THE INFLUENCE OF COLLISIONAL DAMPING ON SURFACE PLASMON-POLARITON DISPERSION

HAL Id: jpa-00224144

https://hal.archives-ouvertes.fr/jpa-00224144

Submitted on 1 Jan 1984

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THE INFLUENCE OF COLLISIONAL DAMPING ON SURFACE PLASMON-POLARITON DISPERSION

A. Boardman, P. Egan

To cite this version:

A. Boardman, P. Egan. THE INFLUENCE OF COLLISIONAL DAMPING ON SURFACE PLASMON-POLARITON DISPERSION. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-179- C5-190. �10.1051/jphyscol:1984526�. �jpa-00224144�

JOURNAL DE PHYSIQUE

Colloque C5, supplément a u n04, Tome 45, avril 1984 page C5-179

T H E INFLUENCE OF COLLISIONAL DAMPING ON SURFACE PLASMON-POLARITON DISPERSION

A . D . Boardman and P . Egan

Department of Pure and AppZied physics, U n i v e r s i t y of SaZford, SaZford M5 4YT, U . K .

Résumé - Nous p r é s e n t o n s une a n a l y s e d é t a i l l é e , a n a l y t i q u e e t numérique, de l ' i n f l u e n c e c o n j o i n t e d e l ' a m o r t i s s e m e n t dû aux c o l l i s i o n s e t de l a d i s p e r s i o n s p a t i a l e s u r l e s p l a s m o n s - p o l a r i t o n s d e s u r f a c e . On t r o u v e que l e s courbes de d i s p e r s i o n p r é s e n t e n t p l u s i e u r s branches, d o n t l ' u n e e s t l e mode couramment admis comme é t a n t à i n v e r s i o n de c o u r b u r e e t q u i c o r r e s p o d à une fréquence r é e l l e e t un nombre d ' o n d e complexe. On montre qu'une i n t e r a c t i o n i m p o r t a n t e e n t r e l ' a m o r - t i s s e m e n t dû aux c o l l i s i o n s e t l a d i s p e r s i o n s p a t i a l e p e u t supprimer c e t t e i n v e r s i o n de c o u r b u r e .

A b s t r a c t - A d e t a i l e d a n a l y t i c a l and n u m e r i c a l a n a l y s i s i s g i v e n o f t h e j o i n t i n f l u e n c e o f c o l l i s i o n a l damping and s p a t i a l d i s p e r s i o n on s u r f a c e p l a s m o n - p o l a r i t o n s . I t i s shown t h a t t h e d i s p e r s i o n has s e v e r a l branches, one o f w h i c h i s t h e c u r r e n t l y a c c e p t e d bend-back mode t h a t a r i s e s u n d e r r e a l f r e q u e n c y and complex wave number assumptions. I t i s p r o v e d t h a t an i m p o r t a n t i n t e r a c t i o n between c o l l i s i o n a l damping and s p a t i a l d i s p e r s i o n o c c u r s t h a t can cause t h i s bend-back t o be suppressed.

1 - INTRODUCTION

The i n c l u s i o n o f c o l l i s i o n a l dampin? i n t o a d e s c r i p t i o n o f s u r f a c e /1,2,3,4/

p l a s m o n - p o l a r i t o n s has been d i s c u s s e d s e v e r a l t i m e s i n t h e l i t e r a t u r e /1,2/.

I t i s n o t a s t r a i g h t f o r w a r d r n a t t e r , e s p e c i a l l y i f t h e model i s r e l a t e d t o a t t e n u a t e d t o t a l r e f l e c t i o n (ATR) experiments /2,5/.

A t t h e b a s i c l e v e l , i f c o l l i s i o n s a r e accounted f o r i n a model o f t h e e l e c t r o n gas t h e n t h e r e s u l t i n g s u r f a c e p o l a r i t o n d i s p e r s i o n e q u a t i o n becomes complex. The wave number k and a n g u l a r f r e q u e n c y ^UJ a r e then, i n p r i n c i p l e , b o t h complex. The s o l u t i o n o f t h e d i s p e r s i o n e q u a t i o n w i t h complex k and r e a l ^UJ o r complex ^UJ and r e a l k a r e n o t t r i v i a l l y d i s t i n c t p o s s i b i l i t i e s . Indeed, i f s p a t i a l d i s p e r s i o n i s i g n o r e d t h e n t h e c u r r e n t p o s i t i o n ernbraces t h e f o l l o w i n g s i t u a t i o n s . I f UJ i s r e a l and k i s complex, and equal t o kr+iki, a p l o t o f UJ a g a i n s t kr f o l l o w s t h e u s u a l c o l l i s i o n l e s s s u r f a c e p o l a r i t o n d i s p e r s i o n c u r v e u p t o t h e r e g i o n ^{UJ UJ} / J 2 , where ^UJ i s

