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MODELLING STUDIES OF XUV LASERS
G. Pert
To cite this version:
G. Pert. MODELLING STUDIES OF XUV LASERS. Journal de Physique Colloques, 1986, 47 (C6),
pp.C6-177-C6-185. �10.1051/jphyscol:1986625�. �jpa-00225868�
JOURNAL DE PHYSIQUE,
Colloque C6, suppl6ment au n o 10, Tome 47, octobre 1986
MODELLING STUDIES OF XUV LASERS
G.J. PERT
Department of Applied Physics, university of Hull, GB-Hull HU6 7RX, Great-Britain
Abstract
-
A d i s c u s s i o n of t h e general requirements of c o n d i t i o n s necessary t o generate high g a i n i n expansion cooled recombination l a s e r s w i l l be given.Numerical modelling allows s c a l i n g laws f o r t h e c o n d i t i o n s of optimum gain t o be derived, and hence matched t o plasma generated from l a s e r heated carbon f i b r e s . Comparison between experiment and theory i s good. These s c a l i n g r e l a - t i o n s may be used t o optimise g a i n . G e n e r a l i s a t i o n of t h i s approach using f o i l s and composite t a r g e t allows a range of wavelengths t o be generated.
I n t r o d u c t i o n
Recombination l a s e r s were f i r s t proposed by Gudzenko and s h e l e p i n l i n 1963, although t h e p h y s i c a l b a s i s f o r t h e i r a c t i o n can be c l e a r l y seen i n t h e work of Bates, Kingston and ~ c W h i r t e r ~ , and McWhirter and ~ e a r n 3 . The extension o f t h e scheme t o s h o r t e r wavelengths is a t t r a c t i v e a s a consequence of t h e g e n e r a l s c a l i n g of atomic parameters and r a t e s . The s i m p l e s t such scheme involves hydrogen-like i o n s f o r which a c c u r a t e t r a n s i t i o n r a t e s and energy l e v e l d a t a a r e a v a i l a b l e . I n g e n e r a l t h e h i g h e s t g a i n non-self-terminating t r a n s i t i o n is t h e n = 3 t o n = 2 t r a n s i t i o n (Ha) a t wavelength 6563/z2 1(. I n consequence t h e i s o e l e c t r o n i c t r a n s i t i o n s of t h i s sequence have been much s t u d i e d , with carbon p a r t i c u l a r l y popular.
The recombination l a s e r i n hydrogenic i o n s may be e a s i l y p i c t u r e d i n terms of t h e recombination of f u l l y s t r i p p e d i o n s i n t o high l y i n g ( l a r g e n) s t a t e s , which subsequently decay by cascade t o t h e ground s t a t e . The population i n v e r s i o n i s formed d u r i n g t h e cascade by t h e r a p i d d e p l e t i o n of t h e n = 2 l e v e l due t o t h e s t r o n g n = 2 t o n = 1 (La) r a d i a t i v e t r a n s i t i o n . The upper ( n = 3 ) l e v e l i s s t r o n g l y populated by c o l l i s i o n s from above. Clearly two c o n d i t i o n s must be s a t i s f i e d t o generate a l a r g e i n v e r s i o n d e n s i t y i n a given plasma: namely t h e La l i n e must be o p t i c a l l y t h i n , and t h e n = 3 l e v e l must be a t o r j u s t below t h e LTE l i m i t ( o r c o l l i s i o n l i m i t ) .
It has been f r e q u e n t l y a s s e r t e d t h a t due t o s e l f - a b s o r p t i o n of t h e resonance l i n e , recombination l a s e r s have i n h e r e n t l y low gain. I n f a c t t h i s l i m i t a t i o n may be avoided i n plasmas of small dimensions. This a r i s e s p r i n c i p a l l y due t o t h e motional Doppler s h i f t a c r o s s t h e expanding plasma whereby r a d i a t i o n emitted a t t h e c e n t r e of t h e plasma is not i n frequency resonance with t h a t a t t h e edge.
The e f f e c t i v e a b s o r p t i o n width o f t h e plasma i s t h e r e f o r e much l e s s than t h e p h y s i c a l width. On t h e o t h e r hand motion p a r a l l e l t o t h e a x i s of a c y l i n d r i c a l plasma is n e g l i g i b l e , s o t h a t motion induced changes of t h e gain l e n g t h a r e non-existent. With c a r e f u l design t a k i n g t h i s e f f e c t i n t o account g a i n s of about 10cm-l may be achieved.
