BSTJ 50: 3. March 1971: Time Dispersion in Dielectric Waveguides. (Personick, S.D.)

6.8 MB·English

‘Time Dispersion in Dielectric Waveguides
By §.D. PERSONICK

(taxon ave Oetber 16, 0)

In didlectss ovaries opaing at optical frepuancies, the primary
saute of time dirpecnin of nartow puluer can be made comorion. In
this paper tw drgue that under ceva aarumpions a dilcric waweuide
tls au Fnear ate Se imteenty. That tice the ite inp the
{nlonety output ie equal the snp cmsceed sth nm sey ep
respimae, We show that contrary to éntuston, the with of the émpulse
Tespomon gos wacrouse whan coupling bxtwezn guided modes tereese

ing fle peturttion retuts of Di. Meroise, sv olasn an sessing
ne of exe propagation doun simpectct guides, Te eomae thatthe
nent reypise wih Secraiea a8 Ce pare fet of the guide loath for
tacit ny ie and ape penn say for ety

We sonatude from the faery sa the diperaion as Hider wave
vides nag bw rs oj ayers Blog that which was previa ex
eed tn gies of wise ty oth Aang propels contol tare
mune of made conserion. Teas theoretical seeuts aie wt gal been
toed exporimontaly

Th nultivune dvbetrie waveguides apeseting at opstnl freqiensies
Ue prieaary cause of ine dinpssion of narron pales etn be mode
tenversion. Int geotetielig pereet nie wit more than a single
‘noe, energy italy Iounehed ia u sven rele reraine in chat mode
te it pmpagaten don the guide. Physies) guides have imperfections
fora perfeet gaotoetri sage (eg, rwiglotas at the cerecladding is
terface of a nominally right circular eylindviel que) whieh allows
cecky to couple ntween tees during propeestion Gown the aude
Since eroup vrlocitos differ in genesal sanangst the roe, pe of
fenorgy inilly leuoched in a single made er combination uf morte
Sil be broadvnnt de to the spmeud of propaaation times of diferent
parte af tbe nergy
SE naming suocmse ramus sovan, want

In thie papar we angue that unde certain sesumplion, a élloneie
waveguide arts ze a lissaraystem in istensiy os wall asin vole
"Thet ie, we show that che reltionehip between the input intensity and
uepst intent ofthe yoda is ined in Lee uf wo ontensity impulse
responce, We arguc that this intenciey impale resus, for safety
Tong guides, bas a sean-ecuare widta about ite mean whieh inercases
only linearly with length. Further in the lit of very long guides, we
langue that Be response shape i gausian. We also show that the
reaver the oonpling belween wed, the lew the time dispersion —a
‘cult which wt First conttiet intiticn, Fully, we obtain quantite
{ivr reals uo interesting rand of as peal guide under eestain
seruntion®

‘We conshide from the theory that the dispersion in dieuetrie wave
sides may be order: cf magsitude below thnt weich was previouty
fapectod guides oeuoins'y lg leata having propery ee rolea
large amounes af mods conversion. These cheoreieal results have
tot yet been voided experimentally

Wo new line the sto of te deviations te fll, oe that the
sender ean soll ther uf thew

‘We ec with the feor Ute Ue option) guige yw Suan eysteana
voltage. That ia fe separ the it seed in saial nade eed
expan! the extpat eon! mi tho sare cnaes, vn the time varying
codfiems of sho mua at the outpe: are relat tothe eoficients ab
the iaput by a eet of voltage impulse zezpenees, We then mke an ne-
sitnption about 2 seseiatc et of trameter ‘uretions (Wourertrame=
oem of the impslse sesponses} whieh allows us fo argue thatthe set
sf eveiego out iteration and the act of mp intenstice ao slo
elated by a set of anpuisereepanees, hus te gi is leo lear in
Intensity under the assumptions

‘We next argue thet for suficiently lang guides, these intensity im
pulke rezponses coupling a choses input mode eaccient und w chosen
ftp mode coefficient are siernt to the ries ehoeen except par=
Tips for m nngaitude seal Tatar:

insly, this allows so shaw Dit fer ies longer Wan the above
scale, the intensity pulse response whieh 39 now im eoramon for al
input-output pare bus a mean-fcuere deviatios about ite meas whieh
rows lineal in ergth, and which apzeenskss the gacssian saapo in
the lat of log guide,
uoone warReeroe wt

1 he Rondon Channel

We now ange Usk) user a spl seauipon, the expos vie
ef the intensity at thw ontnut of tutto shane! ie seated to Te
Intensity of the Jul tr “te channel by a smpe eowvolntion wih
din inlenwity ispulee reepone. We sagt wth inp. hareband signal
ith. We tse att) to knows wunlslate a easier v(t} which may
be cahennt oF stafinnary ie seme) ram proses eesterel ab
fequeney fe We vero dhat Yao resle (4) = al2hm(?, bes bun

uti Pr iy is sporty esinnds Irmn BQ + fy to BR + fs

‘We pots v0) Hhagugh a time Swenrinnt for with a eundom jmp
response ROG} repiswnting the eau, sealing im the Anal ontpt
VLE We defing & Fourier trafur rlaionshipetmeen the fonetion

