Negative conductivity in a neon plasma

Volume 26A, number 9

PHYSICS

NEGATIVE

LETTERS

CONDUCTIVITY

25 March 1968

IN A NEON

PLASMA

O. D E B A R B I E R I and C. M A R O L I Istituto di Science F i s i c h e , Universit~ di Milano, Milano, Italy Gruppo Nazionale di Elettronica Quantistica e P l a s m t del CNR, Sezione di Milano, Italy Received 11 January 1968

It is shown that electron-molecule inelastic collisions can lead to a negative high-frequency conductivity in a neon plasma.

In a r e c e n t p a p e r [1], a d e t a i l e d e x a m i n a t i o n h a s b e e n m a d e of t h e e f f e c t of t h e v e l o c i t y d e p e n d e n c e of t h e e l e c t r o n - m o l e c u l e c o l l i s i o n f r e q u e n c y on t h e e x p r e s s i o n s of t h e h i g h - f r e q u e n c y c o n d u c t i v i t y e and of the s u s c e p t i b i l i t y X of a s l i g h t l y i o n i z e d p l a s m a . H e r e we t a k e e x p l i c i t l y into a c c o u n t t h e e f f e c t of i n e l a s t i c c o l l i s i o n s . We c o n s i d e r a n e o n g a s p l a c e d in a c o n s t a n t and u n i f o r m m a g n e t i c f i e l d B , a c t e d on by an e l e c t r i c f i e l d of t h e f o l l o w i n g t y p e : E = / ~ u e o s w t + E o C O s ( ~ t + 5), w h e r e co ~ a ~ coB = l e l B / m c and I E p l >> [ E o l . T h e m a g n i t u d e of E p is t a k e n to be of t h e o r d e r of the f i e l d s n e c e s s a r y to p r o d u c e a b r e a k d o w n and (or) to s u s t a i n a d i s c h a r g e . By e m p l o y i n g t h e t e c h n i q u e u t i l i z e d in [2,3], we h a v e , f o r ~ z z and

-7

~)eff.oI0

t

1.s. ~- ' I " ~ ' T ~ ' -

.--- ~ _

~T:~..~:.~" •

.1o

~-~

"-~

,,~'"

0

Xzz

3 o

×zz

=

f o

d v v 3 O~

w2 a¢ v(v) _ p av v 2 ( v ) + a 2 - - ~ - ~ - ~ a ( a ) ,

u-g-(-(-(-(-(-(-(-(~=-

(1)

~.16 7

[8]

442

"'°

~,x(~), (2)

where v(v) is the elastic electron-molecule coll i s i o n f r e q u e n c y f o r m o m e n t u m t r a n s f e r [4]. T h e f u n c t i o n ~ ( v ) o b e y s eq. (9) of r e f . 5 , w h e r e now t h e f u n c t i o n Vex i s c o n n e c t e d with t h e t o t a l i n e l a s t i c c r o s s - s e c t i o n Q i n , d e f i n e d as t h e d i f f e r e n c e b e t w e e n t h e t o t a l c r o s s - s e c t i o n [6] and t h e e l a s t i c one [7]. T h e f u n c t i o n ~p(v) i s o b v i o u s l y f a r f r o m a M a x w e l l i a n function. E q u a t i n g (1) and (2), a s in r e f . 1, to t h e e l e m e n t a r y e x p r e s s i o n s

~ 2 , eff Veff %z = 4n Ve2ff+a2 ' X z z =

°"1/ / / '

~2, elf 1 4n v 2 f f + a 2 (3)

0

1

2

3

4

Fig. 1. we can o b t a i n ~(~) = ~ 2 , eff/0o2p and Veff , in t e r m s of ~ ( a ) and ~x(~). We h a v e c o n s i d e r e d a n e o n p l a s m a w i t h a n e u t r a l p r e s s u r e p = 4.5 × 1 0 - 2 m m H g and w i t h co = coB = 2 n 8 4 × 1 0 6 H z . In fig. 1 we p l o t the v a l u e s of ~/(a) and Yell(a) o b t a i n e d f o r E p = 1 v o l t / c m and Ep = 10 v o l t / c m (with T 1 = 8.94 eV, T 2 = 24.31 eV r e s p e c t i v e l y ) . W e h a v e a l s o g i v e n t h e v a l u e s of ~?(a) a n d Veff(a) f o r two M a x w e l l l a n d i s t r i b u t i o n s at t h e s a m e t e m p e r a t u r e s . It i s

Volume 26A, number 9

PHYSICS L E T T E R S

i m p o r t a n t to s t r e s s that for s m a l l v a l u e s of a, ~(a) is a p p r e c i a b l y d i f f e r e n t f r o m unity when i n e l a s t i c c o l l i s i o n s a r e taken into account. F o r t h e s e f r e q u e n c i e s Veff b e c o m e s n e g a t i v e (so that, b e c a u s e of eqs. (3), ~zz(a) too i s n e g a t i v e [9]). This i m p l i e s , in p r i n c i p l e , a m p l i f i c a t i o n of the signal at the f r e q u e n c y a. One might i n t e r p r e t these r e s u l t s by s a y i n g that t h e r e is a t r a n s f e r of energy f r o m the s t r o n g wave at the f r e q u e n c y 0: to the s m a l l s i g n a l at the f r e q u e n c y ~ F u r t h e r c a l c u l a t i o n s a r e in p r o g r e s s , to c l a r i f y t h e s e results. 1. R. F. Whitmer and G. H. Herrmann, Phys. Fluids 9 (1966) 768.

