BSTJ 60: 8. October 1981: Electromagnetic Fields, Field Confinement, and Energy Flow in Dispersionless Single-Mode Lightguides With Graded-Index Profiles. (Paek, U.C.; Peterson, G.E.; Carnevale, A.)

6 MB·English

Electromagnetic Fields, Field Confinement,
and Energy Flow in Dispersionless Single-
Mode Lightguides With Graded-Index Profiles

By U.C. PAEK. G. E PETERSON, and A. CARNEVALE
toruscristreclved February 19,1081)

1 te shown by numerical solation of Maxwells equations that, for
‘a gicun enorlengy the degree of confinement of ta electromagnetic
Fil to the core of m geudns-inde,singlecmode, optical fiber can be
‘optimised by the proper cice of the rudiat variation of the index
Such confinement of the energy 20 the core helps alleviate loss. The
‘Bers considered use aera total dispersion bandits in excess of
100 GHte-Kir, a wovelengteebetceen 13 un and 1.55

1. werRoDUETION

(ur caver snk deseribed a method of designing single-mode light
suides with zero total dispersion by varying the index profile in the
fore In the range of wivelengll betweow 1S pn xed 1.5m
Danderdehs in exes of 109 CHa: Kr are altinabte by balancing,
sacerial dispersion with waveguide dispersion. However, one of Une
eruus lifts with singe toe fihers in microbending lowe, We
now thal the microbending los forthe case of the stepindex ange
mole ber is proportional to X/4"" Hore, rofers to the operating
sravelengt and Ly the eolLive ile differs which x Hen

a

‘Now the design of a single-mode fiber mast be such chat ie prevents
{he fed af the fundamental med (HIE) fom extending well into the
cladding. In ocher words, che eleeromgnetic Feld must be tghily
confined othe core. To this end wo methods can be considered: one
4 to increase, whichis « commen and prevailing method, and the
tothe iso chang hidex profie in Chere. This work wil cao
the lier cage by maiming at inde pile where A= Nil ~ Beh
‘Reding microbending by profile design mighe be advancageous be

B= Mae = Mad
‘mse the me and 1x mode can be mini wel heyond the eutaét
point oft step inden, singlicmode fier and the manufacturing toler.
lances are relaxed, Nove that Ny = Ne apd Ns= May

ora single-mode lighigulde having a radially inhomagensous vore
itis usually not yensble In obisin analytical solutions of dosed form
for Maxwell's eyontions, Hence, co atain veetoroleelromagneti eld
tistributions of the Hs, mode and to determine accurate propagation
thareterisien af single-mode fiber, we used « numerical method to
solve the governing equations."*

4 THEORY

ur method uf alvin Maxwells equations for Tighoruides has been,
esrined in our walier publontions. However, we didnot consider the
‘halding fields in much deta. For Une work to be described in this
paper, thin is eaantial. Thus, we develop che neceasary mihurtiral
expressions

Tn an optical ber having a peemitirigy € and permeilty u, we
asmmo that the outer diameter ie ruck lager than the core

17725 THE RELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1981
diameter 2a. Toa eslindtical coordinate aystem, fora postion vector
Prhaving as its components (7, =), he vorrespondiag components
fof electric und mgaeie fields canbe written as F~ (En, By Ee) ad
HE = (Hy, He, 1) (are Fig 1). However, in obtaining 8 complete set of
‘yctor notions iv only neceasary to Gnd the tangential components
(By F.) and (H, Ha} since the radial componente lin and Ly are
Tinea combinations of the other components. In particaar.
Za gy, Bae
Faw Sy Bem, ei

3

In sition, the tangencialfeld components are continuous through
the vorechulding interface axl Unk siplifie the mathematics of the
boundary value problem,

Tithe above equation, Z ia the wave impedance defined by (u/¢
snd isthe index of refaction and a funetion of 2. The effective
Fefractve index N. is defined ly te quantities, nd 2 where fi.
the propagation ranslant slong the Ser ax and km 2a 9 ia a
Aimensionless quantity defined by Rb. The fundamental HE mode
propagnlee in the Tiber when che angular mode number Af equals
Moreover, P< Vetinet be ati, ete Viv Ine rae Pewency
tnd V; isthe cwiof frequency fora single-mode propagasion. ‘The ¥
‘alu is dined by

“

The input dats, ss it the opti cove radius chat will give re
total dispersion forgiven, 3 and A

Th the most general cxee there ary tive posibeslutions lo Max
wes equationa fore guided mode in a ightquide. A general solution
‘ill he rum of there two vector calutions. We introduce variable T
{oeatablish the following relationship wth th tangential Cl vectors
(Bu By) and (Hh ID).

n 5
Tl en, 8
in
n

rom £9. (6) ve denote the (wo vliione i worm Ty aad a. Since

cour earlier work gave a detailed description of the computation of
land Tg, we will avoid repotition of the provedure for ive elution in
‘the core region In the elading rege, Une (ve woltiona designated
by Py and Dg are given by the following expressions”

Mine)
veer] He o
Q
aot
°
raewer| 8 |. o
ato
wore) the dcr contn in the ein and
fo 0h ato
wag gine ne)

yuelsenienmuel |,
where Kita moditied Besse function of the second kind and its prime
‘denotes differentiation will resort to & (Nate that o, i the value of

pat the interface
“Ths total solution can be written inthe core and cladding region

separately.
In the core region, Tis expres hy

VeAta + Ade, o
and in the cladding, Pic express by

Pea + Ade fo)

where A ia on arbitrary constant, j~ 1,2, 3,4

"To eaeulate the field function I, we requize basically four input
data namely, A the aptinuin core zadiue ag, Nand N. Among howe
parameters, calculation of has boen described in detal in Ref. 1
‘The maverial dispersion effec i incorporated wich Maxwell's equa
tions to achieve a high degre of accuracy for Ny. Tia is necded to
"neque the precise eigenfunctions from eg 9 and 10

