BSTJ 60: 4. April 1981: Numerical Calculation of Optimum a for a Germania-Doped Silica Lightguide. (Peterson, G.E.; Carnevale, A.; Paek, U.C.; Fleming, J.W.)

4.8 MB·English

THE BELL SYSTEM
TECHNICAL JOURNAL

Vetucet par Somber

Numerical Calculation of Optimum a for a
Germania-Doped Silica Lightquide

YG. E. PETERSON, A. CARNEVALE, U.C. PAEK, andJ. W. FLEMING

Morurrit recived Septmber 1, 1980)

Using an exact numerical solution to Maxwells equations, we
Aesnrmiins the aptinm refractive ind profite paraoneter a fi
(germina Lihue. Onur resale. agren lool wih the
‘arlior work of Olshansky and Rech and withthe work of Marca.
Hiroe, liferencs esi that may ba important a the manfactare
of ery high Bandwidth lighteuides,

INTRODUCTION
Lis now well enable! thal a igheguide with a near parabolic
ines profile ean have single Gequency bandwidths of | GH ken or
sresies!"" Thic i expected to happen ifthe pratle is smooth, the
ental dip is natraw iesmll the profile parameter properly chosen,
land modes that havea signifi umount of tir energy transported,
inthe elalding are eliminated" Our undersanding of these facts
thas largely come trem approximate eolions to Maxwell's equations
for lignegudea. The avsilaaliy af eau sfloare Tor exsce mime
fenl solutions to Maxwell's wquntions allows win norsarine sere
these effete wd Lo lok olher tneore thal nay influence Eghtguide
Dendvideha, In this paper, we deters opie power le pra
perumetere (he, w parameters) aa 3 function of wavelength fer a
orewania-dopod silica litle,

‘We etudy a lighesuide sith a 13.8 mole petcens germania-doped
silica ente aivl a pure silica eladding. The eare ie assumed Lo be

465
perfectly cylindrical with a diameter of 60 microns, The sadaing is
considered to be eatontially infinite, The doping profile i auch that
the sefactive index ea power law function of disance ‘Thus,

where Ns is the refractive index nt the center ofthe coro, Nz isthe
Fefracrve index m the cladding, Myo tho radius uf the ore, wlohe
the profile parameter, We assume tha thie fanetional form hold a all
‘wavelength of interest.

‘We have accurstaly detarmined (he wavelength variation of the
refractive infor of sea and some doped slcas, Figure 1 gives the
data for 12.5 mole percent rermania-doped lien and pure alia. Asis
wll knnam, the reactive index falls of alowdy with wavelength Cor
Toth of thede glasses! We express the data hy tho following media
Selle formula”

N= Ga +OA+ GAA + Gyo? — 0.096)

+ cya? — 0085)? + cya! ~ 0.0851,

fgg ane 80 ee pee et ati
‘where Aisin microns andi = 1, 2.‘Table Hist the coefficient for
(2) for both 13. male percene germaniacdoned sien and ales.

‘Our numerical procedure to solve Maxwell's equations bas been
deseribed in detail in an earlier paper! Thus, we only give a very brie
‘desrincion here. The method i similar to that described by Vigants
and Schlesinger” and Vassol!" We write Maxwell's cqumtions in
‘cylindrical ponedinates of

a

ab
‘whore Pip) i column veetur whase four ements ure related vo the
Langential enmpenenta of che electromagnetic field. They are aa fl-
lowe:

p “AIC o

Nak,
T= inka,
Tek,

Ty BL,

where Zs i the wave impedance of fre space and the variable p is
defined a8 KB.
"The 1% 4 mates Alph can be waiten 65

Co eee a

pana 0 aN, 0
er ee eT

uN, oO Mine 0

‘This marie dearieeuhe progetto the medi
Werle eg [5] by an opinial fourth-order Runge-Kutta proce
‘dure This iva higher-aecoracy formula than ured in our eaviar werk,
"The moet apartane cormpucationa vaults are the effective indices N,
for che varios meres in the Fightguide
"The genp ice Nansen fm the eflecve ines by
he following Fore

ses Ny ~ MANel ©

Table IGootficionts for 89, (2)
THM Fre Se

NUMERICAL CALCULATION OF OFIIMUM 9457
‘he derivative is obtained from mumerioel ealelatins of N, at: three
closely spaced wavelengths.

