BSTJ 50: 6. July-August 1971: Crosstalk Caused by Scattering in Slab Waveguides. (Marcuse, D.)

6.9 MB·English

Crosstalk Caused by Scattering in
Slab Waveguides

By D, MARCUSE

Light senitered iv one apn’ flee cos be spattered back ante @
wid ste un neighboring optioat Me. Thix tyye of satterng
Erovatal ix nvcaigeted in tha popes for slab wavenaides, ewer,
the veld forthe ah soveguite one ave wpe Luts en the ewe
betwee oo apteal flere, Sooterinynoweall s ezpraced i forms
ff gulet mate radiation tose whose conae ie the rence eetterng
‘ners, Tn the not wecessary fo kro te etal af the seatter-

fener, fst whieh ie mezonenble

Sey weshanos we tong 2
iqeentty, is eown. En aiiition 1 the titel rubation Use the erat
tk deyands ou the width af the vaiaticn tbe a sbiok whe azattesed
rong cere from thes nent The aes fae Pr
orton ta the width of the waiaton lobe at tha i larger shen

he coda le 8 narso. Seatering aovatalk cnn become serous if
oth ides hese sue ovat= sinvatal oyettertion of the aame
Ineckanicct frequency resuiiny sh w radiation tube af early 2670
Iridth te the absence of ¢ tgsewalie imwsnita innefertoe i com
Eeconclied that svatrniyy ecsorate between distetis waveguides
option siers) is wptigile if the radiation tases ave tolerable. Ds

Hs case shappeare enaesiary 2 evpgvese erat hy ie of @
loeny medium fetceen the ive opie,

1b 0 avley panes diene: the enoeeal oteen option in
Laie waveguides eed on the dneetonel enuplerrarheaisoy Ta hat
fag, ecoetalle we ese Ly he tt tt he expen Mery
fii af ane ate reaches the ron of a acighboring guide. This tne
or caesatalic ade eve Hf hoch oe ear emia epee
Aisles eylinder.

‘The presea sper is devoted to across rebenise: pf a eferent
$818 uy gnu, pvsmme raceme. Toma, JULS-AUTT om

type. A dcletric waveguide or aptin! Shor loses power by endstion
Tithe guiding slrcuare i npesiect in any wey. ‘This ligmretion
Inay canst nf deviations jn te gerfot eylindsiral geometry of the
tonedling itera oe it aa eonact ia ruin ssretions of
fefraotive inde digtibaton, Sone of the radiation, causing (oss to
the minded waste ef ane ber, sary teach a seihooring fiber and ex
there be seater! hack iato one of ite guides mutes. This tyne of
Crees f The eaxaed 2p wash impencetions, Hower, jst
the case of the direction rospier elect tie possbie to lnk pe
tsiers of the ellcivenser of this Ore af erotik seelanism to
the loses eatsts to the guise. Thi mae i pombe wo give
‘ery imple ules Thar fsk erecta to radiation lose (see eustion
fs) It ture oul thie sostosing eros i m0 eerous problem
rove cher the wadation lose ot the guiced made ithas ie couse
Ip che fame rethicisr) evans win acweptshle lm

‘Our teratmaent of the veautering crsstalle problem is simplified hy
limiting it 20 fhe east n° the slay mvegdice. Tis inuizey lew
thine the erst bereren slab weveguis wos be stronger the
frosik braeen reese upton ‘ther. 1 Suet, the coer between
pevallel slab waveguides Ss independent of teeie soparnion while tae
frustlk beieeen round ‘thecs mis vary Everly proportional to
their distance Furor, our realm of erastalke ecar:s ont By
esurcing «defile seaiering rechonis. "That, we assis that
the core of eile gue varie in wih Hewever this spo'u aes y=
tion allows slo secre the mane weal case No matter what the
Inochart, sellering at Egbe ean be autzuted to Uke ncivhual
Fourier compensa either ue cory willl variations oof the d=
pre of any orler cuantity som ice porte value. Regard:see of
te prtiular racchensm each Fousiar eompnnent canses 4 narivwy
tuition lobe to expe iv «definite ezeeion, ‘he sauce erzanien
{at prwduced the radiation Tole iv ce guid is rsponibie for the
fapaure ef woe at tie racecion i Ube neigtboring ie. Te thas
mmateral by was meelanism te radesion and seonversion was
preduer, Tn parties, HT we rxptene the strona of the avettering
fucchuis iy the radistion lo sat i sausoe “0 the gifed| made,
‘fo have a deccristion of the eroasall eect thet is inependent of
the tetualcetiermg rshanier, Reshnatien o” these fe beso fue
gently srpfies ee trntmes of the ceatering oro probes,