25 March 1968

2. C. Maroli, Nuovo Cimento 41B (1966) 208. 3. P. Caldirola, O. De Barhieri and C. Maroli, Nuovo Cimento 42B (1966) 266. 4. D. Barbiere, Phys. Rev. 84 (195i) 653. 5. A. Airoldi Crescentini, P. Caldirola, C. Maroli and E. Sindoni, Proc. Eight Intern. Conf. on Phenomena in ionized gases, Wien (1967). 6. C. Ramsauer, Ann. Physik 66 (1921) 546. C. Ramsauer and R. Kollath, Ann. Physik 3 (1929) 536. 7. H.S.W. Massey and E. H. S. Burhop, Electronic and ionic impact phenomea (Oxford University Press, London, 1952). 8. V. L. Ginzburg, Propagation of electromagnetic waves in plasma (Gordon and Breach, New York, 1961). 9. D.H. Sampson and J. Enoch, Phys. Fluids 6 (1963) 28.

* * * * *

A POSTERIORI HOLOGRAPHIC SHARP-FOCUS IMAGE RESTORATION FROM ORDINARY BLURRED PHOTOGRAPHS OF THREE-DIMENSIONAL OBJECTS PHOTOGRAPHED IN O R D I N A R Y W H I T E L I G H T G. W. STROKE, G. INDEBETOUW and C. PUECH

State University of New York, Stony Brook, New York 11790, USA Received 27 February 1968

Application of the Stroke and Zech holographic Fourier-transform division methods has permitted us to restore into sharp focus images of three-dimensional objects from ordinary, blurred, out-of-focus photographs, obtained in ordinary, incoherent white light.

F i g u r e l b shows an " i n - f o c u s " i m a g e f ( x , y ) which we have obtained by a p o s t e r i o r i holographic i m a g e r e s t o r a t i o n f r o m the d e l i b e r a t e l y o u t - o f focus photograph g ( x , y ) = f ( x , y) ® h(x,y) of a t h r e e - d i m e n s i o n a l object, shown in fig. l a , where h(x,y) is the " o u t - o f - f o c u s " s p r e a d function, and @ a spatial i n t e n s i t y convolution. The o u t - o f - f o c u s photograph g ( x , y ) was obtained in o r d i n a r y , i n c o h e r e n t white light, in a c o n v e n t i o n a l photographic camera. A t h e o r y for such holographic i m a g e r e s t o r a tion, b a s e d on holographic F o u r i e r - t r a n s f o r m division, was f i r s t p r o p o s e d in 1967 by Stroke and Zech [1] and s h o r t l y t h e r e a f t e r e x p e r i m e n t a l l y v e r i f i e d by L o h m a n n and W e r l i c h [2] for a single point and a p l a n e , single " b l a c k - o n - w h i t e " l e t t e r "T". The i m a g e - d e b l u r r i n g r e s u l t which we show in fig. 1 f o r a t h r e e - d i m e n s i o n a l c o n t i n u o u s - t o n e object, h a s been obtained a c c o r d i n g to a f o r m of o u r method of ref. 1, f i r s t p r e s e n t e d by one of u s [GWS] on 5 S e p t e m b e r 1967 at the

Colloquium on Modern Optics in Quebec, and to a p p e a r in ref. 3. In a g e n e r a l way, i m a g e r e s t o r a t i o n methods a r e e x t e n s i o n s of the "spatial f i l t e r i n g " p r i n c i p l e s , f i r s t d e s c r i b e d by Mar~chal and C r o c e in 1953 [4], which may now be c o n s i d e r e d to a p p e a r g e n e r a l l y r e a l i z a b l e with the aid of c o m p l e x f i l t e r s r e a l i z e d in the f o r m s of h o l o g r a m s , a s f i r s t i n t r o d u c e d by Dennis Gabor in 1948 [5]. S e v e r a l c o m p a r a b l e , r e m a r k a b l e , n o n - h o l o g r a p h i c i m a g e - r e s t o r a t i o n e x a m p l e s have b e e n d e s c r i b e d , notably by T s u j i u c h i [6], u s i n g s e p a r a t e m a t e r i a l i z a t i o n s of the c o m p l e x c o m p o n e n t s (amplitude and phase) of the f i l t e r s , and by L o h m a n n et al. [7], u s i n g c o m p u t e r - g e n e r a t e d f i l t e r s , as well as in apodisation and F o u r i e r t r a n s f o r m s p e c t r o s c o p y ( r e f e r e n c e s in ref. 1). One of u s (GWS) has long noted the i m p o r t a n c e of " d e - b l u r r i n g " (deconvolution) image r e s t o r a t i o n , such as that which we d e s c r i b e , in a p p l i c a t i o n s r a n g i n g f r o m a s t r o n o m y and r a d i o - a s t r o n o m y , to i m a g i n g with o t h e r r a d i a t i o n s , for i n s t a n c e m i c r o waves, u l t r a s o n i c s and X - r a y s . 443