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A.D. Boardman and P. Egan ... mode that arises under real frequency and complex wave number .... orking to 0(n) this means that the true decay functions can ...
JOURNAL DE PHYSIQUE Colloque C5, supplément a u n04, Tome 45, avril 1984

page C5-179

THE INFLUENCE OF COLLISIONAL DAMPING ON SURFACE PLASMON-POLARITON DISPERSION A.D.

Boardman and P . Egan

Department of Pure and AppZied physics, SaZford M5 4YT, U . K .

University of SaZford,

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

i n p r i n c i p l e , b o t h complex. complex k and r e a l

UJ

UJ

a r e then,

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

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 . and equal t o kr+iki,

a plot of

UJ

If

UJ

i s r e a l and k i s complex,

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 /J2, 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 / .

IfUJ 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

J O U R N A L DE PHYSIQUE

C5-180

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 diagram

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 P 0 . ? < ~ ~ < 0 . 6 a n d 0 . 6 < 2 ~ < 0 . 9 T. n t h e f i r s t a n d

O8x10-~,

Hence i f

127a3d3(4b3+27ad)l

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

1

N2 and N3 a r e g i v e n b y

so t h a t t h e t h r e e r o o t s NI, 3(2)%aN1=2%b2tp%,

(3.2)

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 N3Zl a r e small 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 =

Plasma:

a 2 = 1+K2-fi2

~

2

-

~

2

y2 = ( [ 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 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

so

n2a-(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

cannot

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 . For N we o b t a i n 2/ 3 ='fi

2

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

Now u s i n g p o s i t i v e s i g n s f o r a

2/3

2%

1

(3.15)

a substitution i n t o the and 6 2/3

dispersion equation r e s u lt 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 region

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 equation (3.11).This

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)> that

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

JOURNAL DE PHYSIQUE

This choice leads 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 (6)> 0 o n l y changes t h e

@(a)

cancel out.

i m a g i n a r y p a r t s o f a and 6 w h i c h t h e n

Hence N2 i s a l s o n o t a s u r f a c e wave.

T h i s means t h a t i n

t h e c o l l i s i o n l e s s s p a t i a l l y d i s p e r s i v e plasma o n l y N3 shown i n F i g . 1 r e p r e s e n t s a s u r f a c e wave i n t h e r e g i o n 1 below R k 0 . 5 w h i l e NI 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

I I I above ~fi0.5. The above

t e c h n i q u e w i l l now be used t o a n a l y s e t h e much more d i f f i c u l t case when c o l l i s i o n a l damping i s i n c l u d e d .

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

Fig. 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 nc~ -, l - ~ ~ The i ~ 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

0

lf12c

-50

- 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

t h a t c o n v e r t t h e r e a l and irnaginary p a r t s

o f t h e d i s p e r s i o n e q u a t i o n , a f t e r w r i t i n g a=a' t i a " , 6 = 6 ' + i 6" y=y'+iYu

and K=K'+iKU

,

,

into

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

Fig. 3

Fig. 4

-

-

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) c o l l i s i o n a l damping g e t s s t r o n g e r .

as t h e

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 close t o 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.

&65

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 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.

5 =O, 79 O

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 equations (4.4)

These e q u a t i o n s show t h a t f o r i s s a t i s f i e d o n l y when 2n2>1.

( a 1 ) > 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 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