To t a k e advantage of t h i s e f f e c t t h e t a r g e t must expand r a p i d l y from a n i n i t i a l l y small c r o s s s e c t i o n : a t y p i c a l t a r g e t being a t h i n c y l i n d r i c a l r o d ,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986625
JOURNAL DE PHYSIQUE
f o r example a c a r b o n f i b r e 4 . I n t h i s c a s e t h e c o o l i n g n e c e s s a r y t o i n d u c e p o p u l a t i o n i n v e r s i o n c a n be g e n e r a t e d a d i a b a t i c a l l y by t h e expansion w i t h o u t r e c o u r s e t o a d d i t i o n a l d i s s i p a t i o n such a s r a d i a t i v e h e a t l o s s . D e s p i t e t h e i n i t i a l s m a l l d i a m e t e r o f t h e f i b r e r e q u i r e d , t y p i c a l l y a b o u t lym, t h e a c t u a l width o f t h e a i n zone is r e l a t i v e l y l a r g e , 100vm. due t o t h e maximum e l e c t r o n d e n s i t y (2.1O1$ cm-3 f o r C VI) a t which i n v e r s i o n s c a n be formed. A s a r e s u l t t h e plasma d e n s i t y g r a d i e n t s a r e s m a l l . The r e f r a c t i o n l i m i t a t i o n o f g a i n l e n g t h is t h e r e f o r e r e l a t i v e l y unimportant. I n f a c t s i m p l e p r a c t i c a l c o n s t r a i n s on f i b r e s due t o s t i f f n e s s , alignment e t c . l i m i t t h e l e n g t h t o a few c e n t i m e t r e s . P r a c t i c a l s y s t e m s based o n t h i s approach t h e r e f o r e r e q u i r e g a i n o f a b o u t 10cm-I c o n s i s t e n t w i t h t h a t a c h i e v a b l e .
Carbon F i b r e Laser
These g e n e r a l r e q u i r e m e n t s have been i n v e s t i g a t e d f o r t h e CVI Ha l i n e a t 1 8 2 1 u s i n g s i m p l e c a r b o n f i b r e s i r r a d i a t e d by Nd g l a s s l a s e r p u l s e s . The pumping l a s e r p u l s e o f d u r a t i o n a b o u t l o o p s h e a t s and i o n i s e s t h e carbon atoms i n t h e f i b r e t o b a r e n u c l e i . During t h e s u b s e q u e n t expansion c o o l i n g and r e c o m b i n a t i o n i n t o hydrogen-like i o n s l e a d s t o p o p u l a t i o n i n v e r s i o n and gain5. Experiments have confirmed c o m p u t a t i o n a l modelling t h a t s i g n i f i c a n t g a i n can be g e n e r a t e d i n t h i s way provided t h e mass o f t h e expanding plasma is s u f f i c i e n t l y small6. I n p r a c t i c e , however, t h i s r e q u i r e s s i m p l e f i b r e s o f d i a m e t e r a b o u t 2pm which is t o o s m a l l t o b e u s a b l e due t o p r a c t i c a l l i m i t a t i o n s o f o b s e r v a b i l i t y , s t i f f n e s s e t c .
I n a l l t h e e x p e r i m e n t s performed t o d a t e t h e c a r b o n plasma h a s been formed by t h e incomplete burn o f a r e l a t i v e l y t h i c k f i b r e . A l a y e r o f t y p i c a l l y a b o u t lum t h i c k is a b l a t e d from t h e s u r f a c e o f a f i b r e 5ym d i a m e t e r . S i n c e t h e g a i n o c c u r s a t dimensions much l a r g e r t h a n t h e i n i t i a l f i b r e d i a m e t e r , t h e c o l d c o r e o f t h e f i b r e p l a y s l i t t l e r o l e i n l i m i t i n g g a i n e x c e p t w i t h i n a c i r c u l a r r e g i o n s u r r o u n d i n g t h e a x i s . Indeed t h e expansion o f t h e h o t plasma l e a d i n g t o g a i n t a k e s p l a c e n e a r l y i n d e p e n d e n t l y o f t h e c o r e , behaving s i m i l a r l y t o a s i m p l e f i b r e o f t h e same mass6. We may t h e r e f o r e c o n v e n i e n t l y a n a l y s e t h e behaviour o f t h e s e s y s t e m s by s e p a r a t i n g t h e t r a j e c t o r y i n t o two p a r - t s , namely a b l a t i o n and expansion. This approximation a l l o w s d e t a i l e d i n v e s t i g a t i o n o f a n a l y t i c s c a l i n g laws.