"Phe eter eere eonteving theory ear. ewily heentended ta ae'ade
statstval assinapione abt the settering mechanisas. We are thus
Gio Lo espves the senttering cuesiak is tore: tee oacnalers
the scetcting, mode lee, the rate of waveguide length to iver space
sexmenas nessa 1819

be,

oy of We ged bitin sd the wk of the sedation

As lseussed in the introdsetion, we ase bning Tae seestment of
sea:teving onstalk an the mvehaaisza of ore wil verintione. TO
Tit with, we cuncetace on efsestalle aed hy sineide) ania
tions of the eer of both sla wavegider ‘The sone genera ease of
ruineary ener wilt vetatione ca be treated by deromning the
Sricucy Colca ita Founce sete avd tiiing the roar o> a
Single sine sinve epenent. Finally, we expnses the erostalin torms
Df rasation‘osste snore rar pide mula and aie tans ahe#0
Trea tees ofthe pertielaesetterng meeanis

Tee well owe, shat livkctre warveaiies puss Lopes of
nots Ke given aperasing frnoney teene se Fite meebo of
tiled moe aul a-sontiza of raghtion modes, ‘The eoaping
Between thesa to types of roe gassed bye vnwimions of ie tne
Usek iow Be een eplerod fa eeror papers ° We en “ive sels

wig ~ Cae will io tae ®

sl!
wa) oe all

anal

seem

Ae ubial, che fector expCh at) bas heen appeesed, The phases
pein g in (1) azal (2 exe given by

@
ena

a
ith = wean)

a

and
1800 Tw rua wevcnae MEOUSTEAT Soman, FOLA-AUEUED samt

come pretoomes cane

Tie, tomer of te mo ay mavegidr Pe Hie ewe eda
Tr uses soe ae gs et

"The sla hef-width i @ m,n. my ae the reactive indies of core
ta ielding The power earriod hy she radition mode ie deine! hy
the ortigomality conuibar th vadirien moder

8 [” gig) de = Ph tea a
Ef wou Piet o am
tole Krone apo we Bp) Dine la ine
1 Amen ged mnt cowl cen ution noes
tes Asial arian of evr neste era
errs o
Acoing Rel 8 eration El ene hy he Hse hike
SSSvarblc ie sheen ven bp
tin 96 — 9b ad

“(ua 2)

sone eon ale, = 24), oe same 2 ay ew

dhe soee [eet ad | oan dl a
sexsi eavisratas tet

‘This formula holde for @ <2 < ee. The savin a bs gives by
a8 Gow 10)

‘The peremetar fie Ese proywpicon verstan: of Oe even guided TE
made. The oe parameters yeti tn (9) are

5 =e — ph uy
cal

aa aa

‘There axe we difezenses between (9) and eution (27) of Ret. &
We exe no desling wi ener thickness ves ‘oms ‘mle of vari
ten of only eof fae fagee a the sah eae. The ge length fe ee
bbe seprace with che length coonfinate + xinee we assue that the
side fe very fig and “aah we ar Tag wel loge tothe Biber
ove and at st # 3 Linn far from taints lly perturbed
‘section of length £. Finally, ne have replace the lferentil dp of
on (22) ReC3

tp ~ Bas, a3}
For larg values of « the fier (Js) (Gin 42/2)/a epproaches w dsl

fonesion. We are thus pernited to vale slow'y varying Funes et
‘i he iegral alae efit

ean

‘The Finite on he
hes uot chug 4 value se oly (he ‘mmiedite visiaity of A ~ 0
ives a contetnsion, The depotdonsy uf p unl 6 una follows from
1B) and (00) The pmnagation eastant Fie gun by d= ae — 2 for
2 = 0}. 12 nurs coreaeal, however, to exszie it in tac 0 the
engl coe whieh the radiation i onving the sla ore? We have

vu cuse as)

sn, Sian,

sesin a ay
122 map way sveveat upeuiNTOAr, oURNAL, JULY-ATIST As

‘The eadioton Fld thet is caused ly scattering ef
even TE guided moe hy a sinoidal vniatinn of cho eo
Js tha spy pane wave

