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SURFACE POLARITON SOLITONS
A. Boardman, G. Cooper, P. Egan
To cite this version:
A. Boardman, G. Cooper, P. Egan. SURFACE POLARITON SOLITONS. Journal de Physique
Colloques, 1984, 45 (C5), pp.C5-197-C5-205. �10.1051/jphyscol:1984528�. �jpa-00224146�
JOURNAL DE PHYSIQUE,
Colloque C5, supplenient a u n04, Tome 45, a v r i l 1984 page C5-197
SURFACE POLAR 1 TON
SOLI
TONSA.D. Boardman. G.S. Cooper and P. Egan
Department of Pure and AppZied Physics, University of SaZford, SaZford /VI:, W l ' , Y . K .
Résunié
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Le problème de l a p r o p a g a t i o n de p u l s e s o p t i q u e s i n t e n s e s s u r l e s s u r f a c e s planes d i é l e c t r i q u e s f a i b l e m e n t non l i n é a i r e s e s t c o n s i d é r é . II e s t m o n t r é que des " s o l i t o n s b r i l l a n t s " peuvent e x i s t e r seulement s i l a d i s p e r s i o n de groupe e s t n é g a t i v e . P u i s q u ' e n l ' a b s e n - ce d ' a t t é n u a t i o n , l a d i s p e r s i o n de groupe e s t p o s i t i v e p o u r l e s p o l a - r i t o n s - p l a s m o n s de s u r f a c e , il e s t c o n c l u que l ' e x i s t e n c e des s o l i t o n s e s t c o n d i t i o n n é e p a r une courbure i n v e r s e i n d u i t e p a r l a c o l l i s i o n . La p r o p a g a t i o n dans des c o n d i t i o n s de d i s p e r s i o n normale c o n d u i t l e p l u s probablement à des p u l s e s pouvant ê t r e u t i l i s e s pour l a compres- s i o n d ' i m p u l s i o n s . Ces p u l s e s p r é s e n t e n t l e s p r o p r i é t e s s u i v a n t e s : sommet p l a t , é l a r g i s s e m e n t en fréquence e t c a r a c t è r e a i g u .A b s t r a c t
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The problem o f p r o p a g a t i n g i n t e n s e o p t i c a l p u l s e s on p l a n e weakly n o n l i n e a r d i e l e c t r i c s u r f a c e s i s considered. I t i s p o i n t e d o u t t h a t b r i g h t s o l i t o n s c a n e x i s t on1 y i f t h e group d i s p e r s i o n i s n e g a t i v e . Since, i n t h e absence o f dampinq, qroup d i s p e r s i o n i s ~ o s i t i v e f o r s u r f a c e lasm mon-~olaritons i t i s con- c l u d e d t h a t c o l l i s i o n - i n d u c e d 'bend-back' i s r e q u i r e d f o r s o l i t o n s t o e x i s t . P r o p a g a t i o n under normal d i s o e r s i o n c o n d i t i o n s i s much more l i k e l y l e a d i n g t o f l a t - t o p ~ e d , frequency broadened and pos- i t i v e l y c h i r p e d p u l s e s t h a t c o u l d be used f o r p u l s e compression.1
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INTRODUCTIONN o n l i n e a r e l e c t r o m a g n e t i c s u r f a c e and g u i d e d wave phenomena a r e c u r r e n t l y o f g r e a t i n t e r e s t / 1 - 8 / . Many o f t h e r e s u l t s were o b t a i n e d w i t h o u t assuming t h e n o n l i n e a r i t y t o be weak b u t i n a l 1 cases t h e source o f t h e n o n l i n e a r i t y i s a dependence o f t h e d i e l e c t r i c f u n c t i o n on t h e power c a r r i e d b y t h e wave.
