GAIN CONDITION FOR LONG-LASER-PULSE PRODUCED PLASMA

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GAIN CONDITION FOR LONG-LASER-PULSE PRODUCED PLASMA

W. Brunner, Th. Schlegel

To cite this version:

W. Brunner, Th. Schlegel. GAIN CONDITION FOR LONG-LASER-PULSE PRO- DUCED PLASMA. Journal de Physique Colloques, 1986, 47 (C6), pp.C6-99-C6-108.

�10.1051/jphyscol:1986613�. �jpa-00225856�

JOURNAL DE PHYSIQUE,

Colloque C6, suppl6ment au n o 10, Tome 47, octobre 1986

GAIN CONDITION FOR LONG-LASER-PULSE PRODUCED PLASMA

W. BRUNNER and Th. SCHLEGEL

Zentralinstitut fiir Optik und Spektroskopie der Akademie der Wissenschaften der DDR, Rudower Chaussee 6 , DDR-1199 Berlin, D.R.G.

A b s t r a c t :

The i n v e r s i o n c o n d i t i o n f o r laser-produced plasmas i s analysed t h e o r e t i c a l l y f o r carbon f i b r e t a r g e t s i n r e l a t i o n t o the i n t r i n s i c i n t e n s i t y and d u r a t i o n o f h e a t i n g l a s e r pulses.

The s p a t i a l and temporal dependence o f t h e g a i n f o r t h e 3-2 and 4- 3 t r a n s i t i o n s i n c l u d i n g r a d i a t i o n t r a p p i n g e f f e c t s w i l l be discussed.

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

The t h e o r e t i c a l p r e d i c t i o n o f t h e p o s s i b i l i t y t o produce a m p l i f i - c a t i o n o f XUV o r s o f t X-ray r a d i a t i o n i n laser-produced plasmas [I]

s t i m u l a t e d i n t e n s e e f f o r t s i n laser-plasma i n t e r a c t i o n r e s e a r c h f o r over t h e l a s t decade. C a l c u l a t i o n s [2-41 have shown t h a t under a p p r o p r i a t e plasma c o n d i t i o n s l a r g e g a i n s and a m p l i f i c a t i o n can be' achieved. Recently, s c o n c l u s i v e demonstration o f such an a m p l i f i - c a t i o n conducted a t t h e Novette l a s e r - t a r g e t i r r a d i a t i o n f a c i l i t y was r e p o r t e d on [5,6J. P o p u l a t i o n i n v e r s i o n s i n n e o n l i k e i o n s obtained i n c i t e d above c a l c u l a t i o n s and experiments were e x p l a i n e d by use of a l a s e r - d r i v e n electron-collisional-excitation scheme.

Other pump mechanisms[7] a r e charge-exchange 8 1 , i n t e r m e d i a t e o p t i c a l pumping [ 9 ] o r t h e recombination scheme 110-14

5 .

I n t h i s paper we present some t h e o r e t i c a l r e s u l t s concerning t h e g e n e r a t i o n o f p o p u l a t i o n i n v e r s i o n s i n laser-produced plasmas through recombination pumping. A one-dimensional gasdynamic code w i t h n o n l i n e a r heat conduction s i m u l a t e s t h e i n t e r a c t i o n o f l a s e r l i g h t (1,- = 1.06 ,urn, IL = 1011

-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986613

JOURNAL DE PHYSIQUE

2. Gas dynamics and heat f l o w

The plasma i s assumed t o c o n s i s t o f a c h a r g e - n e u t r a l m i x t u r e o f e l e c t r o n s ( e ) and v a r i o u s species o f carbon i o n s ( i ) .