The Gain Adiabat
I n expansion c o o l e d recombination l a s e r s t h e o n s e t o f p o p u l a t i o n i n v e r s i o n is determined by two e f f e c t s , namely:
1. The La l i n e is o p t i c a l l y t h i n .
2. The c o l l i s i o n l i m i t l i e s a t a b o u t t h e s t a t e n = 3.
I n o r d e r t o maximise t h e i n v e r s i o n must o n s e t a t t h e h i g h e s t d e n s i t y , i e . a s e a r l y a s p o s s i b l e i n t h e expansion. C l e a r l y t h i s w i l l o c c u r when b o t h c o n d i t i o n s a r e s a t i s f i e d s i m u l t a n e o u s l y . I f t h e absorbed energy is t o o s m a l l , e x c e s s r e c o m b i n a t i o n g i v e s r i s e t o a h i g h ( n = 1 ) ground s t a t e p o p u l a t i o n and' s u b s t a n t i a l La s e l f - a b s o r p t i o n , l i m i t i n g g a i n o n s e t . On t h e o t h e r hand i f t h e energy is t o o l a r g e t h e p o p u l a t i o n s o f t h e upper s t a t e s a r e reduced, and g a i n o n s e t is a c c o r d i n g l y delayed. C l e a r l y t h e r e f o r e t h e r e e x i s t s a n optimum energy f o r t h e expansion o f g i v e n plasma body t o g e n e r a t e maximum g a i n . T h i s c o n d i t i o n may b e c o n s i d e r e d t o g e n e r a t e a n o p t i m i s e d g a i n a d i a b a t .
The c o n d i t i o n s f o r o p t i m i s e d g a i n a d i a b a t s were examined i n d e t a i l i n r e f . 5 by means o f a n e x t e n s i v e n u m e r i c a l s t u d y . It was found t h a t f o r a plasma c y l i n d e r o f mass M (gm/cm) p e r u n i t l e n g t h and r a d i u s A (cm) t h e r m a l energy E (J/cm) p e r u n i t l e n g t h was r e q u i r e d f o r optimum
t h e g a i n G (/cm) b e i n g
G = 3.9 x 1 0 - 6 M - 1 / 2 ~ - 1
A s a rough g u i d e l i n e i t was found t h a t a g a i n window of f 50% corresponded t o a thermal energy v a r i a t i o n o f
*
25%.I n o r d e r t o r e l a t e t h e s e g a i n a d i a b a t s t o t h e p r a c t i c a l s i t u a t i o n i n v o l v i n g a plasma mass M h e a t e d by a l a s e r p u l s e of f i n i t e time d u r a t i o n , i t is necessary t o i d e n t i f y a n e q u i v a l e n t v a l u e of t h e r a d i u s A a t which t h e a d i a b a t i c expansion s t a r t s . From simple c o n s i d e r a t i o n s we may expect
t h a t
7 i is a time determined by t h e l a s e r p u l s e c h a r a c t e r ; f o r a s i n g l e p u l s e t h e time-to-peak. Making u s e o f t h i s r e s u l t we o b t a i n :
and
G = 1.6 10-12 M-17/22 q - 8 / 1 1
t h e numerical c o n s t a n t s b e i n g a d j u s t e d t o g i v e good agreement w i t h s i m i l a t i o n . I n t h i s form e q u a t i o n s ( 4 ) and ( 5 ) a c c u r a t e l y reproduce an e x t e n s i v e body o f numerical c a l c u l a t i o n f o r l a r g e masses. For s m a l l masses t h e g a i n f a l l s o f f as t h e c h a r a c t e r i s t i c expansion time becomes s m a l l e r t h a n t h e l a s e r h e a t i n g time, and t h e c o n d i t i o n s o f v a l i d i t y o f t h e model a r e n o t s a t i s f i e d , f i g . 1 . Remarkably, however, i n t h i s r e g i o n e q u a t i o n ( 4 ) remains reasonably a c c u r a t e .
