THEORY OF A CW SUPERSONIC NITROGEN RECOMBINATION LASER

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THEORY OF A CW SUPERSONIC NITROGEN

RECOMBINATION LASER

W. Schall

To cite this version:

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JOURNAL DE PHYSIQUE CoZZoque C9, suppZ6ment a u n O l l , Tome 41, novembre 1980, page C9-471

THEORY OF A CW SUPERSONIC NITROGEN RECOMBINATION LASER

W. Schall.

I n s t i t u t fiir T e c h i s c h e P h y s i k , DFVLR, 0-7000 S t u t t g a r t 80, Fed. Rep. Germany.

Abstract.- Quasi-two dimensional model calculations show the possibility to realize a cw atomic ni- trogen laser, pumped by the recombination process in rapidly expanded plasma flows. Sufficient gain for lasing is demonstrated for stagnation conditions of 13 000 ,< To< 24 000 K and 2

<

p o 6 30 kPa. Location and size of the gain regimes in the flow direction depend on the plasma starting condition. At a wavelength of 0.91 um values of small signal gain up to 0.6 m-l can be attained. The model is described and some selected results are presented. The effect of the electron heat conduction is dis cussed.

1 . I n t r o d u c t i o n

The i d e a t o pump a l a s e r by a r a p i d recom- combination o f expanding plasmas, some b i n a t i o n o f e l e c t r o n s i n t o t h e h i g h l y ex- y i e l d i n g v e r y promising r e s u l t s . (A more c i t e d e l e c t r o n i c s t a t e s of a n e u t r a l o r r e c e n t c o m p i l a t i o n may b e found i n Ref. 6 ) . i o n i c s p e c i e s i s r a t h e r o l d . ' Meanwhile t h e Also many e x p e r i m e n t s have been performed mechanism o f many p u l s e d l a s e r s h a s been and t h e e x i s t e n c e o f p o p u l a t i o n i n v e r s i o n s

e x p l a i n e d by t h i s p r o c e s s . The s u g g e s t i o n has been proven r e p e a t e d l y . But o n l y Camp- t o o p e r a t e a r e c o m b i n a t i o n l a s e r by ex-

p l o i t i n g t h e r a p i d c o o l i n g r a t e s , and hence r e c o m b i n a t i o n r a t e s , i n a s u p e r s o n i c plasma expansion flow h a s been made n e a r l y a s l o n g Though b a s i n g on a d i f f e r e n t p h y s i - c a l p r i n c i p l e , i t h a s always been t e m p t i n g t o r e g a r d such a plasmadynamic l a s e r a s t h e n a t u r a l e x t e n s i o n o f t h e powerful i n f r a r e d gasdynamic l a s e r s i n t o t h e v i s i b l e o r even u l t r a v i o l e t wavelength r e g i o n . S i n c e recom- b i n a t i o n i s a g e n e r a l p h y s i c a l mechanism t o p r e f e r e n t i a l l y p o p u l a t e h i g h l y i n g e l e c - t r o n i c l e v e l s , t h e method s h o u l d be a p p l i - c a b l e t o pump l a s e r t r a n s i t i o n s i n a l m o s t any i o n i z a b l e element o r molecule and t h u s open a wide spectrum o f l a s e r wavelengths. Reviews and more d e t a i l e d d i s c u s s i o n s may b e found i n Refs. 4 and 5. Q u i t e a few t h e - o r e t i c a l models have been developed t o d e s - c r i b e t h e p o p u l a t i o n dynamics by t h e r e -

b e l l e t a l .

'

have been a b l e t o a c t u a l l y de- m o n s t r a t e 1 m s l a s e r a c t i o n i n a n a r g p n plasma, expanded from a p u l s e d MPD a r c s o u r c e . However, i n an a t t e q p t by Hoffmann and ~ i i ~ e l ~ t o e x t e n d t h e s e r e s u l t s t o cw o p e r a t i o n a t e q u a l plasma p a r a m e t e r s ab- s o r p t i o n was found i n t h e l a s e r t r a n s i - t i o n s . Today i t i s n o t known which s t e a d y s t a t e p r o c e s s e s p r e v e n t t h e r e c o m b i n a t i o n l a s e r a c t i o n dnd t o what e x t e n t e x i s t i n g models do n o t d e s c r i b e t h e r e a l p h y s i c s a c c u r a t e l y enough. I n t h i s paper a f u r t h e r t h e o r e t i c a l model i s d e s c r i b e d and some c a l c u l a t e d r e s u l t s a r e p r e s e n t e d f o r a n i t r o g e n plasma a s a p o t e n t i a l l a s e r c a n d i d a t e . The model con- s i s t s o f two main p a r t s t o compute t h e r e - combination and e x c i t a t i o n k i n e t i c s simul- t a n e o u s l y w i t h t h e gasdynamics a l o n g t h e

