theory of optical wavwguides

8/8/2019 Theory of Optical Wavwguides

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Optical and Quantum E lectronics 11 (1979 ) 283 -28 6

ReviewT h eo ry o f optica l w aveguides

L . B . F E L S E NPolytechnic Inst i tute o f New York, Farmingdale, New York, USAReceived 2 6 February 1979

This repo rt summarizes the act iv it ies at the 3 rd International Workshop on Optical Waveguide The ory,held at the Scuola Superiore Guglielmo Reiss Ro mo li, L 'Aqu ila, I taly, on 9-11 September, 1978.

1 . In t roduct ionThe 3 rd In t e rna t iona l W orkshop on O pt i ca l Wave-g u id e T h e o r y f o l l o w e d th e p r e c e d e n t a n d f o r m a to f t h e t w o p r e c e d in g W o r k s h op s a t L a n n i o n ,F rance (1976) and R e i sensburg , Wes t Ge rma ny(1977) . E ach W orkshop , he ld jus t p r io r t o theE u r o p e a n C o n f e re n c e o n O p t i c a l C o m m u n i c a t i o n ,has sough t to address cu r r en t t heore t i ca l p rob-lems in opt ical guided-wave the ory , to c lar i fyf u n d a m e n t a l c o n c e p ts , a n d t o i d e n t i f y p r o b l e mareas need ing fu r ther a t t en t ion .

T h e W o r k s ho p i n L ' A q u i l a w a s a t te n d e d b y26 par t i c ipan t s f rom n ine coun t r i es (Aus t r a l ia ,Den mark , F ran ce , Grea t Br i t a in , I t a ly , Japan ,Nether l ands , Un i t ed S ta t es o f Am er ica , Wes tGermany) . Some 25 shor t con t r ibu t ions werepresen ted b u t subs tan t i a l t ime w as spen t in d is -cuss ion , bo th fo rmal and in fo rmal . S ince thecon t r ibu t ions dea l t t o a la rge ex ten t wi th con t inu-ing r esearch on . sub jec t s i n t rodu ced a t t hepreced ing work shops , t he sum ma ry here wi l l bebr i e f bu t con ta ins an upd a ted l i s t o f re f e r ences .I n th i s l is t [ 1 - 2 8 ] , t h e n a m e o f t h e a u t h o r w h opresen ted the mate r i a l a t t he W orkshop i s i n bo ldlet ters . For m ore detai l , the reader i s refer red tot h e r e p o rt o n t h e 2 n d W o r k s h o p b y P e t e rm a n na n d S o m e da [ 2 9 ] .2 . Techn ica l p rogramThe t echn ica l p rogram a t t he W orkshop compr i sedfou r sessions ent i t le d:

1 . Waveguide and in t eg ra t ed op t i cs the ory2 . M o d a l m e t h o d s a n d m o n o m o d e f i br e s3 . R a y m e t h o d s a n d m u l t i m o d e f i br e s

4 . F ib re imp er f ec t ions and coup l ing phen om enaRather th an fo l low the session fo rma t in thedescr ip t ion o f the var ious con t r ibu t ions , i t i s moreappropr i a t e to g roup them accord ing to top ica lc o n t e n t .2 . 1 . ' E x a c t ' s o l u t io n s2. 1.1. Form ulat ionUsing dyad ic Green ' s f unc t io n metho ds , vec to rmo dal metho ds , spec t ra l t heory , i n teg ra lequa t ions, e tc . , i t i s possible to form ulate uniq uelythe p rob lem of the source-exc i t ed e l ec t romagne t i cf i e ld in a r e l a t ive ly a rb i t r a ry env i ronm ent . Suchprocedures have been em ploye d to dea l wi thgu ided p ropaga t ion wh en the w avegu ide hasarbi t rary perm it t iv i ty and cross-section, and isemb edded in one o r more ex te rna l med ia [1 ] .2.2.2. SolutionTo r educe the genera l f o rmu la t ion to t r ac t ab lefo rm , even by num er ica l means , i t is necessa ry tospecial ize . Par t icular c onf igura t ions that werediscussed include radial ly inho m oge nou s dielectr icwaveguides [2 , 3] , in tegra ted opt ical waveguidesem bed ded in a p lanar hal f-space subst rate , andcascades o f d ielect r ic waveguide steps to sy nthesizegra ting coup le rs o r t r ans i t ions in p l anar ge om et ry[4 ] . Var ious ana ly t i ca l mode l s and num er ica lm e t h o d s w e r e e m p l o y e d t o d e a l w i t h t h e s est ructures: p iecewise con stan t layers for variableprof i les [2] sur face and volum e integralequa t ions [1, 3 ] , d i sc r e ti zed Four i e r tr ansfo rms[5] , and discret ized overal l t reatment of f ie ldsinc lud ing bo th d i sc re t e and con t inuous spec t r a [4 ] .

