NONLINEAR SURFACE AND GUIDED POLARITONS OF A GENERAL LAYERED DIELECTRIC STRUCTURE

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NONLINEAR SURFACE AND GUIDED POLARITONS OF A GENERAL LAYERED

DIELECTRIC STRUCTURE

A. Boardman, P. Egan

To cite this version:

A. Boardman, P. Egan. NONLINEAR SURFACE AND GUIDED POLARITONS OF A GENERAL LAYERED DIELECTRIC STRUCTURE. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-291- C5-303. �10.1051/jphyscol:1984543�. �jpa-00224161�

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J O U R N A L D E PHYSIQUE

Colloque C5, supplément au n04, Tome 45, avril 1984 page C5-291

NONLINEAR SURFACE AND GUIDED POLARITONS OF A GENERAL LAYERED D I E L E C T R I C STRUCTURE

A.D. Boardman and P. Egan

Department of Physics, U n i v e r s i t y o f SaZford, SaZford M 5 4WT, U . K .

Résumé - Les modes de s u r f a c e e t l e s modes guidés, à l a f o i s TE e t TM, sont é t u d i é s pour une s t r u c t u r e générale où un f i l m non l i n é a i r e e s t p r i s en sandwich e n t r e deux milieux d i f f é r e n t s non l i n é a i r e s . Q u ' e l l e s admettent des modes du type TE ou TM, on montre que pour t o u t e s c e s s t r u c t u r e s l ' é q u a t i o n de d i s p e r s i o n a l a même forme, à l a f o i s géné- r a l e e t compacte. Ceci v i e n t du f a i t que l e paramètre physique e n t r a n t dans l e s équations de d i s p e r s i o n e s t l ' a m p l i t u d e du champ é l e c t r i q u e à l ' u n e des l i m i t e s du f i l m . A t i t r e de comparaison, nous présentons a u s s i 1 ' a u t r e approche du problème qui e s t c e l l e développée h a b i t u e l - lement dans l a l i t t é r a t u r e . Des cas p a r t i c u l i e r s sont e n s u i t e é t u d i é s numériquement à p a r t i r d'une équation pour l e f l u x de puissance.

Cette équation montre a u s s i , en termes physiques, l e s t r o i s types de s o l u t i o n s u s c e p t i b l e s d ' e x i s t e r dans l e s s t r u c t u r e s l a m e l l a i r e s .

Abstract - The s u r f a c e and guided modes of a general s t r u c t u r e , c o n s i s t i n g of a nonlinear plane s l a b bounded by d i s s i m i l a r non- l i n e a r media, a r e analysed f o r both TE and TM modes. I t i s shown t h a t a l 1 s t r u c t u r e s , independently of whether i f they a r e support- ing TE o r TM waves, a r e governed by a compact generic d i s p e r s i o n equation w i t h t h e same appearance. This i s proved by emphasising t h a t t h e physical e n t r y parameter t o t h e d i s p e r s i o n equations i s t h e e l e c t r i c f i e l d amplitude a t one s l a b boundary. The a l t e r n a - t i v e approach t h a t corresponds t o t h e c u r r e n t l i t e r a t u r e i s a l s o developed f o r comparison. S p e c i a l i s a t i o n s a r e then made t o generate some numerical examples through a power flow equation. This equation a l s o shows, i n physical terms, t h e t h r e e c l a s s e s of s o l u t i o n t h a t a r e possible f o r layered systems.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984543

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1 - INTRODUCTION

The g e n e r a l p r i n c i p l e s u n d e r l y i n g t h e p r o p a g a t i o n o f s t r o n g l y n o n l i n e a r s u r f a c e waves on d i e l e c t r i c media were f i r s t e s t a b l i s h e d f o r s and p - p o l a r i s e d waves i.e. (TE and TM modes) t r a v e l l i n g a l o n g t h e i n t e r f a c e between two s e m i - i n f i n i t e media /1-6/. I n most cases one o f t h e media i s a vacuum. The immediate and s u r p r i s i n g f e a t u r e t h a t emerges f r o m these c a l c u l a t i o n s i s t h e f a c t t h a t i n t h e n o n l i n e a r s t a t e TE modes a r e s u p p o r t e d i n t h e v i c i n i t y o f t h e i n t e r f a c e . As t h e n o n l i n e a r i t y i s s h u t down t h e y d i s a p p e a r . These modes are, perhaps, n o t s u r f a c e waves i n t h e t r u e sense because t h e y a r e l o c a l i s e d a t a s m a l l d i s t a n c e from t h e s u r f a c e . They a r e , i n f a c t , s e l f - t r a p p e d i n t h a t t h e i n t e r f a c e c o u l d be removed t o + ^-/2/. The TM ( o r p - p o l a r i s e d ) n o n l i n e a r modes a r e more con- v e n t i o n a l i n t h a t t h e y do have a l i n e a r l i m i t .

