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Magnetically confined discharge CO laser
Optics & Laser Technology 33 (2001) 475–478
www.elsevier.com/locate/optlastec
Magnetically con$ned discharge CO laser Tao Wang ∗ , Qingmin Cheng State Key Laboratory of Laser Technology, Huazhong University of Science And Technology, Wuhan, Hubei 430074, People’s Republic of China Received 10 July 2000; received in revised form 9 July 2001; accepted 18 July 2001
Abstract Under the condition of magnetic con$nement, the paper analyzes the following discharge processes from CO gas: (1) the variation of electron energy distribution; (2) the impact excitation rates of electron for the vibrational CO molecules; and (3) the vibrational-state population of CO molecule and small-signal gain. Our results show that, by adding magnetic $eld, the electron density increases greatly in the energy region corresponding to the maximum electron cross sections of CO molecule, while the electron impact on excitation rates also increases. At the same time, the vibrational-state population of CO molecule and the laser small-signal gain of every vibrational level c 2001 Published by Elsevier Science Ltd. are also higher than that without magnetic $eld. Keywords: CO laser; Magnetic con$nement; Small-signal gain
1. Introduction In the recent years, there has been considerable interest in the electric-discharge laser based on the vibrational– rotational transitions from the ground state of carbon monoxide (CO). These lasers have higher quantum conversion e
Corresponding author. Tel.: 86-27-8754-33555; fax: 86-27-8755-6188. E-mail address: [email protected] (T. Wang). c 2001 Published by Elsevier Science Ltd. 0030-3992/01/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 1 ) 0 0 0 6 2 - 7
the force of an electron moved by an external $eld will be * * * * expressed as F = − e( V × B + E ) and the electron-diCuse factors will change in diCerent directions. In order to simplify the case, we will discuss the eCective electric-$eld density, which includes the inDuence of the magnetic $eld on the electric $eld under the condition that collision frequency has no relation with the energy. We shall restrict ourselves to dealing with a weakly ionized gas in which the elastic collisions between electrons and neutral particles play a dominant role. Under the condition of DC discharge, the electron energy obtained from the electric $eld in the unit volume will be * * E2 ne2 2 = 0 EeC ; (1) J · E = Re( ⊥ )E 2 = 2 mvm 1 + !c2 =vm where vm and !c are the electron-collision frequency and the electron-gyro frequency (eB=m), respectively. n is the density of the electron; e is the electron charge; and m is the mass of the electron. EeC is the eCective electric $eld. 0 = ne2 =mvm is the conductivity under the condition of DC discharge without magnetic $eld. The above formula shows that the electron is moved by the force perpendicular to electric and magnetic $elds, so electron velocity in the direction of electric $eld decreases. The energy obtained from the electric $eld decreases because the eCective electric-$eld density decreases (1 + !c2 =v2 )1=2 times. In the weakly ionized gas, the electron-energy distribution is presented with Druyvesteyn distribution function and the
476
T. Wang, Q. Cheng / Optics & Laser Technology 33 (2001) 475–478 -9
0.45 E / N = 2.14 × 10
_20
8.0x10
Vm
2 -9
7.0x10
-9
B=0.3 T
6.0x10 -1
0.35
EXCITATION RATE (cm sec )
0.30
-3
NORMALIZED DISTRIBUTION FUNCTION
0.40
0.25 B=0.1 T
0.20 0.15
B=0 T
-9
5.0x10
-9
4.0x10
B=0.3T
-9
3.0x10
-9
2.0x10
B=0T
-9
1.0x10
0.10
0.0
0.05
2
4
6
8
10
VIBRATIONAL LEVEL (V)
0.00 0
2
4
6
8
10
ELECTRON ENERGY (ev)
Fig. 1. Electron-energy distribution function for a CO–He (532–10640 Pa) laser mixture with magnetic $eld B = 0:0; 0:1 and 0:3 T.