0 3 0 6 - 8 9 1 9 / 7 9 / 0 4 0 2 8 3 - 0 4 5 0 2 . 4 0 / 0 9 1979 Chapm an and Hal l L td . 283


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L. B. Felsen

The ava i labi l i ty of r igorous solut ions is ofp r i m e i m p o r t a n c e n o t o n l y per se bu t a l so fo ra s ses sme nt o f the q ua l i ty o f a pprox ima te s c he me s .T he quo ta t ion ma rks on the s e c t ion t i t l e r e fe r tothe f a c t th a t a c ha ra c te r iz a t ion o f so lu t ions a s'ex ac t ' requires , a t th e ve ry leas t, a re liable es t i-ma te o f the e r ro r . Suc h e s t ima te s a re d i f f i c u l t toob ta in bu t un le s s th i s is done , the u t i l i ty o f theso lu t ions i s s e rious ly impa i re d . C le a r ly , mo re w orkis needed on this essent ia l aspec t .2.2 . Asympto t i c so lu t ionsH i g h f r e q u e n c y a s y m p t o t i c m e t h o d s a r e a t t ra c t iv ea t op t ica l f requen c ies since th e scale of var ia t ionof e n v i ronm e nta l f a c to r s i s ge ne ra l ly sma l l c om-pa re d to the loc a l wa ve le ng th . Asym pto t icsolut ion s a re genera l ly pre sent ed as se r ies expan -s ions in inve r se in te g ra l o r f r a c t ion a l pow e rs o ft h e ( l a rg e ) w a v e n u m b e r k . A l t h o u g h a s y m p t o t i cse ries genera l ly diverge , the y a re useful fo re va lua t ion o f the f i e ld p rov ide d tha t th e s er ie s ist runca ted before success ive te rms begin to increase .L oo se ly spe a k ing , the t ru nc a t io n e r ro r i s e qua l tot h e f i r st o m i t t e d t e r m . E v i d e n t l y , t h e q u a l i t y o fthe a pprox ima t ion i s the be t te r the more in i t i a l lyde c re a s ing te rms c a n b e inc lude d . In v ie w of th i sa s p e ct , it i s i m p o r t a n t t o e m p l o y a s y m p t o t i cp roc e d ure s tha t gua ra n te e the ge ne ra t ion o f thee xa c t e xpa ns ion c oe f f ic ie n t s . T he c or re spond ingso lu t ions the n s erve a s r e fe re nc e s fo r c o mp a r i sonwi th r e su lt s tha t a re de r ive d f rom phy s ic a l lymo t iva te d o r s im i la r c ons ide ra t ions . In s tud ie s o fop t im um re f ra c t ive - inde x p ro f i l e s tha t e l im ina teor min imiz e g roup de la y d i f f e re nc e s be tw e e nmod e s in a mul t im ode f ib re , the a va i la b i l ity o f thee xa c t c oe f f ic ie n t s in the a sy mp to t ic e xpa ns ion o fthe p ropa ga t ion c ons ta n t i s e spe c ia lly s ign i f ic an t .