A n a t u r a l e x t e n s i o n o f t h i s e a r l y work i s t o i n c l u d e g u i d e d and s u r f a c e modes o f a l a y e r e d s t r u c t u r e /7-IO/. The s t u d y of such systems i s m o t i v a t e d by t h e search f o r useful i n t e g r a t e d o p t i c s d e v i c e s /Il/ t h a t may e x h i b i t o p t i c a l b i s t a b i l i t y /7,8,11/ o r o t h e r o p t i c a l i n f o r m a t i o n p r o c e s s i n g c a p a b i l i t i e s . The c a l c u l a t i o n s t h a t e x i s t i n t h e l i t e r a t u r e d e a l w i t h b o t h TE and TM modes.

I n t h e symmetric TE /7/ and t h e asymmetric TE /8/ cases, c o n f i g u r a t i o n s c o n s i s t i n g o f a l i n e a r l a y e r bounded by s e m i - i n f i n i t e n o n l i n e a r media, have been c o n s i d e r e d i n d e t a i l f o r f r e q u e n c y independent d i e l e c t r i c f u n c t i o n s . I n t h e s e c a l c u l a t i o n s an a n a l y s i s o f t h e power f l o w l e a d s t o s u g g e s t i o n s o f o p t i c a l b i s t a b i l i t y .

I n t h i s paper we c o n s i d e r a more general s t r u c t u r e t h a t can embrace a l 1 t h e known r e s u l t s . I t c o n s i s t s o f a n o n l i n e a r l a y e r bounded by d i f f e r e n t non- l i n e a r s e m i - i n f i n i t e media. We show t h a t a compact g e n e r i c f o r m o f t h e d i s p e r s i o n c u r v e can be o b t a i n e d t h a t has the.same appearance f o r b o t h TE and TM modes, p r o v i d e d t h a t t h e parameters a r e c a l c u l a t e d a c c o r d i n g l y . The c u r r e n t c o n v e n t i o n a l approach i s a l s o discussed, as a r e some general power f l o w f o r m u l a e . The t h e o r y i s t h e n i l l u s t r a t e d t h r o u g h t h e development o f s p e c i f i c a n a l y t i c a l and numerical examples.

II - GENERAL THEORY FOR THREE NONLINEAR MEDIA

We c o n s i d e r a u n i a x i a l n o n l i n e a r d i e l e c t r i c l a y e r , w i t h boundaries t h a t l i e i n t h e x - y plane, t h a t i s bounded b y d i s s i m i l a r s e m i - i n f i n i t e n o n l i n e a r u n i a x i a l d i e l e c t r i c s . I n g e n e r a l t h e e l e c t r i c and magnetic f i e l d i n each medium a r e

i ( k x -

E ⁼( E x ( z ) , Ey(z), E,(z))e ut) H = ( H x ( z ) , Hy(z), H,(z))e i ( k x - u t ) (2.1 These f i e l d s r e p r e s e n t a s u r f a c e o r g u i d e d wave t h a t i s p r o p a g a t i n g a l o n g t h e x a x i s w i t h wave number k and a n g u l a r f r e q u e n c y m. I n each medium i t

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i s assumed t h a t t h e d i e l e c t r i c t e n s o r has t h e f o r m

where E ~ ( w ) , cll(w) a r e t h e usual l i n e a r d i e l e c t r i c f u n c t i o n s and ^ai s a frequency independent non1 i n e a r c o e f f i c i e n t .

I n each medium t h e f o l l o w i n g b a s i c e q u a t i o n s a r e t h e n o b t a i n e d TE Modes:

where KE ⁼^{k 2}^- 2 EL

TM Modes:

where ^K^: = k 2 - c l 1 c2

A l 1 t h e development f o r t h e TE case now uses o n l y t h e E f i e l d component and f o r t h e TM case o n l y t h e Ex f i e l d component. We now c o n s i d e r t h e TE Y and TM modes s e p a r a t e l y .