electric $eld is presented with eCective electric $eld. The electron-energy distributions under various magnetic $elds are shown in Fig. 1. The electron-energy distribution function moves to the low-energy area and the electron average energy decreases under the inDuence of the magnetic $eld. At the same time, the distribution function becomes narrow and concentrated. Various electron-energy distributions are obtained by choosing diCerent magnetic $eld densities. The electron impact cross sections of CO molecule have maximum values in the energy region of 1.5 –2:5 eV. By choosing the magnetic-$eld density, the electron density in this energy region could increase greatly, than that without magnetic $eld. The bigger electron-excitation rate could be obtained, and so could the vibrational-state populations of CO molecule and the small-signal gain. 3. The inuence of magnetically conned discharge on electron excitation rate From the solution of electron-energy distribution, the rates at which electrons pump CO molecules from vibrational level v to w; may be obtained from [4] ∞ 1=2 Kv→w = (2=m) duf(u)u1=2 v→w (u); (2) 0
where f is the electron-energy distribution function. v→w (u) is the cross section for the excitation process with an incident electron energy u, which is based on the experimental data of Schulz. In the model of CO-laser discharge, we consider only electron-excitation pumping to and from the ground vibrational level of the CO molecule and the transitions from v ¿ 0 molecules induced by electron collision have been entirely neglected. This approximation can also be supported for the higher vibrational states (v ¿ 8)
Fig. 2. Electron-excitation rates of CO molecules from ground-state v = 0 to vibrational level v under magnetic $elds B = 0:3 and 0 T for E=N value of 2:14 × 10−20 Vm2 .
on theoretical grounds and by observation of the CO-laser performance [5]. So we believe that the v = 0 excitation process and the dominant electron-coupling transition have been included in the model. Fig. 2 shows calculated electron-excitation rates under magnetic $elds B = 0 and 0:3 T. CO molecules are pumped from ground vibrational level (v = 0) to vibrational level v (1–10). In the energy region corresponding to the maximum electron-impact cross sections of CO molecule, electron density increases under the inDuence of magnetic $eld for E=N = 2:14 × 10−20 Vm2 , so that the electron excitation rates of various vibrational levels of CO molecule increase. The electron excitation rates have bigger values, when B is about 0:3 T and they will decrease if the magnetic $eld density has a higher value. 4. The vibrational-state populations of CO molecule and small-signal gain The vibrational-state populations of CO molecule, with and without magnetic $eld, are calculated from the formula [5]. As we see in Fig. 3, there is a long plateau region created in v ≈ 5–30, which decreases only very slowly with the increasing quantum number. This plateau is, de$nitely, the result of the pumping eCect of the V –V process. It can also be noted that, among these v 6 5 states, the population distribution falls oC much more rapidly than a Boltzmann distribution—this is a consequence, primarily, of electron unexcitation and the radiative losses. Finally, for states beyond the plateau region, the population of these states is quite small due to V –T unexcitation and radiative losses. Since magnetic $eld that causes increasing electron excitation rates is added, the vibrational-state populations increase among these v 6 10 states. At the same time, the CO molecules are excited from a lower vibrational state to a
T. Wang, Q. Cheng / Optics & Laser Technology 33 (2001) 475–478 1E14
_4
7.0 x 10
Te=15000 K Pco=532 Pa Phe=3990 Pa 15 -3 ne=2.5*10 m
Te=15000 K Pco=532 Pa Phe=3990 Pa _ ne=2.5*1015 m 3
_4
6.0 x 10
1E13 B=0.3 T
B=0.3 T
_1
MAXIMUM GAIN cm
VIBRATIONAL POPULATION cm-3
477
B=0 T 1E12
_4
5.0 x 10
_4
4.0 x 10
B=0 T 1E11 0
10
20
30
_4
3.0 x 10
VIBRATIONAL QUANTUM NUMBER V
Fig. 3. CO state populations versus vibrational quantum number: eCect of magnetic $eld.
_4
2.0 x 10
0
10
20
30
VIBRATIONAL QUANTUM NUMBER V
higher vibrational state due to the V –V exciting process. The vibrational-state population on every level of CO molecule has a higher value than that without the magnetic $eld. The small-signal gain is calculated using the Dopplerbroadened expression for a single P-branch rotational line [5]: 83 C 4 |Rv; v |2 v J; v J +1 = (J + 1) 3kT (2kT=M )1=2 ×{Bv Nv exp[ − Fv (J )hc=kT ] − Bv Nv exp[ − Fv (J + 1)hc=kT ]}:
(3)
Here, we refer to the transition v; J → v ; J + 1; where v is the vibrational quantum number and J is the rotational quantum number. Positive v J; v J +1 corresponds to gain. We have used the following expressions for the spectroscopic constants: Fv (J ) = Bv J (J + 1) − Dv J 2 (J + 1)2 ; 2 Bv = Be − e v + 12 + !e v + 12 ; Dv = De + Be v + 12 ;
(4)
where the values of De ; Be ; e ; "e , and !e (cm−1 ) are the vibrational and rotational constants. Fig. 4 shows the maximum small-signal gain occurring on each vibrational band for the electron density ne = 2:5 × 1015 m−3 and the gas temperature T = 175 K. From this $gure; we can see that the small-signal gain with magnetic $eld is higher than that without magnetic $eld between vibrational levels v = 7 and 15. These vibrational bands just correspond to the left part of the plateau, where CO molecules have higher inverse populations.