31

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~ 9 - 4 7 2 JOURNAL DE PHYSIQUE

axis of a two-dimensional plane nozzle flow and is qualitatively described in the next sections. The numerical details will be de-

scribed elsewhere. 9

2. Theory

2.1 Gasdynamics

For a maximum recombination rate, an expansion as fast as possible is desired, being inherently more-dimensional. To con- siderably reduce the computational needs for a complete two-dimensional treatment and yet take account for such a geometry within realistically specified boundary conditions, the following compromise has been adopted.

The sharp corner nozzle is known to produce the fastest nozzle expansion possible. Flow parameters, i.e. effective area ratios and Mach numbers can be infered at any point of the flow field for an isentropic expansion by the method of characteristics. A streamline can then be considered as a one-dimensional streamtube with completely defined boundary conditions. To find the effect of the non-isentropic nature of the flow, brought about by the recombination reaction, the full set of one-dimensional flow equations for the macroscopic state variables is then integrated along the streamtube. As a particular and simple streamtube the flow along the axis of the nozzle is chosen for the present calculations. In the method of characteristics the isentropy of the flow is expressed by

the ratio of the specific heats J"= Cp/Cv.

After integrating the full equations a

variable

8

can be assigned to any point of

the flow. With this distribution of local- ly adjusted isentropic coefficients the calculation of the characteristics can be redone to yield a new set of boundary conditions. In general, convergence is reached within less than ten iterations.

Following Appleton and

a ray'^

the flow is

represented by a three component, quasi- neutral plasma consisting of neutral particles and of singly-ionized ions and electrons, that can recombine. Maxwellian velocity distributions are assumed. The electrons are allowed to have a temperature different from the heavy particles, but do flow at the same velocity. Hence, the set of conservation equations comprises an equation for the conservation of momentum, for total energy, electron energy, total mass and number density of the electrons ne, from which velocity w, heavy particle temperature T, electron temperature Te, number density n are calculated. Further conservation equations for atoms in the various excited states can be added as is considered necessary. In particular, equations for the three ground states of the chosen species nitrogen are provided.