TE Modes

As s t a t e d above o n l y t h e E f i e l d component o f f i e l d e n t e r s i n t o t h e problern.

Y

T h i s w i l l now be denoted by Ei where i=1,2,3 r e f e r t o t h e l o w e r medium, t h e s l a b and t h e upper medium r e s p e c t i v e l y . . 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 z w i l l be w r i t t e n as Ei. E q u a t i o n (2.3) t h e n i n t e g r a t e s t o

'2 2 ² ²

Ei - ^{( K ~}- AiEi)Ei = ci , i = 1,2,3 (2.5)

where hi = _2c2 and ci i s t h e c o n s t a n t o f i n t e g r a t i o n . F i r s t fi. ti

-

^O^as

z + + ^m, ⁱ⁼1,3, t h e r e f o r e f o r t h e o u t e r s e m i - i n f i n i t e n o n l i n e a r media

Suppose now t h a t t h e n o n l i n e a r s l a b has i t s b o u n d a r i e s a t z=O and z=d and t h a t t h e e l e c t r i c f i e l d a t z=O i s Eo and a t z=d i s Ed. The c o n t i n u i t y o f Ei and Ei a t a boundary t h e r e f o r e ensures, f r o m e q u a t i o n s (2.5) and (2.6), * t h a t

2 2 2 2 2 2

Eo(rl - AIE ) = c2 + E 0 ( ~ 2 - A2EO)

O (2.7)

Hence, a f t e r s e t t i n g E A ~ = E~ we o b t a i n

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and

The f i e l d s i n s i d e t h e s l a b a r e found by s o l v i n g e q u a t i o n (2.5) w i t h c 2 # 0.

The p a r t i c u l a r s o l u t i o n chosen depends upon t h e s i g n s and magnitudes o f

K 2 ) a2 and c2. The s o l u t i o n , i n t h e g e n e r a l case c 2 # O, i s a J a c o b i e l l i p t i c f u n c t i o n . T h i s need n o t be s p e c i f i e d f o r t h e moment and t h e v a r i o u s o p t i o n s a r e k e p t open by w r i t i n g i t a s T ( u 1 m ) where u i s t h e argument and m t h e modulus. The general s o l u t i o n of (2.5) i n t h e s l a b i s , t h e r e f o r e ,

where p, q and a r e e x p r e s s i o n s c o n t a i n i n g K2, a 2 and c 2 and, s i n c e z=O a t E2=Eo,

I f , as i s common p r a c t i c e , t h e i s dropped f r o m t h e argument o f t h e J a c o b i f u n c t i o n and i n a d d i t i o n 3;l i s d e f i n e d as t h e i n v e r s e o f t h e J a c o b i f u n c t i o n t h e n t h e f i e l d i n t h e s l a b becomes

Now e q u a t i o n (2.10) i s a q u a r t i c i n Ed so t h a t i t has t h e f o l l o w i n g s o l u t i o n s

L

Ed = (E3 - E 2 f J ( E ~ - € 3 1 2 + 2 ( a 2 - a 3 ) c 2 c 2 / ~ ' ) / ( a2 - "3) = a+/- (2.14) Equations (2.13) and (2.14) t h e n i m p l y t h a t t h e d i s p e r s i o n e q u a t i o n f o r TE modes i s i n g e n e r a l ,

Here t h e d i s p e r s i o n e q u a t i o n i s expressed e n t i r e l y i n terms o f t h e i n i t i a l parameters (al, a2, a3, ^{E ~ ,}c2, ^{E ~ ,}k, Eo). F o r m a l l y t h e r e a r e f o u r s o l u t i o n s t o e q u a t i o n ( 2 . 1 5 ) b u t t h e r e l a t i v e values o f ai, ^{E ~ ,}Eo may r e n d e r some o f them i n a d m i s s i b l e . F o r i n s t a n c e E: must be r e a l so t h e e x p r e s s i o n under t h e r o o t s i g n must be > O and a l s o t h e whole r i g h t - h a n d s i d e o f (2.14) must be > O t o keep E: > 0.