1G) ari the fuetion 2

wf

"ee FeO a

0
AQOND,
ysl near sys ae
Xie,
HOH, a
Yip vit
shen
YU = XuuEeD.
‘Define tae envelopes of 212] smd 2) by
a0) = 92 Re be en (ei o
(O92 Re tal0 exp (ni
“The ineensity of che inpst and outpet rignals ste defined as
1 P= oo Ima a)

Pal) = WA,
tionarty of Whaunel tranfer funeon

MFT + a) ~ Uh
provided fy BRS ai ma < ht BE

snore
Assumption:
86a sveraue recsca, oman, ascot 1

The asaumsion, while ayparellyaeitenry ie erentin’ tothe rant
which follow. Posturhaton srealte af Marens tobe dissed
Beolion 4.2, idiate chat weuntiw (5) ay be elite! for the inpute
oat Ceripral sesneferfureinn of « iver npatial eigenmode of a
ia waveguile with mechanical Inspertectiony, provided

eonnsiats ala to be discard

‘Define for aay fanelion Ba

Usa) ~ Ue) a0, ©
-0 wen
“The tebe shown ht oe Aopen ©)
baton +2 [” Y. + Pte de ©
‘Ther sess
bored [xu kang FeMtaEt de 8

Ung equation (5) we obtai

wiorrean Fane oxtintn

EROP 0 PL)
where
Tall) Ml)
‘Thue
an) = lt 40 ao)
where

Oa ry,
‘Thue we have «lines system relationship between the channel input
‘and onrput intensities,

‘a3 atnaion to Veutor Chased
Suppose we have a veetnr chanve! (euresponting to malliode
guide) consieting of vestor of Zinpat funcions

7 Tha raton

+ ite sonra

send [ak wt
museamac sevens ar

—_ "]

Lea

snd Fenty fuetions

xo =

“The ings andl onto fine ace atl by em Lx Le mast of
smpulen responses

= Bilt an

dQ 240 ro

Ye = 04 # x00,

Now consider a ean of tiv verlarsliannels having poles espana
snatrices'9() and "(The inp posses fest through channel and
then throwga ekaacel2,‘The wlsut of hanna 2 ie given by

om

qu oi # XU a3

Define the envelop oy), gal 8) we know that

0? 602 J Viele + AVEO de,

H+ yd + alKali +a)

us)

narnef CELL,

TG

od
Assumption a. Stationaity of Mde ‘Teansfr function,
1. Made ‘vsnafer funetions uncorrelated.
CH + Hal + of REO HC
= OPA Ae Bn ae 8)
4S suman svernm vvcuemious URAL, WAREH LOT

Sour ® Broneck delta
fee {f+ 4} einput sigeal bee

‘We ac ip yr shut eunctr-nea, expecially in plese, ences corre>
lation between the transfer functions of diferent modes, The validity
(ofthis eeimpeien “or eof! geien wth amre as oe ai wl be
Aisexsed in Section 1

Te then follows thot

anaorses2 [SS Ya ar wtnait 4 aX de

oo
dwona $ Ferner
Taff) 60 |.
enor - £ Era n+ eo

hese

ral & Pelt

‘orming she sowie “GLY wth enw pelts sbnilaly GU)
so! ijt aad output satestis ate eelgled by

te

[P= "at #0) 1x as

“Thus the verte ehanne is a veto Tina asters im ineensity a wll
si valle [srnpare enon (78) Lo evant (18)

13 A Limit Tivorom Jer Coad of Weowr Channels

Naw oo a casrade of 4 Teige number AF of velar channel,
fash behaving es dereribed in Setions 2.1 ané 82, fe. 38. ¥.(0 ie
the vesor af she average intensity weapons ab the ety, y+ he
input intensity vector; we have

rave (2 044) «maa ~ oF mor
nuuunexiue wavioeine si0

Gem 5 OM = Gye Gy 2

We would like to anus now tht for eufsin!ly lxge Af, all the
lerzente of Cy a2e Menten is wave, lilering a mine: eon
Sant. Tn other word, we would Hike to argue tal the shape of the
Iivensity svepenae erseen any inpat mods aad any outa swale
indierent. to the ebeiee of input andl onepat modes, for suTeien
long, guile, except perzape for che antgnitude of the seponse
‘We shall prove our rmlte for a lslees two-mede gue. De
rol, b) a the j >t tenity impulee response for b secon of
jo, Wo ub (oon Append A for devivacion)

pe) = nt = onal 20
vane
OF val tel) RB — Limes
wo
auto) =H
Sow seme ttle ok
Lina + torte ey
Purther define: 7
ll = ev
[ntnaat, veer os