E l e c t r o n s w i l l be t r e a t e d as one subsystem, w i t h i n t e r n a l energy &,

per u n i t mass, temperature Te, pressure p and so on. The i o n s form the second subsystem w i t h a correspo8ding s e t o f thermodynamic v a r i a b l e s i n c l u d i n g s p e c i f i c volume V = 1/9 = l/Nimi (Ni

-

d e n s i t y , m i

-

Besides work o f t h e summary pressure 3v the e q u a t i o n o f the -P 3t

e l e c t r o n i n t e r n a l energy c o n t a i n s e l e c t r o n heat f l o w

( w i t h c o e f f i c i e n t o f the classical e l e c t r o n heat conduction x.= %T%)~

exchange o f energy between i o n s and e l e c t r o n s (aei

-

-

-

I n our c a l c u l a t i o n s o n l y a b s o r p t i o n by t h e i n v e r s e Bremsstrahlung mechanism was taken i n t o c o n s i d e r a t i o n . Long i n t e n s e l a s e r pulses (tL- i o n s , EL * 5

-

I n d i f f e r e n c e t o t h e MEDUSA code we s o l v e t h e gas-dynamic equations by means o f a complete i m p l i c i t d i f f e r e n c e scheme on an inhomo- geneous l a g r a n g i a n mesh 1161. I t i s a complete c o n s e r v a t i v e and

F i g . 1 F i g . 2

P l a s m a d e n s i t y N a s f u n c t i o n P l a s m a t e m p e r a t u r e Te a s f u n c - o f t h e d i s t a n c e f r o m t h e t a r - t i o n o f t h e d i s t a n c e f r o m t h e g e t a x i s f o r s e v e r a l times t a r g e t a x i s f o r s e v e r a l times

,,---s*.

c",,ndr,rol r.r- on1.R.t

100 WOO 1)OW

R lpml -*

F i g . 3 F i g . 4

P l a s m a d e n s i t y N a n d t e m p e r a - A s i n F i g . 3 , l a s e r i n t e n s i t y t u r e T, a s f u n c t i o n o f t h e IL = 4 . 0 . 1 0 1 2 W

d i s t a n c e f r o m t a r g e t a x i s f o r cm 2 d i f f e r e n t t i m e s i n t h e i n t e r -

e s t i n g s p a t i a l r e g i o n ; l a s e r i n t e n s i t y 1,=1.7*1013 g

cm 2

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F i g . 5

Temporal b e h a v i o u r o f t h e d e n s i t y f o r d i f f e r e n t d i s t a n c e s f r o m t h e t a r g e t f o r l a s e r i n t e n s i t i e s 1 ,-=6,7.1013 \V/cm2 and

1.7 1013 \l/cm2

F i g . 6

T e m p o r a l b e h a v i o u r o f t h e tempe- r a t u r e f o r d i f f e r e n t d i s t a n c e s f r o m t h e t a r g e t f o r l a s e r i n t e n - s i t i e s fL=6 ,7.1013~/cm2 and 1 . 7 . 1 0 ~ ~ by/cm2

F i g . 7

R e l a t i v e p o p u l a t i o n d e n s i t y of t h e pump l e v e l ,

p4=

N a s a f u n c t i o n o f t h k d e n s i t y and t h e t e m p e r a t u r e Te

-owing t o the i n t r o d u c t e d a r t i f i c i a l v i s c o u s i t y - a homogeneous scheme[17]. Complete c o n s e r v a t i o n means t h a t n o t o n l y c o n s e r v a t i o n laws f o r mass, impuls and f u l l energy b u t a l s o a l o t o f o t h e r impor- t a n t p h y s i c a l r e l a t i o n s l i k e equations f o r i n t e r n a l and k i n e t i c energies o r t h e law o f volume change a r e preserved i n t h e space o f mesh f u n c t i o n s ,