“48 Qug~ noma end
fsa - © ig aad 02 4 stein ata
allow Er cate+ view)

tho phase terms ive by (8, the parsmetcr 5 follows from (0)
bye roplecemon: of with # of 15}

‘This oaleuation prove sie srs the Sap aning-wave radiation
rwulen sellin traveling wv when properly superimpose

“We have co ar ignoral the fete wich of the slang. We cok
cusly take redeetions and infection wt the cladding bound ‘nko
ecouk. However, our dheory i mcent as an orderof-mngnitine
nial of eosrtale eo thatthe ma, rfl Tsses athe cnn
tourdary wil simply be Snore

"As th radiation zeaches the ene of the nelboring slab waveguide,
reflection wad sviracion nro tang plage. Ibis infuiive'y clea that
the plone wave impinging on Tie cote af the aeighboringelsb wave
tide cnn be extseed so soperpedtion of en even and sm ocd
Fadetion nice 0: the guide Th fae, ena be sow thot che plane
‘rave impinging on Ye seoned guide ea be expresed yy

Roa

vith thm even and odd ragistion modes (1) und (2) being expresed
J anordinsta epatouy ventened at ts core of ube second guide. The
iit is eiven by

eel ob nad
feat 2) ar att a

{2s the distaree earcen the exaters uf the nse wavewuide. We rom
Ievow bow the nutisionosigineting atthe fs: guido exetes the radis~
tion modes ofthe eseond guide. 1) we assume that the eee guide
Slo ee ro wer inper‘eetine chan ‘iekiwse change of she
fn if we are eoncernel with the exten of nether even TI’ mod
tly even radi finn wnodee ofthe atwrd guide ean contribute to the
‘vassal: problem:

“The uration of how an even radiation mode exeites an even guid
"TW anade hae beco salvo in oriiple) in Kel. 2 The exeitation of
the guided mde fo expressed iy a1 exltaton coon! cfs}. How

ere gy
eATFFAING CDGHTNLAL 1838

ever, we are intersted only in the value of eat ¢ = Z, To the ft
fonder af pertration rhe aban

0) = ie Ft

c.g = 1 oma em de!
Alor: 2Ylou + 2) Ge cat aa + ant 2h

[re fain

em

Seer sew pommelers have bis intraucel fa 20) tose Fig, 1,
Ris tie sesaration betwen the ener contre af the to guider. The
“inst expareatial fastor of (20) account forthe phase shit atthe plane
‘nove iovide the eludings of che two guides with eladcing thikess
D— 4. The reson exposenta Zamar aeccunte for the Phaze <hite
in the aim a index, betwen the neo guides (Gepertion R — 22),
‘The vale of in the mtn bette wien wa devoted yy
trv {¥6 Fig. The eubsorph dione: the real peat of ws The inane
Ing pet of op aemtie We pare wae til inverse the spe
Iutineen the Oro gute, We have wosuned Ua he care and lang
Dt both guides ae leslie ut al. Zr the poaniity seat the median
Separating the tao ede rag Be Tsay, The lve by Be plane
‘nave i te ign bot nee Use ues by te Cand extn
vetion in 20). "The we part of che sefeaat ve index ef the median
Tkwers the psides i835, the amplitude ot eile the wave
ie gy The rain ny cunts for the slant angle ef the wave and
ReaD ie the seyienion af Uke two gulch felwiingtoneul ding
Aisne), The suiaerple 1 and 2 refer tu aumntites elanging 9
bide I and 2. Pauly, we lave adel the wsnplitade lee eefiets
(2 ad at agcoust Sethe ess muered by each guided mde. Theee
losses cat be thought of as tae rum of bea Lees in each guido piss
rovinion lasce ontsed by the variation of the eon thickness, ‘The
factors my at inloabe Cre amas of he sims theese
varition ef the vores af eush guide. However tis assumed chat the
Detft goometry andthe tefmutive indines of hath gnides ave iene
The mecianiea frequoasies 0, 2xd 0, sro not indepondent ef each
buher. Oniy hea the propagctisn eoastont dy of the guided mode of
the snsond guide and the meshanieal requaney By of the thieknose
‘action ui suce ase ated to eel ter ey Ue equi