P P

t h e plasma f r e q u e n c y . I t t h e n bends back towards t h e l i g h t l i n e . T h i s has been c o n f i r m e d e x p e r i m e n t a l l y f o r s i l v e r / 6 / . If ^UJ i s complex, and e q u a l Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984526

C5-180 J O U R N A L DE PHYSIQUE

t o wr+iwi and k i s r e a l t h e n a p l o t o f lur a g a i n s t k does n o t show t h i s bend-back and f o l l o w s t h e c o l l i s i o n l e s s s u r f a c e p o l a r i t o n c u r v e . The f i r s t p o s s i b i l i t y corresponds t o an ATR measurement i n w h i c h t h e e x c i t a t i o n f r e q u e n c y i s f i x e d and t h e a n g l e o f i n c i d e n c e i s scanned w h i l e t h e second c o r r e s p o n d s t o a f i x e d a n g l e o f i n c i d e n c e w i t h an e x c i t a r i o n f r e q u e n c y scan.

I t i s t h e bend-back t h a t i s o f t h e g r e a t e s t t h e o r e t i c a l i n t e r e s t and i t i s i m p o r t a n t t o d i s c o v e r what happens when s p a t i a l d i s p e r s i o n i s i n c l u d e d . P a r t of t h e answer suggests i t s e l f i m m e d i a t e l y . A t h i g h enough wave numbers i n t h e e l e c t r o s t a t i c r e g i o n t h e (w,kr) d i s p e r s i o n d i a g r a m must have a s p a t i a l d i s p e r s i o n branch even if a t l o w e r wave numbers i t

i s suspected t h a t bend-back can s t i l l o c c u r . The d e t a i l e d answer i s a c o m p l i c a t e d one. I t i s shown i n t h i s paper t h a t bend-back can s t i l l o c c u r b u t , even though t h e wave numbers i n t h e r e g i o n o v e r w h i c h i t o c c u r s a r e s m a l l , i t can be s t r o n g l y i n f l u e n c e d b y s p a t i a l d i s p e r s i o n . I n d e e d if t h i s i n f l u e n c e i s s t r o n g enough i t can i s o l a t e and suppress t h e phenomenon.

The p r e s e n t a t i o n h e r e c o n c e n t r a t e s on t h r e e (w/w ^{) l} r e g i o n s namely O < ~ ' < 0 . 4 , R=w/w 0 . ? < ~ ~ < 0 . 6 a n d 0 . 6 < 2 ~ < 0 . 9 . T n t h e f i r s t a n d P l a s t r e g i o n a n a l y t i c a l r e s u l t s a r e o b t a i n e d . P' These t h e n p r o v i d e a n i n t e r p r e t a t i o n f o r t h e c e n t r e r e g i o n t h a t fias t o be i n v e s t i g a t e d c o r n p u t a t i o n a l l y . The o u t s f d e r e g i o n s , t h e r e f o r e , c o n t i n u e i n

terms o f l a b e l l i n g i n t o t h e c e n t r e r e g i o n . The p r o b l e m i s c o n v e n i e n t l y a n a l y s e d i n terms o f I4'=c2k2/u2, where c i s t h e v e l o c i t y o f 1 i g h t i n vacuo. The c o n c l u s i o n s drawn a r e t h e n f i n a l l y t r a n s l a t e d i n t o

w,kr p l o t s i n which t h e f e a t u r e s d i s c u s s e d above a r e c l e a r l y d i s p l a y e d . II - THE DISPERSION EQUATION

I f b o t h s p a t i a l d i s p e r s i o n and damping a r e accounted f o r i n a hydrodynamic d e s c r i p t i o n o f t h e e l e c t r o n gas t h e n t h e b u l k t r a n s v e r s e wave number kT and t h e b u l k l o n g i t u d i n a l wave number k L a r e , r e s p e c t i v e l y , g i v e n by /1 ,4/

and

where t h e t i m e dependence i s assumed t o be e-iwt,w i s t h e a n g u l a r f r e q u e n c y , 3 2

v i s a c o n s t a n t e l e c t r o n c o l 1 i s i o n f r e q u e n c y and^^= ^{V F} where VF i s t h e Fermi v e l o c i t y . The s u r f a c e mode d i s p e r s i o n e q u a t i o n f o r a vacuum-bounded s e m i - i n f i n i t e plasma, d e r i v e d e a r l i ~ r i n a d i f f e r e n t way by Clemmow and E l g i n /1/ f o r a gaseous e l e c t r o n plasma, i s

