with the simulated ones (b) and (d), respectively.
by a homemade 3-stage Marx generator. Lower and higher discharge voltages were respectively involved with energy and glow discharge instabilities. 3. Results and discussion A MATLAB computer program based on Runge-Kutta method has been employed to solve the system of coupled differential Eqs. (1)–(6) and (8) for the five energy densities and light intensity. In order to evaluate the time evolution of the rotational population variation, the Eq. (8) must be solved simultaneously to obtain the population inversion density between rotational sublevels. The initial values chosen for the mentioned differential equations are: Ti (t = 0) = 300K, i = 1,2,3,4,5 I (t = 0) = 30erg/cm2 · s At first, the measured and simulated discharge current and voltage waveforms are typically shown in Fig. 3. The first spike in Fig. 3-a represents the prior pre-ionization discharge [16]. In overall, a good agreement between simulated and measured pulse shapes can be found from the qualitative as well as quantitative points of view.
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Fig. 4. The measured laser pulse shapes for 45 (a) and 35 kV (c) discharge voltages respectively together with the simulated ones (b) and (d) based on different models. The discharge is assumed to be initiated at t = 0.
Table 1 Various characteristic values of the simulated and measured pulse shapes at different discharge voltages. Data Source
Vd (kV)
tb (s)
Ep (J)
p (ns)
Pmax (MW)
Es /Ep a
VRM with Empirical Ne (t) VRM with Empirical Ne (t) VM with Calculated Ne (t) VM with Calculated Ne (t) VRM with Calculated Ne (t) VRM with Calculated Ne (t) Experimental Experimental
45 35 45 35 45 35 45 35
0.9 1.2 0.7 1 0.7 1 0.75 1
6.26 2.1 5.3 1.8 5.17 1.7 5 1.7
56.2 79.2 39 61 47.5 70 110 75
38.14 14 57 15 45 12.7 26 12
0.43 0.70 0.50 0.63 0.54 0.53 0.56 0.46
a b
Calculated by measuring the areas under the pulse spike and tail sections in the simulated and measured pulse shapes [17]. Defined as the time interval between discharge initiation (t = 0 at Fig. 4) and laser pulse starting.
Typical laser pulse shapes for discharge voltages of 45 and 35 kV and CO2 :N2 :He 1:1:8 gas mixture, traced by THORLABS PM100D photon drag and Tektronix TDS 2022B digital oscilloscope are shown in Fig. 4. The associated simulated pulse shapes based on different models such as vibrational model (VM), vibrational-rotational model (VRM) and VRM with empirical electron density equation are also depicted in Fig. 4. Different characteristic values of the measured and simulated (by the above mentioned methods) laser pulses are given in Table 1. This table contains some overall parameters such as onset delay time t and total pulse energy Ep and also some sectional characteristic values related to different parts of the pulse structure such as spike duration p , peak power Pmax and energies ratio of the pulse spike Es to of the total pulse Ep . Since it regards the overall characteristics of the laser pulses, the best agreements between measured and simulated data for both discharge voltages are respectively due to VRM and VM models with calculated Ne (t). On other hand, the worst coincidences are seen between the measured and all simulated pulse durations for 45 kV discharge voltage that are somewhat reflected in the associated peak powers. This disparity that is known as a common problem in such simulation results [4–7] is significantly abated at 35 kV voltage in which, again, the VRM and VM models with calculated Ne (t) respectively show the most reasonable coincidences with experimental data. These results clearly indicate the advantage of taking account the V-R transitions in the simulation procedure.