Peet en
ean any amount of pawer be coupled from ane guide to the other
asa my mee exeres aeusre

From (10) (with @ = Band A ~ 0) sal (21) we obtain the condition
a8 Goat @
"The range of @, values eontsbnting to ration i given by?
B98 <6 8 +m @
‘nthe mone general ente of»: Fouser spectra of mechanivsl fre-
siuccine of the core tiles vaeation, we seo thet rweh Fousier
Component of gue 2 vrafinn witha verti Fourier canzone of
tulde tte provide erostak

Tlaving determined the amplitude coeisent of the guided made in
ile 2 thet i exited Uy purly sinusoial verietios of Ihe thekness
Of the fiber care me ex inuzoiate™ geascai to the ense of arbix
{rary core thickness vasinlion. We explnd she eore thickness of wath
iid in a Fouter eerie:

en04 (3) Banco tn a
aoe (2) Bousinae v9

swith 8, = vols We choose the Lourie vine serion Deeansa we alrouly
Ienow the roll for ainglesizusoidal thisanen vavition, ae x=
passione (24) and (25) iavolve a complete nt of omhopanal Functions
By replacing

Ya. 8)

wi
oto en

fad suring over ¥ we eonvert (20) co the goxeral ease, Orly one
fimunation ie requied sine euch Fourie component of gulde 1 com
Vines with axly one defite Mosel corsponaa: of gue 2. Hueyer,
Tefore ariting cow she revulling formula we wil calke two more
ttuages Pb, we ean conver: the sty lan intggrel with the help of
‘tho rlacon

3)
seaToRat cnassra 1895

Second, we intraduos che Fousier eoieent g(t) that was iu
fapove the radiation lopses in Ref. 3, BY def ion me bere

a0 Ff tae ae em

‘The wlasion between §(0) and ay Je xiven by the following equation

ee )

9) = Im leleal = —

‘oo portion La indiccts he imagnsy st, Tho we Joma she ratio
of par at the end of quide2, CZ} athe power et she ed af id,
a=

With the Bel of (21) the integral in (20) cap cay be pesfomed
nd we finally ain

Paty) — 9B ented a a
® (sae Sor +%)

quation (22) se the epeesion forthe eos etme or dential
sinh waveguides, Tae eospling reshocion: ie the thiskes variation
of the two waveguide cores. The even guided "TE modes in ether
segue need nat he the aan,

Tz geveal fora she crorialk ratio ig net very wf singe the
see) fanetions gy an gone allyl knw, Lweven, fe is
howble provide estima of the evoestale by using te vadieion
oe sist soode Chay eanee’ bythe eamne mecheran sha,
Iavidee the crosstalk, "he sealtaring hae wae given i Bele 3
ai

Lenton
1806 rae wt sistmMe SeoMSieAL JOrwNAL, SUL-AUGUBE rk

[there is en ort in saesion (te), Ref, & The leftchand se
should reed SPAH4PL),] Equation (0) SMe sam equation (14)
ff Ref. 3 ainoe we hive bere aesursd thut te on thickness varies
‘wile i Ref. # eas sssunned ha ony one side of Une ance elading
Inter in cistorte,

For sipieity me wider nae
cup ganda are Sdenteal acl

see that the Leo modes af che
fly ony rive ware component

tf the Rourir apeetzum ens, We dbus beve x = #2
IO = Lovie = grat MO 0 )
and
feet = Os = Fe xo- mm, my
fF (321 nd (334 wih ws =
Lpyetnaine, oT

‘he inex + fe mwan: to Indioave that wim te radio Toe. Im thi
serial east he roseale i snp seeporsaral to cho sare of the
thode pier iss oodicert 27 nee the poster Low tht the plane
‘ae ae going ifm guide Iso ue 2 Uhrough ce lessy meson
betmnea she guides,

“he venue inition Zo i spy obened Hy cpl pl
srr te coe average {PtH 18). Obaining the ene
‘rorag of (al) ea le sr tpi To snp donnon
Srwtie y hngal mpenin wsncig in (820 on the allen

co)

= 7 Loreen
ish

=~ GBD, en
“With the help of (90) we oun also wre for the ensemble average of