{ w 2 - w P / ( i t i v / w ) } ( c 2 k 2 - w 2 ) ~ t u ' ( c ' k 2 - ~ P ' / ( i + i v / w ) ) 1

This d i s p e r s i o n equation involves r a d i c a l s and i s beyond ordinary a n a l y t i c a l methods of determining t h e r o o t s . As i s shown below t h e

' s q u a r i n g up' of t h e equation, very f o r t u n a t e l y , l e a d s t o a cubic e q u a t i o n . The r o o t s of t h i s equation can, of c o u r s e , be w r i t t e n down but may o r may not be piiysically admissible. I t i s necessary then t o develop a r o o t lacceptance' c r i t e r i o n . P a r t of t h e algorithm can be p h y s i c a l ; t h e main approach however, i s t o check whether t h e r o o t s a t i s f i e s t h e o r i g i n a l d i s p e r s i o n e q u a t i o n .

K2

~ f N = K=ck/w and .Q=w/w , then

P P

where

a , 4 c 2 ~ 3 ( Q + i q ) , b=c2n2+2Q2-1+2iQq, d = l - ~ ~ - i Q q ( 2 e 5 ) and q=v/w ,c ^=B/c. I f N = X ~ % b i s introduced t o equation ( 2 . 4 ) i t reduces even f u r t h e r t o P

x3+qx+r=0 (2.6)

where

q=

-&

The cube r o o t s must now be chosen t o make y z = -

2

1 1

t h e two complex r o o t s of 1 i . e . - ( 1 - i J 3 ) and - - Z ( l t i J 3 j . The general s o l u t i o n i s t h e r e f o r e

3v'2a~=2bb2+(2b3(b3t9ad)+27a2d2+ ~ 2 7 a ~ d ~ ( 4 1 i ~ + 2 7 a d ) l ' ~ + { 2 b 3 ( b 3 t 9 a d ) t 2 7 a 2 d 2 - ~ 7 a 3 d " ( 4 b 3 t 2 7 a d ) } y 3 ( 2 . 9 ) This equation does not involve any approximation and can t h e r e f o r e be used t o study t h e damped and undamped s u r f a c e modes.

I I I - ZERO DAMPING

Although t h i s c a s e has been discussed i n a general form before, i t i s t r u e t o Say t h a t t h e d i s c u s s i o n was very l i m i t e d and t h a t a f u l l and unambiguous development has n ~ t been previously given. Here t h e meaning o f various branches of t h e d i s p e r s i o n diagrarn i s derived and a c l e a r i n d i c a t i o n of which of t h e curves can be l a b e l l e d a s s u r f a c e waves i s given. The

C5-182 JOURNAL DE _PHYSIQUE

r e p r e s e n t a t i o n adopted here i s a u s e f u l and r e v e a l i n g one t h a t a l l o w s t h e r o l e o f c o l l i s i o n s and s p a t i a l d i s p e r s i o n t o be u n d e r s t o o d q u i t e c l e a r l y . L e t us f i r s t c o n s i d e r t h e s o l u t i o n s i n t h e r e g i o n 1 i . e . O<n2<0.4.

T h i s r e g i o n i s chosen t o make a b s o l u t e l y c e r t a i n t h a t t h e a n a l y t i c a l work i s v a l i d b u t i n p r a c t i c e t h e upper l i m i t c o u l d be r a i s e d . I n t h i s r e g i o n we s k i r t t o t h e l e f t of t h e p r o b l e m a t i c p o i n t ~ k 0 . 5 t h a t would be s i n g u l a r i f E+O and q 4 . I n t h i s r e g i o n l b 3 1>8x10-~, a d 4 4 ~ 1 0 - ~ and ( a b d ) 3 / 2 Hence i f

q'= 127a3d3(4b3+27ad)l ( 3 . 2 )

t h e n p>>q' and we have f o r t h e r o o t s i n s i d e { 1

so t h a t t h e t h r e e r o o t s NI, N2 and N3 a r e g i v e n b y

3(2)%aN1=2%b2tp%, 3(2)%a% =2zb2-p't/- 1 - ~ ~ ~ / ( 3 ~ ' ) ( 3 . 6 ) where 2/3 r e f e r s t o +/-. Hence

Here N g i v e s t h e l a r g e K a p p r o x i m a t i o n and NZl3 a r e s m a l l p e r t u r b a t i o n s 1

o f O(() f r o m t h e c o l d plasma case and a r e , i n f a c t , t h e f i r s t two terms o f an expansion, g i v e n i n a d i f f e r e n t form, by Clemmow and E l g i n /1/.