I n t h e s e m i - i n f i n i t e media t h e f i e l d s o l u t i o n s a r e

-

C -1

E3 = i J'$ ^;K ~ [ C O S ~ [ K ~ ( Z + l o 3 ) 1 ) , z ) d, o 3 > O, x 3 r e a l (2.17) where zoi a r e i n t e g r a t i o n c o n s t a n t s . The v a l u e s o f t h e s e c o n s t a n t s

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determine the p r o f i l e o f t h e f i e l d i n the s e m i - i n f i n i t e media e.g. i f

zol > O then t h e r e i s a maximum o f El a t z = - zol and so on.

Suppose, f o r example, t h a t i n the c e n t r e l a y e r a2 > O, c2 > O then 2(C2 - E ~ ) > (al - a2)E0. 2 Hence i f Eo i s r e a l €1 > s 2 + @ 2 > al.

I n p a r t i c u l a r

1 2 1

where z o 2 = t - _qcn-l( _q2₊_"7), ^p= + K2/Zq2 and q = d-2.

Note now t h a t (2.18) i s a c6mp1ete s o l u t i o n o f equation (2.5). There i s , apparently, another s o l u t i o n p " ~ d [ ~ ( z + ~ " ~ ~ ) l u l , say, b u t t h i s can be expressed i n terms o f (2.18) through ze2 = z o 2 - K/p where K i s t h e com- p l e t e e l l i p t i c a l i n t e g r a l o f the f i r s t kind. Al1 t h e f i e l d s i n t h i s general TE case are now known, s i n c e

, ZO2 = + ^.-1 ^cn-1

Zol = + [COS~[EY/K:]]-' '4

q2 + "2

and the s p e c i f i c form o f t h e d i s p e r s i o n equation i s

TM Modes

I n t h i s case b o t h Ex and Ez are i n v o l v e d but, i n p r a c t i c e , because o f t h e u n i a x i a l form o f t h e d i e l e c t r i c f u n c t i o n the problem reduces t o working w i t h Ex along. It i s t h i s t h a t i s now designated as Ei where, i n general ,

KI ( i ) a l

- - [EL ⁺- ^{E ~ ]}E. = C .

( i 1 ² ¹ ¹

EH

u2 ( i ) . Once again ci=O f o r i = l and 3 c o r r e s ~ o n d i n g t o t h e where r : = k - ^c

c 2

o u t e r s e m i - i n f i n i t e media. The boundary c o n d i t i o n s now a r e

The a p p l i c a t i o n o f these gives

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and

2 ( 3 ) 2 ( 3 ) & j 3 )

[%);,,- ( ^{2 K}3% p l ^E; ⁺⁴^c¹( ~ 3 5 , 2 ( 2 ) ) 2

E q u a t i o n (2.21) has f o r m a l l y t h e same s t r u c t u r e as e q u a t i o n (2.5) i n t h a t i t s s o l u t i o n i s , once again, a Jacobi e l l i p t i c f u n c t i o n .

T h e r e f o r e even i n t h e TM case t h e f i e l d component i n t h e n o n l i n e a r s l a b i s s i m p l y

E2 = t p ' n 8 ( z + 20:) l u 8 ) (2.26)

Here pl, q' and p' are, o f course, d i f f e r e n t f r o m t h o s e t h a t a r o s e i n t h e TE case. Nevertheless, f o r TM modes, t h e general d i s p e r s i o n e q u a t i o n has t h e same g e n e r i c f o r m as f o r TE modes i . e .

where a+/- i s EG o b t a i n e d f r o m e q u a t i o n (2.25). T h i s i s a v e r y i n t e r e s t i n g forma1 e q u i v a l e n c e o f appearance f o r b o t h t h e TE and TM d i s p e r s i o n equa- t i o n s . Indeed a l 1 t h e known cases i n t h e l i t e r a t u r e s h o u l d emerge from t h e same g e n e r i c formula.

For t h e t h r e e n o n l i n e a r media system analysed here w i t h , f o r example, a2 < O, c2 > O and t h e TM forms o f p', q' and u ' , t h e s p e c i f i c f o r m o f t h e d i s p e r s i o n e q u a t i o n i s

where

111 - ALTERNATIVE O R M OF DISPERSION EQUAJTIF

The method developed i n s e c t i o n II has t h e m e r i t o f p r o d u c i n g a g e n e r i c f o r m o f d i s p e r s i o n e q u a t i o n t h a t i s a p p l i c a b l e t o a l 1 s t r u c t u r e s and b o t h TE and TM modes. It i s a v e r y c o n v e n i e n t a l g o r i thm f o r computational purposes i n which t h e a p p r o p r i a t e c a n d i d a t e from any o f t h e t w e l v e Jacobi f u n c t i o n s can be r e a d i l y i n s e r t e d . I n t h e l i t e r a t u r e , however, a more forma1 method o f e n q u i r y has been adopted r e s u l t i n g i n separate and

a p p a r e n t l y v e r y d i s s i m i l a r developments f o r TE and TM modes. T h i s approach