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4. Conclusion The process of dynamics in TEA CO2 lasers has been simulated by a six-temperature vibrational-rotational model (6TVRM) together with transmission line method (TLM) for electron density distribution calculation. At first, it has been shown that remarkable corrections in the simulated data are available using calculated electron density instead of the prevalent empirical function. Significant improvements have also been obtained in prediction of overall as well as sectional characteristics of the laser pulse shapes at different discharge voltages via taking account the molecular vibrational-rotational transitions in the laser gas mixture. Acknowledgements The authors wish to thank Mr. K. Silakhori from Laser and Optics Research School for his useful discussion and Mr. B. Kazemi and Mr. A. Pezh from Malek-Ashtar University of Technology for their helpful technical support. References [1] N. Menyuk, P.F. Moulton, Development of a high-repetition rate mini-TEA CO2 laser, Rev. Sci. Instrum. 51 (1980) 216–220. [2] I. Chis, A. Ciura, V. Draganescu, D. Dragulinescu, D. Grigorescu, C. Grigoriu, M.L. Pascu, V.G. Velculescu, M. Abrudean, D. Axente, A. Baldea, M. Gligan, Design and performance of a high repetition rate TEA CO2 laser, J. Phys. E: Sci. Instrum. 21 (1988) 393–396. [3] R. Torabi, H. Saghafifar, A.M. Koushki, A.A. Ganjovi, Simulation and initial experiments of a high power pulsed TEA CO2 laser, Phys. Scr. 91 (2016) 015501. [4] K.R. Manes, H.J. Seguin, Analysis of the CO2 TEA laser, J. Appl. Phys. 43 (1972) 5073. [5] K. Smith, R.M. Thomson, Computer Modeling of Gas Lasers, Plenum Press, New York, 1978. [6] B. Abdul Ghani, TEA CO2 laser simulator: a software tool to predict the output pulse characteristics of TEA CO2 laser, Comput. Phys. Commun. 171 (2005) 93–106. [7] A.M. Koushki, S. Jelvani, K. Silakhori, H. Saeedi, Kinetic modeling of a pulsed CO2 laser, Lasers Eng. 21 (2011) 265–280. [8] A.M. Koushki, K. Silakhori, S. Jelvani, Kinetic modeling of a slow flow CW CO2 laser, Opt. Quantum Electron. 43 (2012) 23–33. [9] S. Al-Hawat, K. Al-Mutaib, Numerical modeling of a fast-axial-flow CW-CO2 laser, Opt. Laser Technol. 39 (2007) 610–615. [10] Manoj Kumar, Jai Khare, A.K. Nath, Numerical solution of Boltzmann tranport equation for TEA CO2 laser having nitrogen-lean gas mixtures to predict laser characteristics and gas lifetime, Opt. Laser Technol. 39 (2007) 86–93. [11] Yanchen Qu, Deming Ren, Liii Zhang, Xiaoyong Hu, Fengmei Liu, Five temperature mathematical modeling of TEA CO2 laser, Proc. SPIE 6263 (2006), 62630J-1-7. [12] (a) P.K. Cheo, R.L. Abrams, Appl. Phys. Lett. 14 (1969) 47; (b) R.L. Abrams, P.K. Cheo, Appl. Phys. Lett. 15 (1969) 177. [13] E.A. Ballik, B.K. Garside, J. Reid, Appl. Phys. Lett. 26 (1975) 380. [14] Tie-Jun Wang, Jin-Yue Gao, Qiong-Yi He, Tao Ma, Yun Jiang, Zhi-Hui Kang, Analysis of the dynamics of a mechanical Q-switched CO2 laser: six-temperature model, J. Appl. Phys. 98 (2005) 073102. [15] Ralph R. Jacobs, Kenneth J. Petti piece, Scott J. Thomas, Rotational relaxation rate constants for CO2 , Appl. Phys. Lett. 24 (1974) 375. [16] W.J. Witteman, The CO2 Laser, in: K. Shimoda (Ed.), Springer-Verlag, 1987. [17] K. Silakhori, S. Jelvani, F. Ghanavati, B. Sajad, M. Talebi, M.R. Sadr, A small size 1–3 atm pulsed CO2 laser with series-connected spark-gaps ultraviolet pre-ionization, Rev. Sci. Instrum. 85 (2014) 013109.