BSTJ 50: 8. October 1971: Bending Losses of the Asymmetric Slab Waveguide. (Marcuse, D.)

4.8 MB·English

Bending Losses of the Asymmetric
Slab Waveguide

By D. MARCUSE

wing Toses of Be enprinsuic lad waepwie are compte
‘Phe compat se baad an the Peay of th ex for 0f te acation
of Maced? natiane ofthe tant seta on te eBid naan pina
ti le fab ee he bt wanegide con be mprrisnatet hy

Straight nye, he result of tha Bary goad arsine we
oeting theory. ft appears the the Peng te fon ea Be ae 4
‘ruinate the bonding totes ofthe roel iad

rer the ferme ave replaced she crrngonling mde
te ron! fer. We present cwoes that aw

the hbo ose af te feat ote een TE
usa

A.J. Marois shorn that a bentslab waveguide lees power by
radition* Tis analysis i betel an slution of the eigenvalue ion
fof the Beak eucequicr. 12 ie peaible to detive the experi far te
bending loess Iron, ni pyran thoory taut is maceh simp than
thesclation of ho cigensa'as cquatias. We us thiseneth derive the
female for tho Inn ‘ces at agymumetic lab wovepuide, The
fymmctaie sab wwe hy suse, ese a thie treatment 8
Fimiting coe, ‘The rele of this approximate thoory. om ery qd
saresnen with the Reory of Maseat i. Furthermore, thr pararseters
bf the HB, sod of tho round eptieal fiber nee used inthe lab waves
tide form, lee value re obi shat ngree well with experimental
Tons als fn this ede

"The bending la sheory prseted in this paper ie bai on te Zl-
Towing in, A bent. lah wevegide est otvesisatly be deers} ina
lindrieal eoontinale spniom sluae axa osineides ith fhe eoter uf
curvntureo dhe wnvagside (Fig, 1). he ohtion of Maxwell's equations
ip ee yon wh eine ad tance hs

in the oslindigal coordi water is hown so that the dupe of the
fei distebotion of the aurved warrguide is noth exept oe oa
Altermined amplitude fier and for the value ofthe onfernuraber of
the eyinderfonotion ofthe eotion. Both of thee waknown garamsters
an be obtained i we eattue thatthe Gl inthe vieity of the waver
ide mint be sinilar to fe field ofthe stra guide ns long athe
‘dius of ourvatc ie Lang, The py eosin elton i uted eloalats
the pomer thet is maatd from the waveguie so hb the pos oss
per ut length caused by the waved rurvatine ea be elerained
‘This procedure leads to a simple equation forthe cureature lew. The
‘theory breaks down whoo the curvature isco severe thatthe fd near
‘the waveguide can xo longer be apprasimated by the field ofthe straight
guido, ‘Tas limits of applicability of the curvature los Uwory ean be
fexpreewed by inequities forthe waveguide panamscere

"Tho Fld of tho straight asymmetric lao waveguide ie ube athe
‘ulation of stmightforward bouaury vale problera, Te geometry of
thesiruolure shown Fig 2. We assure thet there se ld variation
in they ditration an tha the waves ofthe stuetare aro simple TT and
‘TMF mes. We iit nr dseesion a TI mes, ‘The et hen giver
by the flowing eauntions®

Bae farce, (Wy

sinsie— dS sd, Gb)
Swavronie meswiny sues 2558

B, ~ Aw dat + Tain 2d) -esrs-1 Ud

"Pho umplicute of the Sl ean be exoresed by the poner P ented by
the felt

A faotorexp [at ~ dl ns bees suppressed. The constanss and para
ees woman in equations (1) gl (2) ate delined as fellows:

2ef aan feequsney,
pniver eared bythe mode,

= nse permenbilty of Foe apace,
1 aectio permittivity a foe span,

15 = propagation eamstant,

2 ~ ala thiknce,

= GR = 0, @
yao = ae, @
=e — ee, @
f= oteadh ©

= refrotive index in the region <2
thy = lective index in he region —a <2 < dl
hy retrotive iden ine region ae ze —

“Te magnetic eld eoneponens wre obtained from the equations

n--h a
2M nie mk sere moxxaeL Jou, ocr

an,
n,- 5% °

‘The boundary conditions (tequirement of continuity of the transverse
‘eevie and magnetie eld component at the two interface) led ta the
‘Sxcuve equation forthe propagation coustant 8,

wn ad - =, ®

‘Tho solution of Maxell’ equations inthe region (ie }-d) <r-< 2
can bo expressed es fallow:

By = BID od 0)

Tho Hankel function of the swond kind and of order » represints an
cexaet sulin of Moxos equatns nthe eaorioate svete shown i
‘Mg. 1. Thare fs no field variation ‘a the direction ofthe exia of the
‘eliadzieal polar coordiuato system of Fig 1, nth enerdinato ayeora
tellin Fig 2 the dvetion of th polar esate is woul ey) The
rll istanoe ria mesure from tne center? curvntare of he bent
rovenuide, The Henkel fuselina of the retard Kind fs cei snap at
infinite stance, >=, an onward traveling weve Inst roll. With
‘aur time dependence, exp (at), te Feld of ation (10) sino thi