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w i l l be used i n t h i s s e c t i o n f o r comparison w i t h t h e above a n a l y s i s and t o make c o n t a c t w i t h t h e e x i s t i n g l i t e r a t u r e .

TE Modes --

F o r ni > O t h e f i e l d components a r e t h o s e g i v e n by e q u a t i o n s (2.16), (2.17) and (2.18). The a p p l i c a t i o n o f t h e boundary c o n d i t i o n s t h e n g i ves t h e forma1 d i s p e r s i o n e q u a t i o n

where zoi have been d e f i n e d e a r l i e r i n t h e t e x t . TM Modes

( ' ) > O , t h e a ~ o l i c a t i o n o f t h e F o r TM modes, u s i n g a i < O, ci < 0, ~2 3

boundary g i v e s t h e forma1 d i s p e r s i o n e q u a t i o n .

Equations ( 3 . 1 ) and (3.2) reduce t o t h o s e t h a t e x i s t i n t h e l i t e r a t u r e . F o r example, i f a l + O, a3 + O so t h a t zo, -> and t a n h ( ~ ~ z 6 ~ ) + 1 and t a n h r 3 ( d - 183) + 1 as a3 -> O t h e n f o r c i 1 ) = ^E, and c j 3 ) = c i 3 ) = c3 t h e TM d i s p e r s i o n e q u a t i o n becomes

€ l K 3 - - ~ n ( q ' ~ h 2 ) d n ( q ' z 6 2 ) c n ( q ' d - q ' z b 2 )

€ 3 ~ 1 s n ( q ' d - q ' z o 2 ) d n ( q t d - q ' z 0 2 ) c n ( q ' z 8 2 ) as o b t a i n e d by Fedynanin and Mihalache / I O / . I V - THIN SLAB APPROXIMATE FORMULAE : Tb1 MODES

F o r TM modes i n a t h i n n o n l i n e a r s l a b between two d i s s i m i l a r l i n e a r media, p r o v i d e d t h a t sl = K ~ , s3 / = ~K~ ~ and s2 = K2 ~ / , ~t h e t h i c k n e s s o f I ~ ~

/ E 3

t h e s l a b i s g i v e n by

d e p'2s2(sl + - Z u t ) + Z V ' E ~ I - ~ ' ~ 1 2 1 2 (4.1) I n t h e e v e n t t h a t t h e l a y e r a l s o becomes l i n e a r s 2

-

^r2 ^,^q1

^+&

^~2 ^,

sn(qlzo,) ⁺1 and U '

-

1 so t h a t ^Et1

T h i s i s e x a c t l y t h e s m a l l c 2 d 1 i m i t o f

as d e r i v e d by K l i e w e r and Fuchs / 1 2 / . F o r t h e t h r e e n o n l i n e a r media system

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where q' and ^{p 1}r e f e r t o the i n n e r n o n l i n e a r l a y e r and

Thin s l a b formulae f o r TE modes can be s i m i l a r l y derived.

V - POWER FLOW

For a l i n e a r guide t h e r e i s a unique ( ~ , k ) p l o t so i f we imagine a three- dimensional p l o t o f power f l o w as a f u n c t i o n o f w and k t h e l o c i i n a l 1

W, k planes f o r constant powers would be i d e n t i c a l . As P t h e power l e v e l increases, however, t h e o p t i c a l n o n l i n e a r i t y expressed through ai causes the ( ~ , k ) r e l a t i o n s h i p t o change. An i n t e r e s t i n g surface i n (P, ^U, k ) space then develops and v a r i o u s s e c t i o n s through i t w i l l g i v e v i t a l i n f o r m a t i o n . For instance, i f f i x e d P contour maps o f t h e P, :, k t e r r - a i n a r e produced then these are, i n f a c t , power dependent n o n l i n e a r system d i s p e r s i o n curves. A s e c t i o n through (P,w,k) f o r f i x e d w may g i v e i n f o r m a t i o n on p o s s i b l e b i s t a b l e behaviour /7,8/.