The a t t e n u a t i o n c o e f f i c i e n t s o f a vacuum bounded s e m i - i n f i n i t e plasma a r e a, y and 6 where / 4 /

Vacuum: 62 = ^{~ 2 - ~ 2}

Plasma: a 2 = 1+K2-fi2

y 2 = ( [ 2 ~ 2 + l - ~ 2 ) / c 2 I f N1 i s used t h e n

where t h e s i g n s a r e a r b i t r a r y and independent. F o r a s u r f a c e wave a d i s p e r s i o n e q u a t i o n i s a r r i v e d by r e q u i r i n g t h a t al>O,cl>O s o t h e n e g a t i v e s i g n s must be chosen i n e q u a t i o n ( 3 . 1 1 ) .

The d i s p e r s i o n e q u a t i o n i n t h e c o l l i s i o n l e s s case r e d u c e s t o

n 2 a - ( 1 - n 2 ) 6 = ~ ' / y (3.12) Hence on s u b s t i t u t i o n o f al and 61 t h e l e f t - h a n d s i d e o f ( 3 . 1 2 ) g i v e s

and t h e r i g h t hand s i d e g i v e s

B u t we o n l y o b t a i n Rl< O when yl< O. T h i s means t h a t NI c a n n o t r e p r e s e n t a s u r f a c e wave s o l u t i o n .

F o r N we o b t a i n 2 / 3

2 %

2 1-2n2)f p + / - ( ( l - n 2 ) ~ / ( l - 2 n 1 ( 3 . 1 5 )

='fi (

Now u s i n g p o s i t i v e s i g n s f o r a and 6 a s u b s t i t u t i o n i n t o t h e

2/3 2/3

d i s p e r s i o n e q u a t i o n r e s u l t s i n

T h i s i s < O f o r N2 and > O f o r N3, so o n l y N3 r e p r e s e n t s a s u r f a c e wave i n t h i s r e g i o n . N o t e t h a t N3 l i e s below t h e c o l d plasma s o l u t i o n . We t u r n now t o t h e a n a l y t i c a l l y secure r e g i o n I I I i . e . 0.6 < N2< 0.9 where, once a g a i n , t h e l o w e r l i m i t i s approximate and can be l o w e r e d .

I n t h i s r e g i o n

F o r NI, al, 6 , > O w i t h t h e p o s i t i v e s i g n i n e q u a t i o n ( 3 . 1 1 ) . T h i s makes L1'o w h i c h matches R1 f o r yl'O. Hence NI r e p r e s e n t s a s u r f a c e wave s o l u t i o n . F o r r e g i o n III we a l s o o b t a ~ n

F o r N3,@(o)> O,@(&) > O p r o v i d e d t h a t t h e p o s i t i v e r o o t i s t a k e n so t h a t

JOURNAL DE PHYSIQUE

T h i s c h o i c e l e a d s t o

and

(3.25)

5 0 t h a t N3 c a n n o t r e p r e s e n t a s u r f a c e wave. I n t h e same way t a k i n g n e g a t i v e s i g n s f o r t h e r o o t s i n q u e s t i o n (3.19) and (3.22) t o keep

@ ( a )

,a

I V - EFFECT OF COLLISIONAL DAMPING

The e f f e c t s o f damping may be i n t r o d u c e d t h r o u g h t h e t r a n s f o r m a t i o n s

F i g . 1 - The r e a l p a r t o f t h e r o o t s o f e q u a t i o n s (2.4) f o r t h e c o l l i s i o n l e s s case w i t h s p a t i a l d i s p e r s i o n .