Some t y p i c a l r e s u l t s o f t h e n u m e r i c a l c a l c u l a t i o n s a r e presented i n Figs. 1-4. I n a d d i t i o n t o s p a t i a l d i s t r i b u t i o n s o f t h e i o n d e n s i t y and e l e c t r o n temperature f o r d i f f e r e n t times t h e temporal behaviour o f these plasma parameters f o r d e f i n e d d i s t a n c e s from the t a r g e t a x i s i s shown i n Figs. 5-6. I t can be seen t h a t f o r l o n g e r l a s e r p u l s e s (some ns) t h e d e n s i t y d i s t r i b u t i o n -as i s w e l l known-

s t r o n g l y decreases w i t h i n c r e a s i n g d i s t a n c e from the t a r g e t . Compa- r e d t o i t , t h e temperature r i s e s near t h e c r i t i c a l i n t e r f a c e , where t h e maximum o f l a s e r energy w i l l be deposited i n t h e plasma, and h o l d s approximately constant w i t h growing distance. When t h e l a s e r h e a t i n g breaks o f f , t h e temperature begins t o f a l l o f f i n t h e h i g h - d e n s i t y r e g i o n o f plasma caused by the remaining s t r o n g heat flow.

Concerning t h e temporal behaviour o f plasma p r o p e r t i e s d u r i n g t h e f r e e expansion regime, we can conclude from Figs. 1-4 t h a t i n r e g i o n s n o t so f a r from t h e t a r g e t a x l s ( 4 1 mm) t h e d e n s i t y keeps constant i n time, whereas t h e temperature r a p i d e l y decreases.

3. Gain c a l c u l a t i o n s

The g a i n c o e f f i c i e n t concerning t h e t r a n s i t i o n q

-

where A i s t h e spontaneous emission r a t e and $ t h e t r a n s i t i o n

9 P qP

frequency. The f a c t o r g ( 3

-

Therefore, t h e g a i n c o e f f i c i e n t f o r resonant t r a n s i t i o n becomes

An a n a l y t i c a l c a l c u l a t i o n o f t h e p o p u l a t i o n d e n s i t e s N (q=1,2,3), assuming a s i m p l i f i e d f o u r - l e v e l system and q u a s i s t a t i o n a r i t y g i v e s q

C6-104 JOURNAL DE PHYSIQUE

where w.& denote c o l l i s i o n a l e x c i t a t i o n and d e e x c i t a t i o n r a t e s , Z , -

i s t h e t;ansition t i m e from t h e hydrogen

-

-

s t a t e ,

I t can be seen immediately t h a t t h e g a i n f o r t r a n s i t i o n s 3-2 and 4 -3 i s p r o p o r t i o n a l t o t h e populat_ion d e n s i t y N, o f t h e pump

7

l e v e l and the i n v e r s i o n f a c t o r (1

-

f o r c o l l i s i o n a l and r a d i a t i v e procesges g i v e n e x p l i c i t e l y as func- t i o n s o f t h e plasma s t a t e (N,Te,Z) we o b t a i n , f o r example f o r N3:

Then, f o r the i n t e r e s t i n g g a i n f a c t o r s we can w r i t e

f o r t h e t r a n s i t i o n 3-2 and

f o r the t r a n s i t i o n 4 -3.

The term

describes the e f f e c t o f t h e Ld - a b s o r p t i o n (L

-

fl = N1/N - r e l a t i v e p o p u l a t i o n d e n s i t y o f the ground l e v e l ) .

Carbon 3-2

1 ~ - l . 7 x l o ~ ~ ~ ~

Carbon 3-2

n.2mo #.

15 20 25

tlnsi -+

F i g . 8 F i g . 9

Temporal b e h a v i o u r o f t h e g a i n As i n F i g . 8, t h e l a s e r c o e f f i c i e n t f o r t h e t r a n s i t i o n i n t e n s i t y i s 4.0*1012 w/cm2 3-2 a t d i f f e r e n t d i s t a n c e s

from t h e t a r e t . Laser i n t e n s i t y IL 5 1 , 7 ' 1 0 l9 w/cm2

Carbon 4-3 1

-

Carbon 4-3

I , . 4 . 0 ~ 1 0 ' ~ ~ tm

F i g . 10

Temporal b e h a v i o v r o f t h e g a i n f o r t h e 4-3 t r a n s i t i o n a t d i f f e r e n t d i s t a n c e s from t h e t a r g e t . L a s e r i n t e n s i t y