The TM mode power f l o w through a n o n l i n e a r l a y e r bounded by l i n e a r media has already been d e r i v e d / I O / . I f t h i s i s w r i t t e n as P=Pout+Pin, where

Peut i s the power f l o w i n the o u t e r s e m i - i n f i n i t e media and Pin i s t h e power f l o w i n t h e i n n e r media, then the o n l y general m o d i f i c a t i o n needed t o describe TM modes i s a r e d e f i n i t i o n o f Pout t o açcount f o r , i n o u r general s t r u c t u r e , the existence o f o u t e r n o n l i n e a r media.

Since o u r numerical examples concern TE modes a t t e n t i o n w i l l now be r e s t r i cted t o them.

For TE modes, then, w i t h a l , a3 > O t h e power f l o w s i n medium 1 and 3 are

It t h e f i e l d s a t t h e boundaries o f t h e l a y e r are Ebl a t z=O, and Eb2 a t z=d then the t o t a l o u t e r power f l o w becomes Pout=P1+P2 where

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Note here t h a t e q u a t i o n ( 5 . 2 ) does n o t reduce, d i r e c t l y , t o t h e case where t h e o u t s i d e media a r e l i n e a r . F o r a l + O, a3 + 0, and Ai ⁺O t h e two n e g a t i v e s i g n s must be t a k e n so t h a t Pout remains f i n i t e . T h i s makes ( 5 . 2 ) i n d e t e r m i n a t e . A l s o i n e q u a t i o n (5.2) t h e + s i g n i s chosen f o r i n c r e a s i n g f i e l d s a t t h e boundary and t h e -ve s i g n f o r d e c r e a s i n g f i e l d s .

I n t h e i n n e r n o n l i n e a r l a y e r t h e power f l o w i s

F o r c o m p u t a t i o n a l purposes (5.3) may be used d i r e c t l y , o r i t may be expressed i n terms o f E, e l l i p t i c a l i n t e g r a l s o f t h e second k i n d , e.g

I n o r d e r t o i l l u s t r a t e t h e t h e o r y we now s p e c i a l i s e t o a l i n e a r l a y e r , c o n f i n e d between two n o n l i n e a r media. The r e l a t i o n s h i p between Ed and Eo i s g i v e n , i n g e n e r a l , by e q u a t i o n (2.10). I t i s c l e a r f r o m t h i s t h a t a number o f p o s s i b i l i t i e s e x i s t . The t h r e e c h o i c e s a r e Ed=?Eo and EdCEo. T h i s can be seen more e m p h a t i c a l l y , and perhaps more s u r p r i s i n g l y , f r o m t h e case E~ = E ~ , a1 = a3 t h a t reduces e q u a t i o n (2.10) t o

2 2

(Eo - Ed) ^[El - € 2 + (-1

2 ^(E; + E;)] = O (5.5)

Here t h e t h r e e p o s s i b i l i t i e s a r e c l e a r l y d i s p l a y e d and f u r t h e r m o r e i m p l y

2 2 2 2

t h a t Ed + Eo = 2 ( ~ i - € 2 ) , i f EO=Ed. These a r e t h e symmetric, a n t i -

("2)

symmetric and asymmetrlc s o l u t i o n s d e r i v e d i n a c o m p l e t e l y d i f f e r e n t manner by Akmediev / 7 / .

The e x i s t e n c e o f t h e EO~E; s o l u t i o n i s due t o t h e n o n l i n e a r i t y and can be seen f r o m a ' c r o s s - t e r m ' t h a t a r i s e s i n t h e power f l o w f o r m u l a f o r a l i n e a r l a y e r between two d i s s i m i l a r n o n l i n e a r media. T h i s f o r m u l a i s /8/

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C5:300 JOURNAL DE PHYSIQUE

Here s e t t i n g Eo=kEd g i v e s even and odd modes power f l o w s and t h e f i n a l t e r m i s e l i m i n a t e d . I f E ~ # E ; t h e f i n a l t e r m remains and we o b t a i n t h e t h i r d k i n d o f mode. As t h e n o n l i n e a r i t y d i s a p p e a r s t h i s ' c r o s s - t e r m ' d i s a p p e a r s and f o r g u i d e d modes o n l y t h e c o n v e n t i o n a l even and odd modes a r e p o s s i b l e . F i g . 1 shows a p l o t o f TE mode power, i n a r b i t r a r y u n i t s , a g a i n s t n and K f o r a l i n e a r c e n t r e l a y e r w i t h a d i e l e c t r i c f u n c t i o n E ( M ) = E~ +