The t o t a l absorbed energy per u n i t mass, E (J/cm) must i n c l u d e t h e i o n i s a t i o n energy E i (J/cm) of v a l u e
namely
A number of t h e experimental c o n f i g u r a t i o n s have used non-symmetric l a s e r i r r a d i a t i o n , e i t h e r s i n g l e beam one s i d e d o r two beam double s i d e d focussing. I n t h i s c a s e we may e x p e c t t h a t t h e r e s u l t a n t plasma may n o t b e p e r f e c b l y c i r c u l a r i n i t s g e n e r a t i o n . I n o r d e r t o examine t h e e f f e c t s of e l l i p t i c plasma g e n e r a t i o n w e have compared s i m u l a t i o n s of t h e g a i n from plasmas o f varying i n i t i a l e c c e n t r i c i t i e s , such a s might b e formed by t h e a b l a t i o n from t h e s u r f a c e of a f i b r e whose diameter is much l a r g e r t h a n t h e a b l a t i o n depth. Fig. 2 shows a t y p i c a l set o f r e s u l t s . I t can be s e e n t h a t t h e e n e r g y h a s s s c a l i n g f o r peak g a i n is independent o f t h e e c c e n t r i c i t y , a l t h o u g h t h e g a i n d e c r e a s e s a s t h e e c c e n t r i c i t y i n c r e a s e s ( f i g . 3 ) . T h i s r e s u l t is n o t e n t i r e l y unexpected f o r we r e q u i r e t h a t t h e l a s e r energy p e r i n p u t over times l a r g e compared t o t h e c h a r a c t e r i s t i c expansion time. Flows w i t h such slow h e a t i n g aresknown t o behave n e a r l y independently o f t h e i n i t i a l s t a t e 7 , i e . c i r c u l a r l y t h i s is due t o t h e f a c t t h a t t h e s m a l l dimension expands r a p i d l y t o approach t h e l a r g e r d u r i n g t h e i n i t i a l s t a g e s o f h e a t i n g , s o t h a t most of t h e h e a t is i n p u t t o a n e a r l y c i r c u l a r body.
Ablation o f C y l i n d r i c a l Systems
The d i f f e r e n t i r r a d i a t i o n geometries used i n t h e experiments performed t o d a t e have ranged from s i n g l e beam one s i d e d i l l u m i n a t i o n t o a quasitsymmetric f o u r beam arrangement. Despite t h i s v a r i a t i o n t h e concensus o f experimental and modelling d a t a i n d i c a t e s t h a t t h e o v e r a l l burn from s m a l l f i b r e s is approximately symmetric, and t h e r e s u l t a n t plasma n e a r l y i s o t h e r m a l . I n consequence we may i d e n t i f y a s p e c i f i c a b l a t e d mass f o r a given energy d e p o s i t i o n : t h e p r e c i s e dependence o f t h e s e q u a n t i t i e s v a r y i n g a c c o r d i n g t o t h e l a s e r and f i b r e
C6-180 JOURNAL DE PHYSIQUE
parameters. Dimensional a n a l y s i s a l l o w s t h e g e n e r a l f o r m u l a t i o n o f t h e problem i n terms o f simple r e l a t i o n s h i p i n v o l v i n g numerical c o n s t a n t s of o r d e r u n i t y , which may be expected t o vary o n l y s l i g h t l y i n d i f f e r e n t i r r a d i a t i o n geometries.
The burn r e l a t i o n s may t h e r e f o r e be d e r i v e d t o r e a s o n a b l e accuracy, f o r t h i s p a r t i c u l a r c a s e , by c o n s i d e r i n g c y l i n d r i c a l symmetric flows.