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t r i b u t i o n i s needed. Again, i t i s o b t a i n e d t h r e e Zp3 ground s t a t e s t h e t r a n s i t i o n pro- i n t h e s u c c e s s i v e c o u r s e o f i t e r a t i o n s . b a b i l i t i e s a r e t a k e n from Smith, e t . a l . 13 The c a l c u l a t i o n i s s t a r t e d w i t h a s t e a d y The p o p u l a t i o n d e n s i t y ni o f e v e r y e n e r g y s t a t e s o l u t i o n o f a l l c o n s e r v a t i o n equa- l e v e l h i g h e r t h a n t h o s e o f t h e ground t i o n s a t t h e p o i n t of s i n g u l a r i t y , i . e . a t s t a t e s i s computed from t h e a l g e b r a i c s y s - t h e t h r o a t o f t h e n o z z l e . E q u a l i t y o f e l e c - tem o f t h e s t e a d y s t a t e r a t e e q u a t i o n s t r o n - and heavy p a r t i c l e t e m p e r a t u r e i s a s - ( A i = 0). Because of t h e s h o r t c h a r a c t e r - sumed a t t h i s p o i n t , b u t i s n o t mandatory. i s t i c t i m e s o f t h e l e v e l s t o a c q u i r e t h e The c a l c u l a t i o n i s performed up t o an a r - s t e a d y s t a t e v a l u e s t h i s a p p r o x i m a t i o n b i t r a r y Mach number, t y p i c a l l y o f t h e o r d e r p r o v e s t o b e j u s t i f i e d everywhere i n t h e o f 5 t o 6 , w i t h t h e d i s t a n c e from t h e e x p a n s i o n , e x c e p t f o r t h e v e r y f i r s t s t e p s t h r o a t , x , b e i n g t h e i n d e p e n d e n t v a r i a b l e . o f computation. T h i s i s n o t t h e c a s e f o r 2.2 E x c i t a t i o n k i n e t i c s A s t h e e l e m e n t o f i n v e s t i g a t i o n r e p o r t e d h e r e i n a t o m i c n i t r o g e n h a s been chosen, b u t o t h e r e l e m e n t s l i k e c a r b o n , oxygen e t c . have been o r may b e computed a s w e l l . 38 l e v e l s f o r e l e c t r o n i c e x c i t a t i o n a r e con- s i d e r e d i n t h e model, o u t o f which 24 a r e t r e a t e d i n s e p a r a t e e q u a t i o n s . The k i n e t i c of r e c o m b i n a t i o n i s t r e a t e d by t h e c o l l i - s o n a l - r a d i a t i v e decay mechanism, o r i g i n a l l y 11 i n t r o d u c e d by S a t e s e t . a l . : The r a t e e q u a t i o n s a c c o u n t f o r two and t h r e e body r e c o m b i n a t i o n ( i o n i z a t i o n ) i n t o a n a r b i - t r a r y l e v e l i , r a d i a t i v e d e e x c i t a t i o n , and t r a n s i t i o n s induced by e l e c t r o n c o l l i s i o n s . Re-absorption o f r e s o n a n c e r a d i a t i o n i s t h e t h r e e ground s t a t e s w i t h e n e r g i e s of 2 0,eV ( 4 ~ ) , 2.38 eV ( 2 ~ ) and 3.75 eV ( P). The number d e n s i t i e s o f t h e s e l e v e l s n l , n 2 , n3, t o g e t h e r w i t h t h e number d e n s i t y and t e m p e r a t u r e o f t h e e l e c t r o n s , ne and T,, e n t e r t h e s y s t e m o f r a t e e q u a t i o n s a s t h e p r i m a r y , i n d e p e n d e n t v a r i a b l e s . The system c a n b e s o l v e d a s a s e p a r a t e u n i t t o compute r e g i m e s o f p o s s i b l e i n v e r s i o n s i n f u l l g e n e r a l i t y o r , more i n t e r e s t i n g l y , f u l l y c o u p l e d t o t h e i n t e g r a t i o n of t h e f l o w e q u a t i o n s . I n t h e l a t t e r c a s e l e v e l p o p u l a t i o n and p o p u l a t i o n i n v e r s i o n s a t any s t e p of i n t e g r a t i o n a l o n g t h e n o z z l e a x i s c a n be i n f e r e d w i t h r e g a r d t o l o c a - t i o n , s i z e and magnitude o f t h e s m a l l s i g - n a l g a i n . t a k e n i n t o a c c o u n t by m u l t i p l y i n g t h e spon- t a n e o u s t r a n s i t i o n p r o b a b i l i t i e s w i t h an F i g . 1 shows a s e c t i o n o f t h e G r o t r i a n d i - escape p r o b a b i l i t y f a c t o r f o r t h e agram the nitrogen d u b l e t w i t h t r a n s i - c a l c u l a t e d from t h e l o c a l o p t i c a l t h i c k n e s s . tions expected to significant

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JOURNAL DE PHYSIQUE

Fig.1 Section of the electronic level

structure of dublett nitrogen with laser transitions indicated.

3. Numerical results

Calculations were performed for a plane,

sharp-edge nozzle with 1 mm throat height.

The parameter range for the stagnation temperature and pressure of the starting con-

ditions were 13 333

"

To" 24 000 K and

2 r4 p o L 30 kPa. This comprises a degree of

ionization from 0 . 0 5 to 0 . 9 9 .

Fig.2 Variation of the quasi-isentropic

coefficient

8

along the flow direc-

tion for various stagnation conditions at the starting point.

Fig.2 demonstrates the non-isentropic nature of the flow, expressed by the variation of the effective isentropic coeffi- cientf along the flow direction. The local $is recalculated from the actual local Mach number and the heavy particle temperature. It is seen that due to the different heat release from the recombination at var-

ious starting conditions

8

drops rapidly to

values of 1.3 to 1 . 5 after a flow distance

of 1 cm. Fig. 3 shows, how the recombina-

tion of electrons in the strongly non-equilibrium flow tends to freeze the population density of an arbitrary level placed at

1 2 eV compared to its equilibrium density. Differences in the population or depopula- tion mechanisms of two levels, connected by an allowed transition can lead to a tempo- rary inversion of the population densities.