‘The order number » need note an integer in tia ease sie we ned
not reitepevndietyof the fellas wfunetion of the pula seg. TE
sre were inereted in an exit ealution of the problem of the mode
saveling along a ourved waveguide we wld abtain the vas of as ths
lation ofan eigenvalae equacins, nna epproxinsave treatment. we
secure that the fold acne the waveguide cen sil be epprosinstely
Alexorbed by tho dof tho sraigt surceture. Ween ure the econdinste
system ofthe straigne guide (Fig. 2) to decribe the carved guide. "The =

of tho etrrght coordinataayetam booouiea Deut aud we ave the
relation

co ie ay

‘The function exp (—ind) is equivalent to the propagation factor exp
($182) ofthe straight waveguide so that we have the spproximace
‘warnston Rr e 2585,

relation

®

ay

‘Siuoe fe much Inger thar the suvs“ength af hs Fold,» very nego
snarber, Using» se define by (12) in (10) eueittes the est peor
ination, Ie reins to abun ¢eeletua betwoea the empltade factor B
‘nd she power thst i eared by the guided modo. To achive tia we use
‘0 pewimation forthe Hark! Panetion that ie valid for very ange
fonder umber vi the eon

vende as
‘The desired approximation ean be found in Mat.

an
with
a9)
‘We obtain the hyperbue Lames ow hy he tain
[n—(wog}
sony [eo lug
o sm tet —at [—loeg)] a6)

cores 7

“Equntion (12) wae wd to roplnne» by No the axa the waveguide
swe oan ute the approximation r/R ~ 1 so that we obtain from (8),
(2), ad (10)

ah a = ol an

“Thie approximation it edeoquats for the denominator of (MH). The ox
preteion in the exponent rust be aporaximated more wecuruey. We

tue the x enrdinets to describe the redial diatance fm the center ofthe
‘waveyuide core and site
het-%, os

"Thies eontinetecorsond diet the xi of Me straight guide
sa sown in Fig. 2, Uri (4) se (15) le
216 aH M.A wHHMOAT, JOR, OCHRE TE

ay

a tank 4m (LES) oo
‘eg a be ex nent wr
v= GY oY a6) ee
Schein f(t se
¥45G) +5G)+
-HVak ie @ }
‘he it einen os beep get an

econ part of (25) olsine a simple gpomatee zene, Wa thus obtain
‘he approximation

co)
14
= tha B)_1 22, oy
on poz see 88
a

“The approsimations (12), (14, (17), sad (25) allow us to expe the By
component (10) in tke fllowing appresizna frm
vwavesuine assis Lasse: 2a

‘Phe faretional dependener of (25) om the enordinnts = ooinses with
a.

Tor large values of 2 the fed ofthe eurved structure neu the wave-
suide is approximately equal to that of the stright guide. The factor
‘xp (ie) appearing in (21) was omitted frm (La) Compaaon of (In)
ead (26) allows, wilh the help of (2, to deermine the wmpltude
‘oofcont of the Hd,

Jomo?

am

ory fr tam hs won > Rds Fa fiona bs
tame by inna lage rae eth se
ther
2
Fee FAL
Leqoton (28) 4 wry inten. esta i a om the vse
Uh Seid is ver nn ron hel foe sigh srt Were
the fed of te cures racete deers expooetaliy ser tke neNe-
tide icine he fom of sealer she uid The
Ecevit the vet sad rohton (10) of tho eaves eorare xtne
sieved cure wane poe lai

prone nna eo

ince we know the Hild ar fan ube wavegnide i i sow easy #0
cleat the power ase are ie Ce He ia ese adie aay
Thor the wavtguie. Tens of ta waitin Geld independent
Ute senate, The eee Fao is (28H seers onl the
Ses ofthe el, Te jr Ine llered By Ue Gill at given pion»
fan thus be ealelsted ay 2b radial porter fox atthe sumo seston
fren though tae eanisatinn tn hos taditiom may bve came frum
Dama, with «| <e, because each legta elemen: of she guide contribroe
‘nequa emevint of rediation. An lomenc of itlengh the exo Cie
‘waved is pened ova elie a length

wok ea
2588 nay mnuL sins wucnMnAT soMmNAT, aesomN Te

tw istanos + — A en the waveguide, The power I
Tength of waveguide is thus

a por unit

rh,
RE co)

‘Sia ther component uf the Poynting wert,

~~ pats = tna LE on

and P is the power cari! by the mods, Using @7), and G2) we
obtain ftom (20)
owe

a
oe wapnfars te

wit

39)