It i s now o f g r e a t i n t e r e s t t o e n q u i r e about t h e b e h a v i o u r p a t t e r n o f an i n t e n s e o p t i c a l p u l s e p r o p a g a t i n q a l o n g Say, t h e p l a n e s u r f a c e o f a semi- c o n d u c t o r o r a m e t a l . I n p a r t i c u l a r , i t i s i m p o r t a n t t o know i f such a p u l s e can e v o l v e t o a s o l i t o n .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984528
C5-198 JOURNAL DE PHYSIQUE
Some s t u d i e s i n t h i s d i r e c t i o n /3,9/ have a l r e a d y been performed b u t i n d u c e a s o l i t o n - l i k e b e h a v i o u r t h r o u g h a dopant o r t r a n s i t i o n p l a c e d a t t h e i n t e r - face between two d i e l e c t r i c media. T h i s i n t r o d u c e s a p o l a r i s a t i o n c u r r e n t i n t o t h e boundary c o n d i t i o n on t h e t a n g e n t i a l component o f t h e magnetic f i e l d . I t was shown i n one i n v e s t i a a t i o n t h a t s e l f - i n d u c e d t r a n s p a r e n c y /3/ may occur, t h r o u g h t h e d e v i c e s t a t e d above, and t h a t o p t i c a l s o l i t o n b e h a v i o u r i s a p o s s i b i l i t y t h r o u g h t h e e x i s t e n c e o f a sine-Gordon t y p e of e q u a t i o n . Both of these s t u d i e s , i n f a c t , use r e s o n a n t l y a b s o r b i n g media t h a t suddenly becomes t r a n s p a r e n t as t h e power i n t h e o ~ t i c a l p u l s e r i s e s above a c e r t a i n t h r e s h o l d .
Obviously, i t i s expected t h a t s u r f a c e plasmons can e x h i b i t i n t r i n s i c n o n l i n e a r i t y . T h i s i s d e t e c t a b l e b u t weak / I O / . N e v e r t h e l e s s an o p t i c a l K e r r e f f e c t o c c u r s t h a t o u g h t t o be s u f f i c i e n t t o s u s t a i n an o p t i c a l p u l s e t h a t may be a s o l i t o n . Because o f t h e weakness of t h e e f f e c t a t h e o r y t h a t r e l i e s upon t h i s assumption s h o u l d be v e r y s u i t a b l e f o r m a t e r i a l s l i k e semiconductors and m e t a l s . Such a t h e o r y has been d e v e l - oped and used w i t h g r e a t e f f e c t i v e n e s s f o r o p t i c a l f i b r e s /11 ,12,13/.
I t g i v e s t h e p u l s e envelope e q u a t i o n as a m o d i f i e d n o n l i n e a r S c h r o d i n g e r e q u a t i o n and i s , q u i t e g e n e r a l l y , a p p l i c a b l e t o a l 1 weakly n o n l i n e a r guided wave systems /12/.
I n o r d e r t o a p p l y t h i s t h e o r y i t i s necessary t o d e t e r m i n e
a7k, ,
where kr i s t h e r e a l p a r t o f t h e l i n e a r wave number and w i s t h e a n g u l a r frequency. au2 The p o s s i b i l i t y o f o p t i c a l s o l i t o n s t h e n depends o n l y upon t h e shape o f t h e 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 curve.I I - ENVELOPE SOLITON THEORY
Suppose t h a t @ ( x , t ) i s t h e t o t a l e l e c t r i c f i e l d o f e l e c t r o m a g n e t i c wave guided i n t h e x - d i r e c t i o n by a system t h a t , i n t h e f i r s t i n s t a n c e , w i l l be assumed t o be o p t i c a l l y l i n e a r . Furthermore, l e t us assume t h a t any non- l i n e a r i t y t h a t can o c c u r i s s u f f i c i e n t l y weak f o r t h e t r a n s v e r s e inhomogen- e i t y t o be i n c l u d e d through t h e use o f t h e t o t a l d i s p e r s i o n e q u a t i o n of t h e waves. The d i s p e r s i o n e q u a t i o n o f t h e s e guided waves can then be d e f i n e d as k 2 = f ( w ) , where k i s t h e g u i d e d wave number, and we can i n t r o d u c e an e f f e c t i v e ( m a t e r i a l p l u s waveguide) d i e l e c t r i c f u n c t i o n ceff(w) and a f u n c t i o n where
Here, a denotes t h e F o u r i e r t r a n s f o r m , O corresponds t o a d i s v i a c e m e n t and
&eff
c o r r e s ~ o n d s t o an e f f e c t i v e r e f r a c t i v e i n d e x . F o r t h e surface guided wave 4 and O a r e t h e n r e l a t e d t h r o u g h t h e one-dimensional wavee q u a t i o n .