The a n a l y t i c theory o f c y l i n d r i c a l laser-plasma formation can be t r e a t e d i n two l i m i t s 8 :
i ) S e l f - r e g u l a t i n g flows I n t h i s l i m i t l a s e r r a d i a t i o n is absorbed by i n v e r s e bremsstrahlung w i t h i n a n extended plasma atmosphere. Due t o slow d e n s i t y f a l l - o f f i n c y l i n d r i c a l geometry i t is t h e dominant p r o c e s s f o r h i g h energy/long p u l s e i r r a d i a t i o n . S i g n i f i c a n t a b s o r p t i o n w i t h i n t h e atmosphere g i v e s r i s e t o a n e a r l y i s o t h e r m a l plasma. I n p r i n c i p a l we may c o n s i d e r two regimes. For s h o r t p u l s e s
and f o r long :
M
=
2.6 x 10-3 y Z-1 A719 A-419 ro2/9 I3519 t 4 / 9where E is t h e absorbed energy (J/cm), t t h e l a s e r p u l s e width ( s e c ) , A t h e l a s e r wavelength (pm), ro f i b r e r a d i u s (cm), A atomic mass number and Z i o n i s a t i o n number: Y is a p u l s e shape f a c t o r o f v a l u e t y p i c a l l y 1.18.
The c h a r a c t e r i s t i c time f o r t h e change of regime is g i v e n by t h e e q u a l i t y of t h e two v a l u e s o f M. I n p r a c t i c e t h e s h o r t p u l s e f l o w is never observed i n t h e s e experiments: f l o w s s a t i s f y i n g t h i s c o n d i t i o n being dominated by thermal conduction.
i i ) Thermal conduction flows I n t h i s c a s e l a s e r r a d i a t i o n is absorbed a t o r c l o s e t o t h e c r i t i c a l d e n s i t y . Energy t r a n s f e r w i t h i n t h e plasma is dominated by thermal conduction. S i n c e t h e plasma atmosphere must be r e l a t i v e l y s m a l l , t h i s behaviour is c h a r a c t e r i s t i c o f low e n e r g y / s h o r t p u l s e i r r a d i a t i o n . I n t h i s l i m i t we must a l s o d i s t i n g u i s h two regimes. For s h o r t p u l s e s
and f o r l o n g
t h e p u l s e shape f a c t o r Y i n t h i s c a s e being approximately 1.43.
The domains o f t h e s e v a r i o u s s o l u t i o n s a r e r e a d i l y e s t a b l i s h e d on p h y s i c a l grounds. A t low absorbed energy a propagating thermal burn s o l u t i o n ( 1 0 ) is v a l i d , where t h e l a s e r energy is absorbed n e a r t h e t a i l of t h e expansion. A t h i g h e r e n e r g i e s , t h e plasma dimensions d u r i n g i r r a d i a t i o n i n c r e a s e and t h e energy is predominantly absorbed w i t h t h e plasma body a t t h e c r i t i c a l d e n s i t y , e q u a t i o n (11) being a p p r o p r i a t e . F u r t h e r energy i n c r e a s e c r e a t e s a plasma o f s u f f i c i e n t l y l a r g e dimensions t h a t i n v e r s e bremsstrahlung w i t h i n t h e plasma body is important, and t h e a b l a t i o n is d e s c r i b e d by ( 9 ) ( f i g . 4). Other h i g h energy regimes a r e n o t a p p l i c a b l e w i t h i n t h e range of experimental parameters evisaged i n c u r r e n t and f u t u r e experiments, and w i l l n o t t h e r e f o r e be d i s c u s s e d .
I n o r d e r t o check t h e s e numerical c a l c u l a t i o n s and t h e a n a l y t i c r e l a t i o n s two s e t s o f experimental measurements o f t h e a b l a t e d mass/energy s c a l i n g s have been c a r r i e d o u t . One s e t using s i n g l e s i d e d i r r a d i a t i o n under t h e c o n d i t i o n s a t which g a i n was observed, were performed a t low energy i n t h e regime of propagating burn [ e q u a t i o n ( l o ) ] . Fig. 5 shows a comparison of t h e experimental d a t a w i t h e q u a t i o n (10) and w i t h r e s u l t s from 2d computer s i m u l a t i o n . I n t h e
second s e t o f experiments f o u r beam i r r a d i a t i o n a t energy took p l a c e i n t h e s e l f - r e g u l a t i n g regime. Comparison of one- and two-dimensional simulations with t h e s c a l i n g law ( 9 ) shows good agreement f o r both Z = 5 and 6 ( f i g . 6 ) . One dimensional c a l c u l a t i o n s with a s e l f - c o n s i s t e n t s t e a d y s t a t e value of i o n i s a t i o n g i v e s values between t h e s e two l i m i t s c I n both c a s e s t h e experimental d a t a is c o n s i s t e n t w i t h i n experimental e r r o r with t h e c a l c u l a t e d values.