- ase
|

‘Pho approximation, on the rihe‘awl aie of (9) holds fur 4/6 & 1
"hoelation ea = vt — A wa Islay (82). The rage
of valiity uf the beng hee formas (82) canna Ie given peels
We have already eneounterod she inayunlty (13) thet yas weessiry
tor the appronimtion (14) to Tw, A snslar inequality mabe sere
for fhe fld expressed hy Brel add Neamann fetins inde of the
swuvepude, In urder woo uble li expres: the fl inside of Hie our
‘evepuie by smininatespossons that rotate ta the sine nein
Fonetionsappeting in fh) fv Uke Tit of bay ving of eureatare,

vent oo
seeps noo he eeu nts
soa
2 ;
semis) 0
7

pentli$) oo
reaneneune ssine wns 2539

‘The vulidive of our uLeony becomes deulbttl it one or both of these
Inequalities are viooted, Logever, a eqrapezinm with Maree
‘heey eons Ut var ayant i il pie good enn ops
‘ete (86) zo longer eis,

"The simple expression (325 sir the nto Ios of an ayorantain
flea waveguide an be se ny i ie vals «7, 6 and 238" Roan
Acie of course, on's neneeany lo detarune one of theso patanelars
fom Ue eienueeenatina (singe ee ee al utereoumeted by te
equctone () Vorougt

‘tis uae Vo point wu thal Lae Lge equation @2) eames to be ap-
ligula wor yas of seapice tan tae ane for whisa it eae
Sleived, Leave compara experimerial values? uncles ua
rou ape fiber sit ths loss rodictod by $82), Por sul: eomparcon
{ia noeossary toe she perametom oy, te al apply tothe wes
si to vba the forma et be voted, Tense of th en te
erainctors of the TIE, ral wore ands enpule Use Dealing ace
From 93). "Mhe rewum for his chain? panamesere the tet that the
eremmtar decors the dec avi Af ie Hed ona of the
‘waveguide 1 ory ingests al Une groper Hehl deca is ued 0
‘hati ence gic. to use ey sls of Ue rasa fiber itst of A
sie cng fram (8, if G2) eee be used eo compure che being
Tone ef the roa Sl

“tiv eaiouy tack Ut tn hs Sanaa G3) eee Be abate
vthout use ofthe Fake! Maweion appearing iw (10) 3° e use a fl
he fae

uh m
[op on
th

es

Lae vabiity of this esi cor sul he here by portaeng is
Jeet. Vhs futon font of the exon fant eeomertnat
srltruey. Hissever, fe rxjonea sia Fanti feel admits of @ pel

"The nght seaveeuie as ed chat, eta of te core, vests
seooning tv exp (mee) Tr He curved agen a fs netaraly replaced
boy oT we eons er hat he progr uf hg Ie emg seh
19 Teed nl toe listrien of the phate frm ee an yt 6esebe
te sea ef emi truley o'r deen warelengc
(sis the wvelengel of the enh eed,
MOA aw
“The propane mtn is yall to the wavelongta by the relation

(a

Dy roplsing tae peopagation oman i (4 with (we obtain (8)
Since vans hese a oat cs naeal cp zeplaes oF By y oF
fd thusaies wl ara Ueexponie al inetinn appearing i 2)
Ty uring (97) insread of (20) and prossting vnctly wy shawn in this
sapere obrain (32) once moro. Tt is latorstioe Ut we have thus
niblaived an eppizortion for tho Theil fnesion that held far Sue
elon waace the anor nue evry mel ena the ncaa
spell ce fr he rei cere Une nnn in mc eer Ua ender
tiambec, Equavien (08) Powe eles Cash 9(3 wanes from ta
inca vice nero

One aight Type Fiat w xinalurprowrlare woud alle ua able
sppnonimate express fr the herding lve of the nail otc Ree
Lewover, such allel lewd iran hal ne tn aos

‘eth expeciment, Ov Cre wl Fane ae hs Gc C2} agrsen el
‘vith experimunt if he pacunnten of he oun Her ane use
"Th hw fron (22) his fel wale ef th Ttentive Sse
sav ny sit nde anes epi. Sava ide isrees

{io no eee Fe £22) lo be ald

HHecause of ta lang? numbes variables involve i ia noe pose to
vide jaspbie dip fore. porsble eppiertions. ‘The los formal
(92) suticency amp (aeape forthe ae us ctuing ve vege
paemsers yet.) thot los vitucs fer eee iene ean ea De
Ealoulatad. We provide carves rh lin enmpatog Hs beg a
Sho even PE made nt Femur ai serge

inure & ie oonraariniey ue shear wll ie meat ot Bel

‘Tye ondiraty 4 on Hnetion HOR sale (N8}224,7 i posted on
he abc The earl e dein a ng — ay Ge ae ing
fy — tal The expression BneAN assumes the oouant value 50

fo the otro of Lig. 3. The agreement sith Stare thn