Th the desig ofa single-mode ber, Ais wily specified a an input
daca, Tis rather smal, ranging from 0,002 to 0008, since che cladding
fof the fher ia generally made of n high-silica glass. The dispersive
‘character of the cladding is well known Therelare, fora given 8, the
index of the core conter NY, can be expressed in terms of Ns by

LM

aa
“The dispersive properties of the Ns in eq (11) can be described by &
modified Sellmeier forruls!”

14790 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1981

x ay
wi
where = 0.85. "The coeticientsC; are given in our previous work.
‘Por the index profile, we use a well-known formula thet is particu

vata
[ay] oo

Finally, the diapersion ofthe index NY wail be determined by aubstiot
ing ine (Eth a. (HT) an (12.

"hc the cove-cladding interface, must eatin the continuity condi-
‘tom ofthe cangentil field ecmponents, Consequently, this yield a set
of simuleancous equations

ye

Ally + Ady Adn + A
Ala Awe ADa + Aly on

At Ae Ads Al

Alt Ades Ada + Ail
‘To compare the fel disrilations, i is eanwenient to intra the

following normalized variables into eg (3.

as)
an

“Tao

‘We bein our scudy by considering a germania doped silica Hight-
guide with A= 002 and ~ 138 am, The profile parameters examied
fre @ = 100, 2, and 1. Thus, we span the range from rectangular
‘through parabolic to cianguler inex functions. In Table Ie calculate
‘tha values fr the radi to male these Ldhtguides dispersion fo.

SINGLE-MODE LIGHTGUDE 1721
Tabla Idi values tor
siaparsion-tre lighten

"The normale! electromagnetic fields 95 2 fanetion of normalin
radi for thene three cates ar shun in Figs. 2a, 2b, nd 2c, Note that
inal cases the 2 elds re rovch smaller than the other field voapo~
tenis ln fac, thoeo fields are lea chan 3 percent of the tangential
Fields, The magnitmti of the R and componenls of the electric and
Imagneti feds fir w given w are all beverly the came. This is rot
Lika f the index profile hacnmas mora sorpea. For example, a profile

Fig 2 Rowan ld arn Hae ilo Iie,
Tc eaadee ae

1752 HE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1081
‘ontnining a central “burn out" and ripples, which would be charac.
teri of modified chemical vapor deposition (Mev) profiles, would
not have ich simple velationship between field components

‘An intereatng observation from Fig. ia thatthe slopes of the field
‘components atthe corecladding interface change with a. For « = 100,
the feld distribution near the interface form 9 eusp, but it ound
progressively ae a dorreases, This ve salely to the index discribucion
{nthe eore ofthe single-tade her

‘Figure D shows the normalizad teaneverse components ofthe eloc-
‘tromagnetic fled az a funccion of radial distaney for the thr axles.
‘The curves are estentally identical wp lo R= Sm. However, beyond
that distance they deviate appreciably. Also shown, by the vertical
lines, are the optimum rac

somucmrenn i, a

SINGLE MODE LIGHTGUIDE 1799
foto n)
om Fes Aad hon
= a
2 we
i a a
fn an
—/!
|
|encoveae wn on
i
bo

fit Mead han is he a mods ini me ei,

IV, ENERGY FLOW FOR THE ME,, MODE IN DISPERSIONLESS SINGLE.
MODE LIGHTGUIDES.

‘So far we have only consider the fed fn the fiber: However in
cxpaviniental practice iis more convenient to know the Geld intensity,
tchich is the amount of energy flowing through the crose-section ofthe
£ber. This canbe calculate! fom the Poynting vector § in W/cm"
‘The Poynting vector inthe 2 direction, S, ie given by,

Seo MBA ~ BIT, on
‘where *indiences the camplen sonjugate othe variable
‘We define the normalized Poynting vector I by
T= 898.0, (3)

Figure « shows the curves of J versns the normalized radial coordinate

1794 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1081
Ra for thnee diferent «values. "The normalised Poynting vector (eld
intensity) falls off move rapidly with normalized radius for lower «
‘luce Asin Section LL, when « ~ 100 che Poynting vector develops
2 cuap atthe corecading inlarnve. Wer ne nati w nenrieity of
the a= Land = 2 curves, This Une lwo have neatly the ane
focusing power.

\, DEGREE OF FIELD CONFINEMENT

‘Wo know that the degeee of field confinement ina fiber related co
its micrnbending lose” Inthe fabrication of single-mode fiber cables
for underwen applications, mierobending loss has been one of the
faovors thar deverminea the performance of the cable.

SINQLEMODE UGHIGULE 1735
Figures and 4 indicace that the field or power distribution islargely
depenent on the index profile, Ta quantity the focusing power or
‘omfinetnent ofa ight, we inlraduoe a parameter @ defined by:

| " s.ran

J SRUR

‘The paramecer @ yepresenc the degree of puwer confined lo the core
with reapect to the fatal propagating power. This ratio is plotted in
Fig. 5, along with dp, a2 a function of «We sw that ay. increases

4796 THE BELL SYSTEM IECHMICAL JOURNAL, OCTOBER 1051