Fig.3 Population density variation of an

arbitrary level placed at 12 eV by a

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d e n s i t y n , , t h a t have t o be a t t a i n e d s i m u l - t a n e o u s l y d u r i n g t h e expansion. Fig.4 shows t h e s e r a n g e s , v a l i d f o r t h e most prominent t r a n s i t i o n s 3sI2D

-

3 p I 2 ~ O a t 918.8 nm and lo" I I I 0 a2 0.4 0.6 a8 10 1.2 1.4 r.6 Te [CV] F i g . 5 Change o f s t a t e of t h e e l e c t r o n s d u r i n g e x p a n s i o n . examples o f t h e e n s u i n g a x i a l g a i n d i s t r i b u - t i o n a l o n g t h e f l o w d i r e c t i o n a r e d i s p l a y e d i n F i g s . 6 and 7. Depending on t h e p r e s s u r e , t h e g a i n t e n d s t o a p p e a r and a t t a i n i t s Fig.4 Ranges o f s m a l l s i g n a l g a i n f o r t h e

3 p ~ - 3 s ~ transition (steady state ap- maximum r a t h e r a b r u p t l y and t h e n t o f a d e p r o x i m a t i o n f o r l e v e l s i > l ) .

away a s t h e g a s t h i n s o u t i n t h e c o n t i n h e d 3 s ' 'D

-

3p1 2 ~ 0 a t 904.8 nm. The c u r v e s i n - _expansion. _In _a_practical _application

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C9-476 JOURNAL DE PHYSIQUE

though, a f u r t h e r e x p a n s i o n of t h e g a s m a t e l y 0.6 m - I - A s e x p e c t e d , . o t h e r t r a n s i - seems n o t a p p r o p r i a t e a f t e r t h e maximum o f t i o n s e x h i b i t g a i n a s w e l l and w i t h t h e g a i n i s reached. The g a i n r e g i o n i s s h i f t e d same g e n e r a l b e h a v i o r . However, t h e p a r a - downstream a s t h e p r e s s u r e i s r a i s e d , w h i l e meter r a n g e , f o r more t h a n 0 . 1 m-' i s n a r - t h e g a i n maximum r i s e s w i t h i n c r e a s i n g tem- rower and t h e r e g i o n s a r e somewhat s h i f t e d

0 1 I 1 2 16 20 2 L 1 0 4 MKI F i g . 8 V a r i a t i o n o f maximum s m a l l s i g n a l g a i n w i t h t h e s t a r t i n g c o n d i t i o n s . p e r a t u r e . A s r e c o g n i z e d from t h e s e g r a p h s and a l s o from F i g . 8 , where t h e v a r i a t i o n o f t h e g a i n maximum i s p l o t t e d a s a func- t i o n o f t h e s t a r t i n g c o n d i t i o n s , t h i s r a - t h e r g e n e r a l b e h a v i q r i s n o t v a l i d f o r ex- t r e m e l y low s t a r t i n g t e m p e r a t u r e s . The f u r - , t h e r and c o n s i d e r a b l e r i s e o f g a i n a t t h e s e c o n d i t i o n s may be e x p l a i n e d from t h e expan- s i o n c u r v e s i n F i g . 5: t h e a l r e a d y low s t a r t i n g t e m p e r a t u r e and t h e a s s o c i a t e d s m a l l d e g r e e o f i o n i z a t i o n , w i t h t h e s m a l l amount o f a v a i l a b l e i o n i z a t i o n e n e r g y a l - lows t h e e a r l i e r achievement o f a somewhat s m a l l e r e l e c t r o n t e m p e r a t u r e a t com- p a r a t i v l y h i g h e r e l e c t r o n d e n s i t i e s , and hence a d e e p e r p e n e t r a t i o n i n t o t h e i n v e r - s i o n regime o c c u r s . The maximum s m a l l s i g n a l g a i n f o r t h i s p a t r t i c u l a r t r a n s i t i o n amounts t o a p p r o x i -

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about 1 % after 2 mm. Since this occurs well before the gain attains its maximum, the effect will be little, if present at all. In fact, because of the reduced recombination rate at the higher electron temperature the electron state distribution along the flow direction remains very near- ly unchanged.

4. Experiment

An effort has been made to verify the results in an experiment and to achieve laser

oscillation. For this purpose the 5 cathode

arc heater, described in Ref. 15 has been operated with pure nitrogen. The results have been negative. In contrast to earlier findings with a plasma of predominantly ar- gon, the flow became extremely inhomogene- ous in temperature in the optical direction. Small regions with very hot plasma interchanged with broad regions of cold gas.

5. Conclusions

A computer model has been established to

describe the non-equilibrium plasma expansion in a two-dimensional nozzle. Suffi- cient small signal gain has been found in the computation of a nitrogen model, that should allow the achievement of cw laser

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oscillation even in a smaller device. An experimental test was not successful, be-

cause the plasma source did not produce