The c o e f f i c i e n t s o f t h e c u b i c e q u a t i o n become a = 4 F , 2 ~ 3 ( ~ + i q ) , ~ = C ~ R ~ + 2 ~ ~ - i + ? i n ~ , c - l - ~ ~ i ~ ~ . The development i s now r e s t r i c t e d t o f i r s t o r d e r i n 5 and w h i c h i s a v a l i d a p p r o x i m t i o n p r o v i d e d t h a t fi2 does n o t approach t o o c l o s e l y t o 1 i n r e g i o n I I I . I n b o t h r e g i o n s 1 and III

100

50

lc

- 1 00

0.42 0.44 0.46 0.48p0.50 0.52 0.54 0.56

F i g . 2 - E f f e c t o f modest c o l l i s i o n a l damping on t h e r e a l p a r t o f t h e r o o t s o f e q u a t i o n ( 2 . 4 ) . Note t h e ' k i n k ' i n t h e V I branch.

al = + { ( 2 ~ ~ - 1 ) / 2 n ~ + i ~ / c ) , ~ = i { ( 2 ~ ~ - 1 ) / 2 ~ ~ + i n / 5 ) , ~ ~ = 11/2% ( 4 . 3 ) Here, f o r canvenience, we i n t r o d u c e t h e a r t i f i c i a l t r a n s f o r m a t i o n s a c *

6+@ 6 ,

y+pl

y = y ' + i Y u and K=K'+iKU , i n t o

S i n c e we a r e \.)orking t o 0 ( n ) t h i s means t h a t t h e t r u e decay f u n c t i o n s c a n

J O U R N A L DE PHYSIQUE

F i g . 3 - Development o f t h e r e a l p a r t s o f t h e r o o t s o f e q u a t i o n (2.4) as t h e c o l l i s i o n a l damping g e t s s t r o n g e r .

F i g . 4 - D i s p e r s i o n diagram showing k a s a f u n c t i o n o f Kr t h e r e a l p a r t o f K.

H e r e q i s c l o s e t o &65 so t h e r e i s s t r o n g i n t e r a c t i o n between t h e c o l l i s i o n a l and t h e s p a t i a l d i s p e r s i o n branch.

F i g . 5 - A complete p r o g r e s s i o n o f d i s p e r s i o n diagramr from 5 ^=O,

79

showing t h e u l t i m a t e s u p p r e s s i o n of t h e bend-back.

be r e c o v e r e d a f t e r m u l t i p l y i n g t h r o u g h w i t h a f a c t o r ( 1 - 2 i n / R ) . The p r e s e n c e o f t h i s f a c t o r does n o t , i n any way, a l t e r t h e p r o c e s s o f s o r t i n g o u t w h i c h branch r e p r e s e n t s a s u r f a c e wave.

F o r t h e r o o t N1 we have f r o m t h e l e f t (LI ) and r r i g h t (RI) hand s i d e o f e q u a t i o n s ( 4 . 4 )

These e q u a t i o n s show t h a t f o r (a1)> 0, ^{( 6} 1 ) > O t h e d i s p e r s i o n e q u a t i o n i s s a t i s f i e d o n l y when 2n2>1. T h e r e f o r e , as i n t h e c o l l i s i o n l e s s case

C5-188 J O U R N A L DE PHYSIQUE

N1 r e p r e s e n t s a s u r f a c e wave o n l y i n r e g i o n III.

I n r e g i o n 1 (0<Q2<0.4)

The s u b s t i t u t i o n o f t h e r e a l and i m a g i n a r y p a r t s o f these e q u a t i o n s i n t o t h e r e a l p a r t o f t h e d i s p e r s i o n e q u a t i o n shows t h a t o n l y N3 r e p r e s e n t s a s u r f a c e wave.

I n r e g i o n III f o r 0 . 6 ~ ~ ~ $ 0 . 9 , N ~ , as d i s c u s s e d e a r l i e r , r e p r e s e n t s a s u r f a c e wave b u t i t is now necessary t o see i f N2 and/or N3 a l s o r e p r e s e n t s u r f a c e waves. The e x p r e s s i o n s r e q u i r e d a r e

Note here t h a t t h e decay f u n c t i o n i n t h e vacuum i s , i n f a c t ,

~ ( K ' - Q ' ) $ and n o t $ [ ( ~ ' - ~ ' ) [fi$]'that has been used f o r convenience u p t o now. T h i s decay f u n c t i o n , f o r N2,3, becomes

The use o f e q u a t i o n ( 4 . 1 3 ) l e a d s t o t h e s t r o n g e s t r e q u i r e n e n t , however, w i t h t h e a u t o m a t i c f u l f i l l m e n t o f ( 4 . 1 5 ) . T h i s r e q u i r e m e n t i s f o r t h e r e a l p a r t o f e q u a t i o n ( 4 . 1 3 ) w i t h t h e p o s i t i v e s i g n , and t h e use o f Ne, t o be g r e a t e r t h a n z e r o . I t i s n o t d i f f i c u l t now t o show t h a t t h e N3 r o o t does n o t r e p r e s e n t a s u r f a c e wave b u t Np does. T h e r e f o r e i n r e g i o n III i n a d d i t i o n t o N1 t h e r e i s a branch c o r r e s p o n d i n g t o N2.

