‘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;