Now suppose t h a t a p u l s e i s p r o p a g a t i n g and t h a t i t c o n s i s t s o f a c a r r i e r frequency wo modulated by a s l o w l y v a r y i n q envelope A ( x , t ) i n t h e form
@ ( x , t ) = A(x,t)expi [kox - wot] (2.3)
where k 2 = k Z o = f (r u O ) and
and
(2.7) Hence f o r a l i n e a r medium t h e s l o w l y v a r y i n g p u l s e envelope e q u a t i o n i s t o t h e o r d e r i m p l i e d i n e q u a t i o n ( 2 . 4 ) .
T h i s e q u a t i o n n e g l e c t s t h i r d - o r d e r terms and dampinq. Damping, o f course, can be q u i t e i m p o r t a n t and w i l l be i n t r o d u c e d below. T h i r d - o r d e r terms a r e o f importance c l o s e t o k z = O and c a r e must be t a k e n t o i n c l u d e them i f t h i s i s t h e case. I n a n o n l i n e a r medium t h e d i s p e r s i o n e q u a t i o n becomes k 2 = f o ( ~ , / @ 1 2 ) where now
and a i s t h e n o n l i n e a r c o e f f i c i e n t which i s made n o n d i s p e r s i v e b y mak- i n g i t depend upon a f i x e d c a r r i e r frequency wO. Since t h i s i s t h e case c e f f ( t - t ' ) , i n e q u a t i o n ( 2 . 1 ) , can be r e p l a c e d by ~ ( t - t a ) + a ! m ! 2 6 ( t - t ' ) t o g i v e , a f t e r d e n o t i n g
a L
as t h e l i n e a r p a r t o f a ,2 A
F o r a p u l s e s o l u t i o n ! $ i 2 = A l i and iT < -
-
i w 0 eiwrt so t h a tJ O U R N A L DE PHYSIQUE
Thus t h e f i n a l envelope e q u a t i o n becomes
where i t has been assumed t.hqt. i t i s ?*n?roximately t r u e , f o r a s l o w l y v a r y i n g envelope, t h a t
I n t h e d i m e n s i o n l e s s c o o r d i n a t e s
where v =l/k,', i s a c t u a l l y t h e v e l o c i t y o f a n o n - v a r y i n g envelope p u l s e , t h e f i n a l envelope e q u a t i o n reduces t o t h e now f a m i l i a r s o - c a l l e d non- 4 1 in e a r Schrodinger e q u a t i o n
I t i s i n t e r e s t i n g t o n o t e h e r e t h a t t h e ( ç , ~ ) c o o r d i n a t e s may be s c a l e d i n any way d e s i r e d t o f i t any p a r t i c u l a r s o l u t i o n t o an a c t u a l g u i d e d wave. T h i s s c a l i n g has t h e e f f e c t o f a l t e r i n q v t h e r e l a t i o n s h i p o f p u l s e
¶ '
h e i g h t t o w i d t h and t h e s e t t l i n g times o f any t r a n s i e n t e f f e c t s .
e i 6 x ~ o s e c h [ ( t - x k ~ ) / A ] i s a s o l u t i o n o f e q u a t i o n ( 2 . 1 2 ) where A ii a h a l f - w i d t h . On s u b s t i t u t i o n i n t o (2.12) t h i s g i v e s
where F = ( t - x k A ) / ~ . E q u a t i o n (2.16)shows t h a t t h e a m p l i t u d e o f t h e ~ u l s e has t h e f o r m
T h i s s o l u t i o n i s o n l y p o s s i b l e when k:<O and i s known as a b r i g h t s o l i t o n . Dark s o l i t o n s o f t h e form ak:tanh(F) can a l s o e x i s t when kO>O. B r i g h t s o l i t o n s c o n s i s t o f a s t a b l e i n t e n s e p u l s e o f l i g h t movinq a g a i n s t a dark background and propaqate i n r e g i o n s o f anomalous d i s p e r s i o n (kO<O). Dark s o l i t o n s c o n s i s t o f an i l l u m i n a t e d backqround s u o o o r t i n g an i n t e n s e p r o p a g a t i n g b l a c k h o l e . The l a t t e r i s u n l i k e l y t o be o f any p r a c t i c a l i n t e r e s t .
I f damping e x i s t s i n t h e medium t h e n $ ( x , t ) i s m o d i f i e d t o e x p ( i k x - i w t ) e x p ( - y x ) . Then, p r o v i d e d t h a t ki i s n o t t o o l a r g e compared t o kr, t h e e f f e c t o f
damping on t h e p u l s e envelope can b e a p p r o x i m a t e l y i n t r o d u c e d t h r o u g h t h e a d d i t i o n o f i ( y / k o ) q t o t h e non1 i n e a r S c h r o d i n g e r e q u a t i o n .