( E ~ - ~,)/[1 - L I ~ / U ~ ] t h a t corresponds t o a d i a t o m i c c u b i c c r y s t a l . The two o u t s i d e media a r e n o n l i n e a r d i e l e c t r i c s w i t h f r e q u e n c y independent d i e l e c t r i c f u n c t i o n s . The f r e q u e n c y i s n = w/wT and t h e wave number i s K = c/wTk. T h i s f i g u r e shows a c l e a r d i v i s i o n between any s u r f a c e and guides modes. A s u r f a c e wave r e g i o n i s t h e smooth p a r t o f t h e p l o t beyond t h e 'mountain peak' r e g i o n . T h i s l a t t e r r e g i o n becomes v e r y pronounced as w ⁺^WT, as would be expected. The q u i d e d wave r e g i o n shows t h e a l l o w e d modes c a r r y i n g more power. F i g . 2 i s a p l o t f o r t h e same d a t a b u t reaches down t o much l o w e r n. Here i t can be seen t h a t t h e s u r f a c e mode s u r f a c e i s a c t u a l l y t h e dominant f e a t u r e o f t h e t o t a l power p l o t .

Fig. 1 - Even TE modes. Lineat- c e n t r a l frequency dependent l a y e r . c01=2.5, Eo3=2.0, LC,=l .92, &,=9.27,@ k 1.5, %,&, =1.2. Power i s n o r m a l i s e d w i t h bnd,/c .

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Fig. 2 - Even TE modes. Data a s f o r F i g . 1 .

Fig. 3 - Even TE modes. Linear c e n t r a l frequency independent c e n t r a l l a y e r . G 2 = 4 . 0 . Other d a t a a s Fig. 1 .

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C5-302 _JOURNALDE _PHYSIQUE

I f t h e c e n t r a l l i n e a r l a y e r i s replaced by a l i n e a r l a y e r w i t h a c o n s t a n t d i e l e c t r i c function / 7 , 8 / , then the peaked region of Fig. 1 disappears t o be replaced by a much smaller s t r u c t u r e i n t h e lower corner of t h e p l o t . This i s shown, i n d e t a i l , i n Fig. 3. Here t h e peaks a r e q u i t e small but t h e s u r f a c e waves a r e s t i l l a dominant f e a t u r e . The c r o s s - s e c t i o n near t o n s 1 corresponds t o /8/ and a r e s i m i l a r t o / 7 / . Note t h a t i n t h e c o r n e r of Fig. 3 t h e waveguide cut-off locus i s shown, c o r r e s ~ o n d i n g to

K~ and K~ becoming complex. F i n a l l y Fig. 4 shows t h e contour p l o t t h a t

Fig. 4 - Contour p l o t of Fig. 1. a'= ^(0-0.8)^$50,~ < ~ ~ I ( - ~ ) x o . I . Labels on t h e contours a r e a r b i t r a r y giving only t h e r e l a t i v e power magnitudes.

a r i s e s from Fig. 1 . This p l o t can give d i r e c t l y t h e d i s p e r s i o n curves o f t h e nonlinear system. I t can be seen t h a t they would be p r a c t i c a l l y l i n e a r .

VI - CONCLUSION

A general theory of s u r f a c e and guided p o l a r i ton propagation i n a layered s t r u c t u r e has been derived. I t i s shown t h a t a compact formula e x i s t s t h a t accounts i n forma1 terms f o r both TE and TM modes. S p e c i f i c formulae f o r TM and TE modes f o r a number of systems a r e given and t h e theory i s i l l u s t r a t e d with s e l e c t e d numerical examples t h a t r e f e r t o TE modes on s p e c i a l layered s t r u c t u r e s . In p a r t i c u l a r a l i n e a r l a y e r ,

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w i t h a frequency dependent d i e l e c t r i c f u n c t i o n , between two n o n l i n e a r d i s s i m i l a r media i s considered i n terms o f three-dimensional power diagrams. These a r e t h e n contoured t o g e t t h e d i s p e r s i o n e q u a t i o n as a f u n c t i o n o f power l e v e l .

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