The

Tor the Tass sid, and Deuficensl Lange,
for Reb. Tumhonnor « onvolution of

du) * o8f°NO is aponosimataly equal to yl 8 —~ 2} for R-& >
Etoe the seponee yt,8 = A), wish 32a convolution of Feber
Tue nuprow spetron compared tothe other Ie term eonvelution for
25> R.uethermone ask" fr # lenge, Ths feb eaientls lar

“rol 8) © voll Baas ED ~ an er
85 om mec sree coctentea sonata scant 1m
Similusly we bave

Batts =) + lO # ot

yall,

lt DE = lt, 9

(nee oy, + oy, = 1). Thus

oa neeym FY TF" |e gama, gn
4

I5ih i

where 9 = oft, and fl BY A nue W/E lh Ba Pally we
‘blaine rns a a ule of ED seri

GH) = pH DA,

Note thet A js idempotent, ic, AY = A and p( b) is w positive unit

4 Application to Lang Optica! Guile

For n mulling lowes af ye sion Yong Heng, P= RE,
wilh Seite eouping Helwers all modes, we can generalise tion
27} to eonchude that the tensity mpc reaponwe Beieen ab input
fa output noe a tant Hiroe ise pele nit tea fasetion
ptt, L)-convolvi! with ilselt 17Z Limes when Zina seale on whieh
fepiaion £27) Ine in Ce goneratned ame Gngze Uwe kw nodes)
Since the cent Tint thaorem seat that she eavaluton of large
umber of unit arex positive ‘unetions approaches « gtuesian shepe.®
wwe oxnelude she: fhe ipuleo ceepouss should approach a euuscian
hope in the limit of long guides. Further, the impulse zesponce's
econ mouueat wb ts mon Feros FnenrlywithSnereaing ide
Terath dor guides longer then Ts! Tht ie, the weord morcent ebout
he man of he response

BE) — MADE. es
Tire is he “ferential dalay” (Seoprmater) of propagation in ube

1 Provided at, 9, Eta «<n, hen we ase independet random
se ih a dns ie pl ey af Tat ich
Tha mend ce ome of & “nvelaion the vem of he fail
swe as

lowest mies ard ny the diflern:] delay in the festet mode, ten

any.
4

Therefore,
oe 8 a
ane A)
Jor wy f where equation (27) hols [for an N we ease we have an
Ao. mate maltlging otf, 01

a Pertahatio Theory

‘We mall now apply the above result te the ease of a Toslns lab
ieee waves previously tated by Mbareuse? We axpazd the
Int field eo the gue ws

2.0) = Deh Owais) ‘80

whore = isthe etor-setional potion parameter ancl the yyls) are
The eigeamorte of the guide, The Bell cinance 2 dawn the suite is

Had = Dale Onto ow

‘We base a fineer voltage Snpulse respurne ronship Debween che
veetar of input votegrs [el }| and the veetor of sutpul voltages
[itt 5}. Defining Blo.) 04 the Fource transform of (hw have

Bil = E Cole, DB, 0 (33)

Maneuze has shows thal petashtion theory sotion for the C1)
is given by

Caf aii a G3

dap exp 1,62" [of op IRI) —

svhere ty ea sonstant aly dependant upon a and g2) isthe wall
perturbation frm setaghones, Tt ie sumed & x6 j (Por k= ji
Giles) = Mh We have therfore

Siig et st fn, (2 he nrc tiny el oe
“mpl of eeu st tbe owpet
852mm mun sremme recumnteas sous, senor 197.

feted
= Nas exp 1iG,6) — 860+ ont) ff obieter
vio + a dee. 8)

120 [sG,6o} — Plate — lo 3) —
‘Defining the oovrelston funetion

as)

(ove nature g(2) is ¢ wide seme stationary process) ne obtain

Ceuto Neale + 0,0) = rarer tensa f [aes

9 a) — 02 ey 8 oe de
whee

sa = (Hi) — ale + 9} ~ G9

28, = Bia) 8604 9)
1 le — 29 arp que a fe — 4} Ban neo rh
the’ a he spear

ae fo + 01) = ARISE) ~ 86
oo G ann BGs gy

vir 540) = Voir Tattr of 8), a tse tobe
Commas 2 fain of Be) a) for within he sao
Bandnad Fork = n=»

w= eae

@)

‘That ie we eesume o is small encrgh so that there a negligible digper=
tion of enogy sravlling in a single se. Thi, the intensity ila
responee batean input j and output kis (ace Fig. 1)

204 Dy

” oo)

radl be Sala) — Belt —

a)
where

KO -= 1,140, Sra
= 0, otherwise;