Matching Conditions
The incomplete burn mode removes one degree of freedom from t h e design o f a n experimental c o n f i g u r a t i o n . The f i b r e r a d i u s o r plasma l i n e d e n s i t y can no longer be chosen independently of t h e t o t a l plasma energy. The choice of parameters t o match t h e expansion t o an optimum a d i a b a t is t h e r e f o r e l i m i t e d t o l a s e r pulse energy and d u r a t i o n , l a s e r wavelength and t h e i n i t i a l f i b r e r a d i u s .
The c o n d i t i o n f o r optimum g a i n r e q u i r e s t h a t t h e absorbed energy E be a prescribed f u n c t i o n ( 7 ) of t h e plasma mass, namely EG(M). On t h e o t h e r hand t h a t mass is i t s e l f determined by t h e a p p r o p r i a t e a b l a t i o n r e l a t i o n ( 9 ) , (10) o r ( 1 1 )
,
from t h e absorbed energy MB(E). Thus f o r optimum o p e r a t i o n we must chose t h e a v a i l a b l e parameters s o t h a tI n p r a c t i c e t h e s e o p e r a t i n g p o i n t s a r e most conveniently i d e n t i f i e d g r a p h i c a l l y by p l o t t i n g M = MB(E) and E = EG(M) on a "burn diagramw: t h e i n t e r a c t i o n of t h e s e two l i n e s being t h e optimum ( f i g . 1 ).
One of t h e a t t r a c t i v e f e a t u r e s of t h i s scheme can be deduced from t h e c l o s e p a r a l l e l i s m of t h e burn and g a i n l i n e s i n t h e neighbourhood of t h e optimum i n t h e dominant s h o r t p u l s e ( 9 ) and long p u l s e (11) regimes ( f i g . 1 ) . A s a r e s u l t t h e device is i n s e n s i t i v e t o v a r i a t i o n s i n absorbed l a s e r energy f o r o p e r a t i o n i n t h e neighbourhood of t h e optimum. Indeed a range of energy of n e a r l y an o r d e r of magnitude could be t o l e r a t e d without t o o severe a degradation i n gain.
Gain h a s been observed i n two s e t s of experiments under t h e c o n d i t i o n s of f i g s . 2-4 and 5. In g e n e r a l it i s found ( f i g s . 1 and 3) t h a t s u b s t a n t i a l gain is only developed i f t h e mass, and corresp0ndin.g absorbed energy is r e l a t i v e l y small, t y i c a l l y 2-3 J/cm. Reasonably l a r g e g a i n s a r e only observed under t h i s c o n d i t i o n
E
s8. I n c o n t r a s t i n experiments under c o n d i t i o n s s i m i l a r t o f i g * 5 where t h e absorbed energy was about 20J/cm showed only low gaing.I t would appear from f i g s . 2-4 and 5 t h a t n e i t h e r experimental c o n f i g u r a t i o n is optimally matched. I n t h e most r e c e n t experiments f i g . 3 would i n d i c a t e t h a t some improvement i n t h e measured g a i n c o e f f i c i e n t of about 4/cm could be improved. I n t h e e a r l i e r ones, however, t h e measured value 15/cm is about twice t h e peak c a l c u l a t e d . Preliminary c a l c u l a t i o n s with a one dimensional time dependent i o n i s a t i o n code suggest t h a t both t h e s e d i s c r e p a n c i e s may be r e s o l v e d by a complete hydrodynamic d e s c r i p t i o n of both t h e burn and t h e gain.
Thin F o i l Targets
Thus f a r we have examined symmetric systems which g i v e a good approximation t o e x i s t i n g f i b r e experiments. However, a s we have seen, only r e l a t i v e l y small changes i n o p e r a t i n g behaviour occur i f t h e i n i t i a l plasma is e l l i p t i c r a t h e r than c i r c u l a r . This r e s u l t s u g g e s t s t h a t we may u s e plasma of l a r g e e c c e n t r i c i t y formed by i r r a d i a t i n g t h i n f o i l s r a t h e r than f i b r e s : t h e plasma mass per u n i t l e n g t h i n both c a s e s being t h e same. 'In t h i s c a s e t h e mass is determined by t h e t h i c k n e s s of t h e f o i l and width of t h e f o c a l l i n e modified by any l a t e r a l h e a t t r a n s p o r t . I n p r i n c i p l e such t a r g e t s r e t a i n t h e f l e x i b i l i t y of design of f u l l y burnt f i b r e s , b u t i n p r a c t i c e u n c e r t a i n t i e s about t h e l a t e r a l width considerably complicate c a l c u l a t i o n s .