Fig. 4. Maximum small-signal gain occurring between vibrational bands (v → v − 1) versus vibrational quantum number: eCect of magnetic $eld.
5. Conclusion Details of magnetically con$ned CO laser are investigated in this paper. By adding a magnetic $eld perpendicular to the electric $eld, the electron movement can be con$ned and an eCective electric $eld is obtained. In this way, the electron-energy distribution could be adjusted. On the basis of the calculated results presented in Section 4, the paper analyses the following processes: (i) The impact on electron-excitation rates for the vibrational CO molecules is calculated and discussed. It is obvious that the impact on excitation rates from the vibrational level of v = 1–10 increases with the existence of magnetic $eld. (ii) The inDuence of magnetic $eld on vibrational-state populations of CO molecules was investigated. The vibrational-state populations increase under the magnetic $eld. (iii) The small-signal gain is larger for the magnetically con$ned discharge CO laser on these vibrational levels, which may emit laser (v = 7–15). The results from this paper may be signi$cant in investigating other infrared-radiant molecule laser-used part-inverse mechanism.
References [1] Jianguo X, Wang Zh, Wentao J. Radio frequency discharge excited diCusively cooled kilowatt carbon monoxide slab waveguide laser with a three mirror resonator. Appl Phys Lett 1999;75(10):1369–70.
478
T. Wang, Q. Cheng / Optics & Laser Technology 33 (2001) 475–478
[2] Lavrov AV, Volchkova GN, Bungova TA. Numerical modelling of power and spectral characteristics of subsonic discharge CO laser. SPIE 1996;2714:119. [3] Konev Yu B, Kochetov IV, Kurnosov AK, Mirzakarimov BA. A kinetic model of multi-quantum vibrational exchange in CO. J Phys 1994;27(10):2054 –9.
[4] Milonni PM, Paxton AH. Model for the unstable-resonator carbon monoxide electric discharge laser. J Appl Phys 1978;49(3): 1012–27. [5] Rich Joseph W. Kinetic modeling of the high-power carbon monoxide laser. J Appl Phys 1971;42(7):2719–30.
www.elsevier.com/locate/optlastec
Magnetically con$ned discharge CO laser Tao Wang ∗ , Qingmin Cheng State Key Laboratory of Laser Technology, Huazhong University of Science And Technology, Wuhan, Hubei 430074, People’s Republic of China Received 10 July 2000; received in revised form 9 July 2001; accepted 18 July 2001
Abstract Under the condition of magnetic con$nement, the paper analyzes the following discharge processes from CO gas: (1) the variation of electron energy distribution; (2) the impact excitation rates of electron for the vibrational CO molecules; and (3) the vibrational-state population of CO molecule and small-signal gain. Our results show that, by adding magnetic $eld, the electron density increases greatly in the energy region corresponding to the maximum electron cross sections of CO molecule, while the electron impact on excitation rates also increases. At the same time, the vibrational-state population of CO molecule and the laser small-signal gain of every vibrational level c 2001 Published by Elsevier Science Ltd. are also higher than that without magnetic $eld. Keywords: CO laser; Magnetic con$nement; Small-signal gain
1. Introduction In the recent years, there has been considerable interest in the electric-discharge laser based on the vibrational– rotational transitions from the ground state of carbon monoxide (CO). These lasers have higher quantum conversion e
Corresponding author. Tel.: 86-27-8754-33555; fax: 86-27-8755-6188. E-mail address: [email protected] (T. Wang). c 2001 Published by Elsevier Science Ltd. 0030-3992/01/$ - see front matter PII: S 0 0 3 0 - 3 9 9 2 ( 0 1 ) 0 0 0 6 2 - 7
the force of an electron moved by an external $eld will be * * * * expressed as F = − e( V × B + E ) and the electron-diCuse factors will change in diCerent directions. In order to simplify the case, we will discuss the eCective electric-$eld density, which includes the inDuence of the magnetic $eld on the electric $eld under the condition that collision frequency has no relation with the energy. We shall restrict ourselves to dealing with a weakly ionized gas in which the elastic collisions between electrons and neutral particles play a dominant role. Under the condition of DC discharge, the electron energy obtained from the electric $eld in the unit volume will be * * E2 ne2 2 = 0 EeC ; (1) J · E = Re( ⊥ )E 2 = 2 mvm 1 + !c2 =vm where vm and !c are the electron-collision frequency and the electron-gyro frequency (eB=m), respectively. n is the density of the electron; e is the electron charge; and m is the mass of the electron. EeC is the eCective electric $eld. 0 = ne2 =mvm is the conductivity under the condition of DC discharge without magnetic $eld. The above formula shows that the electron is moved by the force perpendicular to electric and magnetic $elds, so electron velocity in the direction of electric $eld decreases. The energy obtained from the electric $eld decreases because the eCective electric-$eld density decreases (1 + !c2 =v2 )1=2 times. In the weakly ionized gas, the electron-energy distribution is presented with Druyvesteyn distribution function and the