T h e m o s t c o m m o n l y e m p l o y e d a s y m p t o t ict e c h n i q u e i s t h e r a y ( o r W K B ) m e t h o d w h i c husua l ly p rov ide s the c o r re c t l e a d ing te rm of thea sym pto t ic e xp a ns ion a nd a lso g ives po igna n tphys ic a l ins igh t in to the p rop a ga t ion me c ha n ism.I t i s ba se d on the t r a c k ing o f loc a l p lane wa ve sw i t h r e a l p h a s e. T h e m e t h o d m u s t b e a u g m e n t e dby ' un i fo rm re pre se n ta t ions ' in t r a ns i t ion r e g ionssur round ing c a us tic s o f the r a y sys te m, a nd i t mus ta lso be mod i f ie d to a c c ou n t fo r e va ne sc e n t f i eldson the 'da rk ' s ide of caus t ics . Recent genera l-i z a t ions ha ve ta ke n in to a c c ou n t the pos s ib lele a kage , due to tunne l l ing , a nd a l so the l a te ra l284

sh i f t , wh e n a r a y i s to ta l ly r e f le c te d a t a c o re -c l ad d i n g b o u n d a r y , t h e r e b y e x t e n d i n g g e o m e t r ic a lop t ic s to inc lude p re v ious ly ne g le c te d phe no me na[6 , 7 ] . Con t r ibu t ions a t the Workshop de a l t w i tha pp l ic a t ions o f the r a y m e th od to s labs a nd f ib re swi th g ra de d- inde x p ro f i le s [8 ] , to f ib re s w i the l l ip t ic c ross-sec t ions [9] , to curved f ibres [10] , tora ndomly d i s to r te d f ib re s [11 , 12] a nd to thes t u d y o f i n t eg r a t e d o p t i cs c o u p l e rs [ 1 3 ] .

Fo r p rop a ga t ion in g ra de d- inde x s la bs a ndf ib res , the e va ne sc e n t wa ve me tho d , ba se d ont ra c k ing o f loc a l p la ne wa ves wi th c om ple x pha se ,p rov ide s a n a l t e rna t ive s c he me th a t e l im ina te s thepre se nc e o f c a us t ics a nd fu rn i she s a phys ic al mode ld i f f e r ing f rom tha t a s soc ia te d wi th the c on ve n-t io n a l r a y m e t h o d [ 1 4 - 1 6 ] .2 .3 . A p p r o x i m a t e m o d e l sT o r e d u c e t h e c o m p l e x i t y an d i n t r a c t a b i l it y o fr i g o ro u s l y f o r m u l a t e d p r o b l e m s a n d t o p r o v i d ea na ly t ic a l r e su l ts use fu l to the e x pe r ime n ta l i s t a ndsys te m de s igne r, a ppro x im a te mo de ls a re o fspe c ia l imp or ta nc e . S uc h mode ls ma y be ba se d onphys ic a l ins igh t, ma the m a t ic a l a nd (o r ) nume r ic a ls impl ic i ty , e xpe r im e n ta l da ta , e tc . A t the Work-s h o p , a p p r o x i m a t e m o d e l s w e r e p r o p o s e d t ~ d e a lw i t h p u l s e d is p e rs i on , m o d e c o u p l i n g [ 1 7 - 1 9 ] ,s ignal c ohe re nc e , a nd r a ndo m imp e r fe c t ions ins ingle f ibres [20, 21] , and with spl ices and cascad-ing in f ib re l inks [2 1- 23 ] . E spe c ia l ly s ign i f ic a n ti s the no t ion o f the c o l le ct ive be ha v iou r o f g roupsof mode s in mu l t imo de f ib re s sinc e the d e sc r ip t ionof mod e c o up l ing e f fe c t s , wh e n p re se n t , is sub-s ta n t ia lly s impl i f i e d the re by .

I f t h e y a r e t o b e u s e f u l, a p p r o x i m a t e m o d e l sshou ld c om bine s impl ic i ty w i th b roa d a pp l ic a b i l ity .T o te s t the i r va l id i ty a nd l im i ta t ions , the y m us tbe c om pa re d w i th va r ious spe c ial t e s t p rob le msfor w hic h a c c ura te so lu t ions a re a va i la b le, o r w i thre l ia b le me a sure m e nts . Som e o f the ide a s pu tfo rw a rd a t the Worksho p , whi le p romis ing , r e qu i reth i s subs ta n t ia t ion be fo re the i r a dop t ion c a n ber e c o m m e n d e d .

A n i n t e r e st i n g ex t e n s i o n o f t h e o p t i m u ma -prof i l e fo r w ide -ba nd long d i s ta nc e t r a nsmis s ionis the mul t ip le c ~ -prof il e [24 ] . T he in t rod uc t io nt h e r e b y o f a d d i t i o n a l d e g re e s o f f r e e d o m p e r m i t sthe b e t te r m ode l l ing o f phys ic a l p rope r t i e s o f thef ib re , a nd a be t te r e qu a l iz a t ion o f moda l g rou pde lays .