IL = 1,791013 w/cm2

t l n s l +

F i g . 11

As i n F i g . 1 0 , laser i n t e n s i t y i s 4.0.1012 w/cm2

C6-106 JOURNAL DE PHYSIQUE

The r e l a t i v e p o p u l a t i o n d e n s i t y o f t h e pump l e v e l i n our s i m p l i f i e d model

was c a l c u l a t e d n u m e r i c a l l y s o l v i n g the f u l l system of r a t e equations f o r p o p u l a t i o n d e n s i t i e s i n t h e q u a s i s t a t i o n a r y approach. R e s u l t i n g f u n c t i o n a l dependence on d e n s i t y and temperature f o r i s shown i n

F i g . 7. f 4

I f we know t h e values f o r d e n s i t y and temperature i n dependence on the space c o o r d i n a t e and time from gas-dynamic c a l c u l a t i o n s , we can determine by use o f F i g . 7 and f i n a l l y e s t i m a t e g a i n c o e f f i c i e n t s

f 4

w i t h formulas (14) and (15). Some r e s u l t s a r e presented i n Figs.8-11.

4. D i s c u s s i o n

Corresponding t o equations (14) and (15) t h e g a i n i s p r o p o r t i o n a l

-

as t h e most i m p o r t a n t dependence

-

( 2 0.2) a r e necessary f o r an o p t i m a l gain.

As we can see from F i g . 7, t h e r e l a t i v e p o p u l a t i o n d e n s i t y o f t h e pump l e v e l increases

-

-

r 4

M and decreesing temperature Te. Considering t h e i n v e r s i o n l i m i t , t h e r e e x i s t a maximum v a l u e f i m O x )

-

f 4 ( m a X ) = l . 2 . 1 0 - ~ f o r t h e 4

-

Reabsorption o f the resonance l i n e L& lowers t h e i n v e r s i o n l i m i t o f the 3 -+2 t r a n s i t i o n , and t h e r e f o r e , (max) w i l l be decrea- sed too.

The g a i n curves i n Figs. 8-11 can be understood i n connection w i t h s p a t i a l and temporal v a r i a t i o n s o f d e n s i t y and temperature i n t h e plasma heated by l o n g l a s e r pulses. So we can see t h a t t h e r e e x i s t a s p a t i a l r e g i o n from approximately 300 t o 3000pm measured from t h e a x i s o f t h e c y l i n d e r i c a l t a r g e t , i n which t h e i n v e r s i o n c o n d i t i o n i s o p t i m a l l y performed. This o p t i m a l s i t u a t i o n occurs a f t e r t h e l a s e r pulses ( 2 10 n s ) and w i l l e x i s t f o r a r e l a t i v e l y l o n g time ( 5 t o 10 ns d u r a t i o n ) .

3 - 2 t r a n s i t i o n

The s p a t i a l r e g i o n w i t h h i g h g a i n i s s i n e l l (200 t o 300 p m ) f o r t h e t r a n s i t i o n 3-2. This i s caused by t h e f a c t t h a t i n v e r s i o n c o u l d be reached a t s u f f i c i e n t l y l o w temperatures, which w i l l be o b t a i n e d a f t e r t h e l a s e r p u l s e i n a s m a l l s p a t i a l r e g i o n n o t so f a r from t h e t a r g e t . The maximum g a i n v a l u e s f o r t h e 3 - 2 t r a n s i t i o n a r e 5 cm-1 However, t a k e i n t o account t h e i n f l u e n c e of t h e Ld - a b s o r p t i o n , t h i s v a l u e i s v a l i d o n l y f o r l a t i o n o f t h e ground s t a t e o f

I1

el

For e t h i c k n e s s o f 100pm we get a g a i n o f 1 cm-1 f o r a t i m e i n t e r - v a l o f 80.0 ns. However. because fl ^{increases i n time (} r 0.1)~ t h e s h o r t e r pulses. e f f e c t o f L,L - a b s o r p t i o n on the 3 - 2 t r a n s i t i o n i educed f o r

E

To summarize t h e t h e o r e t i c a l r e s u l t s concerning t h e X-ray g a i n i n c y l i n d r i c a l carbon plasmas f o r t h e 3 - 2 t r a n s i t i o n u s i n g r e l a t i - v e l y l o n g l a s e r pulses (t, -- 1 0 n s )

.