476
T. Wang, Q. Cheng / Optics & Laser Technology 33 (2001) 475–478 -9
0.45 E / N = 2.14 × 10
_20
8.0x10
Vm
2 -9
7.0x10
-9
B=0.3 T
6.0x10 -1
0.35
EXCITATION RATE (cm sec )
0.30
-3
NORMALIZED DISTRIBUTION FUNCTION
0.40
0.25 B=0.1 T
0.20 0.15
B=0 T
-9
5.0x10
-9
4.0x10
B=0.3T
-9
3.0x10
-9
2.0x10
B=0T
-9
1.0x10
0.10
0.0
0.05
2
4
6
8
10
VIBRATIONAL LEVEL (V)
0.00 0
2
4
6
8
10
ELECTRON ENERGY (ev)
Fig. 1. Electron-energy distribution function for a CO–He (532–10640 Pa) laser mixture with magnetic $eld B = 0:0; 0:1 and 0:3 T.
electric $eld is presented with eCective electric $eld. The electron-energy distributions under various magnetic $elds are shown in Fig. 1. The electron-energy distribution function moves to the low-energy area and the electron average energy decreases under the inDuence of the magnetic $eld. At the same time, the distribution function becomes narrow and concentrated. Various electron-energy distributions are obtained by choosing diCerent magnetic $eld densities. The electron impact cross sections of CO molecule have maximum values in the energy region of 1.5 –2:5 eV. By choosing the magnetic-$eld density, the electron density in this energy region could increase greatly, than that without magnetic $eld. The bigger electron-excitation rate could be obtained, and so could the vibrational-state populations of CO molecule and the small-signal gain. 3. The inuence of magnetically conned discharge on electron excitation rate From the solution of electron-energy distribution, the rates at which electrons pump CO molecules from vibrational level v to w; may be obtained from [4] ∞ 1=2 Kv→w = (2=m) duf(u)u1=2 v→w (u); (2) 0
where f is the electron-energy distribution function. v→w (u) is the cross section for the excitation process with an incident electron energy u, which is based on the experimental data of Schulz. In the model of CO-laser discharge, we consider only electron-excitation pumping to and from the ground vibrational level of the CO molecule and the transitions from v ¿ 0 molecules induced by electron collision have been entirely neglected. This approximation can also be supported for the higher vibrational states (v ¿ 8)
Fig. 2. Electron-excitation rates of CO molecules from ground-state v = 0 to vibrational level v under magnetic $elds B = 0:3 and 0 T for E=N value of 2:14 × 10−20 Vm2 .
on theoretical grounds and by observation of the CO-laser performance [5]. So we believe that the v = 0 excitation process and the dominant electron-coupling transition have been included in the model. Fig. 2 shows calculated electron-excitation rates under magnetic $elds B = 0 and 0:3 T. CO molecules are pumped from ground vibrational level (v = 0) to vibrational level v (1–10). In the energy region corresponding to the maximum electron-impact cross sections of CO molecule, electron density increases under the inDuence of magnetic $eld for E=N = 2:14 × 10−20 Vm2 , so that the electron excitation rates of various vibrational levels of CO molecule increase. The electron excitation rates have bigger values, when B is about 0:3 T and they will decrease if the magnetic $eld density has a higher value. 4. The vibrational-state populations of CO molecule and small-signal gain The vibrational-state populations of CO molecule, with and without magnetic $eld, are calculated from the formula [5]. As we see in Fig. 3, there is a long plateau region created in v ≈ 5–30, which decreases only very slowly with the increasing quantum number. This plateau is, de$nitely, the result of the pumping eCect of the V –V process. It can also be noted that, among these v 6 5 states, the population distribution falls oC much more rapidly than a Boltzmann distribution—this is a consequence, primarily, of electron unexcitation and the radiative losses. Finally, for states beyond the plateau region, the population of these states is quite small due to V –T unexcitation and radiative losses. Since magnetic $eld that causes increasing electron excitation rates is added, the vibrational-state populations increase among these v 6 10 states. At the same time, the CO molecules are excited from a lower vibrational state to a
T. Wang, Q. Cheng / Optics & Laser Technology 33 (2001) 475–478 1E14
_4
7.0 x 10
Te=15000 K Pco=532 Pa Phe=3990 Pa 15 -3 ne=2.5*10 m
Te=15000 K Pco=532 Pa Phe=3990 Pa _ ne=2.5*1015 m 3
_4
6.0 x 10
1E13 B=0.3 T
B=0.3 T
_1
MAXIMUM GAIN cm
VIBRATIONAL POPULATION cm-3
477
B=0 T 1E12
_4
5.0 x 10
_4
4.0 x 10
B=0 T 1E11 0
10
20
30
_4
3.0 x 10
VIBRATIONAL QUANTUM NUMBER V
Fig. 3. CO state populations versus vibrational quantum number: eCect of magnetic $eld.