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Theory o f opt ical waveguides

LT h o s e w h o o r eno t Th eoristsTh eorists in App liedElectromognetics

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Testproblems~ " {wi th erro r bounds)/ \! i I . o o o r ,c o ,3 . G e n e r a l o b s e r v a t i o n sT h e p a p e r s p r e s e n t e d a t t h e 3 r d W o r k s h o pp r o v i d e d a f a i r l y r e p r e s e n ta t i v e , t h o u g h f a r f r o mexhaus t ive , c ros s - s ec t ion o f cu r ren t p rob lem s inop t ica l wavegu ide theo ry . P rogress had c lea r lyb e e n m a d e i n t h e u n d e r s t a n d i n g o f c e r ta i n f u n d a -m e n t a l a s p e c t s t h a t w e r e p o s e d a t t h e 1 st W o r k s h o pt w o y e a r s a g o . R e f e r r i n g t o a s y m p t o t i c m e t h o d s ,th i s inc ludes r ays and m odes , the g loba l (m od a l )ver sus loca l desc r ip t ion o f r ay f i e lds , the ro le o fe v a n e s c e n t w a ve s a n d c o m p l e x r a y s , a n d t h er e l a t io n b e t w e e n r a y a n d W K B m e t h o d s . E f f e c t s o fe l l ip t i c i ty , m i c r o b e n d i n g a n d o t h e r i m p e r f e c t i o n s ,e t c . , a r e b e t t e r u n d e r s t o o d . W i t h th e r e f i n e m e n to f e x p e r i m e n t a l t e c h n i q u e s a n d t h e r a p i d l y in c r e as -i n g a v a i la b i li ty o f m e a s u r e d d a t a [ 2 5 - 2 7 ] i m p r o v e -m e n t s a r e n o w n e e d e d i n t h e p h y s i c a l m o d e l s o ff ib res used as the bas is o f theo re t i ca l p red ic t ion s .T h i s r e q u i re s i n p u t s f r o m e x p e r i m e n t a l i st s a n df i b r e m a n u f a c t u r e r s o n t h e p h y s i c a l c o m p o s i t i o n o fthe f ibre as wel l as on mec han ical s t resses , an iso-

t rop ies , s t rong f i e ld e f f ec t s , e t c . , tha t m ay a f f e c te lec t r i ca l p rope r t i es , no t on l y in f ib res used asg u id i n g e l e m e n t s b u t i n a n e w g e n e r a t i o n o f i n t e -g r a t e d o p t i c a l c o m p o n e n t s [ 2 8 ] . F i b r e d e si gn i sa l so in f luenced by @s tem cons idera t ions , wh ichm a y i m p o s e c o n s t r a in t s o n p r o p o s e d m o d e l s o r o nt h e d e g r ee o f d e t a i l r e q u i r e d f r o m a c a l c u l a ti o n .

W h e n t h e o r e t i c a l s t u d ie s a r e p e r f o r m e d w i t ht h e s e c o n s i d e r a t io n s i n m i n d , t h e i r u t i l i t y t o t h eu s e r c o m m u n i t y w i l l b e e n h a n c e d s u b s ta n t ia l l y:O n e m a y h o p e i n t h is w a y t o t r a n s f o r m t h ediscipl ine of applied e l e c t ro m a g n e t i c s f o r o p t i c a lp ropaga t ion in to a d i s c ip l ine o f applicable e lec t ro -m agne t ics . T he b ox d iag ram in F ig . 1 il lu s t ra tes ,w i t h s o m e f a c e t i o u s r e f e r e n c e t o t h e c o m p o s i t i o no f t h e t h e o r e t i c a l c o m m u n i t y , t h e i n t e r a c t iv ep r o c e s s a n d t h e n e e d f o r v e r i f i c at i o n o f m o d e l sby com par i son w i th r igo rou s ly based r esu l ts .

Succes s in th i s end eavo ur can be ach ieved on lyt h r o u g h m o r e i n te n s i v e d i al o g u e a n d i n t e r a c t i o nb e t w e e n t h e o r e t i c ia n s a n d t h e v a r i o u s u s e r

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theory of optical wavwguides

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