or a lightguide with the materials and profile deseribed earlier, x
simple opprovimate rlatonship ete between group index Ny and
fective index N for certain modes In particular, fora given profile
Devameter a, angular mode nomber Mf, wovelangth A, and for modes
far from ctf, the effective indices are nently Tnesrly relate to the
‘roup indices, Ryders for eh can he abana fran Fig. 2 uve a)
In this case, we have A ~ 06228 micron, MT = 0, and a = 2 The aod
line is the linear Tenst-oquaves fi to the data when the last ro medes

ee

vk \

van \ 1

ayers tions ne ese
ere Cees
neglected. “

450 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1081
eet taepestaoee fis ras awaits

are deleted, (The ast max too far off scale tobe plot) Obviously
‘neatly linear relafonehy existe beewoon Ny and Ne. In addition,

fore i.a very slight aystetatio devintinn from the least-square line
‘This nuggets that additional rr are needed to completely dear
the data. However this wil be of no consequence tothe analysis to be
described here

“The Int two modes (those Una deviate strongly from the line) have
a sinifcwnt fraction oftheir rey heing transported in tho lading

NUMERICAL CALCULATION OF OPTIMUM «459
“Tims, they are not tightly bound to the core and normally sre not
considered in an optimum a calelation

‘An optimum profile parameter a, fora particular family of modes,
fs one that minimizes the spread in group indices for that family at a
particular wavelength. For example, we might consider all the modes
sscociated witha specified Mf value. inthis case, the optimum « would
be the one that gives the smallest slope to the least-equares Tine
relating N, and Ny

Pig. Part fe ao

460 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1981
~ if |

“Te

wt |
\

igure 2, curve tb) ehows « near optimum value of «= 22 for M=
0 and 2 = 046908 micron. We see chat the least-squares line is now
‘nearly horizontal and that evidece exiaia for che necessity of higher-
ter terms o esely ft the date, The strong dvop-off ofthe lat few.
‘modes is til evi

"A very good overall uptimuin a for all M values will be meazly
identical to the optimum for M— 0. Ax wll be auown shorly, this
eosin exellene equalization of the lowar-order modes and a good

NUMERICAL CALCULATION OF OPTIMUM « 461
‘aualication of the higher-order modes. For mest enginoering appl
‘ations, optimum on the wavelength range of 0.65 to L55 microns
tre need

un, ResuLts

Wigures 3 through 7 show plots of versus 1. for various a
parameters and wavelengths for M = 0, Figure 4 shows part of the

“

Fig, 6—Potof peu index vr ct ine rh 12 ana and N= 0,

482. THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1051
ve ilo agro inless fltve ies r= mon a =

A of Fig. on an expanded scale, Note that in all w nearly Tinear
relationship exits hetoreen and N,, except for the Tt. couple of
odes, AS mentioned bofore, {hee medea are nenring cutol anv wns
‘Dewlly not consider in aw eptimium cx eterenination.

"ALeneh wavelength, we see that there ia value, ag that makes
the slope of the Hine nearly 2er0, Table TT Tiss ag values for wave
lengths of engineerin imparcance and also fora fe other mavelengtha
sae wel

‘The sata of Figs 3-7 pertaining to the infared wavolongths are

NUMERICAL GALCULATION OF OPIUM «483
Table Optimum aipha values
"ra a

summariged in Figs 8 ad §. Figure 6 shows the slope of the Hine
‘lating, to N, ax function of wavelength for various @ values Wo
Seo that most inrared wavelengths of interes require an w vale in
the range from 14 to 21. Figure 9 shows che same daca, but this time
ne a foretion of e for various wavelengths. The aame conclusion it
tbo

‘Figure 10 displays «calculation of the group indices forall the bound

“

ig Poof help of sr 8 a fain ovo a aes

404 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1981