_4
2.0 x 10
0
10
20
30
VIBRATIONAL QUANTUM NUMBER V
higher vibrational state due to the V –V exciting process. The vibrational-state population on every level of CO molecule has a higher value than that without the magnetic $eld. The small-signal gain is calculated using the Dopplerbroadened expression for a single P-branch rotational line [5]: 83 C 4 |Rv; v |2 v J; v J +1 = (J + 1) 3kT (2kT=M )1=2 ×{Bv Nv exp[ − Fv (J )hc=kT ] − Bv Nv exp[ − Fv (J + 1)hc=kT ]}:
(3)
Here, we refer to the transition v; J → v ; J + 1; where v is the vibrational quantum number and J is the rotational quantum number. Positive v J; v J +1 corresponds to gain. We have used the following expressions for the spectroscopic constants: Fv (J ) = Bv J (J + 1) − Dv J 2 (J + 1)2 ; 2 Bv = Be − e v + 12 + !e v + 12 ; Dv = De + Be v + 12 ;
(4)
where the values of De ; Be ; e ; "e , and !e (cm−1 ) are the vibrational and rotational constants. Fig. 4 shows the maximum small-signal gain occurring on each vibrational band for the electron density ne = 2:5 × 1015 m−3 and the gas temperature T = 175 K. From this $gure; we can see that the small-signal gain with magnetic $eld is higher than that without magnetic $eld between vibrational levels v = 7 and 15. These vibrational bands just correspond to the left part of the plateau, where CO molecules have higher inverse populations.
Fig. 4. Maximum small-signal gain occurring between vibrational bands (v → v − 1) versus vibrational quantum number: eCect of magnetic $eld.
5. Conclusion Details of magnetically con$ned CO laser are investigated in this paper. By adding a magnetic $eld perpendicular to the electric $eld, the electron movement can be con$ned and an eCective electric $eld is obtained. In this way, the electron-energy distribution could be adjusted. On the basis of the calculated results presented in Section 4, the paper analyses the following processes: (i) The impact on electron-excitation rates for the vibrational CO molecules is calculated and discussed. It is obvious that the impact on excitation rates from the vibrational level of v = 1–10 increases with the existence of magnetic $eld. (ii) The inDuence of magnetic $eld on vibrational-state populations of CO molecules was investigated. The vibrational-state populations increase under the magnetic $eld. (iii) The small-signal gain is larger for the magnetically con$ned discharge CO laser on these vibrational levels, which may emit laser (v = 7–15). The results from this paper may be signi$cant in investigating other infrared-radiant molecule laser-used part-inverse mechanism.
References [1] Jianguo X, Wang Zh, Wentao J. Radio frequency discharge excited diCusively cooled kilowatt carbon monoxide slab waveguide laser with a three mirror resonator. Appl Phys Lett 1999;75(10):1369–70.
478
T. Wang, Q. Cheng / Optics & Laser Technology 33 (2001) 475–478
[2] Lavrov AV, Volchkova GN, Bungova TA. Numerical modelling of power and spectral characteristics of subsonic discharge CO laser. SPIE 1996;2714:119. [3] Konev Yu B, Kochetov IV, Kurnosov AK, Mirzakarimov BA. A kinetic model of multi-quantum vibrational exchange in CO. J Phys 1994;27(10):2054 –9.
[4] Milonni PM, Paxton AH. Model for the unstable-resonator carbon monoxide electric discharge laser. J Appl Phys 1978;49(3): 1012–27. [5] Rich Joseph W. Kinetic modeling of the high-power carbon monoxide laser. J Appl Phys 1971;42(7):2719–30.