F o r any g u i d i n g system, then, t h a t i s weakly n o n l i n e a r i t i s a q e n e r a l f e a t u r e t h a t an e x a m i n a t i o n o f t h e d i s p e r s i o n curves and t h e damping r a t e s w i l l show t h e r e g i o n s o f s o l i t o n p r o p a g a t i o n . T h i s i s done i n t h i s paper f o r a model o f p l a s m o n - p o l a r i t o n s i n a semiconductor. The r e s u l t s a r e t h e n i l l u s t r a t e d w i t h t y p i c a l p u l s e development p i c t u r e s t h a t can be e x p e c t e d f o r b o t h anomalous and normal d i s p e r s i o n regimes.
III
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SURFACE POLARITON MODE DISPERSION I N A SEMICONDUCTORA s u i t a b l e model d i e l e c t r i c f u n c t i o n f o r a semiconductor w i t h s u f f i c i e n t l y h i g h c a r r i e r d e n s i t y i s
S ( Q ) = E - 1
n(st + i n ) (3.7
where EL i s t h e h i g h f r e q u e n c y l a t t i c e d i e l e c t r i c c o n s t a n t . The dimension- l e s s frequency i s n = U J / U where UJ i s t h e plasma frequency d e f i n e d
P' P
w i t h o u t cL and t h e d i m e n s i o n l e s s c o l l i s i o n f r e q u e n c y i s n = v/w where P v i s a c o n s t a n t . I f t h e s u r f a c e mode wave number k i s d e f i n e d as k =
2 K
t h e n t h e 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 i sC
The r e a l and i m a g i n a r y o a r t s o f K a r e Kr and Ki g i v e n by
I f d i f f e r e n t i a t i o n w i t h r e s p e c t t o n i s denoted, f o r example, by
KEK'
t h e nand
K" r s
?
r =(ak
r+ . i ~ r +
bKi+
bKi-
2(2~,k, t 2 ~ ~ i ~ ) k ~ ) / (~IKF+
KPI) ( 3 . 5 ) Now a=N/D, b=nQ3/D so t h a tThe n o n l i n e a r i t y i s i n t r o d u c e d t h r o u g h t h e r e l a t i o n s h i p
JOURNAL DE PHYSIQUE
where K;(W) i s t h e l i n e a r d i s p e r s i o n law. This means t h a t a = J ~ ~ C ? / ( E ( 0 )
+
1 ) 2O O ( 3 . 5 )
f o r n o , t h e dimensionless c a r r i e r frequency. E, i n equation ( 3 . 5 )
a r i s e s because t h e d i e l e c t r i c function of t h e material i s E ~ ( w ) + t 2 ) ~ I 2 .
I f t h e r e f r a c t i v e index of t h e material i s n = n o + n 2 ( E I 2 then i t i s aoprox- imately t r u e t h a t c2=nonz. I t has been assumed here t h a t measurements / I O / of n, a t a c e r t a i n wavelength a r e v a l i d over a much wider wave- length range. Typically no%4, n?0J10-7
-
a t 5 Sm wavelength /S/ f o r InSb.A p l o t of
Kr
a g a i n s t il and ri i s shown i n F i g . 1 f o r a range of frequency c l o s e t o the ( c L + 1 ) - 4 region f o r InSb. Note t h a t i n t h e small q p a r t of t h e p l o tKr
->-
because t h i s corresponds t o a pole i n the (Kr,n)curve t h a t appears a s c o l l i s i o n a l dampinq d i e s away. This pole corresponds t o t h e s u r f a c e plasmon resonance region. As r- i n c r e a s e s the s u r f a c e plasmon-polariton curve, f o r a r e a l frequency and complex wave number s o l u t i o n , exhibi t s 'bend-back'
.