JOURNAL
DE
PHYSIQUEFor a n e f f e c t i v e width of 40pm t h e f o i l s must only be a b o u t 10001( t h i c k . For simple carbon f o i l s t h e g a i n a d i a b a t f o l l o w s t h e same e n e r g y h a s s r e l a t i o n s h i p a s f o r c y l i n d e r s independently of t h e e c c e n t r i c i t y . The g a i n a l s o behaves i n a similar f a s h i o n showing t h e same s c a l i n g b u t with d e c r e a s i n g c o n s t a n t a s t h e e c c e n t r i c i t y is i n c r e a s e d ( f i g s . 2 and 3 ) .
Carbon f o i l s a r e r e l a t i v e l y d i f f i c u l t t o c o n s t r u c t , b u t polymers a r e simpler.
Examination of t h e behaviour o f Formvar f o i l s (C11 Hi8 05) shows t h a t t h e same s e a l i n g r e l a t i o n s a r e obeyed a s f o r pure carbon, b u t t h a t t h e numerical c o n s t a n t s d i f f e r , f i g . 2. I n p a r t i c u l a r c o n s i d e r a b l y more energy is r e q u i r e d t o d r i v e t h e Formvar f o i l , a l t h o u g h t h e a t t a i n a b l e g a i n is comparable w i t h carbon ( f i g . 3 ) . S c a l i n g t o S h o r t e r Wavelengths
The i s o e l e c t r o n i c s c a l i n g o f hydrogenic i o n s a s t h e i o n i s a t i o n , 2, i n c r e a s e s may b e used t o d e s i g n s h o r t e r wavelength l a s e r s based on t h e carbon f i b r e model.
The d e t a i l e d s c a l i n g of such expansion was developed i n r e f . 5. I t was found t h a t f o r Z
<
12 (magnesium/aluminium) some d e g r e e of pre-expans'ion t o match t h e a d i a b a t was necessary. I n p r a c t i c e t h i s a c c o u n t s f o r t h e a b l a t i o n / i o n i s a t i o n phase. Hence w e may a n t i c i p a t e t h a t r e l a t i v e l y simple e x t r a p o l a t i o n o f f i b r e desigfis t o t a r g e t s c o n t a i n i n g n i t r o g e n , oxygen, f l u o r i n e o r neon is p o s s i b l e . Numerical s i m u l a t i o n s confirm t h i s h y p o t h e s i s . Such t a r g e t s may b e formed i n two convenient ways by e i t h e r s e e d i n g o r c o a t i n g carbon f i b r e s . I n t h e former a r e l a t i v e l y low s e e d d e n s i t y o f t h e l a s a n t element ( - 10%) is i o n implanted w i t h i n t h e f i b r e . I n t h e l a t t e r a t h i n o v e r c o a t of polymer o r o t h e r s u i t a b l e m a t e r i a l is overcoated t o p r o v i d e a n a b l a t i o n medium.For h e a v i e r i o n s s u c h . a s magnesium and aluminium t a r g e t d e s i g n r e q u i r e s d e p a r t u r e from t h e simple carbon a d i a b a t s l O . The d e s i g n of s u i t a b l e t a r g e r t s was d i s c u s s e d i n r e f . 10 where a g a i n i t was Suggested t h a t s u i t a b l e t a r g e t s could be e i t h e r c o a t e d f i b r e s o r r i b b o n coated t h i n f o i l s . A t t h e p r e s e n t time t h e r e l a t i v e m e r i t s o f each s u g g e s t i o n a r e unknown.