T h i s branch o n l y e x i s t s p r o v i d e d t h a t

This r e s t r i c t i o n on n and 5 c o n t r o l s t h e appearance on t h e d i s p e r s i o n diagram o f t h e f a m i l i a r bend-back loop, i n c o m p e t i t i o n w i t h s p a t i a l d i s p e r s i o n . T y p i c a l l y i n the bend-back regionmû.71+0.72 so t h a t t h e c r i t e r i o n (4.16) reduces t o t h e simple r u l e rl26:65. This, assuming t h a t i t i s q u a l i t a t i v e l y useful even c l o s e t o t h e $20.5, g i v e s a u s e f u l guide as t o whether t h e bend-back l o o p can e x i s t o r n o t . (4.16) i s of course always s a t i s f i e d when c= ^O and i s never s a t i s f i e d when q = 0.

V - NUMERICAL ANALYSIS

Region II i n t h e v i c i n i t y o f ~ ~ 2 0 . 5 does n o t y i e l d t o a n a l y t i c a l a n a l y s i s so i t has t o be i n v e s t i g a t e d c d m p u t a t i o n a l l y . I n Fig.1,2 and 3 are shown computer-generated graphs o f t h e r e a l p a r t s o f NI, N2 and N3. These correspond t o c o n t i n u a t i o n s i n t o r e g i o n II o f t h e previous a n a l y t i c a l a n a l y s i s . A p a r t i c u l a r f e a t u r e i n F i g . 2 i s t h e ' k i n k ' developed i n NI as rl increases. T h i s shows c l e a r l y the i n t e r a c t i o n between the s p a t i a l d i s p e r s i o n branch, which always e x i s t s , and the c o l l i s i o n - i n d u c e d bend-back curve.

Fig.3 shows what happens when the c o l l i s i o n s p l a y a more dominant r o l e.

The t r a n s l a t i o n of t h e a n a l y t i c a l and computational a n a l y s i s i n t o (R,Kr) diagrams i s shown i n F i g s . 4 and 5 f o r t y p i c a l values. Here on d i s p l a y i s t h e f a c t t h a t , when t h e c r i t e r i o n r i 2 6 5 breaks down, t h e bend-back l o o p i s i s o l a t e d , weakened and e v e n t u a l l y suppressed l e a v i n g o n l y t h e N1 s p a t i a l d i s p e r s i o n branch i n e x i s t e n c e .

V I - CONCLUSION

We have presented a v e r y d e t a i l e d a n a l y s i s o f t h e e f f e c t o f c o l l i s i o n s on t h e s p a t i a l l y d i s p e r s i v e plasmon-polariton mode. Through a

combination o f a n a l y t i ' c a l and computational a n a l y s i s i t i s shown t h a t although the c o m o n l y accepted bend-back can occur i t can a l s o be suppressed by t h e s p a t i a l d i s p e r s i o n . It i s hoped t h a t t h i s paper w i l l form t h e b a s i s f o r a wider d i s c u s s i o n o f t h i s ~ r o b i e m .

VI1 - REFERENCES

1. CLEMMOW P.C. and ELGIN J . , J.Plasma Phys. - 12 (1974) 91.

2. KOVENER G.S., ALEXANDER R.W. and BELL R.J., Phys.Rev.Bl4 - (1976)1458.

3. HALEVI P., "Electrornagnetl'c Surface Modes", John Wiley, Ed.

A.D.Boardman (1982) 271.

4. BOARDMAN A. D. , "Electromagneti c Surface Modes", John N i l e y ,

A.D.Boardrnan (1982) 25.

CS-190 JOURNAL DE P H Y S I Q U E

5. FERGUSON P., WALLIS R.F., BELAKHOVSKY M. and TOMKINSON J., Surf.Science - 76.

6. OTTO A., Z.Phys. 216 (1968) 398.