As t h i s occurs, Kr swi tches from normal (K,.,O) t o anomalous ( $ - < O ) group d i s p e r s i o n . For q = O only normal group d i s p e r s i o n i s p o s s i b l e . I t i s c a l l e d qroup d i s p e r s i o n becauseF i g . 1
-
Group d i s p e r s i o n K r of a s u r f a c e p o l a r i t o n on InSb. E L = 1 6 . Al1 f r e q u e n c i e s scaled with U,P '
Kr i s t h e r a t e o f change w i t h frequency o f t h e grouo v e l o c i t y . F i g . 1 shows t h a t b r i g h t s o l i t o n s a r e c o l l i s i o n - i n d u c e d and t h a t t h e r e i s t h e n a p e n a l t y t o pay through damping i n t r o d u c e d b y Ki. Ki i s d i s o l a y e d i n F i g . 2 where i t i s i n t e r e s t i n g t h a t , a l t h o u g h Ki=O a t Q=O, i t can become q u i t e l a r g e even as q+O. I n t h i s r e g i o n of (Ki ,2,n) space t h e group v e l o c i t y i s changing v e r y r a p i d l y which can account f o r t h i s ' r e s o n a n t ' i n c r e a s e i n Ki.
A t y p i c a l development o f a damped ( 2 s e c h ( ~ ) i n p u t ) s o l i t o n i s g i v e n i n F i g . 3. Such a f i g u r e can, as p o i n t e d o u t above, be s c a l e d t o f i t any system so i t i s g i v e n h e r e i n a r b i t r a r y u n i t s m e r e l y t o show t h e p u l s e
Ill k
F i g . 2
-
Ki t h e i m a g i n a r y p a r t o f t h e wave v e c t o r f o r InSb w i t h E L = l 6.development i n a damped system w i t h anomalous d i s p e r s i o n . I t a p p l i e s t o r e g i o n s o f F i g . 1 f o r which Kr<O. F o r such s u r f a c e p o l a r i t o n s i t s p r o p a g a t i o n frequency range i s, a p o a r e n t l y , v e r y s m a l l and q u i t e p r o b a b l y i n v o k e s severe c o l l i s i o n a l damping i n o r d e r t o e x i s t . As t h e p u l s e progresses damping causes t h e p e r i o d t o ' s t r e t c h ' /10,11,12/ and i t a c q u i r e s a shape s i m i l a r t o t h e i n p u t b u t w i t h a d i m i n i s h e d a m p l i t u d e . F i g . 4 shows t h e p u l s e development f o r normal d i s n e r s i o n and w i t h damping.
T h i s i s a more p r o b a b l e s i t u a t i o n f o r s u r f a c e p o l a r i t o n s i n which t h e p u l s e a c q u i r e s a f l a t - t o p appearance. I t i s n o t a s o l i t o n b u t , s i n c e t h e p r o p a g a t i o n i n t h e n o n l i n e a r g u i d e broadens and c h i r p s t h e pulse, due t o p o s i t i v e group v e l o c i t y d i s ~ e r s i o n and s e l f - p h a s e modulation, i t can be a means o f p u l s e compression. T h i s c a n b e done b y s e n d i n g t h i s now l i n e a r l y frequency swept p u l s e t h r o u g h a l i n e a r l y d i s p e r s i v e d e l a y l i n e /14/.
JOURNAL DE PHYSIQUE
2 N 3 ORDER S O L I T O N W l T H D A M P l Y G
F i g . 3 - Anornalous d i s p e r s i o n . Development of 2sech(T) i n p u t pulse in a r b i t r a r y u n i t s showing t h e r o l e of damping.
NORMAL D I S P E R S I O N U l T H D A M P l N G
F i g . 4 - Nornial d i s p e r s i o n . Developnent of 2 s e c h ( T ) i n p u t pulse i n a r b i t r a r y uni t s .
I V
-
CONCLUSIONI t i s concluded t h a t b r i q h t s o l i t o n s can e x i s t o n l y i n c o l l i s i o n a l s o l i d s t a t e plasmas and t h e n o n l y i n a narrow f r e q u e n c y range about CO P 1-1
.
Dark s o l i t o n s may e x i s t o v e r t h e r e s t o f t h e frequency range b u t t h e s e a r e o f t h e o r e t i c a l i n t e r e s t o n l y . The m a j o r i t y o f a s u r f a c e plasmon- p o l a r i t o n d i s p e r s i o n c u r v e i s a r e g i o n o f normal group d i s p e r s i o n w i t h K,>O. I n t h i s r e g i o n an o p t i c a l p u l s e w i l l a c q u i r e a f l a t - t o p o e d appearance and be frequency broadened and p o s i t i v e l y c h i r p e d . T h i s i n i t s e l f c o u l d be o f g r e a t o r a c t i c a l s i g n i f i c a n c e suggesting, as i t does, a scheme f o r p u l s e compression.
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