Conclusions
Q u a s i - a n a l y t i c modelling a i d e d by computer s i m u l a t i o n h a s allowed t h e development of s c a l i n g r e l a t i o n s f o r t h e carbon f i b r e l a s e r scheme. These simple formula have been shown t o be i n agreement w i t h experiment, and have allowed easy i d e n t i f i c a t i o n o f working parameters. More i m p o r t a n t l y t h e r a t h e r complicated p h y s i c a l r e l a t i o n s h i p s amongst experimental v a r i a b l e s can be understood. This u n d e r l y i n g t h e o r y p r o v i d e s t h e b a s i s f o r t h e development o f s h o r t e r wavelength l a s e r s , now p r o j e c t e d e x p e r i m e n t a l l y .
References
1. L.I. Gudzenko and L.A. S h e l e p i n , Sov. Phys. Doklady,
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147 (1965).2. D.R. Bates, A.E. Kingston and R.W.P. McWhirter, Proc. Roy. Soc. A ,
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297, (1962)i A .
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155 (1962).3. R.W.P. Mckmirter and A.G. Hearn, Proc. Phys. Soc.
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641 (1963).4. R.J. Dewhurst, D. Jacoby, G.J. P e r t and S.A. Ramsden, Phys. Rev. L e t t . 37, 1265 (19761.
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P e r t , J. Phys. B,9,
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2067 (1979).6. D. Jacoby, G.J. P e r t , L.D. Shorrock and G.J. T a l l e d t s , J. ~ h y s . B, 15. 3557 (1982).
7. G . x P e r t , J . F l u i d Mech.,
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257 (1980).8. G.J. P e r t , J . Plasma PhyS: i n p r e s s .
9. E. Mahoney, C.L.S. Lewis, M.J. Lamb, L.D. Shorrock, G.J. P e r t , M.H. Key, C. Hooker and R.B. Eason, C e n t r a l Laser F a c i l i t y Annual Report, Rutherford
Appleton LaboPatory, RL-83-043, p7.2 (1983).
10. G.J. P e r t , Plasma Phys.,
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1427 (1985).Fig. 1 Burn diagram f o r symmetric i r r a d i a t i o n of a 5um diameter carbon f i b r e by a 0.53pm l a s e r p u l s e of 200ps d u r a t i o n i n a f o c a l s p o t 40um wide. The l i n e s show t h e a n a l y t i c e q u a t i o n s (7) and ( 9 1 , and t h e p o i n t s computer s i m u l a t i o n g e n e r a t e d v a l u e s of burn mass ( x ) and g a i n energy ( 0 ) . The peak g a i n computed
( 0 ) is compared w i t h t h e a n a l y t i c e x p r e s s i o n (5).
Fig. 2 Mass energy r e l a t i o n c a l c u l a t i o n s f o r f i b r e s of v a r y i n g e l l i p t i c widths heated by a Tops, 0 . 5 3 ~ ~ 1 l a s e r p u l s e i n a s p o t 40um wide. C i r c u l a r f i b r e s a r e denoted ( o ) , e l l i p t i c of width a c r o s s t h e f o c a l s p o t , 10um ( 0 ) . 20pm
( 0 ) .
40um ( 0 ) and 60um ( x ) . Also shown a r e c a l c u l a t i o n s f o r Formvar s t r i p s of t h i c k n e s s 0.5pm
(El
and 1.0pm ((0. The f u l l l i n e is g i v e n by e q u a t i o n 17).JOURNAL DE PHYSIQUE
Fig. 3 Calculations of gain for the configurations represented in fig. 2.
full line is given by equation (5).
The
Fig. 4 Calculations of the ablation mass for the configuration of fig. 2 using one.dimensiona1 ( V ) and two dimensional ( A ) codes, for Z = 5, and one dimensional (t) for Z = 6. The lines represent the analytic expression ( 1 0 ) and (11) and the dashed line (9) for Z = 6. The two dimensional calculations treat two sided illumination.
Fig.5 Burn diagram for the expe- riments of ref. 6, involving one sided heating of 4um fibres by 1..06um laser radiation of duration laOps with pre-pulse. Experimen- tally measured (e) and computer simulated mass points (x) are shown compared with analytic for- mulae (7) and (10).
Fig.6 Ablation mass calculations for the experiments of ref. 9 in- volving four beam symmetric hea- ting of 5pm fibres by 0.53vm ra- diation of duration 200ps with pre-pulse. Experimental measure- ments (.) are compared with one
(V) and two dimensional (A) simu- lations for Z = 5, one dimensional ( 0 ) for Z = 6 and self-consistent ionisation calculations (x), and with analytic prediction, equa- tion (9).