-1 ^L

c 5 cm i n a s m a l l s p a t i a l r e g i o n ( 4 100,wn) a t d i s t a n c e s from t h e t a r g e t o f 500 t o 1000 /urn. Maximum g a i n w i l l be obtained a f t e r t h e end o f t h e l a s e r pulse. I t c o u l d be d e t e c t e d d u r i n g a t i m e i n t e r v a l o f approximately 5 ns.

4

-

The i n v e r s i o n c o n d i t i o n N = Ninv(Te) shows a n o n l i n e a r dependence on the temperature f o r T e h 1 0 eV. Therefore, h i g h e r temperatures seems t o be favoured t o g i v e an o p t i m a l gain. Moreover, f o r l o n g l a s e r pulses t h e temperature i s r e l a t i v e l y h i g h and n e a r l y constant over a l a r g e s p a t i a l range f a r from the t a r g e t . So a h i g h g a i n f o r t h e 4-3 t r a n s i t i o n w i t h values o f = 0.05 cm'i (GmaX = 0.08 cm'l) w i l l be reached f o r a wide s p a t i a l r e g i o n (1000 t o 3000,um) i n a time i n t e r v a l o f about 10 ns beginning a t t h e end o f t h e l a s r pulse.

~ h e ' i n f l u e n c e o f t h e l a s e r i n t e n s i t y (1012 t o 1013 W/cm ) i s smell. 3

I t g i v e s r i s e t o o n l y s l i g h t l y d i f f e r e n t s p a t i a l and temporal behaviour.

We conclude t h a t for t h e 4 -+3 t r a n s i t i o n l o n g e r l a s e r pulses csn generate a g a i n i n a wide s p a t i a l range f o r l o n times. However, the maximum g a i n v a l u e reached i s s m a l l (s 0.08 smog).

To t e s t our t h e o r e t i c a l model we compared t h e r e s u l t s w i t h some 17 e x p e r i m e n t a l data115i. For - N = 5'1016 cm-3, T, = 9 eV and M = 10 cm", Te = 14 eV, t h e e x p e r i m e n t a l g a i n f a c t o r s a r e 0.01 and 0.02 crn-' r e s p e c t i v e l y , Corrss onding t h e o r e t i c a l values 0.018 and

0.025 cn-l c a l c u l a t e d w i t ! h e l p of Eq. 15 a r e i n good agreement w i t h t h e experimental f i n d i n g s . For t h e p o p u l a t i o n d e n s i t y o f t h e pump l e v e l N4 a v a l u e of ~ ~ ( e x p . 1 , 1.2-1.01~- cm-3 was measured. T h e o r e t i - c a l l y we get from ~ i g . 7 ~ ~ ( t h e 0 y . 1 a 1 . 6 . 1 0 ~ ~ c m - ~ .

I n c o n c l u s i o n we n o t e t h a t analogous i n v e s t i g a t i o n s f o r A l ( 2 = 13) and h i g h e r Z values are i n p r e p a r a t i o n .

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Kvant , Elek t r o n . (Moscow) 2 (1981) 1621 [ 8 ] R.H. Dixon, I.F. Seely, R.C. E l t o n ,

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[16] N.N. Demchenko, Ph. D. t h e s i s , Lebedev P h y s i c a l I n s t i t u t e (Moscow), 1980

[17] A.A. Samarsky, Yu. Popov, D i f f e r e n c e schemes i n gas dynamics ( i n Russian), Moscow, 1975